Discrete cosine transform

13374: 12924: 13369:{\displaystyle {\begin{bmatrix}6.1917&-0.3411&1.2418&0.1492&0.1583&0.2742&-0.0724&0.0561\\0.2205&0.0214&0.4503&0.3947&-0.7846&-0.4391&0.1001&-0.2554\\1.0423&0.2214&-1.0017&-0.2720&0.0789&-0.1952&0.2801&0.4713\\-0.2340&-0.0392&-0.2617&-0.2866&0.6351&0.3501&-0.1433&0.3550\\0.2750&0.0226&0.1229&0.2183&-0.2583&-0.0742&-0.2042&-0.5906\\0.0653&0.0428&-0.4721&-0.2905&0.4745&0.2875&-0.0284&-0.1311\\0.3169&0.0541&-0.1033&-0.0225&-0.0056&0.1017&-0.1650&-0.1500\\-0.2970&-0.0627&0.1960&0.0644&-0.1136&-0.1031&0.1887&0.1444\\\end{bmatrix}}} 9020: 7868: 9644: 7842:, however, there exist other methods to compute the same thing while performing the computations in a different order (i.e. interleaving/combining the algorithms for the different dimensions). Owing to the rapid growth in the applications based on the 3-D DCT, several fast algorithms are developed for the computation of 3-D DCT-II. Vector-Radix algorithms are applied for computing M-D DCT to reduce the computational complexity and to increase the computational speed. To compute 3-D DCT-II efficiently, a fast algorithm, Vector-Radix Decimation in Frequency (VR DIF) algorithm was developed. 7858: 9015:{\displaystyle {\begin{array}{lcl}{\tilde {x}}(n_{1},n_{2},n_{3})=x(2n_{1},2n_{2},2n_{3})\\{\tilde {x}}(n_{1},n_{2},N-n_{3}-1)=x(2n_{1},2n_{2},2n_{3}+1)\\{\tilde {x}}(n_{1},N-n_{2}-1,n_{3})=x(2n_{1},2n_{2}+1,2n_{3})\\{\tilde {x}}(n_{1},N-n_{2}-1,N-n_{3}-1)=x(2n_{1},2n_{2}+1,2n_{3}+1)\\{\tilde {x}}(N-n_{1}-1,n_{2},n_{3})=x(2n_{1}+1,2n_{2},2n_{3})\\{\tilde {x}}(N-n_{1}-1,n_{2},N-n_{3}-1)=x(2n_{1}+1,2n_{2},2n_{3}+1)\\{\tilde {x}}(N-n_{1}-1,N-n_{2}-1,n_{3})=x(2n_{1}+1,2n_{2}+1,2n_{3})\\{\tilde {x}}(N-n_{1}-1,N-n_{2}-1,N-n_{3}-1)=x(2n_{1}+1,2n_{2}+1,2n_{3}+1)\\\end{array}}} 19355: 19345: 12897: 22531: 22541: 2913: 25: 22520: 22551: 6759: 3007:(based on the type-IV DCT), the boundary conditions are intimately involved in the MDCT's critical property of time-domain aliasing cancellation. In a more subtle fashion, the boundary conditions are responsible for the "energy compactification" properties that make DCTs useful for image and audio compression, because the boundaries affect the rate of convergence of any Fourier-like series. 10024: 11839: 7829: 7296: 12208: 2920:=11 data points (red dots), for the four most common types of DCT (types I-IV). Note the subtle differences at the interfaces between the data and the extensions: in DCT-II and DCT-IV both the end points are replicated in the extensions but not in DCT-I or DCT-III (and a zero point is inserted at the sign reversal extension in DCT-III). 11827:

the most important factor. Therefore, although the above proposed 3-D VR algorithm does not achieve the theoretical lower bound on the number of multiplications, it has a simpler computational structure as compared to other 3-D DCT algorithms. It can be implemented in place using a single butterfly and possesses the properties of the

9467: 6151: 9654: 258:(MDCT), which is based on a DCT of overlapping data. Multidimensional DCTs (MD DCTs) are developed to extend the concept of DCT to multidimensional signals. A variety of fast algorithms have been developed to reduce the computational complexity of implementing DCT. One of these is the integer DCT (IntDCT), an 10714: 10489: 12694:, and this method in hindsight can be seen as one step of a radix-4 decimation-in-time Cooley–Tukey algorithm applied to the "logical" real-even DFT corresponding to the DCT-II. Because the even-indexed elements are zero, this radix-4 step is exactly the same as a split-radix step. If the subsequent size 401:

and Jaswant R. Jain further developed motion-compensated DCT video compression, also called block motion compensation. This led to Chen developing a practical video compression algorithm, called motion-compensated DCT or adaptive scene coding, in 1981. Motion-compensated DCT later became the standard

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also uses a real-data FFT. So, there is nothing intrinsically bad about computing the DCT via an FFT from an arithmetic perspective – it is sometimes merely a question of whether the corresponding FFT algorithm is optimal. (As a practical matter, the function-call overhead in invoking a separate FFT

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While DCT algorithms that employ an unmodified FFT often have some theoretical overhead compared to the best specialized DCT algorithms, the former also have a distinct advantage: Highly optimized FFT programs are widely available. Thus, in practice, it is often easier to obtain high performance for

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The conventional method to calculate MD-DCT-II is using a Row-Column-Frame (RCF) approach which is computationally complex and less productive on most advanced recent hardware platforms. The number of multiplications required to compute VR DIF Algorithm when compared to RCF algorithm are quite a few

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The figure to the adjacent shows the four stages that are involved in calculating 3-D DCT-II using VR DIF algorithm. The first stage is the 3-D reordering using the index mapping illustrated by the above equations. The second stage is the butterfly calculation. Each butterfly calculates eight points

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DCTs of types I–IV treat both boundaries consistently regarding the point of symmetry: they are even/odd around either a data point for both boundaries or halfway between two data points for both boundaries. By contrast, DCTs of types V-VIII imply boundaries that are even/odd around a data point for

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DCT is used in most of the compression systems standardized by the Moving Picture Experts Group (MPEG), is the dominant technology for image compression. In particular, it is the core technology of MPEG-2, the system used for DVDs, digital television broadcasting, that has been used for many of the

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We can compute the MD DCT-IV using the regular row-column method or we can use the polynomial transform method for the fast and efficient computation. The main idea of this algorithm is to use the Polynomial Transform to convert the multidimensional DCT into a series of 1-D DCTs directly. MD DCT-IV

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The main consideration in choosing a fast algorithm is to avoid computational and structural complexities. As the technology of computers and DSPs advances, the execution time of arithmetic operations (multiplications and additions) is becoming very fast, and regular computational structure becomes

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is generally discontinuous). This is why DCTs, and in particular DCTs of types I, II, V, and VI (the types that have two even boundaries) generally perform better for signal compression than DFTs and DSTs. In practice, a type-II DCT is usually preferred for such applications, in part for reasons of

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of a function, respectively, in order to talk about its "smoothness".) However, the implicit periodicity of the DFT means that discontinuities usually occur at the boundaries: any random segment of a signal is unlikely to have the same value at both the left and right boundaries. (A similar problem

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Since its introduction in 1974, there has been significant research on the DCT. In 1977, Wen-Hsiung Chen published a paper with C. Harrison Smith and Stanley C. Fralick presenting a fast DCT algorithm. Further developments include a 1978 paper by M. J. Narasimha and A. M. Peterson, and a 1984 paper

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of multiplications associated with the 3-D DCT VR algorithm is less than that associated with the RCF approach by more than 40%. In addition, the RCF approach involves matrix transpose and more indexing and data swapping than the new VR algorithm. This makes the 3-D DCT VR algorithm more efficient

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two-dimensional DCT. Each step from left to right and top to bottom is an increase in frequency by 1/2 cycle. For example, moving right one from the top-left square yields a half-cycle increase in the horizontal frequency. Another move to the right yields two half-cycles. A move down yields two

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of the Fourier series, so that more sinusoids are needed to represent the function with a given accuracy. The same principle governs the usefulness of the DFT and other transforms for signal compression; the smoother a function is, the fewer terms in its DFT or DCT are required to represent it

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In fact, even the DCT algorithms using an ordinary FFT are sometimes equivalent to pruning the redundant operations from a larger FFT of real-symmetric data, and they can even be optimal from the perspective of arithmetic counts. For example, a type-II DCT is equivalent to a DFT of size

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For example, a two-dimensional DCT-II of an image or a matrix is simply the one-dimensional DCT-II, from above, performed along the rows and then along the columns (or vice versa). That is, the 2D DCT-II is given by the formula (omitting normalization and other scale factors, as above):

7315: 6782: 2987:(DSTs). Each boundary can be either even or odd (2 choices per boundary) and can be symmetric about a data point or the point halfway between two data points (2 choices per boundary), for a total of 2 × 2 × 2 × 2 = 16 possibilities. Half of these possibilities, those where the 3504: 11937: 13387:

On the left is the final image. In the middle is the weighted function (multiplied by a coefficient) which is added to the final image. On the right is the current function and corresponding coefficient. Images are scaled (using bilinear interpolation) by factor

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sizes) are typically closely related to FFT algorithms – since DCTs are essentially DFTs of real-even data, one can design a fast DCT algorithm by taking an FFT and eliminating the redundant operations due to this symmetry. This can even be done automatically

5134: 6754:{\displaystyle {\begin{aligned}X_{k_{1},k_{2}}&=\sum _{n_{1}=0}^{N_{1}-1}\left(\sum _{n_{2}=0}^{N_{2}-1}x_{n_{1},n_{2}}\cos \left\right)\cos \left\\&=\sum _{n_{1}=0}^{N_{1}-1}\sum _{n_{2}=0}^{N_{2}-1}x_{n_{1},n_{2}}\cos \left\cos \left.\end{aligned}}} 9176: 2694:, caused by DCT blocks. In a DCT algorithm, an image (or frame in an image sequence) is divided into square blocks which are processed independently from each other, then the DCT blocks is taken within each block and the resulting DCT coefficients are 11208:

butterflies to be computed. Each butterfly requires seven real multiplications (including trivial multiplications) and 24 real additions (including trivial additions). Therefore, the total number of real multiplications needed for this stage is

10019:{\displaystyle X(k_{1},k_{2},k_{3})=\sum _{n_{1}=1}^{{\tfrac {N}{2}}-1}\sum _{n_{2}=1}^{{\tfrac {N}{2}}-1}\sum _{n_{1}=1}^{{\tfrac {N}{2}}-1}{\tilde {x}}_{ijl}(n_{1},n_{2},n_{3})\cos(\varphi (2k_{1}+i)\cos(\varphi (2k_{2}+j)\cos(\varphi (2k_{3}+l))} 226:

coefficients of a periodically and symmetrically extended sequence whereas DFTs are related to Fourier series coefficients of only periodically extended sequences. DCTs are equivalent to DFTs of roughly twice the length, operating on real data with

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Algorithmic performance on modern hardware is typically not principally determined by simple arithmetic counts, and optimization requires substantial engineering effort to make best use, within its intrinsic limits, of available built-in hardware

11549: 10846: 5665: 4268: 10497: 10272: 12625:) for the DCT. Because the algorithms for DFTs, DCTs, and similar transforms are all so closely related, any improvement in algorithms for one transform will theoretically lead to immediate gains for the other transforms as well ( 6763:

The inverse of a multi-dimensional DCT is just a separable product of the inverses of the corresponding one-dimensional DCTs (see above), e.g. the one-dimensional inverses applied along one dimension at a time in a row-column

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Multidimensional DCTs (MD DCTs) have several applications, mainly 3-D DCTs such as the 3-D DCT-II, which has several new applications like Hyperspectral Imaging coding systems, variable temporal length 3-D DCT coding,

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These different boundary conditions strongly affect the applications of the transform and lead to uniquely useful properties for the various DCT types. Most directly, when using Fourier-related transforms to solve

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Potluri, U. S.; Madanayake, A.; Cintra, R. J.; Bayer, F. M.; Rajapaksha, N. (17 October 2012). "Multiplier-free DCT approximations for RF multi-beam digital aperture-array space imaging and directional sensing".

7824:{\displaystyle x_{n_{1},n_{2},n_{3}}=\sum _{k_{1}=0}^{N_{1}-1}\sum _{k_{2}=0}^{N_{2}-1}\sum _{k_{3}=0}^{N_{3}-1}X_{k_{1},k_{2},k_{3}}\cos \left\cos \left\cos \left,\quad {\text{for }}n_{i}=0,1,2,\dots ,N_{i}-1.} 7291:{\displaystyle X_{k_{1},k_{2},k_{3}}=\sum _{n_{1}=0}^{N_{1}-1}\sum _{n_{2}=0}^{N_{2}-1}\sum _{n_{3}=0}^{N_{3}-1}x_{n_{1},n_{2},n_{3}}\cos \left\cos \left\cos \left,\quad {\text{for }}k_{i}=0,1,2,\dots ,N_{i}-1.} 6135:

Multidimensional variants of the various DCT types follow straightforwardly from the one-dimensional definitions: they are simply a separable product (equivalently, a composition) of DCTs along each dimension.

1554:. As of 2019, AVC is by far the most commonly used format for the recording, compression and distribution of video content, used by 91% of video developers, followed by HEVC which is used by 43% of developers. 12856: 3277: 9100: 2630:

algorithms, adaptive video coding and 3-D Compression. Due to enhancement in the hardware, software and introduction of several fast algorithms, the necessity of using MD DCTs is rapidly increasing.

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and the total number of real additions i.e. including the post-additions (recursive additions) which can be calculated directly after the butterfly stage or after the bit-reverse stage are given by

3104: 12203:{\displaystyle X_{k,\ell }=\sum _{n=0}^{N-1}\;\sum _{m=0}^{M-1}\ x_{n,m}\cos \left(\ {\frac {\,(2m+1)(2k+1)\ \pi \,}{4N}}\ \right)\cos \left(\ {\frac {\,(2n+1)(2\ell +1)\ \pi \,}{4M}}\ \right)~,} 6156: 12363: 9165: 11831:

in 3-D. Hence, the 3-D VR presents a good choice for reducing arithmetic operations in the calculation of the 3-D DCT-II, while keeping the simple structure that characterize butterfly-style

3743: 12286: 9462:{\displaystyle X(k_{1},k_{2},k_{3})=\sum _{n_{1}=1}^{N-1}\sum _{n_{2}=1}^{N-1}\sum _{n_{3}=1}^{N-1}{\tilde {x}}(n_{1},n_{2},n_{3})\cos(\varphi k_{1})\cos(\varphi k_{2})\cos(\varphi k_{3})} 4937: 3206: 12560: 12469: 3264: 11132: 1531:

element (top-left) is the DC (zero-frequency) component and entries with increasing vertical and horizontal index values represent higher vertical and horizontal spatial frequencies.

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compression, or the small DCTs (or MDCTs) typically used in audio compression. (Reduced code size may also be a reason to use a specialized DCT for embedded-device applications.)

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arises for the DST, in which the odd left boundary condition implies a discontinuity for any function that does not happen to be zero at that boundary.) In contrast, a DCT where

13653: 12783: 3693: 381:(FFT), developing inter-frame hybrid coders for both, and found that the DCT is the most efficient due to its reduced complexity, capable of compressing image data down to 0.25- 12908: 12604:

extra operations (see below for an exception). However, even "specialized" DCT algorithms (including all of those that achieve the lowest known arithmetic counts, at least for

11908: 5230: 4677: 12916: 6084:, the normalization factor in front of these transform definitions is merely a convention and differs between treatments. For example, some authors multiply the transforms by 3596: 4713: 4567: 4445: 6024:

algorithms for even-length DFTs (e.g. the simplest radix-2 algorithms are only for even lengths), and this increased intricacy carries over to the DCTs as described below.

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has gained popularity for its applications in fast implementation of real-valued polyphase filtering banks, lapped orthogonal transform and cosine-modulated wavelet bases.

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data points. The obvious distinction between a DCT and a DFT is that the former uses only cosine functions, while the latter uses both cosines and sines (in the form of

15521: 4518: 4397: 1469: 14855:; Ahmed, Nasir; Magotra, Neeraj (17 April 1995). Rodriguez, Arturo A.; Safranek, Robert J.; Delp, Edward J. (eds.). "DCT-based scheme for lossless image compression". 13598:

The precise count of real arithmetic operations, and in particular the count of real multiplications, depends somewhat on the scaling of the transform definition. The

11725: 11287: 5827: 5458: 234:

There are eight standard DCT variants, of which four are common. The most common variant of discrete cosine transform is the type-II DCT, which is often called simply

16567: 11621: 11206: 3656: 3567: 10709:{\displaystyle +(-1)^{i}{\tilde {x}}\left(n_{1}+{\frac {n}{2}},n_{2},n_{3}\right)+(-1)^{i+j}{\tilde {x}}\left(n_{1}+{\frac {n}{2}}+{\frac {n}{2}},n_{2},n_{3}\right)} 10484:{\displaystyle +(-1)^{j}{\tilde {x}}\left(n_{1},n_{2}+{\frac {n}{2}},n_{3}\right)+(-1)^{j+l}{\tilde {x}}\left(n_{1},n_{2}+{\frac {n}{2}},n_{3}+{\frac {n}{2}}\right)} 5927: 5857: 5784: 5470: 5418: 4800: 4358: 4096: 4084: 3789: 231:

symmetry (since the Fourier transform of a real and even function is real and even), whereas in some variants the input or output data are shifted by half a sample.

9638: 5747: 5381: 5300: 5167: 4870: 4763: 4643: 4054: 3996: 3623: 3534: 1529: 16685: 13493: 12886: 12676: 12618: 6055: 5355: 4925: 2895: 2826: 16209:. It is currently the most widely used international video compression standard for video telephony on ISDN (Integrated Services Digital Network) telephone lines. 13545: 13519: 12718: 5326: 4896: 4022: 3815: 11084: 5973: 5950: 4328: 4301: 3967: 1550:

video coding format for coding still images. AVC uses 4 x 4 and 8 x 8 blocks. HEVC and HEIF use varied block sizes between 4 x 4 and 32 x 32

547:

property. In typical applications, most of the signal information tends to be concentrated in a few low-frequency components of the DCT. For strongly correlated

5892: 3944: 2866: 2846: 2654:(DSP), as well as digital signal processing software. Many companies have developed DSPs based on DCT technology. DCTs are widely used for applications such as 1497: 500:

from 8:1 to 14:1 for near-studio-quality, up to 100:1 for acceptable-quality content. DCT compression standards are used in digital media technologies, such as

13902: 37: 14589:

Proceedings of the International Computer Conference 2006 on Wavelet Active Media Technology and Information Processing: Chongqing, China, 29-31 August 2006

5975:(for DCT-IV). The four additional types of discrete cosine transform correspond essentially to real-even DFTs of logically odd order, which have factors of 21795: 14617:

Princen, John P.; Johnson, A.W.; Bradley, Alan B. (1987). "Subband/Transform coding using filter bank designs based on time domain aliasing cancellation".

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is typically 8 and the DCT-II formula is applied to each row and column of the block. The result is an 8 × 8 transform coefficient array in which the

14810: 11026:{\displaystyle +(-1)^{i+j+l}{\tilde {x}}\left(n_{1}+{\frac {n}{2}},n_{2}+{\frac {n}{2}},n_{3}+{\frac {n}{2}}\right){\text{ where }}i,j,l=0{\text{ or }}1.} 21245: 14730:

Britanak, V. (2011). "On Properties, Relations, and Simplified Implementation of Filter Banks in the Dolby Digital (Plus) AC-3 Audio Coding Standards".

20831: 19601: 15020: 10259:{\displaystyle {\tilde {x}}_{ijl}(n_{1},n_{2},n_{3})={\tilde {x}}(n_{1},n_{2},n_{3})+(-1)^{l}{\tilde {x}}\left(n_{1},n_{2},n_{3}+{\frac {n}{2}}\right)} 9475: 18108: 16187: 12678:

with real-even symmetry whose even-indexed elements are zero. One of the most common methods for computing this via an FFT (e.g. the method used in

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sequences, two issues arise that do not apply for the continuous cosine transform. First, one has to specify whether the function is even or odd at

14531:. Joint Video Team (JVT) of ISO/IEC MPEG & ITU-T VCEG (ISO/IEC JTC1/SC29/WG11 and ITU-T SG16 Q.6). July 2002. pp. 11, 24–9, 33, 40–1, 53–6 14447:

Roese, John A.; Robinson, Guner S. (30 October 1975). Tescher, Andrew G. (ed.). "Combined Spatial And Temporal Coding Of Digital Image Sequences".

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The most efficient algorithms, in principle, are usually those that are specialized directly for the DCT, as opposed to using an ordinary FFT plus

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Muchahary, D.; Mondal, A. J.; Parmar, R. S.; Borah, A. D.; Majumder, A. (2015). "A Simplified Design Approach for Efficient Computation of DCT".

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leads to minimal-multiplication algorithms for the DFT, albeit generally at the cost of more additions, and a similar algorithm was proposed by (

5264:(see above for the corresponding change in DCT-II), so that the DCT-II and DCT-III are transposes of one another. This makes the DCT-III matrix 20715: 20506: 19902: 19432: 16647:

Schnell, Markus; Schmidt, Markus; Jander, Manuel; Albert, Tobias; Geiger, Ralf; Ruoppila, Vesa; Ekstrand, Per; Bernhard, Grill (October 2008).

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Because it is the inverse of DCT-II up to a scale factor (see below), this form is sometimes simply referred to as "the inverse DCT" ("IDCT").

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The DCT, and in particular the DCT-II, is often used in signal and image processing, especially for lossy compression, because it has a strong

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with FFT-based algorithms. Specialized DCT algorithms, on the other hand, see widespread use for transforms of small, fixed sizes such as the

20253: 19396: 18179: 555:(which is optimal in the decorrelation sense). As explained below, this stems from the boundary conditions implicit in the cosine functions. 398: 361: 22503: 22475: 22470: 21495: 16883:

Abousleman, G. P.; Marcellin, M. W.; Hunt, B. R. (January 1995), "Compression of hyperspectral imagery using 3-D DCT and hybrid DPCM/DCT",

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An example showing eight different filters applied to a test image (top left) by multiplying its DCT spectrum (top right) with each filter.

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2000 IEEE International Symposium on Circuits and Systems. Emerging Technologies for the 21st Century. Proceedings (IEEE Cat No.00CH36353)

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and better suited for 3-D applications that involve the 3-D DCT-II such as video compression and other 3-D image processing applications.

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Technically, computing a two-, three- (or -multi) dimensional DCT by sequences of one-dimensional DCTs along each dimension is known as a

566:, where the different variants of the DCT correspond to slightly different even and odd boundary conditions at the two ends of the array. 21125: 20687: 20144: 18847: 18658: 16774: 13679:

factor. Additional multiplications can be saved if one permits the outputs of the transform to be rescaled individually, as was shown by

317:. Ahmed developed a practical DCT algorithm with his PhD students T. Raj Natarajan, Wills Dietrich, and Jeremy Fries, and his friend Dr. 20541: 16031: 16006: 14769: 15487: 3499:{\displaystyle X_{k}={\frac {1}{2}}(x_{0}+(-1)^{k}x_{N-1})+\sum _{n=1}^{N-2}x_{n}\cos \left\qquad {\text{ for }}~k=0,\ \ldots \ N-1~.} 20719: 20502: 19906: 19436: 18547: 15990:

Variable-sized DCT (square or rectangular from 2x2 to 256x256) serves as a fast approximation of the optimal decorrelating transform.

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Chen, Wen-Hsiung; Smith, C. H.; Fralick, S. C. (September 1977). "A Fast Computational Algorithm for the Discrete Cosine Transform".

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DFTs of real data, two of which are zero, and two of which are equal to one another by the even symmetry. Hence giving a single size

16854: 12791: 21061: 20289: 19515: 19053: 18876: 18670: 17798: 15974: 12728:), then the resulting algorithm actually matches what was long the lowest published arithmetic count for the power-of-two DCT-II ( 9027: 21238: 20359: 18361: 17003:

Tai, S.-C; Gi, Y.; Lin, C.-W. (September 2000), "An adaptive 3-D discrete cosine transform coder for medical image compression",

16468:. In Thampi, Sabu M.; Pérez, Gregorio Martínez; Westphall, Carlos Becker; Hu, Jiankun; Fan, Chun I.; Mármol, Félix Gómez (eds.). 13806: 273:

DCT compression, also known as block compression, compresses data in sets of discrete DCT blocks. DCT blocks sizes including 8x8

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Proceedings of the 1998 IEEE International Conference on Acoustics, Speech and Signal Processing, ICASSP '98 (Cat. No.98CH36181)

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Wen-Hsiung Chen; Smith, C.; Fralick, S. (September 1977). "A Fast Computational Algorithm for the Discrete Cosine Transform".

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Chan, S.C.; Liu, W.; Ho, K.I. (2000). "Perfect reconstruction modulated filter banks with sum of powers-of-two coefficients".

238:. This was the original DCT as first proposed by Ahmed. Its inverse, the type-III DCT, is correspondingly often called simply 22615: 22492: 22482: 22462: 22264: 19946: 18081: 17919: 17868: 17269: 15698: 15667: 15601: 15392: 14960: 2250: 15454: 3059: 21283: 20427: 18788: 17601:

Shao, Xuancheng; Johnson, Steven G. (2008). "Type-II/III DCT/DST algorithms with reduced number of arithmetic operations".

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Shao, Xuancheng; Johnson, Steven G. (2008). "Type-II/III DCT/DST algorithms with reduced number of arithmetic operations".

16580: 16062: 14703: 14423: 2368: 21629: 18141:. Free C & FORTRAN libraries for computing fast DCTs (types II–III) in one, two or three dimensions, power of 2 sizes. 13791: 2756:

Like any Fourier-related transform, discrete cosine transforms (DCTs) express a function or a signal in terms of a sum of

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count is for the DCT-II definition shown here; two multiplications can be saved if the transform is scaled by an overall

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by B. G. Lee. These research papers, along with the original 1974 Ahmed paper and the 1977 Chen paper, were cited by the

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In order to apply the VR DIF algorithm the input data is to be formulated and rearranged as follows. The transform size

22580: 22259: 21231: 21187: 21152: 21130: 20329: 20304: 19165: 18903: 18842: 18653: 18603: 18426: 16105: 14104: 13846: 13777: 13437: 13432: 5716: 3004: 2138: 2078: 406: 360:. The DST was described in the 1974 DCT paper by Ahmed, Natarajan and Rao. A type-I DST (DST-I) was later described by 255: 211: 3702: 22353: 21273: 20927: 19978: 19389: 18286: 18271: 18172: 17365: 17301: 17212: 17151:

Chan, S. C.; Luo, L.; Ho, K. L. (1998). "M-Channel compactly supported biorthogonal cosine-modulated wavelet bases".

17096: 16739: 16522: 16485: 16195: 16070: 15737: 15563: 15428: 15358: 15322: 15226: 15188: 15142: 14999: 14919: 14711: 14648:

Princen, J.; Bradley, A. (1986). "Analysis/Synthesis filter bank design based on time domain aliasing cancellation".

14601: 14570: 14501: 14431: 7839: 5129:{\displaystyle X_{k}={\tfrac {1}{2}}x_{0}+\sum _{n=1}^{N-1}x_{n}\cos \left\qquad {\text{ for }}~k=0,\ \ldots \ N-1~.} 3011: 1847: 622: 12888:

but this is an implementation rather than an algorithmic question since it can be solved by unrolling or inlining.)

22138: 21686: 21488: 21447: 21428: 20969: 20523: 20422: 20212: 19614: 19278: 13910: 12217: 4732: 3923:(even symmetry), divided by two. (In contrast, DCT types II-IV involve a half-sample shift in the equivalent DFT.) 2695: 2411: 1472: 805: 334: 322: 1328: 22249: 21369: 21076: 20803: 20667: 20510: 20134: 20088: 19988: 19288: 19126: 18977: 18896: 18690: 1207: 17700:

Sorensen, H.; Jones, D.; Heideman, M.; Burrus, C. (June 1987). "Real-valued fast Fourier transform algorithms".

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Basis functions of the discrete cosine transformation with corresponding coefficients (specific for our image).

6116:

so that the inverse does not require any additional multiplicative factor. Combined with appropriate factors of

4731:, the scaling is arbitrary because scale factors can be combined with a subsequent computational step (e.g. the 3150: 22244: 21066: 20749: 20154: 19983: 19261: 18881: 18675: 18463: 17199:

Katsaggelos, Aggelos K.; Babacan, S. Derin; Chun-Jen, Tsai (2009). "Chapter 15 - Iterative Image Restoration".

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is a free Matlab/Octave toolbox with interfaces to the FFTW implementation of the DCTs and DSTs of type I-IV.

15905: 15876: 14522: 12380: 5674:(and thus, being clearly symmetric, its own inverse) if one further multiplies by an overall scale factor of 2996: 2780:). However, this visible difference is merely a consequence of a deeper distinction: a DCT implies different 2358: 1387: 974: 20241: 19694: 19023: 16505:. In Le, Dac-Nhuong; Kumar, Raghvendra; Mishra, Brojo Kishore; Chatterjee, Jyotir Moy; Khari, Manju (eds.). 3864: 452:

of images. It is a modification of the original DCT algorithm, and incorporates elements of inverse DCT and

22590: 22544: 22305: 22202: 21745: 21540: 21512: 21410: 21209: 20811: 20732: 20442: 20149: 20139: 20048: 19911: 19586: 19508: 19441: 19412: 19382: 19348: 19251: 18793: 18351: 18165: 16250: 14206: 3019:

accurately, and the more it can be compressed. (Here, we think of the DFT or DCT as approximations for the

2667: 2000: 1625: 1601: 1543: 1539: 1398: 1113: 16811: 16281: 4735:

step in JPEG), and a scaling can be chosen that allows the DCT to be computed with fewer multiplications.

2721:

makes use of DCT-based compression artifacts in her glitch art, particularly the DCT blocks found in most

1411:

The DCT-II is an important image compression technique. It is used in image compression standards such as

22534: 22041: 21481: 21390: 20319: 18341: 18336: 16507:

Cyber Security in Parallel and Distributed Computing: Concepts, Techniques, Applications and Case Studies

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Wang, Hanli; Kwong, S.; Kok, C. (2006). "Efficient prediction algorithm of integer DCT coefficients for

13601: 13380:

Each basis function is multiplied by its coefficient and then this product is added to the final image.

12731: 3661: 552: 329:. It described what is now called the type-II DCT (DCT-II), as well as the type-III inverse DCT (IDCT). 22620: 22600: 22595: 22393: 22315: 22254: 21961: 21363: 21340: 21192: 21167: 20657: 20294: 20101: 19553: 19283: 19210: 19048: 19028: 18972: 18630: 18421: 18224: 18131: 17559:

Feig, E.; Winograd, S. (July 1992a). "On the multiplicative complexity of discrete cosine transforms".

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for the standard DCT, and varied integer DCT sizes between 4x4 and 32x32 pixels. The DCT has a strong

22585: 22575: 22165: 22126: 21971: 21871: 21800: 21733: 21254: 20672: 20596: 20515: 19750: 19293: 19234: 18291: 16700: 16663: 16386: 14293:"T.81 – Digital compression and coding of continuous-tone still images – Requirements and guidelines" 13586: 12721: 2651: 2643: 1271: 1093: 813: 716: 188: 17714: 16946: 15729: 15719: 4523: 4401: 3003:, the boundary conditions are directly specified as a part of the problem being solved. Or, for the 444:

Nasir Ahmed also developed a lossless DCT algorithm with Giridhar Mandyam and Neeraj Magotra at the

22524: 21766: 21701: 21654: 21614: 21304: 21103: 21056: 20890: 20844: 20706: 20642: 20379: 19780: 19298: 18871: 18665: 18366: 17957: 17318: 15350:

Cosine-/Sine-Modulated Filter Banks: General Properties, Fast Algorithms and Integer Approximations

14350:

Smith, C.; Fralick, S. (1977). "A Fast Computational Algorithm for the Discrete Cosine Transform".

14211: 13835:

Discrete Cosine and Sine Transforms: General Properties, Fast Algorithms and Integer Approximations

11544:{\displaystyle ~\underbrace {\left} _{\text{Real}}+\underbrace {\left} _{\text{Recursive}}=\left~.} 10841:{\displaystyle +(-1)^{i+l}{\tilde {x}}\left(n_{1}+{\frac {n}{2}},n_{2},n_{3}+{\frac {n}{3}}\right)} 4682: 4572: 4450: 3820: 1650: 1097: 966: 657: 445: 414: 17120:

Queiroz, R. L.; Nguyen, T. Q. (1996). "Lapped transforms for efficient transform/subband coding".

16984:

Song, J.; SXiong, Z.; Liu, X.; Liu, Y., "An algorithm for layered video coding and transmission",

9640:

and are considered, the general formula for the calculation of the 3-D DCT-II can be expressed as

5978: 4303:

real inputs of even symmetry where the even-indexed elements are zero. That is, it is half of the

4273:

The DCT-II is probably the most commonly used form, and is often simply referred to as "the DCT".

3109: 22363: 22348: 22192: 22143: 22066: 21966: 21644: 21530: 21525: 21299: 21049: 21016: 20932: 20870: 20839: 20399: 20159: 19239: 18610: 18497: 18453: 18266: 18249: 18239: 18134:) C library that can compute fast DCTs (types I-IV) in one or more dimensions, of arbitrary size. 17877:

Martucci, S. A. (May 1994). "Symmetric convolution and the discrete sine and cosine transforms".

16229: 15095: 14831: 14240: 13658: 9576: 5172: 2984: 2599: 1394: 1275: 594: 574: 345: 306: 247: 184: 43:

Editors: please remove this warning only after the diffs listed ] have been checked for accuracy.

19648: 17658:

Narasimha, M.; Peterson, A. (June 1978). "On the Computation of the Discrete Cosine Transform".

17524:

Guoan Bi; Gang Li; Kai-Kuang Ma; Tan, T.C. (2000). "On the computation of two-dimensional DCT".

15870:

Hudson, Graham; Léger, Alain; Niss, Birger; Sebestyén, István; Vaaben, Jørgen (31 August 2018).

15416:

Beyond VoIP Protocols: Understanding Voice Technology and Networking Techniques for IP Telephony

5677: 4805: 413:

in 1987, following earlier work by Princen and Bradley in 1986. The MDCT is used in most modern

22285: 22071: 21886: 21831: 21826: 21639: 21604: 21147: 20880: 20727: 20354: 19591: 19532: 19520: 19374: 18864: 18615: 18399: 18244: 17952: 17830:

Feig, E.; Winograd, S. (September 1992b). "Fast algorithms for the discrete cosine transform".

17709: 17228: 16941: 13983: 12472: 6087: 6021: 6017: 5660:{\displaystyle X_{k}=\sum _{n=0}^{N-1}x_{n}\cos \left\qquad {\text{ for }}k=0,\ \ldots \ N-1~.} 5235: 4484: 4363: 4263:{\displaystyle X_{k}=\sum _{n=0}^{N-1}x_{n}\cos \left\qquad {\text{ for }}~k=0,\ \dots \ N-1~.} 3054: 2906: 2699: 2341: 2267: 2215: 2181: 2170: 1709: 1448: 1293: 1279: 1251: 1233: 1181: 602: 497: 426: 378: 364:

in 1976, and a type-II DST (DST-II) was then described by H.B. Kekra and J.K. Solanka in 1978.

298: 282: 228: 82: 19136: 17256: 16780: 16502: 16465: 15657: 15553: 15458: 15378: 15348: 15164: 15130:

Scalable Continuous Media Streaming Systems: Architecture, Design, Analysis and Implementation

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A royalty-free raster-graphics file format that supports both lossy and lossless compression.

22187: 21991: 21956: 21876: 21856: 21778: 21666: 21587: 21110: 20954: 20744: 19963: 19941: 19919: 19581: 19268: 17509:

Alshibami, O.; Boussakta, S. (July 2001). "Three-dimensional algorithm for the 3-D DCT-III".

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in number. The number of Multiplications and additions involved in RCF approach are given by

11137: 3628: 3539: 2691: 2198: 1865: 1685: 1535: 1237: 1133: 570: 533: 489: 430: 286: 136: 21520: 17479:

Chan, S.C.; Ho, K.L. (1990). "Direct methods for computing discrete sinusoidal transforms".

17382: 16794: 15308: 14556: 14387:"Comparative Analysis for Discrete Sine Transform as a suitable method for noise estimation" 7857: 5897: 5832: 5752: 5386: 4768: 4333: 4059: 3759: 22101: 22061: 22031: 21788: 21723: 21545: 21423: 21352: 21335: 21083: 20770: 19931: 19862: 19730: 18952: 18414: 18376: 18197: 17998: 17944: 17886: 17839: 17810: 17774: 17620: 17533: 17423: 17160: 16933: 16892: 16731: 16684:

Lutzky, Manfred; Schuller, Gerald; Gayer, Marc; Krämer, Ulrich; Wabnik, Stefan (May 2004).

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High Definition Television: The Creation, Development and Implementation of HDTV Technology

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boundary is even, correspond to the 8 types of DCT; the other half are the 8 types of DST.

2788: 1907: 1502: 1316: 1185: 1117: 1029: 982: 841: 824: 796: 752: 702: 521: 481: 449: 410: 16755: 13469: 12862: 12652: 6065:

Using the normalization conventions above, the inverse of DCT-I is DCT-I multiplied by 2/(

6034: 5331: 4901: 2956:

boundary. There are two sensible possibilities: either the data are even about the sample

2871: 2802: 8: 22111: 22051: 21810: 21772: 21570: 21555: 21071: 20739: 20531: 20437: 20384: 20096: 19891: 19867: 19815: 19571: 19474: 19421: 19183: 19074: 19033: 19018: 18987: 18982: 18891: 18798: 18731: 18700: 18685: 18468: 18026:

Cheng, L. Z.; Zeng, Y. H. (2003). "New fast algorithm for multidimensional type-IV DCT".

15798: 15761: 15384: 13524: 13498: 12697: 11849:

The image to the right shows a combination of horizontal and vertical frequencies for an

6016:

However, these variants seem to be rarely used in practice. One reason, perhaps, is that

5305: 4875: 4001: 3794: 3015: 2777: 2379: 2290: 1376: 1259: 1166: 1078: 1053: 934: 371: 357: 180: 18002: 17948: 17890: 17843: 17814: 17778: 17624: 17537: 17427: 17164: 17045:

Yeo, B.; Liu, B. (May 1995), "Volume rendering of DCT-based compressed 3D scalar data",

16937: 16896: 16359: 15963: 14868: 14460: 14261: 11044: 5955: 5932: 4310: 4283: 3949: 22338: 22295: 22226: 22096: 22026: 22001: 21936: 21783: 21504: 21380: 21375: 21347: 21162: 20677: 20662: 20202: 20174: 19857: 19820: 19256: 19226: 19205: 19111: 19043: 18937: 18625: 18441: 18431: 18326: 18306: 18301: 18014: 17983: 17970: 17636: 17610: 17439: 17413: 17102: 17028: 16629: 16575: 16528: 16501:

Srivastava, Saurabh Ranjan; Dube, Sachin; Shrivastaya, Gulshan; Sharma, Kavita (2019).

16390: 16222: 16135: 15941: 15852: 15810: 15180: 14966: 14925: 14880: 14777: 14747: 14630: 14472: 14134: 14082: 14041: 14009: 13876: 11916: 5877: 3929: 2851: 2831: 2781: 2675: 2055: 1791: 1742: 1713: 1482: 1379: 1365: 1320: 1215: 1203: 1083: 1057: 1048: 998: 883: 653: 517: 394: 140: 21120: 20021: 18837: 16381:

Valin, Jean-Marc; Maxwell, Gregory; Terriberry, Timothy B.; Vos, Koen (October 2013).

16154: 2944:

boundaries in the definitions below, respectively). Second, one has to specify around

22378: 22300: 22214: 22197: 22160: 22006: 21836: 21805: 21671: 21565: 21437: 21395: 21385: 21309: 21278: 21088: 21039: 20491: 20334: 19924: 19775: 19449: 19200: 19188: 19170: 19038: 18922: 18859: 18705: 18620: 18576: 18537: 18219: 18119: 18077: 17915: 17908: 17864: 17822: 17786: 17361: 17332: 17297: 17265: 17208: 17092: 17020: 16967: 16735: 16532: 16518: 16481: 16191: 16119: 16111: 16101: 16066: 15814: 15802: 15733: 15694: 15663: 15597: 15559: 15424: 15388: 15354: 15318: 15222: 15184: 15138: 15104: 15087: 14995: 14956: 14945:

2015 Fifth International Conference on Communication Systems and Network Technologies

14915: 14707: 14619:

ICASSP '87. IEEE International Conference on Acoustics, Speech, and Signal Processing

14597: 14566: 14497: 14427: 14386: 14367: 14269: 14118: 14110: 14100: 14045: 14033: 14025: 14017: 13880: 13860: 13852: 13842: 13783: 13773: 13403: 11843: 7873: 6124: 5671: 5265: 4716: 3696: 2898: 2575: 1823: 1811: 1634: 1616:

A graphic format that supports the lossy compression of digital images. Developed by

1434: 1416: 1255: 1247: 1121: 1089: 1070: 990: 948: 944: 853: 792: 760: 744: 683: 663: 646: 493: 469: 314: 196: 192: 86: 22046: 17032: 16633: 16183:

IEEE WESCANEX 97: communications, power, and computing : conference proceedings

16139: 15054: 14970: 14929: 14884: 14634: 14476: 14166: 14138: 22383: 22343: 22323: 22290: 22219: 22177: 22091: 21946: 21931: 21906: 21881: 21841: 21691: 21550: 21535: 21268: 21137: 21034: 21026: 20912: 20865: 20793: 20760: 19973: 19639: 19491: 19459: 19408: 19175: 19131: 19104: 19099: 18957: 18942: 18852: 18761: 18756: 18585: 18318: 18296: 18188: 18087: 18056: 18035: 18018: 18006: 17974: 17962: 17894: 17847: 17818: 17782: 17747: 17719: 17688: 17667: 17628: 17568: 17541: 17488: 17431: 17176: 17168: 17106: 17084: 17076: 17054: 17012: 16959: 16951: 16900: 16621: 16605:

Bleidt, R. L.; Sen, D.; Niedermeier, A.; Czelhan, B.; Füg, S.; et al. (2017).

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Alakuijala, Jyrki; Sneyers, Jon; Versari, Luca; Wassenberg, Jan (22 January 2021).

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National Association of Broadcasters Engineering Handbook: NAB Engineering Handbook

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Shishikui, Yoshiaki; Nakanishi, Hiroshi; Imaizumi, Hiroyuki (October 26–28, 1993).

15088:"Real-time software MPEG video decoder on multimedia-enhanced PA 7100LC processors" 14948: 14907: 14899: 14872: 14739: 14657: 14622: 14593: 14464: 14359: 14332: 14265: 14092: 14001: 2647: 2571: 2528: 2458: 1476: 1305: 1285: 1146: 1018: 978: 962: 956: 809: 788: 784: 679: 671: 563: 473: 453: 90: 18138: 17679:

Makhoul, J. (February 1980). "A fast cosine transform in one and two dimensions".

17640: 17443: 15659:

The Filmmaker's Handbook: A Comprehensive Guide for the Digital Age: Fifth Edition

14751: 12617:

are most common, but any other FFT algorithm is also applicable. For example, the

4715:(see below for the corresponding change in DCT-III). This makes the DCT-II matrix 409:(MDCT), was developed by John P. Princen, A.W. Johnson and Alan B. Bradley at the 22011: 21866: 21634: 21609: 21597: 21115: 20907: 20016: 19993: 19094: 18908: 18832: 18813: 18783: 18751: 18717: 18276: 18214: 17632: 17435: 16477: 16181: 16127: 16056: 15625: 15587: 15212: 15086:

Lee, Ruby Bei-Loh; Beck, John P.; Lamb, Joel; Severson, Kenneth E. (April 1995).

14126: 13868: 3000: 2773: 2419: 2322: 2219: 2174: 2127: 1919: 1843: 1746: 1717: 1357: 1297: 1289: 1063: 922: 845: 837: 833: 829: 675: 525: 513: 485: 200: 152: 20477: 16503:"Smartphone Triggered Security Challenges - Issues, Case Studies and Prevention" 16470:

Security in Computing and Communications: 5th International Symposium, SSCC 2017

14626: 9564:{\displaystyle \varphi _{i}={\frac {\pi }{2N}}(4N_{i}+1),{\text{ and }}i=1,2,3.} 3143:. There are several variants of the DCT with slightly modified definitions. The 22605: 22209: 22081: 22056: 22016: 21986: 21861: 21696: 21649: 21624: 21582: 21197: 21044: 20885: 20754: 19847: 18886: 18680: 18409: 18404: 18261: 18234: 18206: 17731: 17723: 17692: 17204: 16861: 16695: 16658: 16514: 14911: 14661: 14292: 14078: 13838: 13758: 6776:

in three dimensional space and mathematically can be calculated by the formula

6027:(The trivial real-even array, a length-one DFT (odd length) of a single number 4276:

This transform is exactly equivalent (up to an overall scale factor of 2) to a

3020: 2792: 2703: 2543: 2489: 2227: 2115: 1911: 1887: 1770: 1361: 1340: 1301: 1177: 1041: 1014: 986: 970: 930: 875: 739: 731: 712: 606: 548: 529: 501: 457: 390: 223: 98: 18060: 17966: 17752: 17735: 17671: 17080: 16546: 15890: 15871: 15848: 15589:

Digital cinema: the revolution in cinematography, postproduction, distribution

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MPEG-4 Enhanced Low Delay AAC - A New Standard for High Quality Communication

16625: 15806: 15108: 14371: 14021: 14013: 13864: 13787: 12562:

methods to compute DCTs are known as fast cosine transform (FCT) algorithms.

11931: dimensional domain. The 2-D DCT-IV of a matrix or an image is given by 3140: 3024: 2733: 2722: 2707: 2687: 2477: 2462: 2435: 2392: 2318: 2314: 2299: 2259: 2223: 2211: 2162: 2102: 1895: 1834: 1762: 1738: 1344: 1267: 1229: 1219: 1197: 1189: 1141: 1033: 1023: 1002: 952: 926: 820: 774: 667: 650: 633: 628: 509: 505: 477: 418: 176: 168: 156: 128: 124: 110: 94: 21223: 20457: 18039: 18010: 16367: 16123: 14096: 14005: 13872: 22236: 22076: 22021: 21951: 21916: 21851: 21750: 21740: 21592: 21442: 21433: 20917: 20682: 20472: 20467: 20324: 20299: 19795: 19314: 18281: 18256: 18157: 18113: 17024: 16971: 16442: 15872:"JPEG-1 standard 25 years: past, present, and future reasons for a success" 15025: 14130: 14037: 13589:, once the trivial and / or duplicate parts are eliminated and / or merged. 12605: 4612: 4611:

signal followed by a multiplication by half shift. This is demonstrated by

3752:

The DCT-I is exactly equivalent (up to an overall scale factor of 2), to a

2718: 2659: 2627: 1962: 1955: 1903: 1777: 1642: 1637:

based on HEVC compression. It improves compression over JPEG, and supports

1585: 1368: 1243: 1193: 1160: 1125: 1105: 994: 766: 735: 598: 374: 251: 18114:

Implementation of MPEG integer approximation of 8x8 IDCT (ISO/IEC 23002-2)

17762: 16131: 14952: 11731:

TABLE 1 Comparison of VR DIF & RCF Algorithms for computing 3D-DCT-II

589:

The DCT is widely used in many applications, which include the following.

577:

of arbitrary functions by series of Chebyshev polynomials, for example in

22436: 22086: 21996: 21981: 21941: 21901: 21760: 21330: 20981: 20922: 20860: 20462: 20225: 20220: 19805: 19725: 19454: 19273: 19151: 18947: 18823: 18773: 17233: 16986:

Proc. Fourth Int. Conf./Exh. High Performance Comput. Asia-Pacific Region

15593: 13429:

Contains a potentially easier to understand example of DCT transformation

5867:

one boundary and halfway between two data points for the other boundary.

3129: 2761: 2741: 2166: 2131: 1725: 1372: 1352: 1348: 1263: 1067: 1006: 858: 694: 368: 219: 70: 18145: 17058: 17016: 12896: 2916:

Illustration of the implicit even/odd extensions of DCT input data, for

2799:

of that function outside the domain. That is, once you write a function

2787:

The Fourier-related transforms that operate on a function over a finite

907:

operations such as the inverse operation between display color formats (

22441: 22148: 21846: 21755: 21711: 21681: 21659: 21462: 20895: 20788: 20197: 19968: 19837: 19800: 19735: 19619: 19496: 19405: 19330: 19121: 19116: 19003: 18962: 18768: 17736:"Fast and numerically stable algorithms for discrete cosine transforms" 17181: 17088: 16568:"Dolby AC-4: Audio Delivery for Next-Generation Entertainment Services" 15492: 13951: 2714: 2579: 2399: 2326: 2275: 2243: 1819: 1223: 1129: 1037: 770: 756: 706: 690: 18144:

Tim Kientzle: Fast algorithms for computing the 8-point DCT and IDCT,

17929: 17898: 17851: 17572: 17545: 17172: 16963: 16955: 16904: 14876: 14468: 11911:

half-cycles horizontally and a half-cycle vertically. The source data

5232:

term) and multiply the resulting matrix by an overall scale factor of

325:

in 1973. They presented their results in a January 1974 paper, titled

22426: 21891: 21706: 21473: 20964: 20566: 20369: 20349: 20169: 20164: 19827: 19785: 16089: 15725: 15690: 15314: 15304: 15218: 15208: 14070: 13987: 13830: 7861:

The four basic stages of computing 3-D DCT-II using VR DIF Algorithm.

2765: 2663: 2334: 2283: 1734: 1638: 1312: 1109: 892: 778: 723: 318: 302: 78: 16836: 12911:

Original size, scaled 10x (nearest neighbor), scaled 10x (bilinear).

3946:

less than 2, while all other DCT types are defined for any positive

2952:

of four equally spaced data points, and say that we specify an even

402:

coding technique for video compression from the late 1980s onwards.

22421: 22411: 22328: 22153: 21976: 21418: 20942: 20821: 20452: 20344: 19770: 19624: 19244: 19089: 18746: 17794: 17763:"Fast fourier transforms: A tutorial review and a state of the art" 17615: 17418: 16395: 16346:

Herre, J.; Dietz, M. (2008). "MPEG-4 high-efficiency AAC coding ".

16159: 15915: 15630: 15529: 15172: 14991: 14852: 14236: 6020:

algorithms for odd-length DFTs are generally more complicated than

2983:

These choices lead to all the standard variations of DCTs and also

2757: 2655: 2470: 2407: 2388: 2351: 2147: 2031: 1923: 1883: 1799: 1334: 1324: 1137: 1101: 904: 801: 727: 641: 15961: 14314: 14087: 13856: 5711:

A variant of the DCT-IV, where data from different transforms are

4679:

and multiply the rest of the matrix by an overall scale factor of

1790:

Storage and handling of digital images in broadcast applications,

22416: 22401: 21619: 21157: 21098: 21011: 20765: 20652: 20611: 20606: 20591: 20586: 20581: 20576: 20571: 20432: 20374: 20314: 20078: 20058: 20043: 19832: 19760: 19740: 19013: 18487: 18436: 16255: 15497: 15377:

Jones, Graham A.; Layer, David H.; Osenkowsky, Thomas G. (2013).

14699:

Mobile Multimedia Broadcasting Standards: Technology and Practice

13759:"Reminiscences of the Early Work in DCT: Interview with K.R. Rao" 13383: 12679: 11727:

respectively. From Table 1, it can be seen that the total number

6069: − 1). The inverse of DCT-IV is DCT-IV multiplied by 2/ 2912: 2516: 2500: 2493: 2330: 2123: 2063: 2059: 1951: 1915: 1891: 1663: 1593: 1534:

The integer DCT, an integer approximation of the DCT, is used in

618: 367:

In 1975, John A. Roese and Guner S. Robinson adapted the DCT for

263: 259: 19404: 18151: 16115: 14114: 14029: 12377:

Although the direct application of these formulas would require

2948:

the function is even or odd. In particular, consider a sequence

1542:(HEVC), introduced in 2013. The integer DCT is also used in the 22446: 22406: 21728: 20959: 20417: 20111: 19951: 19872: 19852: 19790: 19527: 19503: 19481: 19469: 19464: 18527: 18068:

Press, WH; Teukolsky, SA; Vetterling, WT; Flannery, BP (2007),

16919:"Variable temporal-length 3-D discrete cosine transform coding" 16815: 16500: 15910: 14075:

Discrete Cosine Transform: Algorithms, Advantages, Applications

12907: 12423:

operations, it is possible to compute the same thing with only

4724: 3050: 2897:

was not specified. The DFT, like the Fourier series, implies a

2698:. This process can cause blocking artifacts, primarily at high 2449: 2306: 2235: 2017:

standard, having substantially improved compression capability

1984: 1930: 1873: 1753: 1617: 1551: 1438: 1383: 900: 896: 871: 434: 274: 172: 74: 17523: 17356:

Pennebaker, William B.; Mitchell, Joan L. (31 December 1992).

15837:

IEEE Transactions on Circuits and Systems for Video Technology

12851:{\displaystyle ~{\tfrac {17}{9}}N\log _{2}N+{\mathcal {O}}(N)} 3036:

yields a continuous extension at the boundaries (although the

22431: 22368: 21676: 21006: 20937: 20780: 20556: 20519: 20404: 20106: 20073: 20068: 20063: 20053: 20038: 20033: 20028: 20011: 20003: 19765: 19745: 19576: 19566: 19561: 19549: 19537: 19362: 18967: 18560: 18507: 17702:

IEEE Transactions on Acoustics, Speech, and Signal Processing

17681:

IEEE Transactions on Acoustics, Speech, and Signal Processing

14650:

IEEE Transactions on Acoustics, Speech, and Signal Processing

14527: 14299: 3037: 2563: 2550: 2524: 2509: 2372: 2051: 2022: 1992: 1852: 1830: 1699: 1442: 1426: 1421: 1172: 888: 879: 867: 456:. It is a more effective lossless compression algorithm than 386: 267: 183:). DCTs are also important to numerous other applications in 164: 160: 119: 15783: 15721:

Handbook of Internet and Multimedia Systems and Applications

12915: 9095:{\displaystyle 0\leq n_{1},n_{2},n_{3}\leq {\frac {N}{2}}-1} 2972:

and the previous point, in which case the even extension is

1584:

The most widely used image compression standard and digital

468:

The DCT is the most widely used transformation technique in

22373: 20947: 20692: 20647: 20637: 20627: 20601: 20561: 20551: 20447: 20389: 20309: 20272: 20267: 20184: 20126: 20116: 19877: 19755: 19629: 19609: 18517: 18371: 18356: 18346: 18109:

The Discrete Cosine Transform (DCT): Theory and Application

18046: 17699: 16604: 16220: 16094:

Techniques and Standards for Image, Video, and Audio Coding

15448: 15446: 15444: 15442: 15440: 15413:

Hersent, Olivier; Petit, Jean-Pierre; Gurle, David (2005).

14863:. International Society for Optics and Photonics: 474–478. 14857:

Digital Video Compression: Algorithms and Technologies 1995

14732:

IEEE Transactions on Audio, Speech, and Language Processing

14558:

Standard Codecs: Image Compression to Advanced Video Coding

14455:. International Society for Optics and Photonics: 172–181. 14165:

Barbero, M.; Hofmann, H.; Wells, N. D. (14 November 1991).

12725: 12683: 12642: 12471:

complexity by factorizing the computation similarly to the

6123:(see above), this can be used to make the transform matrix 5870:

In other words, DCT types I–IV are equivalent to real-even

4728: 2745: 2726: 2620: 2586: 2485: 2111: 1988: 1943: 1899: 1795: 1608: 1576: 1547: 1430: 1412: 338: 148: 144: 114: 106: 102: 17198: 16646: 15162: 14942: 11838: 5894:

is even or odd), since the corresponding DFT is of length

3699:, if one further multiplies by an overall scale factor of 2936:

the left and right boundaries of the domain (i.e. the min-

2772:(DFT), a DCT operates on a function at a finite number of 1026:

assessment — DCT-based quality degradation metric (DCT QM)

20986: 20974: 20816: 20632: 20536: 20339: 19936: 19678: 19673: 19668: 19663: 19658: 19653: 18492: 18458: 16683: 16464:

Hazra, Sudip; Mateti, Prabhaker (September 13–16, 2017).

16410: 16380: 15635: 4723:

of half-shifted input. This is the normalization used by

3791:

real numbers with even symmetry. For example, a DCT-I of

2730: 2466: 2231: 2202: 2154: 2119: 2047: 2038: 1975: 1803: 1066:— medical image compression, image fusion, watermarking, 916: 912: 908: 438: 422: 382: 132: 17914:(2nd ed.), Upper Saddle River, N.J: Prentice Hall, 17047:

IEEE Transactions on Visualization and Computer Graphics

16607:"Development of the MPEG-H TV Audio System for ATSC 3.0" 15869: 15437: 15021:"What Is a JPEG? The Invisible Object You See Every Day" 14167:"DCT source coding and current implementations for HDTV" 12475:(FFT). One can also compute DCTs via FFTs combined with 11927:

The M-D DCT-IV is just an extension of 1-D DCT-IV on to

5268:, but breaks the direct correspondence with a real-even 4719:, but breaks the direct correspondence with a real-even 3971:

Thus, the DCT-I corresponds to the boundary conditions:

2828:

as a sum of sinusoids, you can evaluate that sum at any

551:, the DCT can approach the compaction efficiency of the 492:

requirements, which is significantly reduced by the DCT

16:

Technique used in signal processing and data compression

18127: 16882: 16438:"WhatsApp laid bare: Info-sucking app's innards probed" 16323: 16301: 15485: 12904:

Consider this 8x8 grayscale image of capital letter A.

1766: 1274:, local distortion detection probability (LDDP) model, 18076:(3rd ed.), New York: Cambridge University Press, 16383:

High-Quality, Low-Delay Music Coding in the Opus Codec

16186:. University of Manitoba, Winnipeg, Manitoba, Canada: 13772:. Tampere International Center for Signal Processing. 12933: 12799: 11225: 11150: 11097: 9816: 9773: 9730: 6090: 5680: 5593: 5565: 5537: 5238: 5056: 5029: 4955: 4685: 4190: 4163: 3745:

but breaks the direct correspondence with a real-even

3709: 3411: 3099:{\displaystyle f:\mathbb {R} ^{N}\to \mathbb {R} ^{N}} 472:, and by far the most widely used linear transform in 448:

in 1995. This allows the DCT technique to be used for

85:

in 1972, is a widely used transformation technique in

17982:

Boussakta, Said.; Alshibami, Hamoud O. (April 2004).

17906:

Oppenheim, Alan; Schafer, Ronald; Buck, John (1999),

16268:

The first videos to receive YouTube's AV1 transcodes.

16100:: Chapter 9; MPEG-1: Chapter 10; MPEG-2: Chapter 11. 15938:"HEIF Comparison - High Efficiency Image File Format" 15488:"More Efficient Mobile Encodes for Netflix Downloads" 14616: 13661: 13604: 13553: 13527: 13501: 13472: 12927: 12865: 12794: 12734: 12700: 12655: 12571: 12520: 12481: 12429: 12383: 12294: 12220: 11940: 11858: 11629: 11560: 11295: 11215: 11140: 11092: 11047: 10857: 10725: 10500: 10275: 10038: 9657: 9619: 9579: 9478: 9179: 9112: 9030: 7871: 7318: 6785: 6154: 6037: 5981: 5958: 5935: 5900: 5880: 5835: 5792: 5755: 5728: 5473: 5426: 5389: 5362: 5334: 5308: 5281: 5199: 5175: 5148: 4940: 4904: 4878: 4851: 4808: 4771: 4744: 4651: 4624: 4575: 4526: 4487: 4453: 4404: 4366: 4336: 4313: 4286: 4099: 4062: 4030: 4004: 3977: 3952: 3932: 3867: 3861:

is exactly equivalent to a DFT of eight real numbers

3823: 3797: 3762: 3705: 3664: 3631: 3604: 3575: 3542: 3515: 3280: 3218: 3153: 3112: 3062: 2874: 2854: 2834: 2805: 1505: 1485: 1451: 17905: 16242: 16223:"AV1 Bitstream & Decoding Process Specification" 15376: 15165:"An HDTV Coding Scheme using Adaptive-Dimension DCT" 13766:

Reprints from the Early Days of Information Sciences

281:

property, capable of achieving high quality at high

17511:

Proc. Sixth Int. Symp. Commun., Theory Applications

16295: 15085: 14851: 14829: 14164: 6073:. The inverse of DCT-II is DCT-III multiplied by 2/ 262:approximation of the standard DCT, used in several 18074:Numerical Recipes: The Art of Scientific Computing 17981: 17907: 17799:"How I came up with the discrete cosine transform" 17508: 17355: 17132: 16248: 16000: 15998: 15551: 15452: 15302: 15206: 14770:"The Use of FFT and MDCT in MP3 Audio Compression" 14241:"How I Came Up With the Discrete Cosine Transform" 13671: 13647: 13578: 13539: 13513: 13487: 13368: 12880: 12850: 12777: 12712: 12670: 12596: 12554: 12506: 12463: 12415: 12358:{\displaystyle ~~\ell =0,\ 1,\ 2,\ \ldots \ M-1~.} 12357: 12280: 12202: 11902: 11719: 11615: 11543: 11281: 11200: 11126: 11078: 11025: 10840: 10708: 10483: 10258: 10018: 9632: 9605: 9563: 9461: 9160:{\displaystyle c(\varphi _{i})=\cos(\varphi _{i})} 9159: 9106:together as shown in the figure just below, where 9094: 9014: 7823: 7290: 6753: 6108: 6049: 6005: 5967: 5944: 5921: 5886: 5851: 5821: 5778: 5741: 5701: 5659: 5452: 5412: 5375: 5349: 5320: 5294: 5256: 5224: 5185: 5161: 5128: 4919: 4890: 4864: 4838: 4794: 4757: 4727:, for example, see. In many applications, such as 4707: 4671: 4637: 4599: 4561: 4512: 4474: 4439: 4391: 4352: 4322: 4295: 4262: 4078: 4048: 4016: 3990: 3961: 3938: 3915: 3853: 3809: 3783: 3737: 3687: 3650: 3617: 3590: 3561: 3528: 3498: 3258: 3200: 3120: 3098: 2901:extension of the original function. A DCT, like a 2889: 2860: 2840: 2820: 1523: 1491: 1463: 289:can appear when heavy DCT compression is applied. 17657: 17588:Fast Fourier transform and convolution algorithms 16509:. John Wiley & Sons. pp. 187–206 (200). 16188:Institute of Electrical and Electronics Engineers 15412: 14385:Dhamija, Swati; Jain, Priyanka (September 2011). 13982: 13757:Stanković, Radomir S.; Astola, Jaakko T. (2012). 12687: 12369:also has several applications in various fields. 3926:Note, however, that the DCT-I is not defined for 348:(DST) was derived from the DCT, by replacing the 34:may misquote or misrepresent many of its sources. 22567: 17380: 16326:"Bringing AV1 Streaming to Netflix Members' TVs" 14320: 13680: 3738:{\displaystyle {\sqrt {{\tfrac {2}{N-1\,}}\,}},} 1886:recording, compression and distribution format, 250:(DST), which is equivalent to a DFT of real and 17760: 17481:IEE Proceedings F - Radar and Signal Processing 15995: 14832:"Vorbis I specification - 1.1.2 Classification" 14725: 14723: 14449:Efficient Transmission of Pictorial Information 14391:IJCSI International Journal of Computer Science 13990:(1 January 1974). "Discrete Cosine Transform". 13756: 12720:real-data FFT is also performed by a real-data 12626: 9647:The single butterfly stage of VR DIF algorithm. 2748:artifacts as the basis of the picture's style. 17194: 17192: 16317: 15552:Ochoa-Dominguez, Humberto; Rao, K. R. (2019). 14825: 14823: 14797: 14795: 14647: 13903:"Beyond resolution: Rosa Menkman's glitch art" 13828: 13824: 13822: 13820: 13818: 13816: 9170:The original 3-D DCT-II now can be written as 7309:and can be computed from the formula given by 4607:DCT-II transformation is also possible using 2 2795:, can be thought of as implicitly defining an 1641:with much more efficient compression than the 895:and color differences, color formats (such as 573:, and fast DCT algorithms (below) are used in 341:'s lossy image compression algorithm in 1992. 21489: 21253: 21239: 19390: 18173: 17829: 17558: 17119: 16779:. Event occurs at 65 minutes. Archived from 15581: 15579: 15577: 15575: 14897: 14446: 12788:A recent reduction in the operation count to 12622: 12281:{\displaystyle ~~k=0,\ 1,\ 2\ \ldots \ N-1~~} 6013:in the denominators of the cosine arguments. 5275:The DCT-III implies the boundary conditions: 3049:Formally, the discrete cosine transform is a 313:in 1972. The DCT was originally intended for 18187: 17927: 17730: 16983: 16719: 16717: 16547:"Open Source Software used in PlayStation 4" 15830: 15716:"Content-Based Video Browsing And Retrieval" 15655: 15158: 15156: 15154: 14720: 14349: 13752: 13750: 13748: 13746: 13744: 13742: 13740: 13738: 13736: 13734: 13732: 13730: 13728: 13726: 13724: 13722: 13720: 12610: 12514:pre- and post-processing steps. In general, 5722:The DCT-IV implies the boundary conditions: 4738:The DCT-II implies the boundary conditions: 2637: 705:, fingerprint-based biometric watermarking, 22504:Global telecommunications regulation bodies 18137:Takuya Ooura: General Purpose FFT Package, 17928:Frigo, M.; Johnson, S. G. (February 2005). 17600: 17403: 17358:JPEG: Still Image Data Compression Standard 17189: 16539: 16463: 16324:Netflix Technology Blog (9 November 2021). 16302:Netflix Technology Blog (5 February 2020). 15619: 15617: 15516: 15514: 15346: 15342: 15340: 15338: 15336: 15334: 15202: 15200: 15015: 15013: 15011: 14820: 14801: 14792: 14550: 14548: 14546: 14384: 14065: 14063: 14061: 14059: 14057: 14055: 13942: 13813: 13718: 13716: 13714: 13712: 13710: 13708: 13706: 13704: 13702: 13700: 22540: 21496: 21482: 21246: 21232: 19397: 19383: 18180: 18166: 17590:(1st ed.). New York: Springer-Verlag. 17585: 16829: 16766: 16560: 16494: 16345: 15572: 15481: 15479: 15477: 15475: 15408: 15406: 15404: 15081: 15079: 14763: 14761: 13896: 13894: 13892: 13890: 11987: 3201:{\displaystyle ~x_{0},\ \ldots \ x_{N-1}~} 2784:from the DFT or other related transforms. 393:scene with image quality comparable to an 18025: 17956: 17751: 17713: 17614: 17417: 17381:Arai, Y.; Agui, T.; Nakajima, M. (1988). 17180: 17150: 17070: 16945: 16723: 16714: 16394: 16279: 15889: 15682: 15555:Discrete Cosine Transform, Second Edition 15486:Netflix Technology Blog (19 April 2017). 15455:"Inside iPhone 4: FaceTime video calling" 15310:Discrete Cosine Transform, Second Edition 15298: 15296: 15294: 15292: 15290: 15288: 15286: 15284: 15282: 15280: 15278: 15276: 15274: 15272: 15270: 15268: 15266: 15264: 15262: 15260: 15258: 15214:Discrete Cosine Transform, Second Edition 15151: 14936: 14906:. Vol. 3. pp. 1769–1772 vol.3. 14563:Institution of Engineering and Technology 14160: 14158: 14156: 14154: 14152: 14150: 14148: 14086: 12555:{\displaystyle ~{\mathcal {O}}(N\log N)~} 12464:{\displaystyle ~{\mathcal {O}}(N\log N)~} 12174: 12131: 12094: 12051: 11842:Two-dimensional DCT frequencies from the 6031: , corresponds to a DCT-V of length 5609: 5552: 5544: 5540: 5535: 5193:instead of by 2 (resulting in an overall 5075: 5036: 5032: 5027: 4832: 4668: 4665: 4209: 4170: 4166: 4161: 3729: 3724: 3681: 3678: 3584: 3581: 3445: 3441: 3437: 3432: 3422: 3409: 3259:{\displaystyle X_{0},\,\ldots ,\,X_{N-1}} 3239: 3232: 3114: 3086: 3071: 3010:In particular, it is well known that any 1406: 377:. They experimented with the DCT and the 17930:"The Design and Implementation of FFTW3" 17876: 17861:Signal Processing with Lapped Transforms 17761:Duhamel, P.; Vetterli, M. (April 1990). 17467: 17333:"Discrete cosine transform - MATLAB dct" 17002: 16429: 16403: 16249:YouTube Developers (15 September 2018). 16088: 15623: 15614: 15585: 15511: 15372: 15370: 15331: 15256: 15254: 15252: 15250: 15248: 15246: 15244: 15242: 15240: 15238: 15197: 15063:École Polytechnique Fédérale de Lausanne 15053:Pessina, Laure-Anne (12 December 2014). 15008: 14729: 14554: 14543: 14052: 13978: 13976: 13974: 13972: 13970: 13968: 13900: 13697: 13382: 12914: 12906: 12895: 11837: 11134:butterflies. The whole 3-D DCT requires 11127:{\displaystyle ~{\tfrac {1}{8}}\ N^{3}~} 11036: 9642: 7856: 6130: 2911: 2681: 1471:blocks are computed and the results are 17678: 17561:IEEE Transactions on Information Theory 17316: 17254: 17248: 17201:The Essential Guide to Image Processing 17038: 16996: 16977: 16916: 16910: 16640: 16280:Brinkmann, Martin (13 September 2018). 16054: 15826: 15824: 15656:Ascher, Steven; Pincus, Edward (2012). 15472: 15401: 15347:Britanak, Vladimir; Rao, K. R. (2017). 15076: 15052: 15048: 15046: 15044: 14767: 14758: 14691: 14231: 14229: 14207:"Video on demand: Research Paper 94/68" 14077:. Signal, Image and Speech Processing. 14073:; Yip, Patrick C. (11 September 1990). 13938: 13936: 13934: 13932: 13930: 13928: 13887: 12859:routine might be significant for small 12691: 12416:{\displaystyle ~{\mathcal {O}}(N^{2})~} 2964:, or the data are even about the point 2686:A common issue with DCT compression in 2614: 2608:Mobile telephony, VoIP, wideband audio 2558:VoIP, wideband audio, mobile telephony 2072: 1445:. There, the two-dimensional DCT-II of 1009:level estimation, mirroring, rotation, 584: 22568: 21503: 18028:IEEE Transactions on Signal Processing 17991:IEEE Transactions on Signal Processing 17879:IEEE Transactions on Signal Processing 17858: 17832:IEEE Transactions on Signal Processing 17526:IEEE Transactions on Signal Processing 17504: 17502: 17478: 17456: 17292:Ruff, Thomas (May 31, 2009). "jpegs". 17138: 17044: 16876: 16772: 16677: 16598: 16435: 16304:"Netflix Now Streaming AV1 on Android" 16282:"How to enable AV1 support on YouTube" 16273: 16221:Peter de Rivaz; Jack Haughton (2018). 16084: 16082: 15863: 15751: 15749: 15626:"Copy protections could come to JPegs" 15122: 15120: 15118: 14900:"Reversible discrete cosine transform" 14689: 14687: 14685: 14683: 14681: 14679: 14677: 14675: 14673: 14671: 14517: 14515: 14513: 14489: 14287: 14285: 14283: 14281: 14279: 14145: 14069: 3916:{\displaystyle a\ b\ c\ d\ e\ d\ c\ b} 2960:, in which case the even extension is 460:. Lossless DCT is also known as LDCT. 21477: 21227: 19378: 18161: 17793: 16855:"Enhanced Voice Services (EVS) Codec" 16804: 16748: 16457: 16374: 16339: 16205:is similar to, but more complex than 16152: 16063:Springer Science & Business Media 15713: 15367: 15235: 14704:Springer Science & Business Media 14440: 14424:Springer Science & Business Media 14415: 14235: 14200: 14198: 14196: 14194: 14192: 13965: 13829:Britanak, Vladimir; Yip, Patrick C.; 13797:from the original on 30 December 2021 6060: 5874:of even order (regardless of whether 2251:High-Efficiency Advanced Audio Coding 2046:An open source format based on VP10 ( 22550: 17291: 16727:VoIP Voice and Fax Signal Processing 16549:. Sony Interactive Entertainment Inc 16472:. Springer. pp. 286–299 (290). 16029: 16004: 15821: 15041: 14983: 14621:. Vol. 12. pp. 2161–2164. 14343: 14226: 13943:Thomson, Gavin; Shah, Athar (2017). 13925: 11919:of these 64 frequency squares. 11041:The whole 3-D DCT calculation needs 7845: 3044: 2909:extension of the original function. 2751: 2369:China Multimedia Mobile Broadcasting 2058:; used by content providers such as 1036:auto inspection, image restoration, 558:DCTs are widely employed in solving 397:requiring 2-bit per pixel. In 1979, 222:. The DCTs are generally related to 18: 18049:IEEE Transactions on Communications 17984:"Fast Algorithm for the 3-D DCT-II" 17740:Linear Algebra and Its Applications 17660:IEEE Transactions on Communications 17499: 16847: 16214: 16155:"The H.320 Recommendation Overview" 16079: 16058:Data Compression in Digital Systems 15955: 15746: 15624:Baraniuk, Chris (15 October 2015). 15453:Daniel Eran Dilger (June 8, 2010). 15126: 15115: 14987:The Electrical Engineering Handbook 14898:Komatsu, K.; Sezaki, Kaoru (1998). 14695: 14668: 14510: 14352:IEEE Transactions on Communications 14324:IEEE Transactions on Communications 14276: 14204: 14000:(1). IEEE Computer Society: 90–93. 13579:{\displaystyle ~{\mathcal {O}}(N)~} 12597:{\displaystyle ~{\mathcal {O}}(N)~} 12507:{\displaystyle ~{\mathcal {O}}(N)~} 2650:. The DCT is widely implemented in 1858:Integrated Services Digital Network 1546:(HEIF), which uses a subset of the 77:functions oscillating at different 13: 21188:Comparison of audio coding formats 18070:"Section 12.4.2. Cosine Transform" 17651: 16096:. Prentice Hall. JPEG: Chapter 8; 15787:Measurement Science and Technology 15181:10.1016/B978-0-444-81844-7.50072-3 14830:Xiph.Org Foundation (2009-06-02). 14585: 14579: 14189: 13648:{\displaystyle ~2N\log _{2}N-N+2~} 13559: 13466:The radix-4 step reduces the size 13438:Modified discrete cosine transform 13433:List of Fourier-related transforms 12891: 12834: 12778:{\displaystyle ~2N\log _{2}N-N+2~} 12577: 12526: 12487: 12435: 12389: 5717:modified discrete cosine transform 4618:Some authors further multiply the 3688:{\displaystyle 1/{\sqrt {2\,}}\,,} 3509:Some authors further multiply the 3266:according to one of the formulas: 2670:. DCTs are also commonly used for 2139:Adaptive Transform Acoustic Coding 2079:Modified discrete cosine transform 407:modified discrete cosine transform 309:. The concept was proposed to the 256:modified discrete cosine transform 14: 22632: 18101: 17383:"A fast DCT-SQ scheme for images" 17264:. Institute of Network Cultures. 17005:IEEE Trans. Inf. Technol. Biomed. 16724:Nagireddi, Sivannarayana (2008). 16614:IEEE Transactions on Broadcasting 16466:"Challenges in Android Forensics" 16032:"Video Coding Standards: Part II" 15906:"The JPEG image format explained" 13901:Alikhani, Darya (April 1, 2015). 13683:for the size-8 case used in JPEG. 11903:{\displaystyle (~N_{1}=N_{2}=8~)} 9573:If the even and the odd parts of 5225:{\displaystyle x_{0}/{\sqrt {2}}} 4672:{\displaystyle 1/{\sqrt {N\,}}\,} 3598:and correspondingly multiply the 3132:), or equivalently an invertible 2924:However, because DCTs operate on 1848:public switched telephone network 623:time-domain aliasing cancellation 246:. Two related transforms are the 69:) expresses a finite sequence of 22549: 22539: 22530: 22529: 22518: 22139:Free-space optical communication 21284:Nyquist–Shannon sampling theorem 19354: 19353: 19344: 19343: 17594: 17579: 17552: 17517: 17472: 17461: 17450: 17397: 17374: 17349: 17317:Colberg, Jörg (April 17, 2009). 16885:IEEE Trans. Geosci. Remote Sens. 16795:"CELT codec presentation slides" 16687:A guideline to audio codec delay 16436:Leyden, John (27 October 2015). 16417:(Home page). Xiph.org Foundation 16007:"Video Coding Standards: Part I" 14816:from the original on 2017-02-13. 13681:Arai, Agui & Nakajima (1988) 13592: 13460: 13450: 11086:stages, and each stage involves 3591:{\displaystyle {\sqrt {2\,}}\,,} 2427: 2412:ultra-high-definition television 2083: 1677: 1557: 496:technique, capable of achieving 335:Joint Photographic Experts Group 323:University of Texas at Arlington 23: 21370:Discrete-time Fourier transform 17910:Discrete-Time Signal Processing 17325: 17310: 17285: 17221: 17144: 17113: 17075:. Vol. 2. pp. 73–76. 17064: 16348:IEEE Signal Processing Magazine 16190:. May 22–23, 1997. p. 30. 16174: 16146: 16092:; Hwang, J. J. (18 July 1996). 16048: 16023: 15980:from the original on 2 May 2021 15930: 15898: 15777: 15757:"Apple ProRes 422 Codec Family" 15707: 15676: 15649: 15545: 14977: 14891: 14845: 14802:Brandenburg, Karlheinz (1999). 14641: 14610: 14483: 14409: 14378: 12688:Narasimha & Peterson (1978) 7840:multidimensional FFT algorithms 7762: 7229: 5615: 5081: 4215: 3451: 2642:DCT plays an important role in 2367:China national audio standard, 1970:Professional video production. 1798:, cable, satellite, high-speed 1538:(AVC), introduced in 2003, and 1208:digital television broadcasting 463: 297:The DCT was first conceived by 20750:ISO/IEC base media file format 17319:"Review: jpegs by Thomas Ruff" 17255:Menkman, Rosa (October 2011). 16917:Chan, Y.; Siu, W. (May 1997), 16776:Presentation of the CELT codec 16153:Davis, Andrew (13 June 1997). 15799:10.1088/0957-0233/23/11/114003 14523:"History of Video Compression" 13993:IEEE Transactions on Computers 13570: 13564: 12845: 12839: 12588: 12582: 12546: 12531: 12498: 12492: 12455: 12440: 12407: 12394: 12372: 12165: 12150: 12147: 12132: 12085: 12070: 12067: 12052: 11897: 11859: 11070: 11051: 10898: 10871: 10861: 10760: 10739: 10729: 10628: 10607: 10597: 10529: 10514: 10504: 10403: 10382: 10372: 10304: 10289: 10279: 10191: 10176: 10166: 10160: 10121: 10115: 10103: 10064: 10046: 10013: 10010: 9988: 9982: 9973: 9951: 9945: 9936: 9914: 9908: 9899: 9860: 9842: 9700: 9661: 9529: 9507: 9456: 9440: 9431: 9415: 9406: 9390: 9381: 9342: 9336: 9222: 9183: 9154: 9141: 9129: 9116: 9005: 8939: 8930: 8855: 8849: 8836: 8776: 8767: 8704: 8698: 8685: 8625: 8616: 8553: 8547: 8534: 8480: 8471: 8420: 8414: 8401: 8341: 8332: 8269: 8263: 8250: 8196: 8187: 8136: 8130: 8117: 8063: 8054: 8003: 7997: 7984: 7936: 7927: 7888: 7882: 5916: 5904: 4708:{\textstyle {\sqrt {{2}/{N}}}} 4562:{\displaystyle y_{4N-n}=y_{n}} 4440:{\displaystyle y_{2n+1}=x_{n}} 3778: 3766: 3355: 3330: 3320: 3304: 3081: 2997:partial differential equations 2884: 2878: 2815: 2809: 2791:, such as the DFT or DCT or a 1518: 1506: 1212:standard-definition television 560:partial differential equations 205:partial differential equations 203:for the numerical solution of 1: 21315:Statistical signal processing 17734:; Tasche, M. (January 2005). 15877:Journal of Electronic Imaging 15522:"Video Developer Report 2019" 14768:Guckert, John (Spring 2012). 13690: 12785:real-arithmetic operations). 6139: 5861: 4600:{\displaystyle 0<n<2N.} 4475:{\displaystyle 0\leq n<N,} 3854:{\displaystyle a\ b\ c\ d\ e} 3695:which makes the DCT-I matrix 2713:DCT blocks are often used in 2359:Dynamic Resolution Adaptation 1938:Internet video, web browsers 975:content-based image retrieval 81:. The DCT, first proposed by 22616:Lossy compression algorithms 22525:Telecommunication portal 22306:Telecommunications equipment 17823:10.1016/1051-2004(91)90086-Z 17787:10.1016/0165-1684(90)90158-U 17633:10.1016/j.sigpro.2008.01.004 17436:10.1016/j.sigpro.2008.01.004 16478:10.1007/978-981-10-6898-0_24 14270:10.1016/1051-2004(91)90086-Z 11922: 6006:{\displaystyle N\pm {1}/{2}} 3121:{\displaystyle \mathbb {R} } 2668:analog-to-digital conversion 2001:High Efficiency Video Coding 1626:High Efficiency Image Format 1602:Open XML Paper Specification 1544:High Efficiency Image Format 1540:High Efficiency Video Coding 1114:analog-to-digital conversion 569:DCTs are closely related to 7: 22042:Alexander Stepanovich Popov 16579:. June 2015. Archived from 15686:Image Processing for Cinema 15683:Bertalmio, Marcelo (2014). 15662:. Penguin. pp. 246–7. 15303:Ochoa-Dominguez, Humberto; 15207:Ochoa-Dominguez, Humberto; 14627:10.1109/ICASSP.1987.1169405 14555:Ghanbari, Mohammed (2003). 14175:European Broadcasting Union 13945:"Introducing HEIF and HEVC" 13672:{\displaystyle {\sqrt {2}}} 13392: 12627:Duhamel & Vetterli 1990 12613:). Algorithms based on the 11833:Cooley–Tukey FFT algorithms 9606:{\displaystyle k_{1},k_{2}} 5702:{\textstyle {\sqrt {2/N}}.} 5186:{\displaystyle {\sqrt {2}}} 3041:computational convenience. 2631: 2191:Digital audio radio service 2091:Audio compression standard 1983:A video codec developed by 709:identification/recognition 311:National Science Foundation 10: 22637: 21746:Telecommunications history 21364:Discrete Fourier transform 21341:Matched Z-transform method 21193:Comparison of video codecs 19703:(GB/T 33475.2,GY/T 299.1) 19235:Compressed data structures 18557:RLE + BWT + MTF + Huffman 18225:Asymmetric numeral systems 17724:10.1109/TASSP.1987.1165220 17693:10.1109/TASSP.1980.1163351 17153:IEEE Trans. Signal Process 17122:IEEE Trans. Signal Process 16926:IEEE Trans. Image Process. 16515:10.1002/9781119488330.ch12 16251:"AV1 Beta Launch Playlist" 15558:. CRC Press. p. 186. 14912:10.1109/ICASSP.1998.681802 14662:10.1109/TASSP.1986.1164954 14490:Cianci, Philip J. (2014). 13399:Discrete wavelet transform 12615:Cooley–Tukey FFT algorithm 11829:Cooley–Tukey FFT algorithm 6109:{\textstyle {\sqrt {2/N}}} 5952:(for DCT-II & III) or 5670:The DCT-IV matrix becomes 5257:{\textstyle {\sqrt {2/N}}} 4929: 4839:{\displaystyle n=N-1/2\,;} 2770:discrete Fourier transform 2706:effect, commonly found in 2702:. This can also cause the 2672:high-definition television 2658:, decoding, video, audio, 2618: 2264:digital audio broadcasting 2076: 1712:. Used primarily in older 1658:Based on HEVC compression 1565:Image compression standard 1292:, 3D-DCT motion analysis, 1011:just-noticeable distortion 939:high-dynamic-range imaging 862:transient electromagnetics 699:facial recognition systems 638:Digital Audio Broadcasting 579:Clenshaw–Curtis quadrature 538:high-definition television 292: 216:discrete Fourier transform 22581:Digital signal processing 22513: 22455: 22392: 22354:Public Switched Telephone 22314: 22278: 22235: 22176: 22166:telecommunication circuit 22127:Fiber-optic communication 22110: 21872:Francis Blake (telephone) 21819: 21667:Optical telecommunication 21511: 21409: 21358:Discrete cosine transform 21323: 21292: 21261: 21255:Digital signal processing 21206: 21180: 21025: 20999: 20853: 20830: 20802: 20779: 20714: 20705: 20620: 20501: 20490: 20282: 20234: 20211: 20183: 20125: 20087: 20002: 19901: 19890: 19718: 19687: 19638: 19600: 19548: 19431: 19420: 19339: 19323: 19307: 19225: 19150: 19082: 19073: 18996: 18930: 18921: 18822: 18739: 18730: 18646: 18594:Discrete cosine transform 18584: 18575: 18524:LZ77 + Huffman + context 18477: 18387: 18317: 18205: 18196: 18139:FFT Package 1-dim / 2-dim 18061:10.1109/TCOM.1977.1093941 17967:10.1109/JPROC.2004.840301 17859:Malvar, Henrique (1992), 17803:Digital Signal Processing 17753:10.1016/j.laa.2004.07.015 17672:10.1109/TCOM.1978.1094144 17586:Nussbaumer, H.J. (1981). 17296:. Aperture. p. 132. 17081:10.1109/ISCAS.2000.856261 16701:Audio Engineering Society 16664:Audio Engineering Society 16387:Audio Engineering Society 15891:10.1117/1.JEI.27.4.040901 15849:10.1109/TCSVT.2006.871390 15718:. In Furht, Borko (ed.). 15714:Zhang, HongJiang (1998). 15609:trials of digital cinema. 15353:. Springer. p. 478. 15169:Signal Processing of HDTV 14744:10.1109/TASL.2010.2087755 14496:. McFarland. p. 63. 14397:(5, No. 3): 162–164 (162) 14364:10.1109/TCOM.1977.1093941 14337:10.1109/TCOM.1977.1093941 14249:Digital Signal Processing 12623:Feig & Winograd 1992a 7854:is assumed to be 2. 6772:is only the extension of 5462: 4513:{\displaystyle y_{2N}=0,} 4392:{\displaystyle y_{2n}=0,} 4088: 3208:are transformed into the 3014:in a function reduce the 2744:, which uses intentional 2652:digital signal processors 2644:digital signal processing 2638:Digital signal processing 2406: 2403: 2030:Research video format by 1464:{\displaystyle N\times N} 1098:digital signal processors 1094:digital signal processing 981:, directional block-wise 814:variable bitrate encoding 763:devices, consumer devices 425:(which uses a hybrid DCT- 327:Discrete Cosine Transform 270:international standards. 212:Fourier-related transform 189:digital signal processing 63:discrete cosine transform 22265:Orbital angular-momentum 21702:Satellite communications 21541:Communications satellite 21391:Post's inversion formula 21305:Digital image processing 19299:Smallest grammar problem 17863:, Boston: Artech House, 17769:(Submitted manuscript). 17493:10.1049/ip-f-2.1990.0063 16693:. 116th AES Convention. 16656:. 125th AES Convention. 16626:10.1109/TBC.2017.2661258 16385:. 135th AES Convention. 15586:McKernan, Brian (2005). 15055:"JPEG changed our world" 14212:House of Commons Library 13443: 12611:Frigo & Johnson 2005 11720:{\displaystyle ~\left~,} 11282:{\displaystyle ~\left~,} 5822:{\displaystyle n=N-1/2;} 5453:{\displaystyle k=N-1/2.} 5272:of half-shifted output. 5142:Some authors divide the 3269: 2985:discrete sine transforms 2527:, VoIP, wideband audio, 2387:VoIP, mobile telephony, 2050:'s internal successor), 1331:, watermarking detection 967:digital image processing 891:formatting — formatting 808:, perceptual weighting, 658:voice activity detection 553:Karhunen-Loève transform 446:University of New Mexico 22144:Molecular communication 21967:Gardiner Greene Hubbard 21796:Undersea telegraph line 21531:Cable protection system 21300:Audio signal processing 21017:Alliance for Open Media 19240:Compressed suffix array 18789:Nyquist–Shannon theorem 18040:10.1109/TSP.2002.806558 18011:10.1109/TSP.2004.823472 17937:Proceedings of the IEEE 16812:"Ekiga 3.1.0 available" 16773:Terriberry, Timothy B. 16368:10.1109/MSP.2008.918684 16230:Alliance for Open Media 15096:Hewlett-Packard Journal 14804:"MP3 and AAC Explained" 14419:Image Sequence Analysis 14097:10.1016/c2009-0-22279-3 14006:10.1109/T-C.1974.223784 11616:{\displaystyle ~\left~} 11201:{\displaystyle ~\left~} 3651:{\displaystyle X_{N-1}} 3562:{\displaystyle x_{N-1}} 2740:by German photographer 2700:data compression ratios 2674:(HDTV) encoder/decoder 2600:Enhanced Voice Services 2594:VoIP, mobile telephony 1395:Wireless sensor network 1276:moving object detection 595:Audio signal processing 575:Chebyshev approximation 498:data compression ratios 346:discrete sine transform 307:Kansas State University 283:data compression ratios 248:discrete sine transform 185:science and engineering 22286:Communication protocol 22072:Charles Sumner Tainter 21887:Walter Houser Brattain 21832:Edwin Howard Armstrong 21640:Information revolution 19937:MPEG-1 Layer III (MP3) 16756:"ITU-T Work Programme" 14984:Chen, Wai Kai (2004). 14586:Li, Jian Ping (2006). 13673: 13649: 13580: 13547:FFT of real data plus 13541: 13515: 13489: 13389: 13377: 13370: 12912: 12901: 12882: 12852: 12779: 12726:Sorensen et al. (1987) 12714: 12672: 12619:Winograd FFT algorithm 12598: 12556: 12508: 12473:fast Fourier transform 12465: 12417: 12359: 12282: 12204: 12014: 11986: 11904: 11846: 11721: 11617: 11545: 11283: 11202: 11128: 11080: 11027: 10842: 10710: 10485: 10260: 10020: 9834: 9791: 9748: 9648: 9634: 9607: 9565: 9463: 9329: 9295: 9261: 9161: 9096: 9016: 7862: 7825: 7487: 7446: 7405: 7292: 6954: 6913: 6872: 6755: 6561: 6520: 6282: 6236: 6110: 6051: 6007: 5969: 5946: 5923: 5922:{\displaystyle 2(N-1)} 5888: 5853: 5852:{\displaystyle X_{k}.} 5823: 5780: 5779:{\displaystyle n=-1/2} 5743: 5703: 5661: 5513: 5454: 5414: 5413:{\displaystyle k=-1/2} 5377: 5351: 5322: 5296: 5258: 5226: 5187: 5163: 5130: 5005: 4921: 4892: 4866: 4840: 4796: 4795:{\displaystyle n=-1/2} 4759: 4709: 4673: 4639: 4601: 4563: 4514: 4476: 4441: 4393: 4354: 4353:{\displaystyle y_{n},} 4324: 4297: 4264: 4139: 4080: 4079:{\displaystyle X_{k}.} 4050: 4018: 3992: 3963: 3940: 3917: 3855: 3811: 3785: 3784:{\displaystyle 2(N-1)} 3739: 3689: 3652: 3619: 3592: 3563: 3530: 3500: 3387: 3260: 3202: 3122: 3100: 2921: 2891: 2862: 2842: 2822: 2508:VoIP, wideband audio, 2342:High-Definition Coding 2268:Digital Radio Mondiale 2216:portable media players 2182:Perceptual Audio Coder 2171:portable media players 1991:container format with 1710:video coding standards 1525: 1493: 1465: 1407:Visual media standards 1329:reversible data hiding 1323:, video watermarking, 1294:video content analysis 1280:Multiview Video Coding 1252:video coding standards 1234:digital versatile disc 1182:digital cinematography 605:(lossy and lossless), 603:audio data compression 379:fast Fourier transform 218:(DFT), but using only 22260:Polarization-division 21992:Narinder Singh Kapany 21957:Erna Schneider Hoover 21877:Jagadish Chandra Bose 21857:Alexander Graham Bell 21588:online video platform 20955:QuickTime File Format 19269:Kolmogorov complexity 19137:Video characteristics 18514:LZ77 + Huffman + ANS 17258:The Glitch Moment(um) 16732:John Wiley & Sons 16055:Hoffman, Roy (2012). 15964:"JPEG XL White Paper" 15421:John Wiley & Sons 15221:. pp. 1–3, 129. 15135:John Wiley & Sons 14953:10.1109/CSNT.2015.134 14834:. Xiph.Org Foundation 14696:Luo, Fa-Long (2008). 14416:Huang, T. S. (1981). 14205:Lea, William (1994). 13986:; Natarajan, T. Raj; 13674: 13650: 13581: 13542: 13516: 13490: 13386: 13371: 12918: 12910: 12899: 12883: 12853: 12780: 12722:split-radix algorithm 12715: 12673: 12599: 12557: 12509: 12466: 12418: 12360: 12283: 12205: 11988: 11960: 11905: 11841: 11722: 11618: 11546: 11284: 11203: 11129: 11081: 11037:Arithmetic complexity 11028: 10843: 10711: 10486: 10261: 10021: 9792: 9749: 9706: 9646: 9635: 9633:{\displaystyle k_{3}} 9608: 9566: 9464: 9296: 9262: 9228: 9162: 9097: 9017: 7860: 7826: 7447: 7406: 7365: 7293: 6914: 6873: 6832: 6756: 6521: 6480: 6242: 6196: 6131:Multidimensional DCTs 6111: 6052: 6008: 5970: 5947: 5924: 5889: 5854: 5824: 5781: 5744: 5742:{\displaystyle x_{n}} 5704: 5662: 5487: 5455: 5415: 5378: 5376:{\displaystyle X_{k}} 5352: 5323: 5297: 5295:{\displaystyle x_{n}} 5259: 5227: 5188: 5164: 5162:{\displaystyle x_{0}} 5131: 4979: 4922: 4893: 4867: 4865:{\displaystyle X_{k}} 4841: 4797: 4760: 4758:{\displaystyle x_{n}} 4710: 4674: 4640: 4638:{\displaystyle X_{0}} 4602: 4564: 4515: 4477: 4442: 4394: 4355: 4325: 4298: 4265: 4113: 4081: 4051: 4049:{\displaystyle n=N-1} 4019: 3993: 3991:{\displaystyle x_{n}} 3964: 3941: 3918: 3856: 3812: 3786: 3740: 3690: 3653: 3620: 3618:{\displaystyle X_{0}} 3593: 3564: 3531: 3529:{\displaystyle x_{0}} 3501: 3361: 3261: 3203: 3123: 3101: 2915: 2892: 2863: 2843: 2823: 2736:. Another example is 2692:compression artifacts 2682:Compression artifacts 2199:Advanced Audio Coding 2077:Further information: 1866:Advanced Video Coding 1708:First of a family of 1686:Video coding standard 1536:Advanced Video Coding 1526: 1524:{\displaystyle (0,0)} 1494: 1466: 1220:encoder/decoder chips 1186:digital movie cameras 1134:signal-to-noise ratio 615:feedback cancellation 571:Chebyshev polynomials 534:high-definition video 431:Advanced Audio Coding 287:compression artifacts 93:. It is used in most 73:in terms of a sum of 40:for more information. 22102:Vladimir K. Zworykin 22062:Almon Brown Strowger 22032:Charles Grafton Page 21687:Prepaid mobile phone 21615:Electrical telegraph 21424:Anti-aliasing filter 21353:Constant-Q transform 21336:Advanced z-transform 21214:Compression software 20771:MPEG media transport 19848:Sorenson Video/Spark 19359:Compression software 18953:Compression artifact 18909:Psychoacoustic model 17207:. pp. 349–383. 15835:/AVC optimization". 15385:Taylor & Francis 15175:. pp. 611–618. 14947:. pp. 483–487. 14853:Mandyam, Giridhar D. 14171:EBU Technical Review 13659: 13602: 13551: 13525: 13499: 13488:{\displaystyle ~4N~} 13470: 12925: 12881:{\displaystyle ~N~,} 12863: 12792: 12732: 12698: 12671:{\displaystyle ~4N~} 12653: 12569: 12518: 12479: 12427: 12381: 12292: 12218: 11938: 11915:is transformed to a 11856: 11627: 11558: 11293: 11213: 11138: 11090: 11045: 10855: 10723: 10498: 10273: 10036: 9655: 9617: 9577: 9476: 9177: 9110: 9028: 7869: 7316: 6783: 6152: 6088: 6050:{\displaystyle N=1.} 6035: 5979: 5956: 5933: 5898: 5878: 5833: 5790: 5753: 5726: 5678: 5471: 5424: 5387: 5360: 5350:{\displaystyle n=N;} 5332: 5306: 5279: 5236: 5197: 5173: 5146: 4938: 4920:{\displaystyle k=N.} 4902: 4876: 4849: 4806: 4769: 4742: 4683: 4649: 4622: 4573: 4524: 4485: 4451: 4402: 4364: 4334: 4311: 4284: 4097: 4060: 4028: 4002: 3975: 3950: 3930: 3865: 3821: 3795: 3760: 3703: 3662: 3629: 3602: 3573: 3540: 3513: 3278: 3216: 3151: 3110: 3060: 3032:boundaries are even 2890:{\displaystyle f(x)} 2872: 2852: 2832: 2821:{\displaystyle f(x)} 2803: 2778:complex exponentials 2615:Multidimensional DCT 2444:Common applications 2097:Common applications 2073:MDCT audio standards 1914:, consumer devices, 1908:streaming television 1693:Common applications 1571:Common applications 1503: 1483: 1449: 1317:digital watermarking 1118:compressive sampling 1030:Image reconstruction 983:image representation 842:streaming television 825:digital distribution 797:lossless compression 759:systems, multimedia 753:Consumer electronics 585:General applications 522:streaming television 482:lossless compression 450:lossless compression 411:University of Surrey 22591:Discrete transforms 22052:Johann Philipp Reis 21811:Wireless revolution 21773:The Telephone Cases 21630:Hydraulic telegraph 21212:for techniques and 21210:Compression methods 19706:HDR Vivid(GY/T 358) 19349:Compression formats 18988:Texture compression 18983:Standard test image 18799:Silence compression 18003:2004ITSP...52..992B 17949:2005IEEEP..93..216F 17891:1994ITSP...42.1038M 17844:1992ITSP...40.2174F 17815:1991DSP.....1....4A 17779:1990SigPr..19..259D 17625:2008SigPr..88.1553S 17538:2000ITSP...48.1171B 17428:2008SigPr..88.1553S 17165:1998ITSP...46.1142C 17059:10.1109/2945.468390 17017:10.1109/4233.870036 16938:1997ITIP....6..758C 16897:1995ITGRS..33...26A 16360:2008ISPM...25..137H 15762:Library of Congress 15029:. 24 September 2013 14869:1995SPIE.2419..474M 14461:1975SPIE...66..172R 14262:1991DSP.....1....4A 13833:(6 November 2006). 13540:{\displaystyle ~N~} 13514:{\displaystyle ~N~} 12921:DCT of the image = 12713:{\displaystyle ~N~} 12686:) was described by 11732: 7838:algorithm. As with 5321:{\displaystyle n=0} 4891:{\displaystyle k=0} 4017:{\displaystyle n=0} 3810:{\displaystyle N=5} 3128:denotes the set of 3016:rate of convergence 2868:where the original 2782:boundary conditions 2729:digital images and 2662:, control signals, 2291:Windows Media Audio 1806:video distribution 1380:arithmetic circuits 1260:motion compensation 1167:event data recorder 1112:, control signals, 1079:Pattern recognition 1054:Electrocardiography 935:digital photography 405:A DCT variant, the 358:Dirichlet condition 301:, T. Natarajan and 22250:Frequency-division 22227:Telephone exchange 22097:Charles Wheatstone 22027:Jun-ichi Nishizawa 22002:Innocenzo Manzetti 21937:Reginald Fessenden 21672:Optical telegraphy 21505:Telecommunications 21381:Integral transform 21376:Impulse invariance 21348:Bilinear transform 19257:Information theory 19112:Display resolution 18938:Chroma subsampling 18327:Byte pair encoding 18272:Shannon–Fano–Elias 17387:IEICE Transactions 16576:Dolby Laboratories 16030:Wang, Yao (2006). 16005:Wang, Yao (2006). 15942:Nokia Technologies 15765:. 17 November 2014 15387:. pp. 558–9. 15127:Lee, Jack (2005). 14778:University of Utah 13913:on 19 October 2019 13669: 13645: 13576: 13537: 13511: 13485: 13390: 13378: 13366: 13360: 12913: 12902: 12878: 12848: 12808: 12775: 12710: 12668: 12594: 12552: 12504: 12461: 12413: 12355: 12278: 12200: 11917:linear combination 11900: 11847: 11730: 11717: 11613: 11541: 11453: 11446: 11358: 11351: 11279: 11234: 11198: 11159: 11124: 11106: 11079:{\displaystyle ~~} 11076: 11023: 10838: 10706: 10481: 10256: 10016: 9825: 9782: 9739: 9649: 9630: 9603: 9561: 9459: 9157: 9092: 9012: 9010: 7863: 7821: 7288: 6751: 6749: 6106: 6061:Inverse transforms 6047: 6003: 5968:{\displaystyle 8N} 5965: 5945:{\displaystyle 4N} 5942: 5919: 5884: 5849: 5819: 5776: 5739: 5699: 5657: 5602: 5574: 5550: 5450: 5410: 5373: 5347: 5318: 5292: 5254: 5222: 5183: 5159: 5126: 5065: 5042: 4964: 4917: 4888: 4862: 4836: 4792: 4755: 4705: 4669: 4635: 4597: 4559: 4510: 4472: 4437: 4389: 4350: 4323:{\displaystyle 4N} 4320: 4296:{\displaystyle 4N} 4293: 4260: 4199: 4176: 4076: 4046: 4014: 3988: 3962:{\displaystyle N.} 3959: 3936: 3913: 3851: 3807: 3781: 3735: 3727: 3685: 3648: 3615: 3588: 3559: 3526: 3496: 3435: 3256: 3198: 3118: 3096: 2922: 2887: 2858: 2838: 2818: 1792:digital television 1743:non-linear editing 1714:video conferencing 1521: 1489: 1461: 1419:standards such as 1360:(RF) technology — 1321:image watermarking 1216:high-definition TV 1204:Digital television 1136:(SNR) estimation, 1084:Region of interest 1058:vectorcardiography 1049:Medical technology 999:image segmentation 949:image file formats 884:feature extraction 717:facial recognition 654:speech recognition 518:digital television 372:motion-compensated 285:. However, blocky 195:devices, reducing 141:digital television 22621:Video compression 22601:Indian inventions 22596:Image compression 22563: 22562: 22301:Store and forward 22296:Data transmission 22210:Network switching 22161:Transmission line 22007:Guglielmo Marconi 21972:Internet pioneers 21837:Mohamed M. Atalla 21806:Whistled language 21471: 21470: 21396:Starred transform 21386:Laplace transform 21310:Speech processing 21279:Estimation theory 21221: 21220: 20995: 20994: 20913:DivX Media Format 20701: 20700: 20486: 20485: 19886: 19885: 19776:Microsoft Video 1 19711:AVS3 P2(GY/T 368) 19697:(GB/T 20090.2/16) 19372: 19371: 19221: 19220: 19171:Deblocking filter 19069: 19068: 18917: 18916: 18726: 18725: 18571: 18570: 18120:Steven G. Johnson 18118:Matteo Frigo and 18107:Syed Ali Khayam: 18083:978-0-521-88068-8 17921:978-0-13-754920-7 17899:10.1109/78.295213 17870:978-0-89006-467-2 17852:10.1109/78.157218 17767:Signal Processing 17603:Signal Processing 17573:10.1109/18.144722 17546:10.1109/78.827550 17406:Signal Processing 17337:www.mathworks.com 17271:978-90-816021-6-7 17173:10.1109/78.668566 16956:10.1109/83.568933 16905:10.1109/36.368225 15700:978-1-4398-9928-1 15669:978-1-101-61380-1 15603:978-0-07-142963-4 15394:978-1-136-03410-7 14962:978-1-4799-1797-6 14877:10.1117/12.206386 14469:10.1117/12.965361 14071:Rao, K. Ramamohan 13667: 13644: 13607: 13575: 13556: 13536: 13530: 13510: 13504: 13495:DFT to four size 13484: 13475: 12874: 12868: 12807: 12797: 12774: 12737: 12709: 12703: 12667: 12658: 12593: 12574: 12551: 12523: 12503: 12484: 12460: 12432: 12412: 12386: 12351: 12339: 12333: 12324: 12315: 12300: 12297: 12277: 12274: 12262: 12256: 12250: 12241: 12226: 12223: 12196: 12188: 12184: 12170: 12127: 12108: 12104: 12090: 12047: 12017: 11896: 11864: 11820: 11819: 11713: 11646: 11632: 11612: 11577: 11563: 11537: 11470: 11451: 11378: 11364: 11362: 11356: 11315: 11301: 11299: 11298: 11275: 11251: 11238: 11233: 11218: 11197: 11163: 11158: 11143: 11123: 11110: 11105: 11095: 11075: 11050: 11018: 10992: 10991: where 10982: 10956: 10930: 10901: 10831: 10792: 10763: 10673: 10660: 10631: 10561: 10532: 10474: 10448: 10406: 10349: 10307: 10249: 10194: 10118: 10049: 9845: 9824: 9781: 9738: 9538: 9505: 9339: 9084: 8852: 8701: 8550: 8417: 8266: 8133: 8000: 7885: 7846:3-D DCT-II VR DIF 7766: 7737: 7709: 7661: 7633: 7585: 7557: 7233: 7204: 7176: 7128: 7100: 7052: 7024: 6722: 6694: 6646: 6618: 6448: 6420: 6367: 6339: 6104: 5887:{\displaystyle N} 5694: 5653: 5641: 5635: 5619: 5601: 5573: 5549: 5252: 5220: 5181: 5122: 5110: 5104: 5089: 5085: 5064: 5041: 4963: 4703: 4666: 4256: 4244: 4238: 4223: 4219: 4198: 4175: 3939:{\displaystyle N} 3909: 3903: 3897: 3891: 3885: 3879: 3873: 3847: 3841: 3835: 3829: 3730: 3726: 3679: 3582: 3492: 3480: 3474: 3459: 3455: 3434: 3416: 3302: 3197: 3178: 3172: 3156: 3045:Formal definition 2861:{\displaystyle x} 2841:{\displaystyle x} 2752:Informal overview 2612: 2611: 2576:videoconferencing 2425: 2424: 2070: 2069: 2013:Successor to the 1824:digital cassettes 1675: 1674: 1635:Image file format 1492:{\displaystyle N} 1417:video compression 1397:(WSN) — wireless 1256:motion estimation 1248:video compression 1122:error concealment 1090:Signal processing 1017:masking effects, 991:image enhancement 957:progressive image 945:Image compression 854:Forgery detection 793:lossy compression 761:telecommunication 745:Network bandwidth 684:videoconferencing 664:Digital telephony 647:Speech processing 545:energy compaction 494:lossy compression 470:signal processing 417:formats, such as 415:audio compression 395:intra-frame coder 350:Neumann condition 337:as the basis for 315:image compression 305:while working at 279:energy compaction 197:network bandwidth 193:telecommunication 87:signal processing 59: 58: 44: 22628: 22586:Fourier analysis 22576:Data compression 22553: 22552: 22543: 22542: 22533: 22532: 22523: 22522: 22521: 22394:Notable networks 22384:Wireless network 22324:Cellular network 22316:Types of network 22291:Computer network 22178:Network topology 22092:Thomas A. Watson 21947:Oliver Heaviside 21932:Philo Farnsworth 21907:Daniel Davis Jr. 21882:Charles Bourseul 21842:John Logie Baird 21551:Data compression 21546:Computer network 21498: 21491: 21484: 21475: 21474: 21269:Detection theory 21248: 21241: 21234: 21225: 21224: 20987:VOB, IFO and BUP 20761:Motion JPEG 2000 20712: 20711: 20499: 20498: 20458:aptX Low Latency 20235:Chinese Standard 19899: 19898: 19688:Chinese Standard 19460:Motion JPEG 2000 19429: 19428: 19399: 19392: 19385: 19376: 19375: 19357: 19356: 19347: 19346: 19176:Lapped transform 19080: 19079: 18958:Image resolution 18943:Coding tree unit 18928: 18927: 18737: 18736: 18582: 18581: 18203: 18202: 18189:Data compression 18182: 18175: 18168: 18159: 18158: 18097: 18096: 18095: 18086:, archived from 18064: 18055:(9): 1004–1009. 18043: 18022: 17988: 17978: 17960: 17934: 17924: 17913: 17902: 17885:(5): 1038–1051. 17873: 17855: 17838:(9): 2174–2193. 17826: 17797:(January 1991). 17790: 17757: 17755: 17727: 17717: 17696: 17675: 17645: 17644: 17618: 17609:(6): 1553–1564. 17598: 17592: 17591: 17583: 17577: 17576: 17567:(4): 1387–1391. 17556: 17550: 17549: 17532:(4): 1171–1183. 17521: 17515: 17514: 17506: 17497: 17496: 17476: 17470: 17465: 17459: 17454: 17448: 17447: 17421: 17412:(6): 1553–1564. 17401: 17395: 17394: 17393:(11): 1095–1097. 17378: 17372: 17371: 17353: 17347: 17346: 17344: 17343: 17329: 17323: 17322: 17314: 17308: 17307: 17289: 17283: 17282: 17280: 17278: 17263: 17252: 17246: 17245: 17243: 17241: 17229:"Mosquito noise" 17225: 17219: 17218: 17196: 17187: 17186: 17184: 17159:(2): 1142–1151. 17148: 17142: 17136: 17130: 17129: 17117: 17111: 17110: 17068: 17062: 17061: 17042: 17036: 17035: 17000: 16994: 16993: 16981: 16975: 16974: 16949: 16923: 16914: 16908: 16907: 16880: 16874: 16873: 16871: 16869: 16859: 16851: 16845: 16844: 16833: 16827: 16826: 16824: 16823: 16814:. Archived from 16808: 16802: 16801: 16799: 16791: 16789: 16788: 16770: 16764: 16763: 16752: 16746: 16745: 16721: 16712: 16711: 16709: 16707: 16692: 16681: 16675: 16674: 16672: 16670: 16655: 16644: 16638: 16637: 16611: 16602: 16596: 16595: 16593: 16591: 16585: 16572: 16564: 16558: 16557: 16555: 16554: 16543: 16537: 16536: 16498: 16492: 16491: 16461: 16455: 16454: 16452: 16450: 16433: 16427: 16426: 16424: 16422: 16407: 16401: 16400: 16398: 16378: 16372: 16371: 16343: 16337: 16336: 16334: 16332: 16321: 16315: 16314: 16312: 16310: 16299: 16293: 16292: 16290: 16288: 16277: 16271: 16270: 16265: 16263: 16246: 16240: 16239: 16237: 16236: 16227: 16218: 16212: 16211: 16208: 16204: 16178: 16172: 16171: 16169: 16167: 16150: 16144: 16143: 16099: 16086: 16077: 16076: 16052: 16046: 16045: 16043: 16037:. Archived from 16036: 16027: 16021: 16020: 16018: 16012:. Archived from 16011: 16002: 15993: 15992: 15987: 15985: 15979: 15968: 15959: 15953: 15952: 15950: 15948: 15934: 15928: 15927: 15925: 15923: 15902: 15896: 15895: 15893: 15867: 15861: 15860: 15834: 15828: 15819: 15818: 15781: 15775: 15774: 15772: 15770: 15753: 15744: 15743: 15711: 15705: 15704: 15680: 15674: 15673: 15653: 15647: 15646: 15644: 15642: 15621: 15612: 15611: 15583: 15570: 15569: 15549: 15543: 15542: 15540: 15538: 15526: 15518: 15509: 15508: 15506: 15504: 15483: 15470: 15469: 15467: 15465: 15450: 15435: 15434: 15410: 15399: 15398: 15374: 15365: 15364: 15344: 15329: 15328: 15317:. pp. 1–3. 15300: 15233: 15232: 15204: 15195: 15194: 15160: 15149: 15148: 15124: 15113: 15112: 15092: 15083: 15074: 15073: 15071: 15069: 15050: 15039: 15038: 15036: 15034: 15017: 15006: 15005: 14981: 14975: 14974: 14940: 14934: 14933: 14895: 14889: 14888: 14849: 14843: 14842: 14840: 14839: 14827: 14818: 14817: 14815: 14808: 14799: 14790: 14789: 14787: 14785: 14774: 14765: 14756: 14755: 14738:(5): 1231–1241. 14727: 14718: 14717: 14693: 14666: 14665: 14656:(5): 1153–1161. 14645: 14639: 14638: 14614: 14608: 14607: 14594:World Scientific 14583: 14577: 14576: 14565:. pp. 1–2. 14552: 14541: 14540: 14538: 14536: 14519: 14508: 14507: 14487: 14481: 14480: 14444: 14438: 14437: 14413: 14407: 14406: 14404: 14402: 14382: 14376: 14375: 14358:(9): 1004–1009. 14347: 14341: 14340: 14331:(9): 1004–1009. 14318: 14312: 14311: 14309: 14307: 14302:. September 1992 14297: 14289: 14274: 14273: 14245: 14239:(January 1991). 14233: 14224: 14223: 14221: 14219: 14202: 14187: 14186: 14184: 14182: 14162: 14143: 14142: 14090: 14067: 14050: 14049: 13980: 13963: 13962: 13960: 13958: 13949: 13940: 13923: 13922: 13920: 13918: 13909:. Archived from 13898: 13885: 13884: 13826: 13811: 13810: 13804: 13802: 13796: 13763: 13754: 13684: 13678: 13676: 13675: 13670: 13668: 13663: 13654: 13652: 13651: 13646: 13642: 13623: 13622: 13605: 13596: 13590: 13585: 13583: 13582: 13577: 13573: 13563: 13562: 13554: 13546: 13544: 13543: 13538: 13534: 13528: 13520: 13518: 13517: 13512: 13508: 13502: 13494: 13492: 13491: 13486: 13482: 13473: 13464: 13458: 13454: 13428: 13424: 13419: 13415: 13411: 13407: 13375: 13373: 13372: 13367: 13365: 13364: 12887: 12885: 12884: 12879: 12872: 12866: 12857: 12855: 12854: 12849: 12838: 12837: 12822: 12821: 12809: 12800: 12795: 12784: 12782: 12781: 12776: 12772: 12753: 12752: 12735: 12719: 12717: 12716: 12711: 12707: 12701: 12677: 12675: 12674: 12669: 12665: 12656: 12640: 12636: 12633:general lengths 12603: 12601: 12600: 12595: 12591: 12581: 12580: 12572: 12561: 12559: 12558: 12553: 12549: 12530: 12529: 12521: 12513: 12511: 12510: 12505: 12501: 12491: 12490: 12482: 12470: 12468: 12467: 12462: 12458: 12439: 12438: 12430: 12422: 12420: 12419: 12414: 12410: 12406: 12405: 12393: 12392: 12384: 12364: 12362: 12361: 12356: 12349: 12337: 12331: 12322: 12313: 12298: 12295: 12287: 12285: 12284: 12279: 12275: 12272: 12260: 12254: 12248: 12239: 12224: 12221: 12209: 12207: 12206: 12201: 12194: 12193: 12189: 12186: 12185: 12183: 12175: 12168: 12129: 12125: 12113: 12109: 12106: 12105: 12103: 12095: 12088: 12049: 12045: 12033: 12032: 12015: 12013: 12002: 11985: 11974: 11956: 11955: 11930: 11914: 11909: 11907: 11906: 11901: 11894: 11887: 11886: 11874: 11873: 11862: 11852: 11733: 11729: 11726: 11724: 11723: 11718: 11711: 11710: 11706: 11705: 11704: 11689: 11688: 11667: 11666: 11657: 11656: 11647: 11639: 11630: 11622: 11620: 11619: 11614: 11610: 11609: 11605: 11598: 11597: 11588: 11587: 11578: 11570: 11561: 11550: 11548: 11547: 11542: 11535: 11534: 11530: 11529: 11528: 11513: 11512: 11491: 11490: 11481: 11480: 11471: 11463: 11452: 11449: 11447: 11442: 11438: 11437: 11436: 11421: 11420: 11399: 11398: 11389: 11388: 11379: 11371: 11357: 11354: 11352: 11347: 11343: 11336: 11335: 11326: 11325: 11316: 11308: 11296: 11288: 11286: 11285: 11280: 11273: 11272: 11268: 11261: 11260: 11249: 11248: 11247: 11236: 11235: 11226: 11216: 11207: 11205: 11204: 11199: 11195: 11194: 11190: 11183: 11182: 11173: 11172: 11161: 11160: 11151: 11141: 11133: 11131: 11130: 11125: 11121: 11120: 11119: 11108: 11107: 11098: 11093: 11085: 11083: 11082: 11077: 11073: 11063: 11062: 11048: 11032: 11030: 11029: 11024: 11019: 11016: 10993: 10990: 10988: 10984: 10983: 10975: 10970: 10969: 10957: 10949: 10944: 10943: 10931: 10923: 10918: 10917: 10903: 10902: 10894: 10891: 10890: 10847: 10845: 10844: 10839: 10837: 10833: 10832: 10824: 10819: 10818: 10806: 10805: 10793: 10785: 10780: 10779: 10765: 10764: 10756: 10753: 10752: 10715: 10713: 10712: 10707: 10705: 10701: 10700: 10699: 10687: 10686: 10674: 10666: 10661: 10653: 10648: 10647: 10633: 10632: 10624: 10621: 10620: 10593: 10589: 10588: 10587: 10575: 10574: 10562: 10554: 10549: 10548: 10534: 10533: 10525: 10522: 10521: 10490: 10488: 10487: 10482: 10480: 10476: 10475: 10467: 10462: 10461: 10449: 10441: 10436: 10435: 10423: 10422: 10408: 10407: 10399: 10396: 10395: 10368: 10364: 10363: 10362: 10350: 10342: 10337: 10336: 10324: 10323: 10309: 10308: 10300: 10297: 10296: 10265: 10263: 10262: 10257: 10255: 10251: 10250: 10242: 10237: 10236: 10224: 10223: 10211: 10210: 10196: 10195: 10187: 10184: 10183: 10159: 10158: 10146: 10145: 10133: 10132: 10120: 10119: 10111: 10102: 10101: 10089: 10088: 10076: 10075: 10063: 10062: 10051: 10050: 10042: 10025: 10023: 10022: 10017: 10003: 10002: 9966: 9965: 9929: 9928: 9898: 9897: 9885: 9884: 9872: 9871: 9859: 9858: 9847: 9846: 9838: 9833: 9826: 9817: 9813: 9806: 9805: 9790: 9783: 9774: 9770: 9763: 9762: 9747: 9740: 9731: 9727: 9720: 9719: 9699: 9698: 9686: 9685: 9673: 9672: 9639: 9637: 9636: 9631: 9629: 9628: 9612: 9610: 9609: 9604: 9602: 9601: 9589: 9588: 9570: 9568: 9567: 9562: 9539: 9536: 9522: 9521: 9506: 9504: 9493: 9488: 9487: 9468: 9466: 9465: 9460: 9455: 9454: 9430: 9429: 9405: 9404: 9380: 9379: 9367: 9366: 9354: 9353: 9341: 9340: 9332: 9328: 9317: 9310: 9309: 9294: 9283: 9276: 9275: 9260: 9249: 9242: 9241: 9221: 9220: 9208: 9207: 9195: 9194: 9166: 9164: 9163: 9158: 9153: 9152: 9128: 9127: 9101: 9099: 9098: 9093: 9085: 9077: 9072: 9071: 9059: 9058: 9046: 9045: 9021: 9019: 9018: 9013: 9011: 8998: 8997: 8976: 8975: 8954: 8953: 8923: 8922: 8898: 8897: 8873: 8872: 8854: 8853: 8845: 8835: 8834: 8813: 8812: 8791: 8790: 8766: 8765: 8747: 8746: 8722: 8721: 8703: 8702: 8694: 8678: 8677: 8662: 8661: 8640: 8639: 8609: 8608: 8590: 8589: 8571: 8570: 8552: 8551: 8543: 8533: 8532: 8517: 8516: 8495: 8494: 8470: 8469: 8457: 8456: 8438: 8437: 8419: 8418: 8410: 8394: 8393: 8372: 8371: 8356: 8355: 8325: 8324: 8300: 8299: 8281: 8280: 8268: 8267: 8259: 8249: 8248: 8227: 8226: 8211: 8210: 8186: 8185: 8167: 8166: 8148: 8147: 8135: 8134: 8126: 8110: 8109: 8094: 8093: 8078: 8077: 8047: 8046: 8028: 8027: 8015: 8014: 8002: 8001: 7993: 7983: 7982: 7967: 7966: 7951: 7950: 7926: 7925: 7913: 7912: 7900: 7899: 7887: 7886: 7878: 7830: 7828: 7827: 7822: 7814: 7813: 7777: 7776: 7767: 7764: 7758: 7754: 7753: 7752: 7743: 7739: 7738: 7730: 7725: 7724: 7710: 7708: 7707: 7695: 7682: 7678: 7677: 7676: 7667: 7663: 7662: 7654: 7649: 7648: 7634: 7632: 7631: 7619: 7606: 7602: 7601: 7600: 7591: 7587: 7586: 7578: 7573: 7572: 7558: 7556: 7555: 7543: 7530: 7529: 7528: 7527: 7515: 7514: 7502: 7501: 7486: 7479: 7478: 7468: 7461: 7460: 7445: 7438: 7437: 7427: 7420: 7419: 7404: 7397: 7396: 7386: 7379: 7378: 7361: 7360: 7359: 7358: 7346: 7345: 7333: 7332: 7297: 7295: 7294: 7289: 7281: 7280: 7244: 7243: 7234: 7231: 7225: 7221: 7220: 7219: 7210: 7206: 7205: 7197: 7192: 7191: 7177: 7175: 7174: 7162: 7149: 7145: 7144: 7143: 7134: 7130: 7129: 7121: 7116: 7115: 7101: 7099: 7098: 7086: 7073: 7069: 7068: 7067: 7058: 7054: 7053: 7045: 7040: 7039: 7025: 7023: 7022: 7010: 6997: 6996: 6995: 6994: 6982: 6981: 6969: 6968: 6953: 6946: 6945: 6935: 6928: 6927: 6912: 6905: 6904: 6894: 6887: 6886: 6871: 6864: 6863: 6853: 6846: 6845: 6828: 6827: 6826: 6825: 6813: 6812: 6800: 6799: 6760: 6758: 6757: 6752: 6750: 6743: 6739: 6738: 6737: 6728: 6724: 6723: 6715: 6710: 6709: 6695: 6693: 6692: 6680: 6667: 6663: 6662: 6661: 6652: 6648: 6647: 6639: 6634: 6633: 6619: 6617: 6616: 6604: 6591: 6590: 6589: 6588: 6576: 6575: 6560: 6553: 6552: 6542: 6535: 6534: 6519: 6512: 6511: 6501: 6494: 6493: 6473: 6469: 6465: 6464: 6463: 6454: 6450: 6449: 6441: 6436: 6435: 6421: 6419: 6418: 6406: 6393: 6389: 6388: 6384: 6383: 6382: 6373: 6369: 6368: 6360: 6355: 6354: 6340: 6338: 6337: 6325: 6312: 6311: 6310: 6309: 6297: 6296: 6281: 6274: 6273: 6263: 6256: 6255: 6235: 6228: 6227: 6217: 6210: 6209: 6188: 6187: 6186: 6185: 6173: 6172: 6122: 6121: 6115: 6113: 6112: 6107: 6105: 6100: 6092: 6077:and vice versa. 6056: 6054: 6053: 6048: 6030: 6012: 6010: 6009: 6004: 6002: 5997: 5992: 5974: 5972: 5971: 5966: 5951: 5949: 5948: 5943: 5928: 5926: 5925: 5920: 5893: 5891: 5890: 5885: 5858: 5856: 5855: 5850: 5845: 5844: 5828: 5826: 5825: 5820: 5812: 5785: 5783: 5782: 5777: 5772: 5748: 5746: 5745: 5740: 5738: 5737: 5715:, is called the 5708: 5706: 5705: 5700: 5695: 5690: 5682: 5666: 5664: 5663: 5658: 5651: 5639: 5633: 5620: 5617: 5614: 5610: 5608: 5604: 5603: 5594: 5580: 5576: 5575: 5566: 5551: 5545: 5538: 5523: 5522: 5512: 5501: 5483: 5482: 5459: 5457: 5456: 5451: 5446: 5420:and even around 5419: 5417: 5416: 5411: 5406: 5382: 5380: 5379: 5374: 5372: 5371: 5356: 5354: 5353: 5348: 5327: 5325: 5324: 5319: 5301: 5299: 5298: 5293: 5291: 5290: 5263: 5261: 5260: 5255: 5253: 5248: 5240: 5231: 5229: 5228: 5223: 5221: 5216: 5214: 5209: 5208: 5192: 5190: 5189: 5184: 5182: 5177: 5168: 5166: 5165: 5160: 5158: 5157: 5135: 5133: 5132: 5127: 5120: 5108: 5102: 5087: 5086: 5083: 5080: 5076: 5071: 5067: 5066: 5057: 5043: 5037: 5030: 5015: 5014: 5004: 4993: 4975: 4974: 4965: 4956: 4950: 4949: 4926: 4924: 4923: 4918: 4897: 4895: 4894: 4889: 4871: 4869: 4868: 4863: 4861: 4860: 4845: 4843: 4842: 4837: 4828: 4802:and even around 4801: 4799: 4798: 4793: 4788: 4764: 4762: 4761: 4756: 4754: 4753: 4714: 4712: 4711: 4706: 4704: 4702: 4697: 4692: 4687: 4678: 4676: 4675: 4670: 4667: 4661: 4659: 4644: 4642: 4641: 4636: 4634: 4633: 4610: 4606: 4604: 4603: 4598: 4568: 4566: 4565: 4560: 4558: 4557: 4545: 4544: 4519: 4517: 4516: 4511: 4500: 4499: 4481: 4479: 4478: 4473: 4446: 4444: 4443: 4438: 4436: 4435: 4423: 4422: 4398: 4396: 4395: 4390: 4379: 4378: 4359: 4357: 4356: 4351: 4346: 4345: 4329: 4327: 4326: 4321: 4302: 4300: 4299: 4294: 4269: 4267: 4266: 4261: 4254: 4242: 4236: 4221: 4220: 4217: 4214: 4210: 4205: 4201: 4200: 4191: 4177: 4171: 4164: 4149: 4148: 4138: 4127: 4109: 4108: 4085: 4083: 4082: 4077: 4072: 4071: 4056:; similarly for 4055: 4053: 4052: 4047: 4024:and even around 4023: 4021: 4020: 4015: 3997: 3995: 3994: 3989: 3987: 3986: 3968: 3966: 3965: 3960: 3945: 3943: 3942: 3937: 3922: 3920: 3919: 3914: 3907: 3901: 3895: 3889: 3883: 3877: 3871: 3860: 3858: 3857: 3852: 3845: 3839: 3833: 3827: 3816: 3814: 3813: 3808: 3790: 3788: 3787: 3782: 3744: 3742: 3741: 3736: 3731: 3728: 3725: 3710: 3707: 3694: 3692: 3691: 3686: 3680: 3674: 3672: 3657: 3655: 3654: 3649: 3647: 3646: 3624: 3622: 3621: 3616: 3614: 3613: 3597: 3595: 3594: 3589: 3583: 3577: 3568: 3566: 3565: 3560: 3558: 3557: 3535: 3533: 3532: 3527: 3525: 3524: 3505: 3503: 3502: 3497: 3490: 3478: 3472: 3457: 3456: 3453: 3450: 3446: 3436: 3433: 3420: 3414: 3412: 3397: 3396: 3386: 3375: 3354: 3353: 3338: 3337: 3316: 3315: 3303: 3295: 3290: 3289: 3265: 3263: 3262: 3257: 3255: 3254: 3228: 3227: 3211: 3207: 3205: 3204: 3199: 3195: 3194: 3193: 3176: 3170: 3166: 3165: 3154: 3146: 3139: 3135: 3127: 3125: 3124: 3119: 3117: 3105: 3103: 3102: 3097: 3095: 3094: 3089: 3080: 3079: 3074: 3001:spectral methods 2903:cosine transform 2896: 2894: 2893: 2888: 2867: 2865: 2864: 2859: 2847: 2845: 2844: 2839: 2827: 2825: 2824: 2819: 2725:formats such as 2648:data compression 2572:Teleconferencing 2529:mobile telephony 2459:Mobile telephony 2432: 2431: 2088: 2087: 2016: 2006: 1871: 1855: 1783: 1765:distribution on 1702: 1682: 1681: 1562: 1561: 1530: 1528: 1527: 1522: 1498: 1496: 1495: 1490: 1479:. In this case, 1470: 1468: 1467: 1462: 1424: 1306:video production 1286:Video processing 1149:feature analysis 1147:Complex cepstrum 1086:(ROI) extraction 1019:foveated imaging 979:corner detection 963:Image processing 874:identification, 810:entropy encoding 789:transform coding 785:Data compression 747:usage reducation 680:teleconferencing 672:mobile telephony 564:spectral methods 549:Markov processes 536:(HD video), and 474:data compression 454:delta modulation 201:spectral methods 122: 91:data compression 54: 51: 45: 42: 27: 26: 19: 22636: 22635: 22631: 22630: 22629: 22627: 22626: 22625: 22566: 22565: 22564: 22559: 22519: 22517: 22509: 22451: 22388: 22310: 22274: 22231: 22180: 22172: 22113: 22106: 22012:Robert Metcalfe 21867:Tim Berners-Lee 21815: 21635:Information Age 21507: 21502: 21472: 21467: 21405: 21319: 21288: 21274:Discrete signal 21257: 21252: 21222: 21217: 21202: 21176: 21021: 20991: 20849: 20826: 20798: 20775: 20697: 20616: 20514: 20493: 20482: 20278: 20250:(GB/T 33475.3) 20244:(GB/T 20090.10) 20230: 20207: 20179: 20121: 20083: 19998: 19994:MPEG-H 3D Audio 19920:MPEG-1 Layer II 19910: 19893: 19882: 19714: 19683: 19634: 19596: 19544: 19440: 19423: 19416: 19403: 19373: 19368: 19335: 19319: 19303: 19284:Rate–distortion 19217: 19146: 19065: 18992: 18913: 18818: 18814:Sub-band coding 18722: 18647:Predictive type 18642: 18567: 18534:LZSS + Huffman 18484:LZ77 + Huffman 18473: 18383: 18319:Dictionary type 18313: 18215:Adaptive coding 18192: 18186: 18146:Algorithm Alley 18104: 18093: 18091: 18084: 18067: 17997:(4): 992–1000. 17986: 17932: 17922: 17871: 17715:10.1.1.205.4523 17654: 17652:Further reading 17649: 17648: 17599: 17595: 17584: 17580: 17557: 17553: 17522: 17518: 17507: 17500: 17477: 17473: 17466: 17462: 17455: 17451: 17402: 17398: 17379: 17375: 17368: 17354: 17350: 17341: 17339: 17331: 17330: 17326: 17315: 17311: 17304: 17290: 17286: 17276: 17274: 17272: 17261: 17253: 17249: 17239: 17237: 17227: 17226: 17222: 17215: 17197: 17190: 17149: 17145: 17137: 17133: 17118: 17114: 17099: 17069: 17065: 17043: 17039: 17001: 16997: 16982: 16978: 16947:10.1.1.516.2824 16921: 16915: 16911: 16881: 16877: 16867: 16865: 16857: 16853: 16852: 16848: 16835: 16834: 16830: 16821: 16819: 16810: 16809: 16805: 16797: 16793: 16786: 16784: 16771: 16767: 16754: 16753: 16749: 16742: 16722: 16715: 16705: 16703: 16690: 16682: 16678: 16668: 16666: 16653: 16645: 16641: 16609: 16603: 16599: 16589: 16587: 16583: 16570: 16566: 16565: 16561: 16552: 16550: 16545: 16544: 16540: 16525: 16499: 16495: 16488: 16462: 16458: 16448: 16446: 16434: 16430: 16420: 16418: 16409: 16408: 16404: 16379: 16375: 16344: 16340: 16330: 16328: 16322: 16318: 16308: 16306: 16300: 16296: 16286: 16284: 16278: 16274: 16261: 16259: 16247: 16243: 16234: 16232: 16225: 16219: 16215: 16206: 16202: 16198: 16180: 16179: 16175: 16165: 16163: 16151: 16147: 16108: 16097: 16087: 16080: 16073: 16065:. p. 255. 16053: 16049: 16041: 16034: 16028: 16024: 16016: 16009: 16003: 15996: 15983: 15981: 15977: 15966: 15960: 15956: 15946: 15944: 15936: 15935: 15931: 15921: 15919: 15904: 15903: 15899: 15868: 15864: 15832: 15829: 15822: 15782: 15778: 15768: 15766: 15755: 15754: 15747: 15740: 15712: 15708: 15701: 15681: 15677: 15670: 15654: 15650: 15640: 15638: 15622: 15615: 15604: 15584: 15573: 15566: 15550: 15546: 15536: 15534: 15524: 15520: 15519: 15512: 15502: 15500: 15484: 15473: 15463: 15461: 15451: 15438: 15431: 15411: 15402: 15395: 15375: 15368: 15361: 15345: 15332: 15325: 15301: 15236: 15229: 15205: 15198: 15191: 15161: 15152: 15145: 15125: 15116: 15090: 15084: 15077: 15067: 15065: 15051: 15042: 15032: 15030: 15019: 15018: 15009: 15002: 14994:. p. 906. 14982: 14978: 14963: 14941: 14937: 14922: 14896: 14892: 14850: 14846: 14837: 14835: 14828: 14821: 14813: 14806: 14800: 14793: 14783: 14781: 14772: 14766: 14759: 14728: 14721: 14714: 14706:. p. 590. 14694: 14669: 14646: 14642: 14615: 14611: 14604: 14596:. p. 847. 14584: 14580: 14573: 14553: 14544: 14534: 14532: 14521: 14520: 14511: 14504: 14488: 14484: 14445: 14441: 14434: 14414: 14410: 14400: 14398: 14383: 14379: 14348: 14344: 14319: 14315: 14305: 14303: 14295: 14291: 14290: 14277: 14243: 14234: 14227: 14217: 14215: 14203: 14190: 14180: 14178: 14163: 14146: 14107: 14068: 14053: 13981: 13966: 13956: 13954: 13947: 13941: 13926: 13916: 13914: 13899: 13888: 13849: 13827: 13814: 13800: 13798: 13794: 13780: 13761: 13755: 13698: 13693: 13688: 13687: 13662: 13660: 13657: 13656: 13618: 13614: 13603: 13600: 13599: 13597: 13593: 13558: 13557: 13552: 13549: 13548: 13526: 13523: 13522: 13500: 13497: 13496: 13471: 13468: 13467: 13465: 13461: 13455: 13451: 13446: 13426: 13422: 13417: 13413: 13409: 13405: 13395: 13359: 13358: 13353: 13348: 13340: 13332: 13327: 13322: 13314: 13305: 13304: 13296: 13288: 13283: 13275: 13267: 13259: 13254: 13248: 13247: 13239: 13231: 13226: 13221: 13213: 13205: 13200: 13194: 13193: 13185: 13177: 13169: 13161: 13156: 13151: 13146: 13140: 13139: 13134: 13126: 13121: 13116: 13108: 13100: 13092: 13083: 13082: 13077: 13072: 13064: 13059: 13051: 13043: 13038: 13032: 13031: 13023: 13018: 13010: 13002: 12997: 12992: 12987: 12981: 12980: 12975: 12967: 12962: 12957: 12952: 12947: 12939: 12929: 12928: 12926: 12923: 12922: 12920: 12894: 12892:Example of IDCT 12864: 12861: 12860: 12833: 12832: 12817: 12813: 12798: 12793: 12790: 12789: 12748: 12744: 12733: 12730: 12729: 12699: 12696: 12695: 12654: 12651: 12650: 12641:DCT-II used in 12638: 12634: 12576: 12575: 12570: 12567: 12566: 12525: 12524: 12519: 12516: 12515: 12486: 12485: 12480: 12477: 12476: 12434: 12433: 12428: 12425: 12424: 12401: 12397: 12388: 12387: 12382: 12379: 12378: 12375: 12293: 12290: 12289: 12219: 12216: 12215: 12176: 12130: 12128: 12124: 12120: 12096: 12050: 12048: 12044: 12040: 12022: 12018: 12003: 11992: 11975: 11964: 11945: 11941: 11939: 11936: 11935: 11928: 11925: 11912: 11882: 11878: 11869: 11865: 11857: 11854: 11853: 11850: 11736:Transform Size 11700: 11696: 11684: 11680: 11662: 11658: 11652: 11648: 11638: 11637: 11633: 11628: 11625: 11624: 11593: 11589: 11583: 11579: 11569: 11568: 11564: 11559: 11556: 11555: 11524: 11520: 11508: 11504: 11486: 11482: 11476: 11472: 11462: 11461: 11457: 11448: 11432: 11428: 11416: 11412: 11394: 11390: 11384: 11380: 11370: 11369: 11365: 11363: 11353: 11331: 11327: 11321: 11317: 11307: 11306: 11302: 11300: 11294: 11291: 11290: 11256: 11252: 11243: 11239: 11224: 11223: 11219: 11214: 11211: 11210: 11178: 11174: 11168: 11164: 11149: 11148: 11144: 11139: 11136: 11135: 11115: 11111: 11096: 11091: 11088: 11087: 11058: 11054: 11046: 11043: 11042: 11039: 11015: 10989: 10974: 10965: 10961: 10948: 10939: 10935: 10922: 10913: 10909: 10908: 10904: 10893: 10892: 10874: 10870: 10856: 10853: 10852: 10823: 10814: 10810: 10801: 10797: 10784: 10775: 10771: 10770: 10766: 10755: 10754: 10742: 10738: 10724: 10721: 10720: 10695: 10691: 10682: 10678: 10665: 10652: 10643: 10639: 10638: 10634: 10623: 10622: 10610: 10606: 10583: 10579: 10570: 10566: 10553: 10544: 10540: 10539: 10535: 10524: 10523: 10517: 10513: 10499: 10496: 10495: 10466: 10457: 10453: 10440: 10431: 10427: 10418: 10414: 10413: 10409: 10398: 10397: 10385: 10381: 10358: 10354: 10341: 10332: 10328: 10319: 10315: 10314: 10310: 10299: 10298: 10292: 10288: 10274: 10271: 10270: 10241: 10232: 10228: 10219: 10215: 10206: 10202: 10201: 10197: 10186: 10185: 10179: 10175: 10154: 10150: 10141: 10137: 10128: 10124: 10110: 10109: 10097: 10093: 10084: 10080: 10071: 10067: 10052: 10041: 10040: 10039: 10037: 10034: 10033: 9998: 9994: 9961: 9957: 9924: 9920: 9893: 9889: 9880: 9876: 9867: 9863: 9848: 9837: 9836: 9835: 9815: 9814: 9801: 9797: 9796: 9772: 9771: 9758: 9754: 9753: 9729: 9728: 9715: 9711: 9710: 9694: 9690: 9681: 9677: 9668: 9664: 9656: 9653: 9652: 9624: 9620: 9618: 9615: 9614: 9597: 9593: 9584: 9580: 9578: 9575: 9574: 9537: and 9535: 9517: 9513: 9497: 9492: 9483: 9479: 9477: 9474: 9473: 9450: 9446: 9425: 9421: 9400: 9396: 9375: 9371: 9362: 9358: 9349: 9345: 9331: 9330: 9318: 9305: 9301: 9300: 9284: 9271: 9267: 9266: 9250: 9237: 9233: 9232: 9216: 9212: 9203: 9199: 9190: 9186: 9178: 9175: 9174: 9148: 9144: 9123: 9119: 9111: 9108: 9107: 9076: 9067: 9063: 9054: 9050: 9041: 9037: 9029: 9026: 9025: 9009: 9008: 8993: 8989: 8971: 8967: 8949: 8945: 8918: 8914: 8893: 8889: 8868: 8864: 8844: 8843: 8840: 8839: 8830: 8826: 8808: 8804: 8786: 8782: 8761: 8757: 8742: 8738: 8717: 8713: 8693: 8692: 8689: 8688: 8673: 8669: 8657: 8653: 8635: 8631: 8604: 8600: 8585: 8581: 8566: 8562: 8542: 8541: 8538: 8537: 8528: 8524: 8512: 8508: 8490: 8486: 8465: 8461: 8452: 8448: 8433: 8429: 8409: 8408: 8405: 8404: 8389: 8385: 8367: 8363: 8351: 8347: 8320: 8316: 8295: 8291: 8276: 8272: 8258: 8257: 8254: 8253: 8244: 8240: 8222: 8218: 8206: 8202: 8181: 8177: 8162: 8158: 8143: 8139: 8125: 8124: 8121: 8120: 8105: 8101: 8089: 8085: 8073: 8069: 8042: 8038: 8023: 8019: 8010: 8006: 7992: 7991: 7988: 7987: 7978: 7974: 7962: 7958: 7946: 7942: 7921: 7917: 7908: 7904: 7895: 7891: 7877: 7876: 7872: 7870: 7867: 7866: 7848: 7809: 7805: 7772: 7768: 7763: 7748: 7744: 7729: 7720: 7716: 7715: 7711: 7703: 7699: 7694: 7693: 7689: 7672: 7668: 7653: 7644: 7640: 7639: 7635: 7627: 7623: 7618: 7617: 7613: 7596: 7592: 7577: 7568: 7564: 7563: 7559: 7551: 7547: 7542: 7541: 7537: 7523: 7519: 7510: 7506: 7497: 7493: 7492: 7488: 7474: 7470: 7469: 7456: 7452: 7451: 7433: 7429: 7428: 7415: 7411: 7410: 7392: 7388: 7387: 7374: 7370: 7369: 7354: 7350: 7341: 7337: 7328: 7324: 7323: 7319: 7317: 7314: 7313: 7301:The inverse of 7276: 7272: 7239: 7235: 7230: 7215: 7211: 7196: 7187: 7183: 7182: 7178: 7170: 7166: 7161: 7160: 7156: 7139: 7135: 7120: 7111: 7107: 7106: 7102: 7094: 7090: 7085: 7084: 7080: 7063: 7059: 7044: 7035: 7031: 7030: 7026: 7018: 7014: 7009: 7008: 7004: 6990: 6986: 6977: 6973: 6964: 6960: 6959: 6955: 6941: 6937: 6936: 6923: 6919: 6918: 6900: 6896: 6895: 6882: 6878: 6877: 6859: 6855: 6854: 6841: 6837: 6836: 6821: 6817: 6808: 6804: 6795: 6791: 6790: 6786: 6784: 6781: 6780: 6748: 6747: 6733: 6729: 6714: 6705: 6701: 6700: 6696: 6688: 6684: 6679: 6678: 6674: 6657: 6653: 6638: 6629: 6625: 6624: 6620: 6612: 6608: 6603: 6602: 6598: 6584: 6580: 6571: 6567: 6566: 6562: 6548: 6544: 6543: 6530: 6526: 6525: 6507: 6503: 6502: 6489: 6485: 6484: 6471: 6470: 6459: 6455: 6440: 6431: 6427: 6426: 6422: 6414: 6410: 6405: 6404: 6400: 6378: 6374: 6359: 6350: 6346: 6345: 6341: 6333: 6329: 6324: 6323: 6319: 6305: 6301: 6292: 6288: 6287: 6283: 6269: 6265: 6264: 6251: 6247: 6246: 6241: 6237: 6223: 6219: 6218: 6205: 6201: 6200: 6189: 6181: 6177: 6168: 6164: 6163: 6159: 6155: 6153: 6150: 6149: 6142: 6133: 6119: 6117: 6096: 6091: 6089: 6086: 6085: 6063: 6036: 6033: 6032: 6028: 5998: 5993: 5988: 5980: 5977: 5976: 5957: 5954: 5953: 5934: 5931: 5930: 5929:(for DCT-I) or 5899: 5896: 5895: 5879: 5876: 5875: 5864: 5840: 5836: 5834: 5831: 5830: 5808: 5791: 5788: 5787: 5786:and odd around 5768: 5754: 5751: 5750: 5749:is even around 5733: 5729: 5727: 5724: 5723: 5686: 5681: 5679: 5676: 5675: 5618: for 5616: 5592: 5585: 5581: 5564: 5557: 5553: 5539: 5536: 5534: 5530: 5518: 5514: 5502: 5491: 5478: 5474: 5472: 5469: 5468: 5465: 5442: 5425: 5422: 5421: 5402: 5388: 5385: 5384: 5383:is even around 5367: 5363: 5361: 5358: 5357: 5333: 5330: 5329: 5328:and odd around 5307: 5304: 5303: 5302:is even around 5286: 5282: 5280: 5277: 5276: 5244: 5239: 5237: 5234: 5233: 5215: 5210: 5204: 5200: 5198: 5195: 5194: 5176: 5174: 5171: 5170: 5153: 5149: 5147: 5144: 5143: 5084: for 5082: 5055: 5048: 5044: 5031: 5028: 5026: 5022: 5010: 5006: 4994: 4983: 4970: 4966: 4954: 4945: 4941: 4939: 4936: 4935: 4932: 4903: 4900: 4899: 4898:and odd around 4877: 4874: 4873: 4872:is even around 4856: 4852: 4850: 4847: 4846: 4824: 4807: 4804: 4803: 4784: 4770: 4767: 4766: 4765:is even around 4749: 4745: 4743: 4740: 4739: 4698: 4693: 4688: 4686: 4684: 4681: 4680: 4660: 4655: 4650: 4647: 4646: 4629: 4625: 4623: 4620: 4619: 4608: 4574: 4571: 4570: 4553: 4549: 4531: 4527: 4525: 4522: 4521: 4492: 4488: 4486: 4483: 4482: 4452: 4449: 4448: 4431: 4427: 4409: 4405: 4403: 4400: 4399: 4371: 4367: 4365: 4362: 4361: 4341: 4337: 4335: 4332: 4331: 4312: 4309: 4308: 4285: 4282: 4281: 4218: for 4216: 4189: 4182: 4178: 4165: 4162: 4160: 4156: 4144: 4140: 4128: 4117: 4104: 4100: 4098: 4095: 4094: 4091: 4067: 4063: 4061: 4058: 4057: 4029: 4026: 4025: 4003: 4000: 3999: 3998:is even around 3982: 3978: 3976: 3973: 3972: 3951: 3948: 3947: 3931: 3928: 3927: 3866: 3863: 3862: 3822: 3819: 3818: 3796: 3793: 3792: 3761: 3758: 3757: 3714: 3708: 3706: 3704: 3701: 3700: 3673: 3668: 3663: 3660: 3659: 3636: 3632: 3630: 3627: 3626: 3609: 3605: 3603: 3600: 3599: 3576: 3574: 3571: 3570: 3547: 3543: 3541: 3538: 3537: 3520: 3516: 3514: 3511: 3510: 3454: for 3452: 3421: 3413: 3410: 3408: 3404: 3392: 3388: 3376: 3365: 3343: 3339: 3333: 3329: 3311: 3307: 3294: 3285: 3281: 3279: 3276: 3275: 3272: 3244: 3240: 3223: 3219: 3217: 3214: 3213: 3209: 3183: 3179: 3161: 3157: 3152: 3149: 3148: 3144: 3137: 3133: 3113: 3111: 3108: 3107: 3090: 3085: 3084: 3075: 3070: 3069: 3061: 3058: 3057: 3047: 3012:discontinuities 2873: 2870: 2869: 2853: 2850: 2849: 2833: 2830: 2829: 2804: 2801: 2800: 2760:with different 2754: 2684: 2640: 2623: 2617: 2430: 2420:MPEG-H 3D Audio 2350:Digital radio, 2323:streaming media 2220:streaming media 2175:streaming media 2128:streaming media 2086: 2081: 2075: 2014: 2004: 1920:video telephony 1869: 1851: 1844:Video telephony 1781: 1747:digital cameras 1718:video telephone 1698: 1680: 1560: 1504: 1501: 1500: 1484: 1481: 1480: 1450: 1447: 1446: 1420: 1409: 1404: 1399:acoustic sensor 1358:Radio frequency 1304:, professional 1298:data extraction 1290:motion analysis 1272:3D video coding 1238:high-definition 1064:Medical imaging 1042:visual recovery 1013:(JND) profile, 953:multiview image 931:digital cameras 923:Digital imaging 846:video-on-demand 838:streaming video 834:streaming audio 830:Streaming media 676:video telephony 587: 526:video on demand 514:streaming media 476:. Uncompressed 466: 295: 240:the inverse DCT 214:similar to the 118: 55: 49: 46: 41: 36:Please see the 28: 24: 17: 12: 11: 5: 22634: 22624: 22623: 22618: 22613: 22608: 22603: 22598: 22593: 22588: 22583: 22578: 22561: 22560: 22558: 22557: 22547: 22537: 22527: 22514: 22511: 22510: 22508: 22507: 22500: 22495: 22490: 22485: 22480: 22479: 22478: 22473: 22465: 22459: 22457: 22453: 22452: 22450: 22449: 22444: 22439: 22434: 22429: 22424: 22419: 22414: 22409: 22404: 22398: 22396: 22390: 22389: 22387: 22386: 22381: 22376: 22371: 22366: 22361: 22356: 22351: 22346: 22341: 22336: 22331: 22326: 22320: 22318: 22312: 22311: 22309: 22308: 22303: 22298: 22293: 22288: 22282: 22280: 22276: 22275: 22273: 22272: 22267: 22262: 22257: 22252: 22247: 22245:Space-division 22241: 22239: 22233: 22232: 22230: 22229: 22224: 22223: 22222: 22217: 22207: 22206: 22205: 22195: 22190: 22184: 22182: 22174: 22173: 22171: 22170: 22169: 22168: 22158: 22157: 22156: 22146: 22141: 22136: 22135: 22134: 22124: 22118: 22116: 22108: 22107: 22105: 22104: 22099: 22094: 22089: 22084: 22082:Camille Tissot 22079: 22074: 22069: 22064: 22059: 22057:Claude Shannon 22054: 22049: 22047:Tivadar Puskás 22044: 22039: 22034: 22029: 22024: 22019: 22017:Antonio Meucci 22014: 22009: 22004: 21999: 21994: 21989: 21987:Charles K. Kao 21984: 21979: 21974: 21969: 21964: 21962:Harold Hopkins 21959: 21954: 21949: 21944: 21939: 21934: 21929: 21924: 21919: 21914: 21909: 21904: 21899: 21894: 21889: 21884: 21879: 21874: 21869: 21864: 21862:Emile Berliner 21859: 21854: 21849: 21844: 21839: 21834: 21829: 21823: 21821: 21817: 21816: 21814: 21813: 21808: 21803: 21801:Videotelephony 21798: 21793: 21792: 21791: 21786: 21776: 21769: 21764: 21758: 21753: 21748: 21743: 21738: 21737: 21736: 21731: 21726: 21716: 21715: 21714: 21704: 21699: 21697:Radiotelephone 21694: 21689: 21684: 21679: 21674: 21669: 21664: 21663: 21662: 21652: 21647: 21642: 21637: 21632: 21627: 21622: 21617: 21612: 21607: 21602: 21601: 21600: 21595: 21590: 21585: 21583:Internet video 21575: 21574: 21573: 21568: 21563: 21558: 21548: 21543: 21538: 21533: 21528: 21523: 21517: 21515: 21509: 21508: 21501: 21500: 21493: 21486: 21478: 21469: 21468: 21466: 21465: 21460: 21455: 21450: 21445: 21440: 21431: 21426: 21421: 21415: 21413: 21407: 21406: 21404: 21403: 21398: 21393: 21388: 21383: 21378: 21373: 21367: 21361: 21355: 21350: 21345: 21344: 21343: 21338: 21327: 21325: 21321: 21320: 21318: 21317: 21312: 21307: 21302: 21296: 21294: 21290: 21289: 21287: 21286: 21281: 21276: 21271: 21265: 21263: 21259: 21258: 21251: 21250: 21243: 21236: 21228: 21219: 21218: 21207: 21204: 21203: 21201: 21200: 21198:List of codecs 21195: 21190: 21184: 21182: 21178: 21177: 21175: 21174: 21173: 21172: 21171: 21170: 21165: 21155: 21150: 21145: 21135: 21134: 21133: 21128: 21123: 21118: 21108: 21107: 21106: 21101: 21091: 21086: 21081: 21080: 21079: 21074: 21069: 21064: 21054: 21053: 21052: 21047: 21042: 21031: 21029: 21023: 21022: 21020: 21019: 21014: 21009: 21003: 21001: 21000:Collaborations 20997: 20996: 20993: 20992: 20990: 20989: 20984: 20979: 20978: 20977: 20967: 20962: 20957: 20952: 20951: 20950: 20940: 20935: 20930: 20925: 20920: 20915: 20910: 20905: 20904: 20903: 20893: 20888: 20883: 20878: 20873: 20868: 20863: 20857: 20855: 20851: 20850: 20848: 20847: 20842: 20836: 20834: 20828: 20827: 20825: 20824: 20819: 20814: 20808: 20806: 20800: 20799: 20797: 20796: 20791: 20785: 20783: 20777: 20776: 20774: 20773: 20768: 20766:MPEG-21 Part 9 20763: 20758: 20755:MPEG-4 Part 14 20752: 20747: 20742: 20737: 20736: 20735: 20724: 20722: 20709: 20703: 20702: 20699: 20698: 20696: 20695: 20690: 20685: 20680: 20675: 20670: 20665: 20660: 20655: 20650: 20645: 20640: 20635: 20630: 20624: 20622: 20618: 20617: 20615: 20614: 20609: 20604: 20599: 20594: 20589: 20584: 20579: 20574: 20569: 20564: 20559: 20554: 20549: 20544: 20539: 20534: 20528: 20526: 20496: 20488: 20487: 20484: 20483: 20481: 20480: 20475: 20470: 20465: 20460: 20455: 20450: 20445: 20440: 20435: 20430: 20425: 20420: 20415: 20414: 20413: 20407: 20402: 20397: 20392: 20387: 20382: 20377: 20372: 20367: 20362: 20357: 20352: 20347: 20342: 20337: 20335:Monkey's Audio 20332: 20327: 20322: 20317: 20312: 20307: 20302: 20297: 20292: 20286: 20284: 20280: 20279: 20277: 20276: 20275:(SJ/T 11299.4) 20270: 20265: 20259: 20258: 20257: 20245: 20238: 20236: 20232: 20231: 20229: 20228: 20223: 20217: 20215: 20209: 20208: 20206: 20205: 20200: 20195: 20189: 20187: 20181: 20180: 20178: 20177: 20172: 20167: 20162: 20157: 20152: 20147: 20142: 20137: 20131: 20129: 20123: 20122: 20120: 20119: 20114: 20109: 20104: 20099: 20093: 20091: 20085: 20084: 20082: 20081: 20076: 20071: 20066: 20061: 20056: 20051: 20046: 20041: 20036: 20031: 20026: 20025: 20024: 20019: 20008: 20006: 20000: 19999: 19997: 19996: 19991: 19986: 19981: 19976: 19971: 19966: 19961: 19956: 19955: 19954: 19949: 19939: 19934: 19932:MPEG-1 Layer I 19929: 19928: 19927: 19916: 19914: 19896: 19888: 19887: 19884: 19883: 19881: 19880: 19875: 19870: 19865: 19860: 19855: 19850: 19845: 19840: 19835: 19830: 19825: 19824: 19823: 19818: 19810: 19809: 19808: 19803: 19793: 19788: 19783: 19778: 19773: 19768: 19763: 19758: 19753: 19748: 19743: 19738: 19733: 19728: 19722: 19720: 19716: 19715: 19713: 19712: 19709: 19708: 19707: 19698: 19691: 19689: 19685: 19684: 19682: 19681: 19676: 19671: 19666: 19661: 19656: 19651: 19645: 19643: 19636: 19635: 19633: 19632: 19627: 19622: 19617: 19612: 19606: 19604: 19598: 19597: 19595: 19594: 19589: 19584: 19579: 19574: 19569: 19564: 19558: 19556: 19546: 19545: 19543: 19542: 19541: 19540: 19538:Part 2 / LCEVC 19535: 19525: 19524: 19523: 19513: 19512: 19511: 19501: 19500: 19499: 19494: 19489: 19479: 19478: 19477: 19467: 19462: 19457: 19452: 19446: 19444: 19426: 19418: 19417: 19402: 19401: 19394: 19387: 19379: 19370: 19369: 19367: 19366: 19351: 19340: 19337: 19336: 19334: 19333: 19327: 19325: 19321: 19320: 19318: 19317: 19311: 19309: 19305: 19304: 19302: 19301: 19296: 19291: 19286: 19281: 19276: 19271: 19266: 19265: 19264: 19254: 19249: 19248: 19247: 19242: 19231: 19229: 19223: 19222: 19219: 19218: 19216: 19215: 19214: 19213: 19208: 19198: 19197: 19196: 19191: 19186: 19178: 19173: 19168: 19163: 19157: 19155: 19148: 19147: 19145: 19144: 19139: 19134: 19129: 19124: 19119: 19114: 19109: 19108: 19107: 19102: 19097: 19086: 19084: 19077: 19071: 19070: 19067: 19066: 19064: 19063: 19062: 19061: 19056: 19051: 19046: 19036: 19031: 19026: 19021: 19016: 19011: 19006: 19000: 18998: 18994: 18993: 18991: 18990: 18985: 18980: 18975: 18970: 18965: 18960: 18955: 18950: 18945: 18940: 18934: 18932: 18925: 18919: 18918: 18915: 18914: 18912: 18911: 18906: 18901: 18900: 18899: 18894: 18889: 18884: 18879: 18869: 18868: 18867: 18857: 18856: 18855: 18850: 18840: 18835: 18829: 18827: 18820: 18819: 18817: 18816: 18811: 18806: 18801: 18796: 18791: 18786: 18781: 18776: 18771: 18766: 18765: 18764: 18759: 18754: 18743: 18741: 18734: 18728: 18727: 18724: 18723: 18721: 18720: 18718:Psychoacoustic 18715: 18714: 18713: 18708: 18703: 18695: 18694: 18693: 18688: 18683: 18678: 18673: 18663: 18662: 18661: 18650: 18648: 18644: 18643: 18641: 18640: 18639: 18638: 18633: 18628: 18618: 18613: 18608: 18607: 18606: 18601: 18590: 18588: 18586:Transform type 18579: 18573: 18572: 18569: 18568: 18566: 18565: 18564: 18563: 18555: 18554: 18553: 18550: 18542: 18541: 18540: 18532: 18531: 18530: 18522: 18521: 18520: 18512: 18511: 18510: 18502: 18501: 18500: 18495: 18490: 18481: 18479: 18475: 18474: 18472: 18471: 18466: 18461: 18456: 18451: 18446: 18445: 18444: 18439: 18429: 18424: 18419: 18418: 18417: 18407: 18402: 18397: 18391: 18389: 18385: 18384: 18382: 18381: 18380: 18379: 18374: 18369: 18364: 18359: 18354: 18349: 18344: 18339: 18329: 18323: 18321: 18315: 18314: 18312: 18311: 18310: 18309: 18304: 18299: 18294: 18284: 18279: 18274: 18269: 18264: 18259: 18254: 18253: 18252: 18247: 18242: 18232: 18227: 18222: 18217: 18211: 18209: 18200: 18194: 18193: 18185: 18184: 18177: 18170: 18162: 18156: 18155: 18149: 18142: 18135: 18128:FFTW Home Page 18116: 18111: 18103: 18102:External links 18100: 18099: 18098: 18082: 18065: 18044: 18034:(1): 213–220. 18023: 17979: 17958:10.1.1.66.3097 17943:(2): 216–231. 17925: 17920: 17903: 17874: 17869: 17856: 17827: 17791: 17773:(4): 259–299. 17758: 17746:(1): 309–345. 17728: 17708:(6): 849–863. 17697: 17676: 17666:(6): 934–936. 17653: 17650: 17647: 17646: 17593: 17578: 17551: 17516: 17498: 17471: 17460: 17449: 17396: 17373: 17366: 17348: 17324: 17309: 17302: 17284: 17270: 17247: 17220: 17213: 17205:Academic Press 17188: 17143: 17131: 17112: 17097: 17063: 17037: 17011:(3): 259–263, 16995: 16976: 16932:(5): 758–763, 16909: 16875: 16862:Fraunhofer IIS 16846: 16837:"☏ FreeSWITCH" 16828: 16803: 16765: 16747: 16740: 16734:. p. 69. 16713: 16696:Fraunhofer IIS 16676: 16659:Fraunhofer IIS 16639: 16620:(1): 202–236. 16597: 16586:on 30 May 2019 16559: 16538: 16523: 16493: 16486: 16456: 16428: 16402: 16373: 16354:(3): 137–142. 16338: 16316: 16294: 16272: 16241: 16213: 16196: 16173: 16145: 16107:978-0133099072 16106: 16078: 16071: 16047: 16044:on 2013-01-23. 16022: 16019:on 2013-01-23. 15994: 15954: 15929: 15897: 15862: 15843:(4): 547–552. 15820: 15793:(11): 114003. 15776: 15745: 15738: 15706: 15699: 15693:. p. 95. 15675: 15668: 15648: 15613: 15602: 15596:. p. 58. 15571: 15564: 15544: 15510: 15471: 15436: 15429: 15423:. p. 55. 15400: 15393: 15366: 15359: 15330: 15323: 15234: 15227: 15196: 15189: 15150: 15143: 15137:. p. 25. 15114: 15075: 15040: 15007: 15000: 14976: 14961: 14935: 14920: 14890: 14844: 14819: 14791: 14757: 14719: 14712: 14667: 14640: 14609: 14602: 14578: 14571: 14542: 14509: 14502: 14482: 14439: 14432: 14426:. p. 29. 14408: 14377: 14342: 14313: 14275: 14225: 14188: 14144: 14106:978-0125802031 14105: 14079:Academic Press 14051: 13964: 13924: 13886: 13848:978-0123736246 13847: 13839:Academic Press 13812: 13779:978-9521528187 13778: 13695: 13694: 13692: 13689: 13686: 13685: 13666: 13641: 13638: 13635: 13632: 13629: 13626: 13621: 13617: 13613: 13610: 13591: 13572: 13569: 13566: 13561: 13533: 13507: 13481: 13478: 13459: 13448: 13447: 13445: 13442: 13441: 13440: 13435: 13430: 13401: 13394: 13391: 13363: 13357: 13354: 13352: 13349: 13347: 13344: 13341: 13339: 13336: 13333: 13331: 13328: 13326: 13323: 13321: 13318: 13315: 13313: 13310: 13307: 13306: 13303: 13300: 13297: 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12893: 12890: 12877: 12871: 12847: 12844: 12841: 12836: 12831: 12828: 12825: 12820: 12816: 12812: 12806: 12803: 12771: 12768: 12765: 12762: 12759: 12756: 12751: 12747: 12743: 12740: 12706: 12692:Makhoul (1980) 12664: 12661: 12590: 12587: 12584: 12579: 12548: 12545: 12542: 12539: 12536: 12533: 12528: 12500: 12497: 12494: 12489: 12457: 12454: 12451: 12448: 12445: 12442: 12437: 12409: 12404: 12400: 12396: 12391: 12374: 12371: 12366: 12365: 12354: 12348: 12345: 12342: 12336: 12330: 12327: 12321: 12318: 12312: 12309: 12306: 12303: 12271: 12268: 12265: 12259: 12253: 12247: 12244: 12238: 12235: 12232: 12229: 12211: 12210: 12199: 12192: 12182: 12179: 12173: 12167: 12164: 12161: 12158: 12155: 12152: 12149: 12146: 12143: 12140: 12137: 12134: 12123: 12119: 12116: 12112: 12102: 12099: 12093: 12087: 12084: 12081: 12078: 12075: 12072: 12069: 12066: 12063: 12060: 12057: 12054: 12043: 12039: 12036: 12031: 12028: 12025: 12021: 12012: 12009: 12006: 12001: 11998: 11995: 11991: 11984: 11981: 11978: 11973: 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8397: 8392: 8388: 8384: 8381: 8378: 8375: 8370: 8366: 8362: 8359: 8354: 8350: 8346: 8343: 8340: 8337: 8334: 8331: 8328: 8323: 8319: 8315: 8312: 8309: 8306: 8303: 8298: 8294: 8290: 8287: 8284: 8279: 8275: 8271: 8265: 8262: 8256: 8255: 8252: 8247: 8243: 8239: 8236: 8233: 8230: 8225: 8221: 8217: 8214: 8209: 8205: 8201: 8198: 8195: 8192: 8189: 8184: 8180: 8176: 8173: 8170: 8165: 8161: 8157: 8154: 8151: 8146: 8142: 8138: 8132: 8129: 8123: 8122: 8119: 8116: 8113: 8108: 8104: 8100: 8097: 8092: 8088: 8084: 8081: 8076: 8072: 8068: 8065: 8062: 8059: 8056: 8053: 8050: 8045: 8041: 8037: 8034: 8031: 8026: 8022: 8018: 8013: 8009: 8005: 7999: 7996: 7990: 7989: 7986: 7981: 7977: 7973: 7970: 7965: 7961: 7957: 7954: 7949: 7945: 7941: 7938: 7935: 7932: 7929: 7924: 7920: 7916: 7911: 7907: 7903: 7898: 7894: 7890: 7884: 7881: 7875: 7874: 7847: 7844: 7832: 7831: 7820: 7817: 7812: 7808: 7804: 7801: 7798: 7795: 7792: 7789: 7786: 7783: 7780: 7775: 7771: 7761: 7757: 7751: 7747: 7742: 7736: 7733: 7728: 7723: 7719: 7714: 7706: 7702: 7698: 7692: 7688: 7685: 7681: 7675: 7671: 7666: 7660: 7657: 7652: 7647: 7643: 7638: 7630: 7626: 7622: 7616: 7612: 7609: 7605: 7599: 7595: 7590: 7584: 7581: 7576: 7571: 7567: 7562: 7554: 7550: 7546: 7540: 7536: 7533: 7526: 7522: 7518: 7513: 7509: 7505: 7500: 7496: 7491: 7485: 7482: 7477: 7473: 7467: 7464: 7459: 7455: 7450: 7444: 7441: 7436: 7432: 7426: 7423: 7418: 7414: 7409: 7403: 7400: 7395: 7391: 7385: 7382: 7377: 7373: 7368: 7364: 7357: 7353: 7349: 7344: 7340: 7336: 7331: 7327: 7322: 7299: 7298: 7287: 7284: 7279: 7275: 7271: 7268: 7265: 7262: 7259: 7256: 7253: 7250: 7247: 7242: 7238: 7228: 7224: 7218: 7214: 7209: 7203: 7200: 7195: 7190: 7186: 7181: 7173: 7169: 7165: 7159: 7155: 7152: 7148: 7142: 7138: 7133: 7127: 7124: 7119: 7114: 7110: 7105: 7097: 7093: 7089: 7083: 7079: 7076: 7072: 7066: 7062: 7057: 7051: 7048: 7043: 7038: 7034: 7029: 7021: 7017: 7013: 7007: 7003: 7000: 6993: 6989: 6985: 6980: 6976: 6972: 6967: 6963: 6958: 6952: 6949: 6944: 6940: 6934: 6931: 6926: 6922: 6917: 6911: 6908: 6903: 6899: 6893: 6890: 6885: 6881: 6876: 6870: 6867: 6862: 6858: 6852: 6849: 6844: 6840: 6835: 6831: 6824: 6820: 6816: 6811: 6807: 6803: 6798: 6794: 6789: 6766: 6765: 6761: 6746: 6742: 6736: 6732: 6727: 6721: 6718: 6713: 6708: 6704: 6699: 6691: 6687: 6683: 6677: 6673: 6670: 6666: 6660: 6656: 6651: 6645: 6642: 6637: 6632: 6628: 6623: 6615: 6611: 6607: 6601: 6597: 6594: 6587: 6583: 6579: 6574: 6570: 6565: 6559: 6556: 6551: 6547: 6541: 6538: 6533: 6529: 6524: 6518: 6515: 6510: 6506: 6500: 6497: 6492: 6488: 6483: 6479: 6476: 6474: 6472: 6468: 6462: 6458: 6453: 6447: 6444: 6439: 6434: 6430: 6425: 6417: 6413: 6409: 6403: 6399: 6396: 6392: 6387: 6381: 6377: 6372: 6366: 6363: 6358: 6353: 6349: 6344: 6336: 6332: 6328: 6322: 6318: 6315: 6308: 6304: 6300: 6295: 6291: 6286: 6280: 6277: 6272: 6268: 6262: 6259: 6254: 6250: 6245: 6240: 6234: 6231: 6226: 6222: 6216: 6213: 6208: 6204: 6199: 6195: 6192: 6190: 6184: 6180: 6176: 6171: 6167: 6162: 6158: 6157: 6141: 6138: 6132: 6129: 6103: 6099: 6095: 6062: 6059: 6046: 6043: 6040: 6001: 5996: 5991: 5987: 5984: 5964: 5961: 5941: 5938: 5918: 5915: 5912: 5909: 5906: 5903: 5883: 5863: 5860: 5848: 5843: 5839: 5829:similarly for 5818: 5815: 5811: 5807: 5804: 5801: 5798: 5795: 5775: 5771: 5767: 5764: 5761: 5758: 5736: 5732: 5698: 5693: 5689: 5685: 5668: 5667: 5656: 5650: 5647: 5644: 5638: 5632: 5629: 5626: 5623: 5613: 5607: 5600: 5597: 5591: 5588: 5584: 5579: 5572: 5569: 5563: 5560: 5556: 5548: 5543: 5533: 5529: 5526: 5521: 5517: 5511: 5508: 5505: 5500: 5497: 5494: 5490: 5486: 5481: 5477: 5464: 5461: 5449: 5445: 5441: 5438: 5435: 5432: 5429: 5409: 5405: 5401: 5398: 5395: 5392: 5370: 5366: 5346: 5343: 5340: 5337: 5317: 5314: 5311: 5289: 5285: 5251: 5247: 5243: 5219: 5213: 5207: 5203: 5180: 5156: 5152: 5137: 5136: 5125: 5119: 5116: 5113: 5107: 5101: 5098: 5095: 5092: 5079: 5074: 5070: 5063: 5060: 5054: 5051: 5047: 5040: 5035: 5025: 5021: 5018: 5013: 5009: 5003: 5000: 4997: 4992: 4989: 4986: 4982: 4978: 4973: 4969: 4962: 4959: 4953: 4948: 4944: 4931: 4928: 4916: 4913: 4910: 4907: 4887: 4884: 4881: 4859: 4855: 4835: 4831: 4827: 4823: 4820: 4817: 4814: 4811: 4791: 4787: 4783: 4780: 4777: 4774: 4752: 4748: 4701: 4696: 4691: 4664: 4658: 4654: 4632: 4628: 4596: 4593: 4590: 4587: 4584: 4581: 4578: 4556: 4552: 4548: 4543: 4540: 4537: 4534: 4530: 4509: 4506: 4503: 4498: 4495: 4491: 4471: 4468: 4465: 4462: 4459: 4456: 4434: 4430: 4426: 4421: 4418: 4415: 4412: 4408: 4388: 4385: 4382: 4377: 4374: 4370: 4349: 4344: 4340: 4319: 4316: 4292: 4289: 4271: 4270: 4259: 4253: 4250: 4247: 4241: 4235: 4232: 4229: 4226: 4213: 4208: 4204: 4197: 4194: 4188: 4185: 4181: 4174: 4169: 4159: 4155: 4152: 4147: 4143: 4137: 4134: 4131: 4126: 4123: 4120: 4116: 4112: 4107: 4103: 4090: 4087: 4075: 4070: 4066: 4045: 4042: 4039: 4036: 4033: 4013: 4010: 4007: 3985: 3981: 3958: 3955: 3935: 3912: 3906: 3900: 3894: 3888: 3882: 3876: 3870: 3850: 3844: 3838: 3832: 3826: 3806: 3803: 3800: 3780: 3777: 3774: 3771: 3768: 3765: 3734: 3723: 3720: 3717: 3713: 3684: 3677: 3671: 3667: 3645: 3642: 3639: 3635: 3612: 3608: 3587: 3580: 3556: 3553: 3550: 3546: 3523: 3519: 3507: 3506: 3495: 3489: 3486: 3483: 3477: 3471: 3468: 3465: 3462: 3449: 3444: 3440: 3431: 3428: 3425: 3419: 3407: 3403: 3400: 3395: 3391: 3385: 3382: 3379: 3374: 3371: 3368: 3364: 3360: 3357: 3352: 3349: 3346: 3342: 3336: 3332: 3328: 3325: 3322: 3319: 3314: 3310: 3306: 3301: 3298: 3293: 3288: 3284: 3271: 3268: 3253: 3250: 3247: 3243: 3238: 3235: 3231: 3226: 3222: 3192: 3189: 3186: 3182: 3175: 3169: 3164: 3160: 3116: 3093: 3088: 3083: 3078: 3073: 3068: 3065: 3046: 3043: 3021:Fourier series 2980:is repeated). 2886: 2883: 2880: 2877: 2857: 2837: 2817: 2814: 2811: 2808: 2793:Fourier series 2753: 2750: 2704:mosquito noise 2683: 2680: 2639: 2636: 2616: 2613: 2610: 2609: 2606: 2603: 2596: 2595: 2592: 2589: 2583: 2582: 2569: 2566: 2560: 2559: 2556: 2553: 2547: 2546: 2544:Wideband audio 2541: 2538: 2532: 2531: 2522: 2519: 2513: 2512: 2506: 2503: 2497: 2496: 2490:wideband audio 2483: 2480: 2474: 2473: 2456: 2453: 2446: 2445: 2442: 2439: 2429: 2426: 2423: 2422: 2416: 2415: 2405: 2402: 2396: 2395: 2385: 2382: 2376: 2375: 2365: 2362: 2355: 2354: 2348: 2345: 2338: 2337: 2319:radio stations 2317:distribution, 2312: 2309: 2303: 2302: 2297: 2294: 2287: 2286: 2281: 2278: 2272: 2271: 2257: 2254: 2247: 2246: 2228:mobile devices 2214:distribution, 2209: 2206: 2195: 2194: 2188: 2185: 2178: 2177: 2165:distribution, 2160: 2157: 2151: 2150: 2145: 2142: 2135: 2134: 2116:digital cinema 2109: 2106: 2099: 2098: 2095: 2092: 2085: 2082: 2074: 2071: 2068: 2067: 2044: 2041: 2035: 2034: 2028: 2025: 2019: 2018: 2011: 2008: 1997: 1996: 1981: 1978: 1972: 1971: 1968: 1965: 1959: 1958: 1949: 1946: 1940: 1939: 1936: 1933: 1927: 1926: 1912:mobile devices 1888:Internet video 1880: 1877: 1862: 1861: 1841: 1838: 1827: 1826: 1817: 1814: 1808: 1807: 1788: 1785: 1774: 1773: 1771:Internet video 1760: 1757: 1750: 1749: 1732: 1729: 1722: 1721: 1706: 1703: 1695: 1694: 1691: 1688: 1679: 1676: 1673: 1672: 1669: 1666: 1660: 1659: 1656: 1653: 1647: 1646: 1632: 1629: 1622: 1621: 1614: 1611: 1605: 1604: 1599: 1596: 1590: 1589: 1582: 1579: 1573: 1572: 1569: 1566: 1559: 1556: 1520: 1517: 1514: 1511: 1508: 1488: 1460: 1457: 1454: 1408: 1405: 1403: 1402: 1392: 1391: 1390: 1382:, directional 1362:RF engineering 1355: 1341:Mobile devices 1332: 1327:watermarking, 1310: 1309: 1308: 1302:video browsing 1283: 1268:motion vectors 1241: 1227: 1201: 1178:Digital cinema 1170: 1163: 1158: 1157: 1156: 1150: 1120:, DCT pyramid 1087: 1081: 1076: 1075: 1074: 1073:classification 1061: 1046: 1045: 1044: 1032:— directional 1027: 1021: 1015:spatiotemporal 987:edge detection 971:image analysis 960: 942: 927:digital images 920: 865: 864:(transient EM) 856: 851: 850: 849: 818: 817: 816: 782: 764: 750: 749: 748: 740:Internet video 732:World Wide Web 721: 720: 719: 713:Face detection 688: 687: 686: 661: 644: 631: 607:surround sound 591: 586: 583: 530:digital cinema 506:digital photos 502:digital images 465: 462: 458:entropy coding 391:videotelephone 294: 291: 224:Fourier series 99:digital images 57: 56: 31: 29: 22: 15: 9: 6: 4: 3: 2: 22633: 22622: 22619: 22617: 22614: 22612: 22609: 22607: 22604: 22602: 22599: 22597: 22594: 22592: 22589: 22587: 22584: 22582: 22579: 22577: 22574: 22573: 22571: 22556: 22548: 22546: 22538: 22536: 22528: 22526: 22516: 22515: 22512: 22505: 22501: 22499: 22496: 22494: 22491: 22489: 22486: 22484: 22481: 22477: 22474: 22472: 22469: 22468: 22466: 22464: 22461: 22460: 22458: 22454: 22448: 22445: 22443: 22440: 22438: 22435: 22433: 22430: 22428: 22425: 22423: 22420: 22418: 22415: 22413: 22410: 22408: 22405: 22403: 22400: 22399: 22397: 22395: 22391: 22385: 22382: 22380: 22377: 22375: 22372: 22370: 22367: 22365: 22362: 22360: 22357: 22355: 22352: 22350: 22347: 22345: 22342: 22340: 22337: 22335: 22332: 22330: 22327: 22325: 22322: 22321: 22319: 22317: 22313: 22307: 22304: 22302: 22299: 22297: 22294: 22292: 22289: 22287: 22284: 22283: 22281: 22277: 22271: 22270:Code-division 22268: 22266: 22263: 22261: 22258: 22256: 22255:Time-division 22253: 22251: 22248: 22246: 22243: 22242: 22240: 22238: 22234: 22228: 22225: 22221: 22218: 22216: 22213: 22212: 22211: 22208: 22204: 22201: 22200: 22199: 22196: 22194: 22191: 22189: 22186: 22185: 22183: 22181:and switching 22179: 22175: 22167: 22164: 22163: 22162: 22159: 22155: 22152: 22151: 22150: 22147: 22145: 22142: 22140: 22137: 22133: 22132:optical fiber 22130: 22129: 22128: 22125: 22123: 22122:Coaxial cable 22120: 22119: 22117: 22115: 22109: 22103: 22100: 22098: 22095: 22093: 22090: 22088: 22085: 22083: 22080: 22078: 22075: 22073: 22070: 22068: 22065: 22063: 22060: 22058: 22055: 22053: 22050: 22048: 22045: 22043: 22040: 22038: 22037:Radia Perlman 22035: 22033: 22030: 22028: 22025: 22023: 22020: 22018: 22015: 22013: 22010: 22008: 22005: 22003: 22000: 21998: 21995: 21993: 21990: 21988: 21985: 21983: 21980: 21978: 21975: 21973: 21970: 21968: 21965: 21963: 21960: 21958: 21955: 21953: 21950: 21948: 21945: 21943: 21940: 21938: 21935: 21933: 21930: 21928: 21927:Lee de Forest 21925: 21923: 21922:Thomas Edison 21920: 21918: 21915: 21913: 21912:Donald Davies 21910: 21908: 21905: 21903: 21900: 21898: 21897:Claude Chappe 21895: 21893: 21890: 21888: 21885: 21883: 21880: 21878: 21875: 21873: 21870: 21868: 21865: 21863: 21860: 21858: 21855: 21853: 21850: 21848: 21845: 21843: 21840: 21838: 21835: 21833: 21830: 21828: 21825: 21824: 21822: 21818: 21812: 21809: 21807: 21804: 21802: 21799: 21797: 21794: 21790: 21787: 21785: 21782: 21781: 21780: 21777: 21775: 21774: 21770: 21768: 21765: 21762: 21759: 21757: 21754: 21752: 21749: 21747: 21744: 21742: 21741:Smoke signals 21739: 21735: 21732: 21730: 21727: 21725: 21722: 21721: 21720: 21719:Semiconductor 21717: 21713: 21710: 21709: 21708: 21705: 21703: 21700: 21698: 21695: 21693: 21690: 21688: 21685: 21683: 21680: 21678: 21675: 21673: 21670: 21668: 21665: 21661: 21658: 21657: 21656: 21653: 21651: 21648: 21646: 21643: 21641: 21638: 21636: 21633: 21631: 21628: 21626: 21623: 21621: 21618: 21616: 21613: 21611: 21608: 21606: 21603: 21599: 21596: 21594: 21591: 21589: 21586: 21584: 21581: 21580: 21579: 21578:Digital media 21576: 21572: 21569: 21567: 21564: 21562: 21559: 21557: 21554: 21553: 21552: 21549: 21547: 21544: 21542: 21539: 21537: 21534: 21532: 21529: 21527: 21524: 21522: 21519: 21518: 21516: 21514: 21510: 21506: 21499: 21494: 21492: 21487: 21485: 21480: 21479: 21476: 21464: 21461: 21459: 21458:Undersampling 21456: 21454: 21453:Sampling rate 21451: 21449: 21446: 21444: 21441: 21439: 21435: 21432: 21430: 21427: 21425: 21422: 21420: 21417: 21416: 21414: 21412: 21408: 21402: 21401:Zak transform 21399: 21397: 21394: 21392: 21389: 21387: 21384: 21382: 21379: 21377: 21374: 21371: 21368: 21365: 21362: 21359: 21356: 21354: 21351: 21349: 21346: 21342: 21339: 21337: 21334: 21333: 21332: 21329: 21328: 21326: 21322: 21316: 21313: 21311: 21308: 21306: 21303: 21301: 21298: 21297: 21295: 21291: 21285: 21282: 21280: 21277: 21275: 21272: 21270: 21267: 21266: 21264: 21260: 21256: 21249: 21244: 21242: 21237: 21235: 21230: 21229: 21226: 21215: 21211: 21205: 21199: 21196: 21194: 21191: 21189: 21186: 21185: 21183: 21179: 21169: 21166: 21164: 21161: 21160: 21159: 21156: 21154: 21151: 21149: 21146: 21144: 21141: 21140: 21139: 21136: 21132: 21129: 21127: 21124: 21122: 21119: 21117: 21114: 21113: 21112: 21109: 21105: 21102: 21100: 21097: 21096: 21095: 21092: 21090: 21087: 21085: 21082: 21078: 21075: 21073: 21070: 21068: 21065: 21063: 21060: 21059: 21058: 21055: 21051: 21048: 21046: 21043: 21041: 21038: 21037: 21036: 21033: 21032: 21030: 21028: 21024: 21018: 21015: 21013: 21010: 21008: 21005: 21004: 21002: 20998: 20988: 20985: 20983: 20980: 20976: 20973: 20972: 20971: 20968: 20966: 20963: 20961: 20958: 20956: 20953: 20949: 20946: 20945: 20944: 20941: 20939: 20936: 20934: 20931: 20929: 20926: 20924: 20921: 20919: 20916: 20914: 20911: 20909: 20906: 20902: 20899: 20898: 20897: 20894: 20892: 20889: 20887: 20884: 20882: 20879: 20877: 20874: 20872: 20869: 20867: 20864: 20862: 20859: 20858: 20856: 20852: 20846: 20843: 20841: 20838: 20837: 20835: 20833: 20829: 20823: 20820: 20818: 20815: 20813: 20810: 20809: 20807: 20805: 20801: 20795: 20792: 20790: 20787: 20786: 20784: 20782: 20778: 20772: 20769: 20767: 20764: 20762: 20759: 20756: 20753: 20751: 20748: 20746: 20743: 20741: 20738: 20734: 20731: 20730: 20729: 20726: 20725: 20723: 20721: 20717: 20713: 20710: 20708: 20704: 20694: 20691: 20689: 20686: 20684: 20681: 20679: 20676: 20674: 20671: 20669: 20666: 20664: 20661: 20659: 20656: 20654: 20651: 20649: 20646: 20644: 20641: 20639: 20636: 20634: 20631: 20629: 20626: 20625: 20623: 20619: 20613: 20610: 20608: 20605: 20603: 20600: 20598: 20595: 20593: 20590: 20588: 20585: 20583: 20580: 20578: 20575: 20573: 20570: 20568: 20565: 20563: 20560: 20558: 20555: 20553: 20550: 20548: 20545: 20543: 20540: 20538: 20535: 20533: 20532:CCITT Group 4 20530: 20529: 20527: 20525: 20521: 20517: 20512: 20508: 20504: 20500: 20497: 20495: 20489: 20479: 20476: 20474: 20471: 20469: 20466: 20464: 20463:aptX Adaptive 20461: 20459: 20456: 20454: 20451: 20449: 20446: 20444: 20441: 20439: 20436: 20434: 20431: 20429: 20426: 20424: 20421: 20419: 20416: 20411: 20410: 20408: 20406: 20403: 20401: 20398: 20396: 20393: 20391: 20388: 20386: 20383: 20381: 20378: 20376: 20373: 20371: 20368: 20366: 20363: 20361: 20358: 20356: 20353: 20351: 20348: 20346: 20343: 20341: 20338: 20336: 20333: 20331: 20328: 20326: 20323: 20321: 20318: 20316: 20313: 20311: 20308: 20306: 20303: 20301: 20298: 20296: 20293: 20291: 20288: 20287: 20285: 20281: 20274: 20271: 20269: 20266: 20263: 20260: 20255: 20252: 20251: 20249: 20246: 20243: 20240: 20239: 20237: 20233: 20227: 20224: 20222: 20219: 20218: 20216: 20214: 20213:Bluetooth SIG 20210: 20204: 20201: 20199: 20196: 20194: 20191: 20190: 20188: 20186: 20182: 20176: 20173: 20171: 20168: 20166: 20163: 20161: 20158: 20156: 20153: 20151: 20148: 20146: 20143: 20141: 20138: 20136: 20133: 20132: 20130: 20128: 20124: 20118: 20115: 20113: 20110: 20108: 20105: 20103: 20100: 20098: 20095: 20094: 20092: 20090: 20086: 20080: 20077: 20075: 20072: 20070: 20067: 20065: 20062: 20060: 20057: 20055: 20052: 20050: 20047: 20045: 20042: 20040: 20037: 20035: 20032: 20030: 20027: 20023: 20020: 20018: 20015: 20014: 20013: 20010: 20009: 20007: 20005: 20001: 19995: 19992: 19990: 19987: 19985: 19982: 19980: 19977: 19975: 19972: 19970: 19967: 19965: 19962: 19960: 19959:MPEG Surround 19957: 19953: 19950: 19948: 19945: 19944: 19943: 19940: 19938: 19935: 19933: 19930: 19926: 19923: 19922: 19921: 19918: 19917: 19915: 19913: 19908: 19904: 19900: 19897: 19895: 19889: 19879: 19876: 19874: 19871: 19869: 19866: 19864: 19861: 19859: 19856: 19854: 19851: 19849: 19846: 19844: 19841: 19839: 19836: 19834: 19831: 19829: 19826: 19822: 19819: 19817: 19814: 19813: 19811: 19807: 19804: 19802: 19799: 19798: 19797: 19794: 19792: 19789: 19787: 19784: 19782: 19779: 19777: 19774: 19772: 19769: 19767: 19764: 19762: 19759: 19757: 19754: 19752: 19749: 19747: 19744: 19742: 19739: 19737: 19734: 19732: 19729: 19727: 19724: 19723: 19721: 19717: 19710: 19705: 19704: 19702: 19699: 19696: 19693: 19692: 19690: 19686: 19680: 19677: 19675: 19672: 19670: 19667: 19665: 19662: 19660: 19657: 19655: 19652: 19650: 19647: 19646: 19644: 19641: 19637: 19631: 19628: 19626: 19623: 19621: 19618: 19616: 19613: 19611: 19608: 19607: 19605: 19603: 19599: 19593: 19590: 19588: 19585: 19583: 19580: 19578: 19575: 19573: 19570: 19568: 19565: 19563: 19560: 19559: 19557: 19555: 19551: 19547: 19539: 19536: 19534: 19531: 19530: 19529: 19526: 19522: 19519: 19518: 19517: 19514: 19510: 19509:Part 2 / HEVC 19507: 19506: 19505: 19502: 19498: 19497:Part 33 / IVC 19495: 19493: 19492:Part 10 / AVC 19490: 19488: 19485: 19484: 19483: 19480: 19476: 19473: 19472: 19471: 19468: 19466: 19463: 19461: 19458: 19456: 19453: 19451: 19448: 19447: 19445: 19443: 19438: 19434: 19430: 19427: 19425: 19419: 19414: 19410: 19407: 19400: 19395: 19393: 19388: 19386: 19381: 19380: 19377: 19364: 19360: 19352: 19350: 19342: 19341: 19338: 19332: 19329: 19328: 19326: 19322: 19316: 19313: 19312: 19310: 19306: 19300: 19297: 19295: 19292: 19290: 19287: 19285: 19282: 19280: 19277: 19275: 19272: 19270: 19267: 19263: 19260: 19259: 19258: 19255: 19253: 19250: 19246: 19243: 19241: 19238: 19237: 19236: 19233: 19232: 19230: 19228: 19224: 19212: 19209: 19207: 19204: 19203: 19202: 19199: 19195: 19192: 19190: 19187: 19185: 19182: 19181: 19179: 19177: 19174: 19172: 19169: 19167: 19164: 19162: 19159: 19158: 19156: 19153: 19149: 19143: 19142:Video quality 19140: 19138: 19135: 19133: 19130: 19128: 19125: 19123: 19120: 19118: 19115: 19113: 19110: 19106: 19103: 19101: 19098: 19096: 19093: 19092: 19091: 19088: 19087: 19085: 19081: 19078: 19076: 19072: 19060: 19057: 19055: 19052: 19050: 19047: 19045: 19042: 19041: 19040: 19037: 19035: 19032: 19030: 19027: 19025: 19022: 19020: 19017: 19015: 19012: 19010: 19007: 19005: 19002: 19001: 18999: 18995: 18989: 18986: 18984: 18981: 18979: 18976: 18974: 18971: 18969: 18966: 18964: 18961: 18959: 18956: 18954: 18951: 18949: 18946: 18944: 18941: 18939: 18936: 18935: 18933: 18929: 18926: 18924: 18920: 18910: 18907: 18905: 18902: 18898: 18895: 18893: 18890: 18888: 18885: 18883: 18880: 18878: 18875: 18874: 18873: 18870: 18866: 18863: 18862: 18861: 18858: 18854: 18851: 18849: 18846: 18845: 18844: 18841: 18839: 18836: 18834: 18831: 18830: 18828: 18825: 18821: 18815: 18812: 18810: 18809:Speech coding 18807: 18805: 18804:Sound quality 18802: 18800: 18797: 18795: 18792: 18790: 18787: 18785: 18782: 18780: 18779:Dynamic range 18777: 18775: 18772: 18770: 18767: 18763: 18760: 18758: 18755: 18753: 18750: 18749: 18748: 18745: 18744: 18742: 18738: 18735: 18733: 18729: 18719: 18716: 18712: 18709: 18707: 18704: 18702: 18699: 18698: 18696: 18692: 18689: 18687: 18684: 18682: 18679: 18677: 18674: 18672: 18669: 18668: 18667: 18664: 18660: 18657: 18656: 18655: 18652: 18651: 18649: 18645: 18637: 18634: 18632: 18629: 18627: 18624: 18623: 18622: 18619: 18617: 18614: 18612: 18609: 18605: 18602: 18600: 18597: 18596: 18595: 18592: 18591: 18589: 18587: 18583: 18580: 18578: 18574: 18562: 18559: 18558: 18556: 18551: 18549: 18546: 18545: 18544:LZ77 + Range 18543: 18539: 18536: 18535: 18533: 18529: 18526: 18525: 18523: 18519: 18516: 18515: 18513: 18509: 18506: 18505: 18503: 18499: 18496: 18494: 18491: 18489: 18486: 18485: 18483: 18482: 18480: 18476: 18470: 18467: 18465: 18462: 18460: 18457: 18455: 18452: 18450: 18447: 18443: 18440: 18438: 18435: 18434: 18433: 18430: 18428: 18425: 18423: 18420: 18416: 18413: 18412: 18411: 18408: 18406: 18403: 18401: 18398: 18396: 18393: 18392: 18390: 18386: 18378: 18375: 18373: 18370: 18368: 18365: 18363: 18360: 18358: 18355: 18353: 18350: 18348: 18345: 18343: 18340: 18338: 18335: 18334: 18333: 18330: 18328: 18325: 18324: 18322: 18320: 18316: 18308: 18305: 18303: 18300: 18298: 18295: 18293: 18290: 18289: 18288: 18285: 18283: 18280: 18278: 18275: 18273: 18270: 18268: 18265: 18263: 18260: 18258: 18255: 18251: 18248: 18246: 18243: 18241: 18238: 18237: 18236: 18233: 18231: 18228: 18226: 18223: 18221: 18218: 18216: 18213: 18212: 18210: 18208: 18204: 18201: 18199: 18195: 18190: 18183: 18178: 18176: 18171: 18169: 18164: 18163: 18160: 18153: 18150: 18147: 18143: 18140: 18136: 18133: 18129: 18125: 18121: 18117: 18115: 18112: 18110: 18106: 18105: 18090:on 2011-08-11 18089: 18085: 18079: 18075: 18071: 18066: 18062: 18058: 18054: 18050: 18045: 18041: 18037: 18033: 18029: 18024: 18020: 18016: 18012: 18008: 18004: 18000: 17996: 17992: 17985: 17980: 17976: 17972: 17968: 17964: 17959: 17954: 17950: 17946: 17942: 17938: 17931: 17926: 17923: 17917: 17912: 17911: 17904: 17900: 17896: 17892: 17888: 17884: 17880: 17875: 17872: 17866: 17862: 17857: 17853: 17849: 17845: 17841: 17837: 17833: 17828: 17824: 17820: 17816: 17812: 17808: 17804: 17800: 17796: 17792: 17788: 17784: 17780: 17776: 17772: 17768: 17764: 17759: 17754: 17749: 17745: 17741: 17737: 17733: 17729: 17725: 17721: 17716: 17711: 17707: 17703: 17698: 17694: 17690: 17686: 17682: 17677: 17673: 17669: 17665: 17661: 17656: 17655: 17642: 17638: 17634: 17630: 17626: 17622: 17617: 17612: 17608: 17604: 17597: 17589: 17582: 17574: 17570: 17566: 17562: 17555: 17547: 17543: 17539: 17535: 17531: 17527: 17520: 17512: 17505: 17503: 17494: 17490: 17486: 17482: 17475: 17469: 17468:Martucci 1994 17464: 17458: 17453: 17445: 17441: 17437: 17433: 17429: 17425: 17420: 17415: 17411: 17407: 17400: 17392: 17388: 17384: 17377: 17369: 17367:9780442012724 17363: 17359: 17352: 17338: 17334: 17328: 17320: 17313: 17305: 17303:9781597110938 17299: 17295: 17288: 17273: 17267: 17260: 17259: 17251: 17236: 17235: 17230: 17224: 17216: 17214:9780123744579 17210: 17206: 17202: 17195: 17193: 17183: 17178: 17174: 17170: 17166: 17162: 17158: 17154: 17147: 17140: 17135: 17128:(5): 497–507. 17127: 17123: 17116: 17108: 17104: 17100: 17098:0-7803-5482-6 17094: 17090: 17086: 17082: 17078: 17074: 17067: 17060: 17056: 17052: 17048: 17041: 17034: 17030: 17026: 17022: 17018: 17014: 17010: 17006: 16999: 16991: 16987: 16980: 16973: 16969: 16965: 16961: 16957: 16953: 16948: 16943: 16939: 16935: 16931: 16927: 16920: 16913: 16906: 16902: 16898: 16894: 16890: 16886: 16879: 16863: 16856: 16850: 16842: 16838: 16832: 16818:on 2011-09-30 16817: 16813: 16807: 16796: 16783:on 2011-08-07 16782: 16778: 16777: 16769: 16761: 16757: 16751: 16743: 16741:9780470377864 16737: 16733: 16729: 16728: 16720: 16718: 16702: 16698: 16697: 16689: 16688: 16680: 16665: 16661: 16660: 16652: 16651: 16643: 16635: 16631: 16627: 16623: 16619: 16615: 16608: 16601: 16582: 16578: 16577: 16569: 16563: 16548: 16542: 16534: 16530: 16526: 16524:9781119488057 16520: 16516: 16512: 16508: 16504: 16497: 16489: 16487:9789811068980 16483: 16479: 16475: 16471: 16467: 16460: 16445: 16444: 16439: 16432: 16416: 16412: 16406: 16397: 16392: 16388: 16384: 16377: 16369: 16365: 16361: 16357: 16353: 16349: 16342: 16327: 16320: 16305: 16298: 16283: 16276: 16269: 16258: 16257: 16252: 16245: 16231: 16224: 16217: 16210: 16199: 16197:9780780341470 16193: 16189: 16185: 16184: 16177: 16162: 16161: 16156: 16149: 16141: 16137: 16133: 16129: 16125: 16121: 16117: 16113: 16109: 16103: 16095: 16091: 16085: 16083: 16074: 16072:9781461560319 16068: 16064: 16060: 16059: 16051: 16040: 16033: 16026: 16015: 16008: 16001: 15999: 15991: 15976: 15972: 15965: 15958: 15943: 15939: 15933: 15918:. 31 May 2018 15917: 15913: 15912: 15907: 15901: 15892: 15887: 15883: 15879: 15878: 15873: 15866: 15858: 15854: 15850: 15846: 15842: 15838: 15827: 15825: 15816: 15812: 15808: 15804: 15800: 15796: 15792: 15788: 15780: 15764: 15763: 15758: 15752: 15750: 15741: 15739:9780849318580 15735: 15731: 15727: 15723: 15722: 15717: 15710: 15702: 15696: 15692: 15688: 15687: 15679: 15671: 15665: 15661: 15660: 15652: 15637: 15633: 15632: 15627: 15620: 15618: 15610: 15605: 15599: 15595: 15591: 15590: 15582: 15580: 15578: 15576: 15567: 15565:9781351396486 15561: 15557: 15556: 15548: 15532: 15531: 15523: 15517: 15515: 15499: 15495: 15494: 15489: 15482: 15480: 15478: 15476: 15460: 15456: 15449: 15447: 15445: 15443: 15441: 15432: 15430:9780470023631 15426: 15422: 15418: 15417: 15409: 15407: 15405: 15396: 15390: 15386: 15382: 15381: 15373: 15371: 15362: 15360:9783319610801 15356: 15352: 15351: 15343: 15341: 15339: 15337: 15335: 15326: 15324:9781351396486 15320: 15316: 15312: 15311: 15306: 15299: 15297: 15295: 15293: 15291: 15289: 15287: 15285: 15283: 15281: 15279: 15277: 15275: 15273: 15271: 15269: 15267: 15265: 15263: 15261: 15259: 15257: 15255: 15253: 15251: 15249: 15247: 15245: 15243: 15241: 15239: 15230: 15228:9781351396486 15224: 15220: 15216: 15215: 15210: 15203: 15201: 15192: 15190:9781483298511 15186: 15182: 15178: 15174: 15170: 15166: 15159: 15157: 15155: 15146: 15144:9780470857649 15140: 15136: 15132: 15131: 15123: 15121: 15119: 15110: 15106: 15102: 15098: 15097: 15089: 15082: 15080: 15064: 15060: 15056: 15049: 15047: 15045: 15028: 15027: 15022: 15016: 15014: 15012: 15003: 15001:9780080477480 14997: 14993: 14989: 14988: 14980: 14972: 14968: 14964: 14958: 14954: 14950: 14946: 14939: 14931: 14927: 14923: 14921:0-7803-4428-6 14917: 14913: 14909: 14905: 14901: 14894: 14886: 14882: 14878: 14874: 14870: 14866: 14862: 14858: 14854: 14848: 14833: 14826: 14824: 14812: 14805: 14798: 14796: 14780: 14779: 14771: 14764: 14762: 14753: 14749: 14745: 14741: 14737: 14733: 14726: 14724: 14715: 14713:9780387782638 14709: 14705: 14701: 14700: 14692: 14690: 14688: 14686: 14684: 14682: 14680: 14678: 14676: 14674: 14672: 14663: 14659: 14655: 14651: 14644: 14636: 14632: 14628: 14624: 14620: 14613: 14605: 14603:9789812709998 14599: 14595: 14591: 14590: 14582: 14574: 14572:9780852967102 14568: 14564: 14560: 14559: 14551: 14549: 14547: 14530: 14529: 14524: 14518: 14516: 14514: 14505: 14503:9780786487974 14499: 14495: 14494: 14486: 14478: 14474: 14470: 14466: 14462: 14458: 14454: 14450: 14443: 14435: 14433:9783642870378 14429: 14425: 14421: 14420: 14412: 14396: 14392: 14388: 14381: 14373: 14369: 14365: 14361: 14357: 14353: 14346: 14338: 14334: 14330: 14326: 14325: 14317: 14301: 14294: 14288: 14286: 14284: 14282: 14280: 14271: 14267: 14263: 14259: 14255: 14251: 14250: 14242: 14238: 14232: 14230: 14214: 14213: 14208: 14201: 14199: 14197: 14195: 14193: 14176: 14172: 14168: 14161: 14159: 14157: 14155: 14153: 14151: 14149: 14140: 14136: 14132: 14128: 14124: 14120: 14116: 14112: 14108: 14102: 14098: 14094: 14089: 14084: 14080: 14076: 14072: 14066: 14064: 14062: 14060: 14058: 14056: 14047: 14043: 14039: 14035: 14031: 14027: 14023: 14019: 14015: 14011: 14007: 14003: 13999: 13995: 13994: 13989: 13985: 13979: 13977: 13975: 13973: 13971: 13969: 13953: 13946: 13939: 13937: 13935: 13933: 13931: 13929: 13912: 13908: 13904: 13897: 13895: 13893: 13891: 13882: 13878: 13874: 13870: 13866: 13862: 13858: 13854: 13850: 13844: 13840: 13836: 13832: 13825: 13823: 13821: 13819: 13817: 13808: 13793: 13789: 13785: 13781: 13775: 13771: 13767: 13760: 13753: 13751: 13749: 13747: 13745: 13743: 13741: 13739: 13737: 13735: 13733: 13731: 13729: 13727: 13725: 13723: 13721: 13719: 13717: 13715: 13713: 13711: 13709: 13707: 13705: 13703: 13701: 13696: 13682: 13664: 13639: 13636: 13633: 13630: 13627: 13624: 13619: 13615: 13611: 13608: 13595: 13588: 13567: 13531: 13505: 13479: 13476: 13463: 13457:optimization. 13453: 13449: 13439: 13436: 13434: 13431: 13421: 13402: 13400: 13397: 13396: 13385: 13381: 13361: 13355: 13350: 13345: 13342: 13337: 13334: 13329: 13324: 13319: 13316: 13311: 13308: 13301: 13298: 13293: 13290: 13285: 13280: 13277: 13272: 13269: 13264: 13261: 13256: 13251: 13244: 13241: 13236: 13233: 13228: 13223: 13218: 13215: 13210: 13207: 13202: 13197: 13190: 13187: 13182: 13179: 13174: 13171: 13166: 13163: 13158: 13153: 13148: 13143: 13136: 13131: 13128: 13123: 13118: 13113: 13110: 13105: 13102: 13097: 13094: 13089: 13086: 13079: 13074: 13069: 13066: 13061: 13056: 13053: 13048: 13045: 13040: 13035: 13028: 13025: 13020: 13015: 13012: 13007: 13004: 12999: 12994: 12989: 12984: 12977: 12972: 12969: 12964: 12959: 12954: 12949: 12944: 12941: 12936: 12930: 12917: 12909: 12905: 12898: 12889: 12875: 12869: 12842: 12829: 12826: 12823: 12818: 12814: 12810: 12804: 12801: 12786: 12769: 12766: 12763: 12760: 12757: 12754: 12749: 12745: 12741: 12738: 12727: 12723: 12704: 12693: 12689: 12685: 12681: 12662: 12659: 12646: 12644: 12630: 12628: 12624: 12620: 12616: 12612: 12607: 12585: 12563: 12543: 12540: 12537: 12534: 12495: 12474: 12452: 12449: 12446: 12443: 12402: 12398: 12370: 12352: 12346: 12343: 12340: 12334: 12328: 12325: 12319: 12316: 12310: 12307: 12304: 12301: 12269: 12266: 12263: 12257: 12251: 12245: 12242: 12236: 12233: 12230: 12227: 12213: 12212: 12197: 12190: 12180: 12177: 12171: 12162: 12159: 12156: 12153: 12144: 12141: 12138: 12135: 12121: 12117: 12114: 12110: 12100: 12097: 12091: 12082: 12079: 12076: 12073: 12064: 12061: 12058: 12055: 12041: 12037: 12034: 12029: 12026: 12023: 12019: 12010: 12007: 12004: 11999: 11996: 11993: 11989: 11982: 11979: 11976: 11971: 11968: 11965: 11961: 11957: 11952: 11949: 11946: 11942: 11934: 11933: 11932: 11920: 11918: 11891: 11888: 11883: 11879: 11875: 11870: 11866: 11845: 11840: 11836: 11834: 11830: 11824: 11815: 11812: 11809: 11806: 11804:64 × 64 × 64 11803: 11802: 11798: 11795: 11792: 11789: 11787:32 × 32 × 32 11786: 11785: 11781: 11778: 11775: 11772: 11770:16 × 16 × 16 11769: 11768: 11764: 11761: 11758: 11755: 11752: 11751: 11747: 11744: 11741: 11738: 11735: 11734: 11728: 11714: 11707: 11701: 11697: 11693: 11690: 11685: 11681: 11677: 11674: 11671: 11668: 11663: 11659: 11653: 11649: 11643: 11640: 11634: 11606: 11602: 11599: 11594: 11590: 11584: 11580: 11574: 11571: 11565: 11551: 11538: 11531: 11525: 11521: 11517: 11514: 11509: 11505: 11501: 11498: 11495: 11492: 11487: 11483: 11477: 11473: 11467: 11464: 11458: 11454: 11443: 11439: 11433: 11429: 11425: 11422: 11417: 11413: 11409: 11406: 11403: 11400: 11395: 11391: 11385: 11381: 11375: 11372: 11366: 11359: 11348: 11344: 11340: 11337: 11332: 11328: 11322: 11318: 11312: 11309: 11303: 11276: 11269: 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9625: 9621: 9598: 9594: 9590: 9585: 9581: 9571: 9558: 9555: 9552: 9549: 9546: 9543: 9540: 9532: 9526: 9523: 9518: 9514: 9510: 9501: 9498: 9494: 9489: 9484: 9480: 9451: 9447: 9443: 9437: 9434: 9426: 9422: 9418: 9412: 9409: 9401: 9397: 9393: 9387: 9384: 9376: 9372: 9368: 9363: 9359: 9355: 9350: 9346: 9333: 9325: 9322: 9319: 9314: 9311: 9306: 9302: 9297: 9291: 9288: 9285: 9280: 9277: 9272: 9268: 9263: 9257: 9254: 9251: 9246: 9243: 9238: 9234: 9229: 9225: 9217: 9213: 9209: 9204: 9200: 9196: 9191: 9187: 9180: 9173: 9172: 9171: 9168: 9149: 9145: 9138: 9135: 9132: 9124: 9120: 9113: 9089: 9086: 9081: 9078: 9073: 9068: 9064: 9060: 9055: 9051: 9047: 9042: 9038: 9034: 9031: 9023: 9002: 8999: 8994: 8990: 8986: 8983: 8980: 8977: 8972: 8968: 8964: 8961: 8958: 8955: 8950: 8946: 8942: 8936: 8933: 8927: 8924: 8919: 8915: 8911: 8908: 8905: 8902: 8899: 8894: 8890: 8886: 8883: 8880: 8877: 8874: 8869: 8865: 8861: 8858: 8846: 8831: 8827: 8823: 8820: 8817: 8814: 8809: 8805: 8801: 8798: 8795: 8792: 8787: 8783: 8779: 8773: 8770: 8762: 8758: 8754: 8751: 8748: 8743: 8739: 8735: 8732: 8729: 8726: 8723: 8718: 8714: 8710: 8707: 8695: 8682: 8679: 8674: 8670: 8666: 8663: 8658: 8654: 8650: 8647: 8644: 8641: 8636: 8632: 8628: 8622: 8619: 8613: 8610: 8605: 8601: 8597: 8594: 8591: 8586: 8582: 8578: 8575: 8572: 8567: 8563: 8559: 8556: 8544: 8529: 8525: 8521: 8518: 8513: 8509: 8505: 8502: 8499: 8496: 8491: 8487: 8483: 8477: 8474: 8466: 8462: 8458: 8453: 8449: 8445: 8442: 8439: 8434: 8430: 8426: 8423: 8411: 8398: 8395: 8390: 8386: 8382: 8379: 8376: 8373: 8368: 8364: 8360: 8357: 8352: 8348: 8344: 8338: 8335: 8329: 8326: 8321: 8317: 8313: 8310: 8307: 8304: 8301: 8296: 8292: 8288: 8285: 8282: 8277: 8273: 8260: 8245: 8241: 8237: 8234: 8231: 8228: 8223: 8219: 8215: 8212: 8207: 8203: 8199: 8193: 8190: 8182: 8178: 8174: 8171: 8168: 8163: 8159: 8155: 8152: 8149: 8144: 8140: 8127: 8114: 8111: 8106: 8102: 8098: 8095: 8090: 8086: 8082: 8079: 8074: 8070: 8066: 8060: 8057: 8051: 8048: 8043: 8039: 8035: 8032: 8029: 8024: 8020: 8016: 8011: 8007: 7994: 7979: 7975: 7971: 7968: 7963: 7959: 7955: 7952: 7947: 7943: 7939: 7933: 7930: 7922: 7918: 7914: 7909: 7905: 7901: 7896: 7892: 7879: 7865: 7864: 7859: 7855: 7853: 7843: 7841: 7837: 7818: 7815: 7810: 7806: 7802: 7799: 7796: 7793: 7790: 7787: 7784: 7781: 7778: 7773: 7769: 7759: 7755: 7749: 7745: 7740: 7734: 7731: 7726: 7721: 7717: 7712: 7704: 7700: 7696: 7690: 7686: 7683: 7679: 7673: 7669: 7664: 7658: 7655: 7650: 7645: 7641: 7636: 7628: 7624: 7620: 7614: 7610: 7607: 7603: 7597: 7593: 7588: 7582: 7579: 7574: 7569: 7565: 7560: 7552: 7548: 7544: 7538: 7534: 7531: 7524: 7520: 7516: 7511: 7507: 7503: 7498: 7494: 7489: 7483: 7480: 7475: 7471: 7465: 7462: 7457: 7453: 7448: 7442: 7439: 7434: 7430: 7424: 7421: 7416: 7412: 7407: 7401: 7398: 7393: 7389: 7383: 7380: 7375: 7371: 7366: 7362: 7355: 7351: 7347: 7342: 7338: 7334: 7329: 7325: 7320: 7312: 7311: 7310: 7308: 7304: 7285: 7282: 7277: 7273: 7269: 7266: 7263: 7260: 7257: 7254: 7251: 7248: 7245: 7240: 7236: 7226: 7222: 7216: 7212: 7207: 7201: 7198: 7193: 7188: 7184: 7179: 7171: 7167: 7163: 7157: 7153: 7150: 7146: 7140: 7136: 7131: 7125: 7122: 7117: 7112: 7108: 7103: 7095: 7091: 7087: 7081: 7077: 7074: 7070: 7064: 7060: 7055: 7049: 7046: 7041: 7036: 7032: 7027: 7019: 7015: 7011: 7005: 7001: 6998: 6991: 6987: 6983: 6978: 6974: 6970: 6965: 6961: 6956: 6950: 6947: 6942: 6938: 6932: 6929: 6924: 6920: 6915: 6909: 6906: 6901: 6897: 6891: 6888: 6883: 6879: 6874: 6868: 6865: 6860: 6856: 6850: 6847: 6842: 6838: 6833: 6829: 6822: 6818: 6814: 6809: 6805: 6801: 6796: 6792: 6787: 6779: 6778: 6777: 6775: 6771: 6762: 6744: 6740: 6734: 6730: 6725: 6719: 6716: 6711: 6706: 6702: 6697: 6689: 6685: 6681: 6675: 6671: 6668: 6664: 6658: 6654: 6649: 6643: 6640: 6635: 6630: 6626: 6621: 6613: 6609: 6605: 6599: 6595: 6592: 6585: 6581: 6577: 6572: 6568: 6563: 6557: 6554: 6549: 6545: 6539: 6536: 6531: 6527: 6522: 6516: 6513: 6508: 6504: 6498: 6495: 6490: 6486: 6481: 6477: 6475: 6466: 6460: 6456: 6451: 6445: 6442: 6437: 6432: 6428: 6423: 6415: 6411: 6407: 6401: 6397: 6394: 6390: 6385: 6379: 6375: 6370: 6364: 6361: 6356: 6351: 6347: 6342: 6334: 6330: 6326: 6320: 6316: 6313: 6306: 6302: 6298: 6293: 6289: 6284: 6278: 6275: 6270: 6266: 6260: 6257: 6252: 6248: 6243: 6238: 6232: 6229: 6224: 6220: 6214: 6211: 6206: 6202: 6197: 6193: 6191: 6182: 6178: 6174: 6169: 6165: 6160: 6148: 6147: 6146: 6137: 6128: 6126: 6101: 6097: 6093: 6083: 6080:Like for the 6078: 6076: 6072: 6068: 6058: 6044: 6041: 6038: 6025: 6023: 6019: 6014: 5999: 5994: 5989: 5985: 5982: 5962: 5959: 5939: 5936: 5913: 5910: 5907: 5901: 5881: 5873: 5868: 5859: 5846: 5841: 5837: 5816: 5813: 5809: 5805: 5802: 5799: 5796: 5793: 5773: 5769: 5765: 5762: 5759: 5756: 5734: 5730: 5720: 5718: 5714: 5709: 5696: 5691: 5687: 5683: 5673: 5654: 5648: 5645: 5642: 5636: 5630: 5627: 5624: 5621: 5611: 5605: 5598: 5595: 5589: 5586: 5582: 5577: 5570: 5567: 5561: 5558: 5554: 5546: 5541: 5531: 5527: 5524: 5519: 5515: 5509: 5506: 5503: 5498: 5495: 5492: 5488: 5484: 5479: 5475: 5467: 5466: 5460: 5447: 5443: 5439: 5436: 5433: 5430: 5427: 5407: 5403: 5399: 5396: 5393: 5390: 5368: 5364: 5344: 5341: 5338: 5335: 5315: 5312: 5309: 5287: 5283: 5273: 5271: 5267: 5249: 5245: 5241: 5217: 5211: 5205: 5201: 5178: 5154: 5150: 5140: 5123: 5117: 5114: 5111: 5105: 5099: 5096: 5093: 5090: 5077: 5072: 5068: 5061: 5058: 5052: 5049: 5045: 5038: 5033: 5023: 5019: 5016: 5011: 5007: 5001: 4998: 4995: 4990: 4987: 4984: 4980: 4976: 4971: 4967: 4960: 4957: 4951: 4946: 4942: 4934: 4933: 4927: 4914: 4911: 4908: 4905: 4885: 4882: 4879: 4857: 4853: 4833: 4829: 4825: 4821: 4818: 4815: 4812: 4809: 4789: 4785: 4781: 4778: 4775: 4772: 4750: 4746: 4736: 4734: 4730: 4726: 4722: 4718: 4699: 4694: 4689: 4662: 4656: 4652: 4630: 4626: 4616: 4614: 4594: 4591: 4588: 4585: 4582: 4579: 4576: 4554: 4550: 4546: 4541: 4538: 4535: 4532: 4528: 4507: 4504: 4501: 4496: 4493: 4489: 4469: 4466: 4463: 4460: 4457: 4454: 4432: 4428: 4424: 4419: 4416: 4413: 4410: 4406: 4386: 4383: 4380: 4375: 4372: 4368: 4347: 4342: 4338: 4317: 4314: 4306: 4290: 4287: 4279: 4274: 4257: 4251: 4248: 4245: 4239: 4233: 4230: 4227: 4224: 4211: 4206: 4202: 4195: 4192: 4186: 4183: 4179: 4172: 4167: 4157: 4153: 4150: 4145: 4141: 4135: 4132: 4129: 4124: 4121: 4118: 4114: 4110: 4105: 4101: 4093: 4092: 4086: 4073: 4068: 4064: 4043: 4040: 4037: 4034: 4031: 4011: 4008: 4005: 3983: 3979: 3969: 3956: 3953: 3933: 3924: 3910: 3904: 3898: 3892: 3886: 3880: 3874: 3868: 3848: 3842: 3836: 3830: 3824: 3817:real numbers 3804: 3801: 3798: 3775: 3772: 3769: 3763: 3755: 3750: 3748: 3732: 3721: 3718: 3715: 3711: 3698: 3682: 3675: 3669: 3665: 3643: 3640: 3637: 3633: 3610: 3606: 3585: 3578: 3554: 3551: 3548: 3544: 3521: 3517: 3493: 3487: 3484: 3481: 3475: 3469: 3466: 3463: 3460: 3447: 3442: 3438: 3429: 3426: 3423: 3417: 3405: 3401: 3398: 3393: 3389: 3383: 3380: 3377: 3372: 3369: 3366: 3362: 3358: 3350: 3347: 3344: 3340: 3334: 3326: 3323: 3317: 3312: 3308: 3299: 3296: 3291: 3286: 3282: 3274: 3273: 3267: 3251: 3248: 3245: 3241: 3236: 3233: 3229: 3224: 3220: 3212:real numbers 3190: 3187: 3184: 3180: 3173: 3167: 3162: 3158: 3147:real numbers 3142: 3141:square matrix 3131: 3091: 3076: 3066: 3063: 3056: 3053:, invertible 3052: 3042: 3039: 3035: 3031: 3026: 3025:cosine series 3022: 3017: 3013: 3008: 3006: 3002: 2998: 2992: 2990: 2986: 2981: 2979: 2975: 2971: 2967: 2963: 2959: 2955: 2951: 2947: 2943: 2939: 2935: 2931: 2927: 2919: 2914: 2910: 2908: 2905:, implies an 2904: 2900: 2881: 2875: 2855: 2835: 2812: 2806: 2798: 2794: 2790: 2785: 2783: 2779: 2775: 2771: 2767: 2763: 2759: 2749: 2747: 2743: 2739: 2735: 2734:digital audio 2732: 2728: 2724: 2723:digital media 2720: 2717:. The artist 2716: 2711: 2709: 2708:digital video 2705: 2701: 2697: 2693: 2689: 2688:digital media 2679: 2677: 2673: 2669: 2665: 2661: 2657: 2653: 2649: 2646:specifically 2645: 2635: 2633: 2629: 2622: 2607: 2604: 2601: 2598: 2597: 2593: 2590: 2588: 2585: 2584: 2581: 2577: 2573: 2570: 2567: 2565: 2562: 2561: 2557: 2554: 2552: 2549: 2548: 2545: 2542: 2539: 2537: 2534: 2533: 2530: 2526: 2523: 2520: 2518: 2515: 2514: 2511: 2507: 2504: 2502: 2499: 2498: 2495: 2491: 2487: 2484: 2481: 2479: 2476: 2475: 2472: 2468: 2464: 2463:voice-over-IP 2460: 2457: 2454: 2451: 2448: 2447: 2443: 2440: 2437: 2436:Speech coding 2434: 2433: 2428:Speech coding 2421: 2418: 2417: 2413: 2409: 2401: 2398: 2397: 2394: 2393:PlayStation 4 2390: 2386: 2383: 2381: 2378: 2377: 2374: 2370: 2366: 2363: 2360: 2357: 2356: 2353: 2349: 2346: 2343: 2340: 2339: 2336: 2332: 2328: 2324: 2320: 2316: 2315:Digital audio 2313: 2310: 2308: 2305: 2304: 2301: 2300:Windows Media 2298: 2295: 2292: 2289: 2288: 2285: 2282: 2279: 2277: 2274: 2273: 2269: 2265: 2261: 2260:Digital radio 2258: 2255: 2252: 2249: 2248: 2245: 2241: 2237: 2233: 2229: 2225: 2224:game consoles 2221: 2217: 2213: 2212:Digital audio 2210: 2207: 2204: 2200: 2197: 2196: 2192: 2189: 2186: 2183: 2180: 2179: 2176: 2172: 2168: 2164: 2163:Digital audio 2161: 2158: 2156: 2153: 2152: 2149: 2146: 2143: 2140: 2137: 2136: 2133: 2129: 2125: 2121: 2117: 2113: 2110: 2107: 2104: 2103:Dolby Digital 2101: 2100: 2096: 2093: 2090: 2089: 2084:General audio 2080: 2065: 2061: 2057: 2053: 2049: 2045: 2042: 2040: 2037: 2036: 2033: 2029: 2026: 2024: 2021: 2020: 2012: 2009: 2002: 1999: 1998: 1994: 1990: 1986: 1982: 1979: 1977: 1974: 1973: 1969: 1966: 1964: 1961: 1960: 1957: 1956:Blu-ray Discs 1953: 1950: 1947: 1945: 1942: 1941: 1937: 1934: 1932: 1929: 1928: 1925: 1921: 1917: 1913: 1909: 1905: 1901: 1897: 1896:Blu-ray Discs 1893: 1889: 1885: 1881: 1878: 1875: 1867: 1864: 1863: 1859: 1854: 1849: 1845: 1842: 1839: 1836: 1835:MPEG-4 Part 2 1832: 1829: 1828: 1825: 1821: 1818: 1815: 1813: 1810: 1809: 1805: 1801: 1797: 1793: 1789: 1786: 1779: 1776: 1775: 1772: 1768: 1764: 1763:Digital video 1761: 1758: 1755: 1752: 1751: 1748: 1744: 1740: 1739:video editing 1736: 1733: 1730: 1727: 1724: 1723: 1719: 1715: 1711: 1707: 1704: 1701: 1697: 1696: 1692: 1689: 1687: 1684: 1683: 1678:Video formats 1670: 1667: 1665: 1662: 1661: 1657: 1654: 1652: 1649: 1648: 1644: 1640: 1636: 1633: 1630: 1627: 1624: 1623: 1619: 1615: 1612: 1610: 1607: 1606: 1603: 1600: 1597: 1595: 1592: 1591: 1587: 1583: 1580: 1578: 1575: 1574: 1570: 1567: 1564: 1563: 1558:Image formats 1555: 1553: 1549: 1545: 1541: 1537: 1532: 1515: 1512: 1509: 1486: 1478: 1477:entropy coded 1474: 1458: 1455: 1452: 1444: 1440: 1436: 1432: 1428: 1423: 1418: 1414: 1400: 1396: 1393: 1389: 1388:space imaging 1385: 1381: 1378: 1374: 1370: 1367: 1363: 1359: 1356: 1354: 1350: 1346: 1345:mobile phones 1342: 1339: 1338: 1336: 1333: 1330: 1326: 1322: 1318: 1314: 1311: 1307: 1303: 1299: 1295: 1291: 1287: 1284: 1281: 1277: 1273: 1269: 1265: 1261: 1257: 1253: 1249: 1245: 1242: 1239: 1235: 1231: 1230:Digital video 1228: 1225: 1221: 1218:(HDTV), HDTV 1217: 1213: 1209: 1205: 1202: 1199: 1198:Dolby Digital 1195: 1191: 1190:video editing 1187: 1183: 1179: 1176: 1175: 1174: 1171: 1168: 1164: 1162: 1159: 1155: 1151: 1148: 1145: 1144: 1143: 1142:Wiener filter 1139: 1135: 1131: 1127: 1123: 1119: 1115: 1111: 1107: 1103: 1099: 1095: 1091: 1088: 1085: 1082: 1080: 1077: 1072: 1069: 1065: 1062: 1059: 1055: 1052: 1051: 1050: 1047: 1043: 1039: 1035: 1031: 1028: 1025: 1024:Image quality 1022: 1020: 1016: 1012: 1008: 1004: 1003:interpolation 1000: 996: 992: 988: 984: 980: 976: 972: 968: 964: 961: 958: 955:compression, 954: 950: 946: 943: 941:(HDR imaging) 940: 936: 932: 928: 924: 921: 918: 914: 910: 906: 902: 898: 894: 890: 887: 886: 885: 881: 877: 873: 869: 866: 863: 860: 857: 855: 852: 847: 843: 839: 835: 831: 828: 827: 826: 822: 821:Digital media 819: 815: 811: 807: 804:operations — 803: 800: 799: 798: 794: 790: 786: 783: 780: 776: 775:steganography 772: 768: 765: 762: 758: 754: 751: 746: 743: 742: 741: 737: 733: 729: 725: 722: 718: 714: 711: 710: 708: 704: 700: 697:orientation, 696: 692: 689: 685: 681: 677: 673: 669: 668:voice over IP 665: 662: 659: 655: 652: 651:speech coding 648: 645: 643: 639: 635: 634:Digital radio 632: 630: 629:Digital audio 627: 626: 624: 621:recognition, 620: 616: 612: 611:acoustic echo 608: 604: 600: 596: 593: 592: 590: 582: 580: 576: 572: 567: 565: 561: 556: 554: 550: 546: 541: 539: 535: 531: 527: 523: 519: 515: 511: 510:digital video 507: 503: 499: 495: 491: 487: 483: 479: 478:digital media 475: 471: 461: 459: 455: 451: 447: 442: 440: 436: 432: 428: 424: 420: 419:Dolby Digital 416: 412: 408: 403: 400: 396: 392: 388: 384: 380: 376: 373: 370: 365: 363: 359: 355: 351: 347: 342: 340: 336: 330: 328: 324: 320: 316: 312: 308: 304: 300: 290: 288: 284: 280: 276: 271: 269: 265: 261: 257: 253: 252:odd functions 249: 245: 241: 237: 232: 230: 225: 221: 217: 213: 208: 206: 202: 198: 194: 190: 186: 182: 178: 174: 170: 169:speech coding 166: 162: 158: 157:digital radio 154: 150: 146: 142: 138: 134: 130: 129:Dolby Digital 126: 125:digital audio 121: 116: 112: 111:digital video 108: 104: 100: 96: 95:digital media 92: 88: 84: 80: 76: 72: 68: 64: 53: 39: 35: 32:This article 30: 21: 20: 22237:Multiplexing 22112:Transmission 22077:Nikola Tesla 22067:Henry Sutton 22022:Samuel Morse 21952:Robert Hooke 21917:Amos Dolbear 21852:John Bardeen 21771: 21751:Telautograph 21655:Mobile phone 21610:Edholm's law 21593:social media 21560: 21526:Broadcasting 21448:Quantization 21443:Oversampling 21434:Nyquist rate 21429:Downsampling 21357: 21142: 20264:(GB/T 22726) 19925:Multichannel 19781:MSU Lossless 19695:AVS1 P2/AVS+ 19649:TrueMotion S 19587:H.265 / HEVC 19533:Part 1 / EVC 19521:Part 3 / VVC 19487:Part 2 / ASP 19315:Hutter Prize 19279:Quantization 19184:Compensation 19160: 19008: 18978:Quantization 18701:Compensation 18598: 18593: 18448: 18267:Shannon–Fano 18207:Entropy type 18123: 18092:, retrieved 18088:the original 18073: 18052: 18048: 18031: 18027: 17994: 17990: 17940: 17936: 17909: 17882: 17878: 17860: 17835: 17831: 17806: 17802: 17770: 17766: 17743: 17739: 17705: 17701: 17687:(1): 27–34. 17684: 17680: 17663: 17659: 17606: 17602: 17596: 17587: 17581: 17564: 17560: 17554: 17529: 17525: 17519: 17510: 17484: 17480: 17474: 17463: 17452: 17409: 17405: 17399: 17390: 17386: 17376: 17360:. Springer. 17357: 17351: 17340:. Retrieved 17336: 17327: 17312: 17293: 17287: 17275:. Retrieved 17257: 17250: 17238:. Retrieved 17232: 17223: 17200: 17156: 17152: 17146: 17134: 17125: 17121: 17115: 17072: 17066: 17050: 17046: 17040: 17008: 17004: 16998: 16989: 16985: 16979: 16929: 16925: 16912: 16891:(1): 26–34, 16888: 16884: 16878: 16866:. Retrieved 16864:. March 2017 16849: 16840: 16831: 16820:. Retrieved 16816:the original 16806: 16785:. Retrieved 16781:the original 16775: 16768: 16759: 16750: 16726: 16704:. Retrieved 16694: 16686: 16679: 16667:. Retrieved 16657: 16649: 16642: 16617: 16613: 16600: 16588:. Retrieved 16581:the original 16574: 16562: 16551:. Retrieved 16541: 16506: 16496: 16469: 16459: 16447:. Retrieved 16443:The Register 16441: 16431: 16419:. Retrieved 16414: 16411:"Opus Codec" 16405: 16382: 16376: 16351: 16347: 16341: 16329:. Retrieved 16319: 16307:. Retrieved 16297: 16285:. Retrieved 16275: 16267: 16260:. Retrieved 16254: 16244: 16233:. Retrieved 16216: 16201: 16182: 16176: 16164:. Retrieved 16158: 16148: 16093: 16057: 16050: 16039:the original 16025: 16014:the original 15989: 15982:. Retrieved 15970: 15957: 15945:. Retrieved 15932: 15920:. Retrieved 15909: 15900: 15881: 15875: 15865: 15840: 15836: 15790: 15786: 15779: 15767:. Retrieved 15760: 15720: 15709: 15685: 15678: 15658: 15651: 15641:13 September 15639:. Retrieved 15629: 15607: 15588: 15554: 15547: 15535:. Retrieved 15528: 15501:. Retrieved 15491: 15462:. Retrieved 15459:AppleInsider 15415: 15379: 15349: 15309: 15213: 15168: 15129: 15100: 15094: 15068:13 September 15066:. Retrieved 15058: 15033:13 September 15031:. Retrieved 15026:The Atlantic 15024: 14986: 14979: 14944: 14938: 14903: 14893: 14860: 14856: 14847: 14836:. Retrieved 14782:. Retrieved 14776: 14735: 14731: 14698: 14653: 14649: 14643: 14618: 14612: 14588: 14581: 14557: 14533:. Retrieved 14526: 14492: 14485: 14452: 14448: 14442: 14418: 14411: 14399:. Retrieved 14394: 14390: 14380: 14355: 14351: 14345: 14328: 14322: 14316: 14304:. Retrieved 14253: 14247: 14237:Ahmed, Nasir 14218:20 September 14216:. Retrieved 14210: 14179:. Retrieved 14170: 14074: 13997: 13991: 13984:Ahmed, Nasir 13915:. Retrieved 13911:the original 13906: 13834: 13805:– via 13799:. Retrieved 13769: 13765: 13594: 13462: 13452: 13379: 12903: 12787: 12647: 12631: 12606:power-of-two 12564: 12376: 12367: 11926: 11848: 11825: 11821: 11739:3D VR Mults 11552: 11040: 10028: 9572: 9471: 9169: 9104: 7851: 7849: 7835: 7833: 7306: 7302: 7300: 6773: 6769: 6767: 6143: 6134: 6079: 6074: 6070: 6066: 6064: 6026: 6015: 5869: 5865: 5721: 5712: 5710: 5669: 5274: 5141: 5138: 4737: 4733:quantization 4617: 4275: 4272: 3970: 3925: 3751: 3508: 3130:real numbers 3048: 3033: 3029: 3009: 2993: 2988: 2982: 2977: 2973: 2969: 2965: 2961: 2957: 2953: 2949: 2945: 2941: 2937: 2933: 2929: 2925: 2923: 2917: 2796: 2786: 2755: 2737: 2719:Rosa Menkman 2712: 2685: 2660:multiplexing 2641: 2628:video coding 2624: 1987:used in the 1963:Apple ProRes 1904:web browsers 1902:broadcasts, 1778:MPEG-2 Video 1643:animated GIF 1586:image format 1533: 1410: 1313:Watermarking 1266:prediction, 1244:Video coding 1194:film editing 1161:Surveillance 1126:downsampling 1106:multiplexing 995:image fusion 959:transmission 806:quantization 767:Cryptography 736:social media 703:watermarking 701:, biometric 599:audio coding 588: 568: 557: 544: 542: 467: 464:Applications 443: 429:algorithm), 404: 399:Anil K. Jain 375:video coding 366: 362:Anil K. Jain 353: 343: 331: 326: 296: 278: 272: 243: 239: 235: 233: 220:real numbers 209: 97:, including 66: 62: 60: 47: 38:cleanup page 33: 22437:NPL network 22149:Radio waves 22087:Alfred Vail 21997:Hedy Lamarr 21982:Dawon Kahng 21942:Elisha Gray 21902:Yogen Dalal 21827:Nasir Ahmed 21761:Teleprinter 21625:Heliographs 21331:Z-transform 20982:MOD and TOD 20923:Flash Video 20861:3GP and 3G2 20542:HEIC / HEIF 20494:compression 20254:Audio Vivid 19989:MPEG-D USAC 19984:MPEG-4 CELP 19979:MPEG-4 HVXC 19894:compression 19726:Apple Video 19642:and AOMedia 19592:H.266 / VVC 19582:H.264 / AVC 19424:compression 19409:compression 19274:Prefix code 19127:Frame types 18948:Color space 18774:Convolution 18504:LZ77 + ANS 18415:Incremental 18388:Other types 18307:Levenshtein 17457:Malvar 1992 17234:PC Magazine 17182:10722/42775 17139:Malvar 1992 17089:10722/46174 16590:11 November 15730:83–108 (89) 15728:. pp. 15594:McGraw-Hill 13801:30 December 13587:butterflies 12373:Computation 11745:3D VR Adds 7307:3-D DCT-III 2848:, even for 2768:. Like the 2762:frequencies 2742:Thomas Ruff 2690:are blocky 2327:video games 2167:MP3 players 2132:video games 1726:Motion JPEG 1373:beamforming 1353:videophones 1349:smartphones 1337:technology 1264:inter-frame 1100:(DSP), DSP 1071:compression 1068:brain tumor 1007:image noise 859:Geophysical 695:fingerprint 480:as well as 433:(AAC), and 369:inter-frame 299:Nasir Ahmed 210:A DCT is a 199:usage, and 83:Nasir Ahmed 79:frequencies 71:data points 22570:Categories 22483:Antarctica 22442:Toasternet 22364:Television 21847:Paul Baran 21779:Television 21763:(teletype) 21756:Telegraphy 21734:transistor 21712:Phryctoria 21682:Photophone 21660:Smartphone 21650:Mass media 21463:Upsampling 21324:Techniques 21293:Sub-fields 21216:for codecs 21163:Daubechies 21138:Transforms 21040:Arithmetic 20707:Containers 20412:True Audio 20256:(GY/T 363) 19974:MPEG-4 DST 19969:MPEG-4 SLS 19964:MPEG-4 ALS 19838:SheerVideo 19812:QuickTime 19640:TrueMotion 19406:Multimedia 19331:Mark Adler 19289:Redundancy 19206:Daubechies 19189:Estimation 19122:Frame rate 19044:Daubechies 19004:Chain code 18963:Macroblock 18769:Companding 18706:Estimation 18626:Daubechies 18332:Lempel–Ziv 18292:Exp-Golomb 18220:Arithmetic 18130:. A free ( 18094:2011-08-13 17809:(1): 4–9. 17732:Plonka, G. 17616:cs/0703150 17513:: 104–107. 17487:(6): 433. 17419:cs/0703150 17342:2019-07-11 17277:19 October 17240:19 October 16964:10397/1928 16868:19 October 16841:SignalWire 16822:2019-10-19 16787:2019-10-19 16706:24 October 16669:20 October 16553:2017-12-11 16449:19 October 16396:1602.04845 16331:14 January 16309:14 January 16287:14 January 16262:14 January 16235:2022-01-14 16166:7 November 15769:13 October 15537:5 November 15503:20 October 15493:Medium.com 15305:Rao, K. R. 15209:Rao, K. R. 14838:2009-09-22 14535:3 November 14401:4 November 14256:(1): 4–5. 14181:4 November 14123:1008648293 13955:Retrieved 13952:Apple Inc. 13917:19 October 13907:POSTmatter 13857:2006931102 13831:Rao, K. R. 13691:References 11753:8 × 8 × 8 11742:RCF Mults 7836:row-column 7303:3-D DCT-II 6774:2-D DCT-II 6770:3-D DCT-II 6764:algorithm. 6140:M-D DCT-II 6125:orthogonal 5862:DCT V-VIII 5713:overlapped 5672:orthogonal 5266:orthogonal 4717:orthogonal 3697:orthogonal 2946:what point 2766:amplitudes 2715:glitch art 2619:See also: 2580:voice mail 2452:(LD-MDCT) 2400:Dolby AC-4 2276:Cook Codec 2244:BlackBerry 1820:Camcorders 1720:products. 1240:(HD) video 1224:ultra HDTV 1165:Vehicular 1130:upsampling 1038:inpainting 880:blurriness 781:protection 771:encryption 757:multimedia 707:palm print 691:Biometrics 484:have high 254:, and the 187:, such as 22467:Americas 22456:Locations 22427:Internet2 22188:Bandwidth 21892:Vint Cerf 21789:streaming 21767:Telephone 21707:Semaphore 21598:streaming 21438:frequency 20965:RealMedia 20567:JPEG 2000 20370:RealAudio 20350:OptimFROG 19828:RealVideo 19816:Animation 19786:OMS Video 19413:container 19308:Community 19132:Interlace 18518:Zstandard 18297:Fibonacci 18287:Universal 18245:Canonical 17953:CiteSeerX 17795:Ahmed, N. 17710:CiteSeerX 17053:: 29–43, 16992:: 700–703 16942:CiteSeerX 16533:214034702 16090:Rao, K.R. 15815:119888170 15807:0957-0233 15726:CRC Press 15691:CRC Press 15315:CRC Press 15219:CRC Press 15109:0018-1153 15059:EPFL News 14372:0090-6778 14088:1109.0337 14046:206619973 14022:0018-9340 14014:1557-9956 13988:Rao, K.R. 13881:118873224 13873:18495589M 13865:220853454 13788:1456-2774 13631:− 13625:⁡ 13420:transform 13343:− 13335:− 13317:− 13309:− 13299:− 13291:− 13278:− 13270:− 13262:− 13242:− 13234:− 13216:− 13208:− 13188:− 13180:− 13172:− 13164:− 13129:− 13111:− 13103:− 13095:− 13087:− 13067:− 13054:− 13046:− 13026:− 13013:− 13005:− 12970:− 12942:− 12824:⁡ 12761:− 12755:⁡ 12541:⁡ 12450:⁡ 12344:− 12335:… 12302:ℓ 12267:− 12258:… 12172:π 12157:ℓ 12118:⁡ 12092:π 12038:⁡ 12008:− 11990:∑ 11980:− 11962:∑ 11953:ℓ 11923:MD-DCT-IV 11748:RCF Adds 11675:− 11669:⁡ 11600:⁡ 11499:− 11493:⁡ 11450:Recursive 11444:⏟ 11407:− 11401:⁡ 11349:⏟ 11338:⁡ 11263:⁡ 11185:⁡ 11065:⁡ 10899:~ 10865:− 10761:~ 10733:− 10629:~ 10601:− 10530:~ 10508:− 10404:~ 10376:− 10305:~ 10283:− 10192:~ 10170:− 10116:~ 10047:~ 9986:φ 9980:⁡ 9949:φ 9943:⁡ 9912:φ 9906:⁡ 9843:~ 9828:− 9794:∑ 9785:− 9751:∑ 9742:− 9708:∑ 9495:π 9481:φ 9444:φ 9438:⁡ 9419:φ 9413:⁡ 9394:φ 9388:⁡ 9337:~ 9323:− 9298:∑ 9289:− 9264:∑ 9255:− 9230:∑ 9146:φ 9139:⁡ 9121:φ 9087:− 9074:≤ 9035:≤ 8925:− 8912:− 8900:− 8887:− 8875:− 8862:− 8850:~ 8749:− 8736:− 8724:− 8711:− 8699:~ 8611:− 8598:− 8573:− 8560:− 8548:~ 8440:− 8427:− 8415:~ 8327:− 8314:− 8302:− 8289:− 8264:~ 8169:− 8156:− 8131:~ 8049:− 8036:− 7998:~ 7883:~ 7852:N × N × N 7816:− 7800:… 7765:for 7697:π 7687:⁡ 7621:π 7611:⁡ 7545:π 7535:⁡ 7481:− 7449:∑ 7440:− 7408:∑ 7399:− 7367:∑ 7283:− 7267:… 7232:for 7164:π 7154:⁡ 7088:π 7078:⁡ 7012:π 7002:⁡ 6948:− 6916:∑ 6907:− 6875:∑ 6866:− 6834:∑ 6682:π 6672:⁡ 6606:π 6596:⁡ 6555:− 6523:∑ 6514:− 6482:∑ 6408:π 6398:⁡ 6327:π 6317:⁡ 6276:− 6244:∑ 6230:− 6198:∑ 5986:± 5911:− 5803:− 5763:− 5646:− 5637:… 5542:π 5528:⁡ 5507:− 5489:∑ 5437:− 5397:− 5115:− 5106:… 5034:π 5020:⁡ 4999:− 4981:∑ 4819:− 4779:− 4539:− 4458:≤ 4249:− 4240:… 4168:π 4154:⁡ 4133:− 4115:∑ 4041:− 3773:− 3719:− 3658:terms by 3641:− 3569:terms by 3552:− 3485:− 3476:… 3427:− 3418:π 3402:⁡ 3381:− 3363:∑ 3348:− 3324:− 3249:− 3234:… 3188:− 3174:… 3082:→ 2797:extension 2758:sinusoids 2696:quantized 2664:signaling 2414:(UHD TV) 2335:Knowledge 2284:RealAudio 1735:QuickTime 1639:animation 1473:quantized 1456:× 1154:filtering 1110:signaling 893:luminance 882:measure, 779:copyright 724:Computers 490:bandwidth 319:K. R. Rao 303:K. R. Rao 171:(such as 159:(such as 143:(such as 127:(such as 113:(such as 101:(such as 50:July 2022 22535:Category 22422:Internet 22412:CYCLADES 22329:Ethernet 22279:Concepts 22203:terminal 22154:wireless 21977:Bob Kahn 21820:Pioneers 21645:Internet 21536:Cable TV 21419:Aliasing 21411:Sampling 21084:Lossless 21050:Modified 20943:Matroska 20822:Matroska 20733:MPEG-PES 20345:Musepack 20242:AVS1 P10 19821:Graphics 19771:Lagarith 19294:Symmetry 19262:Timeline 19245:FM-index 19090:Bit rate 19083:Concepts 18931:Concepts 18794:Sampling 18747:Bit rate 18740:Concepts 18442:Sequitur 18277:Tunstall 18250:Modified 18240:Adaptive 18198:Lossless 17294:Aperture 17033:18016215 17025:11026596 16972:18282969 16634:30821673 16421:July 31, 16160:EE Times 16140:56983045 16124:34617596 16116:96015550 15975:Archived 15971:JPEG Org 15947:5 August 15922:5 August 15916:BT Group 15884:(4): 1. 15631:BBC News 15530:Bitmovin 15307:(2019). 15211:(2019). 15173:Elsevier 14992:Elsevier 14971:16411333 14930:17045923 14885:13894279 14811:Archived 14635:58446992 14477:62725808 14139:12270940 14131:2207570M 14115:89029800 14030:75642478 13957:5 August 13792:Archived 13412:Discrete 13393:See also 11844:JPEG DCT 5719:(MDCT). 5169:term by 4645:term by 3055:function 2974:dcbaabcd 2968:between 2940:and max- 2930:discrete 2899:periodic 2774:discrete 2656:encoding 2471:FaceTime 2465:(VoIP), 2438:standard 2408:ATSC 3.0 2389:WhatsApp 2352:HD Radio 2266:(DAB+), 2148:MiniDisc 2141:(ATRAC) 2032:Xiph.org 1924:FaceTime 1884:HD video 1882:Popular 1850:(PSTN), 1800:Internet 1645:format. 1401:networks 1366:aperture 1335:Wireless 1325:3D video 1214:(SDTV), 1206:(DTV) — 1138:transmux 1102:software 1056:(ECG) — 1034:textures 905:decoding 802:Encoding 728:Internet 726:and the 670:(VoIP), 642:HD Radio 640:(DAB+), 540:(HDTV). 421:(AC-3), 244:the IDCT 22555:Commons 22545:Outline 22498:Oceania 22417:FidoNet 22402:ARPANET 22215:circuit 21784:digital 21513:History 21158:Wavelet 21099:DEFLATE 21045:Huffman 21035:Entropy 21027:Methods 21012:MPEG LA 20901:Smacker 20789:H.222.0 20745:MPEG-TS 20740:MPEG-PS 20728:MPEG-ES 20612:TIFF/IT 20607:TIFF/EP 20592:JPEG XT 20587:JPEG XS 20582:JPEG XR 20577:JPEG XL 20572:JPEG-LS 20453:aptX HD 20433:WavPack 20375:RTAudio 20315:Codec 2 20248:AVS2 P3 20175:GSM-EFR 20145:AMR-WB+ 20079:G.729.1 20059:G.723.1 20049:G.722.2 20044:G.722.1 19843:Smacker 19833:RTVideo 19761:Huffyuv 19741:Cinepak 19701:AVS2 P2 19415:formats 19252:Entropy 19201:Wavelet 19180:Motion 19039:Wavelet 19019:Fractal 19014:Deflate 18997:Methods 18784:Latency 18697:Motion 18621:Wavelet 18538:LHA/LZH 18488:Deflate 18437:Re-Pair 18432:Grammar 18262:Shannon 18235:Huffman 18191:methods 18019:3385296 17999:Bibcode 17975:6644892 17945:Bibcode 17887:Bibcode 17840:Bibcode 17811:Bibcode 17775:Bibcode 17621:Bibcode 17534:Bibcode 17424:Bibcode 17161:Bibcode 17107:1757438 16934:Bibcode 16893:Bibcode 16792:, also 16356:Bibcode 16256:YouTube 16132:978319M 15857:2060937 15498:Netflix 15464:June 9, 14865:Bibcode 14784:14 July 14457:Bibcode 14306:12 July 14258:Bibcode 14177:: 22–33 14173:(251). 14038:1799331 12724:(as in 12680:FFTPACK 11913:( 8×8 ) 11816:24.047 11813:24.047 11799:19.594 11796:19.594 11782:15.188 11779:15.188 11765:10.875 11762:10.875 6118:√ 4930:DCT-III 4613:Makhoul 4330:inputs 4307:of the 3106:(where 2966:halfway 2962:dcbabcd 2536:EVRC-WB 2517:G.729.1 2501:G.722.1 2494:G.722.1 2331:Spotify 2253:(AAC+) 2240:Android 2205:Audio) 2201:(AAC / 2193:(DARS) 2124:Blu-ray 2105:(AC-3) 2064:Netflix 2060:YouTube 2003:(HEVC, 1954:media, 1952:Windows 1916:Netflix 1892:YouTube 1860:(ISDN) 1728:(MJPEG) 1664:JPEG XL 1594:JPEG XR 1384:sensing 1377:digital 1236:(DVD), 1226:(UHDTV) 1116:(ADC), 625:(TDAC) 619:phoneme 528:(VOD), 356:with a 321:at the 293:History 264:ISO/IEC 260:integer 236:the DCT 167:), and 22493:Europe 22463:Africa 22447:Usenet 22407:BITNET 22344:Mobile 22220:packet 21729:MOSFET 21724:device 21521:Beacon 21372:(DTFT) 21262:Theory 20960:RatDVD 20854:Others 20621:Others 20423:VMR-WB 20418:TwinVQ 20283:Others 20170:GSM-FR 20165:GSM-HR 20155:EVRC-B 20140:AMR-WB 20112:Vorbis 19952:AAC-LD 19947:HE-AAC 19853:Theora 19796:ProRes 19791:Pixlet 19719:Others 19528:MPEG-5 19516:MPEG-I 19504:MPEG-H 19482:MPEG-4 19475:Part 2 19470:MPEG-2 19465:MPEG-1 19363:codecs 19324:People 19227:Theory 19194:Vector 18711:Vector 18528:Brotli 18478:Hybrid 18377:Snappy 18230:Golomb 18080: 18017: 17973: 17955: 17918: 17867: 17712: 17641:986733 17639: 17444:986733 17442: 17364: 17300: 17268: 17211: 17105: 17095: 17031: 17023: 16970: 16944: 16738: 16632: 16531: 16521: 16484: 16194: 16138: 16130: 16122: 16114: 16104: 16069: 15984:14 Jan 15911:BT.com 15855: 15813: 15805: 15736: 15697: 15666: 15600: 15562: 15533:. 2019 15427: 15391: 15357: 15321: 15225: 15187: 15141: 15107: 14998: 14969: 14959: 14928: 14918: 14883: 14752:897622 14750: 14710: 14633: 14600: 14569: 14500: 14475: 14430: 14370: 14137: 14129: 14121: 14113: 14103: 14044: 14036: 14028: 14020: 14012: 13879: 13871: 13863: 13855: 13845: 13786: 13776: 13643: 13606: 13574: 13555: 13535: 13529: 13509: 13503: 13483: 13474: 13427: 13423: 13418: 13416:cosine 13414: 13410: 13406: 13356:0.1444 13351:0.1887 13346:0.1031 13338:0.1136 13330:0.0644 13325:0.1960 13320:0.0627 13312:0.2970 13302:0.1500 13294:0.1650 13286:0.1017 13281:0.0056 13273:0.0225 13265:0.1033 13257:0.0541 13252:0.3169 13245:0.1311 13237:0.0284 13229:0.2875 13224:0.4745 13219:0.2905 13211:0.4721 13203:0.0428 13198:0.0653 13191:0.5906 13183:0.2042 13175:0.0742 13167:0.2583 13159:0.2183 13154:0.1229 13149:0.0226 13144:0.2750 13137:0.3550 13132:0.1433 13124:0.3501 13119:0.6351 13114:0.2866 13106:0.2617 13098:0.0392 13090:0.2340 13080:0.4713 13075:0.2801 13070:0.1952 13062:0.0789 13057:0.2720 13049:1.0017 13041:0.2214 13036:1.0423 13029:0.2554 13021:0.1001 13016:0.4391 13008:0.7846 13000:0.3947 12995:0.4503 12990:0.0214 12985:0.2205 12978:0.0561 12973:0.0724 12965:0.2742 12960:0.1583 12955:0.1492 12950:1.2418 12945:0.3411 12937:6.1917 12873: 12867: 12796: 12773: 12736: 12708: 12702: 12666: 12657: 12639:8 × 8 12592: 12573: 12550: 12522: 12502: 12483: 12459: 12431: 12411: 12385: 12350: 12338: 12332: 12323: 12314: 12299: 12296: 12276: 12273: 12261: 12255: 12249: 12240: 12225: 12222: 12195: 12187: 12169: 12126: 12107: 12089: 12046: 12016: 11895: 11863: 11851:8 × 8 11790:4.375 11756:2.625 11712: 11631: 11611: 11562: 11536: 11297: 11274: 11250: 11237: 11217: 11196: 11162: 11142: 11122: 11109: 11094: 11074: 11049: 10029:where 9472:where 9024:where 5652: 5640: 5634: 5463:DCT-IV 5121: 5109: 5103: 5088: 4725:Matlab 4360:where 4255: 4243: 4237: 4222: 4089:DCT-II 3908: 3902: 3896: 3890: 3884: 3878: 3872: 3846: 3840: 3834: 3828: 3491: 3479: 3473: 3458: 3415: 3196: 3177: 3171: 3155: 3051:linear 3034:always 2926:finite 2789:domain 2666:, and 2632:DCT-IV 2602:(EVS) 2450:AAC-LD 2361:(DRA) 2344:(HDC) 2307:Vorbis 2293:(WMA) 2270:(DRM) 2236:iTunes 2184:(PAC) 2112:Cinema 1985:Google 1931:Theora 1874:MPEG-4 1868:(AVC, 1754:MPEG-1 1628:(HEIF) 1618:Google 1552:pixels 1439:Theora 1415:, and 1369:arrays 1169:camera 901:YUV411 897:YUV444 872:artist 868:Images 730:— the 486:memory 435:Vorbis 389:for a 275:pixels 173:AAC-LD 75:cosine 22606:H.26x 22476:South 22471:North 22432:JANET 22369:Telex 22359:Radio 22198:Nodes 22193:Links 22114:media 21692:Radio 21677:Pager 21605:Drums 21571:video 21566:image 21556:audio 21366:(DFT) 21360:(DCT) 21181:Lists 21126:ADPCM 21121:µ-law 21116:A-law 21089:Lossy 21062:ACELP 21007:NETVC 20832:SMPTE 20794:T.802 20781:ITU-T 20757:(MP4) 20557:JBIG2 20520:ITU-T 20492:Image 20428:VSELP 20405:SVOPC 20395:Siren 20365:RCELP 20360:QCELP 20305:ATRAC 20290:ACELP 20107:Speex 20074:G.729 20069:G.728 20064:G.726 20054:G.723 20039:G.722 20034:G.719 20029:G.718 20022:µ-law 20017:A-law 20012:G.711 20004:ITU-T 19892:Audio 19766:Indeo 19746:Daala 19602:SMPTE 19577:H.263 19572:H.262 19567:H.261 19562:H.120 19550:ITU-T 19455:MJPEG 19422:Video 19154:parts 19152:Codec 19117:Frame 19075:Video 19059:SPIHT 18968:Pixel 18923:Image 18877:ACELP 18848:ADPCM 18838:μ-law 18833:A-law 18826:parts 18824:Codec 18732:Audio 18671:ACELP 18659:ADPCM 18636:SPIHT 18577:Lossy 18561:bzip2 18552:LZHAM 18508:LZFSE 18410:Delta 18302:Gamma 18282:Unary 18257:Range 18152:LTFAT 18015:S2CID 17987:(PDF) 17971:S2CID 17933:(PDF) 17637:S2CID 17611:arXiv 17440:S2CID 17414:arXiv 17262:(PDF) 17103:S2CID 17029:S2CID 16922:(PDF) 16858:(PDF) 16798:(PDF) 16691:(PDF) 16654:(PDF) 16630:S2CID 16610:(PDF) 16584:(PDF) 16571:(PDF) 16529:S2CID 16391:arXiv 16226:(PDF) 16207:H.261 16203:H.263 16136:S2CID 16098:H.261 16042:(PDF) 16035:(PDF) 16017:(PDF) 16010:(PDF) 15978:(PDF) 15967:(PDF) 15853:S2CID 15833:H.264 15811:S2CID 15525:(PDF) 15103:(2). 15091:(PDF) 14967:S2CID 14926:S2CID 14881:S2CID 14814:(PDF) 14807:(PDF) 14773:(PDF) 14748:S2CID 14631:S2CID 14528:ITU-T 14473:S2CID 14300:CCITT 14296:(PDF) 14244:(PDF) 14135:S2CID 14083:arXiv 14042:S2CID 14010:eISSN 13948:(PDF) 13877:S2CID 13795:(PDF) 13762:(PDF) 13444:Notes 13425:- 13408:- 11807:5.25 3270:DCT-I 3038:slope 2738:Jpegs 2676:chips 2605:2014 2591:2011 2568:2008 2564:G.719 2555:2008 2551:G.718 2540:2007 2525:G.729 2521:2006 2510:G.722 2505:1999 2482:1999 2478:Siren 2455:1999 2441:Year 2404:2015 2384:2012 2373:DVB-H 2364:2008 2347:2002 2311:2000 2296:1999 2280:1998 2256:1997 2208:1997 2187:1996 2159:1993 2144:1992 2108:1991 2094:Year 2052:Daala 2023:Daala 2015:H.264 2005:H.265 1993:HTML5 1870:H.264 1853:H.320 1846:over 1831:H.263 1782:H.262 1756:Video 1700:H.261 1443:Daala 1427:MJPEG 1422:H.26x 1282:(MVC) 1200:audio 1173:Video 1060:(VCG) 889:Color 876:focus 848:(VOD) 660:(VAD) 387:pixel 268:ITU-T 177:Siren 120:H.26x 22611:JPEG 22488:Asia 22374:UUCP 22334:ISDN 21208:See 21153:MDCT 21131:DPCM 21077:WLPC 21067:CELP 20970:RIFF 20948:WebM 20938:M2TS 20928:HEIF 20896:Bink 20876:AIFF 20804:IETF 20693:WebP 20688:WBMP 20683:QTVR 20663:ICER 20658:FLIF 20648:DjVu 20638:AVIF 20628:APNG 20602:TIFF 20562:JPEG 20552:JBIG 20547:HEVC 20524:JPEG 20511:IETF 20478:LLAC 20473:LHDC 20468:LDAC 20448:aptX 20409:TTA 20390:SILK 20330:MELP 20325:Lyra 20320:iSAC 20310:CELT 20300:Asao 20295:ALAC 20273:ExAC 20268:L2HC 20198:AC-4 20193:AC-3 20185:ETSI 20150:EVRC 20127:3GPP 20117:FLAC 20102:iLBC 20097:Opus 20089:IETF 19912:MPEG 19878:YULS 19858:Thor 19806:4444 19756:FFV1 19736:Bink 19630:VC-6 19625:VC-5 19620:VC-3 19615:VC-2 19610:VC-1 19554:VCEG 19442:MPEG 19411:and 19166:DPCM 18973:PSNR 18904:MDCT 18897:WLPC 18882:CELP 18843:DPCM 18691:WLPC 18676:CELP 18654:DPCM 18604:MDCT 18548:LZMA 18449:LDCT 18427:DPCM 18372:LZWL 18362:LZSS 18357:LZRW 18347:LZJB 18124:FFTW 18078:ISBN 17916:ISBN 17865:ISBN 17362:ISBN 17298:ISBN 17279:2019 17266:ISBN 17242:2019 17209:ISBN 17093:ISBN 17021:PMID 16968:PMID 16870:2019 16736:ISBN 16708:2019 16671:2019 16592:2019 16519:ISBN 16482:ISBN 16451:2019 16423:2012 16415:Opus 16333:2022 16311:2022 16289:2022 16264:2022 16192:ISBN 16168:2019 16120:OCLC 16112:LCCN 16102:ISBN 16067:ISBN 15986:2022 15949:2019 15924:2019 15803:ISSN 15771:2019 15734:ISBN 15695:ISBN 15664:ISBN 15643:2019 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9347:n 9343:( 9334:x 9326:1 9320:N 9315:1 9312:= 9307:3 9303:n 9292:1 9286:N 9281:1 9278:= 9273:2 9269:n 9258:1 9252:N 9247:1 9244:= 9239:1 9235:n 9226:= 9223:) 9218:3 9214:k 9210:, 9205:2 9201:k 9197:, 9192:1 9188:k 9184:( 9181:X 9155:) 9150:i 9142:( 9133:= 9130:) 9125:i 9117:( 9114:c 9090:1 9082:2 9079:N 9069:3 9065:n 9061:, 9056:2 9052:n 9048:, 9043:1 9039:n 9032:0 9006:) 9003:1 9000:+ 8995:3 8991:n 8987:2 8984:, 8981:1 8978:+ 8973:2 8969:n 8965:2 8962:, 8959:1 8956:+ 8951:1 8947:n 8943:2 8940:( 8937:x 8934:= 8931:) 8928:1 8920:3 8916:n 8909:N 8906:, 8903:1 8895:2 8891:n 8884:N 8881:, 8878:1 8870:1 8866:n 8859:N 8856:( 8847:x 8837:) 8832:3 8828:n 8824:2 8821:, 8818:1 8815:+ 8810:2 8806:n 8802:2 8799:, 8796:1 8793:+ 8788:1 8784:n 8780:2 8777:( 8774:x 8771:= 8768:) 8763:3 8759:n 8755:, 8752:1 8744:2 8740:n 8733:N 8730:, 8727:1 8719:1 8715:n 8708:N 8705:( 8696:x 8686:) 8683:1 8680:+ 8675:3 8671:n 8667:2 8664:, 8659:2 8655:n 8651:2 8648:, 8645:1 8642:+ 8637:1 8633:n 8629:2 8626:( 8623:x 8620:= 8617:) 8614:1 8606:3 8602:n 8595:N 8592:, 8587:2 8583:n 8579:, 8576:1 8568:1 8564:n 8557:N 8554:( 8545:x 8535:) 8530:3 8526:n 8522:2 8519:, 8514:2 8510:n 8506:2 8503:, 8500:1 8497:+ 8492:1 8488:n 8484:2 8481:( 8478:x 8475:= 8472:) 8467:3 8463:n 8459:, 8454:2 8450:n 8446:, 8443:1 8435:1 8431:n 8424:N 8421:( 8412:x 8402:) 8399:1 8396:+ 8391:3 8387:n 8383:2 8380:, 8377:1 8374:+ 8369:2 8365:n 8361:2 8358:, 8353:1 8349:n 8345:2 8342:( 8339:x 8336:= 8333:) 8330:1 8322:3 8318:n 8311:N 8308:, 8305:1 8297:2 8293:n 8286:N 8283:, 8278:1 8274:n 8270:( 8261:x 8251:) 8246:3 8242:n 8238:2 8235:, 8232:1 8229:+ 8224:2 8220:n 8216:2 8213:, 8208:1 8204:n 8200:2 8197:( 8194:x 8191:= 8188:) 8183:3 8179:n 8175:, 8172:1 8164:2 8160:n 8153:N 8150:, 8145:1 8141:n 8137:( 8128:x 8118:) 8115:1 8112:+ 8107:3 8103:n 8099:2 8096:, 8091:2 8087:n 8083:2 8080:, 8075:1 8071:n 8067:2 8064:( 8061:x 8058:= 8055:) 8052:1 8044:3 8040:n 8033:N 8030:, 8025:2 8021:n 8017:, 8012:1 8008:n 8004:( 7995:x 7985:) 7980:3 7976:n 7972:2 7969:, 7964:2 7960:n 7956:2 7953:, 7948:1 7944:n 7940:2 7937:( 7934:x 7931:= 7928:) 7923:3 7919:n 7915:, 7910:2 7906:n 7902:, 7897:1 7893:n 7889:( 7880:x 7811:i 7807:N 7803:, 7797:, 7794:2 7791:, 7788:1 7785:, 7782:0 7779:= 7774:i 7770:n 7760:, 7756:] 7750:3 7746:k 7741:) 7735:2 7732:1 7727:+ 7722:3 7718:n 7713:( 7705:3 7701:N 7691:[ 7680:] 7674:2 7670:k 7665:) 7659:2 7656:1 7651:+ 7646:2 7642:n 7637:( 7629:2 7625:N 7615:[ 7604:] 7598:1 7594:k 7589:) 7583:2 7580:1 7575:+ 7570:1 7566:n 7561:( 7553:1 7549:N 7539:[ 7525:3 7521:k 7517:, 7512:2 7508:k 7504:, 7499:1 7495:k 7490:X 7484:1 7476:3 7472:N 7466:0 7463:= 7458:3 7454:k 7443:1 7435:2 7431:N 7425:0 7422:= 7417:2 7413:k 7402:1 7394:1 7390:N 7384:0 7381:= 7376:1 7372:k 7363:= 7356:3 7352:n 7348:, 7343:2 7339:n 7335:, 7330:1 7326:n 7321:x 7278:i 7274:N 7270:, 7264:, 7261:2 7258:, 7255:1 7252:, 7249:0 7246:= 7241:i 7237:k 7227:, 7223:] 7217:3 7213:k 7208:) 7202:2 7199:1 7194:+ 7189:3 7185:n 7180:( 7172:3 7168:N 7158:[ 7147:] 7141:2 7137:k 7132:) 7126:2 7123:1 7118:+ 7113:2 7109:n 7104:( 7096:2 7092:N 7082:[ 7071:] 7065:1 7061:k 7056:) 7050:2 7047:1 7042:+ 7037:1 7033:n 7028:( 7020:1 7016:N 7006:[ 6992:3 6988:n 6984:, 6979:2 6975:n 6971:, 6966:1 6962:n 6957:x 6951:1 6943:3 6939:N 6933:0 6930:= 6925:3 6921:n 6910:1 6902:2 6898:N 6892:0 6889:= 6884:2 6880:n 6869:1 6861:1 6857:N 6851:0 6848:= 6843:1 6839:n 6830:= 6823:3 6819:k 6815:, 6810:2 6806:k 6802:, 6797:1 6793:k 6788:X 6745:. 6741:] 6735:2 6731:k 6726:) 6720:2 6717:1 6712:+ 6707:2 6703:n 6698:( 6690:2 6686:N 6676:[ 6665:] 6659:1 6655:k 6650:) 6644:2 6641:1 6636:+ 6631:1 6627:n 6622:( 6614:1 6610:N 6600:[ 6586:2 6582:n 6578:, 6573:1 6569:n 6564:x 6558:1 6550:2 6546:N 6540:0 6537:= 6532:2 6528:n 6517:1 6509:1 6505:N 6499:0 6496:= 6491:1 6487:n 6478:= 6467:] 6461:1 6457:k 6452:) 6446:2 6443:1 6438:+ 6433:1 6429:n 6424:( 6416:1 6412:N 6402:[ 6391:) 6386:] 6380:2 6376:k 6371:) 6365:2 6362:1 6357:+ 6352:2 6348:n 6343:( 6335:2 6331:N 6321:[ 6307:2 6303:n 6299:, 6294:1 6290:n 6285:x 6279:1 6271:2 6267:N 6261:0 6258:= 6253:2 6249:n 6239:( 6233:1 6225:1 6221:N 6215:0 6212:= 6207:1 6203:n 6194:= 6183:2 6179:k 6175:, 6170:1 6166:k 6161:X 6120:2 6102:N 6098:/ 6094:2 6075:N 6071:N 6067:N 6042:= 6039:N 6029:a 6000:2 5995:/ 5990:1 5983:N 5963:N 5960:8 5940:N 5937:4 5917:) 5914:1 5908:N 5905:( 5902:2 5882:N 5847:. 5842:k 5838:X 5817:; 5814:2 5810:/ 5806:1 5800:N 5797:= 5794:n 5774:2 5770:/ 5766:1 5760:= 5757:n 5735:n 5731:x 5697:. 5692:N 5688:/ 5684:2 5655:. 5649:1 5643:N 5631:, 5628:0 5625:= 5622:k 5612:] 5606:) 5599:2 5596:1 5590:+ 5587:k 5583:( 5578:) 5571:2 5568:1 5562:+ 5559:n 5555:( 5547:N 5532:[ 5520:n 5516:x 5510:1 5504:N 5499:0 5496:= 5493:n 5485:= 5480:k 5476:X 5444:/ 5440:1 5434:N 5431:= 5428:k 5408:2 5404:/ 5400:1 5394:= 5391:k 5369:k 5365:X 5345:; 5342:N 5339:= 5336:n 5316:0 5313:= 5310:n 5288:n 5284:x 5250:N 5246:/ 5242:2 5218:2 5212:/ 5206:0 5202:x 5179:2 5155:0 5151:x 5124:. 5118:1 5112:N 5100:, 5097:0 5094:= 5091:k 5078:] 5073:n 5069:) 5062:2 5059:1 5053:+ 5050:k 5046:( 5039:N 5024:[ 5012:n 5008:x 5002:1 4996:N 4991:1 4988:= 4985:n 4977:+ 4972:0 4968:x 4961:2 4958:1 4952:= 4947:k 4943:X 4915:. 4912:N 4909:= 4906:k 4886:0 4883:= 4880:k 4858:k 4854:X 4834:; 4830:2 4826:/ 4822:1 4816:N 4813:= 4810:n 4790:2 4786:/ 4782:1 4776:= 4773:n 4751:n 4747:x 4700:N 4695:/ 4690:2 4663:N 4657:/ 4653:1 4631:0 4627:X 4609:N 4595:. 4592:N 4589:2 4583:n 4577:0 4555:n 4551:y 4547:= 4542:n 4536:N 4533:4 4529:y 4508:, 4505:0 4502:= 4497:N 4494:2 4490:y 4470:, 4467:N 4461:n 4455:0 4433:n 4429:x 4425:= 4420:1 4417:+ 4414:n 4411:2 4407:y 4387:, 4384:0 4381:= 4376:n 4373:2 4369:y 4348:, 4343:n 4339:y 4318:N 4315:4 4291:N 4288:4 4258:. 4252:1 4246:N 4234:, 4231:0 4228:= 4225:k 4212:] 4207:k 4203:) 4196:2 4193:1 4187:+ 4184:n 4180:( 4173:N 4158:[ 4146:n 4142:x 4136:1 4130:N 4125:0 4122:= 4119:n 4111:= 4106:k 4102:X 4074:. 4069:k 4065:X 4044:1 4038:N 4035:= 4032:n 4012:0 4009:= 4006:n 3984:n 3980:x 3957:. 3954:N 3934:N 3911:b 3905:c 3899:d 3893:e 3887:d 3881:c 3875:b 3869:a 3849:e 3843:d 3837:c 3831:b 3825:a 3805:5 3802:= 3799:N 3779:) 3776:1 3770:N 3767:( 3764:2 3733:, 3722:1 3716:N 3712:2 3683:, 3676:2 3670:/ 3666:1 3644:1 3638:N 3634:X 3611:0 3607:X 3586:, 3579:2 3555:1 3549:N 3545:x 3522:0 3518:x 3494:. 3488:1 3482:N 3470:, 3467:0 3464:= 3461:k 3448:] 3443:k 3439:n 3430:1 3424:N 3406:[ 3394:n 3390:x 3384:2 3378:N 3373:1 3370:= 3367:n 3359:+ 3356:) 3351:1 3345:N 3341:x 3335:k 3331:) 3327:1 3321:( 3318:+ 3313:0 3309:x 3305:( 3300:2 3297:1 3292:= 3287:k 3283:X 3252:1 3246:N 3242:X 3237:, 3230:, 3225:0 3221:X 3210:N 3191:1 3185:N 3181:x 3168:, 3163:0 3159:x 3145:N 3138:N 3134:N 3115:R 3092:N 3087:R 3077:N 3072:R 3067:: 3064:f 2978:a 2976:( 2970:a 2958:a 2942:n 2938:n 2918:N 2885:) 2882:x 2879:( 2876:f 2856:x 2836:x 2816:) 2813:x 2810:( 2807:f 2007:) 1876:) 1837:) 1833:( 1784:) 1780:( 1519:) 1516:0 1513:, 1510:0 1507:( 1487:N 1459:N 1453:N 919:) 437:( 65:( 52:) 48:(

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Index