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filter, then using the discrete resampling filter to directly resample the image. For decimation by an integer amount, only a single sampled filter is necessary; for interpolation by an integer amount, different samplings are needed for different phases – for instance, if one is upsampling by a factor of 4, then one sampled filter is used for the half-way point, while a different sampled filter is used for the point 1/4 of the way from one point to another.
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In photography, a great variety of interpolation filters exist, some proprietary, for which opinions are mixed. Evaluation is often subjective, with reactions being varied, and some arguing that at realistic resampling ratios, there is little difference between them, as compared with bicubic, though
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of higher order – the box filter and tent filter being the 0th and 1st order cardinal B-splines. These filters fail to be interpolating filters, since their impulse response do not vanish at all non-zero original sample points – for 1:1 resampling, they are not the identity, but rather blur. On the
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waveform has an infinite response to a signal, in both the positive and negative time directions, which is impossible to perform in real time – as it would require infinite delay. Consequently, real reconstruction filters typically either allow some energy above the
Nyquist rate, attenuate some
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For reconstruction purposes, a variety of kernels are used, many of which can be interpreted as approximating the sinc function, either by windowing or by giving a spline approximation, either by cubics or higher order splines. In the case of windowed sinc filters, the frequency response of the
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For resampling, in principle the analog image is reconstructed, then sampled, and this is necessary for general changes in resolution. For integer ratios of sampling rate, one may simplify by sampling the impulse response of the continuous reconstruction filter to produce a discrete resampling
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reconstruction filter can be understood in terms of the frequency response of the window, as the frequency response of a windowed filter is the convolution of the original response (for sinc, a brick-wall) with the frequency response of the window. Among these, the
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A subtlety in image processing is that (linear) signal processing assumes linear luminance – that doubling a pixel value doubles the luminance of the output. However, images are frequently
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color space, so luminance is not linear. Thus to apply a linear filter, one must first gamma decode the values – and if resampling, one must gamma decode, resample, then gamma encode.
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and a reconstruction filter may be of identical design. For example, both the input and the output for audio equipment may be sampled at 44.1 kHz. In this case, both
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Alternatively, a system may have no reconstruction filter and simply tolerate some energy being wasted reproducing higher frequency images of the primary signal spectrum.
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signal must be bandlimited, to prevent imaging (meaning
Fourier coefficients being reconstructed as spurious high-frequency 'mirrors'). This is an implementation of the
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of the signal is also known, in addition to the amplitude, and conversely that also performing derivative reconstruction can improve signal reconstruction methods.
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133:. In doing so, it reconstructs the continuous time signal (whether originally sampled, or modelled by digital logic) corresponding to the digital time sequence.
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Medical Image
Computing and Computer-Assisted Intervention--MICCAI '99: second international conference, Cambridge, UK, September 19–22, 1999 proceedings
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Practical filters have non-flat frequency or phase response in the pass band and incomplete suppression of the signal elsewhere. The ideal
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may be used to ensure that frequencies of interest are accurately reproduced without excess energy being emitted out of band.
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For the same reason, the output of a DAC requires a low-pass analog filter, called a reconstruction filter - because the
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Reconstruction filters are also used when "reconstructing" a waveform or an image from a collection of
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steps and other output pulses, if unfiltered, would contain spurious high-frequency replicas, "
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These are in increasing order of stopband suppression (anti-aliasing), and decreasing speed
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block as much as possible above 22 kHz and pass as much as possible below 20 kHz.
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Quantitative
Comparison of Sinc-Approximating Kernels for Medical Image Interpolation
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ACM SIGGRAPH International
Conference on Computer Graphics and Interactive Techniques
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Theußl, Thomas; Hauser, Helwig; Gröller, Meister Eduard (October 2000).
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other hand, being nonnegative, they do not introduce any overshoot or
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Filter used to construct a smooth analog signal from a digital input
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372:. Salt Lake City, Utah, United States. pp. 101–108.
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for higher resampling ratios behavior is more varied.
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Mitchell, Don P.; Netravali, Arun N. (August 1988).
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Another class of reconstruction filters include the
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113:While in theory a DAC outputs a series of discrete
477:Meijering, Erik H. W.; Niessen; Pluim; Viergever.
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141:in-band frequencies, or both. For this reason,
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362:Mastering Windows: Improving Reconstruction
101:. This can be achieved by a filter with a '
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413:"Filters for Common Resampling Tasks"
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510:Digital Photo Interpolation Review
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186:Resampling may be referred to as
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93:Ideally, both filters should be
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51:describes why the input of an
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55:requires a low-pass analog
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19:In a mixed-signal system (
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318:Iterative reconstruction
313:Reconstruction algorithm
256:are frequently praised.
521:Interpolation -- Part I
498:dpreview: Interpolation
411:Turkowski, Ken (1990).
79:as a lower frequency).
228:bilinear interpolation
31:, sometimes called an
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378:10.1109/VV.2000.10002
335:Signal reconstruction
234:bicubic interpolation
29:reconstruction filter
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150:anti-aliasing filter
105:' impulse response.
61:anti-aliasing filter
129:(copies) above the
33:anti-imaging filter
330:Signal processing
296:coefficients. In
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304:photos or
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306:MRI scans
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261:Gaussian
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252:and
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