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surface, and the amount of reflection into the camera is measured. Using this information, a look-up table can be created that maps reflected intensities for each light source to a list of possible normal vectors. This puts constraints on the possible normal vectors the surface may have, and reduces the photometric stereo problem to an interpolation between measurements. Typical known surfaces to calibrate the look-up table with are spheres for their wide variety of surface orientations.
38:
60:). It is based on the fact that the amount of light reflected by a surface is dependent on the orientation of the surface in relation to the light source and the observer. By measuring the amount of light reflected into a camera, the space of possible surface orientations is limited. Given enough light sources from different angles, the surface orientation may be constrained to a single orientation or even overconstrained.
678:
Restricting the BRDF to be symmetrical. If the BRDF is symmetrical, the direction of the light can be restricted to a cone about the direction to the camera. Which cone this is depends on the BRDF itself, the normal vector of the surface, and the measured intensity. Given enough measured intensities
674:
Determine the explicit BRDF prior to scanning. To do this, a different surface is required that has the same or a very similar BRDF, of which the actual geometry (or at least the normal vectors for many points on the surface) is already known. The lights are then individually shone upon the known
669:
surfaces. Some techniques have been developed to model (almost) general BRDFs. In practice, all of these require many light sources to obtain reliable data. These are methods in which surfaces with general BRDFs can be measured.
768:
644:. Computer vision followed a similar course with photometric stereo. Specular reflections were among the first deviations from the Lambertian model. These are a few adaptations that have been developed.
652:. The reflected light intensities towards the camera is measured, and the inverse reflectance function is fit onto the measured intensities, resulting in a unique solution for the normal vector.
625:, with perfectly diffuse reflection. This is unrealistic for many types of materials, especially metals, glass and smooth plastics, and will lead to aberrations in the resulting normal vectors.
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Many techniques ultimately rely on modelling the reflectance function of the surface, that is, how much light is reflected in each direction. This reflectance function has to be
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780:
B. K. P. Horn, 1989. Obtaining shape from shading information. In B. K. P. Horn and M. J. Brooks, eds., Shape from
Shading, pages 121–171. MIT Press.
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688:
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surface finishes. Current research aims to make the method work in the presence of projected shadows, highlights, and non-uniform lighting.
665:(BRDF) model, a surface may distribute the amount of light it receives in any outward direction. This is the most general known model for
228:
This model can easily be extended to surfaces with non-uniform albedo, while keeping the problem linear. Taking an albedo reflectivity of
41:
Photometric stereo analyzes multiple images of an object under different lighting conditions to estimate a normal direction at each pixel.
838:
890:
903:
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The technique was originally introduced by
Woodham in 1980. The special case where the data is a single image is known as
17:
864:
640:, the commonly used model to render surfaces started with Lambertian surfaces and progressed first to include simple
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and the resulting light directions, these cones can be approximated and therefore the normal vectors of the surface.
851:
695:. However, such methods are still fairly restrictive in photometric stereo. Better results have been achieved with
790:
744:
447:
841:. In IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. PAMI-3, issue 6, pages 661-669. IEEE.
793:. In IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 25, issue 10, pages 1239-1252. IEEE.
791:
The 4-source photometric stereo technique for 3-dimensional surfaces in the presence of highlights and shadows
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is not square (there are more than 3 lights), a generalisation of the inverse can be obtained using the
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Many methods have been developed to lift this assumption. In this section, a few of these are listed.
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31:
839:
Determining
Surface Orientations of Specular Surfaces by Using the Photometric Stereo Method
825:
453:
893:. In 2011 IEEE Conference on Computer Vision and Pattern Recognition, pages 689-696. IEEE.
854:. In IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 27, no. 8. IEEE.
828:. In IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 13, no. 2. IEEE.
8:
932:
713:
641:
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877:
386:
906:. In IEEE Conference on Computer Vision and Pattern Recognition, 2007, pages 1-8. IEEE.
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Some progress has been made towards modelling an even more general surfaces, such as
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is square (there are exactly 3 lights) and non-singular, it can be inverted, giving:
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Example-Based
Photometric Stereo: Shape Reconstruction with General, Verying BRDFs
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Interreflection
Removal for Photometric Stereo by Using Spectrum-dependent Albedo
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53:
49:
880:. In International Journal of Computer Vision, vol. 6, number 3, pages 173-195.
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826:
A Theory of
Photometric Stereo for a Class of Diffuse Non-Lambertian Surfaces
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Photometric method for determining surface orientation from multiple images
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After which the normal vector and albedo can be solved as described above.
56:
of objects by observing that object under different lighting conditions (
863:
Michael
Holroyd, Jason Lawrence, Greg Humphreys and Todd Zickler, 2008.
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904:
Polarization and Phase-shifting for 3D Scanning of
Translucent Objects
37:
902:
Tongbo Chen, Hendrik Lensch, Christian Fuchs and H.P. Seidel, 2007.
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Bidirectional surface scattering reflectance distribution functions
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The classical photometric stereo problem concerns itself only with
84:
87:— the problem can be solved by inverting the linear equation
802:
867:. In ACM SIGGRAPH Asia 2008 Papers, pages 133:1-133:9. ACM.
865:
A Photometric
Approach for Estimating Normals and Tangents
876:
Shree K. Nayar, Katsushi
Ikeuchi and Takeo Kanade, 1991.
248:, the formula for the reflected light intensity becomes:
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Spatially Varying Bidirectional Distribution Functions
83:, known point-like distant light sources, and uniform
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824:Hemant D. Tagare and Rui J.P. de Figueiredo, 1991.
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426:is the normalised direction of that vector. If
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850:Aaron Hertzmann and Steven M. Seitz, 2005.
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761:
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745:"Radiometry, BRDF and Photometric Stereo"
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450:, by simply multiplying both sides with
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771:. Optical Engineerings 19, I, 139-144.
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534:{\displaystyle L^{T}I=L^{T}k(L\cdot n)}
225:matrix of normalized light directions.
79:Under Woodham's original assumptions —
14:
915:
603:{\displaystyle (L^{T}L)^{-1}L^{T}I=kn}
179:is the (unknown) surface normal, and
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24:
803:Chaman Singh Verma and Mon-Ju Wu.
789:S. Barsky and Maria Petrou, 2003.
25:
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807:. University of Wisconsin-Madison
383:must be the length of the vector
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691:(BSSRDF), and accounting for
285:{\displaystyle I=k(L\cdot n)}
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878:Shape from interreflections
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448:Moore–Penrose pseudoinverse
10:
949:
353:{\displaystyle L^{-1}I=kn}
112:{\displaystyle I=L\cdot n}
29:
750:. Northwestern University
218:{\displaystyle 3\times m}
837:Katsushi Ikeuchi, 1981.
657:General BRDFs and beyond
30:Not to be confused with
617:Non-Lambertian surfaces
139:is a (known) vector of
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32:Stereo photogrammetry
805:"Photometric Stereo"
767:Woodham, R.J. 1980.
642:specular reflections
632:Specular reflections
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27:3D imaging technique
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399:{\displaystyle kn}
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65:shape from shading
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46:Photometric stereo
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18:Shape from shading
923:Computer graphics
661:According to the
638:computer graphics
636:Historically, in
439:{\displaystyle L}
419:{\displaystyle n}
376:{\displaystyle k}
308:{\displaystyle L}
241:{\displaystyle k}
192:{\displaystyle L}
172:{\displaystyle n}
152:{\displaystyle m}
132:{\displaystyle I}
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75:Basic Method
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933:3D imaging
917:Categories
811:2015-03-24
754:2015-03-25
730:References
724:3D scanner
714:Photometry
687:(SVBRDF),
650:invertible
69:Lambertian
58:photometry
743:Ying Wu.
571:−
523:⋅
331:−
274:⋅
210:×
104:⋅
703:See also
477:giving:
119:, where
667:opaque
406:, and
85:albedo
748:(PDF)
295:If
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