Photometric stereo

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Photometric stereo is a technique in computer vision for estimating the surface normals of objects by observing that object under different lighting conditions. 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.[1] 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.

The technique was originally introduced by Woodham in 1980.[2] The special case where the data is a single image is known as shape from shading, and was analyzed by B. K. P. Horn in 1989.[3]

Methods[edit]

Under Woodham's original assumptions — Lambertian reflectance, known point-like distant light sources, and uniform albedo — the problem can be solved by inverting the linear equation I = n \cdot L, where I is a (known) vector of m observed intensities, n is the (unknown) surface normal, and L is a (known) 3 \times m matrix of normalized light directions.

This model can easily be extended to surfaces with non-uniform albedo, while keeping the problem linear.[4] Taking an albedo reflectivity of k, the formula for the reflected light intensity becomes:

I = k (n \cdot L)

If L is square (there are exactly 3 lights) and non-singular, it can be inverted, giving:

k n = L^{-1} I

Since the normal vector is known to have length 1, k must be the length of the vector k n, and n is the normalised direction of that vector. If L is not square (there are more than 3 lights), a generalisation of the inverse can be obtained using the Moore-Penrose pseudoinverse,[5] giving:

k n = (L^T L)^{-1} L^T I

After which the normal vector and albedo can be solved as described above.

Photometric stereo has since been generalized to many other situations, including extended light sources and non-Lambertian surface finishes.[6] Current research aims to make the method work in the presence of projected shadows, highlights, and non-uniform lighting. Surface normals define the local metric, using this observation Bronstein et al. [7] defined a 3D face recognition system based on the reconstructed metric without integrating the surface. The metric of the facial surface is known to be robust to expressions.

See also[edit]

References[edit]

  1. ^ Ying Wu. "Radiometry, BRDF and Photometric Stereo". Northwestern University. Retrieved 2015-03-25. 
  2. ^ Woodham, R.J. 1980. Photometric method for determining surface orientation from multiple images. Optical Engineerings 19, I, 139-144.
  3. ^ 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.
  4. ^ S. Barsky and Maria Petrou, 2003. The 4-source photometric stereo technique for 3-dimensional surfaces in the presence of highlights and shadows. In IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 25, issue 10, pages 1239-1252. IEEE.
  5. ^ Chaman Singh Verma and Mon-Ju Wu. "Photometric Stereo". University of Wisconsin-Madison. Retrieved 2015-03-24. 
  6. ^ "A Photometric Stereo Approach to Face Recognition". University of the West of England. Retrieved 2011-03-27. 
  7. ^ A.M. Bronstein, M.M. Bronstein, A. Spira, R. Kimmel. "Face recognition from facial surface metric". European Conf. Computer Vision-ECCV 2004. Retrieved 2004.