Abstract
In this paper, we present a symmetric shape-from-shading (SFS) approach to recover both shape and albedo for symmetric objects. Lambertian surfaces with unknown varying albedo and orthographic projections are assumed. In our formulation of symmetric SFS, we have two image irradiance equations. One is the standard equation used in SFS, and the other is a self-ratio image irradiance equation. This new image irradiance equation relates the self-ratio image which is defined as the ratio of two-halves of the input image to light source and surface shape. The introduction of the self-ratio image facilitates the direct use of symmetry cue. Based on the self-ratio image, a new model-based symmetric source-from-shading algorithm is also presented. We then propose symmetric SFS algorithms to recover both shape and albedo from a single image and present experimental results.
The new symmetric SFS scheme has one important property: the existence of a unique (global) solution which consists of unique (local) solutions at each point simultaneously obtained using the intensity information at that point and the surrounding local region and the assumption of a C 2 surface. Proofs for the existence of a unique solution in the cases of unknown constant and non-constant albedos are provided.
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References
Atick, J., Griffin, P., and Redlich, N. 1996. Statistical approach to shape from shading: Reconstruction of three-dimensional face surfaces from single two-dimensional images. Neural Computation, 8:1321-1340.
Bertero, M., Poggio, T., and Torre, V. 1987. Ill-posed problems in early vision. MIT Artificial Intelligence Laboratory, Technical Report 924.
Bichsel, M. and Pentland, A. 1992. A simple algorithm for shape from shading. In Proc. Conference on Computer Vision and Pattern Recognition, Urbana/Champaign, IL, pp. 459-465.
Bruss, A.R. 1982. The Eikonal equation: Some results applicable to computer vision. Journal of Mathmetical Physics, 23:890-896.
Dupuis, P. and Oliensis, J. 1992. Direct method for reconstructing shape from shading. In Proc. Conference on Computer Vision and Pattern Recognition, Urbana/Champaign, IL, pp. 453-458.
Frankot, R.T. and Chellappa, R. 1988. A method for enforcing integrability in shape from shading algorithms. IEEE Trans. on PAMI, 10:439-451.
Horn, B.K.P. 1970. Shape from shading: A method for obtaining the shape of a smooth opaque object from one view. PhD Thesis, Massachusetts Institute of Technology.
Horn, B.K.P. 1990. Height and gradient from shading. Int. Journal of Computer Vision, 5:37-75.
Horn, B.K.P. and Brooks, M.J. 1989. Shape from Shading. MIT Press: Cambridge, MA.
Ikeuchi, K. and Horn, B.K.P. 1981. Numerical shape from shading and occluding boundaries. Artificial Intelligence, 17:141-184.
Jacobs, D.W., Belhumeur, P.N., and Basri, R. 1998. Comparing images under variable illumination. In Proc. Conference on Computer Vision and Pattern Recognition, Santa Barbara, CA, pp. 610-617.
Lee, C.H. and Rosenfeld, A. 1989. Improved methods of estimating shape from shading using the light source coordinate system. In Shape from Shading, B.K.P. Horn and M.J. Brooks (Eds.). MIT Press: Cambridge, MA, pp. 323-569.
Nayar, S.K., Ikeuchi, K., and Kanade, T. 1991. Surface reflection: Physical and geometrical perspectives. IEEE Trans. on PAMI, 13:611-634.
Oliensis, J. 1991. Uniqueness in shape from shading. Int. Journal of Computer Vision, 6:75-104.
Onn, R. and Bruckstein, A. 1990. Integrability disambiguates surface recovery in two-image photometric stereo. Int. Journal of Computer Vision, 5:105-113.
Pentland, A.P. 1982. Finding the illumination directions. Journal of the Optical Society of America A, 72:448-455.
Riklin-Raviv, T. and Shashua, A. 1999. The quotient image: Class based rerendering and recognition with varying illuminations. In Proc. of Computer Vision and Pattern Recognition, Fort Collins, CO, pp. 566-571.
Saxberg, B.V.H. 1989. A modern differential geometric approach to shape from shading. MIT Artificial Intelligence Laboratory, Technical Report 1117.
Shimoshoni, I., Moses Y., and Lindenbaum, M. 2000. Shape reconstruction of 3D bilaterally symmetric surfaces. Int. Journal of Computer Vision, 39:97-112.
Tsai, P.S. and Shah, M. 1994. Shape from shading using linear approximation. Journal of Image and Vision Computing, 12:487-498.
Wei, G.Q. and Hirzinger, G. 1997. Parametric shape-from-shading by radial basis functions. IEEE Trans. on PAMI, 19:353-365.
Wolff, L.B. and Angelopoulou, E. 1994. 3-D stereo using photometric ratios. In Proc. European Conference on Computer Vision, Springer: Berlin, pp. 247-258.
Woodham, R. 1980. Photometric method for determining surface orientation from multiple images. Optical Engineering, 19:139-144.
Zhang, R., Tsai, P.S., Cryer, J.E., and Shah, M. 1999. IEEE Trans. on PAMI, 21:690-706.
Zhao, W. 1999. Robust image based 3D face recognition. PhD Thesis, University of Maryland.
Zhao W. and Chellappa, R. 2000. Illumination-insensitive face recognition using symmetric shape-from-shading. In Proc. Conference on Computer Vision and Pattern Recognition, SC, pp. 286-293.
Zheng, Q. and Chellappa, R. 1991. Estimation of illumination direction, Albedo, and shape from shading. IEEE Trans. on PAMI, 13:680-702.
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Zhao, W.Y., Chellappa, R. Symmetric Shape-from-Shading Using Self-ratio Image. International Journal of Computer Vision 45, 55–75 (2001). https://doi.org/10.1023/A:1012369907247
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DOI: https://doi.org/10.1023/A:1012369907247