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A uniqueness result in the theory of stereo vision: Coupling shape from shading and binocular information allows unambiguous depth reconstruction. (English) Zbl 0837.35146
Summary: We study here the mathematical consistency of coupling two classical methods in the theory of vision and surface reconstruction, namely the shape from shading theory and the theory of sterio vision. It is known that each of these approaches by itself is incomplete and leads to ill posed problems and multiple solutions, even under drastic simplifying assumptions.
We show in this paper that, from a mathematical point of view, these ambiguities disappear when both theories are cooperatively implemented. In section 2 we state our assumptions; then part 3 is devoted to the presentation of the binocular vision theory. Section 4 eventually studies, in the one- and two-dimensional case, how introducing the shape from shading tool leads to the uniqueness of the solution. In the annex (section 6), a few mathematical results are explained, and some experiments (in 1D) are presented.

##### MSC:
 35R25 Ill-posed problems for PDEs 35R30 Inverse problems for PDEs 78A05 Geometric optics 68U10 Computing methodologies for image processing
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##### References:
 [1] Crandall, M. G.; Ishii, M.; Lions, P. L., Uniqueness of viscosity solutions of Hamilton equations, revised, J. Math. Soc. Japan, Vol. 39, 4, (1987) [2] A. Gennert, Brightness-Based Stereo Matching, in 2nd International Conference on Computer Vision, 1988, pp. 139-143. [3] Grimson, W. E.L., A computer implementation of a theory of human stereo vision, Philosophical Transactions of the Royal Society of London, Vol. 292, 1058, (May 1981) [4] Horn, K. P., Robot vision, (1986), MIT Press Cambridge MA [5] Lions, P.-L., Generalized solutions of Hamilton-Jacobi equations, (1982), Pitman [6] Lions, P.-L., On the Hamilton-Jacobi-Bellman equations, Acta Applicandae Mathematicae, Vol. 1, 17-41, (1983) · Zbl 0594.93069 [7] March, R., Computation of stereo disparity using regularization, Pattern Recognition Letters, Vol. 8, 181-187, (1988) [8] Marr, D., Vision, (1982), W. H. Freeman [9] A. Pentland, Shape Information from Shading: a Theory About Human Perception, in: 2nd International Conference on Computer Vision, 1988, pp. 404-413. [10] Rouy, E.; Tourin, A., A viscosity solution approach to shape from shading, SIAM J. Numer. Anal, Vol. 29, 3, 867-884, (June 1992) [11] H. Takahashi and F. Tomita, Self-Calibration of Stereo Cameras, in: 2nd International Conference on Computer Vision, 1988, pp. 123-128. [12] J. Weng, N. Ahuja and T. S. Huang, Two-View Matching, in: 2nd international Conference on Computer Vision, 1988, pp. 64-73.
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