# zbMATH — the first resource for mathematics

Instability of the eikonal equation and shape from shading. (English) Zbl 0973.35017
In the shape from shading problem of computer vision one attempts to recover the three-dimensional shape of an object or landscape from the shading on a single image. Under the assumptions that the surface is dusty, distant, and illuminated only from above, the problem reduces to that of solving the eikonal equation $$|Du|= f$$ on a domain in $$\mathbb{R}^2$$. Despite various existence and uniqueness theorems for smooth solution, the authors show that this problem is unstable, which is catastrophic for general numerical algorithms. A detailed analysis is included and the paper exhibits the results in several figures with resolution to $$60\times 40$$. An excellent paper breaking new ground in computer image analysis.

##### MSC:
 35A35 Theoretical approximation in context of PDEs 68U10 Computing methodologies for image processing 68T45 Machine vision and scene understanding
##### Keywords:
instability; numerical analysis; computer image analysis
Geomview
Full Text:
##### References:
 [1] R.A. Adams, Sobolev Space. Academic Press New York (1975). [2] J.M. Ball, A version of the fundamental theorem for Young measure. Lect. Notes Phys. Springer Verlag 344 (1988) 207-215. Zbl0991.49500 · Zbl 0991.49500 [3] A.R. Bruss, results applicable to computer vision. J. Math. Phys.23 (1982) 890-896. Zbl0502.35079 · Zbl 0502.35079 · doi:10.1063/1.525441 [4] J. Chabrowski and K.-W. Zhang, On shape from shading problem Functional Analysis, Approximation Theory and Numerical Analysis, J.M. Rassias Ed., World Scientific (1994) 93-105. Zbl0878.35026 · Zbl 0878.35026 [5] B. Dacorogna Direct Methods in the Calculus of Variations. Springer-Verlag (1989). [6] P. Deift and J. Sylvester, Some remarks on the shape-from-shading problem in computer vision. J. Math. Anal. Appl.84 (1981) 235-248. · Zbl 0485.35081 · doi:10.1016/0022-247X(81)90161-X [7] L.C. Evans and R.F. Gariepy, Measure Theory and Fine Properties of Functions. Stud. in Adv. Math. CRC Press, Boca Raton (1992). · Zbl 0804.28001 [8] I. Ekeland and R. Temam, Analyse convexe et problèmes variationnels. Dunod Paris (1974). · Zbl 0281.49001 [9] L. Gritz, Blue Moon Rendering Tools: Ray tracing software available from ftp://ftp.gwu.edu/pub/graphics/BMRT (1995). [10] B.K.P. Horn, Robot Vision. Engineering and Computer Science Series, MIT Press, MacGraw Hill (1986). [11] B.K.P. Horn and M.J. Brooks, Shape from Shading. Ed. MIT Press Ser. in Artificial Intelligence (1989). · Zbl 0629.65125 [12] B.K.P. Horn and M.J. Brooks, Variational Approach to Shape from Shading in [11] · Zbl 0629.65125 [13] S. Levy, T. Munzner and M. Phillips, Geomview Visualisation software available from ftp.geom.umn.edu or http://www.geom.umn.edu/locate/geomview [14] P.-L. Lions, E. Rouy and A. Tourin, Shape-from-shading, viscosity solutions and edges. Numer. Math.64 (1993) 323-353. Zbl0804.68160 · Zbl 0804.68160 · doi:10.1007/BF01388692 · eudml:133708 [15] Pixar, The RenderMan Interface, version 3.1, official specification. Pixar (1989) [16] M. Phillips, S. Levy and T. Munzner, Geomview: An Interactive Geometry Viewer. Notices Amer. Math. Soc.40 (1993) 985-988. [17] E. Rouy and A. Tourin, A viscosity solution approach to shape-from-shading. SIAM J. Numer. Anal.29 (1992) 867-884. Zbl0754.65069 · Zbl 0754.65069 · doi:10.1137/0729053 [18] E.M. Stein, Singular Integrals and Differentiability Properties of Functions. Princeton University Press (1970). Zbl0207.13501 · Zbl 0207.13501 [19] L. Tartar, Compensated compactness and partial differential equations, in Microstructure and Phase Transitions, D. Kinderlehrer et al. Eds., Springer Verlag (1992). · Zbl 0897.35010
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.