Uniqueness in Shape from Shading Revisited

We analyse the problem of representing solutions of first-orderpartial differential equations in terms of complete integrals and envelopes. In this context, we revisit the uniqueness results alreadyexisting in the shape-from-shading literature that concern eikonalequations corresponding to the images of a Lambertian hemi-sphere and aLambertian plane. We show that the approach adopted by Brooks in [2, 3] isincomplete and subsequently re-establish its uniqueness claims.

[1]  P. Dupuis,et al.  An Optimal Control Formulation and Related Numerical Methods for a Problem in Shape Reconstruction , 1994 .

[2]  Wojciech Chojnacki.  A note on complete integrals , 1995 .

[3]  Wojciech Chojnacki,et al.  Impossible and ambiguous shading patterns , 2004, International Journal of Computer Vision.

[4]  Ryszard Kozera.  A note on existence and uniqueness in shape from shading , 1993, 1993 (4th) International Conference on Computer Vision.

[5]  R. Courant,et al.  Methods of Mathematical Physics , 1962 .

[6]  Michael J. Brooks.  Two Results Concerning Ambiguity in Shape From Shading , 1983, AAAI.

[7]  Wojciech Chojnacki,et al.  Shading without shape , 1992 .

[8]  Anna R. Bruss.  The Eikonal equation: some results applicable to computer vision , 1982 .

[9]  Ryszard Kozera.  On complete integrals and uniqueness in shape from shading , 1995 .

[10]  Ian N. Sneddon.  Elements Of Partial Differential Equations , 1957 .

[11]  G. M.,et al.  Partial Differential Equations I , 2023, Applied Mathematical Sciences.

[12]  P. Deift,et al.  Some remarks on the shape-from-shading problem in computer vision☆ , 1981 .

[13]  Alfred M. Bruckstein,et al.  Integrability disambiguates surface recovery in two-image photometric stereo , 1992, International Journal of Computer Vision.

[14]  Alberto Dou,et al.  Lectures on partial differential equations of first order , 1972 .

[15]  Wojciech Chojnacki,et al.  Direct computation of shape from shading , 1994, Proceedings of 12th International Conference on Pattern Recognition.

[16]  W. Klingenberg.  A course in differential geometry , 1978 .

[17]  John Oliensis,et al.  Uniqueness in shape from shading , 1991, International Journal of Computer Vision.

[18]  R. K. Luneburg,et al.  Mathematical Theory of Optics , 1966 .

[19]  R. J. Bell,et al.  An Elementary Treatise on Coordinate Geometry of Three Dimensions , 2009, Nature.

[20]  E. Rouy,et al.  A viscosity solutions approach to shape-from-shading , 1992 .

[21]  A. P. S. Selvadurai,et al.  Partial differential equations of the first-order , 2000 .

[22]  Ryszard Kozera.  On Shape Recovery from Two Shading Patterns , 1992, Int. J. Pattern Recognit. Artif. Intell..

[23]  Wojciech Chojnacki,et al.  Circularly symmetric eikonal equations and non-uniqueness in computer vision , 1992 .

[24]  P. Klingenberg.  Advances in Biochemical Engineering. Herausgegeben von T. K. Ghose, A. Fiechter und N. Blakebrough. Bd. 4, 1972 Seiten, zahlr. Abb. Springer‐Verlag, Berlin, Heidelberg, New York 1976. Preis: 60,– DM , 1978 .

[25]  Robert J. Woodham,et al.  Photometric method for determining surface orientation from multiple images , 1980 .

[26]  R. Kozera.  Existence and uniqueness in photometric stereo , 1991 .

[27]  Berthold K. P. Horn.  Robot vision , 1986, MIT electrical engineering and computer science series.