# Numerical algorithms for perspective shape from shading

Kybernetika (2010)

• Volume: 46, Issue: 2, page 207-225
• ISSN: 0023-5954

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## Abstract

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The Shape-From-Shading (SFS) problem is a fundamental and classic problem in computer vision. It amounts to compute the 3-D depth of objects in a single given 2-D image. This is done by exploiting information about the illumination and the image brightness. We deal with a recent model for Perspective SFS (PSFS) for Lambertian surfaces. It is defined by a Hamilton–Jacobi equation and complemented by state constraints boundary conditions. In this paper we investigate and compare three state-of-the-art numerical approaches. We begin with a presentation of the methods. Then we discuss the use of some acceleration techniques, including cascading multigrid, for all the tested algorithms. The main goal of our paper is to analyze and compare recent solvers for the PSFS problem proposed in the literature.

## How to cite

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Breuss, Michael, et al. "Numerical algorithms for perspective shape from shading." Kybernetika 46.2 (2010): 207-225. <http://eudml.org/doc/196399>.

@article{Breuss2010,
abstract = {The Shape-From-Shading (SFS) problem is a fundamental and classic problem in computer vision. It amounts to compute the 3-D depth of objects in a single given 2-D image. This is done by exploiting information about the illumination and the image brightness. We deal with a recent model for Perspective SFS (PSFS) for Lambertian surfaces. It is defined by a Hamilton–Jacobi equation and complemented by state constraints boundary conditions. In this paper we investigate and compare three state-of-the-art numerical approaches. We begin with a presentation of the methods. Then we discuss the use of some acceleration techniques, including cascading multigrid, for all the tested algorithms. The main goal of our paper is to analyze and compare recent solvers for the PSFS problem proposed in the literature.},
author = {Breuss, Michael, Cristiani, Emiliano, Durou, Jean-Denis, Falcone, Maurizio, Vogel, Oliver},
journal = {Kybernetika},
keywords = {hyperbolic partial differential equation; Hamilton–Jacobi equation; finite difference method; semi-Lagrangian scheme; Shape-from-Shading; finite difference method; hyperbolic partial differential equation; Hamilton-Jacobi equation; semi-Lagrangian scheme; shape-from-shading},
language = {eng},
number = {2},
pages = {207-225},
publisher = {Institute of Information Theory and Automation AS CR},
title = {Numerical algorithms for perspective shape from shading},
url = {http://eudml.org/doc/196399},
volume = {46},
year = {2010},
}

TY - JOUR
AU - Breuss, Michael
AU - Cristiani, Emiliano
AU - Durou, Jean-Denis
AU - Falcone, Maurizio
AU - Vogel, Oliver
TI - Numerical algorithms for perspective shape from shading
JO - Kybernetika
PY - 2010
PB - Institute of Information Theory and Automation AS CR
VL - 46
IS - 2
SP - 207
EP - 225
AB - The Shape-From-Shading (SFS) problem is a fundamental and classic problem in computer vision. It amounts to compute the 3-D depth of objects in a single given 2-D image. This is done by exploiting information about the illumination and the image brightness. We deal with a recent model for Perspective SFS (PSFS) for Lambertian surfaces. It is defined by a Hamilton–Jacobi equation and complemented by state constraints boundary conditions. In this paper we investigate and compare three state-of-the-art numerical approaches. We begin with a presentation of the methods. Then we discuss the use of some acceleration techniques, including cascading multigrid, for all the tested algorithms. The main goal of our paper is to analyze and compare recent solvers for the PSFS problem proposed in the literature.
LA - eng
KW - hyperbolic partial differential equation; Hamilton–Jacobi equation; finite difference method; semi-Lagrangian scheme; Shape-from-Shading; finite difference method; hyperbolic partial differential equation; Hamilton-Jacobi equation; semi-Lagrangian scheme; shape-from-shading
UR - http://eudml.org/doc/196399
ER -

## References

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1. Bornemann, F., Deuflhard, P., Cascadic multigrid methods, In: Domain Decomposition Methods in Sciences and Engineering (R. Glowinski, J. Periaux, Z. Shi, and O. Widlund, eds.), John Wiley, New York 1997, pp. 205–212. MR1943461
2. Breuß, M., Vogel, O., Weickert, J., Efficient numerical techniques for perspective shape from shading, In: Proc. Algoritmy 2009, Podbanské 2009 (A. Handlovičová, P. Frolkovič, K. Mikula, and D. Ševcovič, eds.), Slovak University of Technology, Bratislava 2009, pp. 11–20.
3. Camilli, F., Prados, E., 10.1016/j.apnum.2006.03.007, Appl. Numer. Math. 56 (2006), 9, 1225–1237. Zbl1096.65059MR2244973DOI10.1016/j.apnum.2006.03.007
4. Courteille, F., Crouzil, A., Durou, J.-D, Gurdjos, P., Towards shape from shading under realistic photographic conditions, In: Proc. 17th Internat. Conf. Patt. Recog. (vol. II), Cambridge 2004, pp. 277–280.
5. Cristiani, E., Fast Marching and Semi-Lagrangian Methods for Hamilton–Jacobi Equations with Applications, Ph.D. Thesis, Dipartimento di Metodi e Modelli Matematici per le Scienze Applicate, Università di Roma “La Sapienza”, Rome 2007.
6. Cristiani, E., Falcone, M., Seghini, A., Numerical solution of the perspective Shape-from-Shading problem, In: Proc. Control Systems: Theory, Numerics and Applications, Rome 2005. Proceedings of Science (CSTNA2005) 008, http://pos.sissa.it/
7. Cristiani, E., Falcone, M., Seghini, A., Some remarks on perspective Shape-from-Shading models, In: Proc. 1st Internat. Conf. Scale Space and Variational Methods in Comput. Vis., Ischia 2007 (F. Sgallari, A. Murli, and N. Paragios, eds., Lecture Notes in Comput. Sci. 4485), Springer, Berlin 2008, pp. 276–287.
8. Durou, J.-D., Falcone, M., Sagona, M., 10.1016/j.cviu.2007.09.003, Comp. Vis. and Image Underst. 109 (2008), 1, 22–43. DOI10.1016/j.cviu.2007.09.003
9. Falcone, M., Ferretti, R., 10.1006/jcph.2001.6954, J. Comput. Phys. 175 (2002), 2, 559–575. Zbl1007.65060MR1880118DOI10.1006/jcph.2001.6954
10. Foley, J. D., Dam, A. van, Feiner, S. K., Hughes, J. F., Computer Graphics: Principles and Practice, Addison–Wesley, Reading 1996.
11. Hoppe, R. H. W., 10.1007/BF01389627, Numer. Math. 49 (1986), 2-3, 239–254. MR0848524DOI10.1007/BF01389627
12. Horn, B. K. P., Obtaining shape from shading information, In: The Psychology of Computer Vision (P. H. Winston, ed.), McGraw-Hill, New York 1975, Ch. 4, pp. 115–155. MR0416135
13. Horn, B. K. P., Robot Vision, MIT Press, Cambridge Mass. 1986.
14. Horn, B. K. P., Brooks, M. J., Shape from Shading, Artificial Intelligence Series, MIT Press, Cambridge Mass.1989. Zbl0629.65125MR1062877
15. Okatani, T., Deguchi, K., 10.1006/cviu.1997.0613, Comp. Vis. and Image Underst. 66 (1997), 2, 119–131. DOI10.1006/cviu.1997.0613
16. Prados, E., Faugeras, O., “Perspective Shape from Shading” and viscosity solutions, In: Proc. 9th IEEE Internat. Conf. Comp. Vis. (vol. II), Nice 2003, pp. 826–831.
17. Prados, E., Faugeras, O., Unifying approaches and removing unrealistic assumptions in Shape From Shading: Mathematics can help, In: Proc. 8th Eur. Conf. Comp. Vis. (vol. IV), Prague 2004, Lecture Notes in Comp. Sci.3024, pp. 141–154. Zbl1098.68844
18. Prados, E., Camilli, F., Faugeras, O., 10.1007/s10851-006-6899-x, J. Math. Imag. and Vis. 25 (2006), 3, 307–328. MR2283609DOI10.1007/s10851-006-6899-x
19. Prados, E., Camilli, F., Faugeras, O., 10.1051/m2an:2006018, M2AN Math. Model. Numer. Anal. 40 (2006), 2, 393–412. Zbl1112.49025MR2241829DOI10.1051/m2an:2006018
20. Rosenfeld, A., Multiresolution Image Processing and Analysis, Springer, Berlin 1984. Zbl0537.68086
21. Rouy, E., Tourin, A., 10.1137/0729053, SIAM J. Numer. Anal. 29 (1992), 3, 867–884. Zbl0754.65069MR1163361DOI10.1137/0729053
22. Tankus, A., Sochen, N., Yeshurun, Y., A new perspective [on] Shape-from-Shading, In: Proc. 9th IEEE Internat. Conf. Comp. Vis. (vol. II), Nice 2003, pp. 862–869.
23. Vogel, O., Breuß, M., Weickert, J., A direct numerical approach to perspective Shape-from-Shading, In: Proc. Vision, Modeling, and Visualization Workshop 2007, Saarbrücken 2007 (H. Lensch, B. Rosenhahn, H.-P. Seidel, P. Slusallek, and J. Weickert, eds.), pp. 91–100.
24. Zhang, R., Tsai, P.-S., Cryer, J. E., Shah, M., 10.1109/34.784284, IEEE Trans. Patt. Anal. Mach. Intell. 21 (1999), 8, 690–706. DOI10.1109/34.784284
25. Zhao, H., 10.1090/S0025-5718-04-01678-3, Math. Comp. 74 (2004), 250, 603–627. Zbl1070.65113MR2114640DOI10.1090/S0025-5718-04-01678-3

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