ECCV 2006: Computer Vision – ECCV 2006 pp 239-250

# Integrating Surface Normal Vectors Using Fast Marching Method

• Jeffrey Ho
• Jongwoo Lim
• Ming-Hsuan Yang
• David Kriegman
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3953)

## Abstract

Integration of surface normal vectors is a vital component in many shape reconstruction algorithms that require integrating surface normals to produce their final outputs, the depth values. In this paper, we introduce a fast and efficient method for computing the depth values from surface normal vectors. The method is based on solving the Eikonal equation using Fast Marching Method. We introduce two ideas. First, while it is not possible to solve for the depths Z directly using Fast Marching Method, we solve the Eikonal equation for a function W of the form W = Z + λf. With appropriately chosen values for λ, we can ensure that the Eikonal equation for W can be solved using Fast Marching Method. Second, we solve for W in two stages with two different λ values, first in a small neighborhood of the given initial point with large λ, and then for the rest of the domain with a smaller λ. This step is needed because of the finite machine precision and rounding-off errors. The proposed method is very easy to implement, and we demonstrate experimentally that, with insignificant loss in precision, our method is considerably faster than the usual optimization method that uses conjugate gradient to minimize an error function.

## References

1. 1.
Georghiades, A., Kriegman, D., Belhumeur, P.: From few to many: Generative models for recognition under variable pose and illumination. IEEE Transactions on Pattern Analysis and Machine Intelligence 23, 643–660 (2001)
2. 2.
Zhang, L., Curless, B., Hertzmann, A., Seitz, S.: Shape and motion under varying illumination: Unifying structure from motion, photometric stereo, and multi-view stereo. In: Proc. Int. Conf. on Computer Vision, pp. 618–625 (2003)Google Scholar
3. 3.
Horn, B.K.P., Brook, M.J.: Shape from Shading. MIT Press, Cambridge (1997)Google Scholar
4. 4.
Frankot, R., Chellappa, R.: A method of enforcing integrability in shape from shading algorithms. IEEE Transactions on Pattern Analysis and Machine Intelligence 10, 439–451 (1988)
5. 5.
Sethian, J.A.: Level Set Methods and Fast Marching Methods. Cambridge Press, Cambridge (1996)Google Scholar
6. 6.
Prado, E., Faugeras, O.: Shape from shading: A well-posed problem? In: Proc. IEEE Conf. on Comp. Vision and Patt. Recog., pp. 158–164 (2005)Google Scholar
7. 7.
Tankus, A., Sochen, N., Yeshurun, H.: Perspective shape from shading by fast marching. In: Proc. IEEE Conf. on Comp. Vision and Patt. Recog., pp. 618–625 (2004)Google Scholar
8. 8.
Trucco, E., Verri, A.: Introductory Techniques for 3D Computer Vision. Prentice-Hall, Englewood Cliffs (1998)Google Scholar
9. 9.
Press, W.H., Teukolsky, S.A., Vetterling, W.T., Flannery, B.P.: Numerical Recipes in C, 2nd edn. Cambridge Press (1992)Google Scholar
10. 10.
Yuille, A., Snow, D.: Shape and albedo from multiple images using integrability. In: Proc. IEEE Conf. on Comp. Vision and Patt. Recog., pp. 158–164 (1997)Google Scholar

## Authors and Affiliations

• Jeffrey Ho
• 1
• Jongwoo Lim
• 2
• Ming-Hsuan Yang
• 2
• David Kriegman
• 3
1. 1.Department of CISEUniversity of FloridaGainesvilleUSA
2. 2.Honda Research InstituteMountain ViewUSA
3. 3.Department of Computer Science and EngineeringUniversity of CaliforniaSan DiegoUSA