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Shape from Shading

  1. Ariel Tankus1,
  2. Nir Sochen2,
  3. Yehezkel Yeshurun3

Published Online: 16 MAR 2009

DOI: 10.1002/9780470050118.ecse628

Wiley Encyclopedia of Computer Science and Engineering

Wiley Encyclopedia of Computer Science and Engineering

How to Cite

Tankus, A., Sochen, N. and Yeshurun, Y. 2009. Shape from Shading. Wiley Encyclopedia of Computer Science and Engineering. 2511–2522.

Author Information

  1. 1

    University of California, Division of Neurosurgery, Los Angeles, California

  2. 2

    Tel-Aviv University, School of Mathematics, Tel-Aviv, Israel

  3. 3

    Tel-Aviv University, School of Computer Science, Tel-Aviv, Israel

Publication History

  1. Published Online: 16 MAR 2009


Shape-from-shading (SfS) is a fundamental problem in computer vision. Its goal is reconstruction of surface depth (i.e., distance from camera plane) based on a single image of the surface. The problem was introduced in the early 1970s by Horn. A very common assumption in this field is that image projection is orthographic. We will present the orthographic shape-from-shading problem and an algorithm for its solution: the fast marching method of Kimmel and Sethian. We shall than reexamine the basis of SfS, which is the image irradiance equation, under a perspective projection assumption. The resultant equation does not depend on the depth function directly, but on its natural logarithm, and as such it is invariant to scale changes of the depth function. A reconstruction method based on the perspective formula is then described; it is a modification of the aforementioned orthographic fast marching method. Then, a comparison of the orthographic fast marching, perspective fast marching, and the perspective algorithm of Prados and Faugeras on synthetic images is are presented. The two perspective methods equate with each other and show better reconstruction results than the orthographic. We then compare the orthographic and perspective versions of the fast marching method on endoscopic images. The perspective algorithm outperformed the orthographic one. These findings suggest that the more realistic set of assumptions of perspective SfS improves reconstruction significantly with respect to orthographic SfS. The findings also provide evidence that perspective SfS can be used for real-life applications in fields such as endoscopy.


  • computer vision;
  • shape from shading;
  • shape from X;
  • perspective projection;
  • orthographic projection;
  • fast marching;
  • surface reconstruction;
  • depth recovery