Journal of Mathematical Imaging and Vision

, Volume 14, Issue 3, pp 237–244 | Cite as

Optimal Algorithm for Shape from Shading and Path Planning

  • Ron Kimmel
  • James A. Sethian


An optimal algorithm for the reconstruction of a surface from its shading image is presented. The algorithm solves the 3D reconstruction from a single shading image problem. The shading image is treated as a penalty function and the height of the reconstructed surface is a weighted distance. A consistent numerical scheme based on Sethian's fast marching method is used to compute the reconstructed surface. The surface is a viscosity solution of an Eikonal equation for the vertical light source case. For the oblique light source case, the reconstructed surface is the viscosity solution to a different partial differential equation. A modification of the fast marching method yields a numerically consistent, computationally optimal, and practically fast algorithm for the classical shape from shading problem. Next, the fast marching method coupled with a back tracking via gradient descent along the reconstructed surface is shown to solve the path planning problem in robot navigation.

fast marching Eikonal equations shape from shading navigation 


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Copyright information

© Kluwer Academic Publishers 2001

Authors and Affiliations

  • Ron Kimmel
    • 1
  • James A. Sethian
    • 2
  1. 1.Department of Computer Science, TechnionIsrael Institute of TechnologyHaifaIsrael
  2. 2.Department of Mathematics and Lawrence Berkeley National LaboratoryUniversity of CaliforniaBerkeleyUSA

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