Optimal Algorithm for Shape from Shading and Path Planning

  • Authors:

  • Ron Kimmel;James A. Sethian

  • Affiliations:

  • Department of Computer Science, Technion, Israel Institute of Technology, Haifa 32000, Israel. ron@cs.technion.ac.il;Department of Mathematics and Lawrence Berkeley National Laboratory, University of California, Berkeley, CA 94720, USA. sethian@math.berkeley.edu

  • Venue:
  • Journal of Mathematical Imaging and Vision
  • Year:
  • 2001

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Abstract

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.