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Instability of the eikonal equation and shape from shading

Published online by Cambridge University Press:  15 April 2002

Ian Barnes  and
Kewei Zhang
Ian Barnes
Affiliation:
Department of Mathematics, Macquarie University, Sydney NSW 2109, Australia. (barnes@maths.mq.edu.au)
Kewei Zhang
Affiliation:
Department of Mathematics, Macquarie University, Sydney NSW 2109, Australia. (kewei@maths.mq.edu.au)
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Abstract

In the shape from shading problem of computer vision one attempts to recover the three-dimensional shape of an object or landscape from the shading on a single image. Under the assumptions that the surface is dusty, distant, and illuminated only from above, the problem reduces to that of solving the eikonal equation |Du|=f on a domain in $\mathbb{R}^2$. Despite various existence and uniqueness theorems for smooth solutions, we show that this problem is unstable, which is catastrophic for general numerical algorithms.

Keywords

Eikonal equationshape from shadinginstabilitynumerical analysis.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 34 , Issue 1 , January 2000 , pp. 127 - 138
Copyright
© EDP Sciences, SMAI, 2000

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