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Instability of the eikonal equation and shape from shading. (English) Zbl 0973.35017
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 solution, the authors show that this problem is unstable, which is catastrophic for general numerical algorithms. A detailed analysis is included and the paper exhibits the results in several figures with resolution to \(60\times 40\). An excellent paper breaking new ground in computer image analysis.

MSC:
35A35 Theoretical approximation in context of PDEs
68U10 Computing methodologies for image processing
68T45 Machine vision and scene understanding
Software:
Geomview
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