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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.

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