Instability of the eikonal equation and shape from shading

• Published in: ESAIM Mathematical Modelling and Numerical Analysis Vol. 34; no. 1; pp. 127 - 138
• Main Authors: Barnes, Ian, Zhang, Kewei
• Format: Journal Article
• Language: English
• Published: Les Ulis EDP Sciences 01.01.2000
• Subjects: 26A16; 54C20; 57M05; 57N05; Eikonal equation; Exact sciences and technology; General topology; instability; Manifolds and cell complexes; Mathematical analysis; Mathematics; numerical analysis; Partial differential equations; Real functions; Sciences and techniques of general use; shape from shading; Topology. Manifolds and cell complexes. Global analysis and analysis on manifolds; Computer vision; Existence theorem; Sobolev space; Lebesgue space; Shading; Partial differential equation; Three dimensional model; Lambert surface; Numerical analysis; Lipschitz boundary; Eikonal equation; Instability; Eikonal approximation; Domain decomposition
• ISSN: 0764-583X; 1290-3841; 1290-3841
• DOI: 10.1051/m2an:2000134

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.