Instability of the eikonal equation and shape from shading
Department of Mathematics, Macquarie University,
Sydney NSW 2109, Australia. (firstname.lastname@example.org)
2 Department of Mathematics, Macquarie University, Sydney NSW 2109, Australia. (email@example.com)
Revised: 19 August 1999
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 . Despite various existence and uniqueness theorems for smooth solutions, we show that this problem is unstable, which is catastrophic for general numerical algorithms.
Mathematics Subject Classification: 57N05 / 57M05 / 26A16 / 54C20
Key words: Eikonal equation / shape from shading / instability / numerical analysis.
© EDP Sciences, SMAI, 2000