Optimal Approximations by Piecewise Smooth Functions and Associated Variational Problems - [ Mumford-Shah ]

"The purpose of this paper is to introduce and study the most basic properties of three new variational problems which are suggested by applications to computer vision. In computer vision, a fundamental problem is to appropriately decompose the domain R of a function g(x, y) of two variables. To explain this problem, we have to start by describing the physical situation whch produces images: assume that a three-dimensional world is observed by an eye or camera from some point P and that g l ( p ) represents the intensity of the light in this world approaching the point P from a direction p. If one has a lens at P focussing this light on a retina or a film-in both cases a plane domain R in which we may introduce coordinates x, y-then let g(x, y) be the strength of the light signal striking R at a point with coordinates ( x , y); g(x, y) is essentially the same as gl( p)-possibly after a simple transformation given by the geometry of the imaging system. The function g(x, y ) defined...

• Type: Journal
• Year: 1989
• Where: Communication on Pure and Applied Math
• Authors: David Mumford and Jayant Shah
• Keywords: Mumford Shah Segmentation