Optimal approximations by piecewise smooth functions and associated variational problems. (English) Zbl 0691.49036

Mumford, David; Shah, Jayant

Commun. Pure Appl. Math. 42, No. 5, 577-685 (1989).

This paper treats the following variational problem: given a function f on a domain D in n-space, find a codimension 1 set S and a function g which is allowed to be discontinuous across S, which minimize a weighted sum of a) the \(L^ 2\)-norm of (f-g), b) the \(L^ 2\)-norm of grad(g) on D-S and c) the n-1-dimensional volume of S. The problem arose in computer vision, where \(n=2\), f is the measured intensity of light coming from a direction x,y, S is the set of ‘edges’ in the perceived scene, i.e. places where there is a discontinuity between the objects producing the scene and g is a ‘cartoon’ simplified signal. The paper derives the Euler equations for this problem, discusses the singularities on S and proves that a solution exists in the limiting case where term b) dominates the others, forcing g to be piecewise constant.

Reviewer: D.Mumford

MSC:

49Q15	Geometric measure and integration theory, integral and normal currents in optimization
49Q20	Variational problems in a geometric measure-theoretic setting
49M15	Newton-type methods

Keywords:

computer vision

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