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Optimal approximations by piecewise smooth functions and associated variational problems. (English) Zbl 0691.49036
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 $L\sp 2$-norm of (f-g), b) the $L\sp 2$-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

49Q15Geometric measure and integration theory, integral and normal currents (optimization)
49Q20Variational problems in a geometric measure-theoretic setting
49M15Newton-type methods in calculus of variations
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