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 $$L^ 2$$-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

computer vision
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References:

 [1] Allard, Inventiones Math. 34 pp 83– (1976) [2] and , eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Mathematics Series, 1965. [3] Buser, Proc. of Symp. in Pure Math. 36 (1980) [4] and , Using Weak Continuity Constraints, Report CSR-186–85, Dept. of Comp. Sci., Edinburgh University, 1985. [5] Functional Integration and Partial Differential Equations, Annals of Math. Studies, Princeton University Press, 1985. · Zbl 0568.60057 [6] Geman, IEEE Trans., PAMI 6 pp 721– (1984) · Zbl 0573.62030 [7] Elliptic Problems in Nonsmooth Domains, Pitman, 1985. · Zbl 0695.35060 [8] Gurtin, Archive Rat. Mech. Anal. 87 pp 187– (1985) [9] Kondratiev, Trans. Moscow Math. Soc. 16 pp 227– (1967) [10] Surface reconstruction preserving discontinuities, Artificial Intelligence Lab. Memo 792, M.I.T., 1984. [11] Stochastic Integrals, Academic Press, 1969. · Zbl 0191.46603 [12] Mathematical analysis in the mechanics of fracture, in Fracture, An Advanced Treatise, ed., Academic Press, 1968. · Zbl 0214.51802 [13] Simon, Proc. Centre for Math. Anal., Australian Nat. Univ. 3 (1983) [14] and , A Course of Modern Analysis, 4th ed., Cambridge University Press, 1952.
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