# zbMATH — the first resource for mathematics

Approximation of functionals depending on jumps by elliptic functionals via $$\Gamma$$-convergence. (English) Zbl 0722.49020
Summary: We show how it is possible to approximate the Mumford-Shah image segmentation functional [D. Mumford and J. Shah, “Boundary detection by minimizing functionals”, in: Proc. IEEE Computer Soc. Conf. Comput. Vision Pattern Recognition, San Francisco/CA 1985, 22-26 (Piscataway/NJ 1985)] ${\mathcal G}(u,K)=\int_{\Omega \setminus K}[| \nabla u|^ 2+\beta (u-g)^ 2]dx\quad +\quad \alpha {\mathcal H}^{n- 1}(K),$
$u\in W^{1,2}(\Omega \setminus K),\quad K\subset \Omega \quad closed\text{ in } \Omega$ by elliptic functionals defined on Sobolev spaces. The heuristic idea is to consider functionals $${\mathcal G}_ h(u,z)$$ with z ranging between 0 and 1 and related to the set K. The minimizing $$z_ h$$ are near to 1 in a neighborhood of the set K, and far from the neighborhood they are very small. The neighborhood shrinks as $$h\to +\infty.$$ For a similar approach to the problem see S. Mitter, T. Richardson and S. R. Kulkarni [in: Signal processing, Part I: Signal processing theory, Proc. Lect., Minneapolis/MN (USA) 1988, IMA Vol. Math. Appl. 22, 189-210 (1990; Zbl 0701.49003)]. The approximation of $${\mathcal G}_ h$$ to $${\mathcal G}$$ takes place in a variational sense, the De Giorgi $$\Gamma$$-convergence.

##### MSC:
 49J45 Methods involving semicontinuity and convergence; relaxation 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems 41A65 Abstract approximation theory (approximation in normed linear spaces and other abstract spaces) 49J27 Existence theories for problems in abstract spaces
Full Text:
##### References:
 [1] Ambrosio, Boll. Un. Mat. It. 3-B pp 857– (1989) [2] Ambrosio, Arch. Rat. Mech. Anal. [3] Ambrosio, Acta Appl. Math. 17 pp 1– (1989) [4] Ambrosio, Proc. AMS. 108 pp 691– (1990) [5] Variational Convergence for Functions and Operators, Pitman, Boston, 1984. [6] Weak Convergence of Measures, Academic Press, New York, 1982. · Zbl 0538.28003 [7] Blake, Pattern Recog. Lett. 6 pp 51– (1987) [8] Brezis, Comm. Math. Phys. 107 pp 649– (1986) [9] Calderon, Rev. Unit. Mat. Arg. 20 pp 102– (1960) [10] , , and , Euler conditions for a minimum problem with free discontinuity surfaces, preprint, 1988. [11] Carriero, Nonl. Anal. [12] and , A general theory of variational functional, in Topics in Functional Analysis 1980-81, Scuola Normale Sup., Pisa, 1981. [13] Dal Maso, Acta Math. [14] De Giorgi, Atti Accad. Naz. Lincei, Rend. Cl. Sci. Fit. Mat. Natur. 58 pp 842– (1975) [15] De Giorgi, Rend. Sem. Mat. Brescia 3 pp 63– (1979) [16] De Giorgi, Rend. Cl. Sci. Fis. Mat. Nat. 2-8 (1988) [17] De Giorgi, Arch. Rat. Mech. Anal. 108 pp 195– (1989) [18] Equilibrium theory of liquid crystals, Advances in Liquid Crystals 2, Academic Press, 1976, pp. 233–298. [19] Geometric Measure Theory, Springer-Verlag, Berlin, 1969. · Zbl 0176.00801 [20] Geman, IEEE Trans. Pattern Anal. Mach. Intell. 6 pp 721– (1984) [21] Minimal Surfaces and Functions of Bounded Variation, Birkhäuser, Boston, 1984. · Zbl 0545.49018 [22] A regularization model for stereo vision with controlled continuity, C.N.R.-I.E. I. Internal Report B4-07, Pisa, 1989. [23] Marr, Proc. Roy. Soc. London. B204 pp 301– (1980) [24] Marroquin, J. Amer. Stat. Assoc. 82 pp 397– (1987) [25] , and , An existence theorem and lattice approximation for a variational problem. Preprint Cent. For Intell. Control Systems, M.I.T., to appear in Proc. Of the Workshop on Signal Processing, Inst. for Math, and Appl., University of Minnesota. [26] Modica, Arch. Rat. Mech. Anal. 98 pp 123– (1987) [27] Modica, Boll. Un. Mat. Ital. 14-B pp 285– (1977) [28] Geometric Measure Theory: A Beginner’s Guide, Academic Press, New York, 1988. · Zbl 0671.49043 [29] and , Boundary detection by minimizing functionals, Proc. IEEE Conf. on Computer Vision and Pattern Recognition, San Francisco, 1985. [30] Mumford, Comm. Pure Appl. Math. 42 pp 577– (1989) [31] and , Digital Picture Processing, Academic Press, New York, 1976. [32] Serrin, Trans. Amer. Mat. Soc. 101 pp 139– (1961) [33] Lectures on Geometric Measure Theory, Proc. Centre for Math. Anal. 3, Australian National University, 1983. [34] Singular Integrals and Differentiability Properties of Functions, Princeton University Press, 1970. · Zbl 0207.13501 [35] Terzopoulos, IEEE Trans. Pattern Anal. Mach. Intell. 8 pp 413– (1988) [36] and , Analysis in classes of discontinuous functions and equations of mathematical physics, Martinus Nijhoff, Dordrecht, 1985. [37] Vol’pert, Math. USSR Sb. 17 pp 255– (1972)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.