×

zbMATH — the first resource for mathematics

A variational method in image segmentation: Existence and approximation results. (English) Zbl 0772.49006
The paper deals with a variational approach to the image segmentation problem, recently proposed by D. Mumford and J. Shah [Common. Pure Appl. Math. 42, No. 5, 577-684 (1989; Zbl 0691.49036)]. Given a bounded domain \(\Omega\subset{\mathbf R}^ 2\) and a function \(g\in L^ \infty(\Omega)\), the problem consists in finding a pair \((\bar u,\overline K)\) minimizing the functional \[ J(u,K)=\int_{\Omega\backslash K} \nabla u^ 2 dx+\int_{\Omega\backslash K} u-g^ 2 dx+{\mathcal H}^ 1(K\cap\Omega) \] among all pairs \((u,K)\) with \(K\subset{\mathbf R}^ 2\) closed and \(u\in C^ 1(\Omega\backslash K)\). Existence of a weak solution of this problem (with \(K\) replaced by \(S_ u\), the jump set of \(u\)) can be obtained in the so-called class \(\text{SBV}(\Omega)\) of special functions of bounded variation in \(\Omega\). The aim of the paper is to prove that any minimizer of the relaxed formulation of the problem actually corresponds to a minimizer of \(J\). This is obtained by showing the existence of \(\gamma>0\) such that \[ {{\mathcal H}^ 1(S_ u\cap B_ \rho(x))\over\rho}<\gamma\Longrightarrow u\in C^ 1(B_{\rho/2}(x)) \] for any minimizer \(u\in\text{SBV}(\Omega)\) and any ball \(B_ \rho(x)\subset\Omega\).
Reviewer: L.Ambrosio (Roma)

MSC:
49J20 Existence theories for optimal control problems involving partial differential equations
PDF BibTeX XML Cite
Full Text: DOI EuDML
References:
[1] Ambrosio, L., A compactness theorem for a new class of functions of bounded variation.Boll. Un. Mat. Ital. B (7), 3 (1989), 857–881. · Zbl 0767.49001
[2] –, Variational problems in SBV and image segmentation.Acta Appl. Math., 17 (1989), 1–40. · Zbl 0697.49004 · doi:10.1007/BF00052492
[3] Amini, A. A., Tehrani, S. &Weymouth, T. E., Using dynamic programming for minimizing the energy of active contours.Second International Conference on Computer Vision (Tampa, Florida, 1988), pp. 95–99. IEEE Computer Society Press, no. 883, Washington, 1988.
[4] Besicovitch, A. S., A general form of the covering principle and relative differentiations of additive functions.Proc. Cambridge Philos. Soc. I, 41 (1945), 103–110. · Zbl 0063.00352 · doi:10.1017/S0305004100022453
[5] Blat, J. & Morel, J. M., Elliptic problems in image segmentation and their relation to fracture theory.Proc. of the International Conference on Nonlinear Elliptic and Parabolic Problems (Nancy, 1988). To appear. · Zbl 0727.35040
[6] Brezis, H., Coron, J. M. &Lieb, E. H., Harmonic maps with defects.Comm. Math. Phys., 107 (1986), 679–705. · Zbl 0608.58016 · doi:10.1007/BF01205490
[7] Carriero, M., Leaci, A., Pallara, D. & Pascali, E., Euler conditions for a minimum problem with free discontinuity surfaces. Preprint Univ. Lecce, Lecce, 1988.
[8] De Giorgi, E., Free discontinuity problems in calculus of variations.Analyse Mathématique et Applications (Paris, 1988). Gauthier-Villars, Paris, 1988. · Zbl 0758.49002
[9] De Giorgi, E. & Ambrosio, L., Un nuovo tipo di funzionale del calcolo delle variazioni.Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur., (1988).
[10] De Giorgi, E., Carriero, M. &Leaci, A., Existence theorem for a minimum proble with free discontinuity set.Arch. Rational Mech. Anal., 108 (1989), 195–218. · Zbl 0682.49002 · doi:10.1007/BF01052971
[11] De Giorgi, E. Colombini, F. & Piccinini, L. C., Frontiere orientate di misura minima e questioni collegate. Quaderno della Scuole Normale Superiore, Pisa, 1972. · Zbl 0296.49031
[12] Ericksen, J. L., Equilibrium theory of liquid crystals.Advances in Liquid Crystals, 233–299. Academic Press, New York, 1976.
[13] Federer, H.,Geometric Measure Theory, Springer-Verlag, New York, 1969. · Zbl 0176.00801
[14] –, Colloquium lectures on geometric measure theory.Bull. Amer. Math. Soc., 84 (1978), 291–338. · Zbl 0392.49021 · doi:10.1090/S0002-9904-1978-14462-0
[15] Geman, S. & Geman, D., Stochastic relaxation, Gibbs distribution and the Bayesian restoration of images.IEEE PAMI, 6 (1984). · Zbl 0573.62030
[16] Giusti, E.,Minimal Surfaces and Functions of Bounded Variation. Birkhäuser, Basel, 1983. · Zbl 0524.35040
[17] Kass, M., Witkin, A. &Terzopoulos, D., Snakes: active contour models.First International conference on Computer Vision (London, 1987), pp. 259–268. IEEE Computer Society Press, no. 77, Washington, 1987.
[18] Massari, U. &Miranda, M.,Miniman Surfaces of Codimension One. Notas de Matematica, North Holland, Amsterdam, 1984. · Zbl 0565.49030
[19] Morel, J. M. &Solimini, S., Segmentation of images by variational methods: a constructive approach.Revista Matematical Universidad Complutense de Madrid, 1 (1988), 169–182. · Zbl 0679.68205
[20] –, Segmentation d’images par méthode variationnelle: une preuve constructive d’existence.C.R. Acad. Sci. Paris Sér I Math., 308 (1989), 465–470. · Zbl 0676.68051
[21] Mumford, D. & Shah, J., Boundary detection by minimizing functionals, I.Proc. IEEE Conf. on Computer Vision and Patter Recognition (San Francisco, 1985) andImage Understanding, 1988.
[22] –, Optimal approximation by piecewise smooth functions and associated variational problems.Comm. Pure Appl. Math., 42 (1989), 577–684. · Zbl 0691.49036 · doi:10.1002/cpa.3160420503
[23] Richardson, T., Existence result for a variational problem arising in computer vision theory. Preprint CICS, P-63, MIT, 1988.
[24] Rosenfeld, A. &Kak, A. C.,Digital Picture Processing. Academic Press, New York, 1982. · Zbl 0564.94002
[25] Simon, L., Lectures on geometric measure theory.Proc. of the Centre for Mathematical Analysis (Canberra, 1983). Australian National University, 3, 1983. · Zbl 0546.49019
[26] Virga, E., Forme di equilibrio di piccole gocce di cristallo liquido. Preprint IAN, Pavia, 1987.
[27] Volpert, A. I. &Hudjaev, S. I.,Analysis in classes of Discontinuous Functions and Equations of Mathematical Physics. Martinus Nijhoff Publishers. Dordrecht, 1985.
[28] Yuille, A. L. &Grzywacz, N. M., The motion coherence theory.Second Internaional Conference of Computer Vision (Tampa, Florida, 1988), IEEE Computer Society Press, no. 883, Washington, 1988.
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.