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Variational problems in SBV and image segmentation. (English) Zbl 0697.49004
Summary: We show how it is possible to prove the existence of solutions of the Mumford-Shah image segmentation functional \[ F(u,K)=\int_{\Omega \setminus K}[| \nabla u|^ 2+\beta (u-g)^ 2]dx+\alpha {\mathcal H}^{n-1}(K),\quad u\in W^{1,2}(\Omega \setminus K),\quad K\subset \Omega \quad closed\quad in\quad \Omega \] [see D. Mumford and J. Shah, “Boundary detection by minimizing functionals. I”, Proc. IEEE Computer Vision and Pattern Recognition, San Francisco 1985, 22-26 (Silver Spring, MD, 1985) (For a review of the entire collection see Zbl 0627.68072)]. We use a weak formulation of the minimum problem in a special class SBV(\(\Omega)\) of functions of bounded variation. Moreover, we also deal with the regularity of minimizers and the approximation of F by elliptic functionals defined on Sobolev spaces. In this paper, we have collected the main results of the author [Boll. Unione Mat. Ital., VII Ser., B3, No.4, 857-881 (1989)], E. De Giorgi, M. Carriero and A. Leaci [Arch. Ration. Mech. Anal. 108, No.3, 195-218 (1989; Zbl 0682.49002)] and M. Carriero, A. Leaci, D. Pallara and E. Pascali [“Euler conditions for a minimum problem with free discontinuity surfaces.” Preprint (Univ. Lecce/Italy 1988)].

MSC:
49J10 Existence theories for free problems in two or more independent variables
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
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[1] Ambrosio, L.: A compactness theorem for a special class of functions of bounded variation, to appear in Boll. Un. Mat. Ital.
[2] Ambrosio, L.: Existence theory for a new class of variational problems, to appear in Arch. Rat. Mech. Anal.
[3] Ambrosio, L. and Tortorelli, V. M.: Approximation of functionals depending on jumps by elliptic functionals via ?-convergence, to appear. · Zbl 0722.49020
[4] Carriero, M., Leaci, A., Pallara, D., and Pascali, E.: Euler conditions for a minimum problem with free discontinuity surfaces, preprint University of Lecce, 1988.
[5] De Giorgi, E. and Ambrosio, L.: Un nuovo tipo di funzionale del Calcolo delle Variazioni, to appear in Atti Accad. Naz. Lincei, Rend. Cl. Sci. Fis. Mat. Natur.
[6] De Giorgi, E., Carriero, M., and Leaci, A.: Existence theorem for a minimum problem with free discontinuity set, to appear in Arch. Rat. Mech. Anal. · Zbl 0682.49002
[7] Kulkarni, S. R.: Minkowski content and lattice approximation for a variational problem, preprint Center for Intelligent Control Systems, MIT, to appear.
[8] Marroquin J., Mitter S., and Poggio T.: Probabilistic solutions of ill posed problems in Computational Vision. J. Am. Statist. Assoc. 82 (1987) 397. · Zbl 0641.65100 · doi:10.2307/2289127
[9] Mumford, D. and Shah, J.: Boundary detection by minimizing functionals, Proc. IEEE Conference on Computer Vision and Pattern Recognition, San Francisco, 1985.
[10] Mumford, D. and Shah, J.: Optimal approximation by piecewise smooth functions and associated variational problems, to appear in Commun. Pure Appl. Math. · Zbl 0691.49036
[11] Richardson, T.: Existence result for a variational problem arising in computer vision theory, preprint Center for Intelligent Control Systems, P-63, MIT, July 1988.
[12] Richardson, T.: Recovery of boundary by a variational method, to appear.
[13] Dal Maso, G., Morel, J. M., and Solimini, S.: A variational method in image segmentation. Existence and approximation results, to appear. · Zbl 0772.49006
[14] Almgren F. J.: Existence and regularity almost everywhere of solutions to elliptic variational problems with constraints, Mem. Amer. Mat. Soc. 4 (1976) 165. · Zbl 0327.49043
[15] De Giorgi E.: Su una teoria generale della misura (r?1)-dimensionale in uno spazio a r dimensioni, Ann. Mat. Pura Appl. 36 (1954) 191-213 · Zbl 0055.28504 · doi:10.1007/BF02412838
[16] De Giorgi E.: Nuovi teoremi relativi alle misure (r?1)-dimensionali in uno spazio a r dimensioni. Ricerche Mat. 4 (1955) 95-113. · Zbl 0066.29903
[17] Giusti E.: Minimal Surfaces and Functions of Bounded Variation, Birkhäuser, Boston, 1984. · Zbl 0545.49018
[18] Federer H.: Geometric Measure Theory, Springer-Verlag, Berlin, 1969. · Zbl 0176.00801
[19] Federer H.: A note on Gauss-Green theorem. Proc. Amer. Mat. Soc. 9 (1958) 447-451. · Zbl 0087.27302 · doi:10.1090/S0002-9939-1958-0095245-2
[20] Federer H.: 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
[21] Morgan F.: Geometric Measure Theory?A Beginner’s Guide, Academic Press, New York, 1988. · Zbl 0671.49043
[22] Simons, L.: Lectures on Geometric Measure Theory, Proc. Centre for Mathematical Analysis, Australian Mathematical University 3, 1983.
[23] Vol’pert A. I. and Huhjaev S. I.: Analysis in Classes of Discontinuous Functions and Equations of Mathematical Physics, Martinus Nijhoff, Dordrecht, 1985.
[24] Vol’pert A. I.: The spaces BV and quasi linear equations. Math. USSR. Sb. 17 (1972) 225-267. · Zbl 0255.35013 · doi:10.1070/SM1972v017n04ABEH001603
[25] Agmon S., Douglis A., and Niremberg L.: Estimates near the boundary for solutions of elliptic partial differential equations satisfying boundary condition. Commun. Pure Appl. Math. 12 (1959) 623-727. · Zbl 0093.10401 · doi:10.1002/cpa.3160120405
[26] Ambrosio L.: Nuovi risultati sulla semicontinuita inferiore di certi funzionali integrali. Atti Accad. Naz. dei Lincei, Rend. Cl. Sci. Fis. Mat. Natur. (79) 5 (1985) 82-89.
[27] Ambrosio L.: New lower semicontinuity results for integral functionals, Rend. Accad. Naz. Sci. XL Mem. Mat. Sci. Fis. Mat. Natur. 105 (1987) 1-42. · Zbl 0642.49007
[28] Ambrosio, L. and Dal Maso, G.: The chain rule for distributional derivative, to appear in Proc. Amer. Math. Soc. · Zbl 0685.49027
[29] Anzellotti G. and Giaquinta M.: Funzioni BV e tracce, Rend. Sem. Mat. Univ. Padova 60 (1978) 1-22. · Zbl 0432.46031
[30] Attouch H.: Variational Convergence for Functions and Operators, Pitman, Boston, 1984. · Zbl 0561.49012
[31] Brezis H.: Analyse Fonctionelle, Masson, Paris, 1983.
[32] Buttazzo G. and Dal Maso G.: ?-limits of integral functionals, J. Anal. Math. 37 (1980) 145-185. · Zbl 0446.49012 · doi:10.1007/BF02797684
[33] Buttazzo G.: Su una definizione generale dei ?-limiti, Boll. Un. Mat. Ital. (5) 14?B (1977) 722-744. · Zbl 0445.49016
[34] Caffarelli, L. A. and Alt, H. W.: Existence and regularity for a minimum problem with free boundary, 1980. · Zbl 0449.35105
[35] Calderon A. P. and Zygmund A.: On the differentiability of functions which are of bounded variation in Tonelli’s sense, Rev. Un. Mat. Argentina 20 (1960) 102-121.
[36] Carbone L. and Sbordone C.: Some properties of ?-limits of integral functionals. Ann. Mat. Pura Appl. (4) 122 (1979) 1-60. · Zbl 0474.49016 · doi:10.1007/BF02411687
[37] Cesari, L.: Sulle funzioni a variazione limitata, Ann. Scuola Norm. Sup. Pisa, Ser. 2, Vol. 5, 1936. · Zbl 0014.29605
[38] Dal Maso G. and Modica L.: A general theory of variational functionals, Topics in Functional Analysis 1980-81, Scuola Normale Superiore, Pisa, 1981. · Zbl 0493.49005
[39] De Giorgi E. and Spagnolo: Sulla convergenza degli integrali dell’energia per operatori ellittici del II ordine. Boll. Un. Mat.Ital. (4) 8 (1973) 391-411. · Zbl 0274.35002
[40] De Giorgi E. and Franzoni T.: Su un tipo di convergenza variazionale, Atti Accad. Naz. Lincei, Rend. Cl. Sci. Fis. Mat. Natur. (8) 58 (1975) 842-850. · Zbl 0339.49005
[41] De Giorgi E. and Franzoni T.: Su un tipo di convergenza variazionale, Rend. Sem. Mat. Brescia 3 (1979) 63-101. · Zbl 0339.49005
[42] Ekeland I. and Temam R.: Convex Analysis and Variational Problems, North-Holland, Amsterdam, 1976. · Zbl 0322.90046
[43] Federer H. and Ziemer W. P.: The Lebesgue set of a function whose distributional derivatives are pth power summable, Indiana Univ. Math. J. 22 (1972) 139-158. · Zbl 0238.28015 · doi:10.1512/iumj.1972.22.22013
[44] Fleming W. H. and Rishel R.: An integral formula for total gradient variation, Arch. Math. 11, (1960) 218-222. · Zbl 0094.26301 · doi:10.1007/BF01236935
[45] Giaquinta, M.: Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems. Princeton University Press, 1983. · Zbl 0516.49003
[46] Ioffe A. D.: On lower semicontinuity of integral functionals I, SIAM J. Cont. Optim. 15 (1977) 521-538. · Zbl 0361.46037 · doi:10.1137/0315035
[47] Ioffe A. S.: On lower semicontinuity of integral functionals II, SIAM J. Cont. Optim. 15 (1977), 991-1000. · Zbl 0379.46022 · doi:10.1137/0315064
[48] Marcellini P. and Sbordone C.: Semicontinuity problems in the calculus of variations, Nonlinear Anal. 4 (1980) 241-257. · Zbl 0537.49002 · doi:10.1016/0362-546X(80)90052-8
[49] Modica L.: The gradient theory of phase transitions and the minimal interface criterion, Arch. Rat. Mech., Analysis 98 (1987) 123-142. · Zbl 0616.76004 · doi:10.1007/BF00251230
[50] Modica L. and Mortola S.: Un esempio di ?-convergenza, Boll. Un. Mat. Ital. 5 14?B (1977) 285-299. · Zbl 0356.49008
[51] Reshetnyak Y. G.: Weak convergence of completely additive vector functions on a set, Siberian Math. J. 9 (1968) 1039-1045 (translation of Sibirsk Mat. Z. 9 (1968) 1386-1394. · Zbl 0176.44402
[52] Serrin J.: A new definition of the integral for non-parametric problems in the Calculus of variations, Acta Math. 102 (1959) 23-32. · Zbl 0089.08601 · doi:10.1007/BF02559566
[53] Serrin J.: On the definition and properties of certain variational integrals, Trans. Amer. Mat. Soc. 101 (1961) 139-167. · Zbl 0102.04601 · doi:10.1090/S0002-9947-1961-0138018-9
[54] Spagnolo S.: Sul limite delle soluzioni dei problemi di Cauchy relativi all’equazione del calore, Ann. Sc. Norm. Sup. Pisa Cl. Sci. (3) 21 (1967) 657-699. · Zbl 0153.42103
[55] Spagnolo S.: Sulla convergenza della soluzioni di equazioni paraboliche ed ellittiche, Ann. Sc. Norm. Sup. Pisa Cl. Sci. (3) 22 (1968) 575-597.
[56] Tonelli, L.: Sulla quadratura della superficie, Rend. Accad. Naz. Lincei 6, No. 3, 1926. · JFM 52.0251.01
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