December  2011, 31(4): 1129-1150. doi: 10.3934/dcds.2011.31.1129

Uniform density estimates for Blake & Zisserman functional

1. 

Università del Salento, Dipartimento di Matematica “Ennio De Giorgi”, 73100 Lecce, Italy, Italy

2. 

Politecnico di Milano, Dipartimento di Matematica “Francesco Brioschi”, 20133 Milano, Italy

Received  November 2009 Revised  March 2010 Published  September 2011

We prove density estimates and elimination properties for minimizing triplets of functionals which are related to contour detection in image segmentation and depend on free discontinuities, free gradient discontinuities and second order derivatives. All the estimates concern optimal segmentation under Dirichlet boundary conditions and are uniform in the image domain up to the boundary.
Citation: Michele Carriero, Antonio Leaci, Franco Tomarelli. Uniform density estimates for Blake & Zisserman functional. Discrete & Continuous Dynamical Systems - A, 2011, 31 (4) : 1129-1150. doi: 10.3934/dcds.2011.31.1129
References:
[1]

L. Ambrosio, L. Faina and R. March, Variational approximation of a second order free discontinuity problem in computer vision,, SIAM J. Math. Anal., 32 (2001), 1171.  doi: 10.1137/S0036141000368326.  Google Scholar

[2]

L. Ambrosio, N. Fusco and D. Pallara, "Functions of Bounded Variations and Free Discontinuity Problems,", Oxford Mathematical Monographs, (2000).   Google Scholar

[3]

L. Ambrosio and V. Tortorelli, Approximation of functionals depending on jumps by elliptic functionals via $\Gamma$-convergence,, Comm. Pure Appl. Math., 43 (1990), 999.  doi: 10.1002/cpa.3160430805.  Google Scholar

[4]

A. Blake and A. Zisserman, "Visual Reconstruction,", The MIT Press Series in Artificial Intelligence, (1987).   Google Scholar

[5]

T. Boccellari and F. Tomarelli, About well-posedness of optimal segmentation for Blake & Zisserman functional,, Istituto Lombardo (Rend. Cl. Sci. Mat. Nat.), 142 (2008), 237.   Google Scholar

[6]

T. Boccellari and F. Tomarelli, Generic uniqueness of minimizer for Blake & Zisserman functional,, Dip. Matematica, QDD 66 (2010), 1.   Google Scholar

[7]

M. Carriero, A. Farina and I. Sgura, Image segmentation in the framework of free discontinuity problems,, in, 14 (2004), 85.   Google Scholar

[8]

M. Carriero and A. Leaci, Existence theorem for a Dirichlet problem with free dicontinuity set,, Nonlinear Analysis, 15 (1990), 661.  doi: 10.1016/0362-546X(90)90006-3.  Google Scholar

[9]

M. Carriero, A. Leaci and F. Tomarelli, Free gradient discontinuities,, in, 18 (1993), 131.   Google Scholar

[10]

M. Carriero, A. Leaci and F. Tomarelli, A second order model in image segmentation: Blake & Zisserman functional,, in, 25 (1994), 57.   Google Scholar

[11]

M. Carriero, A. Leaci and F. Tomarelli, Strong minimizers of Blake & Zisserman functional,, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 25 (1997), 257.   Google Scholar

[12]

M. Carriero, A. Leaci and F. Tomarelli, Density estimates and further properties of Blake & Zisserman functional,, in, 55 (2001), 381.   Google Scholar

[13]

M. Carriero, A. Leaci and F. Tomarelli, Necessary conditions for extremals of Blake & Zisserman functional,, C. R. Math. Acad. Sci. Paris, 334 (2002), 343.   Google Scholar

[14]

M. Carriero, A. Leaci and F. Tomarelli, Local minimizers for a free gradient discontinuity problem in image segmentation,, in, 51 (2002), 67.   Google Scholar

[15]

M. Carriero, A. Leaci and F. Tomarelli, Calculus of variations and image segmentation,, J. of Physiology, 97 (2003), 343.  doi: 10.1016/j.jphysparis.2003.09.008.  Google Scholar

[16]

M. Carriero, A. Leaci and F. Tomarelli, Second order variational problems with free discontinuity and free gradient discontinuity,, in, 14 (2004), 135.   Google Scholar

[17]

M. Carriero, A. Leaci and F. Tomarelli, Euler equations for Blake & Zisserman functional,, Calc. Var. Partial Differential Equations, 32 (2008), 81.   Google Scholar

[18]

M. Carriero, A. Leaci and F. Tomarelli, A Dirichlet problem with free gradient discontinuity,, Adv. Math. Sci. Appl., 20 (2010), 107.   Google Scholar

[19]

M. Carriero, A. Leaci and F. Tomarelli, A candidate local minimizer of Blake & Zisserman functional,, J. Math. Pures Appl., 96 (2011), 58.  doi: 10.1016/j.matpur.2011.01.005.  Google Scholar

[20]

M. Carriero, A. Leaci and F. Tomarelli, Variational approach to image segmentation,, Pure Math. Appl. (Pu.M.A.), 20 (2009), 141.   Google Scholar

[21]

M. Carriero, A. Leaci and F. Tomarelli, About Poincaré inequalities for functions lacking summability,, Note Mat., 31 (2011), 67.   Google Scholar

[22]

M. Carriero, A. Leaci and F. Tomarelli, Free gradient discontinuity and image inpaintig,, Proc. Steklov Inst. Math., (2011).   Google Scholar

[23]

V. Caselles, G. Haro, G. Sapiro and J. Verdera, On geometric variational models for inpainting surface holes,, Computer Vision and Image Understanding, 111 (2008), 351.  doi: 10.1016/j.cviu.2008.01.002.  Google Scholar

[24]

T. Chan, S. Esedoglu, F. Park and A. Yip, Total variation image restoration: Overview and recent developments,, in, (2006), 17.  doi: 10.1007/0-387-28831-7_2.  Google Scholar

[25]

E. De Giorgi, Free discontinuity problems in calculus of variations,, in, (1991), 55.   Google Scholar

[26]

E. De Giorgi and L. Ambrosio, Un nuovo tipo di funzionale del Calcolo delle Variazioni (Italian), [New functionals in the calculus of variations], 82 (1988), 199.   Google Scholar

[27]

R. J. Duffin, Continuation of biharmonic functions by reflection,, Duke Math. J., 22 (1955), 313.  doi: 10.1215/S0012-7094-55-02233-X.  Google Scholar

[28]

S. Esedoglu and J. Shen, Digital inpainting based on the Mumford-Shah-Euler image model,, European J. Appl. Math., 13 (2002), 353.  doi: 10.1017/S0956792502004904.  Google Scholar

[29]

H. Federer, "Geometric Measure Theory,", Die Grundlehren der Mathematischen Wissenschaften, 153 (1969).   Google Scholar

[30]

M. Giaquinta, "Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems,", Ann. Math. Stud., 105 (1983).   Google Scholar

[31]

F. A. Lops, F. Maddalena and S. Solimini, Hölder continuity conditions for the solvability of Dirichlet problems involving functionals with free discontinuities,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 18 (2001), 639.   Google Scholar

[32]

R. March, Visual reconstruction with discontinuities using variational methods,, Image and Vision Computing, 10 (1992), 30.  doi: 10.1016/0262-8856(92)90081-D.  Google Scholar

[33]

J.-M. Morel and S. Solimini, "Variational Methods in Image Segmentation. With Seven Image Processing Experiments,", Progr. Nonlinear Differential Equations Appl., 14 (1995).   Google Scholar

[34]

D. Mumford and J. Shah, Optimal approximation by piecewise smooth functions and associated variational problems,, Comm. Pure Appl. Math., 42 (1989), 577.  doi: 10.1002/cpa.3160420503.  Google Scholar

[35]

J. Verdera, V. Caselles, M. Bertalmio and G. Sapiro, Inpainting surface holes,, Int. Conference on Image Processing, (2003), 903.   Google Scholar

show all references

References:
[1]

L. Ambrosio, L. Faina and R. March, Variational approximation of a second order free discontinuity problem in computer vision,, SIAM J. Math. Anal., 32 (2001), 1171.  doi: 10.1137/S0036141000368326.  Google Scholar

[2]

L. Ambrosio, N. Fusco and D. Pallara, "Functions of Bounded Variations and Free Discontinuity Problems,", Oxford Mathematical Monographs, (2000).   Google Scholar

[3]

L. Ambrosio and V. Tortorelli, Approximation of functionals depending on jumps by elliptic functionals via $\Gamma$-convergence,, Comm. Pure Appl. Math., 43 (1990), 999.  doi: 10.1002/cpa.3160430805.  Google Scholar

[4]

A. Blake and A. Zisserman, "Visual Reconstruction,", The MIT Press Series in Artificial Intelligence, (1987).   Google Scholar

[5]

T. Boccellari and F. Tomarelli, About well-posedness of optimal segmentation for Blake & Zisserman functional,, Istituto Lombardo (Rend. Cl. Sci. Mat. Nat.), 142 (2008), 237.   Google Scholar

[6]

T. Boccellari and F. Tomarelli, Generic uniqueness of minimizer for Blake & Zisserman functional,, Dip. Matematica, QDD 66 (2010), 1.   Google Scholar

[7]

M. Carriero, A. Farina and I. Sgura, Image segmentation in the framework of free discontinuity problems,, in, 14 (2004), 85.   Google Scholar

[8]

M. Carriero and A. Leaci, Existence theorem for a Dirichlet problem with free dicontinuity set,, Nonlinear Analysis, 15 (1990), 661.  doi: 10.1016/0362-546X(90)90006-3.  Google Scholar

[9]

M. Carriero, A. Leaci and F. Tomarelli, Free gradient discontinuities,, in, 18 (1993), 131.   Google Scholar

[10]

M. Carriero, A. Leaci and F. Tomarelli, A second order model in image segmentation: Blake & Zisserman functional,, in, 25 (1994), 57.   Google Scholar

[11]

M. Carriero, A. Leaci and F. Tomarelli, Strong minimizers of Blake & Zisserman functional,, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 25 (1997), 257.   Google Scholar

[12]

M. Carriero, A. Leaci and F. Tomarelli, Density estimates and further properties of Blake & Zisserman functional,, in, 55 (2001), 381.   Google Scholar

[13]

M. Carriero, A. Leaci and F. Tomarelli, Necessary conditions for extremals of Blake & Zisserman functional,, C. R. Math. Acad. Sci. Paris, 334 (2002), 343.   Google Scholar

[14]

M. Carriero, A. Leaci and F. Tomarelli, Local minimizers for a free gradient discontinuity problem in image segmentation,, in, 51 (2002), 67.   Google Scholar

[15]

M. Carriero, A. Leaci and F. Tomarelli, Calculus of variations and image segmentation,, J. of Physiology, 97 (2003), 343.  doi: 10.1016/j.jphysparis.2003.09.008.  Google Scholar

[16]

M. Carriero, A. Leaci and F. Tomarelli, Second order variational problems with free discontinuity and free gradient discontinuity,, in, 14 (2004), 135.   Google Scholar

[17]

M. Carriero, A. Leaci and F. Tomarelli, Euler equations for Blake & Zisserman functional,, Calc. Var. Partial Differential Equations, 32 (2008), 81.   Google Scholar

[18]

M. Carriero, A. Leaci and F. Tomarelli, A Dirichlet problem with free gradient discontinuity,, Adv. Math. Sci. Appl., 20 (2010), 107.   Google Scholar

[19]

M. Carriero, A. Leaci and F. Tomarelli, A candidate local minimizer of Blake & Zisserman functional,, J. Math. Pures Appl., 96 (2011), 58.  doi: 10.1016/j.matpur.2011.01.005.  Google Scholar

[20]

M. Carriero, A. Leaci and F. Tomarelli, Variational approach to image segmentation,, Pure Math. Appl. (Pu.M.A.), 20 (2009), 141.   Google Scholar

[21]

M. Carriero, A. Leaci and F. Tomarelli, About Poincaré inequalities for functions lacking summability,, Note Mat., 31 (2011), 67.   Google Scholar

[22]

M. Carriero, A. Leaci and F. Tomarelli, Free gradient discontinuity and image inpaintig,, Proc. Steklov Inst. Math., (2011).   Google Scholar

[23]

V. Caselles, G. Haro, G. Sapiro and J. Verdera, On geometric variational models for inpainting surface holes,, Computer Vision and Image Understanding, 111 (2008), 351.  doi: 10.1016/j.cviu.2008.01.002.  Google Scholar

[24]

T. Chan, S. Esedoglu, F. Park and A. Yip, Total variation image restoration: Overview and recent developments,, in, (2006), 17.  doi: 10.1007/0-387-28831-7_2.  Google Scholar

[25]

E. De Giorgi, Free discontinuity problems in calculus of variations,, in, (1991), 55.   Google Scholar

[26]

E. De Giorgi and L. Ambrosio, Un nuovo tipo di funzionale del Calcolo delle Variazioni (Italian), [New functionals in the calculus of variations], 82 (1988), 199.   Google Scholar

[27]

R. J. Duffin, Continuation of biharmonic functions by reflection,, Duke Math. J., 22 (1955), 313.  doi: 10.1215/S0012-7094-55-02233-X.  Google Scholar

[28]

S. Esedoglu and J. Shen, Digital inpainting based on the Mumford-Shah-Euler image model,, European J. Appl. Math., 13 (2002), 353.  doi: 10.1017/S0956792502004904.  Google Scholar

[29]

H. Federer, "Geometric Measure Theory,", Die Grundlehren der Mathematischen Wissenschaften, 153 (1969).   Google Scholar

[30]

M. Giaquinta, "Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems,", Ann. Math. Stud., 105 (1983).   Google Scholar

[31]

F. A. Lops, F. Maddalena and S. Solimini, Hölder continuity conditions for the solvability of Dirichlet problems involving functionals with free discontinuities,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 18 (2001), 639.   Google Scholar

[32]

R. March, Visual reconstruction with discontinuities using variational methods,, Image and Vision Computing, 10 (1992), 30.  doi: 10.1016/0262-8856(92)90081-D.  Google Scholar

[33]

J.-M. Morel and S. Solimini, "Variational Methods in Image Segmentation. With Seven Image Processing Experiments,", Progr. Nonlinear Differential Equations Appl., 14 (1995).   Google Scholar

[34]

D. Mumford and J. Shah, Optimal approximation by piecewise smooth functions and associated variational problems,, Comm. Pure Appl. Math., 42 (1989), 577.  doi: 10.1002/cpa.3160420503.  Google Scholar

[35]

J. Verdera, V. Caselles, M. Bertalmio and G. Sapiro, Inpainting surface holes,, Int. Conference on Image Processing, (2003), 903.   Google Scholar

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