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.

[2]

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

[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.

[4]

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

[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.

[6]

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

[7]

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

[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.

[9]

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

[10]

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

[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.

[12]

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

[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.

[14]

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

[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.

[16]

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

[17]

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

[18]

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

[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.

[20]

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

[21]

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

[22]

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

[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.

[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.

[25]

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

[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.

[27]

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

[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.

[29]

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

[30]

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

[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.

[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.

[33]

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

[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.

[35]

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

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.

[2]

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

[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.

[4]

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

[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.

[6]

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

[7]

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

[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.

[9]

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

[10]

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

[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.

[12]

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

[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.

[14]

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

[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.

[16]

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

[17]

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

[18]

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

[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.

[20]

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

[21]

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

[22]

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

[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.

[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.

[25]

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

[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.

[27]

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

[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.

[29]

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

[30]

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

[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.

[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.

[33]

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

[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.

[35]

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

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