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 and Continuous Dynamical Systems, 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), 1171-1197. doi: 10.1137/S0036141000368326.

[2]

L. Ambrosio, N. Fusco and D. Pallara, "Functions of Bounded Variations and Free Discontinuity Problems," Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 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), 999-1036. doi: 10.1002/cpa.3160430805.

[4]

A. Blake and A. Zisserman, "Visual Reconstruction," The MIT Press Series in Artificial Intelligence, MIT Press, Cambridge, MA, 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), 237-265.

[6]

T. Boccellari and F. Tomarelli, Generic uniqueness of minimizer for Blake & Zisserman functional, Dip. Matematica, Politecnico di Milano, QDD 66 (2010), 1-73. Available from: http://www1.mate.polimi.it/biblioteca/qddview.php?id=1390&L=i.

[7]

M. Carriero, A. Farina and I. Sgura, Image segmentation in the framework of free discontinuity problems, in "Calculus of Variations: Topics from the Mathematical Heritage of E. De Giorgi" (ed. D. Pallara), Quad. Mat., 14, Dept. Math., Seconda Univ. Napoli, Caserta, (2004), 85-133.

[8]

M. Carriero and A. Leaci, Existence theorem for a Dirichlet problem with free dicontinuity set, Nonlinear Analysis, 15 (1990), 661-677. doi: 10.1016/0362-546X(90)90006-3.

[9]

M. Carriero, A. Leaci and F. Tomarelli, Free gradient discontinuities, in "Calculus of Variations, Homogeneization and Continuum Mechanics," (eds. G. Buttazzo, G. Bouchitte and P. Suquet) (Marseille, 1993), 131-147, Ser. Adv. Math Appl. Sci., 18, World Sci. Publishing, River Edge, NJ, 1994.

[10]

M. Carriero, A. Leaci and F. Tomarelli, A second order model in image segmentation: Blake & Zisserman functional, in "Variational Methods for Discontinuous Structures," (eds. R. Serapioni and F. Tomarelli) (Como, 1994), 57-72, Progr. Nonlinear Differential Equations Appl., 25, Birkhäuser, Basel, 1996.

[11]

M. Carriero, A. Leaci and F. Tomarelli, Strong minimizers of Blake & Zisserman functional, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 25 (1997), 257-285.

[12]

M. Carriero, A. Leaci and F. Tomarelli, Density estimates and further properties of Blake & Zisserman functional, in "From Convexity to Nonconvexity" (eds. R. Gilbert and P. Pardalos), 381-392, Nonconvex Optim. Appl., 55, Kluwer Acad. Publ., Dordrecht, 2001.

[13]

M. Carriero, A. Leaci and F. Tomarelli, Necessary conditions for extremals of Blake & Zisserman functional, C. R. Math. Acad. Sci. Paris, 334 (2002), 343-348.

[14]

M. Carriero, A. Leaci and F. Tomarelli, Local minimizers for a free gradient discontinuity problem in image segmentation, in "Variational Methods for Discontinuous Structures" (eds. G. Dal Maso and F. Tomarelli), 67-80, Progr. Nonlinear Differential Equations Appl., 51, Birkhäuser, Basel, 2002.

[15]

M. Carriero, A. Leaci and F. Tomarelli, Calculus of variations and image segmentation, J. of Physiology, Paris, 97 (2003), 343-353. 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 "Calculus of Variations: Topics from the Mathematical Heritage of E. De Giorgi" (ed. D. Pallara), 135-186, Quad. Mat., 14, Dept. Math., Seconda Univ. Napoli, Caserta, 2004.

[17]

M. Carriero, A. Leaci and F. Tomarelli, Euler equations for Blake & Zisserman functional, Calc. Var. Partial Differential Equations, 32 (2008), 81-110.

[18]

M. Carriero, A. Leaci and F. Tomarelli, A Dirichlet problem with free gradient discontinuity, Adv. Math. Sci. Appl., 20 (2010), 107-141.

[19]

M. Carriero, A. Leaci and F. Tomarelli, A candidate local minimizer of Blake & Zisserman functional, J. Math. Pures Appl., 96 (2011), 58-87. 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), 141-156.

[21]

M. Carriero, A. Leaci and F. Tomarelli, About Poincaré inequalities for functions lacking summability, Note Mat., 31 (2011), 67-84.

[22]

M. Carriero, A. Leaci and F. Tomarelli, Free gradient discontinuity and image inpaintig, Proc. Steklov Inst. Math., to appear, 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), 351-373. 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 "Handbook of Mathematical Models in Computer Vision" (eds. N. Paragios, Y. Chen and O. Faugeras), 17-31, Springer, New York, 2006. doi: 10.1007/0-387-28831-7_2.

[25]

E. De Giorgi, Free discontinuity problems in calculus of variations, in "Frontiers in Pure and Applied Mathematics" (ed. R. Dautray), 55-62, North-Holland, Amsterdam, 1991.

[26]

E. De Giorgi and L. Ambrosio, Un nuovo tipo di funzionale del Calcolo delle Variazioni (Italian) [New functionals in the calculus of variations], Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur., 82 (1988), 199-210.

[27]

R. J. Duffin, Continuation of biharmonic functions by reflection, Duke Math. J., 22 (1955), 313-324. 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), 353-370. doi: 10.1017/S0956792502004904.

[29]

H. Federer, "Geometric Measure Theory," Die Grundlehren der Mathematischen Wissenschaften, 153, Springer-Verlag New York Inc., New York, 1969.

[30]

M. Giaquinta, "Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems," Ann. Math. Stud., 105, Princeton U. P., Princeton, NJ, 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), 639-673.

[32]

R. March, Visual reconstruction with discontinuities using variational methods, Image and Vision Computing, 10 (1992), 30-38. 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, Birkhäuser Boston, Inc., Boston, MA, 1995.

[34]

D. Mumford and J. Shah, Optimal approximation by piecewise smooth functions and associated variational problems, Comm. Pure Appl. Math., 42 (1989), 577-685. doi: 10.1002/cpa.3160420503.

[35]

J. Verdera, V. Caselles, M. Bertalmio and G. Sapiro, Inpainting surface holes, Int. Conference on Image Processing, (2003), 903-906.

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), 1171-1197. doi: 10.1137/S0036141000368326.

[2]

L. Ambrosio, N. Fusco and D. Pallara, "Functions of Bounded Variations and Free Discontinuity Problems," Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 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), 999-1036. doi: 10.1002/cpa.3160430805.

[4]

A. Blake and A. Zisserman, "Visual Reconstruction," The MIT Press Series in Artificial Intelligence, MIT Press, Cambridge, MA, 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), 237-265.

[6]

T. Boccellari and F. Tomarelli, Generic uniqueness of minimizer for Blake & Zisserman functional, Dip. Matematica, Politecnico di Milano, QDD 66 (2010), 1-73. Available from: http://www1.mate.polimi.it/biblioteca/qddview.php?id=1390&L=i.

[7]

M. Carriero, A. Farina and I. Sgura, Image segmentation in the framework of free discontinuity problems, in "Calculus of Variations: Topics from the Mathematical Heritage of E. De Giorgi" (ed. D. Pallara), Quad. Mat., 14, Dept. Math., Seconda Univ. Napoli, Caserta, (2004), 85-133.

[8]

M. Carriero and A. Leaci, Existence theorem for a Dirichlet problem with free dicontinuity set, Nonlinear Analysis, 15 (1990), 661-677. doi: 10.1016/0362-546X(90)90006-3.

[9]

M. Carriero, A. Leaci and F. Tomarelli, Free gradient discontinuities, in "Calculus of Variations, Homogeneization and Continuum Mechanics," (eds. G. Buttazzo, G. Bouchitte and P. Suquet) (Marseille, 1993), 131-147, Ser. Adv. Math Appl. Sci., 18, World Sci. Publishing, River Edge, NJ, 1994.

[10]

M. Carriero, A. Leaci and F. Tomarelli, A second order model in image segmentation: Blake & Zisserman functional, in "Variational Methods for Discontinuous Structures," (eds. R. Serapioni and F. Tomarelli) (Como, 1994), 57-72, Progr. Nonlinear Differential Equations Appl., 25, Birkhäuser, Basel, 1996.

[11]

M. Carriero, A. Leaci and F. Tomarelli, Strong minimizers of Blake & Zisserman functional, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 25 (1997), 257-285.

[12]

M. Carriero, A. Leaci and F. Tomarelli, Density estimates and further properties of Blake & Zisserman functional, in "From Convexity to Nonconvexity" (eds. R. Gilbert and P. Pardalos), 381-392, Nonconvex Optim. Appl., 55, Kluwer Acad. Publ., Dordrecht, 2001.

[13]

M. Carriero, A. Leaci and F. Tomarelli, Necessary conditions for extremals of Blake & Zisserman functional, C. R. Math. Acad. Sci. Paris, 334 (2002), 343-348.

[14]

M. Carriero, A. Leaci and F. Tomarelli, Local minimizers for a free gradient discontinuity problem in image segmentation, in "Variational Methods for Discontinuous Structures" (eds. G. Dal Maso and F. Tomarelli), 67-80, Progr. Nonlinear Differential Equations Appl., 51, Birkhäuser, Basel, 2002.

[15]

M. Carriero, A. Leaci and F. Tomarelli, Calculus of variations and image segmentation, J. of Physiology, Paris, 97 (2003), 343-353. 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 "Calculus of Variations: Topics from the Mathematical Heritage of E. De Giorgi" (ed. D. Pallara), 135-186, Quad. Mat., 14, Dept. Math., Seconda Univ. Napoli, Caserta, 2004.

[17]

M. Carriero, A. Leaci and F. Tomarelli, Euler equations for Blake & Zisserman functional, Calc. Var. Partial Differential Equations, 32 (2008), 81-110.

[18]

M. Carriero, A. Leaci and F. Tomarelli, A Dirichlet problem with free gradient discontinuity, Adv. Math. Sci. Appl., 20 (2010), 107-141.

[19]

M. Carriero, A. Leaci and F. Tomarelli, A candidate local minimizer of Blake & Zisserman functional, J. Math. Pures Appl., 96 (2011), 58-87. 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), 141-156.

[21]

M. Carriero, A. Leaci and F. Tomarelli, About Poincaré inequalities for functions lacking summability, Note Mat., 31 (2011), 67-84.

[22]

M. Carriero, A. Leaci and F. Tomarelli, Free gradient discontinuity and image inpaintig, Proc. Steklov Inst. Math., to appear, 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), 351-373. 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 "Handbook of Mathematical Models in Computer Vision" (eds. N. Paragios, Y. Chen and O. Faugeras), 17-31, Springer, New York, 2006. doi: 10.1007/0-387-28831-7_2.

[25]

E. De Giorgi, Free discontinuity problems in calculus of variations, in "Frontiers in Pure and Applied Mathematics" (ed. R. Dautray), 55-62, North-Holland, Amsterdam, 1991.

[26]

E. De Giorgi and L. Ambrosio, Un nuovo tipo di funzionale del Calcolo delle Variazioni (Italian) [New functionals in the calculus of variations], Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur., 82 (1988), 199-210.

[27]

R. J. Duffin, Continuation of biharmonic functions by reflection, Duke Math. J., 22 (1955), 313-324. 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), 353-370. doi: 10.1017/S0956792502004904.

[29]

H. Federer, "Geometric Measure Theory," Die Grundlehren der Mathematischen Wissenschaften, 153, Springer-Verlag New York Inc., New York, 1969.

[30]

M. Giaquinta, "Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems," Ann. Math. Stud., 105, Princeton U. P., Princeton, NJ, 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), 639-673.

[32]

R. March, Visual reconstruction with discontinuities using variational methods, Image and Vision Computing, 10 (1992), 30-38. 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, Birkhäuser Boston, Inc., Boston, MA, 1995.

[34]

D. Mumford and J. Shah, Optimal approximation by piecewise smooth functions and associated variational problems, Comm. Pure Appl. Math., 42 (1989), 577-685. doi: 10.1002/cpa.3160420503.

[35]

J. Verdera, V. Caselles, M. Bertalmio and G. Sapiro, Inpainting surface holes, Int. Conference on Image Processing, (2003), 903-906.

[1]

Nuno R. O. Bastos, Rui A. C. Ferreira, Delfim F. M. Torres. Necessary optimality conditions for fractional difference problems of the calculus of variations. Discrete and Continuous Dynamical Systems, 2011, 29 (2) : 417-437. doi: 10.3934/dcds.2011.29.417

[2]

Jianping Zhang, Ke Chen, Bo Yu, Derek A. Gould. A local information based variational model for selective image segmentation. Inverse Problems and Imaging, 2014, 8 (1) : 293-320. doi: 10.3934/ipi.2014.8.293

[3]

Liam Burrows, Weihong Guo, Ke Chen, Francesco Torella. Reproducible kernel Hilbert space based global and local image segmentation. Inverse Problems and Imaging, 2021, 15 (1) : 1-25. doi: 10.3934/ipi.2020048

[4]

Ioan Bucataru, Matias F. Dahl. Semi-basic 1-forms and Helmholtz conditions for the inverse problem of the calculus of variations. Journal of Geometric Mechanics, 2009, 1 (2) : 159-180. doi: 10.3934/jgm.2009.1.159

[5]

Bernard Dacorogna, Giovanni Pisante, Ana Margarida Ribeiro. On non quasiconvex problems of the calculus of variations. Discrete and Continuous Dynamical Systems, 2005, 13 (4) : 961-983. doi: 10.3934/dcds.2005.13.961

[6]

Daniel Faraco, Jan Kristensen. Compactness versus regularity in the calculus of variations. Discrete and Continuous Dynamical Systems - B, 2012, 17 (2) : 473-485. doi: 10.3934/dcdsb.2012.17.473

[7]

Ibrar Hussain, Haider Ali, Muhammad Shahkar Khan, Sijie Niu, Lavdie Rada. Robust region-based active contour models via local statistical similarity and local similarity factor for intensity inhomogeneity and high noise image segmentation. Inverse Problems and Imaging, , () : -. doi: 10.3934/ipi.2022014

[8]

Felix Sadyrbaev. Nonlinear boundary value problems of the calculus of variations. Conference Publications, 2003, 2003 (Special) : 760-770. doi: 10.3934/proc.2003.2003.760

[9]

Fan Jia, Xue-Cheng Tai, Jun Liu. Nonlocal regularized CNN for image segmentation. Inverse Problems and Imaging, 2020, 14 (5) : 891-911. doi: 10.3934/ipi.2020041

[10]

Agnieszka B. Malinowska, Delfim F. M. Torres. Euler-Lagrange equations for composition functionals in calculus of variations on time scales. Discrete and Continuous Dynamical Systems, 2011, 29 (2) : 577-593. doi: 10.3934/dcds.2011.29.577

[11]

Delfim F. M. Torres. Proper extensions of Noether's symmetry theorem for nonsmooth extremals of the calculus of variations. Communications on Pure and Applied Analysis, 2004, 3 (3) : 491-500. doi: 10.3934/cpaa.2004.3.491

[12]

Jacky Cresson, Fernando Jiménez, Sina Ober-Blöbaum. Continuous and discrete Noether's fractional conserved quantities for restricted calculus of variations. Journal of Geometric Mechanics, 2022, 14 (1) : 57-89. doi: 10.3934/jgm.2021012

[13]

Ye Yuan, Yan Ren, Xiaodong Liu, Jing Wang. Approach to image segmentation based on interval neutrosophic set. Numerical Algebra, Control and Optimization, 2020, 10 (1) : 1-11. doi: 10.3934/naco.2019028

[14]

Dominique Zosso, Jing An, James Stevick, Nicholas Takaki, Morgan Weiss, Liane S. Slaughter, Huan H. Cao, Paul S. Weiss, Andrea L. Bertozzi. Image segmentation with dynamic artifacts detection and bias correction. Inverse Problems and Imaging, 2017, 11 (3) : 577-600. doi: 10.3934/ipi.2017027

[15]

Matthew S. Keegan, Berta Sandberg, Tony F. Chan. A multiphase logic framework for multichannel image segmentation. Inverse Problems and Imaging, 2012, 6 (1) : 95-110. doi: 10.3934/ipi.2012.6.95

[16]

Matthew D. Kvalheim, Daniel E. Koditschek. Necessary conditions for feedback stabilization and safety. Journal of Geometric Mechanics, 2022  doi: 10.3934/jgm.2022013

[17]

Piernicola Bettiol, Nathalie Khalil. Necessary optimality conditions for average cost minimization problems. Discrete and Continuous Dynamical Systems - B, 2019, 24 (5) : 2093-2124. doi: 10.3934/dcdsb.2019086

[18]

Francis Clarke. A general theorem on necessary conditions in optimal control. Discrete and Continuous Dynamical Systems, 2011, 29 (2) : 485-503. doi: 10.3934/dcds.2011.29.485

[19]

Alexander Blokh. Necessary conditions for the existence of wandering triangles for cubic laminations. Discrete and Continuous Dynamical Systems, 2005, 13 (1) : 13-34. doi: 10.3934/dcds.2005.13.13

[20]

Shahlar F. Maharramov. Necessary optimality conditions for switching control problems. Journal of Industrial and Management Optimization, 2010, 6 (1) : 47-55. doi: 10.3934/jimo.2010.6.47

2021 Impact Factor: 1.588

Metrics

  • PDF downloads (64)
  • HTML views (0)
  • Cited by (7)

[Back to Top]