A Survey on the Blake–Zisserman Functional


The aim of this work is to provide a concise survey of results about the Blake–Zisserman functional for image segmentation and inpainting. Moreover a refinement of the Almansi decomposition is shown for biharmonic functions in 2-dimensional open disks with crack-tip at the origin.

This is a preview of subscription content, access via your institution.


  1. 1.

    E. Almansi, Sull’integrazione dell’equazione differenziale \({\varDelta^{2n}}\) = 0, Ann. Mat. Pura Appl., III, (1899), 1–51.

  2. 2.

    Alberti G., Bouchitté G., Dal Maso G.: The calibration method for the Mumford- Shah functional and free-discontinuity problems. Calc. Var. Partial Differential Equations, 16, 299–333 (2003)

    MATH  MathSciNet  Article  Google Scholar 

  3. 3.

    Amar M., De Cicco V.: The uniqueness as a generic property for some one dimensional segmentation problems. Rend. Sem. Univ. Padova, 88, 151–173 (1992)

    MATH  MathSciNet  Google Scholar 

  4. 4.

    Ambrosio L., Faina L., March R.: Variational approximation of a second order free discontinuity problem in computer vision. SIAM J. Math. Anal., 32, 1171–1197 (2001)

    MATH  MathSciNet  Article  Google Scholar 

  5. 5.

    L. Ambrosio, N. Fusco, D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Oxford Mathematical Monographs, Oxford University Press, Oxford, 2000.

  6. 6.

    L. Ambrosio, D. Pallara, Partial regularity of free discontinuity sets,Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 24 1 (1997), 1–38.

  7. 7.

    L. Ambrosio, N. Fusco, D. Pallara, Partial regularity of free discontinuity sets II, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 24 1 (1997), 39–62.

  8. 8.

    Ambrosio L., Tortorelli V.M.: Approximation of functionals depending on jumps by elliptic functionals via \({\Gamma}\)-convergence. Comm. Pure Appl. Math., 43, 999–1036 (1990)

    MATH  MathSciNet  Article  Google Scholar 

  9. 9.

    L. Ambrosio, V.M. Tortorelli, On the approximation of free discontinuity problems, Boll. Un. Mat. Ital. B (7), 6 (1992), 105–123.

  10. 10.

    G. Aubert, P. Kornprobst, Mathematical problems in image processing, Partial Differential Equations and the Calculus of Variations, Applied Mathematical Sciences, 147 2nd ed., Springer, New York, 2006.

  11. 11.

    Babadjian J.-F., Chambolle A., Lemenant A.: Energy release rate for non smooth cracks in planar elasticity. Journal de l’Ecole Polytechnique - Mathématiques, 2, 117–152 (2015)

    MathSciNet  Article  Google Scholar 

  12. 12.

    C. Baiocchi, G. Buttazzo, F. Gastaldi, F. Tomarelli General Existence Theorems For Unilateral Problems in Continuum-Mechanics Arch. Rational Mech. Anal., 100 2 (1988), 149–189.

  13. 13.

    G. Bellettini, A. Coscia, Approximation of a functional depending on jumps and corners, Boll. Un. Mat. Ital. B (7), 8 (1994), 151–181.

  14. 14.

    M. Bergounioux, L. Piffet, A full second order variational model for multiscale texture analysis, Comput. Optim. Appl., doi:10.1007/s10589-012-9484-9, 2013.

  15. 15.

    M. Bergounioux, Mathematical Analysis of a Inf-Convolution Model for Image Processing, J. Optim. Theory Appl., doi:10.1007/s10957-015-0734-8, 2015.

  16. 16.

    M. Bertalmío, V. Caselles, S. Masnou, G. Sapiro, Inpainting, in “Encyclopedia of Computer Vision”, Springer, 2011.

  17. 17.

    A. Blake, A. Zisserman, Visual Reconstruction, The MIT Press, Cambridge, 1987.

  18. 18.

    Boccellari T., Tomarelli F.: About well-posedness of optimal segmentation for Blake & Zisserman functional. Istituto Lombardo (Rend. Scienze), 142, 237–266 (2008)

    MathSciNet  Google Scholar 

  19. 19.

    Boccellari T., Tomarelli F.: Generic uniqueness of minimizer for Blake & Zisserman functional. Revista Matematica Complutense, 26, 361–408 (2013)

    MathSciNet  Article  Google Scholar 

  20. 20.

    A. Bonnet, G. David, Crack-tip is a global Mumford-Shah minimizer, Astérisque, 274 (2001).

  21. 21.

    Braides A.: Lower semicontinuity conditions for functionals on jumps and creases. SIAM J. Math. Anal., 26, 1184–1198 (1995)

    MATH  MathSciNet  Article  Google Scholar 

  22. 22.

    A. Braides, A. De Franceschi, E. Vitali, A compactness result for a second-order variational discrete model, ESAIM: Mathematical Modelling and Numerical Analysis, 46 (2012), 389–410.

  23. 23.

    D. Bucur, S. Luckhaus, Monotonicity Formula and Regularity for General Free Discontinuity Problems, Arch. Rational Mech. Anal., 211 2 (2014), 489–511.

  24. 24.

    G. Buttazzo, F. Tomarelli, Compatibility Conditions For Nonlinear Neumann Problems Advances In Mathematics 89 2 (1991), 127–143.

  25. 25.

    L. Calatroni, B. During, C.B. Schonlieb, ADI splitting schemes for a 4th order nonlinear PDE from image processing, Discr.Cont.Dynamical Systems, Series A, Special Issue for Arieh Iserles 65th birthday, 34 (3), March (2014), 931–957.

  26. 26.

    M. Carriero, A. Leaci, F. Tomarelli, Free gradient discontinuities, in “Calculus of Variations, Homogeneization and Continuum Mechanics”, (Marseille 1993), 131–147, Ser. Adv. Math Appl. Sci., 18, World Sci. Publishing, River Edge, NJ, 1994.

  27. 27.

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

  28. 28.

    M. Carriero, A. Leaci, F. Tomarelli, Strong minimizers of Blake & Zisserman functional, Ann. Scuola Norm. Sup. Pisa Cl.Sci. (4), 25, n. 1–2 (1997), 257–285.

  29. 29.

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

  30. 30.

    M. Carriero, A. Leaci, F. Tomarelli, Necessary conditions for extremals of Blake & Zisserman functional, C. R. Math. Acad. Sci. Paris, 334, n.4 (2002), 343–348.

  31. 31.

    M. Carriero, A. Leaci, F. Tomarelli, Calculus of Variations and image segmentation, J. of Physiology, Paris, 97, n. 2-3 (2003), 343–353.

  32. 32.

    M. Carriero, A. Leaci, F. Tomarelli, Euler equations for Blake & Zisserman functional, Calc. Var. Partial Differential Equations, 32, n. 1 (2008), 81–110.

  33. 33.

    M. Carriero, A. Leaci, F. Tomarelli, Uniform density estimates for Blake & Zisserman functional, Discrete Contin. Dyn. Syst., Series A, 31, (4) (2011), 1129–1150.

  34. 34.

    M. Carriero, A. Leaci, F. Tomarelli, A Dirichlet problem with free gradient discontinuity, Advances in Mathematical Sciences and Applications, 20, n. 1 (2010), 107– 141.

  35. 35.

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

    MATH  MathSciNet  Article  Google Scholar 

  36. 36.

    M. Carriero, A. Leaci, F. Tomarelli, Free Gradient Discontinuity and Image Inpainting, J. Math. Sci. (N.Y.), 181, n. 6 (2012), 805–819.

  37. 37.

    M. Carriero, A. Leaci, F. Tomarelli Image inpainting via variational approximation of a Dirichlet problem with free discontinuity, Adv. Calc. Var., 7 (3) (2014), 267–295.

  38. 38.

    M. Carriero, A. Leaci, F. Tomarelli, Corrigendum to “A candidate local minimizer of Blake & Zisserman functional” [J. Math. Pures Appl. 96 (1) (2011), 58–87], J. Math. Pures Appl., to appear, doi:10.1016/j.matpur.2015.03.012

  39. 39.

    M. Carriero, A. Leaci, F. Tomarelli Almansi decomposition around the crack-tip and power series expansion for harmonic, biharmonic and poly-harmonic functions in open sets with a flat crack, to appear.

  40. 40.

    Caselles V., Haro G., Sapiro G., Verdera J.: On geometric variational models for inpainting surface holes. Computer Vision and Image Understanding, 111, 351–373 (2008)

    Article  Google Scholar 

  41. 41.

    T.F. Chan, J. Shen, Variational image inpainting, Comm. Pure Appl. Math., LVIII (2005), 579–619.

  42. 42.

    A. Coscia, Existence result for a new variational problem in one-dimensional segmentation theory, Ann. Univ. Ferrara-Sez. VII - Sc. Mat., XXXVII (1991), 185–203.

  43. 43.

    Dal Maso G., Morel J.M., Solimini S.: A variational method in image segmentation: existence and approximation results. Acta Math., 168, 89–151 (1992)

    MATH  MathSciNet  Article  Google Scholar 

  44. 44.

    G. David, Singular sets of minimizers for the Mumford-Shah functional. Progress in Mathematics, 233, Birkhäuser, Basel, 2005.

  45. 45.

    E. De Giorgi, Free discontinuity problems in calculus of variations, in “Frontiers in Pure & Appl. Math.”, R.Dautray Ed., North–Holland, Amsterdam, (1991), 55–61.

  46. 46.

    E. De Giorgi, Selected papers. Edited by L. Ambrosio, G. Dal Maso, M. Forti, M. Miranda and S. Spagnolo. Reprint of the 2006 edition. Springer Collected Works in Mathematics. Springer, Heidelberg, 2013.

  47. 47.

    De Giorgi E., Ambrosio L.: Un nuovo tipo di funzionale del calcolo delle variazioni. Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. 82, 199–210 (1988)

    MATH  MathSciNet  Google Scholar 

  48. 48.

    De Giorgi E., Carriero M., Leaci A.: Existence theorem for a minimum problem with free discontinuity set. Arch. Rational Mech. Anal., 108, 195–218 (1989)

    MATH  MathSciNet  Article  Google Scholar 

  49. 49.

    E. De Giorgi, T. Franzoni Su un tipo di convergenza variazionale, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8), 58, n. 6 (1975), 842–850.

  50. 50.

    C. De Lellis, M. Focardi Density lower bound estimates for local minimizers of the 2d Mumford-Shah energy, Manuscripta Math., 142, n. 1–2 (2013), 215–232.

  51. 51.

    Duffin R.J.: Continuation of biharmonic functions by reflection. Duke Math. J., 22, 313–324 (1955)

    MATH  MathSciNet  Article  Google Scholar 

  52. 52.

    S. Esedoglu, J.H. Shen, Digital inpainting based on the Mumford-Shah-Euler image model, Eur. J. Appl. Math., 13, n. 4 (2002), 353–370.

  53. 53.

    H. Federer, Geometric Measure Theory, Springer, Berlin, 1969.

  54. 54.

    I. Fonseca, G. Leoni, F. Maggi, M. Morini, Exact reconstruction of damaged color images using a total variation model, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 5 (2010), 1291–1331.

  55. 55.

    M. Fornasier, C.B. Schonlieb, Subspace correction methods for total variation and \({\ell^{1}}\) minimization, SIAM J.Num.An., 47 (5) (2009), 3397–3428.

  56. 56.

    Fusco N.: An Overview of the Mumford-Shah Problem. Milan J. Math., 71, 95–119 (2003)

    MATH  MathSciNet  Article  Google Scholar 

  57. 57.

    F.A. Lops, F. Maddalena, S. Solimini, Hölder continuity conditions for the solvability of Dirichlet problems involving functionals with free discontinuities, Ann. Inst. H. Poincaré Anal. Non Linéaire, 18 n. 6 (2001), 639–673.

  58. 58.

    P.A. Markovich, Applied Partial Differential Equations: a Visual Approach, Springer, New York, 2007.

  59. 59.

    L. Modica, S. Mortola, Un esempio di \({\Gamma}\)-convergenza, Boll. Un. Mat. ltal. 5 14-B (1977), 285–299.

  60. 60.

    J.M. Morel, S. Solimini, Variational Models in Image Segmentation, Progr. Nonlinear Differential Equations Appl., 14, Birkhäuser, Basel, 1995.

  61. 61.

    D. Mumford, J. Shah, Optimal approximation by piecewise smooth functions and associated variational problems, Comm. Pure Appl. Math., XLII (1989), 577–685.

  62. 62.

    D. Pallara, Some new results on functions of bounded variation, Rend. Accad. Naz. delle Scienze (dei XL), (108) XIV (1990), 295–321.

  63. 63.

    G. Savaré, F. Tomarelli, Superposition and chain rule for bounded Hessian functions, Adv.Math., 140, 2 (1998), 237–281.

  64. 64.

    C.-B. Schönlieb, A. Bertozzi, Unconditionally stable schemes for higher order inpainting, Communications in Mathematical Sciences, 9, 2 (2011), 413–457.

  65. 65.

    J. Verdera, V. Caselles, M. Bertalmio, G. Sapiro, Inpainting Surface Holes, In: Int. Conference on Image Processing (2003), 903–906.

  66. 66.

    M. Zanetti, A. Vitti, The Blake-Zisserman model for digital surface models segmentation, In: ISPRS Ann. Photogramm. Remote Sens. Spatial Inf. Sci., II-5/W2, doi:10.5194/isprsannals-II-5-W2-355-2013 (2013), 355–360.

Download references

Author information



Corresponding author

Correspondence to Antonio Leaci.

Additional information

This work is supported by PRIN 2010-2011 M.I.U.R (Progetto “Calculus of Variations”).

Lecture given by A. Leaci in the Seminario Matematico e Fisico di Milano on February 16, 2015

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Carriero, M., Leaci, A. & Tomarelli, F. A Survey on the Blake–Zisserman Functional. Milan J. Math. 83, 397–420 (2015). https://doi.org/10.1007/s00032-015-0246-x

Download citation

Mathematics Subject Classification

  • Primary 49J45
  • Secondary 49K20


  • Calculus of variations
  • image segmentation
  • inpainting
  • free discontinuity
  • Almansi decomposition
  • crack-tip