Annals of PDE

, 4:9 | Cite as

Non-local Functionals Related to the Total Variation and Connections with Image Processing

  • Haïm BrezisEmail author
  • Hoai-Minh Nguyen


We present new results concerning the approximation of the total variation, \(\int _{\Omega } |\nabla u|\), of a function u by non-local, non-convex functionals of the form
$$\begin{aligned} \Lambda _\delta (u) = \int _{\Omega } \int _{\Omega } \frac{\delta \varphi \big ( |u(x) - u(y)|/ \delta \big )}{|x - y|^{d+1}} \, dx \, dy, \end{aligned}$$
as \(\delta \rightarrow 0\), where \(\Omega \) is a domain in \(\mathbb {R}^d\) and \(\varphi : [0, + \infty ) \rightarrow [0, + \infty )\) is a non-decreasing function satisfying some appropriate conditions. The mode of convergence is extremely delicate and numerous problems remain open. De Giorgi’s concept of \(\Gamma \)-convergence illuminates the situation, but also introduces mysterious novelties. The original motivation of our work comes from Image Processing.


Total variation Bounded variation Non-local functional Non-convex functional \(\Gamma \)-Convergence Sobolev spaces 

Mathematics Subject Classification

49Q20 26B30 46E35 28A75 



We are extremely grateful to J. Bourgain for sharing fruitful ideas which led to the joint work [10] with the second author. Some of the techniques developed in [10] served as a source of inspiration for many subsequent works. The first author (H.B.) warmly thanks R. Kimmel and J. M. Morel for useful discussions concerning Image Processing.


  1. 1.
    Allaire, G.: Numerical Analysis and Optimization: An Introduction to Mathematical Modelling and Numerical Simulation. Oxford University Press, Oxford (2007)zbMATHGoogle Scholar
  2. 2.
    Ambrosio, L., Bourgain, J., Brezis, H., Figalli, A.: Perimeter of sets and BMO-type norms. C. R. Acad. Sc. Paris 352, 697–698 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Ambrosio, L., Bourgain, J., Brezis, H., Figalli, A.: BMO-type norms related to the perimeter of sets. Commun. Pure Appl. Math. 69, 1062–1086 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Ambrosio, L., De Philippis, G., Martinazzi, L.: Gamma-convergence of nonlocal perimeter functionals. Manuscr. Math. 134, 377–403 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 6.
    Antonucci, C., Gobbino, M., Migliorini, M., Picenni, N.: Optimal constants for a non-local approximation of Sobolev norms and total variation. Preprint (2017). arXiv:1708.01231
  6. 7.
    Aubert, G., Kornprobst, P.: Can the nonlocal characterization of Sobolev spaces by Bourgain et al. be useful for solving variational problems? SIAM J. Numer. Anal. 47, 844–860 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 8.
    Bourgain, J., Brezis, H., Mironescu, P.: Another look at Sobolev spaces. In: Menaldi, J.L., Rofman, E., Sulem, A. (eds.) Optimal Control and Partial Differential Equations, A Volume in Honour of A. Bensoussan’s 60th Birthday, pp. 439–455. IOS Press, Amsterdam (2001)Google Scholar
  8. 9.
    Bourgain, J., Brezis, H., Mironescu, P.: Limiting embedding theorems for \(W^{s, p}\) when \(s \uparrow 1\) and applications. J. Anal. Math. 87, 77–101 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 10.
    Bourgain, J., Brezis, H., Mironescu, P.: A new function space and applications. J. Eur. Math. Soc. 17, 2083–2101 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 11.
    Bourgain, J., Nguyen, H.-M.: A new characterization of Sobolev spaces. C. R. Acad. Sci. Paris 343, 75–80 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 12.
    Braides, A.: \(\Gamma \)-Convergence for Beginners, Oxford Lecture Series in Mathematics and Its Applications, vol. 22. Oxford University Press, Oxford (2002)Google Scholar
  12. 13.
    Brezis, H.: How to recognize constant functions. Connections with Sobolev spaces, Volume in honor of M. Vishik, Uspekhi Mat. Nauk 57 (2002), 59–74; English translation in Russian Math. Surveys 57 (2002), 693–708Google Scholar
  13. 14.
    Brezis, H.: Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext. Springer, Berlin (2010)CrossRefGoogle Scholar
  14. 15.
    Brezis, H.: New approximations of the total variation and filters in Imaging. Atti. Accad. Naz. Lincei Rend. Lincei Mat. Appl. 26, 223–240 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 16.
    Brezis, H., Mironescu, P.: Sobolev Maps with Values into the Circle. Birkhäuser (in preparation)Google Scholar
  16. 17.
    Brezis, H., Nguyen, H.-M.: On a new class of functions related to VMO. C. R. Acad. Sci. Paris 349, 157–160 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 18.
    Brezis, H., Nguyen, H.-M.: Two subtle convex nonlocal approximation of the \(BV\)-norm. Nonlinear Anal. 137, 222–245 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 19.
    Brezis, H., Nguyen, H.-M.: The BBM formula revisited. Atti. Accad. Naz. Lincei Rend. Lincei Mat. Appl. 27, 515–533 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 20.
    Brezis, H., Nguyen, H.-M.: Non-convex, non-local functionals converging to the total variation. C. R. Acad. Sci. Paris 355, 24–27 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 21.
    Brezis, H., Nguyen, H.-M.: Paper in preparationGoogle Scholar
  21. 22.
    Buades, A., Coll, B., Morel, J.M.: A review of image denoising algorithms, with a new one. Multiscale Model. Simul. 4, 490–530 (2005); Updated version in SIAM Review 52 (2010), 113–147Google Scholar
  22. 23.
    Buades, A., Coll, B., Morel, J.M.: Neighborhood filters and PDE’s. Numer. Math. 105, 1–34 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 24.
    Buades, A., Coll, B., Morel, J.M.: Non-local means denoising. Image Process. Online 1, 208–212 (2011)Google Scholar
  24. 25.
    Bucur, C., Valdinoci, E.: Nonlocal Diffusion and Applications, Lecture Notes of the Unione Matematica Italiana, vol. 20. Springer, Berlin (2016)Google Scholar
  25. 26.
    Caffarelli, L., Roquejoffre, J.M., Savin, O.: Nonlocal minimal surfaces. Commun. Pure Appl. Math. 63, 1111–1144 (2010)MathSciNetzbMATHGoogle Scholar
  26. 27.
    Caffarelli, L., Valdinoci, E.: Uniform estimates and limiting arguments for nonlocal minimal surfaces. Calc. Var. PDE 41, 203–240 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 28.
    Cai, J.-F., Dong, B., Osher, S., Shen, Z.: Image restoration, total variation, wavelet frames, and beyond. J. Am. Math. Soc. 25, 1033–1089 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 29.
    Chambolle, A., Lions, P.-L.: Image recovery via total variation minimization and related problems. Numer. Math. 76, 167–188 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 30.
    Chan, T., Esedoglu, S., Park, F., Yip, A.: Recent developments in total variation image restoration. In: Paragios, N., Chen, Y., Faugeras, O. (eds.) Mathematical Models in Computer Vision, pp. 17–30. Springer, Berlin (2005)Google Scholar
  30. 31.
    Dal Maso, G.: An Introduction to \(\Gamma \)-Convergence, Progress in Nonlinear Differential Equations and their Applications, vol. 8. Birkhäuser Boston Inc., Boston (1993)Google Scholar
  31. 32.
    Davila, J.: On an open question about functions of bounded variation. Calc. Var. Partial Differ. Equ. 15, 519–527 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 33.
    Figalli, A., Fusco, N., Maggi, F., Millot, V., Morini, M.: Isoperimetry and stability properties of balls with respect to nonlocal energies. Commun. Math. Phys. 336, 441–507 (2015)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  33. 34.
    Evans, L.C., Gariepy, R.F.: Measure Theory and Fine Properties of Functions, Studies in Advanced Mathematics. CRC Press, Boca Raton (1992)zbMATHGoogle Scholar
  34. 35.
    Fusco, N.: The quantitative isoperimetric inequality and related topics. Bull. Math. Sci. 5, 517–607 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 36.
    Gilboa, G., Osher, S.: Nonlocal linear image regularization and supervised segmentation. Multiscale Model. Simul. 6, 595–630 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 37.
    Gilboa, G., Osher, S.: Nonlocal operators with applications to image processing. Multiscale Model. Simul. 7, 1005–1028 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 38.
    Haddad, A., Meyer, Y.: An improvement of Rudin–Osher–Fatemi model. Appl. Comput. Harmon. Anal. 22, 319–334 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 39.
    Kindermann, S., Osher, S., Jones, P.W.: Deblurring and denoising of images by nonlocal functionals. Multiscale Model. Simul. 4, 1091–1115 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 40.
    Lee, J.S.: Digital image smoothing and the sigma filter. Comput. Vis. Graph. Image Process. 24, 255–269 (1983)CrossRefGoogle Scholar
  40. 41.
    Leoni, G., Spector, D.: Characterization of Sobolev and BV spaces. J. Funct. Anal. 261, 2926–2958 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 42.
    Leoni, G., Spector, D.: Corrigendum to “Characterization of Sobolev and BV spaces”. J. Funct. Anal. 266, 1106–1114 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 43.
    Nguyen, H.-M.: Some new characterizations of Sobolev spaces. J. Funct. Anal. 237, 689–720 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  43. 44.
    Nguyen, H.-M.: \(\Gamma \)-convergence and Sobolev norms. C. R. Acad. Sci. Paris 345, 679–684 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  44. 45.
    Nguyen, H.-M.: Further characterizations of Sobolev spaces. J. Eur. Math. Soc. 10, 191–229 (2008)MathSciNetzbMATHGoogle Scholar
  45. 46.
    Nguyen, H.-M.: \(\Gamma \)-convergence, Sobolev norms, and BV functions. Duke Math. J. 157, 495–533 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  46. 47.
    Nguyen, H.-M.: Some inequalities related to Sobolev norms. Calc. Var. Partial Differ. Equ. 41, 483–509 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  47. 48.
    Nguyen, H.-M.: Estimates for the topological degree and related topics. J. Fixed Point Theory 15, 185–215 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  48. 49.
    Paris, S., Kornprobst, P., Tumblin, J., Durand, F.: Bilateral filtering: theory and applications. Found. Trends Comput. Graph. Vis. 4, 1–73 (2008)CrossRefzbMATHGoogle Scholar
  49. 50.
    Ponce, A.: A new approach to Sobolev spaces and connections to \(\Gamma \)-convergence. Calc. Var. Partial Differ. Equ. 19, 229–255 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  50. 51.
    Ponce, A.: Personal communication to the authors (2005)Google Scholar
  51. 52.
    Ponce, A., Van Schaftingen, J.: Personal communication to the authors (2005)Google Scholar
  52. 53.
    Rudin, L.I., Osher, S., Fatemi, E.: Nonlinear total variation based noise removal algorithms. Phys. D 60, 259–268 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  53. 54.
    Smith, S.M., Brady, J.M.: SUSAN—a new approach to low level image processing. Int. J. Comput. Vis. 23, 45–78 (1997)CrossRefGoogle Scholar
  54. 55.
    Van Schaftingen, J., Willem, M.: Set Transformations, Symmetrizations and Isoperimetric Inequalities, in Nonlinear Analysis and Applications to Physical Sciences, pp. 135–152. Springer Italia, Milan (2004)zbMATHGoogle Scholar
  55. 56.
    Yaroslavsky, L.P.: Digital Picture Processing. An Introduction. Springer, Berlin (1985)CrossRefGoogle Scholar
  56. 57.
    Yaroslavsky, L.P., Eden, M.: Fundamentals of Digital Optics. Springer, Berlin (1996)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mathematics, Hill Center, Busch CampusRutgers UniversityPiscatawayUSA
  2. 2.Departments of Mathematics and Computer ScienceTechnion, Israel Institute of TechnologyHaifaIsrael
  3. 3.Laboratoire Jacques-Louis Lions UPMCParisFrance
  4. 4.EPFL SB MATHAA CAMALausanneSwitzerland

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