Abstract

The purpose of this paper is to analyze the asymptotic behaviour in the spirit of Γ-convergence of BMO-type functionals related to the total variation of a function u. Moreover, we deal with a minimization problem coming from applications in image processing.

Author information

Contact details are reproduced from the original publication and may be historical.

Roberta Schiattarella

Dip. di Matematica e Applicazioni "R. Caccioppoli", Università di Napoli "Federico II", Napoli, Italy

roberta.schiattarella@unina.it

Keywords

  • Total variation
  • bounded variation
  • BMO
  • Gamma-convergence

Mathematics Subject Classification

49Q20, 49J45, 26B30, 26D10

References

  1. L. Ambrosio, J. Bourgain, H. Brezis, A. Figalli: BMO-type norms related to the perimeter of sets, Comm. Pure Appl. Math. 69 (2016) 1062–1086.https://doi.org/10.1002/cpa.21620Google Scholar
  2. L. Ambrosio, G. Comi: Anisotropic surface measures as limits of volume fractions, Current Res. Nonlinear Anal. 135 (2018) 1–32.https://doi.org/10.1007/978-3-319-89800-1_1Google Scholar
  3. L. Ambrosio, N. Fusco, D. Pallara: Functions of Bounded Variation and Free Discontinuity Problems, Oxford Mathematical Monographs, Oxford University Press, New York (2000).https://doi.org/10.1093/oso/9780198502456.001.0001Google Scholar
  4. A. Arroyo-Rabasa, P. Bonicatto, G. Del Nin: Representation of the total variation as a Γ-limit of BMO-type seminorms, arXiv: 2112.03832 (2021).Google Scholar
  5. G. Aubert, P. Kornprobst: Can the nonlocal characterization of Sobolev spaces by Bourgain et al. be useful for solving variational problems?, SIAM J. Numer. Anal. 47 (2009) 844–860.https://doi.org/10.1137/070696751Google Scholar
  6. J. Bourgain, H. Brezis, P. Mironescu: A new function space and applications, J. Eur. Math. Soc. 17 (2015) 2083–2101.https://doi.org/10.4171/jems/551Google Scholar
  7. J. Bourgain, H. M. Nguyen: A new characterization of Sobolev spaces, C. R. Acad. Sci. Paris 343 (2006) 75–80.https://doi.org/10.1016/j.crma.2006.05.021Google Scholar
  8. H. Brezis, H. M. Nguyen: Two subtle convex nonlocal approximations of the BV-norm, Nonlinear Anal. 137 (2016) 222–245.https://doi.org/10.1016/j.na.2016.02.005Google Scholar
  9. H. Brezis, H. M. Nguyen: Non-local functionals related to the total variation and connections with image processing, Ann. Part. Diff. Equations 4/1 (2018), art. no. 9, 77p.https://doi.org/10.1007/s40818-018-0044-1Google Scholar
  10. T. F. Chan, S. Esedoglu: Aspects of total variation regularized L1 function approximation, SIAM J. Appl. Math. 65/5 (2005) 1817–1837.https://doi.org/10.1137/040604297Google Scholar
  11. G. Dal Maso: An Introduction to Γ-Convergence, Birkhäuser, Boston (1993).https://doi.org/10.1007/978-1-4612-0327-8Google Scholar
  12. G. De Philippis, N. Fusco, A. Pratelli: On the approximation of SBV functions, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 28 (2017) 369–413.https://doi.org/10.4171/rlm/768Google Scholar
  13. F. Demengel, R. Temam: Convex functions of a measure and applications, Indiana Univ. Math. J. 33/5 (1984) 673–709.https://doi.org/10.1512/iumj.1984.33.33036Google Scholar
  14. F. Farroni, N. Fusco, S. Guarino Lo Bianco, R. Schiattarella: A formula for the anisotropic total variation of BV functions, J. Funct. Anal. 278/9 (2020) 108–451.https://doi.org/10.1016/j.jfa.2019.108451Google Scholar
  15. F. Farroni, S. Guarino Lo Bianco, R. Schiattarella: BMO-type seminorms generating Sobolev functions, J. Math. Anal. Appl. 491 (2020), art. no. 124298.https://doi.org/10.1016/j.jmaa.2020.124298Google Scholar
  16. M. Fuchs, J. Muller: A higher order TV-type variational problem related to the denoising and inpainting of images, Nonlinear Anal. 154 (2017) 122–147.https://doi.org/10.1016/j.na.2016.08.004Google Scholar
  17. N. Fusco, G. Moscariello, C. Sbordone: A formula for the total variation of SBV functions, J. Funct. Anal. 270/1 (2016) 419–446.https://doi.org/10.1016/j.jfa.2015.06.010Google Scholar
  18. N. Fusco, G. Moscariello, C. Sbordone: BMO-type seminorms and Sobolev functions, ESAIM Control Optim. Calc. Var. 24/2 (2018) 835–847.https://doi.org/10.1051/cocv/2017023Google Scholar
  19. G. Gilboa, S. Osher: Nonlocal operators with applications to image processing, Multiscale Model. Simul. 7 (2008) 1005–1028.https://doi.org/10.1137/070698592Google Scholar
  20. S. Guarino Lo Bianco, R. Schiattarella: A BMO-type characterization of higher order Sobolev spaces, Potential Anal. 59/3 (2023) 917–932.https://doi.org/10.1007/s11118-022-09987-8Google Scholar
  21. R. L. Jerrard, H. M. Soner: Functions of bounded higher variation, Indiana Univ. Math. J. 51 (2002) 645–677.https://doi.org/10.1512/iumj.2002.51.2229Google Scholar
  22. M. Marcus: Exceptional sets with respect to Lebesgue differentiation of functions in Sobolev spaces, Ann. Sc. Norm. Pisa, Serie VI 1 (1974) 113–130.Google Scholar
  23. L. I. Rudin, S. Osher, E. Fatemi: Nonlinear total variation based noise removal algorithms, Physica D: Nonlinear Phen. 60 (1992) 259–268.https://doi.org/10.1016/0167-2789(92)90242-FGoogle Scholar

Suggested citation

S. Guarino Lo Bianco, R. Schiattarella. “BMO-Type Functionals Related to the Total Variation and Connection to Denoising Models.” Journal of Convex Analysis 31 (2024), No. 2, 671–688. https://doi.org/10.68381/jca31032

Download BibTeXPublished by Heldermann Verlag.

Published by Heldermann Verlag, 2024. Rights now held by Banach Press LLC.