A quantitative variational analysis of the staircasing phenomenon for a second order regularization of the Perona-Malik functional

Gobbino, Massimo; Picenni, Nicola

doi:10.1090/tran/8841

A quantitative variational analysis of the staircasing phenomenon for a second order regularization of the Perona-Malik functional
HTML articles powered by AMS MathViewer

by Massimo Gobbino and Nicola Picenni;

Trans. Amer. Math. Soc. 376 (2023), 5307-5375

DOI: https://doi.org/10.1090/tran/8841

Published electronically: May 9, 2023

HTML | PDF | Request permission

Abstract:

We consider the Perona-Malik functional in dimension one, namely an integral functional whose Lagrangian is convex-concave with respect to the derivative, with a convexification that is identically zero. We approximate and regularize the functional by adding a term that depends on second order derivatives multiplied by a small coefficient.

We investigate the asymptotic behavior of minima and minimizers as this small parameter vanishes. In particular, we show that minimizers exhibit the so-called staircasing phenomenon, namely they develop a sort of microstructure that looks like a piecewise constant function at a suitable scale.

Our analysis relies on Gamma-convergence results for a rescaled functional, blow-up techniques, and a characterization of local minimizers for the limit problem. This approach can be extended to more general models.

References

Giovanni Alberti and Stefan Müller, A new approach to variational problems with multiple scales, Comm. Pure Appl. Math. 54 (2001), no. 7, 761–825. MR 1823420, DOI 10.1002/cpa.1013
Roberto Alicandro, Andrea Braides, and Maria Stella Gelli, Free-discontinuity problems generated by singular perturbation, Proc. Roy. Soc. Edinburgh Sect. A 128 (1998), no. 6, 1115–1129. MR 1664085, DOI 10.1017/S0308210500027256
H. Amann, Time-delayed Perona-Malik type problems, Acta Math. Univ. Comenian. (N.S.) 76 (2007), no. 1, 15–38. MR 2331050
L. Ambrosio, A compactness theorem for a new class of functions of bounded variation, Boll. Un. Mat. Ital. B (7) 3 (1989), no. 4, 857–881 (English, with Italian summary). MR 1032614
Luigi Ambrosio, Nicola Fusco, and Diego Pallara, Functions of bounded variation and free discontinuity problems, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 2000. MR 1857292
Giovanni Bellettini, Antonin Chambolle, and Michael Goldman, The $\Gamma$-limit for singularly perturbed functionals of Perona-Malik type in arbitrary dimension, Math. Models Methods Appl. Sci. 24 (2014), no. 6, 1091–1113. MR 3187192, DOI 10.1142/S0218202513500772
G. Bellettini and G. Fusco, The $\Gamma$-limit and the related gradient flow for singular perturbation functionals of Perona-Malik type, Trans. Amer. Math. Soc. 360 (2008), no. 9, 4929–4987. MR 2403710, DOI 10.1090/S0002-9947-08-04495-4
Giovanni Bellettini, Giorgio Fusco, and Nicola Guglielmi, A concept of solution and numerical experiments for forward-backward diffusion equations, Discrete Contin. Dyn. Syst. 16 (2006), no. 4, 783–842. MR 2257160, DOI 10.3934/dcds.2006.16.783
G. Bellettini, M. Novaga, M. Paolini, and C. Tornese, Convergence of discrete schemes for the Perona-Malik equation, J. Differential Equations 245 (2008), no. 4, 892–924. MR 2427400, DOI 10.1016/j.jde.2008.05.003
Michiel Bertsch, Lorenzo Giacomelli, and Alberto Tesei, Measure-valued solutions to a nonlinear fourth-order regularization of forward-backward parabolic equations, SIAM J. Math. Anal. 51 (2019), no. 1, 374–402. MR 3911715, DOI 10.1137/18M1203821
M. Bertsch, F. Smarrazzo, and A. Tesei, On a class of forward-backward parabolic equations: formation of singularities, J. Differential Equations 269 (2020), no. 9, 6656–6698. MR 4107065, DOI 10.1016/j.jde.2020.05.007
Andrea Braides and Valerio Vallocchia, Static, quasistatic and dynamic analysis for scaled Perona-Malik functionals, Acta Appl. Math. 156 (2018), 79–107. MR 3829576, DOI 10.1007/s10440-018-0155-4
Francine Catté, Pierre-Louis Lions, Jean-Michel Morel, and Tomeu Coll, Image selective smoothing and edge detection by nonlinear diffusion, SIAM J. Numer. Anal. 29 (1992), no. 1, 182–193. MR 1149092, DOI 10.1137/0729012
Maria Colombo and Massimo Gobbino, Slow time behavior of the semidiscrete Perona-Malik scheme in one dimension, SIAM J. Math. Anal. 43 (2011), no. 6, 2564–2600. MR 2873232, DOI 10.1137/100818698
Ennio De Giorgi, Conjectures concerning some evolution problems, Duke Math. J. 81 (1996), no. 2, 255–268 (Italian, with English summary). A celebration of John F. Nash, Jr. MR 1395405, DOI 10.1215/S0012-7094-96-08114-4
Selim Esedoḡlu, An analysis of the Perona-Malik scheme, Comm. Pure Appl. Math. 54 (2001), no. 12, 1442–1487. MR 1852979, DOI 10.1002/cpa.3008
Selim Esedoḡlu, Stability properties of the Perona-Malik scheme, SIAM J. Numer. Anal. 44 (2006), no. 3, 1297–1313. MR 2231865, DOI 10.1137/S0036142903424817
Marina Ghisi and Massimo Gobbino, Gradient estimates for the Perona-Malik equation, Math. Ann. 337 (2007), no. 3, 557–590. MR 2274543, DOI 10.1007/s00208-006-0047-1
Patrick Guidotti, A new nonlocal nonlinear diffusion of image processing, J. Differential Equations 246 (2009), no. 12, 4731–4742. MR 2523301, DOI 10.1016/j.jde.2009.03.017
Patrick Guidotti, A backward-forward regularization of the Perona-Malik equation, J. Differential Equations 252 (2012), no. 4, 3226–3244. MR 2871799, DOI 10.1016/j.jde.2011.10.022
Patrick Guidotti and James V. Lambers, Two new nonlinear nonlocal diffusions for noise reduction, J. Math. Imaging Vision 33 (2009), no. 1, 25–37. MR 2471042, DOI 10.1007/s10851-008-0108-z
Satyanad Kichenassamy, The Perona-Malik paradox, SIAM J. Appl. Math. 57 (1997), no. 5, 1328–1342. MR 1470926, DOI 10.1137/S003613999529558X
Seonghak Kim and Baisheng Yan, On Lipschitz solutions for some forward-backward parabolic equations, Ann. Inst. H. Poincaré C Anal. Non Linéaire 35 (2018), no. 1, 65–100. MR 3739928, DOI 10.1016/j.anihpc.2017.03.001
M. Morini and M. Negri, Mumford-Shah functional as $\Gamma$-limit of discrete Perona-Malik energies, Math. Models Methods Appl. Sci. 13 (2003), no. 6, 785–805. MR 1992879, DOI 10.1142/S0218202503002726
Stefan Müller, Singular perturbations as a selection criterion for periodic minimizing sequences, Calc. Var. Partial Differential Equations 1 (1993), no. 2, 169–204. MR 1261722, DOI 10.1007/BF01191616
P. Perona and J. Malik, Scale-space and edge detection using anisotropic diffusion, Technical Report, EECS Department, University of California, Berkeley, December 1988.
I. Ripoli, Investigation on a Perona-Malik problem, Master Thesis, University of Pisa, 2020.