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Frontiers of Mathematics in China

ISSN 1673-3452

ISSN 1673-3576(Online)

CN 11-5739/O1

Postal Subscription Code 80-964

2018 Impact Factor: 0.565

Front. Math. China    2015, Vol. 10 Issue (6) : 1355-1388    https://doi.org/10.1007/s11464-015-0489-4
RESEARCH ARTICLE
Minimizers of anisotropic Rudin-Osher-Fatemi models
Ruiling JIA,Meiyue JIANG()
LMAM, School of Mathematical Sciences, Peking University, Beijing 100871, China
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Abstract

We give the explicit formulas of the minimizers of the anisotropic Rudin-Osher-Fatemi models

E1φ(u)=Ωφo(Du)dx+λΩ|uf|dx,uBV(Ω),E2φ(u)=Ωφo(Du)dx+λΩ(uf)2dx,uBV(Ω),

where Ω?2 is a domain, φo is an anisotropic norm on ?2, and f is a solution of the anisotropic 1-Laplacian equations.

Keywords Anisotropic Rudin-Osher-Fatemi (ROF) model      Euler-Lagrange equation      φ-curvature')" href="#">φ-curvature     
Corresponding Author(s): Meiyue JIANG   
Issue Date: 12 October 2015
 Cite this article:   
Ruiling JIA,Meiyue JIANG. Minimizers of anisotropic Rudin-Osher-Fatemi models[J]. Front. Math. China, 2015, 10(6): 1355-1388.
 URL:  
https://academic.hep.com.cn/fmc/EN/10.1007/s11464-015-0489-4
https://academic.hep.com.cn/fmc/EN/Y2015/V10/I6/1355
1 Alliney S. A property of the minimum vectors of a regularizing functional defined by means of the absolute norm. IEEE Trans Signal Process, 1997, 45: 913−917
https://doi.org/10.1109/78.564179
2 Ambrosio L, Fusco N, Pallara D. Functions of Bounded Variation and Free Discontinuity Problems. Oxford: Oxford University Press, 2000
3 Andreu-Vaillo F, Ballester C, Caselles V, Mazón J M. Minimizing total variation flow. Differential Integral Equations, 2001, 14: 321−360
4 Andreu-Vaillo F, Caselles V, Mazón J M. Parabolic Quasilinear Equations Minimizing Linear Growth Functionals. Basel-Boston-Berlin: Birkhäuser, 2004
https://doi.org/10.1007/978-3-0348-7928-6
5 Bellettini G. Anisotropic and crystalline mean curvature flow. In: Bao D, Bryant R L, Chern S S, Shen Z M, eds. A Sampler of Riemann-Finsler Geometry. Math Sci Res Inst Publ, 50. Cambridge: Cambridge Press, 2004, 49−82
6 Bellettini G, Caselles V, Novaga M. The total variation flow in RN. J Differential Equations, 2002, 184: 475−525
https://doi.org/10.1006/jdeq.2001.4150
7 Chambolle A, Lions P L. Image recovery via tota<?Pub Caret?>l variation minimization and related problems. Numer Math, 1997, 76: 167−188
https://doi.org/10.1007/s002110050258
8 Chan T F, Esedoglu S. Aspects of total variation regularized L1 function approximation. SIAM J Appl Math, 2005, 65: 1817−1837
https://doi.org/10.1137/040604297
9 Chan T F, Esedoglu S, Nikolova M. Algorithms for finding global minimizers of denoising and segmentation models. SIAM J Appl Math, 2006, 66: 1632−1648
https://doi.org/10.1137/040615286
10 Chan T F, Esedoglu S, Park F, Yip A. Total variation image restoration: overview and recent developments. In: Paragios N, Chen Y M, Faugeras O, eds. Handbook of Mathematical Models in Computer Vision. New York: Springer, 2006, 17−31
https://doi.org/10.1007/0-387-28831-7_2
11 Chang K-C. The spectrum of the 1-Laplace operator. Commun Contemp Math, 2009, 11: 865−894
https://doi.org/10.1142/S0219199709003570
12 Esedoglu S, Osher S. Decomposition of images by the anisotropic Rudin-Osher-Fatemi model. Comm Pure Appl Math, 2004, 57: 1609−1626
https://doi.org/10.1002/cpa.20045
13 Giusti E. On the equation of surfaces of prescribed mean curvature: existence and uniqueness without boundary conditions. Invent Math, 1978, 46: 111−137
https://doi.org/10.1007/BF01393250
14 Jiang M-Y. Eigenfunctions of 1-Laplacian and Minimizers of the Rudin-Osher-Fatemi Functionals. Research Report, No 49. Institute of Mathematics Peking University, 2012
15 Jiang M-Y. Minimizers of the Rudin-Osher-Fatemi functionals in bounded domains. Adv Nonlinear Stud, 2014, 14: 31−46
16 Kawohl B, Schuricht F. Dirichlet problems for the 1-Laplacian operator, including the eigenvalue problem. Commun Contemp Math, 2007, 9: 515−543
https://doi.org/10.1142/S0219199707002514
17 Meyer Y. Oscillating Patterns in Image Processing and Nonlinear Evolution Equations. The Fifteenth Dean Jacqueline B. Lewis Memorial Lectures. Providence: Amer Math Soc, 2001
18 Rudin L, Osher S, Fatemi E. Nonlinear total variation based noise removal algorithms. Phys D, 1992, 60: 259−268
https://doi.org/10.1016/0167-2789(92)90242-F
19 Strong D, Chan T F. Exact solutions to total variation regularization problems. UCLA CAM Report, 1996
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