American Institute of Mathematical Sciences

Advanced
December  2007, 6(4): 917-936. doi: 10.3934/cpaa.2007.6.917

Analysis of the TV regularization and $H^{-1}$ fidelity model for decomposing animage into cartoon plus texture

C.M. Elliott 1, and  S. A. Smitheman 1,

1. 

Department of Mathematics, University of Sussex, Brighton, BN1 9RF, United Kingdom, United Kingdom

Received  January 2006 Revised  March 2007 Published  September 2007

We study the Osher-Solé-Vese model [11], which is the gradient flow of an energy consisting of the total variation functional plus an $H^{-1}$ fidelity term. A variational inequality weak formulation for this problem is proposed along the lines of that of Feng and Prohl [7] for the Rudin-Osher-Fatemi model [12]. A regularized energy is considered, and the minimization problems corresponding to both the original and regularized energies are shown to be well-posed. The Galerkin method of Lions [9] is used to prove the well-posedness of the weak problem corresponding to the regularized energy. By letting the regularization parameter $\epsilon$ tend to $0$, we recover the well-posedness of the weak problem corresponding to the original energy. Further, we show that for both energies the solution of the weak problem tends to the minimizer of the energy as $t \to \infty$. Finally, we find the rate of convergence of the weak solution of the regularized problem to that of the original one as $\epsilon \downarrow 0$.
Keywords: cartoon plus texture, TV and $H^{-1}$ model, Image decomposition, fourth order parabolic equation..
Mathematics Subject Classification: 35K35, 35K55, 35K3.
Citation: C.M. Elliott, S. A. Smitheman. Analysis of the TV regularization and $H^{-1}$ fidelity model for decomposing animage into cartoon plus texture. Communications on Pure and Applied Analysis, 2007, 6 (4) : 917-936. doi: 10.3934/cpaa.2007.6.917
[1]

John B. Greer, Andrea L. Bertozzi. $H^1$ Solutions of a class of fourth order nonlinear equations for image processing. Discrete and Continuous Dynamical Systems, 2004, 10 (1&2) : 349-366. doi: 10.3934/dcds.2004.10.349
[2]

Jian-Feng Cai, Raymond H. Chan, Zuowei Shen. Simultaneous cartoon and texture inpainting. Inverse Problems and Imaging, 2010, 4 (3) : 379-395. doi: 10.3934/ipi.2010.4.379
[3]

Feishe Chen, Lixin Shen, Yuesheng Xu, Xueying Zeng. The Moreau envelope approach for the L1/TV image denoising model. Inverse Problems and Imaging, 2014, 8 (1) : 53-77. doi: 10.3934/ipi.2014.8.53
[4]

Ying Wen, Jiebao Sun, Zhichang Guo. A new anisotropic fourth-order diffusion equation model based on image features for image denoising. Inverse Problems and Imaging, , () : -. doi: 10.3934/ipi.2022004
[5]

Changchun Liu. A fourth order nonlinear degenerate parabolic equation. Communications on Pure and Applied Analysis, 2008, 7 (3) : 617-630. doi: 10.3934/cpaa.2008.7.617
[6]

José A. Carrillo, Ansgar Jüngel, Shaoqiang Tang. Positive entropic schemes for a nonlinear fourth-order parabolic equation. Discrete and Continuous Dynamical Systems - B, 2003, 3 (1) : 1-20. doi: 10.3934/dcdsb.2003.3.1
[7]

Luca Calatroni, Bertram Düring, Carola-Bibiane Schönlieb. ADI splitting schemes for a fourth-order nonlinear partial differential equation from image processing. Discrete and Continuous Dynamical Systems, 2014, 34 (3) : 931-957. doi: 10.3934/dcds.2014.34.931
[8]

Shi Yan, Jun Liu, Haiyang Huang, Xue-Cheng Tai. A dual EM algorithm for TV regularized Gaussian mixture model in image segmentation. Inverse Problems and Imaging, 2019, 13 (3) : 653-677. doi: 10.3934/ipi.2019030
[9]

Huafei Di, Yadong Shang, Xiaoxiao Zheng. Global well-posedness for a fourth order pseudo-parabolic equation with memory and source terms. Discrete and Continuous Dynamical Systems - B, 2016, 21 (3) : 781-801. doi: 10.3934/dcdsb.2016.21.781
[10]

Xu Liu, Jun Zhou. Initial-boundary value problem for a fourth-order plate equation with Hardy-Hénon potential and polynomial nonlinearity. Electronic Research Archive, 2020, 28 (2) : 599-625. doi: 10.3934/era.2020032
[11]

Stephen C. Preston, Ralph Saxton. An $H^1$ model for inextensible strings. Discrete and Continuous Dynamical Systems, 2013, 33 (5) : 2065-2083. doi: 10.3934/dcds.2013.33.2065
[12]

Jianhong (Jackie) Shen, Sung Ha Kang. Quantum TV and applications in image processing. Inverse Problems and Imaging, 2007, 1 (3) : 557-575. doi: 10.3934/ipi.2007.1.557
[13]

Wei Wang, Caifei Li. A variational saturation-value model for image decomposition: Illumination and reflectance. Inverse Problems and Imaging, 2022, 16 (3) : 547-567. doi: 10.3934/ipi.2021061
[14]

Hayden Schaeffer, John Garnett, Luminita A. Vese. A texture model based on a concentration of measure. Inverse Problems and Imaging, 2013, 7 (3) : 927-946. doi: 10.3934/ipi.2013.7.927
[15]

Zongming Guo, Long Wei. A fourth order elliptic equation with a singular nonlinearity. Communications on Pure and Applied Analysis, 2014, 13 (6) : 2493-2508. doi: 10.3934/cpaa.2014.13.2493
[16]

Zongming Guo, Long Wei. A perturbed fourth order elliptic equation with negative exponent. Discrete and Continuous Dynamical Systems - B, 2018, 23 (10) : 4187-4205. doi: 10.3934/dcdsb.2018132
[17]

Amine Laghrib, Abdelkrim Chakib, Aissam Hadri, Abdelilah Hakim. A nonlinear fourth-order PDE for multi-frame image super-resolution enhancement. Discrete and Continuous Dynamical Systems - B, 2020, 25 (1) : 415-442. doi: 10.3934/dcdsb.2019188
[18]

Iurii Posukhovskyi, Atanas G. Stefanov. On the normalized ground states for the Kawahara equation and a fourth order NLS. Discrete and Continuous Dynamical Systems, 2020, 40 (7) : 4131-4162. doi: 10.3934/dcds.2020175
[19]

To Fu Ma. Positive solutions for a nonlocal fourth order equation of Kirchhoff type. Conference Publications, 2007, 2007 (Special) : 694-703. doi: 10.3934/proc.2007.2007.694
[20]

Carlos Banquet, Élder J. Villamizar-Roa. On the management fourth-order Schrödinger-Hartree equation. Evolution Equations and Control Theory, 2020, 9 (3) : 865-889. doi: 10.3934/eect.2020037

2020 Impact Factor: 1.916

Tools

Metrics

  • PDF downloads (75)
  • HTML views (0)
  • Cited by (9)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy