American Institute of Mathematical Sciences

Advanced
  • Previous Article
    A greedy method for reconstructing polycrystals from three-dimensional X-ray diffraction data
  • IPI Home
  • This Issue
  • Next Article
    A positivity principle for parabolic integro-differential equations and inverse problems with final overdetermination
February  2009, 3(1): 43-68. doi: 10.3934/ipi.2009.3.43

Image recovery using functions of bounded variation and Sobolev spaces of negative differentiability

Yunho Kim 1, and  Luminita A. Vese 1,

1. 

Department of Mathematics, University of California, Los Angeles, Los Angeles, CA 90095-1555, United States, United States

Received  September 2008 Revised  November 2008 Published  February 2009

In this work we wish to recover an unknown image from a blurry, or noisy-blurry version. We solve this inverse problem by energy minimization and regularization. We seek a solution of the form $u + v$, where $u$ is a function of bounded variation (cartoon component), while $v$ is an oscillatory component (texture), modeled by a Sobolev function with negative degree of differentiability. We give several results of existence and characterization of minimizers of the proposed optimization problem. Experimental results show that this cartoon + texture model better recovers textured details in natural images, by comparison with the more standard models where the unknown is restricted only to the space of functions of bounded variation.
Keywords: Sobolev spaces, oscillatory functions., Image restoration, bounded variation, variational models.
Mathematics Subject Classification: Primary: 35J85, 49J40, 65M06; Secondary: 65M3.
Citation: Yunho Kim, Luminita A. Vese. Image recovery using functions of bounded variation and Sobolev spaces of negative differentiability. Inverse Problems and Imaging, 2009, 3 (1) : 43-68. doi: 10.3934/ipi.2009.3.43
[1]

Bartomeu Coll, Joan Duran, Catalina Sbert. Half-linear regularization for nonconvex image restoration models. Inverse Problems and Imaging, 2015, 9 (2) : 337-370. doi: 10.3934/ipi.2015.9.337
[2]

Sébastien Gouëzel. An interval map with a spectral gap on Lipschitz functions, but not on bounded variation functions. Discrete and Continuous Dynamical Systems, 2009, 24 (4) : 1205-1208. doi: 10.3934/dcds.2009.24.1205
[3]

Ke Chen, Yiqiu Dong, Michael Hintermüller. A nonlinear multigrid solver with line Gauss-Seidel-semismooth-Newton smoother for the Fenchel pre-dual in total variation based image restoration. Inverse Problems and Imaging, 2011, 5 (2) : 323-339. doi: 10.3934/ipi.2011.5.323
[4]

Petteri Harjulehto, Peter Hästö, Juha Tiirola. Point-wise behavior of the Geman--McClure and the Hebert--Leahy image restoration models. Inverse Problems and Imaging, 2015, 9 (3) : 835-851. doi: 10.3934/ipi.2015.9.835
[5]

Duchao Liu, Beibei Wang, Peihao Zhao. On the trace regularity results of Musielak-Orlicz-Sobolev spaces in a bounded domain. Communications on Pure and Applied Analysis, 2016, 15 (5) : 1643-1659. doi: 10.3934/cpaa.2016018
[6]

Mehdi Bastani, Davod Khojasteh Salkuyeh. On the GSOR iteration method for image restoration. Numerical Algebra, Control and Optimization, 2021, 11 (1) : 27-43. doi: 10.3934/naco.2020013
[7]

Harun Karsli, Purshottam Narain Agrawal. Rate of convergence of Stancu type modified $ q $-Gamma operators for functions with derivatives of bounded variation. Mathematical Foundations of Computing, 2022  doi: 10.3934/mfc.2022002
[8]

Rafail Krichevskii and Vladimir Potapov. Compression and restoration of square integrable functions. Electronic Research Announcements, 1996, 2: 42-49.

[9]

Nicolas Lermé, François Malgouyres, Dominique Hamoir, Emmanuelle Thouin. Bayesian image restoration for mosaic active imaging. Inverse Problems and Imaging, 2014, 8 (3) : 733-760. doi: 10.3934/ipi.2014.8.733
[10]

Amir Averbuch, Pekka Neittaanmäki, Valery Zheludev. Periodic spline-based frames for image restoration. Inverse Problems and Imaging, 2015, 9 (3) : 661-707. doi: 10.3934/ipi.2015.9.661
[11]

Ying Zhang, Xuhua Ren, Bryan Alexander Clifford, Qian Wang, Xiaoqun Zhang. Image fusion network for dual-modal restoration. Inverse Problems and Imaging, 2021, 15 (6) : 1409-1419. doi: 10.3934/ipi.2021067
[12]

Ruiqiang He, Xiangchu Feng, Xiaolong Zhu, Hua Huang, Bingzhe Wei. RWRM: Residual Wasserstein regularization model for image restoration. Inverse Problems and Imaging, 2021, 15 (6) : 1307-1332. doi: 10.3934/ipi.2020069
[13]

Chunlin Wu, Juyong Zhang, Xue-Cheng Tai. Augmented Lagrangian method for total variation restoration with non-quadratic fidelity. Inverse Problems and Imaging, 2011, 5 (1) : 237-261. doi: 10.3934/ipi.2011.5.237
[14]

Andreas Asheim, Alfredo Deaño, Daan Huybrechs, Haiyong Wang. A Gaussian quadrature rule for oscillatory integrals on a bounded interval. Discrete and Continuous Dynamical Systems, 2014, 34 (3) : 883-901. doi: 10.3934/dcds.2014.34.883
[15]

Xiaoman Liu, Jijun Liu. Image restoration from noisy incomplete frequency data by alternative iteration scheme. Inverse Problems and Imaging, 2020, 14 (4) : 583-606. doi: 10.3934/ipi.2020027
[16]

Alina Toma, Bruno Sixou, Françoise Peyrin. Iterative choice of the optimal regularization parameter in TV image restoration. Inverse Problems and Imaging, 2015, 9 (4) : 1171-1191. doi: 10.3934/ipi.2015.9.1171
[17]

Jing Xu, Xue-Cheng Tai, Li-Lian Wang. A two-level domain decomposition method for image restoration. Inverse Problems and Imaging, 2010, 4 (3) : 523-545. doi: 10.3934/ipi.2010.4.523
[18]

Gerard Gómez, Josep–Maria Mondelo, Carles Simó. A collocation method for the numerical Fourier analysis of quasi-periodic functions. I: Numerical tests and examples. Discrete and Continuous Dynamical Systems - B, 2010, 14 (1) : 41-74. doi: 10.3934/dcdsb.2010.14.41
[19]

Gerard Gómez, Josep–Maria Mondelo, Carles Simó. A collocation method for the numerical Fourier analysis of quasi-periodic functions. II: Analytical error estimates. Discrete and Continuous Dynamical Systems - B, 2010, 14 (1) : 75-109. doi: 10.3934/dcdsb.2010.14.75
[20]

Franco Obersnel, Pierpaolo Omari. Multiple bounded variation solutions of a capillarity problem. Conference Publications, 2011, 2011 (Special) : 1129-1137. doi: 10.3934/proc.2011.2011.1129

2020 Impact Factor: 1.639

Tools

Metrics

  • PDF downloads (127)
  • HTML views (0)
  • Cited by (19)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy