Abstract and Applied Analysis

An Alternative Variational Framework for Image Denoising

Elisha Achieng Ogada, Zhichang Guo, and Boying Wu

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We propose an alternative framework for total variation based image denoising models. The model is based on the minimization of the total variation with a functional coefficient, where, in this case, the functional coefficient is a function of the magnitude of image gradient. We determine the considerations to bear on the choice of the functional coefficient. With the use of an example functional, we demonstrate the effectiveness of a model chosen based on the proposed consideration. In addition, for the illustrative model, we prove the existence and uniqueness of the minimizer of the variational problem. The existence and uniqueness of the solution associated evolution equation are also established. Experimental results are included to demonstrate the effectiveness of the selected model in image restoration over the traditional methods of Perona-Malik (PM), total variation (TV), and the D-α-PM method.

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Abstr. Appl. Anal., Volume 2014, Special Issue (2014), Article ID 939131, 16 pages.

First available in Project Euclid: 2 October 2014

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Ogada, Elisha Achieng; Guo, Zhichang; Wu, Boying. An Alternative Variational Framework for Image Denoising. Abstr. Appl. Anal. 2014, Special Issue (2014), Article ID 939131, 16 pages. doi:10.1155/2014/939131. https://projecteuclid.org/euclid.aaa/1412278112

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  • P. Perona and J. Malik, “Scale-space and edge detection using anisotropic diffusion,” IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 12, no. 7, pp. 629–639, 1990.
  • J. J. Koenderink, “The structure of images,” Biological Cybernetics, vol. 50, no. 5, pp. 363–370, 1984.
  • A. Witkin, Scale-Space Filtering. Readings in Computer Vision: Issues, Problems, Principles, and Paradigms, 1987.
  • J. Weickert, Anisotropic Diffusion in Image Processing, vol. 1 ofEuropean Consortium for Mathematics in Industry, Teubner, Stuttgart, Germany, 1998.
  • Z. Guo, J. Sun, D. Zhang, and B. Wu, “Adaptive Perona-Malik model based on the variable exponent for image denoising,” IEEE Transactions on Image Processing, vol. 21, no. 3, pp. 958–967, 2012.
  • Y. Chen and T. Wunderli, “Adaptive total variation for image res-toration in BV space,” Journal of Mathematical Analysis and Applications, vol. 272, no. 1, pp. 117–137, 2002.
  • L. Vese, “A study in the BV space of a denoising-deblurring vari-ational problem,” Applied Mathematics and Optimization, vol. 44, no. 2, pp. 131–161, 2001.
  • L. I. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica D: Nonlinear Phenomena, vol. 60, no. 1–4, pp. 259–268, 1992.
  • S. Osher and L. I. Rudin, “Feature-oriented image enhancement using shock filters,” SIAM Journal on Numerical Analysis, vol. 27, no. 4, pp. 919–940, 1990.
  • G. Aubert and P. Kornprobst, Mathematical Problems in Image Processing: Partial Differential Equations and the Calculus of Variations, vol. 147 of Applied Mathematical Sciences, Springer, New York, NY, USA, 2nd edition, 2006.
  • A. Marquina and S. Osher, “Explicit algorithms for a new timedependent model based on level set motion for nonlinear debl-urring and noise removal,” SIAM Journal on Scientific Computing, vol. 22, no. 2, pp. 387–405, 2000.
  • Y. Chen, S. Levine, and M. Rao, “Variable exponent, linear gro-wth functionals in image restoration,” SIAM Journal on Applied Mathematics, vol. 66, no. 4, pp. 1383–1406, 2006.
  • P. Blomgren, T. F. Chan, and P. Mulet, “Extensions to total vari-ation denoising,” in Advanced Signal Processing: Algorithms, Architectures and Implementations VII, Proceedings of the SPIE 3162, pp. 367–375, San Diego, Calif, USA, July 1997.
  • T. F. Chan and S. Esedoglu, “Aspects of total variation regular-ized ${L}^{1}$ function approximation,” SIAM Journal on Applied Mat-hematics, vol. 65, no. 5, pp. 1817–1837, 2005.
  • D. M. Strong and T. F. Chan, “Spatially and scale adaptive totalvariation based regularization and anisotropic diffusion in image processing,” Diusion in Image Processing, UCLA Math Department CAM Report Cite- seer 1996, 1996.
  • A. Chambolle and P.-L. Lions, “Image recovery via total variation minimization and related problems,” Numerische Mathematik, vol. 76, no. 2, pp. 167–188, 1997.
  • L. Xiao, L.-L. Huang, and B. Roysam, “Image variational den-oising using gradient fidelity on curvelet shrinkage,” Eurasip Journal on Advances in Signal Processing, vol. 2010, Article ID 398410, 2010.
  • T. F. Chan and J. Shen, “Mathematical models for local nontexture inpaintings,” SIAM Journal on Applied Mathematics, vol. 62, no. 3, pp. 1019–1043, 2002.
  • L. Vese, Problemes variationnels et EDP pour l analyse d images et l evolution de courbes [Ph.D. thesis], Universite de Nice Sophia-Antipolis, 1996.
  • T. Chan, A. Marquina, and P. Mulet, “High-order total vari-ation-based image restoration,” SIAM Journal on Scientific Com-puting, vol. 22, no. 2, pp. 503–516, 2000.
  • M. Bildhauer and M. Fuchs, “A variational approach to the den-oising of images based on different variants of the TV-regu-larization,” Applied Mathematics and Optimization, vol. 66, no. 3, pp. 331–361, 2012.
  • S. Osher, M. Burger, D. Goldfarb, J. Xu, and W. Yin, “An iter-ative regularization method for total variation-based image res-toration,” Multiscale Modeling & Simulation, vol. 4, no. 2, pp. 460–489, 2005.
  • X. Zhou, “An evolution problem for plastic antiplanar shear,” Applied Mathematics and Optimization, vol. 25, no. 3, pp. 263–285, 1992.
  • B. Dacorogna, Introduction to the Calculus of Variations, World Scientific, 2004.
  • M. Xu and S. Zhou, “Existence and uniqueness of weak solutions for a generalized thin film equation,” Nonlinear Analysis: Theory, Methods & Applications, vol. 60, no. 4, pp. 755–774, 2005.
  • F. Andreu, V. Caselles, J. I. Díaz, and J. M. Mazón, “Some qual-itative properties for the total variation flow,” Journal of Functional Analysis, vol. 188, no. 2, pp. 516–547, 2002.
  • Q. Liu, Z. Yao, and Y. Ke, “Entropy solutions for a fourth-order nonlinear degenerate problem for noise removal,” Nonlinear Analysis: Theory, Methods & Applications, vol. 67, no. 6, pp. 1908–1918, 2007.
  • Z. Wu, J. Zhao, J. Yin, and H. Li, Nonlinear Diffusion Equations, World Scientific, River Edge, NJ, USA, 2001.
  • Z. Wu, J. Yin, and C. Wang, Elliptic & Parabolic Equations, World Scientific, River Edge, NJ, USA, 2006.
  • L. C. Evans and Y. Yu, “Various properties of solutions of theinfinity-Laplacian equation,” Communications in Partial Differential Equations, vol. 30, no. 7–9, pp. 1401–1428, 2005.
  • Z. Guo, J. Yin, and Q. Liu, “On a reaction-diffusion system app-lied to image decomposition and restoration,” Mathematical and Computer Modelling, vol. 53, no. 5-6, pp. 1336–1350, 2011.
  • L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions, vol. 5 of Studies in Advanced Mathematics, CRC Press, Boca Raton, Fla, USA, 1992.
  • J. Weickert, B. M. Ter Haar Romeny, and M. A. Viergever, “Effi-cient and reliable schemes for nonlinear diffusion filtering,” IEEE Transactions on Image Processing, vol. 7, no. 3, pp. 398–410, 1998.
  • S. Durand, J. Fadili, and M. Nikolova, “Multiplicative noise rem-oval using L1 fidelity on frame coefficients,” Journal of Mathematical Imaging and Vision, vol. 36, no. 3, pp. 201–226, 2010.
  • Z. Wang, A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli, “Image quality assessment: from error visibility to structural similarity,” IEEE Transactions on Image Processing, vol. 13, no. 4, pp. 600–612, 2004.
  • A. Horé and D. Ziou, “Image quality metrics: PSNR vs. SSIM,” in Proceedings of the 20th International Conference on Pattern Recognition (ICPR '10), pp. 2366–2369, Istanbul, Turkey, August 2010.
  • Y. Yu and S. T. Acton, “Speckle reducing anisotropic diffusion,” IEEE Transactions on Image Processing, vol. 11, no. 11, pp. 1260–1270, 2002. \endinput