Abstract and Applied Analysis

New Regularization Models for Image Denoising with a Spatially Dependent Regularization Parameter

Tian-Hui Ma, Ting-Zhu Huang, and Xi-Le Zhao

Full-text: Open access

Abstract

We consider simultaneously estimating the restored image and the spatially dependent regularization parameter which mutually benefit from each other. Based on this idea, we refresh two well-known image denoising models: the LLT model proposed by Lysaker et al. (2003) and the hybrid model proposed by Li et al. (2007). The resulting models have the advantage of better preserving image regions containing textures and fine details while still sufficiently smoothing homogeneous features. To efficiently solve the proposed models, we consider an alternating minimization scheme to resolve the original nonconvex problem into two strictly convex ones. Preliminary convergence properties are also presented. Numerical experiments are reported to demonstrate the effectiveness of the proposed models and the efficiency of our numerical scheme.

Article information

Source
Abstr. Appl. Anal., Volume 2013, Special Issue (2013), Article ID 729151, 15 pages.

Dates
First available in Project Euclid: 26 February 2014

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1393447479

Digital Object Identifier
doi:10.1155/2013/729151

Mathematical Reviews number (MathSciNet)
MR3134166

Zentralblatt MATH identifier
07095291

Citation

Ma, Tian-Hui; Huang, Ting-Zhu; Zhao, Xi-Le. New Regularization Models for Image Denoising with a Spatially Dependent Regularization Parameter. Abstr. Appl. Anal. 2013, Special Issue (2013), Article ID 729151, 15 pages. doi:10.1155/2013/729151. https://projecteuclid.org/euclid.aaa/1393447479


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References

  • L. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica D, vol. 60, pp. 259–268, 1992.
  • E. Giusti, Minimal Surfaces and Functions of Bounded Variation,, Birkhäuser, Boston, Mass, USA, 1984.
  • D. Strong and T. Chan, “Edge-preserving and scale-dependent properties of total variation regularization,” Inverse Problems, vol. 19, no. 6, pp. S165–S187, 2003.
  • W. Ring, “Structural properties of solutions to total variation regularization problems,” ESAIM: Mathematical Modelling and Numerical Analysis, vol. 34, no. 4, pp. 799–810, 2000.
  • J. Weickert, Anisotropic Diffusion in Image Processing, B. G. Teubner, Stuttgart, Germany, 1998.
  • A. Buades, B. Coll, and J. M. Morel, “The staircasing effect in neighborhood filters and its solution,” IEEE Transactions on Image Processing, vol. 15, pp. 1499–1505, 2006.
  • T. Chan, A. Marquina, and P. Mulet, “High-order total variation-based image restoration,” SIAM Journal on Scientific Computing, vol. 22, no. 2, pp. 503–516, 2000.
  • M. Lysaker, A. Lundervold, and X.-C. Tai, “Noise removal using fourth-order partial differential equations with applications to medical magnetic resonance images in space and time,” IEEE Transactions on Image Processing, vol. 12, pp. 1579–1590, 2003.
  • Y.-L. You and M. Kaveh, “Fourth-order partial differential equations for noise removal,” IEEE Transactions on Image Processing, vol. 9, no. 10, pp. 1723–1730, 2000.
  • W. Hinterberger and O. Scherzer, “Variational methods on the space of functions of bounded Hessian for convexification and denoising,” Computing, vol. 76, no. 1-2, pp. 109–133, 2006.
  • M. Lysaker, S. Osher, and X.-C. Tai, “Noise removal using smoothed normals and surface fitting,” IEEE Transactions on Image Processing, vol. 13, no. 10, pp. 1345–1357, 2004.
  • J. B. Greer and A. L. Bertozzi, “Traveling wave solutions of fourth order PDEs for image processing,” SIAM Journal on Mathematical Analysis, vol. 36, no. 1, pp. 38–68, 2004.
  • M. Lysaker and X.-C. Tai, “Iterative image restoration combining total variation minimization and a second-order functional,” International Journal of Computer Vision, vol. 66, pp. 5–18, 2006.
  • F. Li, C.-M. Shen, J.-S. Fan, and C.-L. Shen, “Image restoration combining a total variational filter and a fourth-order filter,” Journal of Visual Communication and Image Representation, vol. 18, pp. 322–330, 2007.
  • F.-C. Yang, K. Chen, and B. Yu, “Efficient homotopy solution and a convex combination of ROF and LLT models for image restoration,” International Journal of Numerical Analysis and Modeling, vol. 9, no. 4, pp. 907–927, 2012.
  • Q.-S. Chang, X.-C. Tai, and L. Xing, “A compound algorithm of denoising using second-order and fourth-order partial differential equations,” Numerical Mathematics Theory Methods and Applications, vol. 2, no. 4, pp. 353–376, 2009.
  • Z.-F. Pang and Y.-F. Yang, “A projected gradient algorithm based on the augmented Lagrangian strategy for image restoration and texture extraction,” Image and Vision Computing, vol. 29, pp. 117–126, 2011.
  • G. Gilboa, N. Sochen, and Y. Y. Zeevi, “Variational denoising of partly textured images by spatially varying constraints,” IEEE Transactions on Image Processing, vol. 15, pp. 2281–2289, 2006.
  • M. Bertalmio, V. Caselles, B. Rougé, and A. Solé, “TV based image restoration with local constraints,” Journal of Scientific Computing, vol. 19, pp. 95–122, 2003.
  • A. Almansa, C. Ballester, V. Caselles, and G. Haro, “A TV based restoration model with local constraints,” Journal of Scientific Computing, vol. 34, no. 3, pp. 209–236, 2008.
  • Y.-Q. Dong, M. Hintermüller, and M. M. Rincon-Camacho, “Automated regularization parameter selection in multi-scale total variation models for image restoration,” Journal of Mathematical Imaging and Vision, vol. 40, no. 1, pp. 82–104, 2011.
  • K. Bredies, Y.-Q. Dong, and M. Hintermüller, “Spatially dependent regularization parameter selection in total generalized variation models for image restoration,” International Journal of Computer Mathematics, vol. 90, no. 1, pp. 109–123, 2013.
  • C.-L. Wu and X.-C. Tai, “Augmented Lagrangian method, dual methods, and split Bregman iteration for ROF, vectorial TV, and high order models,” SIAM Journal on Imaging Sciences, vol. 3, no. 3, pp. 300–339, 2010.
  • D. P. Bertsekas, A. Nedic, and A. E. Ozdaglar, Convex Analysis and Optimization, Athena Scientic, Boston, Mass, USA, 2003.
  • M. K. Ng, P. Weiss, and X. Yuan, “Solving constrained total-variation image restoration and reconstruction problems via alternating direction methods,” SIAM Journal on Scientific Computing, vol. 32, no. 5, pp. 2710–2736, 2010.
  • J. Eckstein and D. P. Bertsekas, “On the Douglas-Rachford splitting method and the proximal point algorithm for maximal monotone operators,” Mathematical Programming, vol. 55, no. 3, pp. 293–318, 1992.
  • T. Goldstein and S. Osher, “The split Bregman method for L1-regularized problems,” SIAM Journal on Imaging Sciences, vol. 2, no. 2, pp. 323–343, 2009.
  • M. Ng, F. Wang, and X. Yuan, “Inexact alternating direction methods for image recovery,” SIAM Journal on Scientific Computing, vol. 33, no. 4, pp. 1643–1668, 2011.
  • B. He and X. Yuan, “On the $O(1/n)$ convergence rate of the Douglas-Rachford alternating direction method,” SIAM Journal on Numerical Analysis, vol. 50, no. 2, pp. 700–709, 2012.