Table of Contents Author Guidelines Submit a Manuscript
Abstract and Applied Analysis
Volume 2013 (2013), Article ID 483791, 19 pages
http://dx.doi.org/10.1155/2013/483791
Research Article

Fractional Partial Differential Equation: Fractional Total Variation and Fractional Steepest Descent Approach-Based Multiscale Denoising Model for Texture Image

1School of Computer Science and Technology, Sichuan University, Chengdu 610065, China
2State Key Laboratory of Networking and Switching Technology, Beijing University of Posts and Telecommunications, Beijing 100876, China
3Université de Paris 12 (LiSSi, E.A. 3956), 61 avenue du Général de Gaulle, 94010 Créteil Cedex, France
4Library of Sichuan University, Chengdu 610065, China

Received 14 June 2013; Accepted 11 August 2013

Academic Editor: Juan J. Trujillo

Copyright © 2013 Yi-Fei Pu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. E. R. Love, “Fractional derivatives of imaginary order,” Journal of the London Mathematical Society, vol. 3, pp. 241–259, 1971. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  2. K. B. Oldham and Spanier, The Fractional Calculus: Integrations and Differentiations of Arbitrary Order, Academic Press, New York, NY, USA, 1974.
  3. A. C. McBride, Fractional Calculus, Halsted Press, New York, NY, USA, 1986.
  4. K. Nishimoto, Fractional Calculus, University of New Haven Press, New Haven, Conn, USA, 1989.
  5. S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach Science Publishers, Yverdon, Switzerland, 1993. View at MathSciNet
  6. K. S. Miller, “Derivatives of noninteger order,” Mathematics Magazine, vol. 68, no. 3, pp. 183–192, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach Science, Yverdon, Switzerland, 1987. View at MathSciNet
  8. N. Engheta, “On fractional calculus and fractional multipoles in electromagnetism,” Institute of Electrical and Electronics Engineers. Transactions on Antennas and Propagation, vol. 44, no. 4, pp. 554–566, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. N. Engheta, “On the role of fractional calculus in electromagnetic theory,” IEEE Antennas and Propagation Magazine, vol. 39, no. 4, pp. 35–46, 1997. View at Publisher · View at Google Scholar
  10. M.-P. Chen and H. M. Srivastava, “Fractional calculus operators and their applications involving power functions and summation of series,” Applied Mathematics and Computation, vol. 81, no. 2-3, pp. 287–304, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. P. L. Butzer and U. Westphal, “An introduction to fractional calculus,” in Applications of Fractional Calculus in Physics, chapter 1, pp. 1–85, World Scientific, Singapore, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. S. Kempfle, I. Schäfer, and H. Beyer, “Fractional calculus via functional calculus: theory and applications,” Nonlinear Dynamics, vol. 29, no. 1–4, pp. 99–127, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. R. L. Magin, “Fractional calculus in bioengineering,” Critical Reviews in Biomedical Engineering, vol. 32, no. 3-4, pp. 195–377, 2004. View at Publisher · View at Google Scholar
  14. A. A. Kilbas, H. M. Srivastava, and J. J. Trujiilo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, The Netherlands, 2006.
  15. J. Sabatier, O. P. Agrawal, and J. A. Tenreiro Machado, Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering, Springer, Dordrecht, The Netherlands, 2007. View at Publisher · View at Google Scholar · View at MathSciNet
  16. R. C. Koeller, “Applications of fractional calculus to the theory of viscoelasticity,” Journal of Applied Mechanics, vol. 51, no. 2, pp. 299–307, 1984. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. Y. A. Rossikhin and M. V. Shitikova, “Applications of fractional calculus to dynamic problems of linear and nonlinear hereditary mechanics of solids,” Applied Mechanics Reviews, vol. 50, no. 1, pp. 15–67, 1997. View at Publisher · View at Google Scholar · View at Scopus
  18. S. Manabe, “A suggestion of fractional-order controller for flexible spacecraft attitude control,” Nonlinear Dynamics, vol. 29, no. 1–4, pp. 251–268, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  19. W. Chen and S. Holm, “Fractional Laplacian time-space models for linear and nonlinear lossy media exhibiting arbitrary frequency power-law dependency,” Journal of the Acoustical Society of America, vol. 115, no. 4, pp. 1424–1430, 2004. View at Publisher · View at Google Scholar · View at Scopus
  20. E. Perrin, R. Harba, C. Berzin-Joseph, I. Iribarren, and A. Bonami, “Nth-order fractional Brownian motion and fractional Gaussian noises,” IEEE Transactions on Signal Processing, vol. 49, no. 5, pp. 1049–1059, 2001. View at Publisher · View at Google Scholar · View at Scopus
  21. C.-C. Tseng, “Design of fractional order digital FIR differentiators,” IEEE Signal Processing Letters, vol. 8, no. 3, pp. 77–79, 2001. View at Publisher · View at Google Scholar · View at Scopus
  22. Y. Q. Chen and B. M. Vinagre, “A new IIR-type digital fractional order differentiator,” Signal Processing, vol. 83, no. 11, pp. 2359–2365, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  23. Y.-F. Pu, Research on application of fractional calculus to latest signal analysis and processing [Ph.D. thesis], Sichuan University, Chengdu, China, 2006.
  24. Y.-F. Pu, X. Yuan, K. Liao, Z.-L. Chen, and J.-L. Zhou, “Five numerical algorithms of fractional calculus applied in modern signal analyzing and processing,” Journal of Sichuan University, vol. 37, no. 5, pp. 118–124, 2005. View at Google Scholar · View at Scopus
  25. Y.-F. Pu, X. Yuan, K. Liao et al., “Structuring analog fractance circuit for 1/2 order fractional calculus,” in Proceedings of the IEEE 6th International Conference on ASIC (ASICON '05), vol. 2, pp. 1136–1139, Shanghai, China, October 2005. View at Publisher · View at Google Scholar · View at Scopus
  26. Y.-F. Pu, X. Yuan, K. Liao, and J.-L. Zhou, “Implement any fractional order neural-type pulse oscillator with net-grid type analog fractance circuit,” Journal of Sichuan University, vol. 38, no. 1, pp. 128–132, 2006. View at Google Scholar · View at Scopus
  27. R. Duits, M. Felsberg, L. Florack et al., “Scale spaces on a bounded domain,” in Proceedings of the 4th International Conference Scale Spaces, pp. 494–510, Isle of Skye, UK, 2003.
  28. S. Didas, B. Burgeth, A. Imiya, and J. Weickert, “Regularity and scale-space properties of fractional high order linear filtering,” in Proceedings of the 5th International Conference on Scale Space and PDE Methods in Computer Vision, Scale-Space, vol. 3459, pp. 13–25, April 2005. View at Scopus
  29. M. Unser and T. Blu, “Fractional splines and wavelets,” SIAM Review, vol. 42, no. 1, pp. 43–67, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  30. B. Ninness, “Estimation of 1/f noise,” IEEE Transactions on Information Theory, vol. 44, no. 1, pp. 32–46, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  31. B. Mathieu, P. Melchior, A. Oustaloup, and C. Ceyral, “Fractional differentiation for edge detection,” Signal Processing, vol. 83, no. 11, pp. 2421–2432, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  32. S.-C. Liu and S. Chang, “Dimension estimation of discrete-time fractional Brownian motion with applications to image texture classification,” IEEE Transactions on Image Processing, vol. 6, no. 8, pp. 1176–1184, 1997. View at Publisher · View at Google Scholar · View at Scopus
  33. Y.-F. Pu, “Fractional calculus approach to texture of digital image,” in Proceedings of the IEEE 8th International Conference on Signal Processing (ICSP '06), pp. 1002–1002, Beijing, China, November 2006. View at Publisher · View at Google Scholar · View at Scopus
  34. Y.-F. Pu, “Fractional differential filter of digital image,” Invention Patent of China, No.ZL200610021702.3, 2006.
  35. Y.-F. Pu, “High precision fractional calculus filter of digital image,” Invention Patent of China, No.ZL201010138742.2, 2010.
  36. Y.-F. Pu, W. Wang, and J.-L. Zhou, “Fractional differential approach to detecting textural features of digital image and its fractional differential filter implementation,” Science in China Series F, vol. 38, no. 12, pp. 2252–2272, 2008. View at Google Scholar · View at Zentralblatt MATH
  37. Y.-F. Pu and J.-L. Zhou, “A novel approach for multi-scale texture segmentation based on fractional differential,” International Journal of Computer Mathematics, vol. 88, no. 1, pp. 58–78, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  38. Y.-F. Pu, J.-L. Zhou, and X. Yuan, “Fractional differential mask: a fractional differential-based approach for multiscale texture enhancement,” IEEE Transactions on Image Processing, vol. 19, no. 2, pp. 491–511, 2010. View at Publisher · View at Google Scholar · View at MathSciNet
  39. T. Chan, S. Esedoglu, F. Park et al., “Recent developments in total variation image restoration,” in Mathematical Models of Computer Vision, Springer, New York, NY, USA, 2005. View at Google Scholar
  40. A. Buades, B. Coll, and J. M. Morel, “A review of image denoising algorithms, with a new one,” Multiscale Modeling & Simulation, vol. 4, no. 2, pp. 490–530, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  41. J. Weickert, Anisotropic Diffusion in Image Processing, European Consortium for Mathematics in Industry, B. G. Teubner, Stuttgart, Germany, 1998. View at MathSciNet
  42. G. Aubert and P. Kornprobst, Mathematical Problems in Image Processing: Partial Differential Equationsand the Calculus of Variations, vol. 147 of Applied Mathematical Sciences, Springer, New York, NY, USA, 2nd edition, 2006. View at MathSciNet
  43. 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. View at Publisher · View at Google Scholar · View at Scopus
  44. L. I. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica D, vol. 60, no. 1–4, pp. 259–268, 1992. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  45. G. Sapiro and D. L. Ringach, “Anisotropic diffusion of multivalued images with applications to color filtering,” IEEE Transactions on Image Processing, vol. 5, no. 11, pp. 1582–1586, 1996. View at Publisher · View at Google Scholar · View at Scopus
  46. P. Blomgren and T. F. Chan, “Color TV: total variation methods for restoration of vector-valued images,” IEEE Transactions on Image Processing, vol. 7, no. 3, pp. 304–309, 1998. View at Publisher · View at Google Scholar · View at Scopus
  47. N. P. Galatsanos and A. K. Katsaggelos, “Methods for choosing the regularization parameter and estimating the noise variance in image restoration and their relation,” IEEE Transactions on Image Processing, vol. 1, no. 3, pp. 322–336, 1992. View at Publisher · View at Google Scholar · View at Scopus
  48. S. Z. Li, “Close-form solution and parameter selection for convex minimization-based edge-preserving smoothing,” IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 20, no. 9, pp. 916–932, 1998. View at Publisher · View at Google Scholar · View at Scopus
  49. N. Nguyen, P. Milanfar, and G. Golub, “Efficient generalized cross-validation with applications to parametric image restoration and resolution enhancement,” IEEE Transactions on Image Processing, vol. 10, no. 9, pp. 1299–1308, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  50. D. M. Strong, J. F. Aujol, and T. F. Chan, “Scale recognition, regularization parameter selection, and Meyer's G norm in total variation regularization,” Multiscale Modeling & Simulation, vol. 5, no. 1, pp. 273–303, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  51. A. M. Thompson, J. C. Brown, J. W. Kay, and D. M. Titterington, “A study of methods of choosing the smoothing parameter in image restoration by regularization,” IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 13, no. 4, pp. 326–339, 1991. View at Publisher · View at Google Scholar · View at Scopus
  52. P. Mrázek and M. Navara, “Selection of optimal stopping time for nonlinear diffusion filtering,” International Journal of Computer Vision, vol. 52, no. 2-3, pp. 189–203, 2003. View at Publisher · View at Google Scholar · View at Scopus
  53. G. Gilboa, N. Sochen, and Y. Y. Zeevi, “Estimation of optimal PDE-based denoising in the SNR sense,” IEEE Transactions on Image Processing, vol. 15, no. 8, pp. 2269–2280, 2006. View at Publisher · View at Google Scholar · View at Scopus
  54. C. R. Vogel and M. E. Oman, “Iterative methods for total variation denoising,” SIAM Journal on Scientific Computing, vol. 17, no. 1, pp. 227–238, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  55. J. Darbon and M. Sigelle, “Exact optimization of discrete constrained total variation minimization problems,” in Proceedings of the 10th nternational Workshop on Combinatorial Image AnalysisI (WCIA '04), pp. 548–557, Auckland, New Zealand, 2004. View at Zentralblatt MATH · View at MathSciNet
  56. J. Darbon and M. Sigelle, “Image restoration with discrete constrained total variation. I. Fast and exact optimization,” Journal of Mathematical Imaging and Vision, vol. 26, no. 3, pp. 261–276, 2006. View at Publisher · View at Google Scholar · View at MathSciNet
  57. J. Darbon and M. Sigelle, “Image restoration with discrete constrained total variation. II. Levelable functions, convex priors and non-convex cases,” Journal of Mathematical Imaging and Vision, vol. 26, no. 3, pp. 277–291, 2006. View at Publisher · View at Google Scholar · View at MathSciNet
  58. B. Wohlberg and P. Rodriguez, “An iteratively reweighted norm algorithm for minimization of total variation functionals,” IEEE Signal Processing Letters, vol. 14, no. 12, pp. 948–951, 2007. View at Publisher · View at Google Scholar · View at Scopus
  59. F. Catté, P. L. Lions, J. M. Morel, and T. Coll, “Image selective smoothing and edge detection by nonlinear diffusion,” SIAM Journal on Numerical Analysis, vol. 29, no. 1, pp. 182–193, 1992. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  60. Y. Meyer, Oscillating Patterns in Image Processing and in Some Nonlinear Evolution Equations, The American Mathematical Society, Providence, RI, USA, 2001.
  61. 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  62. S. Alliney, “A property of the minimum vectors of a regularizing functional defined by means of the absolute norm,” IEEE Transactions on Signal Processing, vol. 45, no. 4, pp. 913–917, 1997. View at Publisher · View at Google Scholar · View at Scopus
  63. M. Nikolova, “A variational approach to remove outliers and impulse noise,” Journal of Mathematical Imaging and Vision, vol. 20, no. 1-2, pp. 99–120, 2004. View at Publisher · View at Google Scholar · View at MathSciNet
  64. T. F. Chan and S. Esedo\=glu, “Aspects of total variation regularized L1 function approximation,” SIAM Journal on Applied Mathematics, vol. 65, no. 5, pp. 1817–1837, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  65. M. Nikolova, “Minimizers of cost-functions involving nonsmooth data-fidelity terms,” SIAM Journal on Numerical Analysis, vol. 40, no. 3, pp. 965–994, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  66. S. Osher, M. Burger, D. Goldfarb, J. Xu, and W. Yin, “An iterative regularization method for total variation-based image restoration,” Multiscale Modeling & Simulation, vol. 4, no. 2, pp. 460–489, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  67. G. Gilboa, Y. Y. Zeevi, and N. Sochen, “Texture preserving variational denoising using an adaptive fidelity term,” in Proceedings of the 2nd IEEE Workshop on Variational, Geometric and Level Set Methods in Computer Vision (VLSM '03), pp. 137–144, Nice, France.
  68. S. Esedoglu and S. J. Osher, “Decomposition of images by the anisotropic Rudin-Osher-Fatemi model,” Communications on Pure and Applied Mathematics, vol. 57, no. 12, pp. 1609–1626, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  69. P. Blomgren, T. F. Chan, and P. Mulet, “Extensions to total variation denoising,” in Proceedings of the SPIE on Advanced Signal Processing: Algorithms, Architectures and Implementations VII, vol. 3162, pp. 367–375, San Diego, Calif, USA, July 1997. View at Publisher · View at Google Scholar · View at Scopus
  70. P. Blomgren, P. Mulet, T. F. Chan, and C. K. Wong, “Total variation image restoration: numerical methods and extensions,” in Proceedings of the International Conference on Image Processing (ICIP '97), pp. 384–387, October 1997. View at Publisher · View at Google Scholar · View at Scopus
  71. 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  72. 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  73. M. Lysaker, A. Lundervold, and X. C. Tai, “Noise removal using fourth-order partial differential equation with applications to medical magnetic resonance images in space and time,” IEEE Transactions on Image Processing, vol. 12, no. 12, pp. 1579–1589, 2003. View at Publisher · View at Google Scholar · View at Scopus
  74. G. Gilboa, N. Sochen, and Y. Y. Zeevi, “Image enhancement and denoising by complex diffusion processes,” IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 26, no. 8, pp. 1020–1036, 2004. View at Publisher · View at Google Scholar · View at Scopus
  75. 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  76. S. Osher, A. Solé, and L. Vese, “Image decomposition and restoration using total variation minimization and the H1 norm,” Multiscale Modeling & Simulation, vol. 1, no. 3, pp. 349–370, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  77. 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, no. 1, pp. 5–18, 2006. View at Publisher · View at Google Scholar · View at Scopus
  78. F. Li, C. Shen, J. Fan, and C. Shen, “Image restoration combining a total variational filter and a fourth-order filter,” Journal of Visual Communication and Image Representation, vol. 18, no. 4, pp. 322–330, 2007. View at Publisher · View at Google Scholar · View at Scopus
  79. 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. View at Publisher · View at Google Scholar · View at MathSciNet
  80. F. Dong, Z. Liu, D. Kong, and K. Liu, “An improved LOT model for image restoration,” Journal of Mathematical Imaging and Vision, vol. 34, no. 1, pp. 89–97, 2009. View at Publisher · View at Google Scholar · View at MathSciNet
  81. P. Guidotti and J. V. Lambers, “Two new nonlinear nonlocal diffusions for noise reduction,” Journal of Mathematical Imaging and Vision, vol. 33, no. 1, pp. 25–37, 2009. View at Publisher · View at Google Scholar · View at MathSciNet
  82. J. Bai and X.-C. Feng, “Fractional-order anisotropic diffusion for image denoising,” IEEE Transactions on Image Processing, vol. 16, no. 10, pp. 2492–2502, 2007. View at Publisher · View at Google Scholar · View at MathSciNet
  83. G. Leitmann, The Calculus of Variations and Optimal Control: An Introduction, Springer, New York, NY, USA, 1981. View at MathSciNet
  84. V. E. Tarasov, “Fractional vector calculus and fractional Maxwell's equations,” Annals of Physics, vol. 323, no. 11, pp. 2756–2778, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  85. W. Rudin, Functional Analysis, 2nd Edition, McGraw-Hill, New York, NY, USA, 1991. View at MathSciNet
  86. C. Tomasi and R. Manduchi, “Bilateral filtering for gray and color images,” in Proceedings of the 1998 IEEE 6th International Conference on Computer Vision, pp. 839–846, January 1998. View at Scopus
  87. M. Zhang and B. K. Gunturk, “Multiresolution bilateral filtering for image denoising,” IEEE Transactions on Image Processing, vol. 17, no. 12, pp. 2324–2333, 2008. View at Publisher · View at Google Scholar · View at MathSciNet
  88. H. Yu, L. Zhao, and H. Wang, “Image denoising using trivariate shrinkage filter in the wavelet domain and joint bilateral filter in the spatial domain,” IEEE Transactions on Image Processing, vol. 18, no. 10, pp. 2364–2369, 2009. View at Publisher · View at Google Scholar · View at MathSciNet
  89. M. N. Do and M. Vetterli, “The contourlet transform: an efficient directional multiresolution image representation,” IEEE Transactions on Image Processing, vol. 14, no. 12, pp. 2091–2106, 2005. View at Publisher · View at Google Scholar · View at Scopus
  90. D. D.-Y. Po and M. N. Do, “Directional multiscale modeling of images using the contourlet transform,” IEEE Transactions on Image Processing, vol. 15, no. 6, pp. 1610–1620, 2006. View at Publisher · View at Google Scholar · View at MathSciNet
  91. G. Y. Chen and T. D. Bui, “Multiwavelets denoising using neighboring coefficients,” IEEE Signal Processing Letters, vol. 10, no. 7, pp. 211–214, 2003. View at Publisher · View at Google Scholar · View at Scopus
  92. A. Buades, B. Coll, and J.-M. Morel, “A non-local algorithm for image denoising,” in Proceedings of the IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR '05), vol. 2, pp. 60–65, San Diego, Calif, USA, June 2005. View at Publisher · View at Google Scholar · View at Scopus
  93. A. Buades, B. Coll, and J.-M. Morel, “Nonlocal image and movie denoising,” International Journal of Computer Vision, vol. 76, no. 2, pp. 123–139, 2008. View at Publisher · View at Google Scholar · View at Scopus
  94. 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. View at Publisher · View at Google Scholar · View at Scopus
  95. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions With Formulas, Graphs and Mathematical Tables, U.S. Department of Commerce, Washington, DC, USA, 1964.
  96. P. R. Halmos, Measure Theory, D. van Nostrand Company, New York, NY, USA, 1950. View at MathSciNet
  97. M. E. Munroe, Introduction to Measure and Integration, Addison Wesley, London, UK, 1953. View at MathSciNet