From Local Kernel to Nonlocal Multiple-Model Image Denoising

  • Vladimir Katkovnik
  • Alessandro Foi
  • Karen Egiazarian
  • Jaakko Astola
Article

Abstract

We review the evolution of the nonparametric regression modeling in imaging from the local Nadaraya-Watson kernel estimate to the nonlocal means and further to transform-domain filtering based on nonlocal block-matching. The considered methods are classified mainly according to two main features: local/nonlocal and pointwise/multipoint. Here nonlocal is an alternative to local, and multipoint is an alternative to pointwise. These alternatives, though obvious simplifications, allow to impose a fruitful and transparent classification of the basic ideas in the advanced techniques. Within this framework, we introduce a novel single- and multiple-model transform domain nonlocal approach. The Block Matching and 3-D Filtering (BM3D) algorithm, which is currently one of the best performing denoising algorithms, is treated as a special case of the latter approach.

Keywords

Image denoising Nonparametric regression Spatially adaptive filters Aggregation Nonlocal means Multiple-model nonlocal estimates 

References

  1. Abramovich, F., & Benjamini, Y. (1996). Adaptive thresholding of wavelet coefficients. Computational Statistics and Data Analysis, 22, 351–361. CrossRefMathSciNetGoogle Scholar
  2. Abramovich, F., Benjamini, Y., Donoho, D., & Johnstone, I. (2006). Adapting to unknown sparsity by controlling the false discovery rate. The Annals of Statistics, 34(2), 584–653. MATHCrossRefMathSciNetGoogle Scholar
  3. Aharon, M., Elad, M., & Bruckstein, A. M. (2006). The K-SVD: An algorithm for designing of overcomplete dictionaries for sparse representation. IEEE Transactions on Signal Processing, 54(11), 4311–4322. CrossRefGoogle Scholar
  4. Alexander, S., Kovacic, S., & Vrscay, E. (2007). A simple model for image self-similarity and the possible use of mutual information. In Proc. 15th Eur. signal process. conf., EUSIPCO 2007, Poznan, Poland, 2007. Google Scholar
  5. Astola, J., & Yaroslavsky, L. (Eds.) (2007). EURASIP book series on signal processing and communications : Vol. 7. Advances in signal transforms: theory and applications. Cairo: Hindawi Publishing. Google Scholar
  6. Barash, D. (2002). A fundamental relationship between bilateral filtering, adaptive smoothing, and the nonlinear diffusion equation. IEEE Transactions on Pattern Analysis and Machine Intelligence, 24(6), 844–847. CrossRefGoogle Scholar
  7. Benjamini, Y., & Liu, W. (1999). A step-down multiple hypotheses testing procedure that controls the false discovery rate under independence. Journal of Statistical Planning and Inference, 82(1–2), 163–170. MATHCrossRefMathSciNetGoogle Scholar
  8. Birge, L. (2006). Model selection via testing: an alternative to (penalized) maximum likelihood estimators. Annales de l’Institut Henri Poincaré (B) Probabilités et Statistiques, 40, 273–325. CrossRefMathSciNetGoogle Scholar
  9. Brown, R. (1963). Smoothing, forecasting and prediction of discrete time series. Prentice-Hall: Englewood Cliffs. Google Scholar
  10. Buades, A., Coll, B., & Morel, J. M. (2005). A review of image denoising algorithms, with a new one. SIAM Multiscale Modeling and Simulation, 4, 490–530. MATHCrossRefMathSciNetGoogle Scholar
  11. Buades, A., Coll, B., & Morel, J. M. (2006a). The staircasing effect in neighborhood filters and its solution. IEEE Transactions on Image Processing, 15, 1499–1505. CrossRefGoogle Scholar
  12. Buades, A., Coll, B., & Morel, J. M. (2006b). Neighborhood filters and PDE’s. Numerische Mathematik, 105(1), 1–34. MATHCrossRefMathSciNetGoogle Scholar
  13. Buades, A., Coll, B., & Morel, J. M. (2008). Nonlocal image and movie denoising. International Journal of Computer Vision, 76(2), 123–139. CrossRefGoogle Scholar
  14. Bunea, F., Tsybakov, A., & Wegkamp, M. (2007). Aggregation for regression learning. Annals of Statistics, 35, 1674–1697. MATHCrossRefMathSciNetGoogle Scholar
  15. Chan, T., & Shen, J. (2005). Image processing and analysis: variational, PDE, wavelet, and stochastic methods. Philadelphia: SIAM. MATHGoogle Scholar
  16. Chatterjee, P., & Milanfar, P. (2008). A generalization of non-local means via kernel regression. In Proc. SPIE conf. on computational imaging, San Jose, January 2008. Google Scholar
  17. Cleveland, W. S., & Devlin, S. J. (1988). Locally weighted regression: an approach to regression analysis by local fitting. Journal of American Statistical Association, 83, 596–610. CrossRefGoogle Scholar
  18. Coifman, R., & Donoho, D. (1995). Translation-invariant de-noising. In A. Antoniadis & G. Oppenheim (Eds.), Lecture notes in statistics. Wavelets and statistics (pp. 125–150). Berlin: Springer. Google Scholar
  19. Dabov, K., Foi, A., Katkovnik, V., & Egiazarian, K. (2007). Image denoising by sparse 3-D transform-domain collaborative filtering. IEEE Transactions on Image Processing, 16(8), 2080–2095. CrossRefMathSciNetGoogle Scholar
  20. Dabov, K., Foi, A., Katkovnik, V., & Egiazarian, K. (2008). A nonlocal and shape-adaptive transform-domain collaborative filtering. In Proc. 2008 int. workshop on local and non-local approximation in image processing, LNLA 2008, Lausanne, Switzerland. Google Scholar
  21. Dabov, K., Foi, A., Katkovnik, V., & Egiazarian, K. (2009). BM3D image denoising with shape-adaptive principal component analysis. In Proc. workshop on signal processing with adaptive sparse structured representations (SPARS’09), Saint-Malo, France, April 2009. Google Scholar
  22. Donoho, D., & Johnstone, I. (1995). Adapting to unknown smoothness via wavelet shrinkage. Journal of American Statistical Association, 90(432), 1200–1224. MATHCrossRefMathSciNetGoogle Scholar
  23. Egiazarian, K., Katkovnik, V., & Astola, J. (2001). Local transform-based image de-noising with adaptive window size selection. In Proc. SPIE image and signal processing for remote sensing VI (Vol. 4170, p. 4170-4), January 2001. Google Scholar
  24. Elad, M. (2002). On the origin of the bilateral filter and ways to improve it. IEEE Transactions on Image Processing, 10(10), 1141–1151. CrossRefMathSciNetGoogle Scholar
  25. Elad, M. (2006). Why shrinkage is still relevant for redundant representations? IEEE Transactions on Information Theory, 52(12), 5559–5569. CrossRefMathSciNetGoogle Scholar
  26. Elad, M., & Aharon, M. (2006). Image denoising via sparse and redundant representations over learned dictionaries. IEEE Transactions on Image Processing, 15(12), 3736–3745. CrossRefMathSciNetGoogle Scholar
  27. Elad, M., & Yavneh, I. (2009). A plurality of sparse representations is better than the sparsest one alone. IEEE Transactions on Information Theory (to appear). Google Scholar
  28. Elmoataz, A., Lezoray, O., Bougleux, S., & Ta, V. T. (2008). Unifying local and nonlocal processing with partial difference operators on weighted graphs. In Proc. 2008 int. workshop on local and non-local approximation in image processing, LNLA 2008, Lausanne, Switzerland, August 2008. Google Scholar
  29. Ercole, C., Foi, A., Katkovnik, V., & Egiazarian, K. (2005). Spatio-temporal pointwise adaptive denoising of video: 3D non-parametric approach. In Proc. of the 1st international workshop on video processing and quality metrics for consumer electronics, VPQM2005, Scottsdale, AZ, January 2005. Google Scholar
  30. Fan, J., & Gijbels, I. (1996). Local polynomial modeling and its application. London: Chapman and Hall. Google Scholar
  31. Foi, A. (2005). Anisotropic nonparametric image processing: theory, algorithms and applications. Ph.D. Thesis, Dip. di Matematica, Politecnico di Milano, ERLTDD-D01290. Google Scholar
  32. Foi, A. (2007). Pointwise shape-adaptive DCT image filtering and signal-dependent noise estimation. D.Sc. Tech. Thesis, Institute of Signal Processing, Tampere University of Technology, Publication 710, December 2007. Google Scholar
  33. Foi, A., Katkovnik, V., Egiazarian, K., & Astola, J. (2004a). A novel anisotropic local polynomial estimator based on directional multiscale optimizations. In Proc. 6th IMA int. conf. math. in signal processing (pp. 79–82), Cirencester (UK). Google Scholar
  34. Foi, A., Katkovnik, V., Egiazarian, K., & Astola, J. (2004b). Inverse halftoning based on the anisotropic LPA-ICI deconvolution. In Proc. int. TICSP workshop spectral meth. multirate signal proc., SMMSP 2004 (pp. 49–56), Vienna, September 2004. Google Scholar
  35. Foi, A., Alenius, S., Trimeche, M., Katkovnik, V., & Egiazarian, K. (2005a). A spatially adaptive Poissonian image deblurring. In IEEE 2005 int. conf. image processing, ICIP 2005, September 2005. Google Scholar
  36. Foi, A., Bilcu, R., Katkovnik, V., & Egiazarian, K. (2005b). Anisotropic local approximations for pointwise adaptive signal-dependent noise removal. In XIII European signal proc. conf., EUSIPCO 2005. Google Scholar
  37. Foi, A., Katkovnik, V., & Egiazarian, K. (2007). Pointwise shape-adaptive DCT for high-quality denoising and deblocking of grayscale and color images. IEEE Transactions on Image Processing, 16(5), 1395–1411. CrossRefMathSciNetGoogle Scholar
  38. Freeman, W. T., & Adelson, E. H. (1991). The design and use of steerable filters. IEEE Transactions on Pattern Analysis and Machine Intelligence, 13(9), 891–906. CrossRefGoogle Scholar
  39. Gilboa, G., & Osher, S. (2007a). Nonlocal linear image regularization and supervised segmentation. SIAM Multiscale Modeling and Simulation, 6(2), 595–630. MATHCrossRefMathSciNetGoogle Scholar
  40. Gilboa, G., & Osher, S. (2007b). Nonlocal operators with applications to image processing, UCLA Computational and Applied Mathematics. Reports cam (07-23), July 2007. http://www.math.ucla.edu/applied/cam.
  41. Goldenshluger, A. (2009). A universal procedure for aggregating estimators. Annals of Statistics, 37(1), 542–568. MATHCrossRefMathSciNetGoogle Scholar
  42. Goldenshluger, A., & Nemirovski, A. (1997). On spatial adaptive estimation of nonparametric regression. Mathematical Methods of Statistics, 6, 135–170. MATHMathSciNetGoogle Scholar
  43. Guleryuz, O. (2007). Weighted averaging for denoising with overcomplete dictionaries. IEEE Transactions on Image Processing, 16(12), 3020–3034. CrossRefMathSciNetGoogle Scholar
  44. Hammond, D. K., & Simoncelli, E. P. (2008). Image modeling and denoising with orientation-adapted Gaussian scale mixtures. IEEE Transactions on Image Processing, 17(11), 2089–2101. CrossRefGoogle Scholar
  45. Hastie, T. J., & Loader, C. (1993). Local regression: automatic kernel carpentry (with discussion). Statistical Science, 8(2), 120–143. CrossRefGoogle Scholar
  46. Hel-Or, Y., & Shaked, D. (2008). A discriminative approach for wavelet denoising. IEEE Transactions on Image Processing, 17(4), 443–457. CrossRefMathSciNetGoogle Scholar
  47. Hirakawa, K., & Parks, T. W. (2006). Image denoising using total least squares. IEEE Transactions on Image Processing, 15(9), 2730–2742. CrossRefGoogle Scholar
  48. Hua, G., & Orchard, M. T. (2003). A new interpretation of translation invariant image denoising. In Proc. asilomar conf. signals syst. comput. (pp. 332–336), Pacific Grove, CA. Google Scholar
  49. Hurvich, C. M., Simonoff, J. S., & Tsai, C.-L. (1998). Smoothing parameter selection in nonparametric regression using an improved Akaike information criterion. Journal of the Royal Statistical Society, Series B, 60, 271–293. MATHCrossRefMathSciNetGoogle Scholar
  50. Hyvärinen, A., Karhunen, J., & Oja, E. (2001). Independent component analysis. New York: Wiley. CrossRefGoogle Scholar
  51. Jolliffe, I. T. (2002). Principal component analysis (2nd edn.). Berlin: Springer. MATHGoogle Scholar
  52. Katkovnik, V. (1976). Linear estimation and stochastic optimization problems. Moscow: Nauka (in Russian). Google Scholar
  53. Katkovnik, V. (1985). Nonparametric identification and smoothing of data (local approximation method). Moscow: Nauka (in Russian). MATHGoogle Scholar
  54. Katkovnik, V. (1999). A new method for varying adaptive bandwidth selection. IEEE Transactions on Signal Processing, 47(9), 2567–2571. CrossRefGoogle Scholar
  55. Katkovnik, V., & Spokoiny, V. (2008). Spatially adaptive estimation via fitted local likelihood techniques. IEEE Transactions on Image Processing, 56(3), 873–886. MathSciNetGoogle Scholar
  56. Katkovnik, V., Foi, A., Egiazarian, K., & Astola, J. (2004). Directional varying scale approximations for anisotropic signal processing. In Proc. XII European signal proc. conf., EUSIPCO 2004 (pp. 101–104), Vienna, September 2004. Google Scholar
  57. Katkovnik, V., Foi, A., Egiazarian, K., & Astola, J. (2005). Anisotropic local likelihood approximations. In Proc. of electronic imaging 2005 (p. 5672-19), January 2005. Google Scholar
  58. Katkovnik, V., Egiazarian, K., & Astola, J. (2006). Local approximation techniques in signal and image processing. Bellingham: SPIE Press. CrossRefGoogle Scholar
  59. Katkovnik, V., Foi, A., & Egiazarian, K. (2007). Mix-distribution modeling for overcomplete denoising. In Proc. 9th workshop on adaptation and learning in control and signal processing (ALCOSP’07), St. Petersburg, Russia, 29–31 August 2007. Google Scholar
  60. Katkovnik, V., Foi, A., Egiazarian, K., & Astola, J. (2008). Nonparametric regression in imaging: from local kernel to multiple-model nonlocal collaborative filtering. In Proc. 2008 int. workshop on local and non-local approximation in image processing, LNLA 2008. Lausanne, Switzerland, August 2008. Google Scholar
  61. Kervrann, C., & Boulanger, J. (2005). Local adaptivity to variable smoothness for exemplar-based image denoising and representation (Research Report INRIA, RR-5624). Google Scholar
  62. Kervrann, C., & Boulanger, J. (2006). Unsupervised patch-based image regularization and representation. In LNCS : Vol. 3954. ECCV 2006, Part IV (pp. 555–567). Berlin: Springer. Google Scholar
  63. Kervrann, C., & Boulanger, J. (2008). Local adaptivity to variable smoothness for exemplar-based image regularization and representation. International Journal of Computer Vision, 79, 45–69. CrossRefGoogle Scholar
  64. Kindermann, S., Osher, S., & Jones, P. W. (2005). Deblurring and denoising of images by nonlocal functionals. SIAM Multiscale Modeling & Simulation, 4(4), 1091–1115. MATHCrossRefMathSciNetGoogle Scholar
  65. Kingsbury, N. G. (2001). Complex wavelets for shift invariant analysis and filtering of signals. Journal of Applied and Computational Harmonic Analysis, 10(3), 234–253. MATHCrossRefMathSciNetGoogle Scholar
  66. Lansel, S., Donoho, D., & Weissman, T. (2009). DenoiseLab: a standard test set and evaluation method to compare denoising algorithms. http://www.stanford.edu/~slansel/DenoiseLab/.
  67. Lee, J. S. (1983). Digital image smoothing and the sigma filter. Computer Vision, Graphics, and Image Processing, 24, 255–269. CrossRefGoogle Scholar
  68. Lepski, O. (1990). One problem of adaptive estimation in Gaussian white noise. Theory of Probability and Its Applications, 35(3), 459–470. CrossRefMathSciNetGoogle Scholar
  69. Lepski, O., Mammen, E., & Spokoiny, V. (1997). Ideal spatial adaptation to inhomogeneous smoothness: an approach based on kernel estimates with variable bandwidth selection. The Annals of Statistics, 25(3), 929–947. MATHCrossRefMathSciNetGoogle Scholar
  70. Lindeberg, T. (1998). Edge detection and ridge detection with automatic scale selection. International Journal of Computer Vision, 30(2), 117–154. CrossRefGoogle Scholar
  71. Lindeberg, T. (2009). Scale-space. In B. Wah (Ed.), Encyclopedia of computer science and engineering (Vol. IV, pp. 2495–2504). Hoboken: Wiley. Google Scholar
  72. Loader, C. (1999). Series Statistics and Computing. Local regression and likelihood. New York: Springer. MATHGoogle Scholar
  73. Lou, Y., Favaro, P., & Soatto, S. (2008a). Nonlocal similarity image filtering. In UCLA computational and applied mathematics. Reports cam (8-26), April 2008. http://www.math.ucla.edu/applied/cam.
  74. Lou, Y., Zhang, X., Osher, S., & Bertozzi, A. (2008b). Image recovery via nonlocal operators. Reports cam (8-35), May 2008. http://www.math.ucla.edu/applied/cam.
  75. Mairal, J., Sapiro, G., & Elad, M. (2008). Learning multiscale sparse representations for image and video restoration. SIAM Multiscale Modeling Simulation, 7(1), 214–241. MATHCrossRefMathSciNetGoogle Scholar
  76. Mallat, S. (1999). A wavelet tour of signal processing. San Diego: Academic Press. MATHGoogle Scholar
  77. Meyer, Y. (2001). Univ. lecture ser. 22. Oscillating patterns in image processing and nonlinear evolution equations. Providence: AMS. MATHGoogle Scholar
  78. Muresan, D., & Parks, T. (2003). Adaptive principal components and image denoising. In Proc. 2003 IEEE int. conf. image process., ICIP 2003 (pp. 101–104), September 2003. Google Scholar
  79. Nadaraya, E. A. (1964). On estimating regression. Theory of Probability and Its Applications, 9, 141–142. CrossRefGoogle Scholar
  80. Öktem, H., Katkovnik, V., Egiazarian, K., & Astola, J. (2001). Local adaptive transform based image de-noising with varying window size. In Proc. IEEE int. conf. image process., ICIP 2001 (pp. 273–276), Thessaloniki, Greece. Google Scholar
  81. Öktem, R., Yaroslavsky, L., Egiazarian, K., & Astola, J. (1999). Transform based denoising algorithms: comparative study. Tampere: Tampere University of Technology. Google Scholar
  82. Osher, S., Sole, A., & Vese, L. (2003). Image decomposition and restoration using total variation minimization and the H−1 norm. SIAM Multiscale Modelling & Simulation, 1, 349–370. MATHCrossRefMathSciNetGoogle Scholar
  83. Perona, P., & Malik, J. (1990). Scale space and edge detection using anisotropic diffusion. IEEE Transactions on Pattern Analysis and Machine Intelligence, 12, 629–639. CrossRefGoogle Scholar
  84. Polzehl, J., & Spokoiny, V. (2000). Adaptive weights smoothing with applications to image restoration. Journal of Royal Statistical Society, Series B, 62, 335–354. CrossRefMathSciNetGoogle Scholar
  85. Polzehl, J., & Spokoiny, V. (2003). Image denoising: pointwise adaptive approach. The Annals of Statistics, 31(1), 30–57. MATHCrossRefMathSciNetGoogle Scholar
  86. Polzehl, J., & Spokoiny, V. (2005). Propagation-separation approach for local likelihood estimation. Probability Theory and Related Fields, 135(3), 335–362. CrossRefMathSciNetGoogle Scholar
  87. Portilla, J., Strela, V., Wainwright, M. J., & Simoncelli, E. P. (2003). Image denoising using scale mixtures of Gaussians in the wavelet domain. IEEE Transactions on Image Processing, 12(11), 1338–1351. CrossRefMathSciNetGoogle Scholar
  88. Roth, S., & Black, M. J. (2005). Fields of experts: A framework for learning image priors. In Proceedings of the IEEE international conference on computer vision and pattern recognition (CVPR) (pp. 860–867), San Diego, CA. Google Scholar
  89. Roth, S., & Black, M. J. (2009). Fields of experts. International Journal of Computer Vision, 82(2), 205–229. CrossRefGoogle Scholar
  90. Rudin, L., Osher, S., & Fatemi, E. (1992). Nonlinear algorithms. Physica D, 60, 259–268. MATHGoogle Scholar
  91. Rudin, L., Osher, S., & Fatemi, E. (1993). Nonlinear total variation based noise removal algorithms. Physica D, 60(2), 259–268. Google Scholar
  92. Ruppert, D. (1997). Empirical-bias bandwidths for local polynomial nonparametric regression and density estimation. Journal of American Statistical Association, 92(439), 1049–1062. MATHCrossRefMathSciNetGoogle Scholar
  93. Savitzky, A., & Golay, M. (1964). Smoothing and differentiation of data by simplified least squares procedures. Analytical Chemistry, 36, 1627–1639. CrossRefGoogle Scholar
  94. Schucany, W. R. (1995). Adaptive bandwidth choice for kernel regression. Journal of American Statistical Association, 90(430), 535–540. MATHCrossRefMathSciNetGoogle Scholar
  95. Selesnick, I. W., Baraniuk, R. G., & Kingsbury, N. G. (2005). The dual-tree complex wavelet transform. IEEE Signal Processing Magazine, 22(6), 123–151. CrossRefGoogle Scholar
  96. Seuhling, M., Arigovindan, M., Hunziker, P., & Unser, M. (2004). Multiresolution moment filters: theory and applications. IEEE Transactions on Image Processing, 13(4), 484–495. CrossRefGoogle Scholar
  97. Simoncelli, E. P., Freeman, W. T., Adelson, E. H., & Heeger, D. J. (1992). Shiftable multi-scale transforms. IEEE Transactions on Information Theory, 38, 587–607. CrossRefMathSciNetGoogle Scholar
  98. Simonoff, J. S. (1998). Smoothing methods in statistics. New York: Springer. Google Scholar
  99. Smith, S. M., & Brady, J. M. (1997). SUSAN—a new approach to low level image processing. International Journal of Computer Vision, 23(1), 45–78. CrossRefGoogle Scholar
  100. Spokoiny, V. (1998). Estimation of a function with discontinuities via local polynomial fit with an adaptive window choice. Annals of Statistics, 26, 1356–1378. MATHCrossRefMathSciNetGoogle Scholar
  101. Spokoiny, V. (2009). Local parametric methods in nonparametric estimation. Berlin: Springer (to appear). http://www.wias-berlin.de/people/spokoiny/adabook/ada.pdf. Google Scholar
  102. Steidl, G., Weickert, J., Brox, T., Mrazek, P., & Welk, M. (2004). On the equivalence of soft wavelet shrinkage, total variation diffusion, regularization and SIDEs. SIAM Journal of Numerical Analysis, 42(2), 686–713. MATHCrossRefMathSciNetGoogle Scholar
  103. Takeda, H., Farsiu, S., & Milanfar, P. (2007a). Higher order bilateral filters and their properties. In Proc. of the SPIE conf. on computational imaging, San Jose, January 2007. Google Scholar
  104. Takeda, H., Farsiu, S., & Milanfar, P. (2007b). Kernel regression for image processing and reconstruction. IEEE Transactions on Image Processing, 16(2), 349–366. CrossRefMathSciNetGoogle Scholar
  105. Tomasi, C., & Manduchi, R. (1998). Bilateral filtering for gray and color images. In Proc. of the sixth int. conf. on computer vision (pp. 839–846), 1998. Google Scholar
  106. Tschumperlé, D., & Brun, L. (2008a). Defining some variational methods on the space of patches: application to multi-valued image denoising and registration (Research Report: Les cahiers du GREYC, no. 08-01). Caen, France. Google Scholar
  107. Tschumperlé, D., & Brun, L. (2008b). Image denoising and registration by PDE’s on the space of patches. In Proc. int. workshop on local and non-local approximation in image processing (LNLA’08), Lausanne, Switzerland, August 2008. Google Scholar
  108. Vansteenkiste, E., Van der Weken, D., Philips, W., & Kerre, E. (2006). Perceived image quality measurement of state-of-the-art noise reduction schemes. In LNCS : Vol. 4179. ACIVS 2006 (pp. 114–124). Berlin: Springer. Google Scholar
  109. Vese, L., & Osher, S. (2003). Modeling textures with total variation minimization and oscillating patterns in image processing. Journal of Scientific Computing, 19, 553–572. MATHCrossRefMathSciNetGoogle Scholar
  110. Wand, M. P., & Jones, M. C. (1995). Monographs on statistics and applied probability : Vol. 60. Kernel smoothing. Boca Raton: Hartman&Hall/CRC. MATHGoogle Scholar
  111. Wang, Z., Bovik, A. C., Sheikh, H. R., & Simoncelli, E. P. (2004). Image quality assessment: From error visibility to structural similarity. IEEE Transactions on Image Processing, 13(4), 600–612. CrossRefGoogle Scholar
  112. Watson, G. S. (1964). Smooth regression analysis. Sankhya, Series A, 26, 359–372. MATHGoogle Scholar
  113. Wei, J. (2005). Lebesgue anisotropic image denoising. International Journal of Imaging Systems and Technology, 15(1), 64–73. CrossRefGoogle Scholar
  114. Weickert, J. (1996). Theoretical foundations of anisotropic diffusion in image processing. Computing, Suppl. 11, 221–236. Google Scholar
  115. Weickert, J. (1998). European consortium for mathematics in industry. Anisotropic diffusion in image processing. Stuttgart: B.G. Teubner. MATHGoogle Scholar
  116. Yaroslavsky, L. (1985). Digital picture processing—an introduction. New York: Springer. MATHGoogle Scholar
  117. Yaroslavsky, L. (1996). Local adaptive image restoration and enhancement with the use of DFT and DCT in a running window. In Proc. SPIE wavelet applications in signal and image process. IV (Vol. 2825, pp. 1–13), 1996. Google Scholar
  118. Yaroslavsky, L., & Eden, M. (1996). Fundamentals of digital optics. Boston: Birkhäuser Boston. MATHGoogle Scholar
  119. Yaroslavsky, L., Egiazarian, K., & Astola, J. (2001). Transform domain image restoration methods: review, comparison and interpretation. In Proc. SPIE, nonlinear image process. pattern anal. XII (Vol. 4304, pp. 155–169). San Jose, CA, 2001. Google Scholar
  120. Zimmer, S., Didas, S., & Weickert, J. (2008). A rotationally invariant block matching strategy improving image denoising with non-local means. In Proc. 2008 int. workshop on local and non-local approximation in image processing, LNLA 2008, Lausanne, Switzerland, August 2008. Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  • Vladimir Katkovnik
    • 1
  • Alessandro Foi
    • 1
  • Karen Egiazarian
    • 1
  • Jaakko Astola
    • 1
  1. 1.Department of Signal ProcessingTampere University of TechnologyTampereFinland

Personalised recommendations