Skip to main content
SpringerLink
Log in

From Local Kernel to Nonlocal Multiple-Model Image Denoising

  • Published:
International Journal of Computer Vision Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Abramovich, F., & Benjamini, Y. (1996). Adaptive thresholding of wavelet coefficients. Computational Statistics and Data Analysis, 22, 351–361.

    Article  MathSciNet  Google Scholar 

  • 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.

    Article  MATH  MathSciNet  Google Scholar 

  • 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.

    Article  Google Scholar 

  • 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.

  • 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 

  • 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.

    Article  Google Scholar 

  • 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.

    Article  MATH  MathSciNet  Google Scholar 

  • 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.

    Article  MathSciNet  Google Scholar 

  • Brown, R. (1963). Smoothing, forecasting and prediction of discrete time series. Prentice-Hall: Englewood Cliffs.

    Google Scholar 

  • 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.

    Article  MATH  MathSciNet  Google Scholar 

  • 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.

    Article  Google Scholar 

  • Buades, A., Coll, B., & Morel, J. M. (2006b). Neighborhood filters and PDE’s. Numerische Mathematik, 105(1), 1–34.

    Article  MATH  MathSciNet  Google Scholar 

  • Buades, A., Coll, B., & Morel, J. M. (2008). Nonlocal image and movie denoising. International Journal of Computer Vision, 76(2), 123–139.

    Article  Google Scholar 

  • Bunea, F., Tsybakov, A., & Wegkamp, M. (2007). Aggregation for regression learning. Annals of Statistics, 35, 1674–1697.

    Article  MATH  MathSciNet  Google Scholar 

  • Chan, T., & Shen, J. (2005). Image processing and analysis: variational, PDE, wavelet, and stochastic methods. Philadelphia: SIAM.

    MATH  Google Scholar 

  • 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.

  • 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.

    Article  Google Scholar 

  • 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 

  • 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.

    Article  MathSciNet  Google Scholar 

  • 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.

  • 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.

  • Donoho, D., & Johnstone, I. (1995). Adapting to unknown smoothness via wavelet shrinkage. Journal of American Statistical Association, 90(432), 1200–1224.

    Article  MATH  MathSciNet  Google Scholar 

  • 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.

  • Elad, M. (2002). On the origin of the bilateral filter and ways to improve it. IEEE Transactions on Image Processing, 10(10), 1141–1151.

    Article  MathSciNet  Google Scholar 

  • Elad, M. (2006). Why shrinkage is still relevant for redundant representations? IEEE Transactions on Information Theory, 52(12), 5559–5569.

    Article  MathSciNet  Google Scholar 

  • Elad, M., & Aharon, M. (2006). Image denoising via sparse and redundant representations over learned dictionaries. IEEE Transactions on Image Processing, 15(12), 3736–3745.

    Article  MathSciNet  Google Scholar 

  • Elad, M., & Yavneh, I. (2009). A plurality of sparse representations is better than the sparsest one alone. IEEE Transactions on Information Theory (to appear).

  • 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.

  • 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.

  • Fan, J., & Gijbels, I. (1996). Local polynomial modeling and its application. London: Chapman and Hall.

    Google Scholar 

  • Foi, A. (2005). Anisotropic nonparametric image processing: theory, algorithms and applications. Ph.D. Thesis, Dip. di Matematica, Politecnico di Milano, ERLTDD-D01290.

  • 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.

  • 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).

  • 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.

  • 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.

  • 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.

  • 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.

    Article  MathSciNet  Google Scholar 

  • 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.

    Article  Google Scholar 

  • Gilboa, G., & Osher, S. (2007a). Nonlocal linear image regularization and supervised segmentation. SIAM Multiscale Modeling and Simulation, 6(2), 595–630.

    Article  MATH  MathSciNet  Google Scholar 

  • 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.

  • Goldenshluger, A. (2009). A universal procedure for aggregating estimators. Annals of Statistics, 37(1), 542–568.

    Article  MATH  MathSciNet  Google Scholar 

  • Goldenshluger, A., & Nemirovski, A. (1997). On spatial adaptive estimation of nonparametric regression. Mathematical Methods of Statistics, 6, 135–170.

    MATH  MathSciNet  Google Scholar 

  • Guleryuz, O. (2007). Weighted averaging for denoising with overcomplete dictionaries. IEEE Transactions on Image Processing, 16(12), 3020–3034.

    Article  MathSciNet  Google Scholar 

  • 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.

    Article  Google Scholar 

  • Hastie, T. J., & Loader, C. (1993). Local regression: automatic kernel carpentry (with discussion). Statistical Science, 8(2), 120–143.

    Article  Google Scholar 

  • Hel-Or, Y., & Shaked, D. (2008). A discriminative approach for wavelet denoising. IEEE Transactions on Image Processing, 17(4), 443–457.

    Article  MathSciNet  Google Scholar 

  • Hirakawa, K., & Parks, T. W. (2006). Image denoising using total least squares. IEEE Transactions on Image Processing, 15(9), 2730–2742.

    Article  Google Scholar 

  • 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.

  • 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.

    Article  MATH  MathSciNet  Google Scholar 

  • Hyvärinen, A., Karhunen, J., & Oja, E. (2001). Independent component analysis. New York: Wiley.

    Book  Google Scholar 

  • Jolliffe, I. T. (2002). Principal component analysis (2nd edn.). Berlin: Springer.

    MATH  Google Scholar 

  • Katkovnik, V. (1976). Linear estimation and stochastic optimization problems. Moscow: Nauka (in Russian).

    Google Scholar 

  • Katkovnik, V. (1985). Nonparametric identification and smoothing of data (local approximation method). Moscow: Nauka (in Russian).

    MATH  Google Scholar 

  • Katkovnik, V. (1999). A new method for varying adaptive bandwidth selection. IEEE Transactions on Signal Processing, 47(9), 2567–2571.

    Article  Google Scholar 

  • Katkovnik, V., & Spokoiny, V. (2008). Spatially adaptive estimation via fitted local likelihood techniques. IEEE Transactions on Image Processing, 56(3), 873–886.

    MathSciNet  Google Scholar 

  • 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.

  • Katkovnik, V., Foi, A., Egiazarian, K., & Astola, J. (2005). Anisotropic local likelihood approximations. In Proc. of electronic imaging 2005 (p. 5672-19), January 2005.

  • Katkovnik, V., Egiazarian, K., & Astola, J. (2006). Local approximation techniques in signal and image processing. Bellingham: SPIE Press.

    Book  Google Scholar 

  • 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.

  • 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.

  • Kervrann, C., & Boulanger, J. (2005). Local adaptivity to variable smoothness for exemplar-based image denoising and representation (Research Report INRIA, RR-5624).

  • 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 

  • 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.

    Article  Google Scholar 

  • Kindermann, S., Osher, S., & Jones, P. W. (2005). Deblurring and denoising of images by nonlocal functionals. SIAM Multiscale Modeling & Simulation, 4(4), 1091–1115.

    Article  MATH  MathSciNet  Google Scholar 

  • 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.

    Article  MATH  MathSciNet  Google Scholar 

  • 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/.

  • Lee, J. S. (1983). Digital image smoothing and the sigma filter. Computer Vision, Graphics, and Image Processing, 24, 255–269.

    Article  Google Scholar 

  • Lepski, O. (1990). One problem of adaptive estimation in Gaussian white noise. Theory of Probability and Its Applications, 35(3), 459–470.

    Article  MathSciNet  Google Scholar 

  • 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.

    Article  MATH  MathSciNet  Google Scholar 

  • Lindeberg, T. (1998). Edge detection and ridge detection with automatic scale selection. International Journal of Computer Vision, 30(2), 117–154.

    Article  Google Scholar 

  • Lindeberg, T. (2009). Scale-space. In B. Wah (Ed.), Encyclopedia of computer science and engineering (Vol. IV, pp. 2495–2504). Hoboken: Wiley.

    Google Scholar 

  • Loader, C. (1999). Series Statistics and Computing. Local regression and likelihood. New York: Springer.

    MATH  Google Scholar 

  • 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.

  • 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.

  • Mairal, J., Sapiro, G., & Elad, M. (2008). Learning multiscale sparse representations for image and video restoration. SIAM Multiscale Modeling Simulation, 7(1), 214–241.

    Article  MATH  MathSciNet  Google Scholar 

  • Mallat, S. (1999). A wavelet tour of signal processing. San Diego: Academic Press.

    MATH  Google Scholar 

  • Meyer, Y. (2001). Univ. lecture ser. 22. Oscillating patterns in image processing and nonlinear evolution equations. Providence: AMS.

    MATH  Google Scholar 

  • 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.

  • Nadaraya, E. A. (1964). On estimating regression. Theory of Probability and Its Applications, 9, 141–142.

    Article  Google Scholar 

  • 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.

  • Öktem, R., Yaroslavsky, L., Egiazarian, K., & Astola, J. (1999). Transform based denoising algorithms: comparative study. Tampere: Tampere University of Technology.

    Google Scholar 

  • 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.

    Article  MATH  MathSciNet  Google Scholar 

  • Perona, P., & Malik, J. (1990). Scale space and edge detection using anisotropic diffusion. IEEE Transactions on Pattern Analysis and Machine Intelligence, 12, 629–639.

    Article  Google Scholar 

  • Polzehl, J., & Spokoiny, V. (2000). Adaptive weights smoothing with applications to image restoration. Journal of Royal Statistical Society, Series B, 62, 335–354.

    Article  MathSciNet  Google Scholar 

  • Polzehl, J., & Spokoiny, V. (2003). Image denoising: pointwise adaptive approach. The Annals of Statistics, 31(1), 30–57.

    Article  MATH  MathSciNet  Google Scholar 

  • Polzehl, J., & Spokoiny, V. (2005). Propagation-separation approach for local likelihood estimation. Probability Theory and Related Fields, 135(3), 335–362.

    Article  MathSciNet  Google Scholar 

  • 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.

    Article  MathSciNet  Google Scholar 

  • 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.

  • Roth, S., & Black, M. J. (2009). Fields of experts. International Journal of Computer Vision, 82(2), 205–229.

    Article  Google Scholar 

  • Rudin, L., Osher, S., & Fatemi, E. (1992). Nonlinear algorithms. Physica D, 60, 259–268.

    MATH  Google Scholar 

  • Rudin, L., Osher, S., & Fatemi, E. (1993). Nonlinear total variation based noise removal algorithms. Physica D, 60(2), 259–268.

    Google Scholar 

  • Ruppert, D. (1997). Empirical-bias bandwidths for local polynomial nonparametric regression and density estimation. Journal of American Statistical Association, 92(439), 1049–1062.

    Article  MATH  MathSciNet  Google Scholar 

  • Savitzky, A., & Golay, M. (1964). Smoothing and differentiation of data by simplified least squares procedures. Analytical Chemistry, 36, 1627–1639.

    Article  Google Scholar 

  • Schucany, W. R. (1995). Adaptive bandwidth choice for kernel regression. Journal of American Statistical Association, 90(430), 535–540.

    Article  MATH  MathSciNet  Google Scholar 

  • Selesnick, I. W., Baraniuk, R. G., & Kingsbury, N. G. (2005). The dual-tree complex wavelet transform. IEEE Signal Processing Magazine, 22(6), 123–151.

    Article  Google Scholar 

  • Seuhling, M., Arigovindan, M., Hunziker, P., & Unser, M. (2004). Multiresolution moment filters: theory and applications. IEEE Transactions on Image Processing, 13(4), 484–495.

    Article  Google Scholar 

  • 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.

    Article  MathSciNet  Google Scholar 

  • Simonoff, J. S. (1998). Smoothing methods in statistics. New York: Springer.

    Google Scholar 

  • 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.

    Article  Google Scholar 

  • Spokoiny, V. (1998). Estimation of a function with discontinuities via local polynomial fit with an adaptive window choice. Annals of Statistics, 26, 1356–1378.

    Article  MATH  MathSciNet  Google Scholar 

  • 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 

  • 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.

    Article  MATH  MathSciNet  Google Scholar 

  • 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.

  • Takeda, H., Farsiu, S., & Milanfar, P. (2007b). Kernel regression for image processing and reconstruction. IEEE Transactions on Image Processing, 16(2), 349–366.

    Article  MathSciNet  Google Scholar 

  • 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.

  • 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.

  • 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.

  • 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 

  • Vese, L., & Osher, S. (2003). Modeling textures with total variation minimization and oscillating patterns in image processing. Journal of Scientific Computing, 19, 553–572.

    Article  MATH  MathSciNet  Google Scholar 

  • Wand, M. P., & Jones, M. C. (1995). Monographs on statistics and applied probability : Vol. 60. Kernel smoothing. Boca Raton: Hartman&Hall/CRC.

    MATH  Google Scholar 

  • 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.

    Article  Google Scholar 

  • Watson, G. S. (1964). Smooth regression analysis. Sankhya, Series A, 26, 359–372.

    MATH  Google Scholar 

  • Wei, J. (2005). Lebesgue anisotropic image denoising. International Journal of Imaging Systems and Technology, 15(1), 64–73.

    Article  Google Scholar 

  • Weickert, J. (1996). Theoretical foundations of anisotropic diffusion in image processing. Computing, Suppl. 11, 221–236.

    Google Scholar 

  • Weickert, J. (1998). European consortium for mathematics in industry. Anisotropic diffusion in image processing. Stuttgart: B.G. Teubner.

    MATH  Google Scholar 

  • Yaroslavsky, L. (1985). Digital picture processing—an introduction. New York: Springer.

    MATH  Google Scholar 

  • 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.

  • Yaroslavsky, L., & Eden, M. (1996). Fundamentals of digital optics. Boston: Birkhäuser Boston.

    MATH  Google Scholar 

  • 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.

  • 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.

Download references

Author information

Authors and Affiliations

  1. Department of Signal Processing, Tampere University of Technology, Tampere, Finland

    Vladimir Katkovnik, Alessandro Foi, Karen Egiazarian & Jaakko Astola

Authors
  1. Vladimir Katkovnik
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Alessandro Foi
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Karen Egiazarian
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Jaakko Astola
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Alessandro Foi.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Katkovnik, V., Foi, A., Egiazarian, K. et al. From Local Kernel to Nonlocal Multiple-Model Image Denoising. Int J Comput Vis 86, 1–32 (2010). https://doi.org/10.1007/s11263-009-0272-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11263-009-0272-7

Keywords

Search

Navigation