Abstract

The search for efficient image denoising methods is still a valid challenge at the crossing of functional analysis and statistics. In spite of the sophistication of the recently proposed methods, most algorithms have not yet attained a desirable level of applicability. All show an outstanding performance when the image model corresponds to the algorithm assumptions but fail in general and create artifacts or remove image fine structures. The main focus of this paper is, first, to define a general mathematical and experimental methodology to compare and classify classical image denoising algorithms and, second, to propose a nonlocal means (NL-means) algorithm addressing the preservation of structure in a digital image. The mathematical analysis is based on the analysis of the "method noise," defined as the difference between a digital image and its denoised version. The NL-means algorithm is proven to be asymptotically optimal under a generic statistical image model. The denoising performance of all considered methods are compared in four ways; mathematical: asymptotic order of magnitude of the method noise under regularity assumptions; perceptual-mathematical: the algorithms artifacts and their explanation as a violation of the image model; quantitative experimental: by tables of L2 distances of the denoised version to the original image. The most powerful evaluation method seems, however, to be the visualization of the method noise on natural images. The more this method noise looks like a real white noise, the better the method.

MSC codes

  1. 62H35

Keywords

  1. image restoration
  2. nonparametric estimation
  3. PDE smoothing filters
  4. adaptive filters
  5. frequency domain filters

Get full access to this article

View all available purchase options and get full access to this article.

References

1.
Luis Alvarez, Pierre‐Louis Lions, Jean‐Michel Morel, Image selective smoothing and edge detection by nonlinear diffusion. II, SIAM J. Numer. Anal., 29 (1992), 845–866
2.
Jean‐François Aujol, Gilles Aubert, Laure Blanc‐Féraud, Antonin Chambolle, Image decomposition into a bounded variation component and an oscillating component, J. Math. Imaging Vision, 22 (2005), 71–88
3.
S. A. Awate and R. T. Whitaker, Image denoising with unsupervised, information‐theoretic, adaptive filtering, in Proceedings of the IEEE International Conference on Computer Vision and Pattern Recognition, 2005, to appear.
4.
A. Buades, B. Coll, and J.‐M. Morel, Procédé de traitement de données d’image, par réduction de bruit d’image, et caméra intégrant des moyens de mise en oeuvre du procédé (Image data process by image noise reduction and camera integrating the means for implementing this process), French patent application, serial number: 0404837.
5.
Francine Catté, Françoise Dibos, Georges Koepfler, A morphological scheme for mean curvature motion and applications to anisotropic diffusion and motion of level sets, SIAM J. Numer. Anal., 32 (1995), 1895–1909
6.
Antonin Chambolle, Pierre‐Louis Lions, Image recovery via total variation minimization and related problems, Numer. Math., 76 (1997), 167–188
7.
T. F. Chan and H. M. Zhou, Total variation improved wavelet thresholding in image compression, in Proceedings of the IEEE International Conference on Image Processing, Vol. 2, Vancouver, BC, Canada, 2000, pp. 391–394.
8.
Anestis Antoniadis, Georges Oppenheim, Wavelets and statistics, Lecture Notes in Statistics, Vol. 103, Springer‐Verlag, 1995ii+411, Papers from the conference held in Villard de Lans, November 16–18, 1994
9.
David Donoho, De‐noising by soft‐thresholding, IEEE Trans. Inform. Theory, 41 (1995), 613–627
10.
David Donoho, Iain Johnstone, Ideal spatial adaptation by wavelet shrinkage, Biometrika, 81 (1994), 425–455
11.
S. Durand and M. Nikolova, Restoration of wavelet coefficients by minimizing a specially designed objective function, in Proceedings of the IEEE Workshop on Variational and Level Set Methods in Computer Vision, 2003.
12.
Sylvain Durand, Jacques Froment, Reconstruction of wavelet coefficients using total variation minimization, SIAM J. Sci. Comput., 24 (2003), 1754–1767
13.
A. Efros and T. Leung, Texture synthesis by non parametric sampling, in Proceedings of the IEEE International Conference on Computer Vision, Vol. 2, Corfu, Greece, 1999, pp. 1033–1038.
14.
F. Guichard, J. M. Morel, and R. Ryan, Image Analysis and P.D.E.’s, preprint.
15.
E. Le Pennec and S. Mallat, Geometrical image compression with bandelets, in Proceedings of the SPIE 2003, Vol. 5150, Lugano, Switzerland, 2003, pp. 1273–1286.
16.
E. Levina, Statistical Issues in Texture Analysis, Ph.D. thesis, UC‐Berkeley, Berkeley, CA, 2002.
17.
M. Lindenbaum, M. Fischer, and A. M. Bruckstein, On Gabor contribution to image enhancement, Pattern Recognition, 27 (1994), pp. 1–8.
18.
Stéphane Lintner, François Malgouyres, Solving a variational image restoration model which involves L constraints, Inverse Problems, 20 (2004), 815–831
19.
F. Malgouyres, A noise selection approach of image restoration, in Proceedings of Wavelet Applications in Signal and Image Processing IX, SPIE Proc. Ser. 4478, SPIE, Bellingham, WA, 2001, pp. 34–41.
20.
Stéphane Mallat, A wavelet tour of signal processing, Academic Press Inc., 1998xxiv+577
21.
B. Merriman, J. Bence, and S. Osher, Diffusion generated motion by mean curvature, in Proceedings of the Geometry Center Workshop, Minneapolis, MN, 1992.
22.
Yves Meyer, Oscillating patterns in image processing and nonlinear evolution equations, University Lecture Series, Vol. 22, American Mathematical Society, 2001x+122, The fifteenth Dean Jacqueline B. Lewis memorial lectures
23.
È. Nadaraja, On non‐parametric estimates of density functions and regression, Teor. Verojatnost. i Primenen., 10 (1965), 199–203
24.
E. Ordentlich, G. Seroussi, S. Verdú, M. Weinberger, and T. Weissman, A discrete universal denoiser and its application to binary images, in Proceedings of the IEEE International Conference on Image Processing, Vol. 1, Barcelona, Spain, 2003, pp. 117–120.
25.
S. Osher, M. Burger, D. Goldfarb, J. Xu, and W. Yin, An iterative regularization method for total variation‐based image restoration, Multiscale Model. Simul., 4 (2005), pp. 460–489.
26.
Stanley Osher, Andrés Solé, Luminita Vese, Image decomposition and restoration using total variation minimization and the H-1 norm, Multiscale Model. Simul., 1 (2003), 349–370
27.
P. Perona and J. Malik, Scale space and edge detection using anisotropic diffusion, IEEE Trans. Patt. Anal. Mach. Intell., 12 (1990), pp. 629–639.
28.
George Roussas, Nonparametric regression estimation under mixing conditions, Stochastic Process. Appl., 36 (1990), 107–116
29.
L. Rudin and S. Osher, Total variation based image restoration with free local constraints, in Proceedings of the IEEE International Conference on Image Processing, Vol. 1, Austin, TX, 1994, pp. 31–35.
30.
L. Rudin, S. Osher, and E. Fatemi, Nonlinear total variation based noise removal algorithms, Phys. D, 60 (1992), pp. 259–268.
31.
Claude Shannon, Warren Weaver, The Mathematical Theory of Communication, The University of Illinois Press, Urbana, Ill., 1949vi+117
32.
A. Sharf, M. Alexa, and D. Cohen‐Or, Context‐based surface completion, in Proceedings of the 31st International Conference on Computer Graphics and Interactive Techniques, Los Angeles, CA, 2004, pp. 878–887.
33.
S. M. Smith and J. M. Brady, SUSAN ‐ a new approach to low level image processing, International Journal of Computer Vision, 23 (1997), pp. 45–78.
34.
Jean‐Luc Starck, Emmanuel Candès, David Donoho, The curvelet transform for image denoising, IEEE Trans. Image Process., 11 (2002), 670–684
35.
Eitan Tadmor, Suzanne Nezzar, Luminita Vese, A multiscale image representation using hierarchical (BV,L2) decompositions, Multiscale Model. Simul., 2 (2004), 554–579
36.
Luminita Vese, Stanley Osher, Modeling textures with total variation minimization and oscillating patterns in image processing, J. Sci. Comput., 19 (2003), 553–572, Special issue in honor of the sixtieth birthday of Stanley Osher
37.
Geoffrey Watson, Smooth regression analysis, Sankhyā Ser. A, 26 (1964), 359–372
38.
T. Weissman, E. Ordentlich, G. Seroussi, S. Verdu, and M. Weinberger, Universal discrete denoising: Known channel, IEEE Trans. Inform. Theory, 51 (2005), pp. 5–28.
39.
L. Yaroslavsky, K. Egiazarian, and J. Astola, Transform domain image restoration methods: Review, comparison and interpretation, in Nonlinear Image Processing and Pattern Analysis, SPIE, Bellingham, WA, 2001, pp. 155–169.
40.
L. Yaroslavsky, Digital picture processing, Springer Series in Information Sciences, Vol. 9, Springer‐Verlag, 1985xii+276, An introduction
41.
L. Yaroslavsky and M. Eden, Fundamentals of Digital Optics, Birkhäuser Boston, Boston, MA, 1996.
42.
L. Yaroslavsky, Local adaptive image restoration and enhancement with the use of DFT and DCT in a running window, in Proceedings of Wavelet Applications in Signal and Image Processing IV, SPIE Proc. Ser. 2825, Denver, CO, 1996, pp. 1–13.

Information & Authors

Information

Published In

cover image Multiscale Modeling & Simulation
Multiscale Modeling & Simulation
Pages: 490 - 530
ISSN (online): 1540-3467

History

Published online: 26 July 2006

MSC codes

  1. 62H35

Keywords

  1. image restoration
  2. nonparametric estimation
  3. PDE smoothing filters
  4. adaptive filters
  5. frequency domain filters

Authors

Affiliations

Metrics & Citations

Metrics

Citations

If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.

Cited By