Abstract

In this paper, a new denoising algorithm to deal with the additive white Gaussian noise model is described. Following the nonlocal (NL) means approach, we propose an adaptive estimator based on the weighted average of observations taken in a neighborhood with weights depending on the similarity of local patches. The idea is to compute adaptive weights that best minimize an upper bound of the pointwise $L_2$ risk. In the framework of adaptive estimation, we show that the “oracle” weights are optimal if we consider triangular kernels instead of the commonly used Gaussian kernel. Furthermore, we propose a way to automatically choose the spatially varying smoothing parameter for adaptive denoising. Under conventional minimal regularity conditions, the obtained estimator converges at the usual optimal rate. The implementation of the proposed algorithm is also straightforward and the simulations show that our algorithm significantly improves the classical NL means and is competitive when compared to the more sophisticated NL means filters, both in terms of peak signal-to-noise ratio values and visual quality.

Keywords

  1. image denoising
  2. image patches
  3. nonparametric estimation
  4. pointwise $L_2$ risk
  5. optimization

MSC codes

  1. 62H35
  2. 68U10

Get full access to this article

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

References

1.
A. Buades, B. Coll, and J. M. Morel, A review of image denoising algorithms, with a new one, Multiscale Model. Simul., 4 (2005), pp. 490--530.
2.
A. Buades, B. Coll, and J.-M. Morel, Non-local means denoising, IPOL J., 1 (2011), pp. 208--212.
3.
T. Buades, Y. Lou, J.-M. Morel, and Z. Tang, A note on multi-image denoising, in Proceedings of the 2009 International Workshop on Local and Non-Local Approximation in Image Processing, Tuusula, Finland, 2009, pp. 1--15.
4.
P. Chatterjee and P. Milanfar, Is denoising dead?, IEEE Trans. Image Process., 19 (2010), pp. 895--911.
5.
P. Chatterjee and P. Milanfar, Patch-based near-optimal image denoising, IEEE Trans. Image Process., 21 (2012), pp. 1635--1649.
6.
M. Colom and A. Buades, Analysis and extension of the percentile method, estimating a noise curve from a single image, IPOL J., 3 (2013), pp. 332--359.
7.
P. Coupé, P. Yger, S. Prima, P. Hellier, C. Kervrann, and C. Barillot, An optimized blockwise non-local means denoising filter for 3D magnetic resonance images, IEEE Trans. Med. Imaging, 27 (2008), pp. 325--441.
8.
K. Dabov, A. Foi, V. Katkovnik, and K. Egiazarian, Image denoising by sparse 3-D transform-domain collaborative filtering, IEEE Trans. Image Process., 16 (2007), pp. 2080--2095.
9.
J. Darbon, A. Cunha, and T. Chan, Fast nonlocal filtering applied to electron cryomicroscopy, in Proceedings of the IEEE International Symposium on Biomedical Imaging (ISBI), Paris, France, 2008, pp. 1331--1334.
10.
C.-A. Deledalle, L. Denis, and F. Tupin, Iterative weighted maximum likelihood denoising with probabilistic patch-based weights, IEEE Trans. Image Process., 18 (2009), pp. 2661--2672.
11.
C.-A. Deledalle, V. Duval, and J. Salmon, Non-local methods with shape-adaptive patches (NLM-SAP), J. Math. Imaging Vision, 43 (2012), pp. 103--120.
12.
D. Donoho and J. Johnstone, Ideal spatial adaptation by wavelet shrinkage, Biometrika, 81 (1994), pp. 425--455.
13.
V. Doré and M. Cheriet, Robust NL means filter with optimal pixel-wise smoothing parameter for statistical image denoising, IEEE Trans. Signal Process., 57 (2009), pp. 1703--1716.
14.
V. Duval, J. Aujol, and Y. Gousseau, A bias-variance approach for the nonlocal means, SIAM J. Imaging Sci., 4 (2011), pp. 760--788.
15.
M. Elad and M. Aharon, Image denoising via sparse and redundant representations over learned dictionaries, IEEE Trans. Image Process., 15 (2006), pp. 3736--3745.
16.
J. Froment, Parameter-free fast pixelwise non-local means denoising, IPOL J., 4 (2014), pp. 300--326.
17.
B. Goossens, Q. Luong, A. Pizurica, and W. Philips, An improved non-local denoising algorithm, in Proceedings of the 2008 International Workshop on Local and Non-Local Approximation in Image Processing, Lausanne, Switzerland, 2008.
18.
J. Immerkaer, Fast noise variance estimation, Comput. Vis. Image Underst., 64 (1996), pp. 300--302.
19.
V. Katkovnik, A. Foi, K. Egiazarian, and J. Astola, From local kernel to nonlocal multiple-model image denoising, Int. J. Comput. Vis., 86 (2010), pp. 1--32.
20.
C. Kervrann, PEWA: Patch-based exponentially weighted aggregation for image denoising, in Proceedings of the Neural Information Processing Systems Conference (NIPS'14), Montréal, Canada, 2014, pp. 2150--2158.
21.
C. Kervrann and J. Boulanger, Optimal spatial adaptation for patch-based image denoising, IEEE Trans. Image Process., 15 (2006), pp. 2866--2878.
22.
C. Kervrann and J. Boulanger, Unsupervised patch-based image regularization and representation, in Proceedings of the 9th European Conference on Computer Vision (ECCV'06), Graz, Austria, 2006, pp. 555--567.
23.
C. Kervrann and J. Boulanger, Local adaptivity to variable smoothness for exemplar-based image regularization and representation, Int. J. Comput. Vis., 79 (2008), pp. 45--69.
24.
C. Kervrann, J. Boulanger, and P. Coupé, Bayesian non-local means filter, image redundancy and adaptive dictionaries for noise removal, in Proceedings of the 1st International Conference on Scale Space and Variational Methods (SSVM'07), Ischia, Italy, 2007, pp. 520--532.
25.
S. Kindermann, S. Osher, and P. Jones, Deblurring and denoising of images by nonlocal functionals, Multiscale Model. Simul., 4 (2005), pp. 1091--1115.
26.
M. Lebrun, A. Buades, and J.-M. Morel, Implementation of the “Non-Local Bayes” (NL-Bayes) image denoising algorithm, IPOL J., 3 (2013), pp. 1--42.
27.
O. Lepski, On a problem of adaptive estimation in Gaussian white noise, Theory Probab. Appl., 35 (1990), pp. 454--466.
28.
O. Lepski, E. Mammen, and V. Spokoiny, Optimal spatial adaptation to inhomogeneous smoothness: An approach based on kernel estimates with variable bandwidth selectors, Ann. Statist., 25 (1997), pp. 929--947.
29.
A. Levin, B. Nadler, F. Durand, and W. Freeman, Patch complexity, finite pixel correlations and optimal denoising, in Proceedings of the 12th European Conference on Computer Vision (ECCV'12), Florence, Italy, 2012, pp. 73--86.
30.
B. Li, Q. Liu, J. Xu, and X. Luo, A new method for removing mixed noises, Sci. China Inf. Sci. 54 (2010), pp. 51--59.
31.
Y. Lou, X. Zhang, S. Osher, and A. Bertozzi, Image recovery via nonlocal operators, J. Sci. Comput., 42 (2010), pp. 185--197.
32.
C. Louchet and L. Moisan, Total variation as a local filter, SIAM J. Imaging Sci., 4 (2011), pp. 651--694.
33.
C. Louchet and L. Moisan, Posterior expectation of the total variation model: Properties and experiments, SIAM J. Imaging Sci., 6 (2013), pp. 2640--2684.
34.
J. Mairal, F. Bach, J. Ponce, G. Sapiro, and A. Zisserman, Non-local sparse models for image restoration, in Proceedings of the 2009 IEEE International Conference on Computer Vision (ICCV'09), Tokyo, Japan, 2009, pp. 2272--2279.
35.
A. Maleki, M. Narayan, and R. Baraniuk, Suboptimality of nonlocal means for images with sharp edges, Appl. Comput. Harmon. Anal., 33 (2012), pp. 370--387.
36.
A. Maleki, M. Narayan, and R. Baraniuk, Anisotropic nonlocal means, Appl. Comput. Harmon. Anal., 35 (2013), pp. 452--482.
37.
P. Milanfar, A tour of modern image filtering, IEEE Signal Process. Mag., 30 (2013), pp. 106--128.
38.
A. Nazin, J. Roll, L. Ljung, and I. Grama, Direct weight optimization in statistical estimation and system identification, in Proceedings of the 7th International Conference on System Identification and Control Problems (SICPRO'08), Moscow, Russia, 2008, pp. 27--67.
39.
L. Pizarro, P. Mrázek, S. Didas, S. Grewenig, and J. Weickert, Generalised nonlocal image smoothing, Int. J. Comput. Vis., 90 (2010), pp. 62--87.
40.
J. Polzehl and V. Spokoiny, Adaptive weights smoothing with applications to image restoration, J. R. Stat. Soc. Ser. B Stat. Methodol., 62 (2000), pp. 335--354.
41.
J. Polzehl and V. Spokoiny, Image denoising: Pointwise adaptive approach, Ann. Statist., 31 (2003), pp. 30--57.
42.
J. Polzehl and V. Spokoiny, Propagation-separation approach for local likelihood estimation, Probab. Theory Related Fields, 135 (2006), pp. 335--362.
43.
I. Ram, M. Elad, and I. Cohen, Image processing using smooth ordering of its patches, IEEE Trans. Image Process., 22 (2013), pp. 2764--2774.
44.
M. Raphan and E. Simoncelli, Empirical Bayes Least Squares Estimation Without an Explicit Prior, Technical report TR2007-900, Courant Institute of Mathematical Sciences, New York University, New York, 2007.
45.
R. Rockafellar, Lagrange multipliers and optimality, SIAM Rev., 35 (1993), pp. 183--238.
46.
J. Roll, Local and Piecewise Affine Approaches to System Identification, Ph.D. dissertation, Linköping University, Linköping, Sweden, 2003.
47.
J. Roll and L. Ljung, Extending the Direct Weight Optimization Approach, Technical report LiTH-ISY-R-2601 Linköping University, Linköping Sweden, 2004.
48.
J. Roll, A. Nazin, and L. Ljung, Nonlinear system identification via direct weight optimization, Automatica J. IFAC, 41 (2005), pp. 475--490.
49.
S. Roth and M. Black, Fields of experts, Int. J. Comput. Vis., 82 (2009), pp. 205--229.
50.
L. Rudin, S. Osher, and E. Fatemi, Nonlinear total variation based noise removal algorithms, Phys. D, 60 (1992), pp. 259--268.
51.
J. Sacks and D. Ylvisaker, Linear estimation for approximately linear models, Ann. Statist., 6 (1978), pp. 1122--1137.
52.
J. Salmon, On two parameters for denoising with non-local means, IEEE Signal Process. Lett., 17 (2010), pp. 269--272.
53.
J. Salmon and E. Le Pennec, NL-means and aggregation procedures, in Proceedings of the 16th IEEE International Conference on Image Processing (ICIP'09), Cairo, Egypt, 2009, pp. 2977--2980.
54.
J. Salmon and Y. Strozecki, Patch reprojections for non-local methods, Signal Process., 92 (2012), pp. 477--489.
55.
J. Salmon, R. Willett, and E. Arias-Castro, A two-stage denoising filter: The preprocessed Yaroslavsky filter, in Proceedings of the 2012 IEEE Statistical Signal Processing Workshop (SSP'12), Ann Arbor, MI, 2012, pp. 464--467.
56.
C. Sutour, C.-A. Deledalle, and J.-F. Aujol, Adaptive regularization of the NL-means: Application to image and video denoising, IEEE Trans. Image Process., 23 (2014), pp. 3506--3521.
57.
C. Sutour, C.-A. Deledalle, and J.-F. Aujol, Estimation of the noise level function based on a nonparametric detection of homogeneous image regions, SIAM J. Imaging Sci., 8 (2015), pp. 2622--2661.
58.
C. Tomasi and R. Manduchi, Bilateral filtering for gray and color images, in Proceedings of 6th IEEE International Conference on Computer Vision (ICCV'98), Bombay, India, 1998, pp. 839--846.
59.
D. Van de Ville and M. Kocher, SURE based non-local means, IEEE Signal Process. Lett., 16 (2009), pp. 973--976.
60.
Y.-Q. Wang and J.-M. Morel, SURE guided Gaussian mixture image denoising, SIAM J. Imaging Sci., 6 (2013), pp. 999--1034.
61.
P. Whittle, Optimization under Constraints: Theory and Applications of Nonlinear Programming, Wiley-Interscience, New York, 1971.
62.
L. P. Yaroslavsky, Digital Picture Processing: An Introduction, Springer-Verlag, Berlin, 1985.
63.
G. Yu and G. Sapiro, DCT image denoising: A simple and effective image denoising algorithm, IPOL J. 1 (2011), pp. 292--296.
64.
D. Zoran and Y. Weiss, From learning models of natural image patches to whole image restoration, in Proceedings of the 2011 IEEE International Conference on Computer Vision (ICCV'11), Barcelona, Spain, 2011, pp. 479--486.

Information & Authors

Information

Published In

cover image SIAM Journal on Imaging Sciences
SIAM Journal on Imaging Sciences
Pages: 1878 - 1920
ISSN (online): 1936-4954

History

Submitted: 20 June 2016
Accepted: 9 May 2017
Published online: 2 November 2017

Keywords

  1. image denoising
  2. image patches
  3. nonparametric estimation
  4. pointwise $L_2$ risk
  5. optimization

MSC codes

  1. 62H35
  2. 68U10

Authors

Affiliations

Funding Information

Natural Science Fund of Inner Magnolia Autonomous Region : 2016MS0107
Scientific Research Projection of Higher Schools of Inner Magnolia : NJZY16017
National Natural Science Foundation of China https://doi.org/10.13039/501100001809 : 61661039, 11571052, 11401590

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.

View Options

View options

PDF

View PDF

Media

Figures

Other

Tables

Share

Share

Copy the content Link

Share with email

Email a colleague

Share on social media