Source: http://aimsciences.org/article/doi/10.3934/ipi.2018036
Timestamp: 2019-04-24 13:50:30+00:00

Document:
We introduce an oracle filter for removing the Gaussian noise with weights depending on a similarity function. The usual Non-Local Means filter is obtained from this oracle filter by substituting the similarity function by an estimator based on similarity patches. When the sizes of the search window are chosen appropriately, it is shown that the oracle filter converges with the optimal rate. The same optimal convergence rate is preserved when the similarity function has suitable errors-in measurements. We also provide a statistical estimator of the similarity which converges at a convenient rate. Based on our convergence theorems, we propose some simple formulas for the choice of the parameters. Simulation results show that our choice of parameters improves the restoration quality of the filter compared with the usual choice of parameters in the original algorithm.
Keywords: Image denoising, oracle filter, Non-Local Means filter, weighted means, convergence, mean squared error.
M. Aharon, M. Elad and A. Bruckstein, rmk-svd: An algorithm for designing overcomplete dictionaries for sparse representation, IEEE Trans. Signal Process., 54 (2006), 4311-4322. doi: 10.1109/TSP.2006.881199.
E. Arias-Castro, J. Salmon and R. Willett, Oracle inequalities and minimax rates for non-local means and related adaptive kernel-based methods, Siam Journal on Imaging Sciences, 5 (2012), 944-992. doi: 10.1137/110859403.
R. C. Bilcu and M. Vehvilainen, Fast nonlocal means for image denoising, In Proc. of SPIE Conf. on Digital Photography III, 6502 (2007), 65020R. doi: 10.1117/12.695079.
J. Boulanger, C. Kervrann, P. Bouthemy, P. Elbau, J. B. Sibarita and J. Salamero, Patch-based nonlocal functional for denoising fluorescence microscopy image sequences, IEEE Transactions on Medical Imaging, 29 (2010), 442-454. doi: 10.1109/TMI.2009.2033991.
A. Buades, B. Coll and J. M. Morel, A review of image denoising algorithms, with a new one, Multiscale Model. Simul., 4 (2005), 490-530. doi: 10.1137/040616024.
A. Buades, B. Coll and J. M. Morel, The staircasing effect in neighborhood filters and its solution, IEEE Trans. Image Process., 15 (2006), 1499-1505. doi: 10.1109/TIP.2006.871137.
T. Buades, Y. Lou, J. M. Morel and Z. Tang, A note on multi-image denoising, In Int. workshop on Local and Non-Local Approximation in Image Processing, pages 1–15, August 2009. doi: 10.1109/LNLA.2009.5278408.
P. Chatterjee and P. Milanfar, A generalization of non-local means via kernel regression, In Proc. of SPIE Conf. on Computational Imaging, Citeseer, 6814 (2008), 6814Op. doi: 10.1117/12.778615.
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), 2080-2095. doi: 10.1109/TIP.2007.901238.
K. Dabov, A. Foi, V. Katkovnik and K. Egiazarian, Bm3d image denoising with shape-adaptive principal component analysis, In Proc. Workshop on Signal Processing with Adaptive Sparse Structured Representations (SPARS 09), volume 49, Citeseer, 2009.
C. A. Deledalle, V. Duval and J. Salmon, Non-local methods with shape-adaptive patches (nlm-sap), Journal of Mathematical Imaging and Vision, 43 (2012), 103-120. doi: 10.1007/s10851-011-0294-y.
D. L. Donoho and J. M. Johnstone, Ideal spatial adaptation by wavelet shrinkage, Biometrika, 81 (1994), 425-455. doi: 10.1093/biomet/81.3.425.
V. Duval, J.-F. Aujol and Y. Gousseau, A bias-variance approach for the nonlocal means, SIAM Journal on Imaging Sciences, 4 (2011), 760-788. doi: 10.1137/100790902.
J. Q. Fan and I. Gijbels, Local Polynomial Modelling and Its Applications, Chapman & Hall, London, 1996.
A. Foi, V. Katkovnik, K. Egiazarian and J. Astola, A novel anisotropic local polynomial estimator based on directional multiscale optimizations, In Proc. 6th IMA Int. Conf. Math. in Signal Process, pages 79–82, Citeseer.
D. K. Hammond and E. P. Simoncelli, Image modeling and denoising with orientation-adapted gaussian scale mixtures, IEEE Trans. Image Process., 17 (2008), 2089-2101. doi: 10.1109/TIP.2008.2004796.
K. Hirakawa and T. W. Parks, Image denoising using total least squares, IEEE Trans. Image Process., 15 (2006), 2730-2742. doi: 10.1109/TIP.2006.877352.
H. Hu, B. Li and Q. Liu, Removing mixture of gaussian and impulse noise by patch-basedweighted means, Journal of Scientific Computing, 67 (2016), 103-129. doi: 10.1007/s10915-015-0073-9.
J. Immerkaer, Fast noise variance estimation, Computer vision and image understanding, 64 (1996), 300-302. doi: 10.1006/cviu.1996.0060.
Q. Jin, I. Grama, C. Kervrann and Q. Liu, Nonlocal means and optimal weights for noise removal, SIAM Journal on Imaging Sciences, 10 (2017), 1878-1920. doi: 10.1137/16M1080781.
Q. Jin, I. Grama and Q. Liu, Removing gaussian noise by optimization of weights in non-local means, preprint, arXiv: 1109.5640.
Q. Jin, I. Grama and Q. Liu, A new poisson noise filter based on weights optimization, Journal of Scientific Computing, 58 (2014), 548-573. doi: 10.1007/s10915-013-9743-7.
V. Karnati, M. Uliyar and S. Dey, Fast non-local algorithm for image denoising, In IEEE International Conference on Image Processing (ICIP), 2009 16th, pages 3873–3876, IEEE, 2009. doi: 10.1109/ICIP.2009.5414044.
V. Katkovnik, A. Foi, K. Egiazarian and J. Astola, Directional varying scale approximations for anisotropic signal processing, In Proc. XII European Signal Proc. Conf., EUSIPCO 2004, Vienna, pages 101–104, 2004.
V. Katkovnik, A. Foi, K. Egiazarian and J. Astola, From local kernel to nonlocal multiple-model image denoising, Int. J. Comput. Vis., 86 (2010), 1-32. doi: 10.1007/s11263-009-0272-7.
C. Kervrann and J. Boulanger, Optimal spatial adaptation for patch-based image denoising, IEEE Trans. Image Process., 15 (2006), 2866-2878. doi: 10.1109/TIP.2006.877529.
C. Kervrann and J. Boulanger, Local adaptivity to variable smoothness for exemplar-based image regularization and representation, Int. J. Comput. Vis., 79 (2008), 45-69. doi: 10.1007/s11263-007-0096-2.
M. Lebrun, A. Buades and J. M. Morel, Implementation of the "Non-Local Bayes" (NL-bayes) image denoising algorithm, Image Processing On Line, 2013 (2013), 1-42.
M. Lebrun, A. Buades and J. M. Morel, A nonlocal bayesian image denoising algorithm, SIAM Journal on Imaging Sciences, 6 (2013), 1665-1688. doi: 10.1137/120874989.
A. Levin and B. Nadler, Natural image denoising: Optimality and inherent bounds, In Computer Vision and Pattern Recognition, pages 2833–2840, 2011. doi: 10.1109/CVPR.2011.5995309.
B. Li, Q. Liu, J. Xu and X. Luo, A new method for removing mixed noises, Science China Information Sciences, 54 (2011), 51-59. doi: 10.1007/s11432-010-4128-0.
Y. Lou, X. Zhang, S. Osher and A. Bertozzi, Image recovery via nonlocal operators, J. Sci. Comput., 42 (2010), 185-197. doi: 10.1007/s10915-009-9320-2.
M. Mahmoudi and G. Sapiro, Fast image and video denoising via nonlocal means of similar neighborhoods, IEEE Signal. Proc. Let., 12 (2005), 839-842. doi: 10.1109/LSP.2005.859509.
A. Maleki, M. Narayan and R. Baraniuk, Suboptimality of Nonlocal Means on Images with Sharp Edges, In Annual Allerton Conference on Communication, Control, and Computing, 2011.
A. Maleki, M. Narayan and R. Baraniuk, Anisotropic nonlocal means denoising, Applied and Computational Harmonic Analysis, 35 (2013), 452-482. doi: 10.1016/j.acha.2012.11.003.
J. Polzehl and V. Spokoiny, Propagation-separation approach for local likelihood estimation, Probab. Theory Rel. Fields, 135 (2006), 335-362. doi: 10.1007/s00440-005-0464-1.
S. Roth and M. J. Black, Fields of experts, Int. J. Comput. Vision, 82 (2009), 205-229. doi: 10.1007/s11263-008-0197-6.
J. Salmon and E. Le Pennec, Nl-means and aggregation procedures, In IEEE Int. Conf. Image Process. (ICIP), pages 2977–2980. IEEE, 2009. doi: 10.1109/ICIP.2009.5414512.
N. A. Thacker, P. A. Bromiley and J. V. Manjonb, A quantitative theory of the non-local means algorithm, In Proc. MIUA 2008, Dundee, Scotland, pages 174–178. Citeseer, 2008.
C. Tomasi and R. Manduchi, Bilateral filtering for gray and color images, In Proc. Int. Conf. Computer Vision, pages 839–846, 1998. doi: 10.1109/ICCV.1998.710815.
D. Van De Ville and M. Kocher, Non-local means with dimensionality reduction and sure-based parameter selection, IEEE Trans. Image Process., 20 (2010), 2683-2690. doi: 10.1109/TIP.2011.2121083.
R. Vignesh, B. T. Oh and C. C. J. Kuo, Fast non-local means (nlm) computation with probabilistic early termination, IEEE Signal. Proc. Let., 17 (2010), 277-280. doi: 10.1109/LSP.2009.2038956.
Y. Q. Wang, The implementation of sure guided piecewise linear image denoising, Image Processing On Line, 2013 (2013), 43-67. doi: 10.5201/ipol.2013.52.
L. P. Yaroslavsky, Digital Picture Processing. An Introduction, In Springer-Verlag, Berlin, 1985. doi: 10.1007/978-3-642-81929-2.

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