Determining neighborhoods of image pixels automatically for adaptive image denoising using nonlinear time series analysis. (English) Zbl 1189.94022

Summary: This paper presents a method determining neighborhoods of the image pixels automatically in adaptive denoising. The neighborhood is named stationary neighborhood (SN). In this method, the noisy image is considered as an observation of a nonlinear time series (NTS). Image denoising must recover the true state of the NTS from the observation. At first, the false neighbors (FNs) in a neighborhood for each pixel are removed according to the context. After moving the FNs, we obtain an SN, where the NTS is stationary and the real state can be estimated using the theory of stationary time series (STS). Since each SN of an image pixel consists of elements with similar context and nearby locations, the method proposed in this paper can not only adaptively find neighbors and determine size of the SN according to the characteristics of a pixel, but also be able to denoise while effectively preserving edges. Finally, in order to show the superiority of this algorithm, we compare this method with the existing universal denoising algorithms.


94A08 Image processing (compression, reconstruction, etc.) in information and communication theory
62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
Full Text: DOI EuDML


[1] N. Weiner, Extrapolation, Interpolation, and Smoothing of Stationary Time Series, John Wiley & Sons, New York, NY, USA, 1949. · Zbl 0138.12302
[2] L. \cSendur and I. W. Selesnick, “Bivariate shrinkage with local variance estimation,” IEEE Signal Processing Letters, vol. 9, no. 12, pp. 438-441, 2002.
[3] M. Mignotte, “Image denoising by averaging of piecewise constant simulations of image partitions,” IEEE Transactions on Image Processing, vol. 16, no. 2, pp. 523-533, 2007. · Zbl 05453719
[4] Z. Dengwen and C. Wengang, “Image denoising with an optimal threshold and neighbouring window,” Pattern Recognition Letters, vol. 29, no. 11, pp. 1694-1697, 2008.
[5] D. Coupier, A. Desolneux, and B. Ycart, “Image denoising by statistical area thresholding,” Journal of Mathematical Imaging and Vision, vol. 22, no. 2, pp. 183-197, 2005. · Zbl 1478.94021
[6] C. Tomasi and R. Manduchi, “Bilateral filtering for gray and color images,” in Proceedings of the IEEE International Conference on Computer Vision, pp. 839-846, Bombay, India, 1998.
[7] M. Zhang and B. K. Gunturk, “Multiresolution bilateral filtering for image denoising,” IEEE Transactions on Image Processing, vol. 17, no. 12, pp. 2324-2333, 2008. · Zbl 1371.94455
[8] H. Yu, L. Zhao, and H. Wang, “Image denoising using trivariate shrinkage filter in the wavelet domain and joint bilateral filter in the spatial domain,” IEEE Transactions on Image Processing, vol. 18, no. 10, pp. 2364-2369, 2009. · Zbl 1371.94443
[9] Y.-L. Liu, J. Wang, X. Chen, Y.-W. Guo, and Q.-S. Peng, “A robust and fast non-local means algorithm for image denoising,” Journal of Computer Science and Technology, vol. 23, no. 2, pp. 270-279, 2008. · Zbl 05347675
[10] Z. Ji, Q. Chen, Q.-S. Sun, and D.-S. Xia, “A moment-based nonlocal-means algorithm for image denoising,” Information Processing Letters, vol. 109, no. 23-24, pp. 1238-1244, 2009. · Zbl 1206.68333
[11] V. Katkovnik, A. Foi, K. Egiazarian, and J. Astola, “From local kernel to nonlocal multiple-model image denoising,” International Journal of Computer Vision, vol. 86, no. 1, pp. 1-32, 2010. · Zbl 1477.94014
[12] X. Wu and N. Memon, “Context-based, adaptive, lossless image code,” IEEE Transactions on Communications, vol. 45, no. 4, pp. 437-444, 1997. · Zbl 1477.94014
[13] M. S. Crouse and R. G. Baraniuk, “Contextual hidden Markov models for wavelet-domain signal processing,” in Proceedings of the 31st Asilomar Conference on Signals, Systems and Computers, Pacific Grove, Calif, USA, November 1997.
[14] G. Fan and X.-G. Xia, “Image denoising using a local contextual hidden Markov model in the wavelet domain,” IEEE Signal Processing Letters, vol. 8, no. 5, pp. 125-128, 2001.
[15] S. G. Chang, B. Yu, and M. Vetterli, “Spatially adaptive wavelet thresholding with context modeling for image denoising,” IEEE Transactions on Image Processing, vol. 9, no. 9, pp. 1522-1531, 2000. · Zbl 0962.94027
[16] Z. Liao and Y. Y. Tang, “Signal denoising using wavelets and block hidden Markov model,” International Journal of Pattern Recognition and Artificial Intelligence, vol. 19, no. 5, pp. 681-700, 2005. · Zbl 02218430
[17] A. Buades, B. Coll, and J.-M. Morel, “Nonlocal image and movie denoising,” International Journal of Computer Vision, vol. 76, no. 2, pp. 123-139, 2008, special section: selection of papers for CVPR 2005, Guest Editors: C. Schmid, S. Soatto and C. Tomasi. · Zbl 05322199
[18] H. Kantz and T. Schreiber, Nonlinear Time Series Analysis, Cambridge University Press, Cambridge, UK, 2nd edition, 2004. · Zbl 1050.62093
[19] M. Li, W.-S. Chen, and L. Han, “Correlation matching method for the weak stationarity test of LRD traffic,” Telecommunication Systems, vol. 43, no. 3-4, pp. 181-195, 2010. · Zbl 05803250
[20] M. Li and J.-Y. Li, “On the predictability of long-range dependent series,” Mathematical Problems in Engineering, vol. 2010, Article ID 397454, 9 pages, 2010. · Zbl 1191.62160
[21] M. Li and W. Zhao, “Representation of a stochastic traffic bound,” IEEE Transactions on Parallel and Distributed Systems. In press.
[22] C. Cattani and A. Kudreyko, “Application of periodized harmonic wavelets towards solution of eigenvalue problems for integral equations,” Mathematical Problems in Engineering, vol. 2010, Article ID 570136, 8 pages, 2010. · Zbl 1191.65175
[23] E. G. Bakhoum and C. Toma, “Mathematical transform of traveling-wave equations and phase aspects of quantum interaction,” Mathematical Problems in Engineering, vol. 2010, Article ID 695208, 15 pages, 2010. · Zbl 1191.35220
[24] C. Keng and H. Bo-Tang, “A survey of state space reconstruction of chaotic time series analysis,” Computer Science, vol. 32, no. 4, pp. 67-70, 2005.
[25] S. Lin, J. Qiao, G. Wang, S. Zhang, and L. Zhi, “Phase space reconstruction of nonlinear time series based on Kernel method,” in Proceedings of the 6th World Congress on Intelligent Control and Automation (WCICA ’06), vol. 1, pp. 4364-4368, Dalian, China, 2006.
[26] M. Casdagli, S. Eubank, J. D. Farmer, and J. Gibson, “State space reconstruction in the presence of noise,” Physica D, vol. 51, no. 1-3, pp. 52-98, 1991. · Zbl 0736.62075
[27] L. Wang, H. Zhang, H. Meng, and X. Wang, “Nonlinear analysis of individual vehicle behavior in car following,” in Proceedings of the 11th International IEEE Conference on Intelligent Transportation Systems (ITSC ’08), pp. 265-268, Beijing, China, October 2008.
[28] J. B. Dingwell and J. P. Cusumano, “Nonlinear time series analysis of normal and pathological human walking,” Chaos, vol. 10, no. 4, pp. 848-863, 2000. · Zbl 1055.70501
[29] B. Dennis, R. A. Desharnais, J. M. Cushing, S. M. Henson, and R. F. Costantino, “Can noise induce chaos?” Oikos, vol. 102, no. 2, pp. 329-339, 2003.
[30] R. Poole, “Is it chaos, or is it just noise,” Science, vol. 243, no. 4887, pp. 25-28, 1989.
[31] M. Li, “Fractal time series-a tutorial review,” Mathematical Problems in Engineering, vol. 2010, Article ID 157264, 26 pages, 2010. · Zbl 1191.37002
[32] http://www.hudong.com/wiki/.
[33] V. Guillemin and A. Pollack, Differential Topology, Prentice-Hall, Englewood Cliffs, NJ, USA, 1974. · Zbl 0361.57001
[34] R. C. Kirby and L. C. Siebenmann, Foundational Essays on Topological Manifolds, Smoothings, and Triangulations, Princeton University Press, Princeton, NJ, USA, 1977. · Zbl 0361.57004
[35] Z. Liao, Image Denoising Based on Wavelet Domian Hidden Markov Models, UESTC Press, Chengdu, China, 2006.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.