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Kernel PCA Based Non-Local Means Method for Speckle Reduction in Medical Ultrasound Images

DOI: 10.4236/oalib.1108618, PP. 1-41

Subject Areas: Biotechnology, Bioengineering

Keywords: US, Speckle Noise, NLM Kernel, OBNLM, PCA

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Abstract

The speckle noise is considered one of the main causes of degradation in ultrasound image quality. Many despeckling filters have been proposed, which are always making a trade-off between noise suppression and loss of information. A class of despeckling methods based Non-Local Means (NLM) algorithm is known to efficiently preserve the edges and all fine details of an image while reducing the noise. The core idea of NLM filter is to estimate the denoised pixel by performing a weighted average of similar patches in the neighborhood around the noisy pixel. However, the presence of noise degrades the similarity measurement process of the NLM and thereby decreases its efficiency. In this work, a novel despeckling scheme for ultrasound images is proposed, by introducing the kernel principal component analysis (PCA) to the NLM and computing the similarity in a high dimension kernel PCA subspace. The kernel representation is robust to the presence of noise and it can give better performance even under high noisy conditions. And it takes into account higher-order statistics of the pixels which can lead to accurate edge preservation. In this work, a novel despeckling scheme for ultrasound images is proposed using the kernel PCA-NLM extended to speckle noise model. The visual inspection and image metrics will show that the proposed filter is very competitive with respect to one of state-of-the-art methods, the Optimized Bayesian Non Local Means filter (OBNLM), in terms of low contrast object detectability, speckle noise suppression, edge’s preservation.

Cite this paper

Salih, M. E. , Zhang, X. and Ding, M. (2022). Kernel PCA Based Non-Local Means Method for Speckle Reduction in Medical Ultrasound Images. Open Access Library Journal, 9, e8618. doi: http://dx.doi.org/10.4236/oalib.1108618.

References

[1]  Loizou, C.P. and Pattichis, C.S. (2008) Despeckle Filtering Algorithms and Software for Ultrasound Imaging. Synthesis Lectures on Algorithms and Software in Engineering, Vol. 1, Morgan & Claypool, San Rafael, 1-166. https://doi.org/10.2200/S00116ED1V01Y200805ASE001
[2]  Abd-Elmoniem, K.Z., Youssef, A. and Kadah, Y.M. (2002) Real-Time Speckle Reduction and Coherence Enhancement in Ultrasound Imaging via Nonlinear Anisotropic Diffusion. IEEE Transactions on Biomedical Engineering, 49, 997-1014. https://doi.org/10.1109/TBME.2002.1028423
[3]  Sanchez, J.R. and Oelze, M. (2009) An Ultrasonic Imaging Speckle-Suppression and Contrast-Enhancement Technique by Means of Frequency Compounding and Coded Excitation. IEEE Transactions on Ultrasonics, Ferroelectrics and Frequency Control, 56, 1327-1339. https://doi.org/10.1109/TUFFC.2009.1189
[4]  Buades, A., Coll, B. and Morel, J.-M. (2005) A Review of Image Denoising Algorithms, with a New One. Multiscale Modeling & Simulation, 4, 490-530. https://doi.org/10.1137/040616024
[5]  Duval, V., Aujol, J.-F. and Gousseau, Y. (2011) A Bias-Variance Approach for the Nonlocal Means. SIAM Journal on Imaging Sciences, 4, 760-788. https://doi.org/10.1137/100790902
[6]  Darbon, J., Cunha, A., Chan, T.F., Osher, S. and Jensen, G.J. (2008) Fast Nonlocal Filtering Applied to Electron Cryomicroscopy. 5th IEEE International Symposium on Biomedical Imaging: From Nano to Macro, Paris, 14-17 May 2008, 1331-1334. https://doi.org/10.1109/ISBI.2008.4541250
[7]  Boukerroui, D., Noble, J.A. and Brady, M. (2003) Velocity Estimation in Ultrasound Images: A Block Matching Approach. In: Information Processing in Medical Imaging, Springer, Berlin, 586-598. https://doi.org/10.1007/978-3-540-45087-0_49
[8]  Loupas, T., McDicken, W. and Allan, P. (1989) An Adaptive Weighted Median Filter for Speckle Suppression in Medical Ultrasonic Images. IEEE Transactions on Circuits and Systems, 36, 129-135. https://doi.org/10.1109/31.16577
[9]  Buades, A., Coll, B. and Morel, J.-M. (2010) Image Denoising Methods. A New Nonlocal Principle. SIAM Review, 52, 113-147. https://doi.org/10.1137/090773908
[10]  Gonzalez, R.C. and Richard, E. (2002) Woods, Digital Image Processing. Prentice Hall Press, Hoboken.
[11]  Guo, Y., Wang, Y. and Hou, T. (2011) Speckle Filtering of Ultrasonic Images Using a Modified Non Local-Based Algorithm. Biomedical Signal Processing and Control, 6, 129-138. https://doi.org/10.1016/j.bspc.2010.10.004
[12]  Kervrann, C. and Boulanger, J. (2006) Optimal Spatial Adaptation for Patch-Based Image Denoising. IEEE Transactions on Image Processing, 15, 2866-2878. https://doi.org/10.1109/TIP.2006.877529
[13]  Brox, T., Kleinschmidt, O. and Cremers, D. (2008) Efficient Nonlocal Means for Denoising of Textural Patterns. IEEE Transactions on Image Processing, 17, 1083-1092. https://doi.org/10.1109/TIP.2008.924281
[14]  Deledalle, C.-A., Denis, L. and Tupin, F. (2009) Iterative Weighted Maximum Likelihood Denoising with Probabilistic Patch-Based Weights. IEEE Transactions on Image Processing, 18, 2661-2672. https://doi.org/10.1109/TIP.2009.2029593
[15]  Goossens, B., Luong, Q., Pizurica, A. and Philips, W. (2008) An Improved Non-Local Denoising Algorithm. 2008 International Workshop on Local and Non-Local Approximation in Image Processing (LNLA 2008), Lausanne, 23-24 August 2008, 143-156.
[16]  Orchard, J., Ebrahimi, M. and Wong, A. (2008) Efficient Nonlocal-Means Denoising Using the SVD. 15th IEEE International Conference on Image Processing, San Diego, 12-15 October 2008, 1732-1735. https://doi.org/10.1109/ICIP.2008.4712109
[17]  Kanevsky, M.B. (2008) Radar Imaging of the Ocean Waves. Elsevier, Amsterdam.
[18]  Loizou, C.P., Pattichis, C.S., Christodoulou, C.I., Istepanian, R.S., Pantziaris, M. and Nicolaides, A. (2005) Comparative Evaluation of Despeckle Filtering in Ultrasound Imaging of the Carotid Artery. IEEE Transactions on Ultrasonics, Ferroelectrics and Frequency Control, 52, 1653-1669. https://doi.org/10.1109/TUFFC.2005.1561621
[19]  Mather, P. and Tso, B. (2010) Classification Methods for Remotely Sensed Data. CRC Press, Boca Raton.
[20]  Lee, J.-S. (1980) Digital Image Enhancement and Noise Filtering by Use of Local Statistics. IEEE Transactions on Pattern Analysis and Machine Intelligence, 2, 165-168. https://doi.org/10.1109/TPAMI.1980.4766994
[21]  Lee, J.-S. (1981) Refined Filtering of Image Noise Using Local Statistics. Computer Graphics and Image Processing, 15, 380-389. https://doi.org/10.1016/S0146-664X(81)80018-4
[22]  Lee, J.-S. (1986) Speckle Suppression and Analysis for Synthetic Aperture Radar Images. Optical Engineering, 25, Article ID: 255636. https://doi.org/10.1117/12.7973877
[23]  Frost, V.S., Stiles, J.A., Shanmugan, K.S. and Holtzman, J.C. (1982) A Model for Radar Images and Its Application to Adaptive Digital Filtering of Multiplicative Noise. IEEE Transactions on Pattern Analysis and Machine Intelligence, PAMI-4, 157-166. https://doi.org/10.1109/TPAMI.1982.4767223
[24]  Kuan, D.T., Sawchuk, A., Strand, T.C. and Chavel, P. (1987) Adaptive Restoration of Images with Speckle. IEEE Transactions on Acoustics, Speech and Signal Processing, 35, 373-383. https://doi.org/10.1109/TASSP.1987.1165131
[25]  Lopes, A., Touzi, R. and Nezry, E. (1990) Adaptive Speckle Filters and Scene Heterogeneity. IEEE Transactions on Geoscience and Remote Sensing, 28, 992-1000. https://doi.org/10.1109/36.62623
[26]  Lopes, A., Nezry, E., Touzi, R. and Laur, H. (1993) Structure Detection and Statistical Adaptive Speckle Filtering in SAR Images. International Journal of Remote Sensing, 14, 1735-1758. https://doi.org/10.1080/01431169308953999
[27]  Saniie, J., Wang, T. and Bilgutay, N.M. (1989) Analysis of Homomorphic Processing for Ultrasonic Grain Signal Characterization. IEEE Transactions on Ultrasonics, Ferroelectrics and Frequency Control, 36, 365-375. https://doi.org/10.1109/58.19177
[28]  Suri, J.S. (2008) Advances in Diagnostic and Therapeutic Ultrasound Imaging. Artech House, Norwood.
[29]  Yu, Y. and Acton, S.T. (2002) Speckle Reducing Anisotropic Diffusion. IEEE Transactions on Image Processing, 11, 1260-1270. https://doi.org/10.1109/TIP.2002.804276
[30]  Elad, M. (2002) On the Origin of the Bilateral Filter and Ways to Improve It. IEEE Transactions on Image Processing, 11, 1141-1151. https://doi.org/10.1109/TIP.2002.801126
[31]  Kervrann, C., Boulanger, J. and Coupé, P. (2007) Bayesian Non-Local Means Filter, Image Redundancy and Adaptive Dictionaries for Noise Removal. In: Scale Space and Variational Methods in Computer Vision, Springer, Berlin, 520-532. https://doi.org/10.1007/978-3-540-72823-8_45
[32]  Coupé, P., Hellier, P., Kervrann, C. and Barillot, C. (2009) Nonlocal Means-Based Speckle Filtering for Ultrasound Images. IEEE Transactions on Image Processing, 18, 2221-2229. https://doi.org/10.1109/TIP.2009.2024064
[33]  De Fontes, F.P.X., Barroso, G.A., Coupé, P. and Hellier, P. (2011) Real Time Ultrasound Image Denoising. Journal of Real-Time Image Processing, 6, 15-22. https://doi.org/10.1007/s11554-010-0158-5
[34]  Uzan, A., Rivenson, Y. and Stern, A. (2013) Speckle Denoising in Digital Holography by Nonlocal Means Filtering. Applied Optics, 52, A195-A200. https://doi.org/10.1364/AO.52.00A195
[35]  Dougherty, G. (2009) Digital Image Processing for Medical Applications. Cambridge University Press, Cambridge. https://doi.org/10.1017/CBO9780511609657
[36]  Marques, O. (2011) Practical Image and Video Processing Using MATLAB. Wiley, Hoboken. https://doi.org/10.1002/9781118093467
[37]  Boulanger, J., Kervrann, C., Bouthemy, P., Elbau, P., Sibarita, J.-B. and Salamero, J. (2010) Patch-Based Nonlocal Functional for Denoising Fluorescence Microscopy Image Sequences. IEEE Transactions on Medical Imaging, 29, 442-454. https://doi.org/10.1109/TMI.2009.2033991
[38]  Zhang, B., Fadili, J.M. and Starck, J.-L. (2008) Wavelets, Ridgelets, and Curvelets for Poisson Noise Removal. IEEE Transactions on Image Processing, 17, 1093-1108. https://doi.org/10.1109/TIP.2008.924386
[39]  Anscombe, F.J. (1948) The Transformation of Poisson, Binomial and Negative-Binomial Data. Biometrika, 35, 246-254. https://doi.org/10.1093/biomet/35.3-4.246
[40]  Dutt, V. (1995) Statistical Analysis of Ultrasound Echo Envelope. Biophysical Sciences—Biomedical Imaging—Mayo Graduate School.
[41]  Jain, A.K. (1989) Fundamentals of Digital Image Processing. Prentice-Hall, Inc., Hoboken.
[42]  Zong, X., Laine, A.F. and Geiser, E.A. (1998) Speckle Reduction and Contrast Enhancement of Echocardiograms via Multiscale Nonlinear Processing. IEEE Transactions on Medical Imaging, 17, 532-540. https://doi.org/10.1109/42.730398
[43]  Odegard, J.E., Guo, H., Lang, M., Burrus, C.S., Wells Jr., R.O., Novak, L.M. and Hiett, M. (1995) Wavelet-Based SAR Speckle Reduction and Image Compression. SPIE’s 1995 Symposium on OE/Aerospace Sensing and Dual Use Photonics, Orlando, 17-21 April 1995, 259-271. https://doi.org/10.1117/12.210843
[44]  Gagnon, L. and Smaili, F.D. (1996) Speckle Noise Reduction of Airborne SAR Images with Symmetric Daubechies Wavelets. In: Aerospace/Defense Sensing and Controls, International Society for Optics and Photonics, Bellingham, 14-24. https://doi.org/10.1117/12.241168
[45]  Bovik, A.C. (2009) The Essential Guide to Image Processing. Academic Press, Cambridge.
[46]  Jain, A. (1989) Fundamental of Digital Image Processing. Prentice-Hall, Englewood Cliffs.
[47]  Zhou, Y., Endres, C.J., Brasic, J.R., Huang, S.-C. and Wong, D.F. (2003) Linear Regression with Spatial Constraint to Generate Parametric Images of Ligand-Receptor Dynamic PET Studies with a Simplified Reference Tissue Model. Neuroimage, 18, 975-989. https://doi.org/10.1016/S1053-8119(03)00017-X
[48]  Wagner, R.F., Smith, S.W., Sandrik, J.M. and Lopez, H. (1983) Statistics of Speckle in Ultrasound B-Scans. IEEE Transactions on Sonics and Ultrasonics, 30, 156-163. https://doi.org/10.1109/T-SU.1983.31404
[49]  Burckhardt, C.B. (1978) Speckle in Ultrasound B-Mode Scans. IEEE Transactions on Sonics and Ultrasonics, 25, 1-6. https://doi.org/10.1109/T-SU.1978.30978
[50]  Chen, Y., Yin, R., Flynn, P. and Broschat, S. (2003) Aggressive Region Growing for Speckle Reduction in Ultrasound Images. Pattern Recognition Letters, 24, 677-691. https://doi.org/10.1016/S0167-8655(02)00174-5
[51]  Balocco, S., Gatta, C., Pujol, O., Mauri, J. and Radeva, P. (2010) SRBF: Speckle Reducing Bilateral Filtering. Ultrasound in Medicine & Biology, 36, 1353-1363. https://doi.org/10.1016/j.ultrasmedbio.2010.05.007
[52]  Chen, S., Yang, X., Yao, L.P. and Sun, K. (2005) Total Variation-Based Speckle Reduction Using Multi-Grid Algorithm for Ultrasound Images. In: Image Analysis and Processing-ICIAP 2005, Springer, Berlin, 245-252. https://doi.org/10.1007/11553595_30
[53]  Zhong, H., Li, Y. and Jiao, L. (2009) Bayesian Nonlocal Means Filter for SAR Image Despeckling. 2nd IEEE Asian-Pacific Conference on Synthetic Aperture Radar, Xian, 26-30 October 2009, 1096-1099. https://doi.org/10.1109/APSAR.2009.5374145
[54]  Bengio, Y., Courville, A. and Vincent, P. (2013) Representation Learning: A Review and New Perspectives. IEEE Transactions on Pattern Analysis and Machine Intelligence, 35, 1798-1828. https://doi.org/10.1109/TPAMI.2013.50
[55]  van der Meij, J. and de Jong, T. (2004) Learning with Multiple Representations. Annual Meeting of the American Educational Research Association, San Diego, 12-16 April 2004, 16.
[56]  Hotelling, H. (1933) Analysis of a Complex of Statistical Variables into Principal Components. Journal of Educational Psychology, 24, 417. https://doi.org/10.1037/h0071325
[57]  Pearson, K. (1901) LIII. On Lines and Planes of Closest Fit to Systems of Points in Space. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 2, 559-572. https://doi.org/10.1080/14786440109462720
[58]  Jolliffe, I. (2005) Principal Component Analysis. Wiley Online Library. https://doi.org/10.1002/0470013192.bsa501
[59]  Ivosev, G., Burton, L. and Bonner, R. (2008) Dimensionality Reduction and Visualization in Principal Component Analysis. Analytical Chemistry, 80, 4933-4944. https://doi.org/10.1021/ac800110w
[60]  Wall, M.E., Rechtsteiner, A. and Rocha, L.M. (2003) Singular Value Decomposition and Principal Component Analysis. In: Berrar, D.P., Dubitzky, W. and Granzow, M., Eds., A Practical Approach to Microarray Data Analysis, Kluwer, Norwell, 91. https://doi.org/10.1007/0-306-47815-3_5
[61]  Smith, L.I. (2002) A Tutorial on Principal Components Analysis. Cornell University, Ithaca, 51-52.
[62]  Du, Q. and Fowler, J.E. (2007) Hyperspectral Image Compression Using JPEG2000 and Principal Component Analysis. IEEE Geoscience and Remote Sensing Letters, 4, 201-205. https://doi.org/10.1109/LGRS.2006.888109
[63]  Turk, M. and Pentland, A. (1991) Eigenfaces for Recognition. Journal of Cognitive Neuroscience, 3, 71-86. https://doi.org/10.1162/jocn.1991.3.1.71
[64]  Golub, G. and Kahan, W. (1965) Calculating the Singular Values and Pseudo-Inverse of a Matrix. Journal of the Society for Industrial & Applied Mathematics, Series B: Numerical Analysis, 2, 205-224. https://doi.org/10.1137/0702016
[65]  Golub, G.H. and Van Loan, C.F. (1996) Matrix Computations. Johns Hopkins University Press, Baltimore, 374-426.
[66]  Chang, S.G., Yu, B. and Vetterli, M. (2000) Spatially Adaptive Wavelet Thresholding with Context Modeling for Image Denoising. IEEE Transactions on Image Processing, 9, 1522-1531. https://doi.org/10.1109/83.862630
[67]  Li, X. and Orchard, M.T. (2000) Spatially Adaptive Image Denoising under Overcomplete Expansion. IEEE International Conference on Image Processing, Vancouver, 10-13 September 2000, 300-303.
[68]  Coifman, R. and Donoho, D. (1995) Wavelets and Statistics, Lecture Notes in Statistics. In: Antoniadis, A. and Oppenheim, G., Eds., Translation-Invariant de-Noising, Springer, Berlin, 125-150. https://doi.org/10.1007/978-1-4612-2544-7_9
[69]  Crouse, M.S., Nowak, R.D. and Baraniuk, R.G. (1998) Wavelet-Based Statistical Signal Processing Using Hidden Markov Models. IEEE Transactions on Signal Processing, 46, 886-902. https://doi.org/10.1109/78.668544
[70]  Krim, H., Tucker, D., Mallat, S. and Donoho, D. (1999) On Denoising and Best Signal Representation. IEEE Transactions on Information Theory, 45, 2225-2238. https://doi.org/10.1109/18.796365
[71]  Hyvarinen, A., Hoyer, P. and Oja, E. (1998) Sparse Code Shrinkage for Image Denoising. 1998 IEEE International Joint Conference on Neural Networks Proceedings. IEEE World Congress on Computational Intelligence, Anchorage, 4-9 May 1998, 859-864.
[72]  Deledalle, C.-A., Salmon, J., Dalalyan, A.S. and Champs-sur-Marne, F. (2011) Image Denoising with Patch Based PCA: Local versus Global. Proceedings of the British Machine Vision Conference, Dundee, 29 August-2 September 2011, 1-10. https://doi.org/10.5244/C.25.25
[73]  Salmon, J., Harmany, Z., Deledalle, C.-A. and Willett, R. (2014) Poisson Noise Reduction with Non-Local PCA. Journal of Mathematical Imaging and Vision, 48, 279-294. https://doi.org/10.1007/s10851-013-0435-6
[74]  Tasdizen, T. (2008) Principal Components for Non-Local Means Image Denoising. 15th IEEE International Conference on Image Processing, San Diego, 12-15 October 2008, 1728-1731. https://doi.org/10.1109/ICIP.2008.4712108
[75]  Abrahamsen, T.J. and Hansen, L.K. (2011) A Cure for Variance Inflation in High Dimensional Kernel Principal Component Analysis. The Journal of Machine Learning Research, 12, 2027-2044.
[76]  Thomas, J.K., Scharf, L.L. and Tufts, D.W. (1995) The Probability of a Subspace Swap in the SVD. IEEE Transactions on Signal Processing, 43, 730-736. https://doi.org/10.1109/78.370627
[77]  Salih, M.E., Zhang, X.M. and Ding, M.Y. (2013) An Improvement of Non-Local Means Denoising Method in the Presence of Large Noise. Applied Mechanics and Materials, 263, 223-226. https://doi.org/10.4028/www.scientific.net/AMM.263-266.223
[78]  Salih, M.E., Zhang, X. and Ding, M. (2013) Two Modifications of Weight Calculation of the Non-Local Means Denoising Method. Engineering, 5, 522. https://doi.org/10.4236/eng.2013.510B107
[79]  Schölkopf, B. and Smola, A.J. (2001) Learning with Kernels: Support Vector Machines, Regularization, Optimization, and Beyond (Adaptive Computation and Machine Learning).
[80]  Turk, M.A. and Pentland, A.P. (1991) Face Recognition Using Eigenfaces. IEEE Computer Society Conference on Computer Vision and Pattern Recognition, Maui, 3-6 June 1991, 586-591.
[81]  Martin, S. (2006) The Numerical Stability of Kernel Methods. ISAIM.
[82]  Wu, J., Wang, J. and Liu, L. (2007) Feature Extraction via KPCA for Classification of Gait Patterns. Human Movement Science, 26, 393-411. https://doi.org/10.1016/j.humov.2007.01.015
[83]  Nguyen, M.H. and De la Torre, F. (2008) Robust Kernel Principal Component Analysis. Conference on Neural Information Processing Systems, Vancouver, 8-11 December 2008, 8 p. https://proceedings.neurips.cc/paper/2008/file/8f53295a73878494e9bc8dd6c3c7104f-Paper.pdf
[84]  Mika, S., Schölkopf, B., Smola, A.J., Müller, K.-R., Scholz, M. and Rätsch, G. (1998) Kernel PCA and De-Noising in Feature Spaces. Conference on Neural Information Processing Systems, Denver, CO, 536-542.
[85]  Hamprecht, P.F. (2012) Pattern Recognition Class (Nonlinear SVM).
[86]  Kim, K.I., Franz, M.O. and Scholkopf, B. (2005) Iterative Kernel Principal Component Analysis for Image Modeling. IEEE Transactions on Pattern Analysis and Machine Intelligence, 27, 1351-1366. https://doi.org/10.1109/TPAMI.2005.181
[87]  Schölkopf, B., Smola, A. and Müller, K.-R. (1998) Nonlinear Component Analysis as a Kernel Eigenvalue Problem. Neural Computation, 10, 1299-1319. https://doi.org/10.1162/089976698300017467
[88]  Jensen, J.A. (1996) Field: A Program for Simulating Ultrasound Systems. 10th Nordic-Baltic Conference on Biomedical Imaging, Vol. 4, 351-353.
[89]  Bernardes, R., Maduro, C., Serranho, P., Araújo, A., Barbeiro, S. and Cunha-Vaz, J. (2010) Improved Adaptive Complex Diffusion Despeckling Filter. Optics Express. 18, 24048-24059. https://doi.org/10.1364/OE.18.024048
[90]  Zhang, F., Yoo, Y.M., Koh, L.M. and Kim, Y. (2007) Nonlinear Diffusion in Laplacian Pyramid Domain for Ultrasonic Speckle Reduction. IEEE Transactions on Medical Imaging, 26, 200-211. https://doi.org/10.1109/TMI.2006.889735
[91]  Ullom, J.S., Oelze, M.L. and Sanchez, J.R. (2012) Speckle Reduction for Ultrasonic Imaging Using Frequency Compounding and Despeckling Filters along with Coded Excitation and Pulse Compression. Advances in Acoustics and Vibration, 2012, Article ID: 474039. https://doi.org/10.1155/2012/474039
[92]  Sprawls, P. (1993) The Physical Principles of Medical Imaging. 2nd Edition, SPRAWLS Education Foundation, Madison, Wisconsin. http://www.sprawls.org/ppmi2/BLUR
[93]  Zhong, H., Li, Y. and Jiao, L. (2011) SAR Image Despeckling Using Bayesian Nonlocal Means Filter with Sigma Preselection. Geoscience and Remote Sensing Letters, 8, 809-813. https://doi.org/10.1109/LGRS.2011.2112331
[94]  Schölkopf, B. (2007) Introduction to Kernel Methods. Max Planck Institute for Biological Cybernetics, Tübingen.
[95]  Kim, K.I., Franz, M. and Schölkopf, B. (2004) Kernel Hebbian Algorithm for Single-Frame Super-Resolution.
[96]  Field, D.J. (1994) What Is the Goal of Sensory Coding? Neural Computation, 6, 559-601. https://doi.org/10.1162/neco.1994.6.4.559
[97]  Kim, K.I., Franz, M.O. and Schölkopf, B. (2003) Kernel Hebbian Algorithm for Iterative Kernel Principal Component Analysis. Max-Planck-Institut fr biologische Kybernetik, Tübingen, Tech. Rep. 109.
[98]  Hiremath, P. and Prabhakar, C. (2006) Acquiring Non Linear Subspace for Face Recognition Using Symbolic Kernel PCA Method. Journal of Symbolic Data Analysis, 4, 15-26. https://doi.org/10.1142/9789812772381_0008
[99]  Yang, M.-H., Ahuja, N. and Kriegman, D. (2000) Face Recognition Using Kernel Eigenfaces. 2000 IEEE International Conference on Image Processing, Vancouver, 10-13 September 2000, 37-40.
[100]  Günter, S., Schraudolph, N.N. and Vishwanathan, S. (2007) Fast Iterative Kernel Principal Component Analysis. Journal of Machine Learning Research, 8, 1893-1918.
[101]  Jackson, J.E. (2005) A User’s Guide to Principal Components. John Wiley & Sons, Hoboken.
[102]  Garcia, E. (2006, September) SVD and LSI Tutorial 3: Computing the Full SVD of a Matrix. https://cs.fit.edu/~dmitra/SciComp/Resources/singular-value-decomposition-fast-track-tutorial.pdf
[103]  Autumn, P.A.N. (2009) Lecture Machine Learning (CS 229).
[104]  Abu-Mostafa, P.Y. (2012) Caltech’s Machine Learning Course.

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