Abstract
When considering fast multiresolution techniques for image denoising problems, there are three important aspects. The first one is the choice of the specific multiresolution, the second one the choice of a proper filter function and the third one the choice of the thresholding parameter. Starting from the classical one, namely, linear wavelet algorithms with Donoho and Johnstone’s Soft-thresholding with the universal shrinkage parameter, the first aim of this paper is to improve it in the three mentioned directions. Thus, a new nonlinear approach is proposed and analyzed. On the other hand, the linear approach of Donoho and Johnstone is related with a well known variational problem. Our second aim is to find a related variational problem, more adapted to the denoising problem, for the new approach. We would like to mention that the analysis of theoretical properties in a nonlinear setting are usually notoriously more difficult. Finally, a comparison with other approaches, including linear and nonlinear multiresolution schemes, SVD-based schemes and filters with a non-multiresolution nature, is presented.
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Amat, S., Aràndiga, F., Cohen, A., Donat, R.: Tensor product multiresolution analysis with error control for compact image representation. Signal Proc. 82(4), 587–608 (2002)
Amat, S., Aràndiga, F., Cohen, A., Donat, R., García, G., von Oehsen, M.: Data compresion with ENO schemes. Appl. Comput. Harmon. Anal. 11, 273–288 (2001)
Amat, S., Donat, R., Liandrat, J., Trillo, J.C.: Analysis of a new nonlinear subdivision scheme. Applications in image processing. Found. Comput. Math. 6(2), 193–225 (2002)
Amat, S., Liandrat, J., Ruiz, J., Trillo, J.C.: On a nonlinear cell-average multiresolution scheme for image compression. Appl. Numer. Math. (2008, submitted)
Aràndiga, F., Belda, A.M.: Weighted ENO interpolation and applications. Commun. Nonlinear Sci. Numer. Simul. 9, 187–195 (2003)
Aràndiga, F., Donat, R.: Nonlinear multi-scale decomposition: the approach of A. Harten. Numer. Algorithms 23, 175–216 (2000)
Bacchelli, S., Papi, S.: Filtered wavelets thresholding methods. J. Comput. Appl. Math. 164–165, 39–52 (2004)
Bertalmío, M., Caselles, V., Rougé, B., Solé, A.: TV based image restoration with local constraints. J. Sci. Comput. 19(1–3), 95–122 (2003). Special issue in honor of the sixtieth birthday of Stanley Osher
Candès, E.J., Donoho, D.L.: Recovering edges in ill-posed inverse problems: optimality of curvelet frames. Ann. Stat. 30(3), 784–842 (2002). Dedicated to the memory of Lucien Le Cam
Chambolle, A., DeVore, R.A., Lee, N., Lucier, B.J.: Nonlinear wavelet image processing: variational problem, compression and noise removal through wavelet shrinkage. IEEE Trans. Image Process. 7, 319–335 (1998)
Chan, T.F., Shen, J., Vese, L.: Variational PDE models in image processing. Not. Am. Math. Soc. 50(1), 14–26 (2003)
Chan, T.F., Zhou, H.-M.: Total variation wavelet thresholding. J. Sci. Comput. 32(2), 315–341 (2007)
Cohen, A., Daubechies, I., Feauveau, J.C.: Biorthogonal bases of compactly supported wavelets. Commun. Pure Appl. Math. 45, 485–560 (1992)
Cohen, A., DeVore, R., Petrushev, P., Xu, H.: Nonlinear approximation and the space BV(R 2). Am. J. Math. 121(3), 587–628 (1999)
Dahmen, W., Micchelli, C.A.: Biorthogonal wavelet expansions. Constr. Approx. 13(3), 293–328 (1997)
Daubechies, I., Teschke, G.: Variational image restoration by means of wavelets: simultaneous decomposition, deblurring, and denoising. Appl. Comput. Harmon. Anal. 19(1), 1–16 (2005)
Devore, R.A., Lucier, B.J.: Fast wavelet techniques for near-optimal image processing. In: Proc. IEEE Military Communications Conference, vol. 3, pp. 1129–1135. IEEE Press, New York (1992)
Donoho, D.L.: Denoising by soft thresholding. IEEE Trans. Inf. Theory 41(3), 613–627 (1995)
Donoho, D.L.: Ridge functions and orthonormal ridgelets. J. Approx. Theory 111(2), 143–179 (2001)
Donoho, D.L., Johnstone, I.: Ideal spatial adaptation by wavelet shrinkage. Biometrika 81, 425–455 (1994)
Harten, A.: Discrete multiresolution analysis and generalized wavelets. J. Appl. Numer. Math. 12, 153–192 (1993)
Harten, A.: Multiresolution representation of data II. SIAM J. Numer. Anal. 33(3), 1205–1256 (1996)
Le Pennec, E., Mallat, S.: Image compression with geometrical wavelets. In: IEEE Conference on Image Processing (ICIP), Vancouver, September, 2000
Matei, B.: Smoothness characterization and stability for nonlinear multiscale representations. C. R. Acad. Sci. Paris, Ser. I 338, 321–326 (2004)
Rabbani, M., Jones, P.W.: Digital image compression techniques. Tutorial text. Society of Photo-Optical Instrumentation Engineers (SPIE), TT07 (1991)
Starck, J.-L., Candès, E.J., Donoho, D.L.: The curvelet transform for image denoising. IEEE Trans. Image Process. 11(6), 670–684 (2002)
Sweldens, W.: The lifting scheme: a construction of second generation wavelets. SIAM J. Math. Anal. 29(2), 511–546 (1998)
Vese, L.A., Osher, S.J.: Modeling textures with total variation minimization and oscillating patterns in image processing. J. Sci. Comput. 19(1–3), 553–572 (2003)
Wongsawat, Y., Rao, K.R., Oraintara, S.: Multichannel SVD-based image denoising. In: IEEE International Symposium on Circuits and Systems (2005)
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Research of S. Amat was supported in part by the Spanish grants MTM2007-62945 and 08662/PI/08.
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Amat, S., Ruiz, J. & Trillo, J.C. Fast Multiresolution Algorithms and Their Related Variational Problems for Image Denoising. J Sci Comput 43, 1–23 (2010). https://doi.org/10.1007/s10915-009-9336-7
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DOI: https://doi.org/10.1007/s10915-009-9336-7