Abstract
Using multiresolution based on Harten’s framework [J. Appl. Numer. Math., 12 (1993), pp. 153–192.] we introduce an alternative to construct a prediction operator using Learning statistical theory. This integrates two ideas: generalized wavelets and learning methods, and opens several possibilities in the compressed signal context. We obtain theoretical results which prove that this type of schemes (LMR schemes) are equal to or better than the classical schemes. Finally, we compare traditional methods with the algorithm that we present in this paper.
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Amat, S., Donat, R., Liandrat, J., Trillo, J.C.: A fully adaptive PPH multi-resolution scheme for image processing. Math. Comput. Modelling 46, 2–11 (2007)
Aràndiga, F., Cohen, A., Yáñez, D.F.: Learning-based multiresolution: Application to compress digital images (in preparation)
Aràndiga, F., Donat, R., Harten, A.: Multiresolution based on weighted averages of the hat function II: Non-linear reconstruction technique. SIAM J. Numer. Anal. 20, 1053–1093 (1999)
Aràndiga, F., Donat, R.: Nonlinear multiscale decompositions: The approach of A. Harten. Numerical Algorithms 23, 175–216 (2000)
Boyd, S., Vandenberghe, L.: Convex Optimization. Cambridge University Press, New York (2004)
Cohen, A., Dahmen, W., Daubechies, I., DeVore, R.: Tree approximation and optimal encoding. Appl. Comp. Harm. Anal. 11, 192–226 (2001)
Cohen, A., Daubechies, I., Feauveau, J.: Biorthogonal bases of compactly supported wavelets. Comm. Pure Appl. Math. 45, 485–560 (1992)
Cohen, A.: Numerical Analysis of Wavelet Methods. Elsevier, Paris (2003)
Donoho, D.L.: Unconditional bases are optimal bases for data compression and for statistical estimation. Appl. Comp. Harm. Annal. 1, 110–115 (1993)
Friedman, J., Hastie, T., Tibshirani, R.: The Elements of Statistical Learning. Springer, New York (2001)
Getreuer, P., Meyer, F.: ENO multiresolution schemes with general discretizations. SIAM J. Numer. Anal. 46, 2953–2977 (2008)
Grant, M., Boyd, S.: Graph implementations for nonsmooth convex programs. In: Blondel, V., Boyd, S., Kimura, H. (eds.) Recent Advances in Learning and Control (a tribute to M. Vidyasagar). LNCIS, vol. 371, pp. 95–110. Springer, Heidelberg (2008)
Grant, M., Boyd, S.: CVX: Matlab software for disciplined convex programming (web page and software) (2009), http://stanford.edu/~boyd/cvx
Harten, A.: Discrete multiresolution analysis and generalized wavelets. J. Appl. Numer. Math. 12, 153–192 (1993)
Harten, A.: Multiresolution Representation of data: General framework. SIAM J. Numer. Anal. 33, 1205–1256 (1996)
Mallat, S.: A wavelet tour of signal processing. Academic Press, London (1998)
Vapnik, V.N.: The Nature of Statistical Learning. Springer, New York (1995)
Yáñez, D.F.: Learning Multiresolution: Transformaciones Multiescala derivadas de la Teoría estadística de Aprendizaje, Phd. Thesis, University of Valencia (2010)
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Aràndiga, F., Cohen, A., Yáñez, D.F. (2012). Design of Multiresolution Operators Using Statistical Learning Tools: Application to Compression of Signals. In: Boissonnat, JD., et al. Curves and Surfaces. Curves and Surfaces 2010. Lecture Notes in Computer Science, vol 6920. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-27413-8_6
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DOI: https://doi.org/10.1007/978-3-642-27413-8_6
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