Non-linear Local Polynomial Regression Multiresolution Methods Using \(\ell ^1\)-norm Minimization with Application to Signal Processing

  • Francesc Aràndiga
  • Pep Mulet
  • Dionisio F. YáñezEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9213)


Harten’s Multiresolution has been developed and used for different applications such as fast algorithms for solving linear equations or compression, denoising and inpainting signals. These schemes are based on two principal operators: decimation and prediction. The goal of this paper is to construct an accurate prediction operator that approximates the real values of the signal by a polynomial and estimates the error using \(\ell ^1\)-norm in each point. The result is a non-linear multiresolution method. The order of the operator is calculated. The stability of the schemes is ensured by using a special error control technique. Some numerical tests are performed comparing the new method with known linear and non-linear methods.


Statistical Multiresolution Multiscale decomposition Signal processing Generalized wavelets 


  1. 1.
    Amat, S., Aràndiga, F., Cohen, A., Donat, R.: Tensor product multiresolution analysis whith error control for compact representation. Signal Process. 82, 587–608 (2002)CrossRefzbMATHGoogle Scholar
  2. 2.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Amat, S., Liandrat, J.: On the stability of the pph nonlinear multiresolution. Appl. Comput. Harmon. Anal. 18(2), 198–206 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Amat, S., Moncayo, M.: Exact error bound for the reconstruction processes using interpolating wavelets. Math. Comput. Simul. 79, 3547–3555 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Aràndiga, F., Belda, A., Mulet, P.: Point-value WENO multiresolution applications to stable image compression. J. Sci. Comput. 43(2), 158–182 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Aràndiga, F., Cohen, A., Yáñez, D.F.: Learning-based multiresolution schemes with application to compression of images. Signal Process. 93, 2474–2484 (2013)CrossRefGoogle Scholar
  7. 7.
    Aràndiga, F., Donat, R.: Nonlinear multiscale descompositions: the approach of A. Harten. Numer. Algorithm. 23, 175–216 (2000)CrossRefzbMATHGoogle Scholar
  8. 8.
    Aràndiga, F., Donat, R.: Stability through synchronization in nonlinear multiscale transformation. SIAM J. Sci. Comput. 29, 265–289 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Aràndiga, F., Donat, R., Harten, A.: Multiresolution based on weighted averages of the hat function I: linear reconstruction technique. SIAM J. Numer. Anal. 36, 160–203 (1998)CrossRefzbMATHGoogle Scholar
  10. 10.
    Aràndiga, F., Donat, R., Harten, A.: Multiresolution based on weighted averages of the hat function II: Nonlinear reconstruction technique. SIAM J. Numer. Anal. 20, 1053–1093 (1999)zbMATHGoogle Scholar
  11. 11.
    Aràndiga, F., Yáñez, D.F.: Generalized wavelets design using kernel methods. Application to signal processing. J. Comput. Appl. Math. 250, 1–15 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Aràndiga, F., Yáñez, D.F.: Cell-average multiresolution based on local polynomial regression. Application to image processing. Appl. Math. Comput. 245, 1–16 (2014)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Boyd, S., Vandenberghe, L.: Convex Optimization. Cambridge University Press, New York (2004)CrossRefzbMATHGoogle Scholar
  14. 14.
    Chan, T.F., Golub, G.H., Mulet, P.: A nonlinear primal-dual method for total variation-based image restoration. SIAM J. Numer. Anal. 20, 1964–1977 (1997)MathSciNetGoogle Scholar
  15. 15.
    Cleveland, W.S., Devlin, S.J.: Locally weighted regression: an approach to regression analysis by local fitting. J. Amer. Stat. Assoc. 83, 596–610 (1988)CrossRefzbMATHGoogle Scholar
  16. 16.
    Cohen, A.: Numerical Analysis of Wavelet Methods. Springer, New York (2003)zbMATHGoogle Scholar
  17. 17.
    Cohen, A., Daubechies, I., Feauveau, J.: Biorthogonal bases of compactly supported wavelets. Comm. Pure Appl. Math. 45, 485–560 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Dahmen, W.: Stability of multiscale transformations. J. Fourier Anal. Appl. 2(4), 341–361 (1996)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Fan, J., Gijbels, I.: Local Polynomials Modelling and its Applications. Chapman and Hall, London (1996)Google Scholar
  20. 20.
    Getreuer, P., Meyer, F.: ENO multiresolution schemes with general discretizations. SIAM J. Numer. Anal. 46, 2953–2977 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    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). LNCS, pp. 95–110. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  22. 22.
    Grant, M., Boyd, S.: CVX: Matlab software for disciplined convex programming (web page and software), (2009)
  23. 23.
    Harten, A.: Discrete multiresolution analysis and generalized wavelets. J. Appl. Numer. Math. 12, 153–192 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Harten, A.: Multiresolution representation of data: General framework. SIAM J. Numer. Anal. 33, 1205–1256 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Hastie, T., Tibshirani, R., Friedman, J.: The Elements of Statistical Learning. Springer, New York (2009)CrossRefzbMATHGoogle Scholar
  26. 26.
    Loader, C.: Local Regression and Likelihook. Springer, New York (1999)Google Scholar
  27. 27.
    MATLAB. version 7.10.0 (R2010a). The MathWorks Inc., Natick (2010)Google Scholar
  28. 28.
    Wand, M.P., Jones, M.C.: Kernel Smoothing. Chapman and Hall, London (1995)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Francesc Aràndiga
    • 1
  • Pep Mulet
    • 1
  • Dionisio F. Yáñez
    • 2
    • 3
    Email author
  1. 1.Department of Matemática AplicadaUniversitat de ValènciaBurjassot (Valencia)Spain
  2. 2.Campus CapacitasUniversidad Católica de ValenciaGodella (Valencia)Spain
  3. 3.Departamento de Matemáticas, CC. NN. y CC. SS. aplicadas a la educaciónUniversidad Católica de ValenciaGodella (Valencia)Spain

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