Reconstruction of Wavelet Coefficients Using Total Variation Minimization

We propose a model to reconstruct wavelet coefficients using a total variation minimization algorithm. The approach is motivated by wavelet signal denoising methods, where thresholding small wavelet coefficients leads to pseudo-Gibbs artifacts. By replacing these thresholded coefficients by values minimizing the total variation, our method performs a nearly artifact-free signal denoising. In this paper, we detail the algorithm based on a subgradient descent combining a projection on a linear space. The convergence of the algorithm is established and numerical experiments are reported.

[1]  Ronald R. Coifman,et al.  Combining the Calculus of Variations and Wavelets for Image Enhancement , 2000 .

[2]  Jacques Froment,et al.  Artifact free signal denoising with wavelets , 2001, 2001 IEEE International Conference on Acoustics, Speech, and Signal Processing. Proceedings (Cat. No.01CH37221).

[3]  C. Vogel,et al.  Analysis of bounded variation penalty methods for ill-posed problems , 1994 .

[4]  David L. Donoho,et al.  De-noising by soft-thresholding , 1995, IEEE Trans. Inf. Theory.

[5]  Lionel Moisan.  Extrapolation de spectre et variation totale pondérée , 2001 .

[6]  L. Rudin,et al.  Nonlinear total variation based noise removal algorithms , 1992 .

[7]  I. Johnstone,et al.  Ideal spatial adaptation by wavelet shrinkage , 1994 .

[8]  Naum Zuselevich Shor,et al.  Minimization Methods for Non-Differentiable Functions , 1985, Springer Series in Computational Mathematics.

[9]  S. Durand,et al.  IMAGE DEBLURRING, SPECTRUM INTERPOLATION AND APPLICATION TO SATELLITE IMAGING , 2000 .

[10]  Tony F. Chan,et al.  High-Order Total Variation-Based Image Restoration , 2000, SIAM J. Sci. Comput..

[11]  Fadil Santosa,et al.  Recovery of Blocky Images from Noisy and Blurred Data , 1996, SIAM J. Appl. Math..

[12]  Tony F. Chan,et al.  Total variation improved wavelet thresholding in image compression , 2000, Proceedings 2000 International Conference on Image Processing (Cat. No.00CH37101).

[13]  D. Dobson,et al.  Convergence of an Iterative Method for Total Variation Denoising , 1997 .

[14]  Mila Nikolova,et al.  Local Strong Homogeneity of a Regularized Estimator , 2000, SIAM J. Appl. Math..

[15]  Jacques Froment,et al.  Analyse multirésolution des signaux aléatoires , 1991 .

[16]  Raymond H. Chan,et al.  Continuation method for total variation denoising problems , 1995, Optics & Photonics.

[17]  D. L. Donoho,et al.  Ideal spacial adaptation via wavelet shrinkage , 1994 .

[18]  Albert Cohen,et al.  Nonlinear Approximation of Random Functions , 1997, SIAM J. Appl. Math..

[19]  S. Mallat.  A wavelet tour of signal processing , 1998 .

[20]  François Malgouyres,et al.  Combining total variation and wavelet packet approaches for image deblurring , 2001, Proceedings IEEE Workshop on Variational and Level Set Methods in Computer Vision.

[21]  Curtis R. Vogel,et al.  Iterative Methods for Total Variation Denoising , 1996, SIAM J. Sci. Comput..

[22]  D. Donoho,et al.  Translation-Invariant De-Noising , 1995 .

[23]  P. Lions,et al.  Image recovery via total variation minimization and related problems , 1997 .

[24]  I. Daubechies,et al.  Wavelets on the Interval and Fast Wavelet Transforms , 1993 .

[25]  Yves Bobichon,et al.  Regularized multiresolution methods for astronomical image enhancement , 1997, Optics & Photonics.