Correspondences between Wavelet Shrinkage and Nonlinear Diffusion

We study the connections between discrete one-dimensional schemes for nonlinear diffusion and shift-invariant Haar wavelet shrinkage. We show that one step of (stabilised) explicit discretisation of nonlinear diffusion can be expressed in terms of wavelet shrinkage on a single spatial level. This equivalence allows a fruitful exchange of ideas between the two fields. In this paper we derive new wavelet shrinkage functions from existing diffusivity functions, and identify some previously used shrinkage functions as corresponding to well known diffusivities. We demonstrate experimentally that some of the diffusion-inspired shrinkage functions are among the best for translation-invariant multiscale wavelet shrinkage denoising.

[1]  P. Morettin.  Wavelets in Statistics , 1997 .

[2]  Bernd Kawohl,et al.  Maximum and comparison principle for one-dimensional anisotropic diffusion , 1998 .

[3]  Richard Kronland-Martinet,et al.  A real-time algorithm for signal analysis with the help of the wavelet transform , 1989 .

[4]  Joachim Weickert,et al.  Anisotropic diffusion in image processing , 1996 .

[5]  Masaaki Kijima,et al.  Markov processes for stochastic modeling , 1997 .

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

[7]  D. Donoho,et al.  Translation-Invariant DeNoising , 1995 .

[8]  Antonin Chambolle,et al.  Nonlinear wavelet image processing: variational problems, compression, and noise removal through wavelet shrinkage , 1998, IEEE Trans. Image Process..

[9]  Preprint Nr,et al.  On the Equivalence of Soft Wavelet Shrinkage, Total Variation Diusion, Total Variation Regularization, and SIDEs , 2003 .

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

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

[12]  F. Malgouyres,et al.  Mathematical analysis of a model which combines total variation and wavelet for image restoration 1 , 2002 .

[13]  R. Stollberger,et al.  Nonlinear anisotropic diffusion filtering for multiscale edge enhancement , 2002 .

[14]  Joachim Weickert,et al.  On Iterations and Scales of Nonlinear Filters , 2003 .

[15]  Ronald R. Coifman,et al.  New Methods of Controlled Total Variation Reduction for Digital Functions , 2001, SIAM J. Numer. Anal..

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

[17]  Joachim Weickert,et al.  Relations between Soft Wavelet Shrinkage and Total Variation Denoising , 2002, DAGM-Symposium.

[18]  V. Caselles,et al.  Minimizing total variation flow , 2000, Differential and Integral Equations.

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

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

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

[22]  Thomas Brox,et al.  On the Equivalence of Soft Wavelet Shrinkage, Total Variation Diffusion, Total Variation Regularization, and SIDEs , 2004, SIAM J. Numer. Anal..

[23]  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.

[24]  Antonin Chambolle,et al.  Interpreting translation-invariant wavelet shrinkage as a new image smoothing scale space , 2001, IEEE Trans. Image Process..

[25]  R. DeVore,et al.  Nonlinear Approximation and the Space BV(R2) , 1999 .

[26]  Emmanuel J. Candès,et al.  New multiscale transforms, minimum total variation synthesis: applications to edge-preserving image reconstruction , 2002, Signal Process..

[27]  Stephen L. Keeling,et al.  Nonlinear anisotropic diffusion filters for wide range edge sharpening , 2000, Medical Imaging: Image Processing.

[28]  Christopher M. Brown.  Advances in computer vision , 1987 .

[29]  Jacques Froment,et al.  Reconstruction of Wavelet Coefficients Using Total Variation Minimization , 2002, SIAM J. Sci. Comput..

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

[31]  Philip M. Carrion.  On stability of 1D exact inverse methods , 1986 .

[32]  Jitendra Malik,et al.  Scale-Space and Edge Detection Using Anisotropic Diffusion , 1990, IEEE Trans. Pattern Anal. Mach. Intell..

[33]  Michel Barlaud,et al.  Two deterministic half-quadratic regularization algorithms for computed imaging , 1994, Proceedings of 1st International Conference on Image Processing.

[34]  Guillermo Sapiro,et al.  Robust anisotropic diffusion , 1998, IEEE Trans. Image Process..

[35]  Joachim Weickert,et al.  A semidiscrete nonlinear scale-space theory and its relation to the Perona - Malik paradox , 1996, TFCV.

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

[37]  A. Bruce,et al.  WAVESHRINK WITH FIRM SHRINKAGE , 1997 .

[38]  Hong-Ye Gao,et al.  Wavelet Shrinkage Denoising Using the Non-Negative Garrote , 1998 .