Saarland University | Mathematical Image Analysis Group | Research

Relations between discontinuity preserving methods

Motivation

We consider a classical task of signal denoising: create an estimate u of an original signal z from its noisy measurement f, where f = z + n and n denotes an additive noise function. Various methods have been proposed to remove the noise from z without sacrificing important structures such as edges, including rank-order filtering, mathematical morphology, stochastic methods, adaptive smoothing, wavelet techniques, partial differential equations (PDEs) and variational methods. Although these classes of methods serve the same purpose, relatively few publications examine their similarities and differences, in order to transfer results from one of these classes to the others, or to design hybrid methods that combine the advantages of different classes. By studying their relations, we have tried to fill this gap and contribute to a better understanding of the methods involved.

Our contributions

Equivalence results for soft wavelet shrinkage, total-variation regularization, nonlinear diffusion with TV diffusivity, and dynamical system SIDEs [5,1,3]. The connections allowed us to desing hybrid methods which combine the speed of wavelet shrinkage and the robustness of iterative methods [2].
Correspondences between discrete Haar wavelet shrinkage and nonlinear diffusion have been studied: the methods can be made equivalent by choosing appropriate diffusivity or shrinkage functions [4,7] in 1D. For the two-dimensional situation, the diffusion inspires us to design Haar wavelet filters with improved rotation invariance [6].

Image smoothing example



Input image and its detail.	Result of classical shift-invariant Haar wavelet shrinkage: typical blocky artefacts and misaligned colour channels due to independent processing of the RGB channels.	Shift-invariant Haar wavelet filtering with diffusion-inspired shrinkage rules improves rotation invariance [6] and avoids colour artefacts.

Related publications

[1]	G. Steidl, J. Weickert: Relations between soft wavelet shrinkage and total variation denoising. In L. Van Gool (Ed.): Pattern Recognition. Lecture Notes in Computer Science, Vol. 2449, Springer, Berlin, 198-205, 2002.
[2]	P. Mrázek, J. Weickert, G. Steidl and M. Welk On iterations and scales of nonlinear filters O. Drbohlav (ed.): Computer Vision Winter Workshop 2003, Valtice, Czech Republic, pp.61-66. Czech Pattern Recognition Society, 2003.
[3]	T. Brox, M. Welk, G. Steidl, J. Weickert: Equivalence results for TV diffusion and TV regularisation. L. D. Griffin, M. Lillholm (Eds.): Scale Space Methods in Computer Vision. Lecture Notes in Computer Science, Vol. 2695, Springer, Berlin, 86-100, 2003.
[4]	P. Mrázek, J. Weickert and G. Steidl Correspondences between wavelet shrinkage and nonlinear diffusion L.D. Griffin and M. Lillholm (Eds.): Scale-Space 2003, LNCS 2695, pp. 101-116. Springer, 2003.
[5]	G. Steidl, J. Weickert, T. Brox, P. Mrázek, M. Welk: On the equivalence of soft wavelet shrinkage, total variation diffusion, total variation regularization, and SIDEs. Preprint no. 94, Department of Mathematics, Saarland University, Saarbrücken, Germany, 2003. Accepted for publication in SIAM Journal on Numerical Analysis.
[6]	P. Mrázek and J. Weickert Rotationally invariant wavelet shrinkage B. Michaelis and G. Krell (Eds.): DAGM 2003, LNCS 2781, pp. 156-163. Springer, 2003.
[7]	P. Mrázek, J. Weickert and G. Steidl Diffusion-inspired shrinkage functions and stability results for wavelet denoising Preprint no. 96, Department of Mathematics, Saarland University, Saarbrücken, Germany, 2003.

Pavel Mrázek, mrazek@mia.uni-saarland.de

Last modified: Jan 13, 2004.