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Since the connection between fully discrete nonlinear diffusion filtering and wavelet shrinkage turned out to be
very fruitful in theone-dimensional setting, it is natural to ask whether we can also gain novel insights in the two-di-mensional(2D) case by exploiting a similar strategy. This is the topic and the main result of the present paper: even in 2D, it is possible to establish an exact relation between wavelet shrinkage (limited to a single scale) and nonlinear diffusion (limited to a single iteration).

In the paper we focus mainly on one important nov-elty which appears in the two-dimensional setting, name-ly on channel coupling and its consequences. We show that the link to nonlinear diffusion requires that the shrinkage function of individual channels of wavelet coefficients is synchronised via a joint estimation of the image gradient. The resulting coupled wavelet shrinkage then inherits some desirable properties from the corre-sponding nonlinear diffusion filter, namely a good approximation of rotation invariance and a straightfor-ward extension to filtering of vector- and tensor-valued images.

The property of rotation invariance is natural for diffu-sion, being inherent in the continuous formulation of the diffusion process. Unfortunately, this quality is usually missing in wavelet shrinkage methods. The relation between the two techniques allows to construct rotation-in-variant wavelet shrinkage methods easily, with very little additional computational cost. Diffusion methods have an established approach to filtering of vector-, or even te nsor-valued data. Through a common estimation of diffusivity, the individual data components evolve in a synchronised manner, and pos-sible artifacts or misalignment of features in the indi-vidual data channels are avoided. Having established the link to coupled wavelet shrinkage, wavelet-based methods suitable for filtering of vector- and tensor-val-ued images can be designed in a straightforward way.

References
[1] A. Chambolle, R.A. DeVore, N. Lee, B.L. Lucier, Nonlinear wavelet image processing: variational problems, compression, and noise removal through wavelet shrinkage, IEEE Transactions on Image Processing 7 (3) (1998) 319–335.
[2] A. Chambolle, B.L. Lucier, Interpreting translationally-invariant wavelet shrinkage as a new image smoothing scale space, IEEE Transactions on Image Processing 10 (7) (2001) 993–1000.
[3] A. Cohen, R. DeVore, P. Petrushev, H. Xu, Nonlinear approximation and the space BV(R2), American Journal of Mathematics 121 (1999) 587–628.
[4] A. Cohen, W. Dahmen, I. Daubechies, R. DeVore, Harmonic analysis in the space BV, Revista https://www.51lunwen.org/uk/ Matematica Iberoamericana 19 (2003) 235–262.
[5] Y. Meyer, Oscillating Patterns in Image Processing and Nonlinear Evolution Equations, vol. 22 of University Lecture Series, AMS, Providence, 2001.
[6] J. Shen, A note on wavelets and diffusion, Journal of Computational Analysis and Applications 5 (1) (2003) 147–159.
[7] J. Shen, G. Strang, On wavelet fundamental solutions to the heat equation – heatlets, Journal of Differential Equations 161 (2) (2000) 403–421.