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I am reading several papers applying anisotropic diffusion in the transform domain like Shearlet, curvelet. All of them have got excellent results.

The physical meaning and principle of applying anisotropic diffusion in space domain is quiet straightforward -- the gradient.

I am confused as to the physical meaning and principle of applying anisotropic diffusion in the transform domain.

Papers, for example:

Shearlet-Based Total Variation Diffusion for Denoising Glenn R. Easley, Member, IEEE, Demetrio Labate, and Flavia Colonna IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 18, NO. 2, FEBRUARY 2009

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Physically, there is usually not much implied meaning (but this may depend on a case-by-case basis). Transforms like shearlets are actually used to embed the data into anisotropic or edge-aware transform domains.

Mathematically, it may make more sense. Isotropic diffusion tends to create blur (smear) in whatever domain it is applied. Thus, if you want to obtain a result that is compact in the considered domain, be it transformed or not, applying non-isotropic diffusion is very tempting.

Another way to see it mathematically, is to understand that isotropic diffusion corresponds to solving a least square problem that will create many small non-zero coefficients by trying to spread the error everywhere. However, a helpful and widely available assumption about signal is their sparsity in the transform domain, i.e., they have few non-zero coefficients in a suitably chosen transform domain that concentrate the energy. This corresponds to solving problems where the exponent of the norm of the prior term is less than 2 (typically 1) and thus the use of anisotropic diffusion to find a solution.

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  • $\begingroup$ Thanks for answering. In spacial domain, Isotropic diffusion tends to create blur -- it is the principle of denoising. Does bluring in transform domaining mean something ? $\endgroup$
    – freealbert
    Jul 8, 2013 at 5:17
  • $\begingroup$ "Blurring" = convolution. Convolution in the frequency domain is equivalent to multiplication in the time domain. So if you are blurring by convolving with a gaussian, then you are multiplying by a gaussian in the transform domain. $\endgroup$
    – Wandering Logic
    Jul 8, 2013 at 10:57
  • $\begingroup$ Anisotropic diffusion is analogous (in the limit) to bilateral filtering, which makes the "it's just convolution with a gaussian" analogy clearer. $\endgroup$
    – Wandering Logic
    Jul 8, 2013 at 11:00
  • $\begingroup$ Well, bilateral filtering is just one possible implementation of anisotropic filtering. The main thing is that 1) anisotropic diffusion is a trendy topic and 2) by avoiding blur you favor solutions with concentrated non-zero coefficients instead of spreading their energy in a wide domain, which is also referred to as sparsity. $\endgroup$
    – sansuiso
    Jul 8, 2013 at 14:46

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