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I am reading the book Noise reduction by wavelet thresholding by Maarten Jansen. About vanishing moments, it reads

To create a really sparse representation, we try to make coefficients that live between point s of singularities as small as possible. In these intervals of smooth behavior , the signal can be locally well approximated by a polynomial.

I don't understand the sentence. If the number of vanishing moments is large, does it mean that the filtered image will be smoother? If I hope to keep edges in images, do I need to use Haar wavelets?

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The initial motivation of a sparse representation for a class of signals or images is to pre-separate components with different morphological behaviors.

A traditional model for images is the augmented cartoon: piece-wise smooth + contours + geometrical textures + "noise (unmodelled)".

Smooth background, contour and texture

The mother wavelets (gradient-like operators) can capture isolated singularities at different scale with a few coefficients (and if they oscillate, a bit of textures as well). If the singularities are mixed with a polynomial trend, a standard gradient may also produce coefficients. With a sufficient number of vanishing moments, wavelets somehow "pre-separate" polynomials and singularities, and the polynomial part is cast, subsampled, to a small set of approximmation coefficients.

All in all, you can get few coefficients for singularities, and few coefficients for polynomials between singularities (a long polynomial can be encoded into a few polynomial factors). But still you preserve all the information, especially about the gradients you want to keep.

Wavelets are, under some theoretical conditions, quite optimal for such 1D signals. In practice on 2D data, other aspects come into play and counterbalance the need for vanishing moments: the nature of singularities in images, their finite size, the other wavelet properties, etc.

Resultingly, 1D wavelets work ok for not-too-complex images, but become limited with complicated data, complicated processing, or high noise conditions.

My experience is that, unless you have specific needs, 2-4 moments suffice in most of the cases, meaning that if finally you use a wavelet with more moments, it will be a by-product of other wavelet properties, and not a strong requirement. There are many options in finding the optimal multiscale geometric representation.

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  • $\begingroup$ I'm learning wavelet for image denoising. I have one question. Hard/soft thresholding will remove small wavelet coefficients. However maybe some such small coefficients may come from image structures. So this operation will influence image structures? $\endgroup$ – Jogging Song Feb 6 '16 at 9:19
  • $\begingroup$ You mention complicated data and complicated processing. What do you mean by complicated data and complicated processing? Do Complicated data mean nature images? Currently I hope to use wavelet for images captured under low light situations. Do you have any suggestions or provide some links to useful papers? $\endgroup$ – Jogging Song Feb 6 '16 at 9:22
  • $\begingroup$ Here they color or multivariate images? $\endgroup$ – Laurent Duval Feb 6 '16 at 9:26
  • $\begingroup$ I am processing color images. $\endgroup$ – Jogging Song Feb 6 '16 at 9:35
  • $\begingroup$ Scalar hard/soft thresholding is limited. You can use vector thresholding across scales and image patches. The problem is to fix thresholds. A complicated image (color, multivariate, textured) denoising approach is proposed in A Nonlinear Stein Based Estimator for Multichannel Image Denoising, IEEE Image Processing, 2008. I have used this approach satisfactorily in low-light conditions (engine cylinder), combined with homomorphic filtering in the wavelet domain $\endgroup$ – Laurent Duval Feb 6 '16 at 9:35

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