Linking wavelet coefficients along scales

The only theoretical result to follow wavelet coefficients along scales is the selection of a particular basis. It can be proved that if one selects a wavelet that is a derivative of a gaussian kernel, maxima poduced by singularities in the wavelet domain can be 'followed' without interruptions [1]. The proof is based on the heat equation. This important result has mainly two limits:

• it limits the choice of the basis for the signal expansion (making unuseful the existence of many bases...)

• it doesn't give any practical way for linking wavelet coefficients (it just proves the existence of continuous time-scale trajectories)

  The limits above gave rise to many empirical and/or theoretical attempts to solve this problem (see for instance [2,3]). Probably the most interesting is the one in [3] based on footprints. They consist of wavelet coefficients generated by signal singularities. The idea is to build a vocabolary of all possible sets of wavelet coefficients, to find them in the wavelet expansion (via a matching pursuit scheme [1]) and to follow them at various scales. Unfortunately, also this approach has some limits:

• it is not able to manage scale levels where footprints 'meet': just the scale levels where footprints are separated can be managed.

• the search of footprints with a large vocabulary is somewhat expensive

Atomic approximation [4,5,6] allows to overcome these limits ...

References:

1. S. Mallat, A Wavelet Tour of Signal Processing, Academic Press, 1998.

2. Kuijper, A., Florack, L.M.J. & Viergever, M.A. (2003). Scale space hierarchy. Journal of mathematical imaging and vision, 18, 169-189.

3. P.L. Dragotti, M. Vetterli, Wavelet Footprints: Theory, Algorithms and Applications, IEEE Transactions on Signal Processing, Vol. 51, No. 5, pp. 1306-1323, May 2003.

4. V. Bruni, D. Vitulano (2006). Wavelet based Signal De-noising via Simple Singularities Approximation. Signal Processing Journal, Elsevier Sci- ence, Vol. 86, pp. 859-876, April 2006.

5. V. Bruni, B. Piccoli, D. Vitulano (2008). A fast computation method for time scale signal denoising. Signal Image and Video Processing, Springer (online version), ISSN 1863-1703 (Print) 1863-1711 (Online), June 2008.

6. V. Bruni, B. Piccoli, D. Vitulano (2008). Wavelets and pde for image de- noising. Electronic Letters on Computer Vision and Image Analysis (ELCVIA), Special Issue on Partial Dierential Equations Methods in Graphics and Vision, vol. 6, no. 2, pp. 36-53, January 2008.