Biomedical Imaging GroupSTI
English only   BIG > Publications > Matched Bases

 Home Page
 News & Events
 Tutorials and Reviews
 Download Algorithms

 PDF not available
 PS not available
 All BibTeX References

Matched Wavelet-Like Bases for the Decoupling of Sparse Stochastic Processes

M. Unser, P. Pad

Proceedings of the Eighth International Conference Curves and Surfaces (ICCS'14), Paris, French Republic, June 12-18, 2014, pp. 68.

Sparse stochastic processes are defined in terms of the generalized innovation models or, equivalently, as solutions of stochastic differential equations driven by white Lévy noise [1]. They are continuous-domain random entities that are specified by an infinite-dimensional measure over S'(ℝd) (the space of tempered distributions). They are characterized by a whitening operator L that shapes their Fourier spectrum, and a Lévy exponent ƒ that controls their intrinsic sparsity. In the scenario where L is a shift-invariant operator (Fourier multiplier) and w is a Lévy noise with an exponent ƒ that is p-admissible, s = L−1 w is a well defined generalized stochastic process in S'(ℝd) provided that L−1* is a continuous linear map from S(ℝd) → Lp(ℝd) [2]. We show that, under those conditions, it is possible to partly decouple s by expanding it in a matched wavelet basis where the wavelets at a given scale are of the form ψi = L* ϕi where ϕiLp(ℝd) is a suitable smoothing kernel. The construction of such wavelet bases is feasible in 1D for any ordinary differential operator L with the help of exponential splines [3], or, in multiple dimensions, using the extended framework described in [4].


  1. M. Unser, P.D. Tafti, An Introduction to Sparse Stochastic Processes, Cambridge University Press, in presss.

  2. M. Unser, P.D. Tafti, Q. Sun, "A Unified Formulation of Gaussian Versus Sparse Stochastic Processes—Part I: Continuous-Domain Theory," IEEE Transactions on Information Theory, vol. 60, no. 3, pp. 1945-1962, March 2014.

  3. I. Khalidov, M. Unser, "From Differential Equations to the Construction of New Wavelet-Like Bases," IEEE Transactions on Signal Processing, vol. 54, no. 4, pp. 1256-1267, April 2006.

  4. I. Khalidov, M. Unser, J.P. Ward, "Operator-Like Wavelet Bases of L2(ℝd)," The Journal of Fourier Analysis and Applications, vol. 19, no. 6, pp. 1294-1322, December 2013.

AUTHOR="Unser, M. and Pad, P.",
TITLE="Matched Wavelet-Like Bases for the Decoupling of Sparse
        Stochastic Processes",
BOOKTITLE="Proceedings of the Eighth International Conference Curves and
        Surfaces ({ICCS'14})",
address="Paris, French Republic",
month="June 12-18,",

© 2014 SMAI-SIGMA. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other works must be obtained from SMAI-SIGMA.
This material is presented to ensure timely dissemination of scholarly and technical work. Copyright and all rights therein are retained by authors or by other copyright holders. All persons copying this information are expected to adhere to the terms and constraints invoked by each author's copyright. In most cases, these works may not be reposted without the explicit permission of the copyright holder.