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.

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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'(ℝ^d) (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'(ℝ^d) provided that L^−1* is a continuous linear map from S(ℝ^d) → L_p(ℝ^d) [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 ψ_i = L^* ϕ_i where ϕ_i ∈ L_p(ℝ^d) 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].

References

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(ℝ^d)," The Journal of Fourier Analysis and Applications, vol. 19, no. 6, pp. 1294-1322, December 2013.

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