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 . 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) . 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 ϕi ∈ Lp(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 , or, in multiple dimensions, using the extended framework described in .
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AUTHOR="Unser, M. and Pad, P.",
TITLE="Matched Wavelet-Like Bases for the Decoupling of Sparse
BOOKTITLE="Proceedings of the Eighth International Conference Curves and
address="Paris, French Republic",
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