Left-inverses of fractional Laplacian and sparse stochastic processes

Abstract

The fractional Laplacian (− ) γ /2 commutes with the primary coordination transformations in the Euclidean space R d : dilation, translation and rotation, and has tight link to splines, fractals and stable Levy processes. For 0 < γ < d, its inverse is the classical Riesz potential I γ which is dilationinvariant and translation-invariant. In this work, we investigate the functional properties (continuity, decay and invertibility) of an extended class of differential operators that share those invariance properties. In particular, we extend the definition of the classical Riesz potential I γ to any noninteger number γ larger than d and show that it is the unique left-inverse of the fractional Laplacian (− ) γ /2 which is dilation-invariant and translationinvariant. We observe that, for any 1 ≤ p ≤ ∞ and γ ≥ d(1 − 1/ p), there exists a Schwartz function f such that I γ f is not p-integrable. We then introduce the new unique left-inverse I γ,p of the fractional Laplacian (− ) γ /2 with the property that I γ,p is dilation-invariant (but not translation-invariant) and that I γ,p f is p-integrable for any Schwartz function f . We finally apply that linear operator I γ,p with p = 1 to solve the stochastic partial differential equation (− ) γ /2 = w with white Poisson noise as its driving term w.

A Unifying Parametric Framework for 2D Steerable Wavelet Transforms

“…A necessary requirement for the above definition to make sense is that the Fourier transform of ψ have a sufficient number of zeros at the origin to compensate for the singularity of the frequency response [28]. This is a condition that is generally met by wavelets that have vanishing moments up to order n. The global effect of (−Δ) − n 2 is that of a lowpass filter with a smoothing strength that increases with n.…”

Wavelet Statistics of Sparse and Self-Similar Images

“…which means that Ls d = w. The conditions under which we can define a left-inverse operator L * −1 such that (ii) is fulfilled is not the subject of this paper and was investigated in [17,47,53]. However, some examples will be detailed below for the case of self-similar processes.…”

“…and extended for noninteger γ > d in [47]. However, the Riesz potential can be unstable, with the consequence that the functional ϕ → P w (I γ ϕ) will generally not be well-defined on S(R d ).…”

Sparse Stochastic Processes and Discretization of Linear Inverse Problems

“…Moreover, the underlying differential system is potentially unstable to allow for selfsimilar models. In the present model, the process s is characterized by the formal solution s = L −1 w, where L −1 is an appropriate right inverse of L. The operator L −1 amounts to some generalized "integration" of the innovation w. The implication is that the correlation structure of the stochastic process s is determined by the shaping operator L −1 , whereas its statistical properties and sparsity structure is determined by the driving term w. As an example in the one-dimensional setting, the operator L can be chosen as the first-order continuous-domain derivative operator L = D. For multidimensional signals, an attractive class of operators is the fractional Laplacian (− ) γ 2 which is invariant to translation, dilation, and rotation in R d [17]. This operator gives rise to "1/ ω γ "-type power spectrum and is frequently used to model certain types of images [18], [19].…”