Invariances, Laplacian-like wavelet bases, and the whitening of fractal processes
IEEE Transactions on Image Processing
Self-Similarity: Part II—Optimal Estimation of Fractal Processes
IEEE Transactions on Signal Processing
Stochastic Models for Sparse and Piecewise-Smooth Signals
IEEE Transactions on Signal Processing
| Hi-index | 0.00 |
The fractional Laplacian $(-\triangle)^{\gamma/2}$ commutes with the primary coordination transformations in the Euclidean space 驴 d : dilation, translation and rotation, and has tight link to splines, fractals and stable Levy processes. For 0驴驴驴d, its inverse is the classical Riesz potential I 驴 which is dilation-invariant 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 驴 to any non-integer number 驴 larger than d and show that it is the unique left-inverse of the fractional Laplacian $(-\triangle)^{\gamma/2}$ which is dilation-invariant and translation-invariant. We observe that, for any 1驴驴驴p驴驴驴驴 and 驴驴驴驴d(1驴驴驴1/p), there exists a Schwartz function f such that I 驴 f is not p-integrable. We then introduce the new unique left-inverse I 驴, p of the fractional Laplacian $(-\triangle)^{\gamma/2}$ with the property that I 驴, p is dilation-invariant (but not translation-invariant) and that I 驴, p f is p-integrable for any Schwartz function f. We finally apply that linear operator I 驴, p with p驴=驴1 to solve the stochastic partial differential equation $(-\triangle)^{\gamma/2} \Phi=w$ with white Poisson noise as its driving term w.