Statistics of wavelet coefficients for sparse self-similar images

Abstract

We study the statistics of wavelet coefficients of non-Gaussian images, focusing mainly on the behaviour at coarse scales. We assume that an image can be whitened by a fractional Laplacian operator, which is consistent with an ∥ω∥ −γ spectral decay. In other words, we model images as sparse and self-similar stochastic processes within the framework of generalised innovation models. We show that the wavelet coefficients at coarse scales are asymptotically Gaussian even if the prior model for fine scales is sparse. We further refine our analysis by deriving the theoretical evolution of the cumulants of wavelet coefficients across scales. Especially, the evolution of the kurtosis supplies a theoretical prediction for the Gaussianity level at each scale. Finally, we provide simulations and experiments that support our theoretical predictions.

Wavelet Compressibility of Compound Poisson Processes

“…The observations of this section are consistent with recent theoretical and empirical results demonstrating that wavelet methods outperform classical Fourier-based methods for the analysis of sparse stochastic processes [8]. In this line of works, we mention the analysis of the Gaussianity of the Haar wavelet coefficients of sparse stochastic processes across scales [63], the demonstration-both theoretically and empirically-that wavelets provide better orthonormal transformations for the independent component analysis (ICA) of non-Gaussian stable AR(1) processes [65], and that wavelets are more suitable to the denoising of non-Gaussian stable processes [66].…”

“…The law of Haar wavelet coefficients of s has been characterized in [63] in terms of their characteristic function. For us, it will be useful to connect the wavelet coefficients to the underlying Lévy white noise.…”