Invariances, Laplacian-Like Wavelet Bases, and the Whitening of Fractal Processes
P.D. Tafti, D. Van De Ville, M. Unser
IEEE Transactions on Image Processing, in press.
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In this contribution we study the notion of affine invariance (specifically, invariance to the shifting, scaling, and rotation of the coordinate system), as a starting point for the development of mathematical tools and approaches useful in the characterization and analysis of multivariate fractional Brownian motion (fBm) fields. In particular, using a rigorous and powerful distribution theoretic formulation, we extend previous results of Blu and Unser (2006) to the multivariate case, showing that polyharmonic splines and fBm processes can be seen as the (deterministic vs stochastic) solutions to an identical fractional partial differential equation that involves a fractional Laplacian operator. We then show that the wavelets derived from polyharmonic splines have a behaviour similar to the fractional Laplacian, which also turns out to be the whitening operator for fBm fields. This fact allows us to study the probabilistic properties of the wavelet transform coefficients of fBm-like processes, leading for instance to ways of estimating the Hurst exponent of a multi-parameter process from its wavelet transform coefficients. We provide theoretical and experimental verification of these results.
To complement the toolbox available for multi-resolution processing of stochastic fractals, we also introduce an extended family of multi-dimensional multi-resolution spaces for a large class of (separable and non-separable) lattices of arbitrary dimensionality.