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Affine Invariance, Splines, Wavelets and Fractional Brownian Fields

M. Unser, P.D. Tafti, D. Van De Ville

Mathematical Image Processing Meeting (MIPM'07), Marseille, French Republic, September 3-7, 2007.



Invariance is an attractive principle for specifying image processing algorithms. In this work, we concentrate on affine—more precisely, shift, scale and rotation—invariance and identify the corresponding class of operators, which are fractional Laplacians. We then specify some corresponding differential equation and show that the solution (in the distributional sense) is either a fractional Brownian field (Mandelbrot and Van Ness, 1968) or a polyharmonic spline (Duchon, 1976), depending on the nature of the system input (driving term): stochastic (white noise) or deterministic (stream of Dirac impulses). The affine invariance of the operator has two remarkable consequences: (1) the statistical self-similarity of the fractional Brownian field, and (2) the fact that the polyharmonic splines specify a multiresolution analysis of L2(ℝd) and lend themselves to the construction of wavelet bases. We prove that these wavelets essentially behave like the operator from which they are derived, and that they are ideally suited for the analysis of multidimensional signals with fractal characteristics (isotopic differentiation, and whitening property).


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