Full inference for the anisotropic fractional Brownian field

Escande, Paul; Richard, Frédéric

doi:10.1090/tpms/1204

Authors: Paul Escande and Frédéric J. P. Richard
Journal: Theor. Probability and Math. Statist. 110 (2024), 13-29
MSC (2020): Primary 62M40; Secondary 78M50
DOI: https://doi.org/10.1090/tpms/1204
Published electronically: May 10, 2024
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Abstract: The anisotropic fractional Brownian field (AFBF) is a non-stationary Gaussian random field which has been used for the modeling of textured images. In this paper, we address the open issue of estimating the functional parameters of this field, namely the topothesy and Hurst functions. We propose an original method which fits the empirical semi-variogram of an image to the semi-variogram of a turning-band field that approximates the AFBF. Expressing the fitting criterion in terms of a separable non-linear least square criterion, we design a minimization algorithm inspired by the variable projection approach. This algorithm also includes a coarse-to-fine multigrid strategy based on approximations of functional parameters. Compared to existing methods, the new method enables to estimate both functional parameters on their whole definition domain. On simulated textures, we show that it has a low estimation error, even when the parameters are approximated with a high precision. We also apply the method to characterize mammograms and sample images with synthetic parenchymal patterns.

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