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Stochastic geometry and topology of non-Gaussian fields

Author:
Beuman, Thomas H., Turner, Ari M., Vitelli, Vincenzo
Source:
Proceedings of the National Academy of Sciences of the United States of America 2012 v.109 no.49 pp. 19943-19948
ISSN:
0027-8424
Subject:
image analysis, statistics, topology
Abstract:
Gaussian random fields pervade all areas of science. However, it is often the departures from Gaussianity that carry the crucial signature of the nonlinear mechanisms at the heart of diverse phenomena, ranging from structure formation in condensed matter and cosmology to biomedical imaging. The standard test of non-Gaussianity is to measure higher-order correlation functions. In the present work, we take a different route. We show how geometric and topological properties of Gaussian fields, such as the statistics of extrema, are modified by the presence of a non-Gaussian perturbation. The resulting discrepancies give an independent way to detect and quantify non-Gaussianities. In our treatment, we consider both local and nonlocal mechanisms that generate non-Gaussian fields, both statically and dynamically through nonlinear diffusion.
Agid:
1735509