Abstract
The Fisher–Rao distance is a measure of dissimilarity between probability distributions, which, under certain regularity conditions of the statistical model, is up to a scaling factor the unique Riemannian metric invariant under Markov morphisms. It is related to the Shannon entropy and has been used to enlarge the perspective of analysis in a wide variety of domains such as image processing, radar systems, and morphological classification. Here, we approach this metric considered in the statistical model of normal multivariate probability distributions, for which there is not an explicit expression in general, by gathering known results (closed forms for submanifolds and bounds) and derive expressions for the distance between distributions with the same covariance matrix and between distributions with mirrored covariance matrices. An application of the Fisher–Rao distance to the simplification of Gaussian mixtures using the hierarchical clustering algorithm is also presented.
Highlights
A proper measure to determine the dissimilarity between probability distributions has been approached in many problems and applications
As in [38], we summarize previous results regarding the Fisher–Rao distance in the space of multivariate normal distributions including closed forms for this distance restricted to submanifolds and general bounds
Rn, method based on the Fisher–Rao distance in the submanifold M D (21) to simplify diagonal Gaussian mixture model (DGMM), and we present an application to image segmentation, complementing what was developed in [52]
Summary
A proper measure to determine the dissimilarity between probability distributions has been approached in many problems and applications. The Fisher–Rao distance is a very special metric for statistical models of probability distributions. This distance is invariant by reparametrization of the sample space and covariant by reparameterization of the parameter space [1]. It was applied to quantization of hyperspectral images [23] and to the space of projected lines in the paracatadioptric images [24] This Fisher–Rao model was used to simplify Gaussian mixtures through the k-means method [25] and a hierarchical clustering technique [26].
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