Fisher Information Geometry for Shape Analysis

Angela De Sanctis,Stefano Antonio Gattone

doi:10.3390/ecea-3-c001

Abstract

The aim of this study is to model shapes from complex systems using Information Geometry tools. It is well-known that the Fisher information endows the statistical manifold, defined by a family of probability distributions, with a Riemannian metric, called the Fisher-Rao metric. With respect to this, geodesic paths are determined, minimizing information in Fisher sense. Under the hypothesis that it is possible to extract from the shape a finite number of representing points, called landmarks, we propose to model each of them with a probability distribution, as for example a multivariate Gaussian distribution. Then using the geodesic distance, induced by the Fisher-Rao metric, we can define a shape metric which enables us to quantify differences between shapes. The discriminative power of the proposed shape metric is tested performing a cluster analysis on the shapes of three different groups of specimens corresponding to three species of flatfish. Results show a better ability in recovering the true cluster structure with respect to other existing shape distances.

Highlights

Shape analysis is a timely and interesting research field
The clustering of shapes is a longstanding challenge in the framework of geometric morphometrics [1], since the recognition of groups of similar morphologies, and of the differences among these groups, is a key step of the analysis when geometric morphometrics protocols are applied
Shapes must be invariant to rotation, scale and translation so that a straightforward way to proceed is first to align the objects by using Procrustes analysis and to apply standard clustering algorithms minimizing a given distance or dissimilarity measure evaluated within each cluster [2,3]

Summary

Introduction

Shape analysis is a timely and interesting research field. Applications of shape analysis have been involved in various areas such as morphometry, image analysis, biology, database retrieval, and so on. The variances reflect the uncertainty in the landmark’s placement and the variability across a family of shapes Within this framework , we derive a distance between two shapes using tools from Information Geometry [5,6], which considers statistical models as Riemannian manifolds with the Fisher–Rao metric. We have three sub-cases [7]: In this case the family can identified with the (p + 1)-dimensional half space parameterized by (μ1, μ2, ..., μp, σ), σ > 0, and the Fisher information matrix is: σ2. The family of all independent multivariate normal distributions is the intersection of half-spaces parameterized by (μ1, σ1, μ2, σ2, ..., μp, σp), σi > 0, so the Fisher information matrix is: σ12. (iii) General Gaussian distributions: Σ any symmetric positive definite covariance matrix The analysis is much more difficult and it is not known a closed form for the associated distance

Modeling of 2-Dimensional Shapes

Shape Metrics Based on Geodesic Distance