Differential Geometry for Model Independent Analysis of Images and Other Non-Euclidean Data: Recent Developments

TL;DR

This paper reviews recent nonparametric methods for analyzing images and non-Euclidean data using geometric tools like Fréchet means, with applications in image analysis and Bayesian density estimation on manifolds.

Contribution

It synthesizes recent developments in geometric and Bayesian nonparametric analysis for non-Euclidean data, emphasizing theoretical and application aspects.

Findings

Discussion of uniqueness and asymptotic properties of Fréchet means

Application of nonparametric Bayes methods to density estimation on manifolds

Highlighting of image analysis applications using geometric data analysis

Abstract

This article provides an exposition of recent methodologies for nonparametric analysis of digital observations on images and other non-Euclidean objects. Fr\'echet means of distributions on metric spaces, such as manifolds and stratified spaces, have played an important role in this endeavor. Apart from theoretical issues of uniqueness of the Fr\'echet minimizer and the asymptotic distribution of the sample Fr\'echet mean under uniqueness, applications to image analysis are highlighted. In addition, nonparametric Bayes theory is brought to bear on the problems of density estimation and classification on manifolds.