Hidden Markov chains and fields with observations in Riemannian manifolds

TL;DR

This paper extends hidden Markov models and fields to handle observations in Riemannian manifolds, enabling their application to complex signals and images beyond Euclidean spaces.

Contribution

It introduces a statistical framework for adapting HMMs and related algorithms to Riemannian manifold observations, broadening their applicability.

Findings

Framework for HMMs on Riemannian manifolds

Extension of Baum-Welch algorithm to manifold data

Potential applications in signal and image modeling

Abstract

Hidden Markov chain, or Markov field, models, with observations in a Euclidean space, play a major role across signal and image processing. The present work provides a statistical framework which can be used to extend these models, along with related, popular algorithms (such as the Baum-Welch algorithm), to the case where the observations lie in a Riemannian manifold. It is motivated by the potential use of hidden Markov chains and fields, with observations in Riemannian manifolds, as models for complex signals and images.