Stochastic metamorphosis with template uncertainties

TL;DR

This paper introduces two stochastic perturbations to the metamorphosis equations in image analysis, enhancing the modeling of uncertainties in deformation maps and template reconstructions within a geometric framework.

Contribution

It develops a novel stochastic geometric framework for metamorphosis, incorporating uncertainties in deformation and template reconstruction, with applications to classical image analysis problems.

Findings

Modeling uncertainties improves image matching robustness

Application to landmarks, images, and curves demonstrates versatility

Framework facilitates functional data analysis with stochastic perturbations

Abstract

In this paper, we investigate two stochastic perturbations of the metamorphosis equations of image analysis, in the geometrical context of the Euler-Poincar\'e theory. In the metamorphosis of images, the Lie group of diffeomorphisms deforms a template image that is undergoing its own internal dynamics as it deforms. This type of deformation allows more freedom for image matching and has analogies with complex fluids when the template properties are regarded as order parameters (coset spaces of broken symmetries). The first stochastic perturbation we consider corresponds to uncertainty due to random errors in the reconstruction of the deformation map from its vector field. We also consider a second stochastic perturbation, which compounds the uncertainty in of the deformation map with the uncertainty in the reconstruction of the template position from its velocity field. We apply this general geometric theory to several classical examples, including landmarks, images, and closed curves, and we discuss its use for functional data analysis.