Approximating Intersections and Differences Between Linear Statistical Shape Models Using Markov Chain Monte Carlo

TL;DR

This paper introduces a novel MCMC-based method to compare linear statistical shape models by approximating their intersection and difference spaces, enabling visualization and quantification beyond traditional performance metrics.

Contribution

The paper presents a new approach to qualitatively and quantitatively compare shape models by estimating their intersection and difference spaces using MCMC and PCA, which was not previously possible.

Findings

Accurately recovers ground-truth intersection spaces.

Effectively visualizes differences between shape models.

Demonstrates applicability on face models with gender and identity variations.

Abstract

To date, the comparison of Statistical Shape Models (SSMs) is often solely performance-based, carried out by means of simplistic metrics such as compactness, generalization, or specificity. Any similarities or differences between the actual shape spaces can neither be visualized nor quantified. In this paper, we present a new method to qualitatively compare two linear SSMs in dense correspondence by computing approximate intersection spaces and set-theoretic differences between the (hyper-ellipsoidal) allowable shape domains spanned by the models. To this end, we approximate the distribution of shapes lying in the intersection space using Markov chain Monte Carlo and subsequently apply Principal Component Analysis (PCA) to the posterior samples, eventually yielding a new SSM of the intersection space. We estimate differences between linear SSMs in a similar manner; here, however, the resulting spaces are no longer convex and we do not apply PCA but instead use the posterior samples for visualization. We showcase the proposed algorithm qualitatively by computing and analyzing intersection spaces and differences between publicly available face models, focusing on gender-specific male and female as well as identity and expression models. Our quantitative evaluation based on SSMs built from synthetic and real-world data sets provides detailed evidence that the introduced method is able to recover ground-truth intersection spaces and differences accurately.