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

arXiv: https://arxiv.org/abs/2211.16314 · HTML: https://mlsanity.com/paper/2211.16314/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 41 references that are in the map.
- First posted on arXiv 29 Nov 2022.

## Builds on (most cited first, 14 of 14)

- [Face2Face: Real-time Face Capture and Reenactment of RGB Videos](https://mlsanity.com/paper/2007.14808/) (2020)
- [Learning an Animatable Detailed 3D Face Model from In-The-Wild Images](https://mlsanity.com/paper/2012.04012/) (2020)
- [3D Morphable Face Models -- Past, Present and Future](https://mlsanity.com/paper/1909.01815/) (2019)
- [Towards Metrical Reconstruction of Human Faces](https://mlsanity.com/paper/2204.06607/) (2022)
- [Morphable Face Models - An Open Framework](https://mlsanity.com/paper/1709.08398/) (2017)
- [A Morphable Face Albedo Model](https://mlsanity.com/paper/2004.02711/) (2020)
- [Combining 3D Morphable Models: A Large scale Face-and-Head Model](https://mlsanity.com/paper/1903.03785/) (2019)
- [Multilinear Wavelets: A Statistical Shape Space for Human Faces](https://mlsanity.com/paper/1401.2818/) (2014)
- [Gaussian Process Morphable Models](https://mlsanity.com/paper/1603.07254/) (2016)
- [Robust Model-based Face Reconstruction through Weakly-Supervised Outlier Segmentation](https://mlsanity.com/paper/2106.09614/) (2021)
- [A Closest Point Proposal for MCMC-based Probabilistic Surface Registration](https://mlsanity.com/paper/1907.01414/) (2019)
- [Identity-Expression Ambiguity in 3D Morphable Face Models](https://mlsanity.com/paper/2109.14203/) (2021)
- [Dynamic multi-object Gaussian process models: A framework for data-driven functional modelling of human joints](https://mlsanity.com/paper/2001.07904/) (2020)
- [Learning the shape of female breasts: an open-access 3D statistical shape model of the female breast built from 110 breast scans](https://mlsanity.com/paper/2107.13463/) (2021)

## Built on by (most cited first, 2 of 2)

- [Facing Asymmetry -- Uncovering the Causal Link between Facial Symmetry and Expression Classifiers using Synthetic Interventions](https://mlsanity.com/paper/2409.15927/) (2024)
- [Electromyography-Informed Facial Expression Reconstruction for Physiological-Based Synthesis and Analysis](https://mlsanity.com/paper/2503.09556/) (2025)

## Abstract (arXiv)

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.