Mesh2SSM++: A Probabilistic Framework for Unsupervised Learning of Statistical Shape Model of Anatomies from Surface Meshes

TL;DR

Mesh2SSM++ advances statistical shape modeling by learning unsupervised, correspondence-based probabilistic shape models directly from surface meshes. It introduces a continuous normalizing flow latent space (with a decoupled representation and sampling space) and a surface-projection–driven training regime, enabling data-informed template updates, efficient sampling, and explicit aleatoric uncertainty estimation. Across five diverse anatomies, Mesh2SSM++ achieves competitive or superior surface and SSM metrics, robust uncertainty calibration, and strong downstream task performance, including group-difference analysis and multi-anatomy classification. The framework offers a scalable, interpretable alternative to traditional and deep-learning SSM approaches, with clear clinical relevance due to uncertainty quantification and improved robustness to noise and artifacts.

Abstract

Anatomy evaluation is crucial for understanding the physiological state, diagnosing abnormalities, and guiding medical interventions. Statistical shape modeling (SSM) is vital in this process. By enabling the extraction of quantitative morphological shape descriptors from MRI and CT scans, SSM provides comprehensive descriptions of anatomical variations within a population. However, the effectiveness of SSM in anatomy evaluation hinges on the quality and robustness of the shape models. While deep learning techniques show promise in addressing these challenges by learning complex nonlinear representations of shapes, existing models still have limitations and often require pre-established shape models for training. To overcome these issues, we propose Mesh2SSM++, a novel approach that learns to estimate correspondences from meshes in an unsupervised manner. This method leverages unsupervised, permutation-invariant representation learning to estimate how to deform a template point cloud into subject-specific meshes, forming a correspondence-based shape model. Additionally, our probabilistic formulation allows learning a population-specific template, reducing potential biases associated with template selection. A key feature of Mesh2SSM++ is its ability to quantify aleatoric uncertainty, which captures inherent data variability and is essential for ensuring reliable model predictions and robust decision-making in clinical tasks, especially under challenging imaging conditions. Through extensive validation across diverse anatomies, evaluation metrics, and downstream tasks, we demonstrate that Mesh2SSM++ outperforms existing methods. Its ability to operate directly on meshes, combined with computational efficiency and interpretability through its probabilistic framework, makes it an attractive alternative to traditional and deep learning-based SSM approaches.

Figures (11)

• Figure 1: Correspondences are sets of ordered points on different shapes representing the same anatomical or geometric feature, thereby establishing a consistent relationship between the shapes. The white highlighted points represent predicted correspondences by Mesh2SSM++ in the right superior pulmonary vein (RSPV) antrum region of the left atrium, consistently located across all shapes. Matching colors across samples indicate additional corresponding points.
• Figure 2: Overview of Mesh2SSM++ Framework: (A) The generative model leverages a decoupled representation and generation process. The latent variable $\mathbf{z}$ is mapped from the generation space $\mathbf{z}_0 \sim p(\mathbf{z}_0)$ to the representation space through the invertible mapping $g_\eta^{-1}(\mathbf{z}_0)$, while $\mathcal{X} \sim p_\theta(\mathcal{X}|\mathbf{z})$ represents data sampled in the data space. Inference is performed via $q_\phi(\mathbf{z}|\mathcal{X})$, which maps the input mesh $\mathcal{X}$ to its latent representation $\mathbf{z}$. (B) The Mesh2SSM++ pipeline. Input surface meshes $\mathcal{X}_i = (\mathcal{V}_i, \mathcal{E}_n)$ are processed by the encoder $q_\phi(\mathbf{z}|\mathcal{X})$ to produce latent representations $\mathbf{z}$. These are combined with prior samples $\mathbf{z}_0 \sim \mathcal{N}(0, I)$ for probabilistic sampling. Based on the latent representation $\mathbf{z}$, the implicit field decoder $p_\theta(\mathcal{X}|\mathbf{z})$ learns how to deform the template point cloud into $\mathbf{C}$ such that it matches the input shape surface while establishing correspondence by deforming the same template for every input mesh.
• Figure 3: Distance Metrics: Boxplots show the error distribution across test sets for each model in mm.
• Figure 4: SSM Metrics: A compactness plot displays the cumulative variance ratio as a function of PCA mode count. Generalization and specificity reconstruction error plotted as a function of PCA mode count. For all datasets, a maximum of 30 modes that account for at least 99% of the total variation are shown.
• Figure 5: Correspondence Quality: Predicted correspondence point for test meshes are overlaid over ground truth meshes for all methods and datasets.