Predicting Shape Development: a Riemannian Method

TL;DR

The paper addresses predicting future anatomical shape development from a single baseline by modeling shapes as points on a Riemannian shape space and learning population-level geodesic trends. It introduces a hierarchical geodesic prediction method that estimates a mean geodesic $\overline{\gamma}$ with start $\overline{\gamma}(0)$ and velocity $\overline{v}$, and predicts future shapes via $q^* = \exp_{p^*}(\Pi^{p^*}_{\overline{\gamma}(0)}(\overline{v}))$, with optional GAN-based refinements for local corrections. The method, validated on hippocampal shapes in Alzheimer's disease and on Dynamic FAUST motion data, outperforms Varifold-based approaches and competitive deep-learning variants while remaining interpretable and largely parameter-free. This approach is particularly effective for geodesic-like shape evolution and offers a pathway to integrate data-driven adjustments when baselines deviate from pure geodesics, enabling practical clinical prognosis and motion analysis.

Abstract

Predicting the future development of an anatomical shape from a single baseline observation is a challenging task. But it can be essential for clinical decision-making. Research has shown that it should be tackled in curved shape spaces, as (e.g., disease-related) shape changes frequently expose nonlinear characteristics. We thus propose a novel prediction method that encodes the whole shape in a Riemannian shape space. It then learns a simple prediction technique founded on hierarchical statistical modeling of longitudinal training data. When applied to predict the future development of the shape of the right hippocampus under Alzheimer's disease and to human body motion, it outperforms deep learning-supported variants as well as state-of-the-art.

Figures (4)

• Figure 1: Depiction of the shape prediction method in a shape space $M$.
• Figure 2: GAN-enhanced prediction method.
• Figure 3: Prediction comparison for a hippocampus from the AD group. Upper left: base shape at $t=0$; upper right: the same hippocampus at $t=1$, i.e., after one year (the ground truth for our prediction); bottom left: varifold-based prediction; bottom right: our prediction. The colors encode the vertex-wise differences to the ground truth according to the color map (0 mm 2.25 mm).
• Figure 4: Prediction of human body motion for FAUST dataset. From left to right: Baseline, ground truth, and predicted shapes using our and the Varifold method rekik2016predicting.