Diffeomorphic Gaussian process shape models and approximation of measure-based shape representations

: (Alternative Format Thesis)

Abstract

This thesis contributes novel theoretical and algorithmic research to diffeomorphic shape modelling, randomised approximation of geometric measure-based shape representations, and shape modelling for the detection of dysplasia of the hip. The chapters of this thesis consist of multiple papers written up for publication, both in preparation and under review.

Our first contribution is presented in Chapter 3, where we develop a novel Gaussian process diffeomorphic statistical shape model (GPDSSM). This work combines multi-scale momentum parametrisations from Large Deformation Diffeomorphic Metric Mapping (LDDMM)theory and the nonparametric Bayesian variational Gaussian process latent variable model (GPLVM). The resulting model features a rigorous variational Bayesian inference framework and is suited to learning from small shape datasets with diverse variation, owing to its topological guarantees, flexibility, and ability to account for both model and data uncertainties in fitting and inference.

The next novel contribution of this work is contained in Chapters 4 and 4A, which introduces a novel theoretical approximation and practical compression algorithm for the embedded currents, varifolds, and normal cycle representations of shape. In particular, through randomised orthogonal projections and ridge leverage score (RLS) sampling of approximating control points, making use of results from the kernel learning literature. The algorithms developed in this work are shown to greatly enhance the practical performance and runtime of LDDMM registration pipelines on real-world large-scale surface data.

Use of randomly sampled control points is not only effective for approximating geometric measures, but also for approximating dense diffeomorphisms on the domain. Chapter 5 contains our contributions to the derivation of novel a priori approximation error bounds for the widely used control-point LDDMM framework. This work has further led to practically impactful initialisation schemes
for both the number and position of spatial control points, which is a key choice to be made in this setting.

Finally, Chapter 6 contains a collaborative work that applies the model of Chapter 3 to modelling shape variation in the acetabular surfaces of the pelvis, and automatically identifying patients with hip dysplasia, a significant risk factor for hip osteoarthritis. Our method is shown to outperform traditional clinical angle based methods in terms of detection accuracy and has the potential to accelerate the diagnostic process, saving valuable time for clinicians.

Date of Award	25 Mar 2026
Original language	English
Awarding Institution	University of Bath
Supervisor	Tony Shardlow (Supervisor) & Neill Campbell (Supervisor)