Abstract
With systems for acquiring shape data becoming evermore commonplace, the capacity to design, process, and analyze this information forms the fundamental building blocks of many applications across a wide range of specialisms from computer vision to biology and medicine. We introduce a novel unified probabilistic shape modelling framework, which leverages the concept of Differential Physical models to address common deficiencies which exist within current solutions to a range of problems within this field.Our approach is grounded in the study of flows, which are akin to the equations of motion used in fluid dynamics. Rather than directly modelling the relationship between the shapes in Euclidean space, the central idea is to consider the space of plausible deformations, i.e. diffeomorphisms, which are bijective and are guaranteed to be free from self-intersections. We compute diffeomorphic transformations by modelling it as a smooth time-dependent flow field, which is governed by Ordinary Differential Equation (ODE). A smooth Gaussian process is used to solve the non-stationary ODE flow equation, which provides a powerful paradigm connecting modern machine learning to classical differential equations.
We are readily able to accommodate hard constraints, such as volume preservation, as well as selective priors. The evolution of surface properties under the flow field can be calculated without the use of discrete geometric calculations. Formulating the problem in terms of a Bayesian non-parametric framework allows our approach to assess and propagate uncertainty, which is critically important in many tasks that utilise shape models.
We illustrate the versatility of this framework by demonstrating solutions to a number of quintessential tasks, including plausible shape interpolation, surface fairing, and fitting parametric surfaces to noisy data. Leveraging this knowledge, we ultimately propose a generative approach to statistical shape modelling, which allows the inclusion of priors and soft constraints. The latter are incorporated through functional objectives on the field or surface (e.g. bending energy or path lengths); which ensure smooth generation to interpolate between, and extrapolate from, the training examples without the need for post-processing or subsequent calculations at test time. Unlike existing approaches, it does not require prior knowledge of exact landmarks, nor relies on a hand-crafted template or arbitrarily promotes a single training example as a reference. Instead, our formulation treats the data in a fully group-wise manner and allows automatic inference of a common template from the data.
| Date of Award | 8 Oct 2025 |
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| Original language | English |
| Awarding Institution |
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| Supervisor | Neill Campbell (Supervisor) & Darren Cosker (Supervisor) |
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