Learning the Geometry of Data: A Mathematical Review of Shape Space Analysis
arXiv:2606. 17022v1 Announce Type: cross Abstract: A central objective of machine learning is to identify structure and patterns in data.
arXiv:2602. 11467v2 Announce Type: replace Abstract: Understanding how anatomical shapes evolve in response to developmental covariates - and quantifying their spatially varying uncertainties - is critical in healthcare research.
arXiv:2606. 17022v1 Announce Type: cross Abstract: A central objective of machine learning is to identify structure and patterns in data.
arXiv:2606. 15837v1 Announce Type: cross Abstract: Deep neural networks (DNNs) frequently fail to generalize to out-of-distribution (OOD) medical images because of variations in scanners and acquisition protocols.
arXiv:2603. 04024v2 Announce Type: replace-cross Abstract: Ambiguous 3D medical image segmentation often involves boundaries where different expert delineations are non-identical yet clinically plausible.
arXiv:2607. 19600v1 Announce Type: cross Abstract: The shape of a planar curve is the geometric information that remains once translation, rotation, scale and reparametrisation are removed and is of interest in many health applications, e.
arXiv:2608. 02306v1 Announce Type: cross Abstract: We introduce a mathematical framework for shape comparison based on mapping functions from the shape domain to a common reference domain.
arXiv:2608. 09182v1 Announce Type: cross Abstract: Accurate landmark localization in medical images is a fundamental step for quantitative clinical measurement and downstream analysis.
arXiv:2608. 14766v1 Announce Type: cross Abstract: Uncertainty estimation is critical for the safe clinical deployment of deep learning in medical image segmentation, with aleatoric uncertainty theoretically designed to capture irreducible data ambiguity.
arXiv:2608. 09938v1 Announce Type: cross Abstract: Hyperelastic deformations are highly sensitive to domain geometry and boundary conditions, making generalization across both a critical capability for neural operators applied to these problems.
arXiv:2608. 00187v1 Announce Type: cross Abstract: Correspondence-based statistical shape modeling (SSM) is vital for population-level morphometric analysis, but conventional pipelines assume clean, fully registered surfaces.
We introduce SMART, a framework for learning a flexible, interpretable, and scalable spatio-temporal brain atlas from longitudinal high-resolution 3D medical images. Existing approaches to spatio-temporal atlas construction rely on black-box generative models that lack flexibility, limit interpretability, and struggle to scale to high-dimensional data.
Optimizing 3D shapes within the latent spaces of deep generative models is fundamental to computer assisted engineering, yet remains prone to a critical failure mode we term manifold drift: the tendency of gradient-based optimization to move latent vectors away from the manifold of valid shapes. This problem is exacerbated in state-of-the-art 3D shape generative models that operate in increasingly high-dimensional latent spaces where valid shapes occupy a vanishingly small fraction of the full space.
arXiv:2608. 09460v1 Announce Type: new Abstract: Cardiac digital twin research is moving from subject-specific anatomical replicas toward virtual cohorts that represent clinically relevant population subgroups.