Mostly Biomathematics Lunchtime Seminar
A Scalable Approach to Riemannian Geometric Statistics for Cardiovascular Shape Modeling
Speaker: Zan Ahmad, PhD Candidate - Department of Applied Mathematics and Statistics, Johns Hopkins University
Location: Warren Weaver Hall 1314
Date: Tuesday, April 21, 2026, 12:45 p.m.
Synopsis:
The geometry of cardiac chambers has been linked to outcomes such as arrhythmia, fibrosis, and stroke risk, yet there is no standardized way of incorporating this shape information into clinical decision making. At the same time, analyzing shape data presents a fundamental mathematical challenge: 3D surfaces cannot be meaningfully represented as points in a Euclidean vector space, and any informative comparison between a pair of shapes ideally should be invariant to parameterization, sampling, and discretization.
In this talk I will present a geometric framework for statistical shape analysis of cardiac surfaces that combines ideas from differential geometry, variational methods, and kernel-based representations. I will first review elastic shape analysis, in which surfaces are modeled as points on an infinite-dimensional manifold and distances are defined by the minimal deformation energy required to transform one shape into another. While this approach provides a principled geometric model, its computational cost limits its use on large imaging datasets.
I will then introduce a scalable alternative based on varifold representations and Sobolev-regularized gradients of a parameterization-invariant discrepancy. This construction yields low-dimensional, interpretable descriptors of surface shape that approximate the local geometry of shape space while scaling efficiently to large patient populations. Applications to cardiac imaging will illustrate how these descriptors enable clustering, statistical association analysis, and integration into predictive models. While we focus on examples of cardiac shape data, the methodologies that will be discussed are generalizable to many domains and data types.