Metamorphosis and Schild's Ladder for One-Dimensional Shapes with Applications to the Classification of Cardiac Stem CellsMS31

Advances in deformable template matching has allowed us to investigate various different biological phenomena, from computational anatomy to kinematics. However, there is still more to be explored in what is capable by these models. In particular, the theory of metamorphosis, proposed by Younes, Trouve, and Miller, can be used to assess the similarity between two shapes by solving a variational problem that mixes the deforming of the domain and the modulating of the range of the shape template. This captures more modes of variation than the other pure deformation models, and can be used as a metric to define a Riemannian space of shapes, allowing for additional insights about shape similiarity. In this talk, I will discuss the foundations of metamorphosis in the one dimensional case, as well as provide two additional insights. The first is theory that allows us to solve for one of the variables in the alternating minimization in closed form. The second is a construction of Schild's ladder in the metamorphosis metric space, allowing us to perform parallel transport and adequately compare tangent vectors in the shape space. Additionally, I will show that by using these insights on the shape space of heart cell action potentials, one can suggest the phenotype (atrial vs ventricular) of a newly differentiated heart muscle cell, as well as determine whether a heart cell has a modification that makes it more/less susceptible to a given drug treatment.

This presentation is part of Minisymposium “MS31 - Variational Approaches for Regularizing Nonlinear Geometric Data (3 parts)
organized by: Martin Storath (Universität Heidelberg) , Martin Holler (École Polytechnique, Université Paris Saclay) , Andreas Weinmann (Hochschule Darmstadt) .

Rene Vidal (Johns Hopkins University)
Giann Gorospe (Johns Hopkins University)
computational anatomy, image registration, integral equations for image analysis, nonlinear optimization, partial differential equation models