Symmetric Algorithmic Components for Shape Analysis with Diffeomorphisms

Abstract

In computational anatomy, the statistical analysis of temporal deformations and inter-subject variability relies on shape registration. However, the numerical integration and optimization required in diffeomorphic registration often lead to important numerical errors. In many cases, it is well known that the error can be drastically reduced in the presence of a symmetry. In this work, the leading idea is to approximate the space of deformations and images with a possibly non-metric symmetric space structure using an involution, with the aim to perform parallel transport. Through basic properties of symmetries, we investigate how the implementations of a midpoint and the involution compare with the ones of the Riemannian exponential and logarithm on diffeomorphisms and propose a modification of these maps using registration errors. This leads us to identify transvections, the composition of two symmetries, as a mean to measure how far from symmetric the underlying structure is. We test our method on a set of 138 cardiac shapes and demonstrate improved numerical consistency in the Pole Ladder scheme.

Learning Interactions Between Cardiac Shape and Deformation: Application to Pulmonary Hypertension

“…Further work will also extend the analysis to non-linear methods, which may better preserve the structure of the data spaces associated to each descriptor. It may also include recent developments for transporting spatiotemporal shape data among a population through di↵eomorphic transformations [22], which explicitly aims at preserving the structure of the space encoding cardiac meshes.…”

Numerical Accuracy of Ladder Schemes for Parallel Transport on Manifolds

“…They showed that at the first order, the constructions were equivalent. [28] latter gave a Taylor approximation of the elementary construction showing that each rung of pole ladder is a third order approximation of parallel transport, and additionally showed that PL is exact in affine (hence in Riemannian) locally symmetric spaces (the proof is reproduced in [11]). We first present the scheme and derive the Taylor approximation using the neighboring log.…”

“…It is not relevant to compare the PL with SL on spheres or SPD matrices as in the previous section, as theses spaces are symmetric and thus the PL is exact [11]. We therefore focus on the Lie group of isometries of R 3 , endowed with a left-invariant metric g with the diagonal matrix at identity:…”

Machine Learning Approaches for Myocardial Motion and Deformation Analysis

“…Temporal differences between sequences can also be normalized by resampling before the motion extraction [e.g., piece-wise linear interpolation (30)]. Recent approaches transport the whole subject-specific trajectory instead of the descriptors of interest, with specific computational considerations (31,32). Automatically estimating multiple templates across the sequence may also be well adapted to the cardiac circular/periodic dynamics (33).…”