Estimating Diffeomorphic Mappings between Templates and Noisy Data: Variance Bounds on the Estimated Canonical Volume Form

TL;DR

This paper develops a method to estimate diffeomorphic mappings between templates and noisy data with reduced variance, using a least-action principle to eliminate unobservable stabilizer directions, and demonstrates improved accuracy over existing methods.

Contribution

It introduces a least-action principle approach to reduce variance in diffeomorphic mapping estimates, removing stabilizer subgroup effects and deriving Cramer-Rao bounds for small deformations.

Findings

Least-action principle reduces estimation variance.

Asymmetric LDDMM methods outperform symmetrized counterparts.

Derived analytical bounds for Jacobian variance in small deformations.

Abstract

Anatomy is undergoing a renaissance driven by availability of large digital data sets generated by light microscopy. A central computational task is to map individual data volumes to standardized templates. This is accomplished by regularized estimation of a diffeomorphic transformation between the coordinate systems of the individual data and the template, building the transformation incrementally by integrating a smooth flow field. The canonical volume form of this transformation is used to quantify local growth, atrophy, or cell density. While multiple implementations exist for this estimation, less attention has been paid to the variance of the estimated diffeomorphism for noisy data. Notably, there is an infinite dimensional un-observable space defined by those diffeomorphisms which leave the template invariant. These form the stabilizer subgroup of the diffeomorphic group acting on the template. The corresponding flat directions in the energy landscape are expected to lead to increased estimation variance. Here we show that a least-action principle used to generate geodesics in the space of diffeomorphisms connecting the subject brain to the template removes the stabilizer. This provides reduced-variance estimates of the volume form. Using simulations we demonstrate that the asymmetric large deformation diffeomorphic mapping methods (LDDMM), which explicitly incorporate the asymmetry between idealized template images and noisy empirical images, provide lower variance estimators than their symmetrized counterparts (cf. ANTs). We derive Cramer-Rao bounds for the variances in the limit of small deformations. Analytical results are shown for the Jacobian in terms of perturbations of the vector fields and divergence of the vector field.