Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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Estimating diffeomorphic mappings between templates and noisy data: Variance bounds on the estimated canonical volume form


Authors: Daniel J. Tward, Partha P. Mitra and Michael I. Miller
Journal: Quart. Appl. Math. 77 (2019), 467-488
MSC (2010): Primary 92C55, 49K40
DOI: https://doi.org/10.1090/qam/1527
Published electronically: November 20, 2018
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Abstract | References | Similar Articles | Additional Information

Abstract: Anatomy is undergoing a renaissance driven by the 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 unobservable 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.


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Additional Information

Daniel J. Tward
Affiliation: Center for Imaging Science, Johns Hopkins University, Baltimore, Maryland 21218
Email: dtward@cis.jhu.edu

Partha P. Mitra
Affiliation: Cold Spring Harbor Laboratory, Cold Spring Harbor, New York 11724
Email: mitra@cshl.edu

Michael I. Miller
Affiliation: Department of Biomedical Engineering, Johns Hopkins University, Baltimore, Maryland 21218
Email: mim@cis.jhu.edu

DOI: https://doi.org/10.1090/qam/1527
Keywords: Computational anatomy, morphometry, cell density, Hamiltonian dynamics
Received by editor(s): September 26, 2018
Received by editor(s) in revised form: October 8, 2018
Published electronically: November 20, 2018
Additional Notes: This work was supported by the National Institutes of Health [P41-EB015909, R01-EB020062, R01-NS102670, R01-MH105660, U19-AG033655, U19-MH114821, U01-MH114824]; National Science Foundation 16-569 NeuroNex contract 1707298; the Kavli Neuroscience Discovery Institute; the Crick-Clay Professorship, CSHL; and the H N Mahabala Chair, IIT Madras.
Dedicated: Twenty years after Computational Anatomy was hatched by Ulf Grenander and Michael Miller at the Division of Applied Mathematics at Brown University, we revisit the core geodesic equations and nuisance non-identifiable stabilizing subgroup of the deformable template model examining the variance in estimating the fundamental form and associated Cramer-Rao Bound in the small deformation limit. This picks up on a tradition started by Ulf’s thesis advisor, which we would like to think Ulf would have appreciated.
Article copyright: © Copyright 2018 Brown University