Anisotropic Distributions on Manifolds: Template Estimation and Most Probable Paths

• Published in: Information Processing in Medical Imaging Vol. 24; pp. 193 - 204
• Main Author: Sommer, Stefan
• Format: Book Chapter Journal Article
• Language: English
• Published: Cham Springer International Publishing 2015
• Series: Lecture Notes in Computer Science
• Subjects: Algorithms; Anisotropy; Computer Simulation; Data Interpretation, Statistical; Diffusion; Frame bundle; Geodesics; Image Enhancement - methods; Image Interpretation, Computer-Assisted - methods; Manifold; Models, Statistical; Most probable paths; Pattern Recognition, Automated - methods; Reproducibility of Results; Sensitivity and Specificity; Statistical Distributions; Template estimation
• ISBN: 9783319199917; 3319199919
• ISSN: 0302-9743; 1011-2499; 1611-3349
• DOI: 10.1007/978-3-319-19992-4_15

We use anisotropic diffusion processes to generalize normal distributions to manifolds and to construct a framework for likelihood estimation of template and covariance structure from manifold valued data. The procedure avoids the linearization that arise when first estimating a mean or template before performing PCA in the tangent space of the mean. We derive flow equations for the most probable paths reaching sampled data points, and we use the paths that are generally not geodesics for estimating the likelihood of the model. In contrast to existing template estimation approaches, accounting for anisotropy thus results in an algorithm that is not based on geodesic distances. To illustrate the effect of anisotropy and to point to further applications, we present experiments with anisotropic distributions on both the sphere and finite dimensional LDDMM manifolds arising in the landmark matching problem.