Abstract
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.
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Notes
- 1.
We use the notation \(\tilde{p}(x_t)\) for path densities and p(y) for point densities.
- 2.
If M has more structure, e.g. being a Lie group or a homogeneous space, additional equivalent ways of defining Brownian motion and heat semi-groups exist.
- 3.
In both cases, the representation is not unique because different matrices W/frames \(X_\alpha \) can lead to the same diffusion. Instead, the MLE can be specified by the symmetric positive matrix \(\varSigma =WW^T\) or, in the manifold situation, be considered an element of the bundle of symmetric positive covariant 2-tensors.
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Sommer, S. (2015). Anisotropic Distributions on Manifolds: Template Estimation and Most Probable Paths. In: Ourselin, S., Alexander, D., Westin, CF., Cardoso, M. (eds) Information Processing in Medical Imaging. IPMI 2015. Lecture Notes in Computer Science(), vol 9123. Springer, Cham. https://doi.org/10.1007/978-3-319-19992-4_15
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