Soliton Dynamics in Computational Anatomy

Abstract

Computational anatomy (CA) has introduced the idea of anatomical structures being transformed by geodesic deformations on groups of diffeomorphisms. Among these geometric structures, landmarks and image outlines in CA are shown to be singular solutions of a partial differential equation that is called the geodesic EPDiff equation. A recently discovered momentum map for singular solutions of EPDiff yields their canonical Hamiltonian formulation, which in turn provides a complete parameterization of the landmarks by their canonical positions and momenta. The momentum map provides an isomorphism between landmarks (and outlines) for images and singular soliton solutions of the EPDiff equation. This isomorphism suggests a new dynamical paradigm for CA, as well as new data representation.

published in NeuroImage

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Cited by in corpus (27)

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• Invariant higher-order variational problems II
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• Geodesic flows on semidirect-product Lie groups: geometry of singular measure-valued solutions
• Variational Principles for Stochastic Soliton Dynamics
• Smooth and Peaked Solitons of the CH equation
• On the Euler-Poincaré equation with non-zero dispersion
• Blow-up, zero $\alpha$ limit and the Liouville type theorem for the Euler-Poincaré equations
• On Noether's theorem for the Euler-Poincaré equation on the diffeomorphism group with advected quantities
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• Discrete momentum maps for lattice EPDiff
• Bayesian data assimilation in shape registration
• Interaction Dynamics of Singular Wave Fronts
• Diffusion bridges for stochastic Hamiltonian systems and shape evolutions
• PDE-constrained LDDMM via geodesic shooting and inexact Gauss-Newton-Krylov optimization using the incremental adjoint Jacobi equations
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• Solitary waves and $N$-particle algorithms for a class of Euler-Poincaré equations
• Long-time existence of Brownian motion on configurations of two landmarks
• Approximating the Weil-Petersson Metric Geodesics on the Universal Teichmüller space by Singular Solutions
• A class of fast geodesic shooting algorithms for template matching and its applications via the $N$-particle system of the Euler-Poincaré equations
• Geometric Discretization of the EPDiff Equations
• A landmark-based algorithm for automatic pattern recognition and abnormality detection
• On the non-integrability of the Popowicz peakon system