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The Euler-Poincar\'e (EP) equations describe the geodesic motion on the
diffeomorphism group. For template matching (template deformation), the
Euler-Lagrangian equation, arising from minimizing an energy function, falls
into the Euler-Poincar\'e theory and can be recast into the EP equations. By
casting the EP equations in the Lagrangian (or chara...
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Citations
... Moreover, if the landmark representation is used, it is natural to use the particle formulation of the EP equations to construct a matching algorithm. Recently a diffeomorphic template matching algorithm that takes advantage of the landmark representation and the particle formulation of the EP equation was introduced by Camassa et al. [6, 7, 25]. We give a brief introduction of this geodesic shooting algorithm in Section 2. Using this as the underlying warping algorithm, in this paper we propose a landmark-based algorithm for finding a robust estimator as an average of a group on a general Riemannian manifold. ...
... This yields a conic shape kernel that has advantages for template matching as discussed in [7]; we will use this later in our numerical experiments. The length of a curve φ : [0, 1] → G is defined by using the right translation ...
... For our applications, landmark positions {x i = q i } N i=1 at t = 0 and t = 1 are known, while the initial momenta {p i } N i=1 at t = 0 are not. A shooting algorithm that searches for the initial momenta is introduced in [7]. In that algorithm, a prediction-correction step: ...
We study a class of mathematical and statistical algorithms with the aim of establishing a computer-based framework for fast and reliable automatic pattern recognition and abnormality detection. Under this framework, we propose a numerical algorithm for finding group averages where an average of a group is an estimator that is said to best represent the properties of interest of that group. A novelty of the proposed landmark-based algorithm is that the algorithm tracks information of the momentum field through the geodesic shooting process. The momentum field provides a local template-based coordinate system and is linear in nature. It is also a dual of the velocity field with respect to an assigned base template, yielding advantages for statistical analyses. We apply this framework to a small brain image database for detecting structure abnormality. The brain structure changes identified by our framework are highly consistent with studies in the literature.
