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LDDMM Framework for Registration

General Setting

The LDDMM approach models CA as a deformation of an initial template image I e V by diffeomorphic transformations g e G, where V = {I : Q ! M} is a vector space of images with domain Q and G is a Lie subgroup of the diffeomorphism group Diff (Q) on Q with Lie algebra g. The deformation of an image I e V by a diffeomorphic transformation g e G is defined by a smooth map:

Inner products of V and g are also defined as V and g as < •, • >V = (•, -)V*xV and < •, • >0= (•, 0яя [94, 95].

LDDMM Diffeomorphic Registration

Given two images I0, I e V, the objective of the diffeomorphic registration is to find a curve t ! ut e g that minimizes the energy

where gi is the endpoint of the flow of ut given by

Deformation fields g and deformed images gl at time slot t = 0, 0.2, 0.4, 0.6, 0.8, 1.0 in a 2D LDDMM image registration

Fig. 2.16 Deformation fields gt and deformed images gtl0 at time slot t = 0, 0.2, 0.4, 0.6, 0.8, 1.0 in a 2D LDDMM image registration

This means I0 is smoothly deformed by gt, t e [0,1] to I1. Figure 2.16 shows an example of a sequence of diffeomorphic registration.

Computing the derivative of the matching energy E(ut) should result in the optimal ut satisfying

where ж = ^(gi/o — /i)b and the b-map on a vector space V is defined by

The momentum map o:T * V ! g* is defined by (I о ж, д)д* X0 = (ж, I)V*XV, and : V ! TV is the fundamental vector field generated by д e g [85, 94].

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