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}= (•, 0_{я}*х_{я} [94, 95].

LDDMM Diffeomorphic Registration

Given two images I_{0}, I e V, the objective of the diffeomorphic registration is to find a curve t ! u_{t} e g that minimizes the energy

where gi is the endpoint of the flow of u_{t} given by

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

This means I_{0} is smoothly deformed by g_{t}, t e [0,1] to I_{1}. Figure 2.16 shows an example of a sequence of diffeomorphic registration.

Computing the derivative of the matching energy E(u_{t}) should result in the optimal u_{t} 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].