# 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}= (•, 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*g that minimizes the energy

_{t}e

where gi is the endpoint of the flow of *u _{t}* given by

Fig. 2.16 Deformation fields *g _{t}* and deformed images

*g*at time slot t = 0, 0.2, 0.4, 0.6, 0.8, 1.0 in a 2D LDDMM image registration

_{t}l_{0}This means *I** _{0}* is smoothly deformed by

*g*

_{t}*, t*e [0,1] to

*I*

*Figure 2.16 shows an example of a sequence of diffeomorphic registration.*

_{1}.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)*

*, and :*

_{V}*_{XV}*V ! TV*is the fundamental vector field generated by д e g [85, 94].