Page - 418 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 425 (d) (e) (f) (g) (h) (i) (j) (k) (l) Figure6. (a–l)With thesetupofFigures4and5,generatedfamiliesofcurvesbyvaryingthevertical V∗FM part of the initialmomentum ξ0 ∈ T∗FM but keeping the base point and frame u0 fixed. Theverticalpartallowsvaryingdegreeof“twisting”of thecurve. 6.2. LDDMMLandmarkEquations Weheregiveaexampleof theMPPequationsusingthefinitedimensional landmarkmanifolds thatarisefromrightinvariantmetricsonsubsetsofthediffeomorphismgroupintheLargeDeformation DiffeomorphicMetricMapping(LDDMM)framework[8]. TheLDDMMmetriccanbeconveniently expressedas a cometric, and, using a rank-deficient innerproduct gFkM asdiscussed in Section 5, wecanobtainareductionof thesystemofequations to2(2N+2Nk)comparedto2(2N+(2N)2)with N landmarks inR2. Let {p1, . . . ,pN} be landmarks in a subsetΩ ⊂ Rd. The diffeomorphism group Diff(Ω) acts on the left on landmarkswith the action ϕ.{p1, . . . ,pN} = {ϕ(p1), . . . ,ϕ(pN)}. In LDDMM, aHilbert spacestructure is imposedona linearsubspaceVofL2(Ω,Rd)usingaself-adjointoperator L :V→V∗⊂L2(Ω,Rd)anddefiningthe innerproduct 〈·, ·〉V by 〈v,w〉V= 〈Lv,w〉L2 . Under sufficient conditions on L,V is reproducing and admits a kernelK inverse to L. K is aGreen’skernelwhenL isadifferentialoperator,orK canbeaGaussiankernel. TheHilbert structure onVgivesaRiemannianmetriconasubsetGV⊂Diff(Ω)bysetting‖v‖2ϕ=‖v◦ϕ−1‖2V; i.e., regarding 〈·, ·〉V aninnerproductonTIdGV andextendingthemetric toGV byright-invariance. ThisRiemannian metricdescends toaRiemannianmetriconthe landmarkspace. 418