Page - 419 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 425 Let M be themanifold M = {(p11, . . . ,pd1, . . . ,p1N, . . . ,pdN)|(p1i , . . . ,pdi ) ∈ Rd}. The LDDMM metric on the landmarkmanifoldM is directly related to thekernelKwhenwrittenas a cometric gp(ξ,η)=∑Ni,j=1ξ iK(pi,pj)ηj. Letting ikdenote the indexof thekthcomponentof the ith landmark, the cometric is in coordinates gi kjl p =K(pi,pj)lk. TheChristoffel symbols canbewritten in termsof derivativesof thecometricgij [31] (recall thatδij= g ikgkj= gjkgki) Γkij= 1 2 gir ( gklgrs,l−gslgrk,l−grlgks,l ) gsj . (18) This relation comes from the fact that gjm,k =−gjrgrs,kgsm gives the derivative of themetric. Thederivativesof thecometric is simplygi kjl ,rq =(δ i r+δ j r)∂pqr K(pi,pj)lk. Using(18),derivativesof the Christoffel symbolscanbecomputed Γkij,ξ = 1 2 gir,ξ ( gklgrs,l−gslgrk,l−grlgks,l ) gsj+ 1 2 gir ( gklgrs,l−gslgrk,l−grlgks,l ) gsj,ξ + 1 2 gir ( gkl,ξg rs ,l+g klgrs,lξ−gsl,ξgrk,l−gslgrk,lξ−grl,ξgks,l−grlgks,lξ ) gsj . Thisprovides the fulldata fornumerical integrationof theevolutionequationsonFkM. InFigure 7 (top row),weplotminimizingnormalMPPson the landmarkmanifoldwith two landmarksandvaryingcovariance in theR2 horizontal andverticaldirection. Theplot shows the landmarkequivalentof theexperiment inFigure5.Notehowaddingcovariance in thehorizontaland verticaldirection, respectively,allowstheminimizingnormalMPPtovarymore in thesedirections because theanisotropically-weightedmetricpenalizeshigh-covariancedirections less. Figure 7 (bottom row) showsfive curves satisfying theMPPequationswith varyingvertical V∗FM initialmomentumsimilarly to theplots inFigure6. Again,weseehowtheextradegreesof freedomallows thepaths to twist, generatingahigher-dimensional family thanclassicalgeodesics withrespect toC. (a) (b) (c) (d) (e) (f) (g) (h) (i) (j) Figure7. (Toprow)Matchingof twolandmarks (green) to twolandmarks (red)by (a) computinga minimizingRiemanniangeodesiconthe landmarkmanifold,and(b–e)minimizingMPPswithadded covariance (arrows) inR2 horizontaldirection(b,c) andvertical (d,e). Theactionof thecorresponding diffeomorphismsonaregulargrid isvisualizedbythedeformedgridwhich iscoloredbythewarp strain. Theaddedcovarianceallowsthepaths tohavemoremovement in thehorizontalandvertical direction,respectively,becausetheanisotropicallyweightedmetricpenalizeshigh-covariancedirections less. (bottomrow, (f–j))Five landmarktrajectorieswithfixedinitialvelocityandanisotropiccovariance butvaryingV∗FMvertical initialmomentumξ0. Changingtheverticalmomentum“twists” thepaths. 419