Overview on Geometries of Shape Spaces, Diffeomorphism Groups, and - PowerPoint PPT Presentation

Overview on Geometries of Shape Spaces, Diffeomorphism Groups, and Spaces of Riemannian Metrics Peter W. Michor University of Vienna, Austria Workshop: Geometric Models in Vision Institut Henri Poincar´ e October 22-24, 2014 Based on collaborations with: M. Bauer, M. Bruveris, P. Harms, D. Mumford

◮ A diagram of actions of diffeomorphism groups ◮ Riemannian geometries of spaces of immersions and shape spaces. ◮ A zoo of diffeomorphism groups on R n ◮ Right invariant Riemannian geometries on Diffeomorphism groups. ◮ Robust Infinite Dimensional Riemannian manifolds, Sobolev Metrics on Diffeomorphism Groups, and the Derived Geometry of Shape Spaces.

� �� � � � � � � � A diagram of actions of diffeomorphism groups. r-acts l-acts Diff( M ) Imm( M , N ) Diff A ( N ) (LDDMM) r-acts l-acts needs ¯ g Diff( M ) (LDDMM) Met( M ) B i ( M , N ) Diff( M ,µ ) r-acts Diff( M ) needs ¯ g � � Vol 1 Met( M ) + ( M ) Met A ( N ) Diff( M ) M compact , N pssibly non-compact manifold Met( N ) = Γ( S 2 + T ∗ N ) space of all Riemann metrics on N ¯ g one Riemann metric on N Diff( M ) Lie group of all diffeos on compact mf M Diff A ( N ) , A ∈ { H ∞ , S , c } Lie group of diffeos of decay A to Id N Imm( M , N ) mf of all immersions M → N B i ( M , N ) = Imm / Diff( M ) shape space Vol 1 + ( M ) ⊂ Γ(vol( M )) space of positive smooth probability densities

� � � � � � �� � � � � r-acts l-acts Diff( S 1 ) Imm( S 1 , R 2 ) Diff A ( R 2 ) (LDDMM) r-acts l-acts r-acts needs ¯ Diff( S 1 ) g (LDDMM) Vol + ( S 1 ) Met( S 1 ) B i ( S 1 , R 2 ) r-acts Diff( S 1 ) needs ¯ g � � � fd θ Vol + ( S 1 ) Met( S 1 ) = � R > 0 Met( R 2 ) � √ gd θ Diff( S 1 ) = Diff( S 1 ) Diff( S 1 ) Lie group of all diffeos on compact mf S 1 Diff A ( R 2 ) , A ∈ {B , H ∞ , S , c } Lie group of diffeos of decay A to Id R 2 mf of all immersions S 1 → R 2 Imm( S 1 , R 2 ) B i ( S 1 , R 2 ) = Imm / Diff( S 1 ) shape space Vol + ( S 1 ) = � f d θ : f ∈ C ∞ ( S 1 , R > 0 ) � space of positive smooth probability densities g d θ 2 : g ∈ C ∞ ( S 1 , R > 0 ) Met( S 1 ) = � � space of metrics on S 1

� � � The manifold of immersions Let M be either S 1 or [0 , 2 π ]. Imm( M , R 2 ) := { c ∈ C ∞ ( M , R 2 ) : c ′ ( θ ) � = 0 } ⊂ C ∞ ( M , R 2 ) . The tangent space of Imm( M , R 2 ) at a curve c is the set of all vector fields along c :   T R 2       T c Imm( M , R 2 ) = ∼ = { h ∈ C ∞ ( M , R 2 ) } h h : π    c  R 2  M  Some Notation: v ( θ ) = c ′ ( θ ) 1 ds = | c ′ ( θ ) | d θ, | c ′ ( θ ) | , n ( θ ) = iv ( θ ) , D s = | c ′ ( θ ) | ∂ θ

� Inducing a metric on shape space Imm( M , N ) π B i := Imm( M , N ) / Diff( M ) Every Diff( M )-invariant metric ”above“ induces a unique metric ”below“ such that π is a Riemannian submersion.

Inner versus Outer

The vertical and horizontal bundle ◮ T Imm = Vert � Hor. ◮ The vertical bundle is Vert := ker T π ⊂ T Imm . ◮ The horizontal bundle is Hor := (ker T π ) ⊥ , G ⊂ T Imm . It might not be a complement - sometimes one has to go to the completion of ( T f Imm , G f ) in order to get a complement.

The vertical and horizontal bundle

� Definition of a Riemannian metric 1. Define a Diff( M )-invariant metric G Imm( M , N ) on Imm. π 2. If the horizontal space is a B i ( M , N ) complement, then T π restricted to the horizontal space yields an isomorphism (ker T f π ) ⊥ , G ∼ = T π ( f ) B i . Otherwise one has to induce the quotient metric, or use the completion. 3. This gives a metric on B i such that π : Imm → B i is a Riemannian submersion .

� Riemannian submersions Imm( M , N ) π B i := Imm( M , N ) / Diff( M ) ◮ Horizontal geodesics on Imm( M , N ) project down to geodesics in shape space. ◮ O’Neill’s formula connects sectional curvature on Imm( M , N ) and on B i . [Micheli, M, Mumford, Izvestija 2013]

� � � � � � L 2 metric � G 0 c ( h , k ) = � h ( θ ) , k ( θ ) � ds . M Problem: The induced geodesic distance vanishes. r-acts l-acts Diff( S 1 ) Imm( S 1 , R 2 ) Diff c ( R 2 ) (LDDMM) l-acts r-acts needs ¯ g Diff( S 1 ) (LDDMM) Met( S 1 ) B i ( S 1 , R 2 ) Movies about vanishing: Diff( S 1 ) Imm( S 1 , R 2 ) [MichorMumford2005a,2005b], [BauerBruverisHarmsMichor2011,2012]

The simplest ( L 2 -) metric on Imm( S 1 , R 2 ) � � G 0 c ( h , k ) = S 1 � h , k � ds = S 1 � h , k �| c θ | d θ Geodesic equation � | c t | 2 c θ 1 1 � c tt = − 2 | c θ | ∂ θ − | c θ | 2 � c t θ , c θ � c t . | c θ | A relative of Burger’s equation. Conserved momenta for G 0 along any geodesic t �→ c ( , t ): � v , c t �| c θ | 2 ∈ X ( S 1 ) reparam. mom. � S 1 c t ds ∈ R 2 linear moment. � S 1 � Jc , c t � ds ∈ R angular moment.

Horizontal Geodesics for G 0 on B i ( S 1 , R 2 ) � c t , c θ � = 0 and c t = an = aJ c θ | c θ | for a ∈ C ∞ ( S 1 , R ). We use functions a , s = | c θ | , and κ , only holonomic derivatives: a t = 1 2 κ a 2 , s t = − a κ s , κ t = a κ 2 + 1 � a θ θ = a κ 2 + a θθ s 2 − a θ s θ � s 3 . s s We may assume s | t =0 ≡ 1. Let v ( θ ) = a (0 , θ ), the initial value for a . Then s t s = − a κ = − 2 a t a , so log( sa 2 ) t = 0, thus s ( t , θ ) a ( t , θ ) 2 = s (0 , θ ) a (0 , θ ) 2 = v ( θ ) 2 , a conserved quantity along the geodesic. We substitute s = v 2 a 2 and κ = 2 a t a 2 to get

− a 5 a 2 a tt − 4 a 2 a − a 6 a θθ 2 v 4 + a 6 a θ v θ t θ = 0 , v 5 v 4 a (0 , θ ) = v ( θ ) , a nonlinear hyperbolic second order equation. Note that wherever v = 0 then also a = 0 for all t . So substitute a = vb . The outcome is ( b − 3 ) tt = − v 2 2 ( b 3 ) θθ − 2 vv θ ( b 3 ) θ − 3 vv θθ b 3 , 2 b (0 , θ ) = 1 . This is the codimension 1 version where Burgers’ equation is the codimension 0 version.

Weak Riem. metrics on Emb( M , N ) ⊂ Imm( M , N ). Metrics on the space of immersions of the form: � g ( P f h , k ) vol( f ∗ ¯ G P f ( h , k ) = ¯ g ) M g is some fixed metric on N , g = f ∗ ¯ where ¯ g is the induced metric on M , h , k ∈ Γ( f ∗ TN ) are tangent vectors at f to Imm( M , N ), and P f is a positive, selfadjoint, bijective (scalar) pseudo differential operator of order 2 p depending smoothly on f . Good example: P f = 1 + A (∆ g ) p , where ∆ g is the Bochner-Laplacian on M induced by the metric g = f ∗ ¯ g . Also P has to be Diff( M )-invariant: ϕ ∗ ◦ P f = P f ◦ ϕ ◦ ϕ ∗ .

Elastic metrics on plane curves Here M = S 1 or [0 , 1 π ], N = R 2 . The elastic metrics on Imm( M , R 2 ) is � 2 π G a , b a 2 � D s h , n �� D s k , n � + b 2 � D s h , v �� D s k , v � ds , ( h , k ) = c 0 with P a , b ( h ) = − a 2 � D 2 s h , n � n − b 2 � D 2 s h , v � v c + ( a 2 − b 2 ) κ � � � D s h , v � n + � D s h , n � v a 2 � n , D s h � n + b 2 � v , D s h � v � � + ( δ 2 π − δ 0 ) .

� � � � � � Sobolev type metrics Advantages of Sobolev type metrics: 1. Positive geodesic distance 2. Geodesic equations are well posed 3. Spaces are geodesically complete for p > dim( M ) + 1. 2 [Bruveris, M, Mumford, 2014] for plane curves. A remark in [Ebin, Marsden, 1970], and [Bruveris, Meyer, 2014] for diffeomorphism groups. Problems: 1. Analytic solutions to the geodesic equation? 2. Curvature of shape space with respect to these metrics? 3. Numerics are in general computational expensive p ≥ 1 / 2 p ≥ 1 p ≥ 1 wellp.: r-acts l-acts Diff( S 1 ) Imm( S 1 , R 2 ) Diff c ( R 2 ) Space: (LDDMM) geod. dist.: +: p > 1 2 , − : p ≤ 1 − : p =0 , +: p ≥ 1 − : p < 1 2 , +: p ≥ 1 2 l-acts r-acts needs ¯ g Diff( S 1) (LDDMM) p ≥ 0 p ≥ 1 wellp.: Met( S 1 ) B i ( S 1 , R 2 ) Space: geod. dist.: +: p ≥ 0 − : p =0 , +: p ≥ 1

� � � � � � Sobolev type metrics Advantages of Sobolev type metrics: 1. Positive geodesic distance 2. Geodesic equations are well posed 3. Spaces are geodesically complete for p > dim( M ) + 1. 2 [Bruveris, M, Mumford, 2013] for plane curves. A remark in [Ebin, Marsden, 1970], and [Bruveris, Meyer, 2014] for diffeomorphism groups. Problems: 1. Analytic solutions to the geodesic equation? 2. Curvature of shape space with respect to these metrics? 3. Numerics are in general computational expensive wellp.: p ≥ 1 p ≥ 1 p ≥ 1 r-acts l-acts Space: Diff( M ) Imm( M , N ) Diff c ( N ) (LDDMM) dist.: +: p > 1 , − : p < 1 − : p =0 , +: p ≥ 1 − : p < 1 2 , +: p ≥ 1 2 l-acts r-acts needs ¯ g Diff( M ) (LDDMM) wellp.: p = k , k ∈ N p ≥ 1 Space: Met( M ) B i ( M , N ) dist.: +: p ≥ 0 − : p =0 , +: p ≥ 1

Geodesic equation. The geodesic equation for a Sobolev-type metric G P on immersions is given by ∇ ∂ t f t =1 2 P − 1 � Adj( ∇ P )( f t , f t ) ⊥ − 2 . Tf . ¯ g ( Pf t , ∇ f t ) ♯ � g ( Pf t , f t ) . Tr g ( S ) − ¯ − P − 1 � � ( ∇ f t P ) f t + Tr g � � g ( ∇ f t , Tf ) ¯ Pf t . The geodesic equation written in terms of the momentum for a Sobolev-type metric G P on Imm is given by:  p = Pf t ⊗ vol( f ∗ ¯ g )    ∇ ∂ t p = 1  Adj( ∇ P )( f t , f t ) ⊥ − 2 Tf . ¯ g ( Pf t , ∇ f t ) ♯ � 2   g ( Pf t , f t ) Tr f ∗ ¯ g ( S )  � ⊗ vol( f ∗ ¯ − ¯ g ) 