Stochastic Solitons in Computational Anatomy Darryl D Holm Imperial College Vienna, 20 Feb 2015 Darryl D Holm Imperial College () Stochastic Solitons in Computational Anatomy Vienna, 20 Feb 2015 1 / 48
Organization of the talk Review: Momentum in images 1 Peakon momentum maps in 1D and 2D 2 Statistical models 3 Fokker-Planck equations for peakons and pulsons 4 Numerical experiments for stochastic landmark motion 5 Stratonovich and Itˆ o Stochastic Euler-Poincar´ e equations 6 Summary 7 Darryl D Holm Imperial College () Stochastic Solitons in Computational Anatomy Vienna, 20 Feb 2015 2 / 48
Review: Dynamics of ‘shapes’ C 1 ( S 1 , R 2 ) Darryl D Holm Imperial College () Stochastic Solitons in Computational Anatomy Vienna, 20 Feb 2015 3 / 48
Review: Momentum in images Most problems in CA can be formulated as finding the time-dependent deformation map ϕ t : M → M with minimal geodesic cost , defined by � 1 � 1 ℓ ( u t ) dt = 1 d ϕ t � u t � 2 Cost ( t �→ ϕ t ) = X ( M ) dt , with = u t ◦ ϕ t 2 dt 0 0 under the constraint that the map ϕ t carries a template I 0 at t = 0 to the target I 1 at t = 1 and � · � X ( M ) is a given Riemannian metric. The variable u t ∈ X ( M ) is called the (Eulerian) velocity , and m t := δℓ � u t � 2 = Lu t for X ( M ) = � Lu t , u t � δ u t is called the momentum for L 2 pairing � · , · � and positive symmetric operator L : X ( M ) → X ( M ) ∗ . Darryl D Holm Imperial College () Stochastic Solitons in Computational Anatomy Vienna, 20 Feb 2015 4 / 48
Clebsch approach ⇒ landmark momentum dynamics Consider Hamilton’s principle for a Lagrangian ℓ ( u ) : X ( M ) → R . Constrain HP by the action of vector fields u ∈ X ( M ) as ˙ q ( t ) = u ( q , t ) for q ∈ M . For N landmarks, q ( t ) = { q a ( t ) , a = 1 , 2 , . . . , N } , we take HP as � �� � 1 � � N p , ˙ 0 = δ S = δ ℓ ( u ) + q − u ( q , t ) dt 0 Stationarity of HP leads to the following equations of motion [3], N � T p = − du m ( x , t ) := δℓ ˙ ˙ q = u ( q , t ) , · p , δ u = p δ ( x − q ) . dq The 1st two eqns imply that the momentum m ( x , t ) evolves by the EPDiff equation, u m = − u · ∇ m − ( ∇ u ) T m − m ( div u ) . ∂ t m = − ad ∗ Darryl D Holm Imperial College () Stochastic Solitons in Computational Anatomy Vienna, 20 Feb 2015 5 / 48
Peakons: m ( x , t ) = p δ ( x − q ) , embeddings C 1 ( Z , R ) � u 2 + u 2 When ℓ ( u ) = 1 2 � u � 2 H 1 = 1 x dx for M = R , then m = u − u xx 2 and ∂ t m = − ( um ) x − mu x produces the 1D solitons called ‘peakons’. Singular peakon (landmark) solutions emerge from smooth initial conditions and form a finite dimensional solution set for EPDiff( H 1 ). Darryl D Holm Imperial College () Stochastic Solitons in Computational Anatomy Vienna, 20 Feb 2015 6 / 48
EPDiff( H 1 ) embeddings C 1 ([ 0 , 1 ] , R 2 ) Here, we have the EPDiff equation with u ∈ X ( R 2 ) , ℓ : X ( R 2 ) → R � N � � � d δℓ δℓ p , ˙ 0 = δ S = δ ℓ ( u ) + q − u ( q , t ) dt , = ⇒ δ u + ad ∗ δ u = 0 . u dt Darryl D Holm Imperial College () Stochastic Solitons in Computational Anatomy Vienna, 20 Feb 2015 7 / 48
Definition: Momentum maps and their evolution Smooth invertible maps ϕ t act on the symplectic manifold T ∗ M by flows of cotangent lifts of vector fields u t = d ϕ t dt ◦ ϕ − 1 ∈ X acting on M . t ϕ t ✲ T ∗ M T ∗ M Equivariant m 0 Momentum Map m t ❄ ❄ Ad ∗ ϕ − 1 ✲ X ∗ X ∗ t The associated momap m : T ∗ M → X ∗ evolves via the EPDiff eqn dt = − ad ∗ dm u m for m = δℓ δ u ∈ X ∗ Darryl D Holm Imperial College () Stochastic Solitons in Computational Anatomy Vienna, 20 Feb 2015 8 / 48
Momentum maps for C 1 embeddings C 1 ( S 1 , R 2 ) Embeddings C 1 ( S 1 , R 2 ) admit a dual pair of momaps Momaps recast processes of shape change and reparametrization as: Right & Left group reductions by Diff ( S 1 ) and Diff ( R 2 ) , respectively, of the canonical Hamiltonian motion on T ∗ C 1 ( S 1 , R 2 ) . Embedding phase space T ∗ C 1 ( S 1 , R 2 ) � ❅ � ❅ Left (changes shape) J Sing J S Right (preserves shape) � ❅ ✠ � ❅ ❘ X ( S 1 ) ∗ (Reparameterizes) (Changes shape) X ( R 2 ) ∗ DDH and JE Marsden Momentum maps and measure valued solutions of the Euler-Poincar´ e equations for the diffeomorphism group Progr Math , 232 203-235 (2004) eprint arXiv:nlin.CD/0312048 Darryl D Holm Imperial College () Stochastic Solitons in Computational Anatomy Vienna, 20 Feb 2015 9 / 48
These momaps help quantify differences in shapes ... via geodesics along diffeomorphisms that map one shape to another; e.g. when “shape” is defined as a closed planar curve, that is, as a C 1 ( S 1 , R 2 ) embedding. Darryl D Holm Imperial College () Stochastic Solitons in Computational Anatomy Vienna, 20 Feb 2015 10 / 48
Left and right transformations of q ∈ C 1 ( S 1 , R 2 ) The Right & Left actions of Lie groups Diff + ( S 1 ) & Diff + ( R 2 ) , respectively, on q ∈ C 1 ( S 1 , R 2 ) commute with each other. The Left action of Diff + ( R 2 ) on q ∈ C 1 ( S 1 , R 2 ) : g ∈ Diff + ( R 2 ) , q �→ L g q = g ◦ q , changes shape : it transforms between inequivalent curves. The Right action of Diff + ( S 1 ) on q ∈ ( S 1 , R 2 ) : η ∈ Diff + ( S 1 ) , q �→ R η q = q ◦ η, preserves shape , which defines an equivalence class of curves. Namely, the ‘shape space’ of C 1 embeddings modulo relabelling, C 1 ( S 1 , R 2 ) / Diff + ( S 1 ) Darryl D Holm Imperial College () Stochastic Solitons in Computational Anatomy Vienna, 20 Feb 2015 11 / 48
Left action (motion) has a singular momentum map MM for left action of Diff + ( R 2 ) on q ∈ C 1 ( S 1 , R 2 ) is singular [3] � � � m = L u ( x ) = S 1 p ( s ) δ x − q ( s ) d s =: J Sing ( q , p ) . This MM obeys the EP equation and is preserved by right action. The paths q ( s ) and their canonical momenta p ( s ) are governed by the canonical Hamiltonian equations for H = 1 2 � m , K ∗ m � T p ( s ) = − ∂ u ˙ ˙ q ( s ) = u ( q ( s )) and · p ( s ) ∂ q The reduced Lagrangian in Hamilton’s principle is � ℓ ( u ) = 1 u · L u d 2 x 2 The momentum is m = δℓ δ u = L u and the velocity is u = K ∗ m , where K is the Green’s function for the momentum operator L . Darryl D Holm Imperial College () Stochastic Solitons in Computational Anatomy Vienna, 20 Feb 2015 12 / 48
The momap for right action is conserved The momentum map for right action (reparametrization) is conserved The cotangent-lifted mapping for the right action (reparameterizing) is � � q ◦ η, p ◦ η ( p , q ) �→ R ( p , q ; η ) = . ∂η ∂ s Conservation of the momentum map for right action allows us to set � � − p t · ∂ q t J S ( q t , p t ) = − p t · d q t ( s ) = ds = 0 , ∂ s Hence we may take p t as normal to the planar curve q t ∈ C 1 ( S 1 , R 2 ) . Darryl D Holm Imperial College () Stochastic Solitons in Computational Anatomy Vienna, 20 Feb 2015 13 / 48
Summary of momentum for embeddings C 1 ( S 1 , R 2 ) The processes of shape change and reparametrization may be recast as evolution of left and right momentum maps, J Sing and J S . J Sing evolves by EPDiff, and J S is conserved. In the case of landmarks, momentum m t ( x ) is characterized by a set of N vectors, p t , for a matching problem with N landmarks, q t . Likewise in the case of C 1 embeddings parameterised by s ∈ S 1 , there is no redundancy in the q t ( s ) , p t ( s ) representation of time-dependent deformations governed by m t = J Sing ( q t , p t ) supported on C 1 ( S 1 , R 2 ) . Next: Statistical models Darryl D Holm Imperial College () Stochastic Solitons in Computational Anatomy Vienna, 20 Feb 2015 14 / 48
Statistical models Statistical models Darryl D Holm Imperial College () Stochastic Solitons in Computational Anatomy Vienna, 20 Feb 2015 15 / 48
Momentum map approach for statistical models Under evolution by EPDiff, statistical models for deformations become statistical models for ( q t , p t ) ; with the advantage of being easier to build, sample and estimate on a linear space. Figure: Here are three deformations of a disc produced by EPDiff for random momentum initial conditions, given by uncorrelated noise on its initially circular boundary. Figures courtesy of [5]. Darryl D Holm Imperial College () Stochastic Solitons in Computational Anatomy Vienna, 20 Feb 2015 16 / 48
Stochastic growth models, Trouv´ e and Vialard [7] Trouv´ e and Vialard [7] studied perturbations of the geodesic equations by adding a random force to the landmark momentum equation, intended to represent a stochastic growth model T p ( s ) = − ∂ u q ( s ) = u ( q ( s )) ˙ and ˙ · p ( s ) + σ B ( t ) ∂ q As we shall see, these are stochastic canonical Hamiltonian equations in the sense of Bismut [1] and Laz´ aro-Cam´ ı and Ortega [6]. Figure: A simulation from [7] showing Kunita flow with 40 points on the unit circle on the left of the figure. The z axis (blue arrow) represents the time. Darryl D Holm Imperial College () Stochastic Solitons in Computational Anatomy Vienna, 20 Feb 2015 17 / 48
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