SINGULAR SOLITONS, MOMENTUM MAPS & COMPUTATIONAL ANATOMY

Transcription

1 SINGULAR SOLITONS, MOMENTUM MAPS & COMPUTATIONAL ANATOMY Darryl D. Holm Los Alamos National Laboratory Mathematics Imperial College London dholm@lanl.gov, d.holm@imperial.ac.uk IMA Shape Spaces Workshop April 6, 2006 I shall speak of things... so singular in their oddity as in some manner to instruct, or at least entertain, without wearying. Lorenzo da Ponte IMA Shape Spaces Workshop, April 6,

2 Collaborators & References for this talk DDH, J. T. Rananather, A. Trouvé and L. Younes, Soliton Dynamics in Computational Anatomy. NeuroImage 23 (2004) S DDH & J. E. Marsden, Momentum maps & measure valued solutions of the Euler-Poincar e equations for the diffeomorphism group. Progr. Math. 232 (2004) DDH, J. E. Marsden and T. S. Ratiu, The Euler-Poincar e equations and semidirect products with applications to continuum theories. Adv. in Math., 137 (1998) IMA Shape Spaces Workshop, April 6,

3 More collaborators & references for this talk C. J. Cotter and DDH, Discrete momentum maps for lattice EPDiff, DDH and M. F. Staley, Interaction Dynamics of Singular Wave Fronts. In preparation, see staley/ R. Camassa and DDH, An Integrable Shallow Water Equation with Peaked Solitons. Phys. Rev. Lett. 71 (1993) Mumford D Pattern theory and vision. In Questions Mathématiques En Traitement Du Signal et de L Image, Chapter 3, pp Paris: Institut Henri Poincaré IMA Shape Spaces Workshop, April 6,

4 Problem & Approach for Computational Anatomy In a very large part of morphology, our essential task lies in the comparison of related forms rather than in the precise definition of each.... This process of comparison, of recognizing in one form a definite permutation or deformation of another,... lies within the immediate province of mathematics and finds its solution in... the Theory of Transformations. D Arcy Thompson, On Growth and Form (1917) IMA Shape Spaces Workshop, April 6,

5 Summary of our discussion today Computational Anatomy (CA) compares shapes (graphical structures) by making a geodesic deformation from one shape to the another. Among these graphical structures, landmarks and image outlines in CA are found to be singular solutions of the geodesic EPDiff equation. A momentum map for singular solutions of EPDiff yields their canonical Hamiltonian formulation, which provides a complete parameterization of the landmarks and image outlines by their canonical positions and momenta. IMA Shape Spaces Workshop, April 6,

6 Summary of discussion (cont) The momentum map provides an isomorphism between landmarks (and outlines) for images and singular (weak) solutions of EPDiff. (These are solitons in 1D.) This isomorphism provides for CA: (1) a complete and non-redundant data representation (2) a dynamical paradigm in which image outlines interact by exchange of momentum (3) methods for numerical simulation & data assimilation Euler-Poincaré theory also provides a framework for unifying and extending the various approaches in CA IMA Shape Spaces Workshop, April 6,

7 The importance of momentum for CA Completes the representation of images (momentum of cartoons) Informs template matching of the possibility of soliton-like collisions and momentum exchange in image outline interactions Encodes the subsequent deformation into the initial locus and momentum of an image outline Provides numerical simulation methods using the momentum map for right action as a data structure. Accomplishes matching and data assimilation via the adjoint linear problem for template matching, using the initial momentum as a control variable. All these momentum properties flow from the EPDiff equation IMA Shape Spaces Workshop, April 6,

8 Outline of the rest of the talk Describe the template matching variational problems of computational anatomy Motivate the EPDiff evolution equation. Describe the singular solutions for the EPDiff equation with diffeomorphism group Diff. Explain why these singular solutions exist (singular momentum map) Discuss the consequences of EPDiff for CA. Give numerical examples (by Colin J Cotter, Imperial College London) IMA Shape Spaces Workshop, April 6,

9 Cost Darryl D. Holm, Imperial College London & LANL Template matching assigns the cost for a given comparison Cost(t ϕ t ) = 1 0 l(u t ) dt as a functional defined on curves ϕ t in a Lie group with tangents dϕ t dt = u t ϕ t, I t = ϕ t I 0 (1) In the following, the function u t l(u t ) = u t 2 g is taken as a squared functional norm on the space of velocity vectors. The vector space of right invariant instantaneous velocities, u t = (dϕ t /dt) ϕ 1 t, forms the tangent space at the identity of the considered Lie group, and is isomorphic to the group s Lie algebra, denoted g. IMA Shape Spaces Workshop, April 6,

10 Problem statement Darryl D. Holm, Imperial College London & LANL Given the cost of a time-dependent deformation process, defined by Cost(t ϕ t ) = 1 most problems in CA can be formulated as: 0 u t 2 g dt, (2) Find the deformation path with minimal cost, under the constraint that it carries the template to the target. Such problems have a remarkable analogy with fluid dynamics. IMA Shape Spaces Workshop, April 6,

11 Mathematical analogy: template matching vs fluid dynamics (I) The frameworks in CA and fluid dynamics both involve a right-invariant stationary principle with action, or cost function 1 A = l(u t ) dt } 0{{} Action The main differences are:, l(u t ) = 1 2 u t 2 g }{{} Lagrangian, u t = (dϕ t /dt) ϕ 1 t }{{} Right invariant Template matching Cost function is designed for the application Optimal control problem Fluid dynamics Cost function is Kinetic energy Initial value problem IMA Shape Spaces Workshop, April 6,

12 Mathematical analogy: CA and fluids (cont1) (II) The geodesic evolution for both template matching and fluid dynamics is governed by EPDiff [HoMaRa1998, Mumford1998, Younes1998], ( t + u ) m + ( u) T m + m(div u) = 0. (3) Here u = G m, where G denotes convolution with the Green s kernel G for the operator L, when m = δl =: Lu δu Here, the operator L is symmetric and positive definite for the Cost(t ϕ t ) = 1 0 l(u t ) dt = 1 0 u t 2 g dt = with L 2 pairing, whenever u t 2 g is a norm. 1 0 u t, Lu t dt IMA Shape Spaces Workshop, April 6,

13 Mathematical analogy: CA and fluids (cont2) (III) The flows in CA and fluid dynamics both evolve under a left group action on a linear representation space, I t = ϕ t I 0 They differ in the roles of their advected quantities, a t = a 0 ϕ 1 t : Template matching Image properties are passive Fluid dynamics Advected quantities may affect fluid flows Images are only swept along Waves may form in fluids for l(u, a) for l(u) since restoring forcing depend on a (IV) Some features of CA are a bit more like fluid dynamics, e.g., metamorphoses have a semidirect-product structure. Can metamorphoses have waves? IMA Shape Spaces Workshop, April 6,

14 How EPDiff emerges in CA Darryl D. Holm, Imperial College London & LANL Choose the cost function for continuously morphing I 0 into I 1 as Cost(t ϕ t ) = 1 0 l(u t ) dt = 1 0 u t 2 g dt, where u t is the velocity of the fluid deformation at time t and u t 2 g = u t, Lu t, and L is a positive symmetric linear operator. Then, the momentum governing the process, m t = Lu t, with Green s function G : u t = G m t satisfies EPDiff. Namely, ( t + u ) m + ( u) T m + m(div u) = 0 This equation arises in both template matching and fluid dynamics, and it informs both fields of endeavor. IMA Shape Spaces Workshop, April 6,

15 More about EPDiff Darryl D. Holm, Imperial College London & LANL EPDiff may be written in a Lagrangian sense as d ( ) m dx d n dx x = 0 along dt dt = u = G m That is, EPDiff convects the one-form density of momentum as ( ) ( ) t + u m t dx t d n x t = 0, with Lie derivative u wrt velocity u. Consequently, the solution of EPDiff is, ) m t dx t d n x t = (m 0 dx 0 d n x 0 for velocity u = G m = ϕ t ϕ 1 t and ϕ t Diff. ϕ 1 t Recognizing this as coadjoint motion will be a useful step. IMA Shape Spaces Workshop, April 6,

16 Deriving EPDiff from Hamilton s principle. Euler-Poincaré Reduction starts with a right (or left) G invariant Lagrangian L : T G R on the tangent bundle of a Lie group G. Right invariance of the Lagrangian may be written as L(g(t), ġ(t)) = L(g(t)h, ġ(t)h), for all h G A G invariant Lagrangian defined on T G possesses a symmetryreduced Hamilton s principle defined on the Lie algebra g T G T G/G g. Stationarity of the symmetry-reduced Hamilton s principle yields the Euler-Poincaré equations on the dual Lie algebra g. For G = Diff, this equation is EPDiff. IMA Shape Spaces Workshop, April 6,

17 Theorem 0.1 (Euler-Poincaré Reduction) Let G be a Lie group, L : T G R a right-invariant Lagrangian, and l := L g : g R be its restriction to the Lie algebra g. For a curve g(t) G, let ξ(t) = ġ(t) g(t) 1 g. Then the following four statements are equivalent: IMA Shape Spaces Workshop, April 6,

18 (i) g(t) satisfies the Euler-Lagrange equations for Lagrangian L defined on G. (ii) The variational principle holds, for variations with fixed endpoints, δ b a L(g(t), ġ(t))dt = 0 (iii) The (right invariant) Euler-Poincaré equation holds: µ = ad δl ξ µ with µ = δξ and where ξ = ġg 1 (t). (If G = Diff, this is EPDiff.) l(ξ) := L(e, ġg 1 (t)) (iv) For an arbitrary path η(t) in g which vanishes at the endpoints, i.e., η(a) = η(b) = 0, the reduced variational principle holds on g, δ b a l(ξ(t))dt = 0, for δξ = η [ξ, η] =: η ad ξ η IMA Shape Spaces Workshop, April 6,

19 Example: Outlines & momentum measures Outline matching: Given two collections of curves c 1,..., c N and C 1,..., C N in Ω, find a time-dependent diffeomorphic process (t ϕ t ) of minimal action (or cost) such that ϕ 0 = id and ϕ 1 (c i ) = C i for i = 1,..., N. The matching problem for the image outlines seeks singular momentum solutions which naturally emerge in the computation of geodesics. IMA Shape Spaces Workshop, April 6,

20 Image outlines as Singular Momentum Solutions of EPDiff For example, in the 2D plane, EPDiff has weak singular momentum solutions that are expressed as [CaHo1993, HoSt2003, HoMa2004] N m(x, t) = P a (t, s)δ ( x Q a (t, s) ) ds, (4) a=1 s where s is a Lagrangian coordinate defined along a set of N curves in the plane moving with the flow by the equations x = Q a (t, s) and supported on the delta functions in the EPDiff solution (4). = PPT Thus, the singular momentum solutions of EPDiff represent evolving wavefronts supported on delta functions defined along curves by (4). These solutions exist in any dimension & provide CA matching for points (landmarks), curves, surfaces, in any combination. IMA Shape Spaces Workshop, April 6,

21 Lagrangian representation of the singular solutions of EPDiff Substituting the singular momentum solution formula (4) for s S into EPDiff (3), then integrating against a smooth test function implies canonical Hamiltonian equations for Lagrangian wavefronts N t Q a(s, t) = P b (s, t) G(Q a (s, t), Q b (s, t) ) ds b=1 = u(q a (s, t), t) = δh N, δp a N t P a(s, t) = P a (s, t) P b (s, t) (5) b=1 Q a (s, t) G( Q a (s, t), Q b (s, t) ) ds = δh N δq a. The s S are Lagrangian coordinates; Q a (s, t) moves with the flow. IMA Shape Spaces Workshop, April 6,

22 Canonical Hamiltonian dynamics The singular momentum solutions (4) satisfy geodesic Hamiltonian phase space dynamics (5) for the canonically conjugate vector parameters Q a (s, t) and P a (s, t) with a = 1, 2... N. The Hamiltonian is H N = 1 2 N a, b=1 ( Pa (s, t) P b (s, t) ) G ( Q a (s, t), Q b (s, t) ) ds ds IMA Shape Spaces Workshop, April 6,

23 Here is the Geometry Leading to the Numerics Basic observation that ties everything together in n dimensions: Theorem (Holm and Marsden, 2004): The singular momentum solutions T Emb(S, R n ) g : (P, Q) m define an equivariant momentum map. The embedded manifold S is the support set of the P s and Q s. The momentum map is for left action of the diffeos on S. The whole system is right invariant. Its momentum map for right action is conserved. These constructions persist for a certain class of numerical schemes They apply for every choice of norm for template matching. IMA Shape Spaces Workshop, April 6,

24 Wait a second, what is a momentum map? Give an example A momentum map J : T Q g is a Hamiltonian for canonical action of a Lie group G on phase space T Q. It is expressed in terms of the pairing, : g g R as J, ξ = p, ξ q =: q p, ξ, where (q, p) Tq Q and ξ q is the infinitesimal generator of the action of the Lie algebra element ξ g on q in the manifold Q. The standard example is ξ q = ξ q for R 3 R 3 R 3, with pairing, given by scalar product of vectors. The momentum map is then J ξ = p ξ q = q p ξ J = q p This is angular momentum, the Hamiltonian for phase-space rotations. IMA Shape Spaces Workshop, April 6,

25 How is the singular solution ansatz a momentum map? A momentum map J : T Q g satisfies the defining relation, J(α q ), ξ }{{}, : g g R On LHS, α q T q Q and ξ g. = α q, ξ Q (q) }{{}, : T q Q T q Q R On RHS, ξ Q is the infinitesimal generator of the action of Lie group G on Q (associated to Lie algebra element ξ g, the Lie algebra g of G.) G acts on T Q from the left by g Q = g Q (composition of functions) Singular solution ansatz does satisfy the momentum map formula, by ( Pi (s) δ (x Q(s)) d k s ) ξ i (x)d n x = (Q, P), ξ Q R n S which verifies that both sides are equal = S P i (s)ξ i (Q(s))d k s IMA Shape Spaces Workshop, April 6,

26 What about the momentum map for relabeling symmetry? Relabeling η : S S is given by the right action Diff(S) : Q η = Q η. The infinitesimal generator of this right action is the vector field X Emb(S,R n )(Q) = d Q η ɛ = T Q ξ dɛ ɛ=0 where ξ X is tangent to the curve η ɛ Diff at ɛ = 0. Thus for right action the infinitesimal generator needed in the momentum map formula is ξ Q (s) = T Q ξ(s) IMA Shape Spaces Workshop, April 6,

27 For right action, the momentum map formula yields: J S (Q, P), ξ = (Q, P), T Q ξ = P i (s) Qi (s) S s m ξm (s) d k s ( ) = ξ P(s) dq(s) d k s S ( = P(s) dq(s) d k s, ξ(s) ) S = P dq, ξ for the pairing of the one-form density P dq with the vector field ξ. Symmetry under relabeling implies the momentum map for right action (relabeling momentum) is conserved on the embedded surface J S (Q, P) = P(s) dq(s) IMA Shape Spaces Workshop, April 6,

28 EPDiff dynamics informs optimal control for CA CA must compare two geometric objects, and thus it is concerned with an optimal control problem. However, the initial value problem for EPDiff also has important consequences for CA applications. When matching two geometric structures, the momentum at time t=0 contains all required information for reconstructing the target from the template. This is done via Hamiltonian geodesic flow. Being canonically conjugate, the momentum has exactly the same dimension as the matched structures, so there is no redundancy. Right invariance mods out the relabeling motions from the optimal solution. This symmetry also yields a conserved momentum map. IMA Shape Spaces Workshop, April 6,

29 Besides being one-to-one, the momentum representation is defined on a linear space, being dual to the velocity vectors. This means one may, for example,: study linear instability of CA processes, take averages and apply statistics to the space of image contours. The advantage is the ease of building, sampling and estimating statistical models on a linear space. IMA Shape Spaces Workshop, April 6,

30 Conclusions so far We have: (1) identified momentum as a key concept in the representation of image data for CA and (2) discussed analogies with fluid dynamics. The fundamental idea transferring from fluid dynamics to CA is the idea of momentum maps corresponding to group actions. Relabeling by right action Emb(S, R n ) G Emb(S, R n ) is a symmetry of the template Hamiltonian, so its momentum map is conserved. (Right action also generates the steady EPDiff solutions.) Left action G Emb(S, R n ) Emb(S, R n ) is not a symmetry. Momentum map J represents the weak solution of EPDiff supported on Emb(S, R n ) whose motion is a coadjoint orbit of the left action. IMA Shape Spaces Workshop, April 6,

31 Examples Let Q 0 and Q 1 be two embeddings of S 1 in R 2 which represent two shapes, each a closed planar curve. We seek a 1-parameter family of embeddings Q(t) : S 1 [0, 1] R 2 so that Q(0) = Q 0 and Q(1) matches Q 1 (up to relabeling). Q(t) is found by minimizing the constrained norm of its velocity. To find the equation for Q we require extremal values of the action A = 0 2 u(t) 2 g d t + P(s, t) ( Q(s, t) u(q(s, t))) d t, 0 S 1 i.e. we seek time-series of vector fields u(t) which are minimized in some norm subject to the constraint that Q is advected by the flow using the Lagrange multipliers P (which we call momentum). IMA Shape Spaces Workshop, April 6,

32 The minimizing solutions are δl δu = P(s, t)δ(x Q(s, t)) d s, S 1 Ṗ(s, t) = P(s, t) u(q(s, t), t), Q(s, t) = u(q(s, t), t), subject to Q(s, 0) = Q 0 (s). Now one must seek initial momentum P(s, 0) which takes shape Q 0 (s) to shape Q 1 (s). This last problem can be solved by using a gradient algorithm, where the gradient of the residual error with respect to P(s, 0) is calculated using the adjoint. Refrain from discussing the adjoint dynamics on T g and T g IMA Shape Spaces Workshop, April 6,

33 In the movie examples we use the Variational Particle-Mesh (VPM) method (Cotter, 2005) to discretize the equations, as follows: Discretize the velocity on an Eulerian grid and approximate u there. Replace S 1 by representing the shape by a finite set of Lagrangian particles {Q β } n p β=1. Interpolate from the grid to the particles using basis functions n g n g u(q β ) = u k ψ k (Q β ), with ψ k (x) = 1, x k=1 The action for the continuous time motion on the grid then becomes 1 1 A = 0 2 u(t) 2 grid + P β ( Q β u k ψ k (Q β )) d t, β k and one can obtain a fully discrete method by discretizing the action in time. The discrete adjoint is then applied in computing the inversion. k=1 IMA Shape Spaces Workshop, April 6,

34 Acknowledgments This work was partially supported by US DOE, under contract W-7405-ENG-36 for Los Alamos National Laboratory, and Office of Science ASCAR/AMS/MICS. IMA Shape Spaces Workshop, April 6,

35 Relation to Metamorphosis Consider the action for Metamorphosis, A = 1 0 l dt = u(t) 2 g + 1 Darryl D. Holm, Imperial College London & LANL σ 2 X t X + u X 2 dt where X is an element of any vector representation space of G, u denotes Lie derivative wrt velocity u and we shall take the norms ( ) 2 to define appropriate pairings, ( ) 2 = ( ), ( ). Stationarity with respect to variations of the velocity u yields δu : Lu = Z X X g where Z X = 1 ( ) σx 2 t X + u X = δl δ( t X) and the operation is defined with any chosen pairing by Z X X, η := Z X, η X =: Z X, Xη IMA Shape Spaces Workshop, April 6,

36 Relation to Metamorphosis (cont) The diamond operation is skew symmetric X Z X, η = Z X X, η and it satisfies the chain rule under Lie derivative, Darryl D. Holm, Imperial College London & LANL ξ (Z X X), η = ( ξ Z X ) X, η + Z X ξ X, η Consequently, one finds ( t + ad u)lu = ( t + u )Lu = ( t + u )(Z X X) = 0 IMA Shape Spaces Workshop, April 6,

37 Euler Poincaré equation via Clebsch relations Stationary variations of the constrained Eulerian action, S = l(ξ, a) + v, a t + ξa dt yield the following Clebsch relations, δξ : δl δξ v a = 0 (momentum map) δa : δl δa v t ξ v = 0 δv : a t + ξ a = 0 (advection relation) The Clebsch relations will recover the Euler Poincaré equations, ( ) δl t + ad ξ δξ = δl δa a and a t + ξa = 0. by using the properties of the diamond operation. (6) IMA Shape Spaces Workshop, April 6,

38 The diamond operation is defined by v a, η v, η a = v, a η. This operation is antisymmetric, v a, η = a v, η, and satisfies the chain rule under the Lie derivative, ξ (v a), η = ( ξ v) a, η + v ( ξ a), η. These two properties of the operation and the Clebsch relations (6) together imply ( ) (v t + ) δl ξ a = δa a This manipulation recovers the EP motion equation, since δl ξ δξ = δl ad ξ δξ, for one-form densities such as δl/δξ = v a. IMA Shape Spaces Workshop, April 6,