The abstract setting for shape deformation analysis and LDDMM methods

28/10/2015
Publication GSI2015
OAI : oai:www.see.asso.fr:11784:14332
DOI : http://dx.doi.org/10.1007/978-3-319-25040-3_18You do not have permission to access embedded form.

Résumé

This paper aims to define a unified setting for shape registration and LDDMM methods for shape analysis. This setting turns out to be sub-Riemannian, and not Riemannian. An abstract definition of a space of shapes in ℝ d is given, and the geodesic flow associated to any reproducing kernel Hilbert space of sufficiently regular vector fields is showed to exist for all time.

The abstract setting for shape deformation analysis and LDDMM methods

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This paper aims to define a unified setting for shape registration and LDDMM methods for shape analysis. This setting turns out to be sub-Riemannian, and not Riemannian. An abstract definition of a space of shapes in ℝ d is given, and the geodesic flow associated to any reproducing kernel Hilbert space of sufficiently regular vector fields is showed to exist for all time.

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The Group Structure of Diffeomorphisms Shape Spaces The General Setting for Shape Deformation Analysis and LDDMM Sylvain Arguill`ere, Johns Hopkins University GSI 2015 Sylvain Arguill`ere, Johns Hopkins University The General Setting for Shape Deformation Analysis and LDDMM The Group Structure of Diffeomorphisms Shape Spaces A Brief Summary of LDDMM Purpose of LDDMM: Compare two “shapes” q0 and q1 in Rd in a way that takes into account their geometric properties (smoothness, self-intersection...). The initial shape q0 is the template, q1 is the target. Examples of shapes: Landmarks: q = (x1, . . . xn), xi ∈ Rd Parametrized embedded curves, surfaces and submanifolds: q ∈ Embk (M, Rd ) Unparametrized embedded curves, surfaces and submanifolds: q ∈ Embk (M, Rd )/Dk (M) (Bauer, Bruveris, Michor) Sylvain Arguill`ere, Johns Hopkins University The General Setting for Shape Deformation Analysis and LDDMM The Group Structure of Diffeomorphisms Shape Spaces A Brief Summary of LDDMM Method: Let (V , ·, · ) be a Hilbert space of vector fields with continuous inclusion in C1 0 (Rd , Rd ). A time-dependent vector field v ∈ L2 (0, 1; V ) admits a unique flow ϕ(·) such that ∂tϕ(t, x) = v(t, ϕ(t, x)), t ∈ [0, 1], x ∈ Rd , which acts on the template q0 (composition on the left), deforming it into q(t) = ϕ(t) · q0, yielding the infinitesimal action ˙q(t) = v(t) · q(t). Sylvain Arguill`ere, Johns Hopkins University The General Setting for Shape Deformation Analysis and LDDMM The Group Structure of Diffeomorphisms Shape Spaces A Brief Summary of LDDMM Then, minimize J(v) = 1 2 1 0 v(t) 2 V dt + g(q(1)). with g data attachment that gets smaller the closer q(1) gets to q1, and q(0) = q0, ˙q(t) = v(t) · q(t). Sylvain Arguill`ere, Johns Hopkins University The General Setting for Shape Deformation Analysis and LDDMM The Group Structure of Diffeomorphisms Shape Spaces We have examples of shapes and shape spaces, but no general definition. What if we want to track a vector field along the shape (muscle fibers)? Or shapes with parts of different dimensions? Key remark: we only need an action of D(Rd ) (that satisfies certain properties) to apply the LDDMM method. Sylvain Arguill`ere, Johns Hopkins University The General Setting for Shape Deformation Analysis and LDDMM The Group Structure of Diffeomorphisms Shape Spaces Plan 1 The Group Structure of Diffeomorphisms 2 Shape Spaces Sylvain Arguill`ere, Johns Hopkins University The General Setting for Shape Deformation Analysis and LDDMM The Group Structure of Diffeomorphisms Shape Spaces Plan 1 The Group Structure of Diffeomorphisms 2 Shape Spaces Sylvain Arguill`ere, Johns Hopkins University The General Setting for Shape Deformation Analysis and LDDMM The Group Structure of Diffeomorphisms Shape Spaces Diffeomorphisms of Sobolev regularity Fix s > d/2 + 1 integer. Then (Bruveris, Vialard) Ds (Rd ) = IdRd + Hs (Rd , Rd ) ∩ D(Rd ) is a Hilbert manifold (open subset of an affine Hilbert space) and a topological group for (ϕ, ψ) → ϕ ◦ ψ. Right composition: Rψ : ϕ → ϕ ◦ ψ is of class C∞ ⇒ for v ∈ Hs (Rd , Rd ), right-invariant “vector field” on Ds (Rd ) v(ϕ) = dRϕ(v) = v ◦ ϕ. Left composition: (ϕ, ψ) → ϕ ◦ ψ (resp. (v, ϕ) → v ◦ ϕ) is of class Ck when restricted to Ds+k (Rd ) × Ds (Rd ) (resp. Hs+k (Rd , Rd ) × Ds (Rd )). Sylvain Arguill`ere, Johns Hopkins University The General Setting for Shape Deformation Analysis and LDDMM The Group Structure of Diffeomorphisms Shape Spaces Control Space For LDDMM Towards LDDMM: Fix (V , ·, · ) Hilbert space of vector fields with continuous inclusion in Hs (Rd , Rd ). Now v ∈ L2 (0, 1; V ) gives rise to a unique flow, a curve ϕ(·) ∈ H1 (0, 1; Ds (Rd )) by solving the control system ϕ(0) = IdRd , ˙ϕ(t) = v(t) ◦ ϕ(t), t ∈ [0, 1]. This flow has energy E(ϕ) = E(v) = 1 2 1 0 v(t) 2 dt. Sylvain Arguill`ere, Johns Hopkins University The General Setting for Shape Deformation Analysis and LDDMM The Group Structure of Diffeomorphisms Shape Spaces Distance =⇒ We can define the length, then the distance dV between two diffeomorphisms induced by this V . This is a Sub-Riemannian structure on Ds (Rd ). Proposition 1 (Trouv´e) The metric space (Ds (Rd ), dV ) is complete. Moreover, if d(ϕ, ψ) < +∞, they can be connected by a geodesic. Sylvain Arguill`ere, Johns Hopkins University The General Setting for Shape Deformation Analysis and LDDMM The Group Structure of Diffeomorphisms Shape Spaces Geodesics Question: what are the geodesics (i.e., minimizers of energy with fixed endpoints): Optimal Control. Problem: no Pontryagin Maximum Principle (PMP) in infinite dimensions: no (workable) necessary condition! However, we still have sufficient conditions. Sylvain Arguill`ere, Johns Hopkins University The General Setting for Shape Deformation Analysis and LDDMM The Group Structure of Diffeomorphisms Shape Spaces Hamiltonian Let KV : V ∗ → V be the Riesz operator. Any P ∈ Hs (Rd , Rd )∗ = H−s (Rd , Rd ) also belongs to V ∗ : KV P ∈ V . KV is given by convolution with a kernel K : Rd × Rd → Md (R): KV P(x) = Rd K(x, y)P(y)dy. Hamiltonian: dR∗ ϕP given by (dR∗ ϕP|v) = (P|v ◦ ϕ). The Hamiltonian of the system is H(ϕ, P) = 1 2 KV dR∗ ϕP 2 = 1 2 Rd ×Rd P(x)·K(ϕ(x), ϕ(y))P(y)dydx. Sylvain Arguill`ere, Johns Hopkins University The General Setting for Shape Deformation Analysis and LDDMM The Group Structure of Diffeomorphisms Shape Spaces Theorem 1 Assume V has continuous inclusion in Hs+2 (Rd , Rd ). Then the Hamiltonian geodesic flow ∂tϕ(t, x) = ∂P H(ϕ(t), P(t))(x) = v(t, ϕ(t, x)), ∂tP(t, x) = −∂ϕH(ϕ(t), P(t))(x) = −dv(t, ϕ(t, x))T P(t, x), where v(t, x) = Rd K(x, ϕ(t, y))P(t, y)dy = KV dR∗ ϕ(t)P(t), is complete, and the corresponding curves t → ϕ(t) are geodesics on small enough intervals. Important remark: some geodesics may not appear in the Hamiltonian flow, so this is a sufficient condition, not a necessary condition. Sylvain Arguill`ere, Johns Hopkins University The General Setting for Shape Deformation Analysis and LDDMM The Group Structure of Diffeomorphisms Shape Spaces Momentum Viewpoint Define the momentum map µ(ϕ, P) = dR∗ ϕP ∈ H−s (Rd , Rd ). Proposition 2 (Euler-Poincarr´e) A curve t → (ϕ(t), P(t)) follows the Hamiltonian flow if and only if its momentum µ(t) = µ(ϕ(t), P(t)) satisfies ˙µ(t) = −ad∗ KV µ(t)µ(t), where adv w = −[v, w], which integrates into µ(t) = ϕ∗µ(0). In particular, the regularity of µ is preserved along the trajectory and supp µ(t) = ϕ(t)(supp µ(0)). For example, if µ(0) is a sum of n vector-valued Dirac masses, so is µ(t) for every time t. Sylvain Arguill`ere, Johns Hopkins University The General Setting for Shape Deformation Analysis and LDDMM The Group Structure of Diffeomorphisms Shape Spaces Remark: In general, the equation cannot be converted into the Arnol’d equation ˙v(t) = −adT v(t)v(t) through v(t) = KV µ(t). For example, different µ(0) can yield the same v(0) but not the same v(t). Sylvain Arguill`ere, Johns Hopkins University The General Setting for Shape Deformation Analysis and LDDMM The Group Structure of Diffeomorphisms Shape Spaces Plan 1 The Group Structure of Diffeomorphisms 2 Shape Spaces Sylvain Arguill`ere, Johns Hopkins University The General Setting for Shape Deformation Analysis and LDDMM The Group Structure of Diffeomorphisms Shape Spaces Definition Let S be a Banach manifold, fix ∈ N, and s0 the smallest integer such that s0 > d/2. Let s = s0 + . A continuous action (ϕ, q) → ϕ · q of Ds (M) on S makes it into a shape space of order if: Sylvain Arguill`ere, Johns Hopkins University The General Setting for Shape Deformation Analysis and LDDMM The Group Structure of Diffeomorphisms Shape Spaces Definition Let S be a Banach manifold, fix ∈ N, and s0 the smallest integer such that s0 > d/2. Let s = s0 + . A continuous action (ϕ, q) → ϕ · q of Ds (M) on S makes it into a shape space of order if: For any q ∈ S fixed, Rq : ϕ → ϕ · q is of class C∞ . We then define the infinitesimal action ξ of v ∈ Hs (Rd , Rd ) by ξqv := dRq(IdRd )v. Sylvain Arguill`ere, Johns Hopkins University The General Setting for Shape Deformation Analysis and LDDMM The Group Structure of Diffeomorphisms Shape Spaces Definition Let S be a Banach manifold, fix ∈ N, and s0 the smallest integer such that s0 > d/2. Let s = s0 + . A continuous action (ϕ, q) → ϕ · q of Ds (M) on S makes it into a shape space of order if: For any q ∈ S fixed, Rq : ϕ → ϕ · q is of class C∞ . We then define the infinitesimal action ξ of v ∈ Hs (Rd , Rd ) by ξqv := dRq(IdRd )v. For every k ≥ 1, the following mappings are Ck : Ds+k (Rd ) × S → S Hs+k (Rd , Rd ) × S → TS (ϕ, q) → ϕ · q (v, q) → ξqv Sylvain Arguill`ere, Johns Hopkins University The General Setting for Shape Deformation Analysis and LDDMM The Group Structure of Diffeomorphisms Shape Spaces Examples All classical examples (landmark spaces, embedding spaces, spaces of submanifolds) are shape spaces of varying order (0 for landmarks, for C -embeddings,...). Ds0+ (Rd ) is itself a shape space of order (in a way, it’s the biggest). So is the product of any two shape spaces (which yields unions of shapes with different dimensions). If S is a shape space of order , TS is a shape space of order + 1 (application: shapes with fibers). The space of images L2 (Rd ) with action ϕ · I = I ◦ ϕ−1 is not a shape space. However, this case can still be treated with our method. Sylvain Arguill`ere, Johns Hopkins University The General Setting for Shape Deformation Analysis and LDDMM The Group Structure of Diffeomorphisms Shape Spaces The general LDDMM problem Fix a Hilbert space of vector fields V with continuous inclusion in Hs (Rd , Rd ). Fix a template q0 and let g : S → R be a data attachment term of class C1 . Sylvain Arguill`ere, Johns Hopkins University The General Setting for Shape Deformation Analysis and LDDMM The Group Structure of Diffeomorphisms Shape Spaces The general LDDMM problem Fix a Hilbert space of vector fields V with continuous inclusion in Hs (Rd , Rd ). Fix a template q0 and let g : S → R be a data attachment term of class C1 . We wish to minimize J(v) = 1 2 1 0 v(t) 2 dt + g(q(1)) over v ∈ L2 (0, 1; V ), where q(·) is the trajectory of the sub-Riemannian control system q(0) = q0, ˙q(t) = ξq(t)v(t) (⇔ q(t) = ϕ(t) · q0)). Sylvain Arguill`ere, Johns Hopkins University The General Setting for Shape Deformation Analysis and LDDMM The Group Structure of Diffeomorphisms Shape Spaces Push-forward of the Hamiltonian flow Theorem 2 Assume that V has continuous inclusion in Hs+1 (Rd , Rd ). Then a control v is a critical point of J if and only if the flow of v is the projection of a Hamiltonian geodesic on Ds (Rd ) with final momentum µ(1) = ξ∗ q(1)dg(q(1)). Moreover, µ(t) = ξ∗ q(t)p(t) for some p(t) ∈ T∗ q(t)S. We get v(t) = KV ξ∗ q(t)p(t) and ˙q(t) = ξq(t)KV ξ∗ q(t)p(t) =: Kq(t)p(t). Sylvain Arguill`ere, Johns Hopkins University The General Setting for Shape Deformation Analysis and LDDMM The Group Structure of Diffeomorphisms Shape Spaces Hamiltonian flow on S Letting HS (q, p) = 1 2 (p|Kqp), we just get p(1) + dg(q(1)) = 0, and ˙q(t) = ∂pHS (q(t), p(t)), ˙p(t) = −∂pHS (q(t), p(t)). Sylvain Arguill`ere, Johns Hopkins University The General Setting for Shape Deformation Analysis and LDDMM The Group Structure of Diffeomorphisms Shape Spaces Reduction Important Application: we can simply minimize 1 2 1 0 (p(t)|Kq(t)p(t))dt + g(q(1)), ˙q(t) = Kq(t)p(t), finite dimensional optimal control problem in numerical applications. Further reductions possible, for example if g and ξ are invariant under the action of some other group G (reparametrizetion of an embedded manifold for example). Sylvain Arguill`ere, Johns Hopkins University The General Setting for Shape Deformation Analysis and LDDMM The Group Structure of Diffeomorphisms Shape Spaces Images When matching a smooth template image I0 onto a target I1, we can simply consider the entire Ds (Rd ) as a shape space, with data attachment g(ϕ) = I0 ◦ ϕ−1 − I1 2 L2 . A change of variable easily shows that g is of class C1 , and we can apply our method. This will reprove the well-known fact that, for LDDMM image matching, the momentum µ(t) at time t is pointwise colinear with the gradient in space of I(t) = I0 ◦ ϕ(t)−1 : µ(t) = (I(t) − I1 ◦ ϕ(1 − t))∂x I(t). Sylvain Arguill`ere, Johns Hopkins University The General Setting for Shape Deformation Analysis and LDDMM The Group Structure of Diffeomorphisms Shape Spaces Thank you for your attention! Sylvain Arguill`ere, Johns Hopkins University The General Setting for Shape Deformation Analysis and LDDMM