Stochastic Solitons in Computational Anatomy Darryl D Holm Imperial - - PowerPoint PPT Presentation

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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:

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SLIDE 1

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

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SLIDE 2

Organization of the talk

1

Review: Momentum in images

2

Peakon momentum maps in 1D and 2D

3

Statistical models

4

Fokker-Planck equations for peakons and pulsons

5

Numerical experiments for stochastic landmark motion

6

Stratonovich and Itˆ

  • Stochastic Euler-Poincar´

e equations

7

Summary

Darryl D Holm Imperial College () Stochastic Solitons in Computational Anatomy Vienna, 20 Feb 2015 2 / 48

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SLIDE 3

Review: Dynamics of ‘shapes’ C1(S1, R2)

Darryl D Holm Imperial College () Stochastic Solitons in Computational Anatomy Vienna, 20 Feb 2015 3 / 48

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SLIDE 4

Review: Momentum in images

Most problems in CA can be formulated as finding the time-dependent deformation map ϕt : M → M with minimal geodesic cost, defined by Cost(t → ϕt) = 1 ℓ(ut) dt = 1 2 1 ut2

X(M) dt ,

with dϕt dt = ut ◦ ϕt under the constraint that the map ϕt carries a template I0 at t = 0 to the target I1 at t = 1 and · X(M) is a given Riemannian metric. The variable ut ∈ X(M) is called the (Eulerian) velocity, and mt := δℓ δut = Lut for ut2

X(M) = Lut , ut

is called the momentum for L2 pairing · , · and positive symmetric

  • perator L : X(M) → X(M)∗.

Darryl D Holm Imperial College () Stochastic Solitons in Computational Anatomy Vienna, 20 Feb 2015 4 / 48

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SLIDE 5

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) = {qa(t), a = 1, 2, . . . , N}, we take HP as 0 = δS = δ 1

  • ℓ(u) +

N

p, ˙ q − u(q, t)

  • dt

Stationarity of HP leads to the following equations of motion [3], ˙ q = u(q, t) , ˙ p = − du dq

T

· p , m(x, t) := δℓ δu =

N

  • p δ(x − q) .

The 1st two eqns imply that the momentum m(x, t) evolves by the EPDiff equation, ∂tm = − ad∗

um = − u · ∇m − (∇u)Tm − m(divu) .

Darryl D Holm Imperial College () Stochastic Solitons in Computational Anatomy Vienna, 20 Feb 2015 5 / 48

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SLIDE 6

Peakons: m(x, t) = p δ(x − q), embeddings C1(Z, R)

When ℓ(u) = 1

2u2 H1 = 1 2

  • u2 + u2

x dx for M = R, then m = u − uxx

and ∂tm = −(um)x − mux produces the 1D solitons called ‘peakons’. Singular peakon (landmark) solutions emerge from smooth initial conditions and form a finite dimensional solution set for EPDiff(H1).

Darryl D Holm Imperial College () Stochastic Solitons in Computational Anatomy Vienna, 20 Feb 2015 6 / 48

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SLIDE 7

EPDiff(H1) embeddings C1([0, 1], R2)

Here, we have the EPDiff equation with u ∈ X(R2), ℓ : X(R2) → R 0 = δS = δ

  • ℓ(u) +

N

p, ˙ q − u(q, t)

  • dt , =

⇒ d dt δℓ δu + ad∗

u

δℓ δu = 0 .

Darryl D Holm Imperial College () Stochastic Solitons in Computational Anatomy Vienna, 20 Feb 2015 7 / 48

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SLIDE 8

Definition: Momentum maps and their evolution

Smooth invertible maps ϕt act on the symplectic manifold T ∗M by flows of cotangent lifts of vector fields ut = dϕt

dt ◦ ϕ−1 t

∈ X acting on M.

✲

Equivariant Momentum Map T ∗M T ∗M ϕt

❄

mt

✲

Ad∗

ϕ−1

t

❄

X∗ X∗ m0 The associated momap m : T ∗M → X∗ evolves via the EPDiff eqn

dm dt = − ad∗ um for m = δℓ δu ∈ X∗

Darryl D Holm Imperial College () Stochastic Solitons in Computational Anatomy Vienna, 20 Feb 2015 8 / 48

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Momentum maps for C1 embeddings C1(S1, R2)

Embeddings C1(S1, R2) admit a dual pair of momaps

Momaps recast processes of shape change and reparametrization as: Right & Left group reductions by Diff(S1) and Diff(R2), respectively, of the canonical Hamiltonian motion on T ∗C1(S1, R2).

Embedding phase space T ∗C1(S1, R2) Left (changes shape) JSing JS Right (preserves shape) (Changes shape) X(R2)∗ X(S1)∗ (Reparameterizes)

  • ✠

❅ ❅ ❅ ❅ ❘

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

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SLIDE 10

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 C1(S1, R2) embedding.

Darryl D Holm Imperial College () Stochastic Solitons in Computational Anatomy Vienna, 20 Feb 2015 10 / 48

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SLIDE 11

Left and right transformations of q ∈ C1(S1, R2)

The Right & Left actions of Lie groups Diff+(S1) & Diff+(R2), respectively, on q ∈ C1(S1, R2) commute with each other. The Left action of Diff+(R2) on q ∈ C1(S1, R2): q → Lgq = g ◦ q, g ∈ Diff+(R2), changes shape: it transforms between inequivalent curves. The Right action of Diff+(S1) on q ∈ (S1, R2): q → Rηq = q ◦ η, η ∈ Diff+(S1), preserves shape, which defines an equivalence class of curves. Namely, the ‘shape space’ of C1 embeddings modulo relabelling, C1(S1, R2)/ Diff+(S1)

Darryl D Holm Imperial College () Stochastic Solitons in Computational Anatomy Vienna, 20 Feb 2015 11 / 48

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Left action (motion) has a singular momentum map

MM for left action of Diff+(R2) on q ∈ C1(S1, R2) is singular [3] m = Lu(x) =

  • S1 p(s) δ
  • x − q(s)
  • ds =: JSing(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

2m , K ∗ m

˙ q(s) = u(q(s)) and ˙ p(s) = − ∂u ∂q

T

· p(s) The reduced Lagrangian in Hamilton’s principle is ℓ(u) = 1 2

  • u · Lu d2x

The momentum is m = δℓ

δu = Lu 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

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SLIDE 13

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 (p, q) → R(p, q; η) =

  • q ◦ η, p ◦ η

∂η ∂s

  • .

Conservation of the momentum map for right action allows us to set JS(qt, pt) = − pt · dqt(s) =

  • − pt · ∂qt

∂s

  • ds = 0 ,

Hence we may take pt as normal to the planar curve qt ∈ C1(S1, R2).

Darryl D Holm Imperial College () Stochastic Solitons in Computational Anatomy Vienna, 20 Feb 2015 13 / 48

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Summary of momentum for embeddings C1(S1, R2)

The processes of shape change and reparametrization may be recast as evolution of left and right momentum maps, JSing and JS. JSing evolves by EPDiff, and JS is conserved. In the case of landmarks, momentum mt(x) is characterized by a set

  • f N vectors, pt, for a matching problem with N landmarks, qt.

Likewise in the case of C1 embeddings parameterised by s ∈ S1, there is no redundancy in the qt(s), pt(s) representation of time-dependent deformations governed by mt = JSing(qt, pt) supported on C1(S1, R2). Next: Statistical models

Darryl D Holm Imperial College () Stochastic Solitons in Computational Anatomy Vienna, 20 Feb 2015 14 / 48

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SLIDE 15

Statistical models

Statistical models

Darryl D Holm Imperial College () Stochastic Solitons in Computational Anatomy Vienna, 20 Feb 2015 15 / 48

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SLIDE 16

Momentum map approach for statistical models

Under evolution by EPDiff, statistical models for deformations become statistical models for (qt, pt); 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

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SLIDE 17

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 ˙ q(s) = u(q(s)) and ˙ p(s) = − ∂u ∂q

T

· p(s) + σB(t) 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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Stochastic growth models, Trouv´ e and Vialard [7]

The perturbation considered in [7] applies a stochastic Brownian force (rate of change of momentum) on the particles, rather than making the particle paths stochastic. ˙ q(s, t) = u(q(s, t)) and ˙ p(s, t) = − ∂u ∂q

T

· p(s, t) + σB(t) With this stochastic forcing, Trouv´ e and Vialard [7] proved that:

1

The solutions for these landmark dynamics do not blow up.

2

In infinite dimensions, the solutions are also defined for all times.

3

Simple additive noise in the momentum equation is general enough to account for correlations between points on the curve during landmark evolution under stochastic forcing.

Darryl D Holm Imperial College () Stochastic Solitons in Computational Anatomy Vienna, 20 Feb 2015 18 / 48

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SLIDE 19

Stochastic canonical Hamiltonian equations (SHEs)

To see that the Trouv´ e and Vialard [7] equations ˙ q(s, t) = u(q(s, t)) and ˙ p(s, t) = − ∂u ∂q

T

· p(s, t) + σB(t) are stochastic canonical Hamiltonian equations, we introduce the following Hamilton–Pontryagin variational principle. S(u, p, q) =

  • ℓ(ut)dt +

p, dq dt − ut(q)

  • dt
  • Lebesque integrals in time t

−

  • s,t
  • i

hi(p, q) ◦ dWi(t)

  • Stratonovich integral

δu : m := δℓ δu =

  • s
  • p δ(x − q) ⇒ u(x, t) = K ∗ m =
  • s
  • p K(x, q)

δp : dq = ut(q)dt +

  • i

{q, hi} ◦ dWi(t) δq : dp = − ∂ut ∂q

  • · p dt +
  • i

{p, hi} ◦ dWi(t)

Darryl D Holm Imperial College () Stochastic Solitons in Computational Anatomy Vienna, 20 Feb 2015 19 / 48

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Stochastic Hamiltonian equations for landmarks

Substituting the momentum map relation for landmarks u(qa) = K ∗

N

  • b=1

pb δ(qa − qb) =

N

  • b=1

pb K(qa, qb) into the δp and δq equations from Hamilton’s principle allows us to recognise them as canonical stochastic Hamilton equations dq = ∂h ∂pdt and dp = − ∂h ∂q dt with h(q, p) dt = 1 2

N

  • a,b=1

papb K(qa, qb) dt +

M

  • i=1

hi(q, p) ◦ dWi(t) The stochastic landmark dynamics of Trouv´ e and Vialard [7] will be recovered when we replace hi(q, p) ◦ dWi(t) → − σqiBi(t) for Itˆ

  • noise.

Darryl D Holm Imperial College () Stochastic Solitons in Computational Anatomy Vienna, 20 Feb 2015 20 / 48

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SLIDE 21

Itˆ

  • version of SHEs

In the Itˆ

  • version of stochastic canonical Hamiltonian equations, the

noise terms have zero mean, but additional drift terms arise. These drift terms are double canonical Poisson brackets (diffusive). δu : m := δℓ δu =

  • s
  • p δ(x − q) ⇒ u(x, t) = K ∗ m =
  • s
  • p K(x, q)

δp : dq = ut(q)dt +

  • i

{q, hi}dWi(t)

  • Itˆ
  • Noise for q

+ 1 2

  • i

{q, hi}, hi}}dt

  • Itˆ
  • Drift for q

δq : dp = − ∂ut ∂q

  • · p dt +
  • i

{p, hi}dWi(t)

  • Itˆ
  • Noise for p

+ 1 2

  • i

{p, hi}, hi}}dt

  • Itˆ
  • Drift for p

The stochastic landmark dynamics of Trouv´ e and Vialard [7] is recovered when we choose hi(p, q) = −σqi, with i = 1, 2, 3, for q ∈ R3.

Darryl D Holm Imperial College () Stochastic Solitons in Computational Anatomy Vienna, 20 Feb 2015 21 / 48

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SLIDE 22

Fokker-Planck equations for peakons and pulsons

Fokker-Planck equations for peakons and pulsons

Darryl D Holm Imperial College () Stochastic Solitons in Computational Anatomy Vienna, 20 Feb 2015 22 / 48

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SLIDE 23

Fokker-Planck equation—single pulson

A stochastic process Xt can be described with the help of a transition density function ρ(t, x; ¯ x) which represents the probability density that the process, initially in the state ¯ x, will reach the state x at time t. For the general N-dimensional Itˆ

  • process

dX i

t = ai(Xt) dt + M

  • m=1

bim(Xt) dW m

t

the transition density function satisfies the Fokker-Planck advection-diffusion equation ∂ρ ∂t +

N

  • i=1

∂ ∂xi

  • ai(x)ρ
  • − 1

2

N

  • i,j=1

∂2 ∂xi∂xj

  • dij(x)ρ
  • = 0,

where d = bbT.

Darryl D Holm Imperial College () Stochastic Solitons in Computational Anatomy Vienna, 20 Feb 2015 23 / 48

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SLIDE 24

Fokker-Planck equation—single pulson

Consider N = 1 pulson subject to one-dimensional (i.e., M = 1) Wiener process, with the stochastic potential h(q, p) = βp, where β is a nonnegative real parameter. The stochastic Hamiltonian equations are (Ito and Stratonovich are equivalent here) dq = p dt + β dWt, dp = 0, so the corresponding Fokker-Planck advection-diffusion equation is ∂ρ ∂t + p ∂ρ ∂q − 1 2β2 ∂2ρ ∂q2 = 0 with the initial condition ρ(0, q, p; ¯ q, ¯ p) = δ(q − ¯ q)δ(p − ¯ p).

Darryl D Holm Imperial College () Stochastic Solitons in Computational Anatomy Vienna, 20 Feb 2015 24 / 48

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SLIDE 25

Fokker-Planck equation—single pulson

The Fokker-Planck advection-diffusion equation ∂ρ ∂t + p ∂ρ ∂q − 1 2β2 ∂2ρ ∂q2 = 0 is easily solved with the help of the fundamental solution for the heat equation, and the solution yields ρβ(t, q, p; ¯ q, ¯ p) = 1 β √ 2πt e

− (q−¯

q−pt)2 2β2t

δ(p − ¯ p). This solution means that the pulson/peakon retains its initial momentum/height ¯

  • p. The position has a Gaussian distribution which

widens with time, and whose maximum is advected with velocity ¯ p.

Darryl D Holm Imperial College () Stochastic Solitons in Computational Anatomy Vienna, 20 Feb 2015 25 / 48

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SLIDE 26

Fokker-Planck solution—two pulsons

Consider N = 2 pulsons subject to two-dimensional (i.e., M = 2) Wiener process, with the stochastic potentials h1(q, p) = β1p1 and h2(q, p) = β2p2, where q = (q1, q2), p = (p1, p2). The Fokker-Planck advection-diffusion equation for two pulsons is

∂ρ ∂t + ∂ ∂q1

  • a1(q, p)ρ
  • +

∂ ∂q2

  • a2(q, p)ρ
  • +

∂ ∂p1

  • a3(q, p)ρ
  • +

∂ ∂p2

  • a4(q, p)ρ
  • − 1

2β2

1

∂2ρ ∂q2

1

− 1 2β2

2

∂2ρ ∂q2

2

= 0

with the initial condition ρ

  • 0, q, p; ¯

q, ¯ p

  • = δ(q1 − ¯

q1)δ(p1 − ¯ p1) + δ(q2 − ¯ q2)δ(p2 − ¯ p2), where a1(q, p) = p1 + p2G(q1 − q2), a3(q, p) = −p1p2G′(q1 − q2), a2(q, p) = p2 + p1G(q1 − q2), a4(q, p) = p1p2G′(q1 − q2).

Darryl D Holm Imperial College () Stochastic Solitons in Computational Anatomy Vienna, 20 Feb 2015 26 / 48

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SLIDE 27

Fokker-Planck equation—two pulsons

Despite its relatively simple structure, F-P for two pulsons does not appear to be solvable analytically. Nevertheless, one may verify that the function ρ(t, q1, q2, p1, p2; ¯ q1, ¯ q2, ¯ p1, ¯ p2) = ρβ1(t, q1, p1; ¯ q1, ¯ p1) + ρβ2(t, q2, p2; ¯ q2, ¯ p2), satisfies the F-P for two pulsons asymptotically as q1 − q2 − → ±∞, assuming the Green’s function and its derivative both decay in that limit. This simple observation shows that stochastic pulsons should behave like individual particles when they are far from each other, just as in the deterministic case.

Darryl D Holm Imperial College () Stochastic Solitons in Computational Anatomy Vienna, 20 Feb 2015 27 / 48

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SLIDE 28

Numerical experiments for stochastic landmark motion

Numerical experiments for q-stochastic landmark motion

Darryl D Holm Imperial College () Stochastic Solitons in Computational Anatomy Vienna, 20 Feb 2015 28 / 48

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SLIDE 29

Numerical experiments for stochastic landmark motion

The action functional for the phase-space Hamilton’s principle is S

  • q(t), p(t)
  • =

T

  • N
  • a=1

pa ˙ qa − H(q, p)

  • dt −

T

M

  • i=1

hi(q, p) ◦ dWi(t), with H(q, p) = 1

2

N

a,b=1 papb K(qa, qb) for N landmarks. We take

N = 2, q ∈ R, and h1(q, p) = βp2, for noise in the q2 equation only. The corresponding discrete action functional is Sd =

K−1

  • k=0
  • N
  • a=1

pk

a

qk+1

a

− qk

a

∆t − H(qk+1, pk)

  • ∆t

−

K−1

  • k=0

M

  • i=1

hi(qk, pk) + hi(qk+1, pk+1) 2 ∆W k

i

Darryl D Holm Imperial College () Stochastic Solitons in Computational Anatomy Vienna, 20 Feb 2015 29 / 48

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SLIDE 30

Overtaking two-body collisions of pulsons & peakons

1

N = 2 Green’s functions for pulsons and peakons:

K(q1 − q2) = e−(q1−q2)2, (pulsons) K(q1 − q2) = e−2|q1−q2|, (peakons).

2

M = 1 dimensional noise with the potential h1(q, p) = βp2

3

Initial conditions:

¯ q1 = 0, ¯ p1 = 8, ¯ q2 = 10, ¯ p2 = 1, ¯ q1 = 0, ¯ p1 = 4, ¯ q2 = 10, ¯ p2 = 1, ¯ q1 = 0, ¯ p1 = 2, ¯ q2 = 10, ¯ p2 = 1, ¯ q1 = 0, ¯ p1 = 1, ¯ q2 = 10, ¯ p2 = 1,

Darryl D Holm Imperial College () Stochastic Solitons in Computational Anatomy Vienna, 20 Feb 2015 30 / 48

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SLIDE 31

Deterministic v Sample paths for Gaussian pulsons, with ¯ p1 = 4 and β = 4 Noise makes a big difference!

5 10 15 20 25 30 35 40 50 100 150 200

q

q1 q2 deterministic

5 10 15 20 25 30 35 40 50 100 150 200

q

5 10 15 20 25 30 35 40

t

50 100 150 200

q

Darryl D Holm Imperial College () Stochastic Solitons in Computational Anatomy Vienna, 20 Feb 2015 31 / 48

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SLIDE 32

Sample paths—Gaussian pulsons, ¯ p1 = 4 and β = 4

20 40 60 80 100 50 100 150 200 250 300 350 400

q

q1 q2

20 40 60 80 100 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5

p

p1 p2 p1 +p2

20 40 60 80 100 50 100 150 200 250 300 350

q

20 40 60 80 100 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5

p

20 40 60 80 100

t

50 100 150 200 250 300 350 400 450

q

20 40 60 80 100

t

1 2 3 4 5 6

p

Figure: Example numerical sample paths for Gaussian pulsons for the simulations with ¯ p1 = 4 and β = 4.

Darryl D Holm Imperial College () Stochastic Solitons in Computational Anatomy Vienna, 20 Feb 2015 32 / 48

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SLIDE 33

Mean paths—Gaussian pulsons, ¯ p1 = 4, vary β

20 40 60 80 100 50 100 150 200 250 300 350

q

E(q1) E(q2) 20 40 60 80 100 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5

p

E(p1) E(p2) E(p1 +p2) 20 40 60 80 100 50 100 150 200 250 300

q

20 40 60 80 100 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5

p

20 40 60 80 100

t

50 100 150 200 250 300 350

q

20 40 60 80 100

t

1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5

p

Figure: Numerical mean paths for Gaussian pulsons for the simulations with ¯ p1 = 4. Results for three example choices of the parameter β are presented: β = 1.5 (top), β = 2.5 (middle), and β = 4.5 (bottom).

Darryl D Holm Imperial College () Stochastic Solitons in Computational Anatomy Vienna, 20 Feb 2015 33 / 48

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SLIDE 34

Sample paths—Gaussian pulsons, ¯ p1 = 1 and β = 5

20 40 60 80 100 −40 −20 20 40 60 80 100

q

q1 q2

20 40 60 80 100 0.8 1.0 1.2 1.4 1.6 1.8 2.0

p

p1 p2 p1 +p2

20 40 60 80 100 −20 20 40 60 80 100 120

q

20 40 60 80 100 0.5 1.0 1.5 2.0

p

20 40 60 80 100

t

−20 20 40 60 80 100 120 140

q

20 40 60 80 100

t

0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

p

Figure: Example numerical sample paths for Gaussian pulsons for the simulations with ¯ p1 = 1 and β = 5.

Darryl D Holm Imperial College () Stochastic Solitons in Computational Anatomy Vienna, 20 Feb 2015 34 / 48

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SLIDE 35

Mean paths—Gaussian pulsons, ¯ p1 = 1, vary β

20 40 60 80 100 20 40 60 80 100 120

q

E(q1) E(q2)

20 40 60 80 100 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2

p

E(p1) E(p2) E(p1 +p2)

20 40 60 80 100 20 40 60 80 100 120

q

20 40 60 80 100 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2

p

20 40 60 80 100

t

20 40 60 80 100 120 140

q

20 40 60 80 100

t

0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2

p

Figure: Numerical mean paths for Gaussian pulsons for the simulations with ¯ p1 = 1. Results for three example choices of the parameter β are presented: β = 0.5 (top), β = 1 (middle), and β = 2 (bottom).

Darryl D Holm Imperial College () Stochastic Solitons in Computational Anatomy Vienna, 20 Feb 2015 35 / 48

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SLIDE 36

Probability of crossing, ¯ p1 = 4, vary β

−600 −400 −200 200 400 600 0.0000 0.0005 0.0010 0.0015 0.0020 0.0025 0.0030

ρ

−600 −400 −200 200 400 600 0.0000 0.0005 0.0010 0.0015 0.0020 0.0025

ρ

−600 −400 −200 200 400 600

∆q

0.0000 0.0005 0.0010 0.0015 0.0020 0.0025 0.0030

ρ

Figure: Numerical probability density ρ of the distance ∆q(t) = q2(t) − q1(t) at time t = 100 for Gaussian pulsons for the simulations with ¯ p1 = 4. Results for three example choices of the parameter β are presented: β = 1.5 (top), β = 2.5 (middle), and β = 4.5 (bottom).

Darryl D Holm Imperial College () Stochastic Solitons in Computational Anatomy Vienna, 20 Feb 2015 36 / 48

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SLIDE 37

Probability of crossing

1 2 3 4 5 6 7 0.0 0.2 0.4 0.6 0.8 1.0

P(∆q <0)

¯ p1 =1 ¯ p1 =2 ¯ p1 =4 ¯ p1 =8

1 2 3 4 5 6 7

β

0.0 0.2 0.4 0.6 0.8 1.0

P(∆q <0)

¯ p1 =1 ¯ p1 =2 ¯ p1 =4 ¯ p1 =8

Figure: The probability of crossing, that is, the probability that q2(t) < q1(t) at time t = 100, as a function of the parameter β for Gaussian pulsons (top) and peakons (bottom).

Darryl D Holm Imperial College () Stochastic Solitons in Computational Anatomy Vienna, 20 Feb 2015 37 / 48

slide-38
SLIDE 38

First crossing time

2 4 6 8 10 12 14

τ

0.00 0.05 0.10 0.15 0.20 0.25 0.30

ρ

Figure: Example probability density of the first crossing time Tc for Gaussian pulsons for the simulations with ¯ p1 = 4 and β = 2.5. More precisely, this is the conditional probability density given that Tc < ∞, i.e., assuming the pulsons do cross, the integral b

a ρ(τ) dτ yields the probability that the first

crossing occurs at time Tc ∈ [a, b].

Darryl D Holm Imperial College () Stochastic Solitons in Computational Anatomy Vienna, 20 Feb 2015 38 / 48

slide-39
SLIDE 39

First crossing time

1 2 3 4 5 6 7 10 20 30 40 50 60

E(Tc)

¯ p1 =1 ¯ p1 =2 ¯ p1 =4 ¯ p1 =8

1 2 3 4 5 6 7

β

20 40 60 80 100

E(Tc)

¯ p1 =1 ¯ p1 =2 ¯ p1 =4 ¯ p1 =8

Figure: The mean first crossing time E(Tc) as a function of the parameter β for Gaussian pulsons (top) and peakons (bottom). More precisely, this is the conditional expectation E(Tc|Tc < ∞) given that the pulsons do cross (i.e., Tc < ∞).

Darryl D Holm Imperial College () Stochastic Solitons in Computational Anatomy Vienna, 20 Feb 2015 39 / 48

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SLIDE 40

Stochastic potential for screened noise

The range of the stochastic effects between the two pulsons may be screened, for example, by applying the stochastic potential h1(q, p) = βp2 exp

  • − (q2 − q1)2

γ2

  • .

The parameter β adjusts the noise intensity, just as before. The parameter γ controls the range of the stochastic effects. Noise contributes now in both the momentum and position equations.

Darryl D Holm Imperial College () Stochastic Solitons in Computational Anatomy Vienna, 20 Feb 2015 40 / 48

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SLIDE 41

Sample paths with screened noise

2 4 6 8 10 5 10 15 20 25 30 35 40 45

q

q1 q2

2 4 6 8 10 1 2 3 4 5 6

p

p1 p2 p1 +p2

2 4 6 8 10

t

5 10 15 20 25 30 35 40

q

2 4 6 8 10

t

1 2 3 4 5 6

p

Figure: Examples of numerical paths for Gaussian pulsons for the simulations with the initial conditions ¯ q1 = 0, ¯ p1 = 4, ¯ q2 = 10, and ¯ p2 = 1, and the screened stochastic potential (40) with the parameters β = 4 and γ2 = 4.

Darryl D Holm Imperial College () Stochastic Solitons in Computational Anatomy Vienna, 20 Feb 2015 41 / 48

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SLIDE 42

Stratonovich and Itˆ

  • Stochastic EP equations

Stochastic Stratonovich and Itˆ

  • Euler-Poincar´

e equations (in general, not just for landmarks)

Darryl D Holm Imperial College () Stochastic Solitons in Computational Anatomy Vienna, 20 Feb 2015 42 / 48

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SLIDE 43

The diamond operation defines momentum maps

Definition (The diamond operation)

On a manifold M, the diamond operation (⋄) : T ∗V → X∗ is defined for a vector space V with (q, p) ∈ T ∗V and vector field ξ ∈ X is given in terms of the Lie-derivative operation Lu by Jξ(q, p) = J(q, p) , ξ X := p , −Lξq V = p ⋄ q , ξ X for the pairings · , · V : T ∗V × TV → R and · , · X : X∗ × X → R with p ⋄ q ∈ X∗.

Darryl D Holm Imperial College () Stochastic Solitons in Computational Anatomy Vienna, 20 Feb 2015 43 / 48

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SLIDE 44

Stratonovich stochastic variational principle [2]

Theorem (Stratonovich Stochastic Euler-Poincar´ e equations)

The action for the stochastic variational principle δS = 0 is,

S(u, p, q) = ℓ(u, q) +

  • p , dq

dt + Luq

  • V
  • dt
  • Lebesgue integral

+

i

p ⋄ q , ξi(x)X ◦ dWi(t)

  • Stratonovich integral

. This leads to the following Stratonovich form of the stochastic Euler–Poincar´ e (SEP) equations dm + Ldxt m − δℓ δq ⋄ q dt = 0 , dq = − Ldxt q , dp = δℓ δq dt + LT

dxt p ,

where dxt = u(x, t) dt −

i ξi(x) ◦ dWi(t) ∈ X

is the Stratonovich stochastic vector field and m := δℓ δu = p ⋄ q ∈ X∗ is the left momentum map, which evolves. The right momentum map is still conserved.

Darryl D Holm Imperial College () Stochastic Solitons in Computational Anatomy Vienna, 20 Feb 2015 44 / 48

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SLIDE 45

Itˆ

  • form of the stochastic Euler-Poincar´

e equations [2]

Corollary (Itˆ

  • form)

Upon transforming to the Itˆ

  • representation, the stochastic

Euler-Poincar´ e equations take the following form with m = p ⋄ q, dm + Ld

xtm dt − δℓ

δq ⋄ q dt = 1 2

  • j

Lξj(x)

  • Lξj(x)m
  • dt =: ∆Liem dt ,

dq + Ld

xtq dt = 1

2

  • j

Lξj(x)

  • Lξj(x)q
  • dt =: ∆Lieq dt ,

dp − LT

d xtp dt − δℓ

δq dt = −1 2

  • j

LT

ξj(x)

  • LT

ξj(x)p

  • dt =: ∆Liep dt ,

with Itˆ

  • stochastic vector field,

d xt = u(x, t) dt −

i ξi(x) dWi(t) .

Note: The right momentum map is conserved for Stratonovich, but not for Itˆ

  • , because of ∆Lie terms. Consequently, the peakon momentum

need not be normal to level sets of image data (q) on the Itˆ

  • path d

xt.

Darryl D Holm Imperial College () Stochastic Solitons in Computational Anatomy Vienna, 20 Feb 2015 45 / 48

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SLIDE 46

Summary

1

Our usual thinking about landmark dynamics is based on dual pairs of momentum maps

2

Previous work on stochastic landmark dynamics was based on additive noise in either the momentum initial condition [5], or the momentum equation of motion [7].

3

The latter is a stochastic Hamilton equation (SHE), and the entire theory of SHEs can be brought to bear on landmark dynamics.

4

For SHEs, placing an additive stochastic process in the landmark momentum equation does not cause topology change in the order

  • f landmarks on the real line, but it does in the position equation.

5

A change in the order of EPDiff landmarks on 2D contours would correspond to “penetration” of one contour through the other.

6

The use of SHEs in the development of stochastic growth models is just beginning!

Darryl D Holm Imperial College () Stochastic Solitons in Computational Anatomy Vienna, 20 Feb 2015 46 / 48

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SLIDE 47

Thanks for listening!

Thanks for listening!

Darryl D Holm Imperial College () Stochastic Solitons in Computational Anatomy Vienna, 20 Feb 2015 47 / 48

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SLIDE 48

References

JM Bismut [1981] M´ ecanique al´ eatoire, Berlin: Springer. DD Holm, Variational Principles for Stochastic Fluid Dynamics, arXiv e-print at http://arxiv.org/pdf/1410.8311.pdf DD Holm, JE Marsden. Progr. Math. 232:203–235 (2005). In The Breadth of Symplectic and Poisson Geometry, A Festschrift for Alan Weinstein, Birkh¨ auser, Boston, MA. DD Holm, JE Marsden, TS Ratiu [1998] The Euler–Poincar´ e equations and semidirect products with applications to continuum theories, Adv. in Math., 137:1-81. http://xxx.lanl.gov/abs/chao-dyn/9801015. DD Holm, J. Tilak Ratnanather, A Trouv´ e, L Younes [2004] Soliton dynamics in computational anatomy, NeuroImage, 23 (S1): S170–S178. JA L´ azaro-Cam´ ı and JP Ortega [2008] Stochastic Hamiltonian dynamical systems, Rep. Math. Phys., 61 (1): 65–122. A Trouv´ e, FX Vialard [2012]. Shape splines and stochastic shape evolutions: a second order point of view. Quart. Appl. Math 70, 21925110.1090/S0033-569X-2012-01250-4

Darryl D Holm Imperial College () Stochastic Solitons in Computational Anatomy Vienna, 20 Feb 2015 48 / 48