Solitons, Momentum

& Wave Fronts in Imaging

Science

Darryl D. HolmLos Alamos National Laboratory

&Math @ Imperial College London

IPAM Summer SchoolMath in Brain Imaging

July 15, 2004

x

( , )x t

)1,( x

( ) ( ( ) )t v t

Variational Template Matching for ImagesMiller, Mumford, Younes, Trouvé, Ratnanather…

Satisfies the EPDiff Equation:0 1( (1))I I

1

0

0 1( , ) || ( ) ||Lv V

D I I inf v t dt

Where else have we seen EPDiff?

Momenta of Images along geodesics obey EPDiff & so do Water Waves!

EPDiff – What an equation!Now, same equations have the same solutions.So, let’s have some technology transfer!

Messages here: (1) Momentum is key & (2) Internal Wave Fronts in the Ocean are

analogs of Landmarks in Imaging Science!

Our Story Today• Background for EPDiff equation• Recent confluence of EPDiff ideas in template matching &

in fluid dynamics (Arnold, Hirani, Marsden, Miller, Mumford, Ratiu, Trouvé, Younes, et al.)

• Top 10 reasons why IVP for EPDiff is good for Imaging Science (with Tilak Rananather)

• Landmarks in Imaging are Singular Solitons in IVP• Singular Solitons! What are those? Invariant manifolds

expressed as Momentum Maps! (Momentum is the key!)• Supported on Points in 1D – Curves in 2D – Surfaces in 3D• Open problems:

(1) Numerics (cf. Hirani & Desbrun) & Stability Issues (2) Reversibility (memory wisps)

Recent confluence of ideas for EPDiff in template matching & ideal fluid dynamics

– Arnold (1966) Euler eqns arise from variational principle: EPDiffvol(L

2) – Ebin & Marsden (1970)

Smooth Euler Solutions exist (for finite time)– Mumford (1998), Younes (1998)

Template matching for brain imaging is an EPDiff eqn too!– Holm, Marsden, Ratiu (1998)

Semidirect-Product EP eqns, including EPDiff for continua– Miller, Trouvé & Younes (2002)

Synthesis of EPDiff approaches for Computational Anatomy– Hirani, Desbrun et al. (2003)

Discrete Exterior Calculus for EPDiff– Holm & Marsden (2004)

Momentum maps & singular solitons of EPDiff

Template Matching (Imaging)& Water Waves Share EPDiff!

• EPDiff is essentially geometrical– Geodesic motion on the smooth maps– Arises from a variational principle – Has both Optimization and IVP solutions– Conserves Momentum

• Momentum is a key concept for both:(1) Interactions of water waves & (2) Initial value problem (IVP) for Imaging Science

Template Matching usually focuses on Optimization for EPDiff. Instead, we shall focus on its IVP.

Two viewpoints of EPDiff,shared concepts & our goal

• EPDiff: Geodesic motion on the smooth maps (diffeos) (1) Optimization: Minimum distance between two images (2) IVP: Evolution of image outlines (curves) & momenta

• (1) Brain Imaging often uses Optimal Template Morphing– Arises from a geodesic variational principle – Template outlines evolve along optimal path

• (2) Water-wave solitons propagate and interact by colliding – Soliton wave fronts collide elastically – Elastic collisions conserve momentum

• Momentum of Singular Solutions is a key SHARED concept – contains information for both applications of EPDiff

• Goal: Transfer momentum ideas from Fluids to Imaging Science

Top 10 Reasons Why Image Science Needs IVP for EPDiff (with Tilak R)1. Provides new singular soliton paradigm for evolution &

interaction of image outlines (cartoons) by collisions

2. Momentum map : TS* –>g* for singular solutions of EPDiff – Related to “landmark dynamics,” but also has momentum– Landmark positions ++ their momenta, define an invariant manifold of the IVP for EPDiff

3. Linearity of g* and of TS* implies we can add momenta of images: This allows noise to be added to images and statistics for images to be derived for the IVP

4. Decomposition:1D sections of 2D evolution show 1D behavior: Cartoon/outline dynamics decomposes into elastic collisions.This recalls contacts in fluids (jets, convergent flows & pulses)

Top 10 Reasons Why Image Science Needs IVP for EPDiff

5. Reconnection, or Merger, of image outlines in 2D (and in 3D) shows reversible changes of topology. Note: Reconnection requires memory for reversible changes of topology. (Note the “memory wisps” in the animations below.)

6. 2D section of 3D evolution shows 2D behavior, so we may build up from image mapping in 2D to growth evolution in 3D, where reconnections are a type of morphogenesis (see 3D animations)

7. New perspectives and insights emerge: For example, momentum transfer in 3D growth evolution leads to jet formation by interacting contact discontinuities

Top 10 Reasons Why Image Science Needs IVP for EPDiff

8. Dynamical optimal landmarks emerge from smooth initial data. In IVP, initial and final states are on the same invariant manifold.

9. These optimal landmarks are EPDiff singular solutions, which evolve as coadjoint orbits of the (left) action of diffeos on smoothly embedded submanifolds Sk in Rn. – This motion preserves topology and is reversible in time.

10. Landmarks are not enough to describe reconnection without supplying the subsidiary data to maintain the memory too. Note: Velocity is not the correct additional variable – Instead, one needs momenta of the image outlines, too!

Solitons along a boundary

Soliton Packets at Gibraltar Strait

Synthetic Aperture Radar Image of Soliton Formation at Gibraltar

Gibraltar Soliton Emerging

Soliton wave train at Gibraltar

Other 2D solitons?

We need a 2D extension of KdV• KP is a known (quasi-1D) 2+1 extension of 1+1 KdV.

But KP is only weakly nonlinear and weakly transverse.

• 1+1 CH extends 1+1 KdV to higher asymptotic orderand is nonlinear to quadratic order.

• Dispersionless CH is EPDiff in any number of dimensions

• Here, we shall discuss singular solutions for EPDiff in 2D and 3D

Solitons at Gibraltar Strait are 2D

We shall show two types of numerics for EPDiff in 2D:

• Eulerian -- Martin F. Staley (T-7)

• Lagrangian -- Shengtai Li (T-7)

• The numerics show emergence of 2D filament solitons and their basic interactions, including reconnection

What did we see? EPDiff solutions in 2D form

Lagrangian momentum filaments• 2D EPDiff eqn (SLCM, small potential energy limit)• Velocity forms coherent “solitons” of width alpha (in

velocity) These “diffeons” move with the fluid in 2D as Lagrangian momentum filaments (coadjoint dynamics)

• Nonlinear interactions between filament “diffeons” locally obey the 1D soliton collision rules

• Reconnection occurs, just as for internal waves, provided the numerical method is adequate -- killer ap!

Summary of EPDiff Diffeons (Lagrangian momentum filaments)

• CH peakons (points on the line) generalize to EPDiff diffeons defined on Sk of Rn

• Diffeons evolve as coadjoint orbits of the diffeos acting (from the left) on smoothly embedded subspaces Sk of Rn with k<n

• Numerical observation: Diffeon dynamics is stable for codimension-one singular solutions

• The reconnection of diffeons occurs reversiblyby the action of diffeos on embedded manifolds.

Open Questions• Why do only diffeons form in the IVP?

– The N-diffeon invariant manifold is a coadjoint orbit– 2D & 3D behavior both mimic 1D peakons: Why?– Does smoothness of geodesic flow break down?– Is geodesic flow on diffeos ill-posed?

• How to encode momentum of outlines for IVP of template dynamics into optimization problem?

• General question: coadjoint dynamics for left action of diffeos on arbitrary distributions embedded in Rn?

• Lagrangian representation of image outlines?

What about diffeons in 3D?

What did we see? EPDiff solutions (diffeons) in 3D

form Lagrangian momentum surfaces• The geodesic evolution of shape involves

momentum

• The momentum map yields embedded surfaces (Landmarks - invariant manifold)

• The Landmarks interact by collisions that may cause mergers, or reconnections

End