Solitons, Momentum & Wave Fronts in Imaging Science

Darryl D. HolmLos Alamos National Laboratory&Math @ Imperial College London

IPAM Summer SchoolMath in Brain ImagingJuly 15, 2004

Variational Template Matching for ImagesMiller, Mumford, Younes, Trouv, Ratnanather Satisfies the EPDiff Equation:

EMBED Equation.3

EMBED Equation.3

EMBED Equation.3

EMBED Equation.3

EMBED Equation.3

EMBED Equation.3

EMBED Equation.3

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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, lets 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 TodayBackground for EPDiff equationRecent confluence of EPDiff ideas in template matching & in fluid dynamics (Arnold, Hirani, Marsden, Miller, Mumford, Ratiu, Trouv, 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 IVPSingular Solitons! What are those? Invariant manifolds expressed as Momentum Maps! (Momentum is the key!)Supported on Points in 1D Curves in 2D Surfaces in 3DOpen 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(L2) 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, Trouv & 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 goalEPDiff: 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 EPDiff5. 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 EPDiff8. 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 KdVKP 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 order and 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 filaments2D 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 rulesReconnection 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

Open QuestionsWhy 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 surfacesThe geodesic evolution of shape involves momentumThe momentum map yields embedded surfaces (Landmarks - invariant manifold)The Landmarks interact by collisions that may cause mergers, or reconnections

End

This is the cover of the TWG submission for the RD100 award. It shows images from CCN-8 of level surfaces of vorticity in the TWGs Big Science 2028-cubed run, simulating a new turbulence decay experiment at JHU. The equations are the EP eqns. Im going to explain how ad* on the LHS contributes to the formation of coherent structures. A hint for understanding ad* comes from soliton theory in 1D and 2D. Then we go to turbulence modeling in 3D.This is the way you usually see solitons in shallow water, scurrying along like mice in the gutter, at a scale of a only a few inches in width and height.

Although they may appear insignificant, these coherent structures stimulated a lot of activity in nonlinear science in the past 35 years!In the absence of dispersion, the CH equation has peakon solutions.We see that only peakons emerge from a confined initial condition. In the absence of linear dispersion, the CH iso-spectrum is purely discrete -- only peakons are present in the solution.A train of rightward moving solitons emerges, each traveling at a speed equal to its height, led by the tallest one. The profile of each one is the Greens function for the relation between velocity and momentum.

The traveling wave shape of the peakon is the profile of the Greens function that appears in the momentum norm for the geodesic motion.

Here we see the soliton interactions of the peakons in overtaking collisions. Their collisions transfer momentum nonlinearly, then they re-emerge unscathed, with only a shift in phase that depends on the relative velocity.At the upper left corner of this Shuttle Picture is the Atlantic Ocean, which is fresher than the Mediterranean Sea. The Med lies on the right (East) side of the Gibraltar Strait, which is about 30 km wide. (This sets the scale for the wavelength.) The high tide drives fresh water Eastward from the Atlantic up over a sill in the topography that sits just outside the Strait. The incoming flow accelerates as it rises over the sill and rides over the outgoing saltier flow from the Med. At the step in the topography the halocline is disturbed, which sends out a train of internal waves Eastward into the Med. The wave trains are separated in bursts about every 12 hours, with the cycles of the tides. The Westward outflow during low tide does not excite these waves. We will need a 2D equation to deal with these wave patterns on the surface. Take a look at the lower right corner at the wave-wave interaction patterns there. We also see a Southward diffraction of the waves when they enter the Med, as these are supported more as they come around the lower boundary of the Strait.The map shows the Gibraltar Strait and the Camarinal Sill.

Tidal flow of the Atlantic over the Camarinal Sill excites internal waves on the halocline between the fresher Atlantic water and saltier, denser, Med water.

These nonlinear internal waves of amplitude 60-100 meters show up as just a slight deflection of the waters surface.This Lagrangian solution ansatz and the canonical Hamiltonian form of the CH equations is a big hint for solitons. It says that in 2D the CH eqn describes evolution of solutions whose momentum is defined on delta-functions along curves that move with the fluid velocity. So we know the CH will support these soliton-like solutions. The first issue is whether they are stable! Their stability analysis is straightforward, because their dynamics conserves a norm. Their solution behavior was investigated numerically. (Much more analysis of their solution behavior remains to be done!) We will need a 2D equation to deal with these patterns. Take a look at the lower right corner at the wave-wave interaction patterns there. We also see diffraction of the waves as these come around the lower boundary of the Strait.