EPDiff is short for ``Euler-Poincar\'e equations on the diffeomorphisms.'' EPDiff first arose as a 1D shallow water wave equation, whose weak solutions are solitons, called ``peakons.'' The initial value problem (IVP) for EPDiff in 2D produces emergent soliton-like weak solutions, supported on curves that evolve in the plane. These curves model internal waves in the ocean. Numerical
simulations show that weak solutions supported on ``peakon filaments'' emerge in the IVP of EPDiff, for any confined smooth initial velocity distribution.
Besides dominating the IVP, the weak solutions of EPDiff have three other interesting dynamical properties:
-- they superpose,
-- they form an invariant manifold and
-- their nonlinear interactions allow them to {\it reconnect} with each other in 2D.
The phenomenon of reconnection seen in the IVP for EPDiff is also observed in oceanic internal waves, as seen from the space shuttle using synthetic aperture radar (SAR). Thus, in accord with their original derivation in 1D, weak solutions of EPDiff provide a simplified 2D description of evolving arrays of interacting internal waves in the Ocean.
Remarkably, the same EPDiff equation {\it also arises in image processing} using template matching, an optimization approach in computational anatomy. Here, for example, a 2D measure-valued EPDiff solution optimally interpolates between the outlines, or ``cartoons," of a planar image and its target image obtained by observations at two times. This is template matching. The nonlinear exchange of momentum seen in the interactions of these ``cartoons" introduces the collison paradigm from soliton dynamics into imaging science. Namely, the optimization problem
for template matching corresponds to an evolutionary problem in which image outlines exchange momentum and may reconnect as their positions evolve. In 3D, measure-valued solutions of EPDiff correspond to suface boundaries in 3D images, representing, say, the sequence of shapes executed in a heartbeat.
The existence of these measure-valued solutions of EPDiff is guaranteed -- for any Sobolev norm, and in any number of spatial dimensions. This holds, because the weak solution ansatz is a momentum map for the (left) action of diffeomorphisms on the measure-valued support set of the solutions.
We review these two contexts for EPDiff and show numerical
and analytical results for its solutions in 1D, 2D and 3D.
(EPDiff -- optimization and evolution -- what an equation!)
