Averaged Template Matching Equations

By exploiting an analogy with averaging procedures in fluid dynamics, we present a set of averaged template matching equations. These equations are analogs of the exact template matching equations that retain all the geometric properties associated with the diffeomorphismgrou p, and which are expected to average out small scale features and so should, as in hydrodynamics, be more computationally efficient for resolving the larger scale features. Froma geometric point of view, the new equations may be viewed as coming from a change in norm that is used to measure the distance between images. The results in this paper represent first steps in a longer termpro gram: what is here is only for binary images and an algorithm for numerical computation is not yet operational. Some suggestions for further steps to develop the results given in this paper are suggested

Momentum Maps and Measure-valued Solutions (Peakons, Filaments and Sheets) for the EPDiff Equation

We study the dynamics of measure-valued solutions of what we call the EPDiff equations, standing for the {\it Euler-Poincar\'e equations associated with the diffeomorphism group (of $\mathbb{R}^n$ or an $n$-dimensional manifold $M$)}. Our main focus will be on the case of quadratic Lagrangians; that is, on geodesic motion on the diffeomorphism group with respect to the right invariant Sobolev $H^1$ metric. The corresponding Euler-Poincar\'e (EP) equations are the EPDiff equations, which coincide with the averaged template matching equations (ATME) from computer vision and agree with the Camassa-Holm (CH) equations in one dimension. The corresponding equations for the volume preserving diffeomorphism group are the well-known LAE (Lagrangian averaged Euler) equations for incompressible fluids. We first show that the EPDiff equations are generated by a smooth vector field on the diffeomorphism group for sufficiently smooth solutions. This is analogous to known results for incompressible fluids--both the Euler equations and the LAE equations--and it shows that for sufficiently smooth solutions, the equations are well-posed for short time. In fact, numerical evidence suggests that, as time progresses, these smooth solutions break up into singular solutions which, at least in one dimension, exhibit soliton behavior. With regard to these non-smooth solutions, we study measure-valued solutions that generalize to higher dimensions the peakon solutions of the (CH) equation in one dimension. One of the main purposes of this paper is to show that many of the properties of these measure-valued solutions may be understood through the fact that their solution ansatz is a momentum map. Some additional geometry is also pointed out, for example, that this momentum map is one leg of a natural dual pair.Comment: 27 pages, 2 figures, To Alan Weinstein on the occasion of his 60th Birthda

Integration of the EPDiff equation by particle methods

The purpose of this paper is to apply particle methods to the numerical solution of the EPDiff equation. The weak solutions of EPDiff are contact discontinuities that carry momentum so that wavefront interactions represent collisions in which momentum is exchanged. This behavior allows for the description of many rich physical applications, but also introduces difficult numerical challenges. We present a particle method for the EPDiff equation that is well-suited for this class of solutions and for simulating collisions between wavefronts. Discretization by means of the particle method is shown to preserve the basic Hamiltonian, the weak and variational structure of the original problem, and to respect the conservation laws associated with symmetry under the Euclidean group. Numerical results illustrate that the particle method has superior features in both one and two dimensions, and can also be effectively implemented when the initial data of interest lies on a submanifold

Geodesic Equations on Diffeomorphism Groups

We bring together those systems of hydrodynamical type that can be written as geodesic equations on diffeomorphism groups or on extensions of diffeomorphism groups with right invariant $L^2$ or $H^1$ metrics. We present their formal derivation starting from Euler's equation, the first order equation satisfied by the right logarithmic derivative of a geodesic in Lie groups with right invariant metrics.Comment: This is a contribution to the Proc. of the Seventh International Conference ''Symmetry in Nonlinear Mathematical Physics'' (June 24-30, 2007, Kyiv, Ukraine), published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA