The large deformation matching (LDM) framework is a method for registration
of images and other data structures, used in computational anatomy.
We show how to reformulate the large deformation matching framework for
registration in a geometric way. The general framework also allows to generalize
the large deformation matching framework to include multiple scales
by using the iterated semidirect product of groups.
An important ingredient in the LDM framework is the choice of a suitable
Riemannian metric on the space of diffeomorphisms. Since the space in
question is infinite-dimensional, not every choice of the metric is suitable.
In particular the geodesic distance, which is defined as the infimum over
the length of all paths connecting two points, may vanish. For the family
of Sobolev-type Hs-metrics on the diffeomorphism groups of R and S1 we
establish that the geodesic distance vanishes for metrics of order 0 ≤ s ≤ [half].
The geodesic distance also vanishes for the L2-metric on the Virasoro-Bott
group, which is a central extension of the diffeomorphism group of the circle.
Vanishing of geodesic distance implies that the length-functional, which
assigns to each curve in the manifold its length, has no global minima, when
restricted to paths with fixed endpoints. We show that for the L2-metric on
the diffeomorphism group of R and the Virasoro-Bott group doesn't have
any local minima either.
The large deformation matching framework is not the only approach to
the registration and shape comparison. For curves and surfaces it is possible
to define a Riemannian metric directly on the space of curves or surfaces and
use geodesics with respect to this metric to measure differences in shape. We
use the family of Sobolev-type metrics on surfaces from [7]. We show how
to discretize the geodesic equations and solve the boundary value problem
via a shooting method on the initial velocity. The discrete equations are
implemented via the finite element method.