38 research outputs found
Collective symplectic integrators
We construct symplectic integrators for Lie-Poisson systems. The integrators
are standard symplectic (partitioned) Runge--Kutta methods. Their phase space
is a symplectic vector space with a Hamiltonian action with momentum map
whose range is the target Lie--Poisson manifold, and their Hamiltonian is
collective, that is, it is the target Hamiltonian pulled back by . The
method yields, for example, a symplectic midpoint rule expressed in 4 variables
for arbitrary Hamiltonians on . The method specializes in
the case that a sufficiently large symmetry group acts on the fibres of ,
and generalizes to the case that the vector space carries a bifoliation.
Examples involving many classical groups are presented
On Projective Equivalence of Univariate Polynomial Subspaces
We pose and solve the equivalence problem for subspaces of ,
the dimensional vector space of univariate polynomials of degree . The group of interest is acting by projective transformations
on the Grassmannian variety of -dimensional
subspaces. We establish the equivariance of the Wronski map and use this map to
reduce the subspace equivalence problem to the equivalence problem for binary
forms
M\"obius Invariants of Shapes and Images
Identifying when different images are of the same object despite changes
caused by imaging technologies, or processes such as growth, has many
applications in fields such as computer vision and biological image analysis.
One approach to this problem is to identify the group of possible
transformations of the object and to find invariants to the action of that
group, meaning that the object has the same values of the invariants despite
the action of the group. In this paper we study the invariants of planar shapes
and images under the M\"obius group , which arises
in the conformal camera model of vision and may also correspond to neurological
aspects of vision, such as grouping of lines and circles. We survey properties
of invariants that are important in applications, and the known M\"obius
invariants, and then develop an algorithm by which shapes can be recognised
that is M\"obius- and reparametrization-invariant, numerically stable, and
robust to noise. We demonstrate the efficacy of this new invariant approach on
sets of curves, and then develop a M\"obius-invariant signature of grey-scale
images
Joint Invariants of Primitive Homogenous Spaces
Joint invariants are motivated by the study of congruence problems in Euclidean geometry, where they provide necessary and sufficient conditions for congruence. More recently joint invariants have been used in computer image recognition problems. This thesis develops new methods to compute joint invariants by developing a reduction technique, and applies the reduction to a number of important examples