Shape Space, Recognition, Minimal Distortion, Vision Groups and Applications
Visual objects are often known up to some ambiguity, depending on the
methods used to acquire them. The first-order approximation to any
transformation is, by definition, affine, and the affine approximation to
changes between images has been used often in computer vision. Thus it is
beneficial to deal with objects known only up to an affine transformation.
For example, feature points on a planar transform projectively between
different views, and the projective transformation can in many cases be
approximated by an affine transformation.
More generally, given two visual objects in a containing Euclidean space
R^k, one may study vision group actions between these two objects often
with an underlying signature which are equivalent under some symmetry or
minimal distortion action with respect to a suitable metric inherited by
this action. For example, Euclidean groups, similarity, Equi-Affine,
projections, camera rotations and video groups.
The study of the space of ordered configurations of n
distinct points in R^k up to similarity transformations was pioneered
by Kendall who coined the name shape space. For
different groups of transformations (rigid, similarity, linear,
affine, projective for example) one obtains different shape spaces.
Moreover, while these formulations allow often global optimal
optimization, e.g. using convex objectives , many of the problems above
require efficient approximation methods which work locally.
This framework has applications to
biological structural molecule reconstruction problems, to recognition
tasks and to matching features across images with minimal distortion”
This talk will discuss various work with collaborators around this circle
of ideas.