Invariant Geometric Curvilinear Optimization with Restricted Evolution Dynamics

Geometric Dynamics of Optimization

This paper investigates a family of dynamical systems arising from an evolutionary re-interpretation of certain optimal control and optimization problems. We focus particularly on the application in image registration of the theory of metamorphosis. Metamorphosis is a means of tracking the optimal changes of shape that are necessary for registration of images with various types of data structur...

Extracting Curvilinear Structures: A Differential Geometric Approach

In this paper a method to extract curvilinear structures from digital images is presented. The approach is based on differential geometric properties of the image function. For each pixel, the second order Taylor polynomial is computed by convolving the image with the derivatives of a Gaussian smoothing kernel. Line points are required to have a vanishing gradient and a high curvature in the di...

Basic geometric invariant theory

Equivalence of Topological Form for Curvilinear Geometric Objects

Given a curvilinear geometric object in R 3 , made up of properly-joined para-metric patches deened in terms of control points, it is of interest to know under what conditions the object will retain its original topological form when the control points are perturbed. For example, the patches might be triangular B ezier surface patches, and the geometric object may represent the boundary of a so...

Geometric invariant theory and flips

Ever since the invention of geometric invariant theory, it has been understood that the quotient it constructs is not entirely canonical, but depends on a choice: the choice of a linearization of the group action. However, the founders of the subject never made a systematic study of this dependence. In light of its fundamental and elementary nature, this is a rather surprising gap, and this pap...