Markov processes invariant under a Lie group action

Metamorphoses Through Lie Group Action

We formally analyze a computational problem which has important applications in image understanding and shape analysis. The problem can be summarized as follows. Starting from a group action on a Riemannian manifold M , we introduce a modification of the metric by partly expressing displacements on M as an effect of the action of some group element. The study of this new structure relates to ev...

Discrete Time Scale Invariant Markov Processes

In this paper we consider a discrete scale invariant Markov process {X(t), t ∈ R} with scale l > 1. We consider to have some fix number of observations in every scale, say T , and to get our samples at discrete points α, k ∈ W, where α is obtained by the equality l = α and W = {0, 1, . . .}. So we provide a discrete time scale invariant Markov (DT-SIM) process X(·) with parameter space {α, k ∈ ...

INVARIANT MEASURES FOR MARKOV PROCESSES(i)

State observers for invariant dynamics on a Lie group

This paper concerns the design of full state observers for state space systems where the state is evolving on a finite dimensional, connected Lie group. Traditional full state observer and filter designs for systems evolving in vector spaces employ the following design principle, going back to the work of Kalman [6] and Luenberger [7] on linear systems. The observer system is designed as a comb...

Distance to the Convex Hull of an Orbit under the Action of a Compact Lie Group

For a real vector space V acted on by a group K and fixed x and y in V , we consider the problem of finding the minimum (resp., maximum) distance, relative to a Kinvariant convex function on V , between x and elements of the convex hull of the K-orbit of y. We solve this problem in the case where V is a Euclidean space and K is a finite reflection group acting on V . Then we use this result to ...