Robert B. Gardner

Ridges for Image Analysis

Representation of object shape by medial structures has been an important aspect of image analysis. Methods for describing objects in a binary image by medial axes are well understood. Many attempts have been made to construct similar medial structures for objects in gray scale images. In particular, researchers have studied images by analyzing the graphs of the intensity data and identifying ridge and valley structures on those surfaces. In this paper we review many of the definitions for ridges. Computational vision models require that medial structures should remain invariant under certain transformations of the spatial locations and intensities. For each ridge definition we point out which invariances the definition satisfies. We also give extensions of the concepts so that we can located-dimensional ridge structures withinn-dimensional images. A comparison of the ridge structures produced by the different definitions is given both by mathematical examples and by an application to a 2-dimensional MR image of a head.

The Method of Equivalence and Its Applications

The ideas of Elie Cartan are combined with the tools of Felix Klein and Sophus Lie to present in this book the only detailed treatment of the method of equivalence. An algorithmic description of this method, is presented for the first time.

The GS algorithm for exact linearization to Brunovsky normal form

An algorithm utilizing the minimal number of integrations for the exact linearization of nonlinear systems to Brunovsky normal form under nonlinear feedback is presented. The tools which are involved are based on classical constructions appearing in the theory of exterior differential systems. >

Feedback equivalence of control systems

Abstract Cartans method of equivalence is applied to the problem of equivalence of 2-state systems with scalar control under feedback. The differential invariants produced by the method completely characterize equivalence classes. The phenomenon of linearizability is associated with the presence of infinite Lie pseudogroups. The generic non-linearizable case has a geometrically defined variational problem which yields a time optimal closed loop control.

Symmetry and the implementation of feedback linearization

Abstract We show that, by making use of the symmetry group of the system, one may find the linearizing feedback for any controllable linearizable system with p controls by solving independent completely integrable Pfaffian systems whose dimensions are the multiplicities of the Kronecker indices, and carrying out purely algebraic calculations. In particular, if the Kronecker indices are distinct, this means solving p independent one dimensional completely integrable Pfaffian systems.

Feedback equivalence for general control systems

Abstract We discuss the solution to the problem of local equivalence of control systems with n states and p controls in a neighbourhood of a generic point, under the Lie pseudo-group of local time independent feedback transformations. We have shown earlier that this problem is identical with the problem of simple equivalence of the time optimal variational problem. Here we indicate the way in which this identification may be used to obtain closed loop time critical controls for general systems. We show that the classification of general nonlinear systems depends on the classification of all n−p dimensional affine subspaces of the space of symmetric forms in p variables and that the case of control linear systems depends on the classification of all n−p dimensional affine subspaces of the space of skew forms in p variables. We show that in the latter case the G-structure is the prolongation of one determined by a state space transformation group. We give a complete list of normal forms for control linear systems in the case p=n−1 .

A geometric isomorphism with applications to closed loop controls

Feedback equivalence of n state,

An algorithm for feedback linearization

n - 1

Generalized spherical functions on projectively flat manifolds

control systems satisfying certain regularity conditions divides such systems into two invariant classes. We show that class one corresponds, via a geometric isomorphism, to classical Lagrangian variational problems. We prove the existence of time critical closed loop controls for systems that satisfy the nondegeneracy condition that the analogue of the Hessian for the Lagrangian problem have full rank. We show that the vanishing of this Hessian characterizes the control linear systems in class one and identify the rank condition for local controllability for such systems as the nonvanishing of a differential invariant. The control linear systems in class two are also characterized by the vanishing of an invariant and the rank condition is identified.

Classical geometries arising in feedback equivalence

Abstract Previous methods for exact linearization by feedback have relied on solving Frobenius systems of partial differential equations of dimensions equal to the Kronecker indices. We will describe an algorithm whereby one may find the linearizing feedback for any controlable linearizable system having distinct Kronecker indices with p -controls by purely algebraic calculations and integration of at most p one-dimensional Frobenius systems. The paper concludes with a concrete example considered by Hunt-Su-Meyer in their paper [3].