Abstract not found

In this paper we develop new Newton and conjugate gradient algorithms on the Grassmann and Stiefel manifolds. These manifolds represent the constraints that arise in such areas as the symmetric eigenvalue problem, nonlinear eigenvalue problems, electronic structures computations, and signal processing. In addition to the new algorithms, we show how the geometrical framework gives penetrating new insights allowing us to create, understand, and compare algorithms. The theory proposed here provides a taxonomy for numerical linear algebra algorithms that provide a top level mathematical view of previously unrelated algorithms. It is our hope that developers of new algorithms and perturbation theories will benefit from the theory, methods, and examples in this paper.

this paper is as follows: in Section 2, we collect some mathematical preliminaries from the literature on controllability of nonlinear systems and on classification of free Lie algebras. These are drawn from classical references in control theory [4, 17, 18, 36, 40] and Lie algebras [15, 43]. In Section 3, using some outstanding results of Brockett on optimal steering of certain classes of systems as motivation [5], we discuss the use of sinusoidal inputs for steering systems of first order, i.e., systems where controllability is achieved after just one level of Lie brackets of the input vector fields. Section 4 attempts to expand the domain of applicability of these results to more complex systems, where several orders of Lie brackets are needed to obtain the full Lie algebra associated with the input distribution. The 4 MURRAY AND SASTRY

We present a new set of algorithms for line-art rendering of smooth surfaces. We introduce an efficient, deterministic algorithm for finding silhouettes based on geometric duality, and an algorithm for segmenting the silhouette curves into smooth parts with constant visibility. These methods can be used to find all silhouettes in real time in software. We present an automatic method for generating hatch marks in order to convey surface shape. We demonstrate these algorithms with a drawing style inspired by A Topological Picturebook by G. Francis.

In this paper, we study the capacity of multiple-antenna fading channels. We focus on the scenario where the fading coefficients vary quickly; thus an accurate estimation of the coefficients is generally not available to either the transmitter or the receiver. We use a noncoherent block fading model proposed by Marzetta and Hochwald. The model does not assume any channel side information at the receiver or at the transmitter, but assumes that the coefficients remain constant for a coherence interval of length symbol periods. We compute the asymptotic capacity of this channel at high signal-to-noise ratio (SNR) in terms of the coherence time , the number of transmit antennas , and the number of receive antennas . While the capacity gain of the coherent multiple antenna channel is min bits per second per hertz for every 3-dB increase in SNR, the corresponding gain for the noncoherent channel turns out to be (1 ) bits per second per herz, where = min 2 . The capacity expression has a geometric interpretation as sphere packing in the Grassmann manifold.

Primal and dual nondegeneracy conditions are defined for semidefinite programming. Given the existence of primal and dual solutions, it is shown that primal nondegeneracy implies a unique dual solution and that dual nondegeneracy implies a unique primal solution. The converses hold if strict complementarity is assumed. Primal and dual nondegeneracy assumptions do not imply strict complementarity, as they do in LP. The primal and dual nondegeneracy assumptions imply a range of possible ranks for primal and dual solutions X and Z. This is in contrast with LP where nondegeneracy assumptions exactly determine the number of variables which are zero. It is shown that primal and dual nondegeneracy and strict complementarity all hold generically. Numerical experiments suggest probability distributions for the ranks of X and Z which are consistent with the nondegeneracy conditions.

Many differential equations of practical interest evolve on Lie groups or on manifolds acted upon by Lie groups. The retention of Lie-group structure under discretization is often vital in the recovery of qualitatively correct geometry and dynamics and in the minimization of numerical error. Having introduced requisite elements of differential geometry, this paper surveys the novel theory of numerical integrators that respect Lie-group structure, highlighting theory, algorithmic issues and a number of applications.

We consider nonlinear systems with input-to-output stable (IOS) unmodelled dynamics which are in the "range" of the input. Assuming the nominal system is globally asymptotically stabilizable and a nonlinear small gain condition is satisfied, we propose a first control law such that all solutions of the perturbed system are bounded and the state of the nominal system is captured by an arbitrarily small neighborhood of the origin. The design of this controller is based on a gain assignment result which allows us to prove our statement via a Small-Gain Theorem [JTP, Theorem 2.1]. However, this control law exhibits a high gain feature for all values. Since this may be undesirable, in a second stage we propose another controller with different characteristics in this respect. This controller requires more a priori knowledge on the unmodelled dynamics, as it is dynamic and incorporates a signal bounding the unmodelled effects. However this is only possible by restraining the IOS property into the exp-IOS property. Nevertheless we show that, in the case of input-to-state stability (ISS) -- the output is the state itself --, ISS and exp-ISS are in fact equivalent properties.

We present a new algorithm to detect humans in still images utilizing covariance matrices as object descriptors. Since these descriptors do not lie on a vector space, well known machine learning techniques are not adequate to learn the classifiers. The space of d-dimensional nonsingular covariance matrices can be represented as a connected Riemannian manifold. We present a novel approach for classifying points lying on a Riemannian manifold by incorporating the a priori information about the geometry of the space. The algorithm is tested on INRIA human database where superior detection rates are observed over the previous approaches.

Abstruct-Zstimating the three-dimensional motion of an object from a sequence of projections is of paramount importance in a variety of applications in control and robotics, such as autonomous navigation, manipulation, servo, tracking, docking, planning, and surveillance. Although “visual motion estimation” is an old problem (the first formulations date back to the beginning of the century), only recently have tools from nonlinear systems estimation theory hinted at acceptable solutions. In this paper we formulate the visual motion estimation lproblem in terms of identification of nonlinear implicit systems with parameters on a topological manifold and propose a dynamic solution either in the local coordinates or in the embedding space of the parameter manifold. Such a formulation has structural advantages over previous recursive schemes, since the estimation of motion is decoupled from the estimation of the structure of