Showing papers by "John B. Moore published in 2004"

A Newton-like method for solving rank constrained linear matrix inequalities

Abstract: This paper presents a Newton?like algorithm for solving systems of rank constrained linear matrix inequalities. Though local quadratic convergence of the algorithm is not a priori guaranteed or observed in all cases, numerical experiments, including application to an output feedback stabilization problem, show the effectiveness of the algorithm.

Pose Estimation via Gauss-Newton-on-manifold

Abstract: We present a Gauss-Newton-on-manifold approach for estimating the relative pose (position and orientation) between a 3D object and its projection on a 2D image plane from a set of point correspondences. The pose estimation problem is formulated as an optimization over three rotation parameters on the intersection of the manifold of the rotation matrices and a cone constraint on these matrices to ensure positive depth parameters. The optimization is based on Newton-type iterations and is locally quadratically convergent. A key feature of the proposed approach, not used in earlier studies, is an analytic geodesic search, alternating between gradient, Gauss-Newton and a random direction, which ensures the escape from local minima and convergence to a global minimum without the need to reinitialize the algorithm. Indeed, for a prescribed number of iterations, the proposed algorithm achieves significantly lower pose estimation errors than earlier methods and it converges to a global minimum in typically 5–10 iterations.

Essential Matrix Estimation via Newton-type Methods

Abstract: In this paper camera parameters are assumed to be known and a novel approach for essential matrix estimation is presented. We estimate the essential matrix from point correspondences between a stereo image pair. The technical approach we take is a generalization of the classical Newton method. It is well-known that the set of essential matrices forms a smooth manifold. Moreover, it is quite natural to minimise a suitable cost function the global minimum of which is the essential matrix we are looking for. In this paper we present several algorithms generalising the classical Newton method in the sense that (i) one of our methods can be considered as an intrinsic Newton method operating on the Riemannian manifold consisting of all essential matrices, (ii) the other two methods approximating the first one but being more efficient from a numerical point of view. To understand the algorithm requires a careful analysis of the underlying geometry of the problem.

Self-concordant functions for optimization on smooth manifolds

Abstract: This paper discusses self-concordant functions on smooth manifolds. In Euclidean space, this class of functions are utilized extensively in interior-point methods for optimization because of the associated low computational complexity. Here, the self-concordant function is carefully defined on a differential manifold. First, generalizations of the properties of self-concordant functions in Euclidean space are derived. Then, Newton decrement is defined and analyzed on the manifold that we consider. Based on this, a damped Newton algorithm is proposed for optimization of self-concordant functions, which guarantees that the solution falls in any given small neighborhood of the optimal solution, with its existence and uniqueness also proved in this paper, in a finite number of steps. It also ensures quadratic convergence within a neighborhood of the minimal point. This neighborhood can be specified by the the norm of Newton decrement. The computational complexity bound of the proposed approach is also given explicitly. This complexity bound is O(- ln(/spl epsi/)), where a is the desired precision. An interesting optimization problem is given to illustrate the proposed concept and algorithm.

Geometric optimization for 3D pose estimation of quadratic surfaces

Abstract: Our task is 3D pose estimation for on-line application in industrial robotics and machine vision. It involves the estimation of object position and orientation relative to a known model. Since most man made objects can be approximated by a small set of quadratic surfaces, in this paper we focus on pose estimation of such surfaces. Our optimization is of an error measure between the CAD model and the measured data. Most existing algorithms are sensitive to noise and occlusion or only converge linearly. Our optimization involves iterative cost function reduction on the smooth manifold of the Special Euclidean Group, SE/sub 3/. The optimization is based on locally quadratically convergent Newton-type iterations on this constraint manifold. A careful analysis of the underlying geometric constraint is required.