A heuristic method has been developed for registering two sets of 3-D curves obtained by using an edge-based stereo system, or two dense 3-D maps obtained by using a correlation-based stereo system. Geometric matching in general is a difficult unsolved problem in computer vision. Fortunately, in many practical applications, some a priori knowledge exists which considerably simplifies the problem. In visual navigation, for example, the motion between successive positions is usually approximately known. From this initial estimate, our algorithm computes observer motion with very good precision, which is required for environment modeling (e.g., building a Digital Elevation Map). Objects are represented by a set of 3-D points, which are considered as the samples of a surface. No constraint is imposed on the form of the objects. The proposed algorithm is based on iteratively matching points in one set to the closest points in the other. A statistical method based on the distance distribution is used to deal with outliers, occlusion, appearance and disappearance, which allows us to do subset-subset matching. A least-squares technique is used to estimate 3-D motion from the point correspondences, which reduces the average distance between points in the two sets. Both synthetic and real data have been used to test the algorithm, and the results show that it is efficient and robust, and yields an accurate motion estimate.

"Structure From Motion" (SFM) refers to the problem of estimating three-dimensional information about the environment from the motion of its two-dimensional projection onto a surface (for instance the retina). We present an analysis of SFM from the point of view of noise. This analysis results in algorithms that are provably convergent and provably optimal with respect to a chosen norm. In particular, we cast SFM as a nonlinear optimization problem and define a bilinear projection iteration that converges to fixed points of a certain cost-function. We then show that such fixed points are "fundamental", i.e. intrinsic to the problem of SFM and not an artifact introduced by our algorithms. We classify and interpret geometrically local extrema, and we argue that they correspond to phenomena observed in visual psychophysics. Finally, we show under what conditions it is possible - given convergence to a local extremum - to "jump" to the valley containing the optimum; this leads us to sugges...

Abstract. A collection of inverse eigenvalue problems are identi ed and classi ed according to their characteristics. Current developments in both the theoretic and the algorithmic aspects are summarized and reviewed in this paper. This exposition also reveals many open questions that deserves further study. An extensive bibliography of pertinent literature is attached.

“Structure From Motion” (SFM) refers to the problem of estimating spatial properties of a threedimensional scene from the motion of its projection onto a two-dimensional surface, such as the retina. We present an analysis of SFM which results in algorithms that are provably convergent and provably optimal with respect to a chosen norm. In particular, we cast SFM as the minimization of a high-dimensional quadratic cost function, and show how it is possible to reduce it to the minimization of a two-dimensional function whose stationary points are in one-to-one correspondence with those of the original cost function. As a consequence, we can plot the reduced cost function and characterize the configurations of structure and motion that result in local minima. As an example, we discuss two local minima that are associated with well-known visual illusions. Knowledge of the topology of the residual in the presence of such local minima allows us to formulate minimization algorithms that, in addition to provably converge to stationary points of the original cost function, can switch between different local extrema in order to converge to the global minimum, under suitable conditions. We also offer an experimental study of the distribution of the estimation error in the presence of noise in the measurements, and characterize the sensitivity of the algorithm using the structure of Fisher’s Information matrix.

Given two vectors a � 2 R n,theSchur-Horn theorem states that a majorizes if and only if there exists a Hermitian matrix H with eigenvalues and diagonal entries a. While the theory is regarded as classical by now, the known proof is not constructive. To construct a Hermitian matrix from its diagonal entries and eigenvalues therefore becomes an interesting and challenging inverse eigenvalue problem. Two algorithms for determining the matrix numerically are proposed in this paper. The lift and projection method is an iterative method which involves an interesting application of the Wielandt-Ho man theorem. The projected gradient method is a continuous method which, besides its easy implementation, o ers a new proof of existence because of its global convergence property. Key words. Schur-Horn theorem, majorization, inverse eigenvalue problem, lift and projection, projected gradient. AMS(MOS) subject classi cations. 65F15, 65H15. 1. Introduction.

Many mathematical problems, such as existence questions, are studied by using an appropriate realization process, either iteratively or continuously. This article is a collection of di erential equations that have been proposed as special continuous realization processes. In some cases, there are remarkable connections betweensmooth ows and discrete numerical algorithms. In other cases, the ow approach seems advantageous in tackling very di cult problems. The ow approach has potential applications ranging from new development ofnumerical algorithms to the theoretical solution of open problems. Various aspects of the recent development and applications of the ow approach are reviewed in this article. 1

"Structure From Motion" (SFM) refers to the problem of estimating three-dimensional information about the environment from the motion of its two-dimensional projection onto a surface (for instance the retina). Noise plays an important role in this problem, but it has been addressed only marginally in more than twenty years of research. We present an analysis of SFM from the point of view of noise. This analysis results in algorithms that are provably convergent and provably optimal with respect to a chosen norm. In particular, we cast SFM as a nonlinear optimization problem and define a bilinear projection iteration that converges to fixed points of a certain cost-function. We then show that such fixed points are "fundamental", i.e. are intrinsic to the problem of SFM and not an artifact introduced by our algorithms. We classify and interpret geometrically local extrema, and we argue that they correspond to phenomena observed in visual psychophysics. Finally, we show under what conditi...

This paper analyzes continuous and discrete versions of the generalized rigid body equations and the role of these equations in numerical analysis, optimal control and integrable Hamiltonian systems. In particular, we present a symmetric representation of the rigid body equations on the Cartesian product SO(n) × SO(n) and study its associated symplectic structure. We describe the relationship of these ideas with the Moser-Veselov theory of discrete integrable systems and with the theory of variational symplectic integrators. Preliminary work on the ideas discussed in the present paper may be found in Bloch, Crouch, Marsden and Ratiu [1998].

Graph matching is a fundamental problem that arises frequently in the areas of distributed control, computer vision, and facility allocation. In this paper, we consider the optimal graph matching problem for weighted graphs, which is computationally challenging due the combinatorial nature of the set of permutations. Contrary to optimization-based relaxations to this problem, in this paper we develop a novel relaxation by constructing dynamical systems on the manifold of orthogonal matrices. In particular, since permutation matrices are orthogonal matrices with nonnegative elements, we define two gradient flows in the space of orthogonal matrices. The first minimizes the cost of weighted graph matching over orthogonal matrices, whereas the second minimizes the distance of an orthogonal matrix from the finite set of all permutations. The combination of the two dynamical systems converges to a permutation matrix which, provides a suboptimal solution to the weighted graph matching problem. Finally, our approach is shown to be promising by illustrating it on nontrivial problems.

In this paper, we study the problem of determining a mathematical description of the surface defined by the shape of a membrane based on an image of it and present an algorithm for reconstructing the surface when the membrane is deformed by unknown external elements. The given data are the projection on an image plane of markings on the surface of the membrane, the undeformed configuration of the membrane, and a model for the membranemechanics. The method of reconstruction is based on the principle that the shape assumedby the membrane will minimize the elastic energy stored in the membrane subject to the constraints implied by the measurements. Energy minimization leads to a set of nonlinear partial differential equations. An approximate solution is found using linearization. The initial motivation, and our first application of these ideas, comes from tactile sensing. Experimental results affirm that this approach can be very effective in this context. KEY WORDS---membrane mechanics,...