Results 1  10
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22
Iterative point matching for registration of freeform curves and surfaces
, 1994
"... A heuristic method has been developed for registering two sets of 3D curves obtained by using an edgebased stereo system, or two dense 3D maps obtained by using a correlationbased stereo system. Geometric matching in general is a difficult unsolved problem in computer vision. Fortunately, in ma ..."
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Cited by 480 (6 self)
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A heuristic method has been developed for registering two sets of 3D curves obtained by using an edgebased stereo system, or two dense 3D maps obtained by using a correlationbased 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 3D 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 subsetsubset matching. A leastsquares technique is used to estimate 3D 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.
Inverse eigenvalue problems
 SIAM Rev
, 1998
"... 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 furth ..."
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Cited by 41 (6 self)
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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.
Optimal Structure From Motion: Local Ambiguities and Global Estimates
, 1998
"... "Structure From Motion" (SFM) refers to the problem of estimating threedimensional information about the environment from the motion of its twodimensional 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 al ..."
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Cited by 33 (1 self)
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"Structure From Motion" (SFM) refers to the problem of estimating threedimensional information about the environment from the motion of its twodimensional 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 costfunction. 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...
Optimal Structure from Motion: Local Ambiguities and Global Estimates
, 2000
"... “Structure From Motion” (SFM) refers to the problem of estimating spatial properties of a threedimensional scene from the motion of its projection onto a twodimensional surface, such as the retina. We present an analysis of SFM which results in algorithms that are provably convergent and provably o ..."
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Cited by 23 (5 self)
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“Structure From Motion” (SFM) refers to the problem of estimating spatial properties of a threedimensional scene from the motion of its projection onto a twodimensional 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 highdimensional quadratic cost function, and show how it is possible to reduce it to the minimization of a twodimensional function whose stationary points are in onetoone 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 wellknown 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.
Constructing a Hermitian matrix from its diagonal entries and eigenvalues
 SIAM J. Matrix Anal. Appl
, 1995
"... Given two vectors a � 2 R n,theSchurHorn 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 ..."
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Cited by 22 (8 self)
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Given two vectors a � 2 R n,theSchurHorn 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 WielandtHo 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. SchurHorn theorem, majorization, inverse eigenvalue problem, lift and projection, projected gradient. AMS(MOS) subject classi cations. 65F15, 65H15. 1. Introduction.
A list of matrix flows with applications
 in Hamiltonian and Gradients Flows, Algorithms and Control
, 1994
"... 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 re ..."
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Cited by 15 (1 self)
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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
Optimal and Suboptimal Structure From Motion
 Proceedings of International Conference on Computer Vision
, 1997
"... "Structure From Motion" (SFM) refers to the problem of estimating threedimensional information about the environment from the motion of its twodimensional projection onto a surface (for instance the retina). Noise plays an important role in this problem, but it has been addressed only marginally i ..."
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Cited by 13 (0 self)
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"Structure From Motion" (SFM) refers to the problem of estimating threedimensional information about the environment from the motion of its twodimensional 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 costfunction. 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...
A Dynamical Systems Approach to Weighted Graph Matching
, 2006
"... 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 se ..."
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Cited by 12 (3 self)
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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 optimizationbased 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.
The symmetric representation of the rigid body equations and their discretization
 141–71 FEDEROV Y N 2005 INTEGRABLE FLOWS AND BACKLUND TRANSFORMATIONS ON EXTENDED STIEFEL VARIETIES WITH APPLICATION TO THE EULER TOP ON THE LIE GROUP SO(3) PREPRINT NLIN.SI/0505045 AQ2 GELFAND I M AND FOMIN S V 2000 CALCULUS OF VARIATIONS TRANSLATED BY R
, 1998
"... 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 pr ..."
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Cited by 11 (6 self)
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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 MoserVeselov 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].
Reconstructing the Shape of a Deformable Membrane from Image Data
, 2000
"... 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 projectio ..."
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Cited by 6 (2 self)
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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 WORDSmembrane mechanics,...