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63
Closedform solution of absolute orientation using unit quaternions
 J. Opt. Soc. Am. A
, 1987
"... Finding the relationship between two coordinate systems using pairs of measurements of the coordinates of a number of points in both systems is a classic photogrammetric task. It finds applications in stereophotogrammetry and in robotics. I present here a closedform solution to the leastsquares pr ..."
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Cited by 719 (3 self)
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Finding the relationship between two coordinate systems using pairs of measurements of the coordinates of a number of points in both systems is a classic photogrammetric task. It finds applications in stereophotogrammetry and in robotics. I present here a closedform solution to the leastsquares problem for three or more points. Currently various empirical, graphical, and numerical iterative methods are in use. Derivation of the solution is simplified by use of unit quaternions to represent rotation. I emphasize a symmetry property that a solution to this problem ought to possess. The best translational offset is the difference between the centroid of the coordinates in one system and the rotated and scaled centroid of the coordinates in the other system. The best scale is equal to the ratio of the rootmeansquare deviations of the coordinates in the two systems from their respective centroids. These exact results are to be preferred to approximate methods based on measurements of a few selected points. The unit quaternion representing the best rotation is the eigenvector associated with the most positive eigenvalue of a symmetric 4 X 4 matrix. The elements of this matrix are combinations of sums of products of corresponding coordinates of the points. 1.
A comparison of four algorithms for estimating 3d rigid transformations
 In Proc. British Machine Vision Conference
, 1995
"... A common need in machine vision is to compute the 3D rigid transformation that exists between two sets of points for which corresponding pairs have been determined. In this paper a comparative analysis of four popular and efficient algorithms is given. Each computes the translational and rotational ..."
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Cited by 61 (1 self)
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A common need in machine vision is to compute the 3D rigid transformation that exists between two sets of points for which corresponding pairs have been determined. In this paper a comparative analysis of four popular and efficient algorithms is given. Each computes the translational and rotational components of the transform in closedform as the solution to a least squares formulation of the problem. They differ in terms of the representation of the transform and the method of solution, using respectively: singular value decomposition of a matrix, orthonormal matrices, unit quaternions and dual quaternions. This comparison presents results of several experiments designed to determine the (1) accuracy in the presence of noise, (2) stability with respect to degenerate data sets, and (3) relative computation time of each approach. 1
Computing the polar decomposition—with applications
 SIAM J. Sci. Stat. Comput
, 1986
"... Abstract. A quadratically convergent Newton method for computing the polar decomposition of a fullrank matrix is presented and analysed. Acceleration parameters are introduced so as to enhance the initial rate ofconvergence and it is shownhow reliable estimates ofthe optimal parametersmaybe compute ..."
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Cited by 55 (22 self)
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Abstract. A quadratically convergent Newton method for computing the polar decomposition of a fullrank matrix is presented and analysed. Acceleration parameters are introduced so as to enhance the initial rate ofconvergence and it is shownhow reliable estimates ofthe optimal parametersmaybe computed in practice. To add to the known best approximation property of the unitary polar factor, the Hermitian polar factorHof a nonsingular Hermitian matrix A is shown to be a good positive definite approximation to A and 1/2(A/H) is shown to be a best Hermitian positive semidefinite approximation to A. Perturbation bounds for the polar factors are derived. Applications of the polar decomposition to factor analysis, aerospace computations and optimisation are outlined; and a new method is derived for computing the square root of a symmetric positive definite matrix. Key words, polar decomposition, singular value decomposition, Newton’s method, matrix square root AMS(MOS) subject classifications. 65F25, 65F30, 65F35 1. Introduction. The
Designing Structured Tight Frames via an Alternating Projection Method
, 2003
"... Tight frames, also known as general WelchBoundEquality sequences, generalize orthonormal systems. Numerous applicationsincluding communications, coding and sparse approximationrequire finitedimensional tight frames that possess additional structural properties. This paper proposes an alterna ..."
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Cited by 44 (6 self)
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Tight frames, also known as general WelchBoundEquality sequences, generalize orthonormal systems. Numerous applicationsincluding communications, coding and sparse approximationrequire finitedimensional tight frames that possess additional structural properties. This paper proposes an alternating projection method that is versatile enough to solve a huge class of inverse eigenvalue problems, which includes the frame design problem. To apply this method, one only needs to solve a matrix nearness problem that arises naturally from the design specifications. Therefore, it is fast and easy to develop versions of the algorithm that target new design problems. Alternating projection will often succeed even if algebraic constructions are unavailable. To demonstrate
Object recognition and full pose registration from a single image for robotic manipulation
 in IEEE ICRA. Kobe: IEEE
, 2009
"... Abstract — Robust perception is a vital capability for robotic manipulation in unstructured scenes. In this context, full pose estimation of relevant objects in a scene is a critical step towards the introduction of robots into household environments. In this paper, we present an approach for buildi ..."
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Cited by 44 (9 self)
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Abstract — Robust perception is a vital capability for robotic manipulation in unstructured scenes. In this context, full pose estimation of relevant objects in a scene is a critical step towards the introduction of robots into household environments. In this paper, we present an approach for building metric 3D models of objects using local descriptors from several images. Each model is optimized to fit a set of calibrated training images, thus obtaining the best possible alignment between the 3D model and the real object. Given a new test image, we match the local descriptors to our stored models online, using a novel combination of the RANSAC and Mean Shift algorithms to register multiple instances of each object. A robust initialization step allows for arbitrary rotation, translation and scaling of objects in the test images. The resulting system provides markerless 6DOF pose estimation for complex objects in cluttered scenes. We provide experimental results demonstrating orientation and translation accuracy, as well a physical implementation of the pose output being used by an autonomous robot to perform grasping in highly cluttered scenes. I.
Iterative quantization: A procrustean approach to learning binary codes
 In Proc. of the IEEE Int. Conf. on Computer Vision and Pattern Recognition (CVPR
, 2011
"... This paper addresses the problem of learning similaritypreserving binary codes for efficient retrieval in largescale image collections. We propose a simple and efficient alternating minimization scheme for finding a rotation of zerocentered data so as to minimize the quantization error of mapping t ..."
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Cited by 41 (6 self)
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This paper addresses the problem of learning similaritypreserving binary codes for efficient retrieval in largescale image collections. We propose a simple and efficient alternating minimization scheme for finding a rotation of zerocentered data so as to minimize the quantization error of mapping this data to the vertices of a zerocentered binary hypercube. This method, dubbed iterative quantization (ITQ), has connections to multiclass spectral clustering and to the orthogonal Procrustes problem, and it can be used both with unsupervised data embeddings such as PCA and supervised embeddings such as canonical correlation analysis (CCA). Our experiments show that the resulting binary coding schemes decisively outperform several other stateoftheart methods. 1.
Registration of Head Volume Images Using Implantable Fiducial Markers
 Ieee Transactions on Medical Imaging
, 1997
"... Abstract—In this paper, we describe an extrinsicpointbased, interactive imageguided neurosurgical system designed at Vanderbilt University, Nashville, TN, as part of a collaborative effort among the Departments of Neurological Surgery, Computer Science, ..."
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Cited by 33 (4 self)
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Abstract—In this paper, we describe an extrinsicpointbased, interactive imageguided neurosurgical system designed at Vanderbilt University, Nashville, TN, as part of a collaborative effort among the Departments of Neurological Surgery, Computer Science,
Matrix nearness problems and applications
 Applications of Matrix Theory
, 1989
"... A matrix nearness problem consists of finding, for an arbitrary matrix A, a nearest member of some given class of matrices, where distance is measured in a matrix norm. A survey of nearness problems is given, with particular emphasis on the fundamental properties of symmetry, positive definiteness, ..."
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Cited by 32 (6 self)
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A matrix nearness problem consists of finding, for an arbitrary matrix A, a nearest member of some given class of matrices, where distance is measured in a matrix norm. A survey of nearness problems is given, with particular emphasis on the fundamental properties of symmetry, positive definiteness, orthogonality, normality, rankdeficiency and instability. Theoretical results and computational methods are described. Applications of nearness problems in areas including control theory, numerical analysis and statistics are outlined.
Recovering a Basic Space from a Set of Issue Scales
, 1996
"... This paper develops a procedure for estimating the basic dimensions underlying a set of issue or attribute scales. A simple HinichOrdeshook spatial theory of voting is used to model Converse's fundamental insight that individuals' positions on issues are bundled together, and the knowledge of one ..."
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Cited by 30 (2 self)
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This paper develops a procedure for estimating the basic dimensions underlying a set of issue or attribute scales. A simple HinichOrdeshook spatial theory of voting is used to model Converse's fundamental insight that individuals' positions on issues are bundled together, and the knowledge of one or two issue positions makes the remaining positions very predictable. The model assumes that individuals' positions on a set issue or attribute dimensions are determined by the individuals' positions on a small number of underlying evaluative or basic dimensions. The procedure developed in this paper for estimating these basic dimensions is, in effect, a method of performing singular value decomposition of a matrix with missing elements. Monte Carlo testing shows that the procedure reliably reproduces the missing elements. Because of this reliability, the estimation procedure can be used to produce EckartYoung matrix lower rank approximations. A number of applications to political data are...
A Review of Medical Image Registration
 Interactive imageguided neurosurgery
, 1993
"... Introduction The ever expanding gamut of medical imaging techniques provides the clinician an increasingly multifaceted view of brain function and anatomy. The information provided by the various imaging modalities is often complementary (i.e. provides separate but useful information) and synergist ..."
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Cited by 24 (0 self)
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Introduction The ever expanding gamut of medical imaging techniques provides the clinician an increasingly multifaceted view of brain function and anatomy. The information provided by the various imaging modalities is often complementary (i.e. provides separate but useful information) and synergistic (i.e. the combination of information provides useful extra information). For example, Xray computed tomography (CT) and magnetic resonance (MR) imaging exquisitely demonstrate brain anatomy but provide little functional information. Positron emission tomography (PET) and single photon emission computed tomography (SPECT) scans display aspects of brain function and allow metabolic measurements but poorly delineate anatomy. Furthermore, CT and MR images describe complementary morphologic features. For example, bone and calcifications are best seen on CT images, while softtissue structures are better differentiated by MR imaging. Clinical diagnosis and therapy planning and evaluatio