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188
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.
Tensor Decompositions and Applications
 SIAM REVIEW
, 2009
"... This survey provides an overview of higherorder tensor decompositions, their applications, and available software. A tensor is a multidimensional or N way array. Decompositions of higherorder tensors (i.e., N way arrays with N â¥ 3) have applications in psychometrics, chemometrics, signal proce ..."
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Cited by 228 (14 self)
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This survey provides an overview of higherorder tensor decompositions, their applications, and available software. A tensor is a multidimensional or N way array. Decompositions of higherorder tensors (i.e., N way arrays with N â¥ 3) have applications in psychometrics, chemometrics, signal processing, numerical linear algebra, computer vision, numerical analysis, data mining, neuroscience, graph analysis, etc. Two particular tensor decompositions can be considered to be higherorder extensions of the matrix singular value decompo
sition: CANDECOMP/PARAFAC (CP) decomposes a tensor as a sum of rankone tensors, and the Tucker decomposition is a higherorder form of principal components analysis. There are many other tensor decompositions, including INDSCAL, PARAFAC2, CANDELINC, DEDICOM, and PARATUCK2 as well as nonnegative variants of all of the above. The Nway Toolbox and Tensor Toolbox, both for MATLAB, and the Multilinear Engine are examples of software packages for working with tensors.
Guaranteed minimumrank solutions of linear matrix equations via nuclear norm minimization
, 2007
"... The affine rank minimization problem consists of finding a matrix of minimum rank that satisfies a given system of linear equality constraints. Such problems have appeared in the literature of a diverse set of fields including system identification and control, Euclidean embedding, and collaborative ..."
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Cited by 218 (15 self)
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The affine rank minimization problem consists of finding a matrix of minimum rank that satisfies a given system of linear equality constraints. Such problems have appeared in the literature of a diverse set of fields including system identification and control, Euclidean embedding, and collaborative filtering. Although specific instances can often be solved with specialized algorithms, the general affine rank minimization problem is NPhard, because it contains vector cardinality minimization as a special case. In this paper, we show that if a certain restricted isometry property holds for the linear transformation defining the constraints, the minimum rank solution can be recovered by solving a convex optimization problem, namely the minimization of the nuclear norm over the given affine space. We present several random ensembles of equations where the restricted isometry property holds with overwhelming probability, provided the codimension of the subspace is sufficiently large. The techniques used in our analysis have strong parallels in the compressed sensing framework. We discuss how affine rank minimization generalizes this preexisting concept and outline a dictionary relating concepts from cardinality minimization to those of rank minimization. We also discuss several algorithmic approaches to solving the norm minimization relaxations, and illustrate our results with numerical examples.
Robust Principal Component Analysis?
, 2009
"... This paper is about a curious phenomenon. Suppose we have a data matrix, which is the superposition of a lowrank component and a sparse component. Can we recover each component individually? We prove that under some suitable assumptions, it is possible to recover both the lowrank and the sparse co ..."
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Cited by 138 (6 self)
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This paper is about a curious phenomenon. Suppose we have a data matrix, which is the superposition of a lowrank component and a sparse component. Can we recover each component individually? We prove that under some suitable assumptions, it is possible to recover both the lowrank and the sparse components exactly by solving a very convenient convex program called Principal Component Pursuit; among all feasible decompositions, simply minimize a weighted combination of the nuclear norm and of the ℓ1 norm. This suggests the possibility of a principled approach to robust principal component analysis since our methodology and results assert that one can recover the principal components of a data matrix even though a positive fraction of its entries are arbitrarily corrupted. This extends to the situation where a fraction of the entries are missing as well. We discuss an algorithm for solving this optimization problem, and present applications in the area of video surveillance, where our methodology allows for the detection of objects in a cluttered background, and in the area of face recognition, where it offers a principled way of removing shadows and specularities in images of faces.
Generalized principal component analysis (GPCA)
 IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE
, 2003
"... This paper presents an algebrogeometric solution to the problem of segmenting an unknown number of subspaces of unknown and varying dimensions from sample data points. We represent the subspaces with a set of homogeneous polynomials whose degree is the number of subspaces and whose derivatives at a ..."
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Cited by 117 (29 self)
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This paper presents an algebrogeometric solution to the problem of segmenting an unknown number of subspaces of unknown and varying dimensions from sample data points. We represent the subspaces with a set of homogeneous polynomials whose degree is the number of subspaces and whose derivatives at a data point give normal vectors to the subspace passing through the point. When the number of subspaces is known, we show that these polynomials can be estimated linearly from data; hence, subspace segmentation is reduced to classifying one point per subspace. We select these points optimally from the data set by minimizing certain distance function, thus dealing automatically with moderate noise in the data. A basis for the complement of each subspace is then recovered by applying standard PCA to the collection of derivatives (normal vectors). Extensions of GPCA that deal with data in a highdimensional space and with an unknown number of subspaces are also presented. Our experiments on lowdimensional data show that GPCA outperforms existing algebraic algorithms based on polynomial factorization and provides a good initialization to iterative techniques such as Ksubspaces and Expectation Maximization. We also present applications of GPCA to computer vision problems such as face clustering, temporal video segmentation, and 3D motion segmentation from point correspondences in multiple affine views.
Matrices, vector spaces, and information retrieval
 SIAM Review
, 1999
"... Abstract. The evolution of digital libraries and the Internet has dramatically transformed the processing, storage, and retrieval of information. Efforts to digitize text, images, video, and audio now consume a substantial portion of both academic and industrial activity. Even when there is no short ..."
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Cited by 108 (1 self)
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Abstract. The evolution of digital libraries and the Internet has dramatically transformed the processing, storage, and retrieval of information. Efforts to digitize text, images, video, and audio now consume a substantial portion of both academic and industrial activity. Even when there is no shortage of textual materials on a particular topic, procedures for indexing or extracting the knowledge or conceptual information contained in them can be lacking. Recently developed information retrieval technologies are based on the concept of a vector space. Data are modeled as a matrix, and a user’s query of the database is represented as a vector. Relevant documents in the database are then identified via simple vector operations. Orthogonal factorizations of the matrix provide mechanisms for handling uncertainty in the database itself. The purpose of this paper is to show how such fundamental mathematical concepts from linear algebra can be used to manage and index large text collections. Key words. information retrieval, linear algebra, QR factorization, singular value decomposition, vector spaces
Kmeans Clustering via Principal Component Analysis
, 2004
"... Principal component analysis (PCA) is a widely used statistical technique for unsupervised dimension reduction. Kmeans clustering is a commonly used data clustering for performing unsupervised learning tasks. Here we ..."
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Cited by 105 (3 self)
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Principal component analysis (PCA) is a widely used statistical technique for unsupervised dimension reduction. Kmeans clustering is a commonly used data clustering for performing unsupervised learning tasks. Here we
ThreeDimensional Face Recognition
, 2005
"... An expressioninvariant 3D face recognition approach is presented. Our basic assumption is that facial expressions can be modelled as isometries of the facial surface. This allows to construct expressioninvariant representations of faces using the bendinginvariant canonical forms approach. The re ..."
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Cited by 103 (22 self)
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An expressioninvariant 3D face recognition approach is presented. Our basic assumption is that facial expressions can be modelled as isometries of the facial surface. This allows to construct expressioninvariant representations of faces using the bendinginvariant canonical forms approach. The result is an efficient and accurate face recognition algorithm, robust to facial expressions, that can distinguish between identical twins (the first two authors). We demonstrate a prototype system based on the proposed algorithm and compare its performance to classical face recognition methods. The numerical methods employed by our approach do not require the facial surface explicitly. The surface gradients field, or the surface metric, are sufficient for constructing the expressioninvariant representation of any given face. It allows us to perform the 3D face recognition task while avoiding the surface reconstruction stage.
Robust Principal Component Analysis for Computer Vision
, 2001
"... Principal Component Analysis (PCA) has been widely used for the representation of shape, appearance, and motion. One drawback of typical PCA methods is that they are least squares estimation techniques and hence fail to account for "outliers" which are common in realistic training sets. In computer ..."
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Cited by 95 (3 self)
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Principal Component Analysis (PCA) has been widely used for the representation of shape, appearance, and motion. One drawback of typical PCA methods is that they are least squares estimation techniques and hence fail to account for "outliers" which are common in realistic training sets. In computer vision applications, outliers typically occur within a sample (image) due to pixels that are corrupted by noise, alignment errors, or occlusion. We review previous approaches for making PCA robust to outliers and present a new method that uses an intrasample outlier process to account for pixel outliers. We develop the theory of Robust Principal Component Analysis (RPCA) and describe a robust Mestimation algorithm for learning linear multivariate representations of high dimensional data such as images. Quantitative comparisons with traditional PCA and previous robust algorithms illustrate the benefits of RPCA when outliers are present. Details of the algorithm are described and a software implementation is being made publically available.
Minimum sumsquared residue coclustering of gene expression data
 In SDM
, 2004
"... Microarray experiments have been extensively used for simultaneously measuring DNA expression levels of thousands of genes in genome research. A key step in the analysis of gene expression data is the clustering of genes into groups that show similar expression values over a range of conditions. Sin ..."
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Cited by 90 (5 self)
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Microarray experiments have been extensively used for simultaneously measuring DNA expression levels of thousands of genes in genome research. A key step in the analysis of gene expression data is the clustering of genes into groups that show similar expression values over a range of conditions. Since only a small subset of the genes participate in any cellular process of interest, by focusing on subsets of genes and conditions, we can lower the noise induced by other genes and conditions — a cocluster characterizes such a subset of interest. Cheng and Church [3] introduced an effective measure of cocluster quality based on mean squared residue. In this paper, we use two similar squared residue measures and propose two fast kmeans like coclustering algorithms corresponding to the two residue measures. Our algorithms discover k row clusters and l column clusters simultaneously while monotonically decreasing the respective squared residues. Our coclustering algorithms inherit the simplicity, efficiency and wide applicability of the kmeans algorithm. Minimizing the residues may also be formulated as trace optimization problems that allow us to obtain a spectral relaxation that we use for a principled initialization for our iterative algorithms. We further enhance our algorithms by an incremental local search strategy that helps avoid empty clusters and escape poor local minima. We illustrate coclustering results on a yeast cell cycle dataset and a human Bcell lymphoma dataset. Our experiments show that our coclustering algorithms are efficient and are able to discover coherent coclusters. Keywords: Geneexpression, coclustering, biclustering, residue, spectral relaxation