Results 1  10
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139
Sparse subspace clustering
 In CVPR
, 2009
"... We propose a method based on sparse representation (SR) to cluster data drawn from multiple lowdimensional linear or affine subspaces embedded in a highdimensional space. Our method is based on the fact that each point in a union of subspaces has a SR with respect to a dictionary formed by all oth ..."
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Cited by 241 (14 self)
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We propose a method based on sparse representation (SR) to cluster data drawn from multiple lowdimensional linear or affine subspaces embedded in a highdimensional space. Our method is based on the fact that each point in a union of subspaces has a SR with respect to a dictionary formed by all other data points. In general, finding such a SR is NP hard. Our key contribution is to show that, under mild assumptions, the SR can be obtained ’exactly ’ by using ℓ1 optimization. The segmentation of the data is obtained by applying spectral clustering to a similarity matrix built from this SR. Our method can handle noise, outliers as well as missing data. We apply our subspace clustering algorithm to the problem of segmenting multiple motions in video. Experiments on 167 video sequences show that our approach significantly outperforms stateoftheart methods. 1.
A benchmark for the comparison of 3D motion segmentation algorithms
 In CVPR
, 2007
"... Over the past few years, several methods for segmenting a scene containing multiple rigidly moving objects have been proposed. However, most existing methods have been tested on a handful of sequences only, and each method has been often tested on a different set of sequences. Therefore, the compari ..."
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Cited by 158 (10 self)
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Over the past few years, several methods for segmenting a scene containing multiple rigidly moving objects have been proposed. However, most existing methods have been tested on a handful of sequences only, and each method has been often tested on a different set of sequences. Therefore, the comparison of different methods has been fairly limited. In this paper, we compare four 3D motion segmentation algorithms for affine cameras on a benchmark of 155 motion sequences of checkerboard, traffic, and articulated scenes. 1.
Robust Subspace Segmentation by LowRank Representation
"... We propose lowrank representation (LRR) to segment data drawn from a union of multiple linear (or affine) subspaces. Given a set of data vectors, LRR seeks the lowestrank representation among all the candidates that represent all vectors as the linear combination of the bases in a dictionary. Unlik ..."
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Cited by 145 (25 self)
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We propose lowrank representation (LRR) to segment data drawn from a union of multiple linear (or affine) subspaces. Given a set of data vectors, LRR seeks the lowestrank representation among all the candidates that represent all vectors as the linear combination of the bases in a dictionary. Unlike the wellknown sparse representation (SR), which computes the sparsest representation of each data vector individually, LRR aims at finding the lowestrank representation of a collection of vectors jointly. LRR better captures the global structure of data, giving a more effective tool for robust subspace segmentation from corrupted data. Both theoretical and experimental results show that LRR is a promising tool for subspace segmentation. 1.
Robust Recovery of Subspace Structures by LowRank Representation
"... In this work we address the subspace recovery problem. Given a set of data samples (vectors) approximately drawn from a union of multiple subspaces, our goal is to segment the samples into their respective subspaces and correct the possible errors as well. To this end, we propose a novel method ter ..."
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Cited by 128 (24 self)
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In this work we address the subspace recovery problem. Given a set of data samples (vectors) approximately drawn from a union of multiple subspaces, our goal is to segment the samples into their respective subspaces and correct the possible errors as well. To this end, we propose a novel method termed LowRank Representation (LRR), which seeks the lowestrank representation among all the candidates that can represent the data samples as linear combinations of the bases in a given dictionary. It is shown that LRR well solves the subspace recovery problem: when the data is clean, we prove that LRR exactly captures the true subspace structures; for the data contaminated by outliers, we prove that under certain conditions LRR can exactly recover the row space of the original data and detect the outlier as well; for the data corrupted by arbitrary errors, LRR can also approximately recover the row space with theoretical guarantees. Since the subspace membership is provably determined by the row space, these further imply that LRR can perform robust subspace segmentation and error correction, in an efficient way.
A geometric analysis of subspace clustering with outliers
 ANNALS OF STATISTICS
, 2012
"... This paper considers the problem of clustering a collection of unlabeled data points assumed to lie near a union of lower dimensional planes. As is common in computer vision or unsupervised learning applications, we do not know in advance how many subspaces there are nor do we have any information a ..."
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Cited by 66 (3 self)
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This paper considers the problem of clustering a collection of unlabeled data points assumed to lie near a union of lower dimensional planes. As is common in computer vision or unsupervised learning applications, we do not know in advance how many subspaces there are nor do we have any information about their dimensions. We develop a novel geometric analysis of an algorithm named sparse subspace clustering (SSC) [11], which significantly broadens the range of problems where it is provably effective. For instance, we show that SSC can recover multiple subspaces, each of dimension comparable to the ambient dimension. We also prove that SSC can correctly cluster data points even when the subspaces of interest intersect. Further, we develop an extension of SSC that succeeds when the data set is corrupted with possibly overwhelmingly many outliers. Underlying our analysis are clear geometric insights, which may bear on other sparse recovery problems. A numerical study complements our theoretical analysis and demonstrates the effectiveness of these methods.
Motion segmentation via robust subspace separation in the presence of outlying, incomplete, or corrupted trajectories
 In IEEE Conference on Computer Vision and Pattern Recognition
, 2008
"... We examine the problem of segmenting tracked feature point trajectories of multiple moving objects in an image sequence. Using the affine camera model, this motion segmentation problem can be cast as the problem of segmenting samples drawn from a union of linear subspaces. Due to limitations of the ..."
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Cited by 53 (7 self)
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We examine the problem of segmenting tracked feature point trajectories of multiple moving objects in an image sequence. Using the affine camera model, this motion segmentation problem can be cast as the problem of segmenting samples drawn from a union of linear subspaces. Due to limitations of the tracker, occlusions and the presence of nonrigid objects in the scene, the obtained motion trajectories may contain grossly mistracked features, missing entries, or not correspond to any valid motion model. In this paper, we develop a robust subspace separation scheme that can deal with all of these practical issues in a unified framework. Our methods draw strong connections between lossy compression, rank minimization, and sparse representation. We test our methods extensively and compare their performance to several extant methods with experiments on the Hopkins 155 database. Our results are on par with stateoftheart results, and in many cases exceed them. All MATLAB code and segmentation results are publicly available for peer evaluation at
Track to the Future: Spatiotemporal Video Segmentation with Longrange Motion Cues
"... Video provides not only rich visual cues such as motion and appearance, but also much less explored longrange temporal interactions among objects. We aim to capture such interactions and to construct a powerful intermediatelevel video representation for subsequent recognition. Motivated by this goa ..."
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Cited by 52 (2 self)
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Video provides not only rich visual cues such as motion and appearance, but also much less explored longrange temporal interactions among objects. We aim to capture such interactions and to construct a powerful intermediatelevel video representation for subsequent recognition. Motivated by this goal, we seek to obtain spatiotemporal oversegmentation of a video into regions that respect object boundaries and, at the same time, associate object pixels over many video frames. The contributions of this paper are twofold. First, we develop an efficient spatiotemporal video segmentation algorithm, which naturally incorporates longrange motion cues from the past and future frames in the form of clusters of point tracks with coherent motion. Second, we devise a new track clustering cost function that includes occlusion reasoning, in the form of depth ordering constraints, as well as motion similarity along the tracks. We evaluate the proposed approach on a challenging set of video sequences of office scenes from feature length movies. 1.
Segmenting motions of different types by unsupervised manifold clustering
 In Proceedings of CVPR
, 2007
"... We propose a novel algorithm for segmenting multiple motions of different types from point correspondences in multiple affine or perspective views. Since point trajectories associated with different motions live in different manifolds, traditional approaches deal with only one manifold type: linear ..."
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Cited by 48 (6 self)
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We propose a novel algorithm for segmenting multiple motions of different types from point correspondences in multiple affine or perspective views. Since point trajectories associated with different motions live in different manifolds, traditional approaches deal with only one manifold type: linear subspaces for affine views, and homographic, bilinear and trilinear varieties for two and three perspective views. As real motion sequences contain motions of different types, we cast motion segmentation as a problem of clustering manifolds of different types. Rather than explicitly modeling each manifold as a linear, bilinear or multilinear variety, we use nonlinear dimensionality reduction to learn a lowdimensional representation of the union of all manifolds. We show that for a union of separated manifolds, the LLE algorithm computes a matrix whose null space contains vectors giving the segmentation of the data. An analysis of the variance of these vectors allows us to distinguish them from other vectors in the null space. This leads to a new algorithm for clustering both linear and nonlinear manifolds. Although this algorithm is theoretically designed for separated manifolds, our experiments demonstrate its performance on real data where this assumption does not hold. We test our algorithm on the Hopkins 155 motion segmentation database and achieve an average classification error of 4.8%, which compares favorably against stateofthe art multiframe motion segmentation methods. 1.
Coarsetofine lowrank structurefrommotion
 in Proc. IEEE Conf. Computer Vision and Pattern Recognition CVPR 2008
"... We address the problem of deformable shape and motion recovery from point correspondences in multiple perspective images. We use the lowrank shape model, i.e. the 3D shape is represented as a linear combination of unknown shape bases. We propose a new way of looking at the lowrank shape model. Ins ..."
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Cited by 43 (3 self)
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We address the problem of deformable shape and motion recovery from point correspondences in multiple perspective images. We use the lowrank shape model, i.e. the 3D shape is represented as a linear combination of unknown shape bases. We propose a new way of looking at the lowrank shape model. Instead of considering it as a whole, we assume a coarsetofine ordering of the deformation modes, which can be seen as a model prior. This has several advantages. First, the high level of ambiguity of the original lowrank shape model is drastically reduced since the shape bases can not anymore be arbitrarily recombined. Second, this allows us to propose a coarsetofine reconstruction algorithm which starts by computing the mean shape and iteratively adds deformation modes. It directly gives the sought after metric model, thereby avoiding the difficult upgrading step required by most of the other methods. Third, this makes it possible to automatically select the number of deformation modes as the reconstruction algorithm proceeds. We propose to incorporate two other priors, accounting for temporal and spatial smoothness, which are shown to improve the quality of the recovered model parameters. The proposed model and reconstruction algorithm are successfully demonstrated on several videos and are shown to outperform the previously proposed algorithms. 1.
A Closed Form Solution to Robust Subspace Estimation and Clustering
"... We consider the problem of fitting one or more subspaces to a collection of data points drawn from the subspaces and corrupted by noise/outliers. We pose this problem as a rank minimization problem, where the goal is to decompose the corrupted data matrix as the sum of a clean, selfexpressive, low ..."
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Cited by 43 (4 self)
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We consider the problem of fitting one or more subspaces to a collection of data points drawn from the subspaces and corrupted by noise/outliers. We pose this problem as a rank minimization problem, where the goal is to decompose the corrupted data matrix as the sum of a clean, selfexpressive, lowrank dictionary plus a matrix of noise/outliers. Our key contribution is to show that, for noisy data, this nonconvex problem can be solved very efficiently and in closed form from the SVD of the noisy data matrix. Remarkably, this is true for both one or more subspaces. An important difference with respect to existing methods is that our framework results in a polynomial thresholding of the singular values with minimal shrinkage. Indeed, a particular case of our framework in the case of a single subspace leads to classical PCA, which requires no shrinkage. In the case of multiple subspaces, our framework provides an affinity matrix that can be used to cluster the data according to the subspaces. In the case of data corrupted by outliers, a closedform solution appears elusive. We thus use an augmented Lagrangian optimization framework, which requires a combination of our proposed polynomial thresholding operator with the more traditional shrinkagethresholding operator. 1.