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134
ThreeDimensional Scene Flow
, 1999
"... Scene flow is the threedimensional motion field of points in the world, just as optical flow is the twodimensional motion field of points in an image. Any optical flow is simply the projection of the scene flow onto the image plane of a camera. In this paper, we present a framework for the computat ..."
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Cited by 128 (9 self)
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Scene flow is the threedimensional motion field of points in the world, just as optical flow is the twodimensional motion field of points in an image. Any optical flow is simply the projection of the scene flow onto the image plane of a camera. In this paper, we present a framework for the computation of dense, nonrigid scene flow from optical flow. Our approach leads to straightforward linear algorithms and a classification of the task into three major scenarios: (1) complete instantaneous knowledge of the scene structure, (2) knowledge only of correspondence information, and (3) no knowledge of the scene structure. We also show that multiple estimates of the normal flow cannot be used to estimate dense scene flow directly without some form of smoothing or regularization. 1
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 120 (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.
A ClosedForm Solution to NonRigid Shape and Motion Recovery
 In European Conference on Computer Vision
, 2004
"... Recovery of three diensWXzm (3D) sD) e and otion of nonsN;m[ s cenes fro a onocular videosdeomWW is i portant forapplications like robot navigation and hu an co puter interaction. If every point in thes cene rando ly oves it is i  posW=J= to recover the nonrigidsr es In practice, any nonrigid o ..."
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Cited by 75 (10 self)
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Recovery of three diensWXzm (3D) sD) e and otion of nonsN;m[ s cenes fro a onocular videosdeomWW is i portant forapplications like robot navigation and hu an co puter interaction. If every point in thes cene rando ly oves it is i  posW=J= to recover the nonrigidsr es In practice, any nonrigid objects e.g. the hu an face under various expres[XFX] defor with certains tructures Theirs hapes can be regarded as a weighted co bination of certains hapebasXJ Shape and otion recovery unders uchs ituations has attracted uch interesX Previous work onthis proble [6, 4, 13] utilized only orthonor ality consWJNm ts on the ca era rotations (ro tation constraints).This paper proves that usJ] only the rotation cons]N]m ts res]N] in a biguous and invalid smWWX];m[ The a biguity arisX fro the fact that thesmX e bas+ are not unique becaus their linear transJW ation is a news et of eligiblebasib To eli inate the a biguity, we propos as et of novel consNXNm ts basis constraints, which uniquely deter ine thesmW e bas;F We prove that, under the weakp ers ective projection odel, enforcing both the bas= and the rotation consW+;m ts leads to a closNm[JF slosNm to the proble of nonrigids hape and otion recovery. The accuracy and robus;Wm[ of ourclos=;m[J slos=; is evaluated quantitatively on sm thetic data and qualitatively on real videoseomWN;JN 1
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 71 (6 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 general framework for motion segmentation: Independent, articulated, rigid, nonrigid, degenerate and nondegenerate
 In ECCV
, 2006
"... Abstract. We cast the problem of motion segmentation of feature trajectories as linear manifold finding problems and propose a general framework for motion segmentation under affine projections which utilizes two properties of trajectory data: geometric constraint and locality. The geometric constra ..."
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Cited by 69 (0 self)
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Abstract. We cast the problem of motion segmentation of feature trajectories as linear manifold finding problems and propose a general framework for motion segmentation under affine projections which utilizes two properties of trajectory data: geometric constraint and locality. The geometric constraint states that the trajectories of the same motion lie in a low dimensional linear manifold and different motions result in different linear manifolds; locality, by which we mean in a transformed space a data and its neighbors tend to lie in the same linear manifold, provides a cue for efficient estimation of these manifolds. Our algorithm estimates a number of linear manifolds, whose dimensions are unknown beforehand, and segment the trajectories accordingly. It first transforms and normalizes the trajectories; secondly, for each trajectory it estimates a local linear manifold through local sampling; then it derives the affinity matrix based on principal subspace angles between these estimated linear manifolds; at last, spectral clustering is applied to the matrix and gives the segmentation result. Our algorithm is general without restriction on the number of linear manifolds and without prior knowledge of the dimensions of the linear manifolds. We demonstrate in our experiments that it can segment a wide range of motions including independent, articulated, rigid, nonrigid, degenerate, nondegenerate or any combination of them. In some highly challenging cases where other stateoftheart motion segmentation algorithms may fail, our algorithm gives expected results. 2 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 68 (5 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.
NonRigid StructureFromMotion: Estimating Shape and Motion with Hierarchical Priors
, 2007
"... This paper describes methods for recovering timevarying shape and motion of nonrigid 3D objects from uncalibrated 2D point tracks. For example, given a video recording of a talking person, we would like to estimate the 3D shape of the face at each instant, and learn a model of facial deformation. ..."
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Cited by 50 (1 self)
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This paper describes methods for recovering timevarying shape and motion of nonrigid 3D objects from uncalibrated 2D point tracks. For example, given a video recording of a talking person, we would like to estimate the 3D shape of the face at each instant, and learn a model of facial deformation. Timevarying shape is modeled as a rigid transformation combined with a nonrigid deformation. Reconstruction is illposed if arbitrary deformations are allowed, and thus additional assumptions about deformations are required. We first suggest restricting shapes to lie within a lowdimensional subspace, and describe estimation algorithms. However, this restriction alone is insufficient to constrain reconstruction. To address these problems, we propose a reconstruction method using a Probabilistic Principal Components Analysis (PPCA) shape model, and an estimation algorithm that simultaneously estimates 3D shape and motion for each instant, learns the PPCA model parameters, and robustly fillsin missing data points. We then extend the model to model temporal dynamics in object shape, allowing the algorithm to robustly handle severe cases of missing data.
Multibody Structure and Motion: 3D Reconstruction of Independently Moving Objects
 In European Conference on Computer Vision
, 2000
"... . This paper extends the recovery of structure and motion to image sequences with several independently moving objects. The motion, structure, and camera calibration are all apriori unknown. The fundamental constraint that we introduce is that multiple motions must share the same camera paramete ..."
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Cited by 47 (0 self)
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. This paper extends the recovery of structure and motion to image sequences with several independently moving objects. The motion, structure, and camera calibration are all apriori unknown. The fundamental constraint that we introduce is that multiple motions must share the same camera parameters. Existing work on independent motions has not employed this constraint, and therefore has not gained over independent staticscene reconstructions. We show how this constraint leads to several new results in structure and motion recovery, where Euclidean reconstruction becomes possible in the multibody case, when it was underconstrained for a static scene. We show how to combine motions of highrelief, lowrelief and planar objects. Additionally we show that structure and motion can be recovered from just 4 points in the uncalibrated, fixed camera, case. Experiments on real and synthetic imagery demonstrate the validity of the theory and the improvement in accuracy obtained usin...
TwoView Multibody Structure from Motion
, 2006
"... We present an algebraic geometric approach to 3D motion estimation and segmentation of multiple rigidbody motions from noisefree point correspondences in two perspective views. Our approach exploits the algebraic and geometric properties of the socalled multibody epipolar constraint and its asso ..."
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Cited by 47 (15 self)
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We present an algebraic geometric approach to 3D motion estimation and segmentation of multiple rigidbody motions from noisefree point correspondences in two perspective views. Our approach exploits the algebraic and geometric properties of the socalled multibody epipolar constraint and its associated multibody fundamental matrix, which are natural generalizations of the epipolar constraint and of the fundamental matrix to multiple motions. We derive a rank constraint on a polynomial embedding of the correspondences, from which one can estimate the number of independent motions as well as linearly solve for the multibody fundamental matrix. We then show how to compute the epipolar lines from the firstorder derivatives of the multibody epipolar constraint and the epipoles by solving a plane clustering problem using Generalized PCA (GPCA). Given the epipoles and epipolar lines, the estimation of individual fundamental matrices becomes a linear problem. The clustering of the feature points is then automatically obtained from either the epipoles and epipolar lines or from the individual fundamental matrices. Although our approach is mostly designed for noisefree correspondences, we also test its performance on synthetic and real data with moderate levels of noise.
S.: Counting crowded moving objects
, 2006
"... In its full generality, motion analysis of crowded objects necessitates recognition and segmentation of each moving entity. The difficulty of these tasks increases considerably with occlusions and therefore with crowding. When the objects are constrained to be of the same kind, however, partitioning ..."
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Cited by 43 (1 self)
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In its full generality, motion analysis of crowded objects necessitates recognition and segmentation of each moving entity. The difficulty of these tasks increases considerably with occlusions and therefore with crowding. When the objects are constrained to be of the same kind, however, partitioning of densely crowded semirigid objects can be accomplished by means of clustering tracked feature points. We base our approach on a highly parallelized version of the KLT tracker in order to process the video into a set of feature trajectories. While such a set of trajectories provides a substrate for motion analysis, their unequal lengths and fragmented nature present difficulties for subsequent processing. To address this, we propose a simple means of spatially and temporally conditioning the trajectories. Given this representation, we integrate it with a learned object descriptor to achieve a segmentation of the constituent motions. We present experimental results for the problem of estimating the number of moving objects in a dense crowd as a function of time. 1