In this paper we address a classification problem that has not been considered before, namely motion segmentation given pairwise matches only. Our contribution to this unexplored task is a novel formulation of motion segmentation as a two-step process. First, motion segmentation is performed on image pairs independently. Secondly, we combine independent pairwise segmentation results in a robust way into the final globally consistent segmentation. Our approach is inspired by the success of averaging methods. We demonstrate in simulated as well as in real experiments that our method is very effective in reducing the errors in the pairwise motion segmentation and can cope with large number of mismatches.
The goal of is to cluster high-dimensional data drawn from multiple low-dimensional subspaces. Existing solutions include Generalized Principal Component Analysis (GPCA) @cite_37 , Local Subspace Affinity (LSA) @cite_49 , Power Factorization (PF) @cite_3 , Agglomerative Lossy Compression (ALC) @cite_27 , Low-Rank Representation (LRR) @cite_45 , Sparse Subspace Clustering (SSC) @cite_22 , Structured Sparse Subspace Clustering (S @math C) @cite_15 , and Robust Shape Interaction Matrix (RSIM) @cite_35 . Motion segmentation can be cast as subspace separation since -- under the affine camera model -- the point trajectories lie in the union of @math subspaces in @math of dimension at most 4, where @math denotes the number of motions and @math denotes the number of images. Subspace separation techniques can also be used to solve motion segmentation in two images under the perspective camera model, since corresponding points undergoing the same motion -- after a proper rearrangement of coordinates -- belong to a subspace of @math of dimension at most 8, as observed in @cite_24 .
@cite_35 @cite_37 @cite_22 @cite_3 @cite_24 @cite_27 @cite_45 @cite_49 @cite_15
2160915541 2056639756 2071422140 1931947167
This paper studies the subspace segmentation problem. Given a set of data points drawn from a union of subspaces, the goal is to partition them into their underlying subspaces they were drawn from. The spectral clustering method is used as the framework. It requires to find an affinity matrix which is close to block diagonal, with nonzero entries corresponding to the data point pairs from the same subspace. In this work, we argue that both sparsity and the grouping effect are important for subspace segmentation. A sparse affinity matrix tends to be block diagonal, with less connections between data points from different subspaces. The grouping effect ensures that the highly corrected data which are usually from the same subspace can be grouped together. Sparse Subspace Clustering (SSC), by using l1-minimization, encourages sparsity for data selection, but it lacks of the grouping effect. On the contrary, Low-Rank Representation (LRR), by rank minimization, and Least Squares Regression (LSR), by l2-regularization, exhibit strong grouping effect, but they are short in subset selection. Thus the obtained affinity matrix is usually very sparse by SSC, yet very dense by LRR and LSR. In this work, we propose the Correlation Adaptive Subspace Segmentation (CASS) method by using trace Lasso. CASS is a data correlation dependent method which simultaneously performs automatic data selection and groups correlated data together. It can be regarded as a method which adaptively balances SSC and LSR. Both theoretical and experimental results show the effectiveness of CASS. We consider the problem of segmenting multiple rigid-body motions from point correspondences in multiple affine views. We cast this problem as a subspace clustering problem in which point trajectories associated with each motion live in a linear subspace of dimension two, three or four. Our algorithm involves projecting all point trajectories onto a 5-dimensional subspace using the SVD, the PowerFactorization method, or RANSAC, and fitting multiple linear subspaces representing different rigid-body motions to the points in ?5 using GPCA. Unlike previous work, our approach does not restrict the motion subspaces to be four-dimensional and independent. Instead, it deals gracefully with all the spectrum of possible affine motions: from two-dimensional and partially dependent to four-dimensional and fully independent. Our algorithm can handle the case of missing data, meaning that point tracks do not have to be visible in all images, by using the PowerFactorization method to project the data. In addition, our method can handle outlying trajectories by using RANSAC to perform the projection. We compare our approach to other methods on a database of 167 motion sequences with full motions, independent motions, degenerate motions, partially dependent motions, missing data, outliers, etc. On motion sequences with complete data our method achieves a misclassification error of less that 5 for two motions and 29 for three motions. The problems of motion segmentation and face clustering can be addressed in a framework of subspace clustering methods. In this paper, we tackle the more general problem of clustering data points lying in a union of low-dimensional linear(or affine) subspaces, which can be naturally applied in motion segmentation and face clustering. For data points drawn from linear (or affine) subspaces, we propose a novel algorithm called Null Space Clustering (NSC), utilizing the null space of the data matrix to construct the affinity matrix. To better deal with noise and outliers, it is converted to an equivalent problem with Frobenius norm minimization, which can be solved efficiently. We demonstrate that the proposed NSC leads to improved performance in terms of clustering accuracy and efficiency when compared to state-of-the-art algorithms on two well-known datasets, i.e., Hopkins 155 and Extended Yale B. Subspace clustering refers to the problem of segmenting data drawn from a union of subspaces. State of the art approaches for solving this problem follow a two-stage approach. In the first step, an affinity matrix is learned from the data using sparse or low-rank minimization techniques. In the second step, the segmentation is found by applying spectral clustering to this affinity. While this approach has led to state of the art results in many applications, it is sub-optimal because it does not exploit the fact that the affinity and the segmentation depend on each other. In this paper, we propose a unified optimization framework for learning both the affinity and the segmentation. Our framework is based on expressing each data point as a structured sparse linear combination of all other data points, where the structure is induced by a norm that depends on the unknown segmentation. We show that both the segmentation and the structured sparse representation can be found via a combination of an alternating direction method of multipliers with spectral clustering. Experiments on a synthetic data set, the Hopkins 155 motion segmentation database, and the Extended Yale B data set demonstrate the effectiveness of our approach.