Robust Structure and Motion Recovery Based on Augmented Factorization

This paper proposes a new strategy to promote the robustness of structure from motion algorithm from uncalibrated video sequences. First, an augmented affine factorization algorithm is formulated to circumvent the difficulty in image registration with noise and outliers contaminated data. Then, an alternative weighted factorization scheme is designed to handle the missing data and measurement uncertainties in the tracking matrix. Finally, a robust strategy for structure and motion recovery is proposed to deal with outliers and large measurement noise. This paper makes the following main contributions: 1) An augmented factorization algorithm is proposed to circumvent the difficult image registration problem of previous affine factorization, and the approach is applicable to both rigid and nonrigid scenarios; 2) by employing the fact that image reprojection residuals are largely proportional to the error magnitude in the tracking data, a simple outliers detection approach is proposed; and 3) a robust factorization strategy is developed based on the distribution of the reprojection residuals. Furthermore, the proposed approach can be easily extended to nonrigid scenarios. Experiments using synthetic and real image data demonstrate the robustness and efficiency of the proposed approach over previous algorithms.22289016157335

Sparse Subspace Clustering: Algorithm, Theory, and Applications

In many real-world problems, we are dealing with collections of high-dimensional data, such as images, videos, text and web documents, DNA microarray data, and more. Often, high-dimensional data lie close to low-dimensional structures corresponding to several classes or categories the data belongs to. In this paper, we propose and study an algorithm, called Sparse Subspace Clustering (SSC), to cluster data points that lie in a union of low-dimensional subspaces. The key idea is that, among infinitely many possible representations of a data point in terms of other points, a sparse representation corresponds to selecting a few points from the same subspace. This motivates solving a sparse optimization program whose solution is used in a spectral clustering framework to infer the clustering of data into subspaces. Since solving the sparse optimization program is in general NP-hard, we consider a convex relaxation and show that, under appropriate conditions on the arrangement of subspaces and the distribution of data, the proposed minimization program succeeds in recovering the desired sparse representations. The proposed algorithm can be solved efficiently and can handle data points near the intersections of subspaces. Another key advantage of the proposed algorithm with respect to the state of the art is that it can deal with data nuisances, such as noise, sparse outlying entries, and missing entries, directly by incorporating the model of the data into the sparse optimization program. We demonstrate the effectiveness of the proposed algorithm through experiments on synthetic data as well as the two real-world problems of motion segmentation and face clustering

Subspace segmentation with a minimal square frobenius norm representation

We introduce a novel subspace segmentation method called Minimal Squared Frobenius Norm Representation (MSFNR). MSFNR performs data clustering by solving a convex optimization problem. We theoretically prove that in the noiseless case, MSFNR is equivalent to the classical Factorization approach and always classifies data correctly. In the noisy case, we show that on both synthetic and real-word datasets, MSFNR is much faster than most state-of-the-art methods while achieving comparable segmentation accuracy.published_or_final_versio

Angles from BB Decays with Charm

Proceedings of the CKM 2005 Workshop (WG5), UC San Diego, 15-18 March 2005.Comment: 62 pages, 55 figures. Proceedings of the CKM 2005 Workshop (WG5), UC San Diego, 15-18 March 200

Parallel accelerated cyclic reduction preconditioner for three-dimensional elliptic PDEs with variable coefficients

We present a robust and scalable preconditioner for the solution of large-scale linear systems that arise from the discretization of elliptic PDEs amenable to rank compression. The preconditioner is based on hierarchical low-rank approximations and the cyclic reduction method. The setup and application phases of the preconditioner achieve log-linear complexity in memory footprint and number of operations, and numerical experiments exhibit good weak and strong scalability at large processor counts in a distributed memory environment. Numerical experiments with linear systems that feature symmetry and nonsymmetry, definiteness and indefiniteness, constant and variable coefficients demonstrate the preconditioner applicability and robustness. Furthermore, it is possible to control the number of iterations via the accuracy threshold of the hierarchical matrix approximations and their arithmetic operations, and the tuning of the admissibility condition parameter. Together, these parameters allow for optimization of the memory requirements and performance of the preconditioner.Comment: 24 pages, Elsevier Journal of Computational and Applied Mathematics, Dec 201

Robust Recovery of Subspace Structures by Low-Rank 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 termed Low-Rank Representation (LRR), which seeks the lowest-rank 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.Comment: IEEE Trans. Pattern Analysis and Machine Intelligenc

Neural Collaborative Subspace Clustering

We introduce the Neural Collaborative Subspace Clustering, a neural model that discovers clusters of data points drawn from a union of low-dimensional subspaces. In contrast to previous attempts, our model runs without the aid of spectral clustering. This makes our algorithm one of the kinds that can gracefully scale to large datasets. At its heart, our neural model benefits from a classifier which determines whether a pair of points lies on the same subspace or not. Essential to our model is the construction of two affinity matrices, one from the classifier and the other from a notion of subspace self-expressiveness, to supervise training in a collaborative scheme. We thoroughly assess and contrast the performance of our model against various state-of-the-art clustering algorithms including deep subspace-based ones.Comment: Accepted to ICML 201