Mining useful clusters from high dimensional data have received significant attention of the computer vision and pattern recognition community in the recent years. Linear and nonlinear dimensionality reduction has played an important role to overcome the curse of dimensionality. However, often such methods are accompanied with three different problems: high computational complexity (usually associated with the nuclear norm minimization), nonconvexity (for matrix factorization methods), and susceptibility to gross corruptions in the data. In this paper, we propose a principal component analysis (PCA) based solution that overcomes these three issues and approximates a low-rank recovery method for high dimensional datasets. We target the low-rank recovery by enforcing two types of graph smoothness assumptions, one on the data samples and the other on the features by designing a convex optimization problem. The resulting algorithm is fast, efficient, and scalable for huge datasets with O(n log(n)) computational complexity in the number of data samples. It is also robust to gross corruptions in the dataset as well as to the model parameters. Clustering experiments on 7 benchmark datasets with different types of corruptions and background separation experiments on 3 video datasets show that our proposed model outperforms 10 state-of-the-art dimensionality reduction models. Our theoretical analysis proves that the proposed model is able to recover approximate low-rank representations with a bounded error for clusterable data.
Abstract Foreground detection is the first step in video surveillance system to detect moving objects. Recent research on subspace estimation by sparse representation and rank minimization represents a nice framework to separate moving objects from the background. Robust Principal Component Analysis (RPCA) solved via Principal Component Pursuit decomposes a data matrix A in two components such that A = L + S , where L is a low-rank matrix and S is a sparse noise matrix. The background sequence is then modeled by a low-rank subspace that can gradually change over time, while the moving foreground objects constitute the correlated sparse outliers. To date, many efforts have been made to develop Principal Component Pursuit (PCP) methods with reduced computational cost that perform visually well in foreground detection. However, no current algorithm seems to emerge and to be able to simultaneously address all the key challenges that accompany real-world videos. This is due, in part, to the absence of a rigorous quantitative evaluation with synthetic and realistic large-scale dataset with accurate ground truth providing a balanced coverage of the range of challenges present in the real world. In this context, this work aims to initiate a rigorous and comprehensive review of RPCA-PCP based methods for testing and ranking existing algorithms for foreground detection. For this, we first review the recent developments in the field of RPCA solved via Principal Component Pursuit. Furthermore, we investigate how these methods are solved and if incremental algorithms and real-time implementations can be achieved for foreground detection. Finally, experimental results on the Background Models Challenge (BMC) dataset which contains different synthetic and real datasets show the comparative performance of these recent methods.