#Accurate rank tracker plus
Although, many efforts have been made to develop methods for the decomposition into low-rank plus additive matrices that perform visually well in foreground detection with reducing their computational cost, no algorithm today seems to emerge and to be able to simultaneously address all the key challenges that accompany real-world videos. Practically, 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. In the application of background/foreground separation, constraints inherent to the specificities of the background and the foreground as the temporal and spatial properties need to be taken into account in the design of the problem formulation. As the problem formulation can be NP-hard in its original formulation, and it can be convex or not following the constraints and the loss functions used, the key challenges concern the design of efficient relaxed models and solvers which have to be with iterations as few as possible, and as efficient as possible. These formulation problems differ from the implicit or explicit decomposition, the loss function, the optimization problem and the solvers. The main goal of these similar problem formulations is to obtain explicitly or implicitly a decomposition into low-rank matrix plus additive matrices. However, similar robust implicit or explicit decompositions can be made in the following problem formulations: Robust Non-negative Matrix Factorization (RNMF), Robust Matrix Completion (RMC), Robust Subspace Recovery (RSR), Robust Subspace Tracking (RST) and Robust Low-Rank Minimization (RLRM).
The most representative problem formulation is the Robust Principal Component Analysis (RPCA) solved via Principal Component Pursuit (PCP) which decomposes a data matrix into a low-rank matrix and a sparse matrix.
Recent research on problem formulations based on decomposition into low-rank plus sparse matrices shows a suitable framework to separate moving objects from the background. Background/foreground separation is the first step in video surveillance system to detect moving objects.
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