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README.md

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-Decomposition into Low-Rank and Sparse Matrices 
+# Decomposition into Low-Rank and Sparse Matrices 
+PCP was proposed and developed by Candes et al ( https://arxiv.org/pdf/0912. 3599.pdf ) via an alternating direction methods to solve RPCA problem:
 
-dataset - Carnegie Mellon Test Images Sequences (Y. Sheikh,Robotics  Institute,  Canergie  Mellon  University,  USA)  (1  video,  GT  images  for  the  se-quence)
+minimize | L | ∗ + | S | 1 subject to L + S = M
+
+where | L | ∗ - is a nuclear norm, | S | 1 - is a L1-norm.
+
+
+
+This approach gives good results for recovering low-rank and the sparse matrices in a background (low rank subspace) and foreground (sparse matrix - outliers) separation. But there are some disadvantages of this algorithm:
+
+• PCP is a very computationally expensive
+
+• PCP is a batch method (stack training frames in the input), so PCP is inappropriate for real-time subtraction (not incremental way, which would be more useful)
+
+• PCP imposed the low-rank component being exactly low-rank and sparse component being exactly sparse but the observations such as in video surveillance are ofter corrupted by noise affecting every entry of the data matrix A lot of people are working on different variations of improved algorithms of PCP - incremental algorithms for updating L and S , real-time implementation. Available methods: Modified PCP with Partial Subspace Knowledge, Inductive PCP, PCP with Free Nuclear Norm and so on.
+
+
+## We implemented Principal Component Pursuit by Alternating Directions with following algorithm:
+![plot](./PCP-algorithm.png)
+
+### Results
+
+<img src="/result.gif" width="600" height="180"/>
+
+### Dataset
+Carnegie Mellon Test Images Sequences 500frames=25seconds (Y. Sheikh,Robotics  Institute,  Canergie  Mellon  University,  USA)  (1  video,  GT  images  for  the  se-quence)
 http://www.cs.cmu.edu/~yaser/new_backgroundsubtraction.htm
+
+