A Unified View of Robust Decomposition in Sparse + Additive Matrices for Computer Vision

Sparse decompositions are similar to low-rank decompositions except that the first matrix is considered to be sparse instead of low-rank. Sparse decompositions are achieved in the different following problem formulations: Sparse dictionary learning, sparse linear approximation and compressive sensing.

Robust Decomposition in Sparse + Additive Matrices

All the decompositions in these different problem formulations of robust subspace learning/tracking can be considered in a unified view that we called Decomposition into Sparse plus Additive Matrices (DSAM). Thus, all the decompositions can be written in a general formulation as follows:

• The first matrix M₁ is a low-rank matrix S.
• The second matrix M₂ is an unconstrained (residual) matrix E.

Key Characteristics

• Minimization problems
• Solvers
• Norms

Applications to Statistical Modeling and Computer Vision

The different robust problem formulations based on the decomposition into sparse plus additive matrices are fundamental in several applications . Indeed, as this decomposition is nonparametric and does not make many assumptions, it is widely applicable to a large scale of problems ranging from:

• Image processing
• Video processing (background and foreground separation)

Author: Thierry BOUWMANS, Associate Professor, Lab. MIA, Univ. Rochelle, France.

Fair Use Policy

As this website gives many information that come from my research, please cite my following survey papers:

T. Bouwmans . A. Sobral, S. Javed, S. Jung, E. Zahzah, "Decomposition into Low-rank plus Additive Matrices for Background/Foreground Separation: A Review for a Comparative Evaluation with a Large-Scale Dataset", Computer Science Review, November 2016. [pdf]

T. Bouwmans, E. Zahzah, “Robust PCA via Principal Component Pursuit: A Review for a Comparative Evaluation in Video Surveillance”, Special Issue on Background Models Challenge, Computer Vision and Image Understanding, CVIU 2014, Volume 122, pages 22–34, May 2014. [pdf]