The goal of this project is to analyze and extract the background information from a timelapse/surveillance video. To solve the dynamic segmentation problems that are encountered in imaging, we can utilize many sparse representation techniques and tools. We utilize the sparsity of a signal to solve an underdetermined system of linear equations. The method we will be utilizing is called RPCA (Robust Principal Component Analysis)^[1] which is a sparse and low rank matrix decomposition problem. We will assume a signal Y ∈ ℝ^m×n that contains static and dynamic components (In our instance it is L= Background; S= Moving objects e.g. cars). Next, we will estimate the values of L and S from the observation Y by solving the following optimization problem:

● Minimize ∥L∥∗+λ∥S∥₁ subject to : L+S=Y

where ║ · ║∗ denotes the nuclear norm.

We’ll be using the Alternating Direction Method of Multipliers for solving the constrained optimization problem.

References: [1] McLean JP, Ling Y, Hendon CP. Frequency-constrained robust principal component analysis:  a sparse representation approach to segmentation of dynamic features in optical coherence  tomography imaging. ​Opt Express​. 2017;25(21):25819–25830. doi:10.1364/OE.25.025819