POD-Based Denoising/Reconstru​ction

Proper Orthogonal Decomposition (POD) is an analysis technique that is able to extract the most energetic modes of a time series. Turns out gaussian noise will spread its energy across all modes in POD, which enables de-noising of video and other N-D time series by reconstructing the time series only from its most energetic modes.

As Matlab's SVD returns the modes in energetic order (mode 1=most energetic), one can reconstruct a time series X with less modes (Modes is an array, could be 1:nModes or anything you fancy!). The series will look blurry and weird, but if enough modes are used without using all modes, noise can be rejected to an extent.

Threshold is found by using Gavish and Donoho's relation (The Optimal Hard Threshold for Singular Values is 4/sqrt(3)), IEEE Transactions on Information Theory, Aug. 2014 - if the user doesn't provide the modes they want to keep.