N. Benjamin Erichson

Randomized low-rank Dynamic Mode Decomposition for motion detection

Fast and robust decomposition of a matrix representing a spatial grid through time.Rapid approximation for robust principal component analysis.Competitive performance in terms of recall and precision for motion detection.GPU accelerated implementation allows faster computation. This paper introduces a fast algorithm for randomized computation of a low-rank Dynamic Mode Decomposition (DMD) of a matrix. Here we consider this matrix to represent the development of a spatial grid through time e.g. data from a static video source. DMD was originally introduced in the fluid mechanics community, but is also suitable for motion detection in video streams and its use for background subtraction has received little previous investigation. In this study we present a comprehensive evaluation of background subtraction, using the randomized DMD and compare the results with leading robust principal component analysis algorithms. The results are convincing and show the random DMD is an efficient and powerful approach for background modeling, allowing processing of high resolution videos in real-time. Supplementary materials include implementations of the algorithms in Python.

Compressed dynamic mode decomposition for background modeling

We introduce the method of compressed dynamic mode decomposition (cDMD) for background modeling. The dynamic mode decomposition is a regression technique that integrates two of the leading data analysis methods in use today: Fourier transforms and singular value decomposition. Borrowing ideas from compressed sensing and matrix sketching, cDMD eases the computational workload of high-resolution video processing. The key principal of cDMD is to obtain the decomposition on a (small) compressed matrix representation of the video feed. Hence, the cDMD algorithm scales with the intrinsic rank of the matrix, rather than the size of the actual video (data) matrix. Selection of the optimal modes characterizing the background is formulated as a sparsity-constrained sparse coding problem. Our results show that the quality of the resulting background model is competitive, quantified by the F-measure, recall and precision. A graphics processing unit accelerated implementation is also presented which further boosts the computational performance of the algorithm.

Multi-resolution Dynamic Mode Decomposition for Foreground/Background Separation and Object Tracking

We demonstrate that the integration of the recently developed dynamic mode decomposition with a multi-resolution analysis allows for a decomposition of video streams into multi-time scale features and objects. A one-level separation allows for background (low-rank) and foreground (sparse) separation of the video, or robust principal component analysis. Further iteration of the method allows a video data set to be separated into objects moving at different rates against the slowly varying background, thus allowing for multiple-target tracking and detection. The algorithm is computationally efficient and can be integrated with many further innovations including compressive sensing architectures and GPU algorithms.

Randomized nonnegative matrix factorization

Abstract Nonnegative matrix factorization (NMF) is a powerful tool for data mining. However, the emergence of ‘big data’ has severely challenged our ability to compute this fundamental decomposition using deterministic algorithms. This paper presents a randomized hierarchical alternating least squares (HALS) algorithm to compute the NMF. By deriving a smaller matrix from the nonnegative input data, a more efficient nonnegative decomposition can be computed. Our algorithm scales to big data applications while attaining a near-optimal factorization, i.e., the algorithm scales with the target rank of the data rather than the ambient dimension of measurement space. The proposed algorithm is evaluated using synthetic and real world data and shows substantial speedups compared to deterministic HALS.