Kavli Affiliate: Eric Miller
| First 5 Authors: Fan Tian, Misha E. Kilmer, Eric Miller, Abani Patra,
| Summary:
Given tensors $boldsymbol{mathscr{A}}, boldsymbol{mathscr{B}},
boldsymbol{mathscr{C}}$ of size $m times 1 times n$, $m times p times 1$,
and $1times p times n$, respectively, their Bhattacharya-Mesner (BM) product
will result in a third-order tensor of dimension $m times p times n$ and
BM-rank of 1 (Mesner and Bhattacharya, 1990). Thus, if an arbitrary $m times p
times n$ third-order tensor can be written as a sum of a small number,
relative to $m,p,n$, of such BM-rank 1 terms, this BM-decomposition (BMD)
offers an implicitly compressed representation of the tensor. In this paper, we
first show that grayscale surveillance video can be accurately captured by a
low BM-rank decomposition and give methods for efficiently computing this
decomposition. To this end, we first give results that connect rank-revealing
matrix factorizations to the BMD. Next, we present a generative model that
illustrates that spatio-temporal video data can be expected to have low
BM-rank. We combine these observations to derive a regularized alternating
least squares (ALS) algorithm to compute an approximate BMD of the video
tensor. The algorithm itself is highly parallelizable since the bulk of the
computations break down into relatively small regularized least squares
problems that can be solved independently. Extensive numerical results compared
against the state-of-the-art matrix-based DMD for surveillance video separation
show our algorithms can consistently produce results with superior compression
properties while simultaneously providing better separation of stationary and
non-stationary features in the data. We then introduce a new type of BM-product
suitable for color video and provide an algorithm that shows an impressive
ability to extract important temporal information from color video while
simultaneously compressing the data.
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