用于视频数据压缩与分析的张量 BM 分解
Tensor BM-Decomposition for Compression and Analysis of Video Data
AI总结：
本文提出低 BM 秩张量分解及其正则化 ALS 算法，用于灰度与彩色视频的压缩、背景/前景分离及时空信息提取，并在实验中优于基于矩阵的 DMD。
摘要：
给定大小分别为 $m \times 1 \times n$、$m \times p \times 1$ 和 $1 \times p \times n$ 的张量 $\boldsymbol{\mathscr{A}}$、$\boldsymbol{\mathscr{B}}$、$\boldsymbol{\mathscr{C}}$，它们的 Bhattacharya-Mesner（BM）乘积将产生一个维度为 $m \times p \times n$ 且 BM 秩为 1 的三阶张量（Mesner 和 Bhattacharya，1990）。因此，如果任意一个 $m \times p \times n$ 三阶张量都可以表示为数量相对于 $m,p,n$ 较少的此类 BM 秩 1 项之和，那么这种 BM 分解（BMD）就提供了该张量的一种隐式压缩表示。本文首先证明灰度监控视频可以由低 BM 秩分解精确刻画，并给出高效计算该分解的方法。为此，我们首先给出将秩揭示矩阵分解与 BMD 联系起来的结果。接下来，我们提出一个生成模型，用来说明时空视频数据可以预期具有较低的 BM 秩。我们结合这些观察结果，推导出一种正则化交替最小二乘（ALS）算法，用于计算视频张量的近似 BMD。该算法本身具有高度可并行性，因为大部分计算可分解为能够独立求解的、规模相对较小的正则化最小二乘问题。大量数值结果与用于监控视频分离的最先进基于矩阵的 DMD 方法进行了比较，结果表明，我们的算法能够持续产生具有更优压缩性能的结果，同时更好地分离数据中的静止和非静止特征。随后，我们引入一种适用于彩色视频的新型 BM 乘积，并提供一种算法；该算法在压缩数据的同时，展现出从彩色视频中提取重要时间信息的出色能力。
英文摘要：
Given tensors $\boldsymbol{\mathscr{A}}, \boldsymbol{\mathscr{B}}, \boldsymbol{\mathscr{C}}$ of size $m \times 1 \times n$, $m \times p \times 1$, and $1\times p \times n$, 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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