In this paper we propose novel methods for completion (from limited samples)
and de-noising of multilinear (tensor) data and as an application consider 3-D
and 4- D (color) video data completion and de-noising. We exploit the recently
proposed tensor-Singular Value Decomposition (t-SVD)[11]. Based on t-SVD, the
notion of multilinear rank and a related tensor nuclear norm was proposed in
[11] to characterize informational and structural complexity of multilinear
data. We first show that videos with linear camera motion can be represented
more efficiently using t-SVD compared to the approaches based on vectorizing or
flattening of the tensors. Since efficiency in representation implies
efficiency in recovery, we outline a tensor nuclear norm penalized algorithm
for video completion from missing entries. Application of the proposed
algorithm for video recovery from missing entries is shown to yield a superior
performance over existing methods. We also consider the problem of tensor
robust Principal Component Analysis (PCA) for de-noising 3-D video data from
sparse random corruptions. We show superior performance of our method compared
to the matrix robust PCA adapted to this setting as proposed in [4].

本研究提出了一种多线性数据完成和去噪的新方法，应用于 3D 和 4D 视频数据的完成和去噪，利用了最近提出的张量奇异值分解（t-SVD）方法，基于 t-SVD，提出了多线性秩和相关张量核范数的概念来表征多线性数据的信息和结构复杂性。在此基础上，我们提出了一个张量核范数惩罚算法，用于从缺失项中完成视频，并且相对于已有方法，该算法表现出更优秀的性能。我们还考虑了从稀疏随机损坏中去噪 3D 视频数据的张量鲁棒主成分分析（PCA）问题，相对于矩阵鲁棒 PCA 的方法，我们的方法表现更优秀。