Generalized visual information analysis via tensorial algebra

Liao, L. and Maybank, Stephen J. (2020) Generalized visual information analysis via tensorial algebra. Journal of Mathematical Imaging and Vision 62 , pp. 560-584. ISSN 0924-9907.

Abstract

High order data is modeled using matrices whose entries are numerical arrays of a fixed size. These arrays, called t-scalars, form a commutative ring under the convolution product. Matrices with elements in the ring of t-scalars are referred to as t-matrices. The t-matrices can be scaled, added and multiplied in the usual way. There are t-matrix generalizations of positive matrices, orthogonal matrices and Hermitian symmetric matrices. With the t-matrix model, it is possible to generalize many well known matrix algorithms. In particular, the t-matrices are used to generalize the SVD (Singular Value Decomposition), HOSVD (High Order SVD), PCA (Principal Component Analysis), 2DPCA (two Dimensional PCA) and GCA (Grassmannian Component Analysis). The generalized t-matrix algorithms, namely TSVD, THOSVD, TPCA, T2DPCA and TGCA, are applied to low-rank approximation, reconstruction and supervised classification of images. Experiments show that the t-matrix algorithms compare favourably with standard matrix algorithms.

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