Lower-Rank Tensor Approximation and Multiway Filtering

Abstract

This paper presents some recent filtering methods based on the lower-rank tensor approximation approach for denoising tensor signals. In this approach, multicomponent data are represented by tensors, that is, multiway arrays, and the presented tensor filtering methods rely on multilinear algebra. First, the classical channel-by-channel SVD-based filtering method is overviewed. Then, an extension of the classical matrix filtering method is presented. It is based on the lower rank-$(K_1,\dots,K_N)$ truncation of the higher order SVD which performs a multimode principal component analysis (PCA) and is implicitly developed for an additive white Gaussian noise. Two tensor filtering methods recently developed by the authors are also overviewed. The first method consists of an improvement of the multimode PCA-based tensor filtering in the case of an additive correlated Gaussian noise. This improvement is specially done thanks to the fourth order cumulant slice matrix. The second method consists of an extension of Wiener filtering for data tensors. The performances and comparative results between all these tensor filtering methods are presented for the cases of noise reduction in color images, multispectral images, and multicomponent seismic data.