About Advances in Tensor Data Denoising Methods

Tensor methods are of great interest since the development of multicomponent sensors. The acquired multicomponent data are represented by tensors, that is, multiway arrays. This paper presents advances on filtering methods to improve tensor data denoising. Channel-by-channel and multiway methods are presented. The first multiway method is based on the lower-rank truncation of the HOSVD. The second one consists of an extension of Wiener filtering to data tensors. When multiway tensor filtering is performed, the processed tensor is flattened along each mode successively, and singular value decomposition of the flattened matrix is performed. Data projection on the singular vectors associated with dominant singular values results in noise reduction. We propose a synthesis of crucial issues which were recently solved, that is, the estimation of the number of dominant singular vectors, the optimal choice of flattening directions, and the reduction of the computational load of multiway tensor filtering methods. The presented methods are compared through an application to a color image and a seismic signal, multiway Wiener filtering providing the best denoising results. We apply multiway Wiener filtering and its fast version to a hyperspectral image. The fast multiway filtering method is 29 times faster and yields very close denoising results.