Signal Processing 85 (2005) 2338–2353 Multidimensional filtering based on a tensor approach Damien Muti à , Salah Bourennane GSM Team, Institut Fresnel UMR CNRS 6133, Universite´Aix-Marseille 3 EGIM Nord, DU de Saint Je´roˆme, 13397 Marseille Cedex 20, France Received 13 June 2003 Available online 12 May 2005 Abstract A new multidimensional modelling of data has recently been suggested, which can be applied in a wide range of signal processing fields. Many studies have proposed new tensorial mathematical tools in order to process multidimensional data. With a view of perfecting this multidimensional model, this paper presents a new tensor approach for multidimensional data filtering. A theoretical expression of n-mode filters is established based on a specific modelling of the desired information. The optimization criterion used in this tensorial filtering is the minimization of the mean square error between the estimated signal and the desired signal. This minimization leads to some estimated n-mode filters which can be considered as an extension of the well-known Wiener filter in a particular mode. An alternating least square algorithm is proposed to determine each n-mode Wiener filter. This new multimode Wiener filtering method is tested for noise reduction in multicomponent seismic data. A comparative study with classical bidimensional filtering methods based on principal component analysis is also proposed and presents encouraging results. r 2005 Elsevier B.V. All rights reserved. Keywords: Multilinear algebra; Tensor; Multiway arrays; Subspace method; Tucker3 decomposition; LRTA; HOSVD; Multidimensional Wiener filtering 1. Introduction Multidimensional models can be used in a large range of fields so diverse as chemometrics, psychology, data analysis or signal processing [1]. In signal processing, tensors are built on vector spaces associated with physical quantities such as length, width, height, time, color channel, etc. Each mode of the tensor is associated with a physical quantity. For instance, in image proces- sing, color images can be modelled as three- dimensional tensors: two dimensions for rows and columns, and one dimension for the color map. In seismic, when a linear antenna composed of multicomponent sensors is used, a three- dimensional modelling of data can be adopted as well: one mode is associated with the spatial ARTICLE IN PRESS www.elsevier.com/locate/sigpro 0165-1684/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.sigpro.2004.11.029 à Corresponding author. Tel.: +33 4 9127 8202. E-mail addresses: damien.muti@fresnel.fr (D. Muti), salah.bourennane@fresnel.fr (S. Bourennane).