Multiway Filtering Based on Fourth-Order Cumulants
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Multiway Filtering Based on Fourth-Order Cumulants Damien Muti Groupe GSM, Institut Fresnel (UMR CNRS 6133), EGIM, Universit´e Aix-Marseille III, D.U. de Saint J´erˆome, 13397 Marseille Cedex 20, France Email: [email protected]
Salah Bourennane Groupe GSM, Institut Fresnel (UMR CNRS 6133), EGIM, Universit´e Aix-Marseille III, D.U. de Saint J´erˆome, 13397 Marseille Cedex 20, France Email: [email protected] Received 31 March 2004; Revised 4 November 2004; Recommended for Publication by Chong-Yung Chi We propose a new multiway filtering based on fourth-order cumulants for the denoising of noisy data tensor with correlated Gaussian noise. The classical multiway filtering is based on the TUCKALS3 algorithm that computes a lower-rank tensor approximation. The presented method relies on the statistics of the analyzed multicomponent signal. We first recall how the well-known lower-rank-(K1 , . . . , KN ) tensor approximation processed by TUCKALS3 alternating least square algorithm exploits second-order statistics. Then, we propose to introduce the fourth-order statistics in the TUCKALS3-based method. Indeed, the use of fourthorder cumulants enables to remove the Gaussian components of an additive noise. In the presented method the estimation of the n-mode projector on the n-mode signal subspace is built from the eigenvectors associated with the largest eigenvalues of a fourthorder cumulant slice matrix instead of a covariance matrix. Each projector is applied by means of the n-mode product operator on the n-mode of the data tensor. The qualitative results of the improved multiway TUCKALS3-based filterings are shown for the case of noise reduction in a color image and multicomponent seismic data. Keywords and phrases: multicomponent signals, tensors, Tucker3 decomposition, HOSVD, cumulant slice matrix, subspace methods.
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INTRODUCTION
In many fields so diverse as chemometric, psychology, data analysis, or signal processing, a multidimensional and multiway modelling can be adopted in which data are represented by higher-order tensors [1]. A tensor of order N is as a multidimensional array whose entries are accessed via N indices. It is noted A ∈ RI1 ×···×IN , where each element is ai1 i2 ···iN , and R is the real manifold. Index in refers to the n-mode of tensor A. In signal processing, tensors are built on vector spaces associated with physical quantities such as length, width, height, time, color channel, and so forth. Each mode of the tensor is associated with a physical quantity. For example, in image processing, color images can be modelled as thirdorder tensors: two dimensions for lines and columns, and one dimension for the color map. In seismic and underwater acoustics, when a linear antenna is used, a three-dimensional modelling of data can be chosen as well: one mode is associated with the spatial sensors of the antenna, one mode with the time, and one mode with the wave polarization components.
In the classical processing based on algebraic methods, multidimensional and multiway data are split
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