Trainee Project
Title:
3D and 4D data processing with quaternion tensor decompositions
Dates:
2022/04/01 - 2022/08/31
Other supervisor(s):
Luciani Xavier (xavier.luciani@univ-tln.fr)
Description:
Context : The efficient processing of 3D and 4D data is pivotal in many applications such as robotics, color and polarization imaging or attitude control,
among others. Such multichannel data is often represented using quaternions - a generalization of complex numbers in four dimensions - in order
to simplify expressions and leverage unique geometric and physical insights offered by this algebraic representation. When 3D/4D data is measured along
multiple diversities (e.g. time, frequency, space, etc.), datasets can be viewed as multidimensional quaternion arrays - also called quaternion tensors.

Summary : While quaternion tensors can encode in a compact and meaningful way 3D/4D datasets, they define a challenging mathematical object for which
little results are currently available. This can be explained by the noncommutativity of quaternion multiplication, which prevents a direct use of the tensor
methods originally developed for real and complex tensors [1]. Yet, motivated by recent results in constrained quaternion matrix factorizations [2], we have
proposed a rigorous framework to perform quaternion tensor decompositions. This M2R internship will take advantage of this new framework and develop
efficient algorithms to perform quaternion tensor decompositions. As a first task, the candidate will focus on the quaternion canonical polyadic
decomposition (CPD), a fundamental tool that allows a decomposition of quaternion tensor in rank-one terms. He will develop and compare the
performances of two categories of algorithms performing quaternion CPD: (i) based on a real-constrained tensor reformulation, inspired by [3], and (ii) full-
quaternion domain algorithms. In addition, he/she will apply the proposed methodology to the study of real datasets from several applications, such as color
imaging (3D data) and polarization imaging (4D) data. One key objective will be to benchmark performances of quaternion tensor decompositions against
standard real-domain tensor decompositions.

[1] T. G. Kolda and B. W. Bader, "Tensor decompositions and applications," SIAM review, vol. 51, no. 3,
pp. 455-500, 2009.
[2] J. Flamant, S. Miron, and D. Brie, "Quaternion non-negative matrix factorization: Definition, uniqueness,
and algorithm," IEEE Transactions on Signal Processing, vol. 68, pp. 1870-1883, 2020.
[3] A. L. De Almeida, X. Luciani, A. Stegeman, and P. Comon, "Confac decomposition approach to blind
identification of underdetermined mixtures based on generating function derivatives," IEEE Transactions
on Signal Processing, vol. 60, no. 11, pp. 5698-5713, 2012.
Department(s): 
Biology, Signals and Systems in Cancer and Neuroscience