System identification under non-negativity constraints - Applications in adaptive filtering and hyperspectral image analysis. (Indentification de système sous la contrainte de non-négativité - Applications dans filtrage adaptatif et analyse dimage hyperspectrale)

In many real-life phenomena due to the inherent physical characteristics of systems under investigation, non-negativity is a desired constraint that can be imposed on the parameters to estimate. The objective of this thesis is to investigate theories and algorithms for system identification under side constraints, in particular the non-negativity constraint and sum-to-one constraint over the vector of parameters to estimate. The first part of the thesis addresses the problem of online system identification subject to non-negativity constraints. The Non-negative Least-Mean-Square algorithm (NNLMS) and its variants are proposed. The stochastic behavior of these algorithms in non-stationnary environments is analytically studied. Finally, coupling these algorithms with optimization allows us to derive an LMS-type algorithm with L1-norm regularization The second part of the thesis focuses on a specific system identifcation problem - nonlinear spectral unmixing, with non-negativity and sum-to-one constraints. We formulate a new kernel-based paradigm that relies on the assumption that the mixing mechanism can be described by a linear mixture of endmember spectra, with additive nonlinear fluctuations defined in a reproducing kernel Hilbert space. A kernel-based algortihm, based on muli-kernel learning, is proposed to determine the fractional abundances of pure materials subject ot constraints. Finally, the spatila correlation between spectral signatures of neighboring pixels is used as prior infromation to improve the performance.