The performance of iterative algorithms aimed at solving a regularized least squares problem typically depends on the value of some regularization parameter. Tuning the regularization parameter value is a fundamental step necessary to control the strength of the regularization and hence ensure a good performance. We address the problem of finding the optimal regularization parameter in such iterative algorithms. We propose to adaptively adjust the regularization parameter throughout the iterations of the algorithm by minimizing an estimate of the current risk, typically the Weighted Stein unbiased risk estimate (WSURE). We then prove that, for the case of the Tikhonov regularization, the proposed ADAptive Parameter Tuning (ADA-PT) strategy provides a stationary point consistent with the risk minimizer. We illustrate the efficiency of ADA-PT on two image deconvolution problems: one with the Tikhonov regularization and one with the weighted ℓ-1 analysis wavelet regularization.

This paper presents a kernel-based nonlinear mixing model for hyperspectral data, where the nonlinear function belongs to a Hilbert space of vector valued functions. The proposed model extends the existing ones by accounting for band-dependent and neighboring nonlinear contributions. The key idea is to work under the assumption that nonlinear contributions are dominant in some parts of the spectrum, while they are less pronounced in other parts. In addition to this, we motivate the need for taking into account nonlinear contributions originating from the ground covers of neighboring pixels by practical considerations, precisely the adjacency effect. The relevance of the proposed model is that the nonlinear function is associated with a matrix valued kernel that allows to jointly model a wide range of nonlinearities and includes prior information regarding band dependences. Furthermore, the choice of the nonlinear function input allows to incorporate neighboring effects. The optimization problem is strictly convex and the corresponding iterative algorithm is based on the alternating direction method of multipliers. Finally, experiments conducted using synthetic and real data demonstrate the effectiveness of the proposed approach.

This communication proposes an unsupervised neighbor dependent nonlinear unmixing algorithm for hyperspectral data. The proposed mixing scheme models the reflectance vector of a pixel as the sum of a linear combination of the endmem-bers plus a nonlinear function acting on neighboring spectra. The nonlinear function belongs to a reproducing kernel Hilbert space. The observations themselves are considered as the endmember candidates, and the group lasso regulariza-tion is used to enable selecting the purest pixels among the candidates. Experiments on synthetic data demonstrate the effectiveness of the proposed approach.

This paper introduces a graph Laplacian regularization in the hyperspectral unmixing formulation. The proposed regularization relies upon the construction of a graph representation of the hyperspectral image. Each node in the graph represents a pixel's spectrum, and edges connect similar pixels. The proposed graph framework promotes smoothness in the estimated abundance maps and collaborative estimation between homogeneous areas of the image. The resulting convex optimization problem is solved using the Alternating Direction Method of Multipliers (ADMM). A special attention is given to the computational complexity of the algorithm, and Graph-cut methods are proposed in order to reduce the computational burden. Finally, simulations conducted on synthetic and real data illustrate the effectiveness of the graph Laplacian regularization with respect to other classical regularizations for hyperspectral unmixing.

This paper addresses the problem of blind and fully constrained unmixing of hyperspectral images. Unmixing is performed without the use of any dictionary, and assumes that the number of constituent materials in the scene and their spectral signatures are unknown. The estimated abundances satisfy the desired sum-to-one and nonnegativity constraints. Two models with increasing complexity are developed to achieve this challenging task, depending on how noise interacts with hyperspectral data. The first one leads to a convex optimization problem and is solved with the alternating direction method of multipliers. The second one accounts for signal-dependent noise and is addressed with a reweighted least squares algorithm. Experiments on synthetic and real data demonstrate the effectiveness of our approach.