We propose a numerical methodology for detecting periodicities in unknown colored noise and for evaluating the ‘significance levels’ (p-values) of the test statistics. The procedure assumes and leverages the existence of a set of time series obtained under the null hypothesis (a null training sample, NTS) and possibly complementary side information. The test statistic is computed from a standardized periodogram, which is a pointwise division of the periodogram of the series under test to an averaged periodogram obtained from the NTS. The procedure provides accurate p-values estimation through a dedicated Monte Carlo procedure. While the methodology is general, our application is here exoplanet detection. The proposed methods are benchmarked on astrophysical data.

Our aim is to devise a detection method for exoplanet signatures (multiple sinusoids) that is both powerful and robust to partially unknown statistics under the null hypothesis. In the considered application, the noise is mostly created by the stellar atmosphere, with statistics depending on the complicated interplay of several parameters. Recent progresses in hydrodynamic (HD) simulations show however that realistic stellar noise realizations can be numerically produced off-line by astrophysicists. We propose a detection method that is calibrated by HD simulations and analyze its performances. A comparison of the theoretical results with simulations on synthetic and real data shows that the proposed method is powerful and robust.

We consider the problem where a large known library of L alternatives is available and we wish to maximize the detection power in a worst case scenario. The considered minimax detection approach relies on a GLR test allied to a sparsity constraint. This approach conditions the optimization of the target subspaces, in number r ≪ L. While the exact solution of the minimax optimization problem can be found for r = 1, the problem for r > 1 is more intricate and we propose two algorithms aimed at finding an approximate solution. The proposed algorithms are illustrated on a face database and on hyperspectral data and are shown to improve on the r = 1 case.

We review several detection strategies that account for the possible sparsity of target sources in data cubes. The considered sparsity can exist in the data acquisition space, or in some transform domain. Theoretical aspects of the detection tests are first described. Emphasis is then put on practical issues that may arise in hyperspectral data, such as spatio-spectral dependencies or very low Signal-to-Noise Ratios. Applications are finally described in the framework of the hyperspectral data of MUSE (Multi-Unit Spectroscopic Explorer) instrument. MUSE is a powerful integral field spectrograph whose observational abilities should provide unprecedented insights about the formation and evolution of galaxies.

In the “Higher Criticism” framework for multiple hypothesis testing, the binomial counts of the empirical CDF of the P-values are standardized through Z-scores. In the asymptotic regime, the distribution of these standardized counts tends to be a standard Gaussian. In the non-asymptotic regime, these distributions depart from normality and generally differ from each other. As a result, comparable values of Z-scores obtained for different P-values reflect in fact different evidence against the joint null hypothesis. We show that such evidence can be computed in closed-form for finite sample size, resulting in an alternate standardization which leads to identically distributed and thus more comparable statistics. Numerical results illustrate some merits of the alternate standardization.

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.

We propose a method aimed at detecting weak, sparse signals in highly noisy three-dimensional (3D) data. 3D data sets usually combine two spatial directions x and y (e.g. image or video frame dimensions) with an additional direction λ (e.g. temporal, spectral or energy dimension). Such data most often suffer from information leakage caused by the acquisition system's point spread functions, which may be different and variable in the three dimensions. The proposed test is based on dedicated 3D dictionaries, and exploits both the sparsity of the data along the λ direction and the information spread in the three dimensions. Numerical results are shown in the context of astrophysical hyperspectral data, for which the proposed 3D model substantially improves over 1D detection approaches.

We consider the problem of detecting a target signature which is known (up to an amplitude factor) to belong to a (possibly very) large library of signatures. Thus we know how each signature to be detected looks like, but we do not know which one is activated under H1. We propose a minimax approach for this problem aimed at maximizing the worst detection performance. Optimization issues and connections with One-Class classifiers are discussed and illustrated geometrically. Numerical results comparing the proposed approach to the classical sparse-coding dictionary learning technique K-SVD are provided on astrophysical hyperspectral data.

This paper deals with the deconvolution of faint diffuse astronomical sources buried in the PSF sidelobes of surrounding bright compact sources, and in the noise. We propose a sparsity promoting restoration model which is based on highly redundant, shift invariant dictionaries, and which is hybrid in its sparsity priors. On one hand, the image to be restored is modelled using sparse synthesis priors as a sum of few atoms (objects) which, as opposed to classical synthesis-based priors, are unknown. On the other hand, these objects are iteratively estimated and deconvolved through analysis-based priors. The faint diffuse source is deconvolved once the data has been cleaned from all brighter sources' contributions. Comparative numerical results show that the method is efficient and fast.

This article deals with a regularized version of the split gradient method (SGM), leading to multiplicative algorithms. The proposed algorithm is available for the optimization of any divergence depending on two data fields under positivity constraint. The SGM-based algorithm is derived to solve the nonnegative matrix factorization (NMF) problem. An example with a Frobenius norm on both the data consistency and the penalty term is developed and applied to hyperspectral data unmixing.