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.

The problem of finding efficient methods for the detection of unknown sparse signals buried in noise is addressed. We present two detection tests adapted to sparse signals, based on the maximum a posteriori (MAP) estimate of the sparse vector of parameters. The first is the posterior density ratio test, which computes the ratio of the a posteriori distribution under each hypothesis of the data model. The second is a likelihood ratio test in which the MAP replaces the maximum likelihood (ML) estimate. The behaviors and the relative differences between these tests are investigated through a detailed study of their structural characteristics. The proposed approaches are compared to the generalized likelihood ratio test (GLR), showing successful results in the case of a simple model first and then for a model in which sparsity is promoted through the use of a highly redundant dictionary.The tests are then applied to massive astrophysical hyperspectral data in the context of the European Southern Observatory's (ESO) Multi Unit Spectroscopic Explorer (MUSE) instrument. Experiments show that the introduced detection methods are more efficient than the GLR, confirming the pertinence of our approaches towards detection goals.

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In an estimation framework, the suspected sparsity of an unknown vector of deterministic parameters is classically accounted for through thresholding functions, some of which being related to Maximum A Posteriori estimates with specific priors. In a detection framework, statistical tests applied to sparse vectors are classically designed so as to limit the focus of the test to a few active components, leading to test statistics of thresholded data. We propose a study of the connections between these two problems. The detection tests we consider are the Generalized Likelihood Ratio Test (GLRT), the Bayes Factor, the Posterior Density Ratio and a LRT using a Maximum A Posteriori estimate with Generalized Gaussian priors. In the case of scalar parameter first, we derive sufficient conditions that any detection test must obey to be equivalent to a GLRT, and prove that the tests above verify them. In the vector case, the LRT with a MAP estimate and the PDR outperform the GLRT and the BF thanks to the thresholding effect of the MAP estimation. The connections between thresholding functions used in estimation and the resulting "thresholded test statistics" are precisely investigated.