Blind inverse problems suffer from scale indeterminacy. The Alternated Minimization using the Optimal ReScaling (AMORS) method addresses this issue by integrating a scalar that compensates for the indeterminacy. Cohen and Leplat correct the indeterminacy by a diagonal matrix in tensor factorization but require positive definite regularizations. We propose to extend these works to matrix compensation without positive definiteness, allowing the use of total variation. We introduce a flexible tuning of the regularizations per column of the estimated matrices. We obtain stable and fast convergence when dealing with dictionary learning in the context of functional connectivity in fMRI.

Whether in medical imaging, astronomy or remote sensing, the data are increasingly complex. In addition to the spatial dimension, the data may contain temporal or spectral information that characterises the different sources present in the image. The compromise between spatial resolution and temporal/spectral resolution is often at the expense of spatial resolution, resulting in a potentially large mixing of sources in the same pixel/voxel. Source separation methods must incorporate spatial information to estimate the contribution and signature of each source in the image. We consider the particular case where the position of the sources is approximately known thanks to external information that may come from another imaging modality or from a priori knowledge. We propose a spatially constrained dictionary learning source separation algorithm that uses e.g. high resolution segmentation map or regions of interest defined by an expert to regularise the source contribution estimation. The originality of the proposed model is the replacement of the sparsity constraint classically expressed in the form of an $\ell _{1}$ penalty on the localisation of sources by an indicator function exploiting the external source localisation information. The model is easily adaptable to different applications by adding or modifying the constraints on the sources properties in the optimisation problem. The performance of this algorithm has been validated on synthetic and quasi-real data, before being applied to real data previously analysed by other methods of the literature in order to compare the results. To illustrate the potential of the approach, different applications have been considered, from scintigraphic data to astronomy or fMRI data.

In this study, a method that aims at detecting small and faint objects in noisy hyperspectral astrophysical images is presented. The particularity of the hyperspectral images that we are interested in is the high dynamics between object intensities. Detection of the smallest and faintest objects is challenging, because their signal-to-noise ratio is low, and if the brightest objects are not well reconstructed, their residuals can be more energetic than faint objects. This paper proposes a marked point process within a nonparametric Bayesian framework for the detection of galaxies in hyperspectral data. The efficiency of the method is demonstrated on synthetic images, and it provides good results for very faint objects in quasi-real astrophysical hyperspectral data.