A mathematical model for image saturation with an application to the restoration of solar images via adaptive sparse deconvolution

In this paper we introduce a mathematical model of the image saturation phenomenon occurring in a charged coupled device (CCD), and we propose a novel computational method for restoring saturated images acquired by the atmospheric imaging assembly (AIA) telescope. The mathematical model takes into account both primary saturation, when the photon-induced charge reaches the CCD full well capacity, and the blooming effect, when the excess charge flows into adjacent pixels. The restoration of AIA saturated images is then formulated as an inverse problem with a forward operator encoding the standard diffraction of light rays by a convolution, the primary saturation by an upper limit to the number of photons and the blooming effect by the conservation of the photon-induced charge spilled over adjacent pixels. As a result of this theoretical formulation we propose an adaptive ℓ 1 regularized inversion method improving the desaturation capabilities of the existing SE-DESAT method [Guastavino S et al 2019 Astrophys. J. 882 109]. We prove that this method has the consistency estimation property also in the case that a fixed unknown background is considered. We test the adaptive method both in the case of synthetic and real data, comparing the performance with the one of the SE-DESAT method, showing that the proposed method avoids edge effects and artifacts in reconstructions even when the background solar activity is particularly intense.