In this paper, we focus on X-ray images of paintings with concealed sub-surface designs (e.g., deriving from reuse of the painting support or revision of a composition by the artist), which include contributions from both the surface painting and the concealed features. In particular, we propose a self-supervised deep learning-based image separation approach that can be applied to the X-ray images from such paintings to separate them into two hypothetical X-ray images. One of these reconstructed images is related to the X-ray image of the concealed painting, while the second one contains only information related to the X-ray of the visible painting. The proposed separation network consists of two components: the analysis and the synthesis sub-networks. The analysis sub-network is based on learned coupled iterative shrinkage thresholding algorithms (LCISTA) designed using algorithm unrolling techniques, and the synthesis sub-network consists of several linear mappings. The learning algorithm operates in a totally self-supervised fashion without requiring a sample set that contains both the mixed X-ray images and the separated ones. The proposed method is demonstrated on a real painting with concealed content, Do\~na Isabel de Porcel by Francisco de Goya, to show its effectiveness.
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