  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.
