Abstract: This paper describes our methods for repairing and restoring images of hidden paintings (paintings that have been painted over and are now covered by a new surface painting) that have been obtained via noninvasive X-ray fluorescence imaging of their canvases. This recently developed imaging technique measures the concentrations of various chemical elements at each two-dimensional spatial location across the canvas. These concentrations in turn result from pigments present both in the surface painting and in the hidden painting beneath. These X-ray fluorescence images provide the best available data from which to noninvasively study a hidden painting. However, they are typically marred by artifacts of the imaging process, features of the surface painting, and areas of information loss. Repairing and restoring these images thus consists of three stages: (1) repairing acquisition artifacts in the dataset, (2) removal of features in the images that result from the surface painting rather than the hidden painting, and (3) identification and repair of areas of information loss. We describe methods we have developed to address each of these stages: a total-variation minimization approach to artifact correction, a novel method for underdetermined blind source separation with multimodal side information to address surface feature removal, and two application-specific new methods for automatically identifying particularly thick or X-ray absorbent surface features in the painting. Finally, we demonstrate the results of our methods on a hidden painting by the artist Vincent van Gogh. (C) 2012 Elsevier B.V. All rights reserved.