Mathematical methods for spectral image reconstruction

We present a method for recovery of damaged parts of old paintings (frescoes), caused by degradation of the pigments contained in the paint layer. The original visible colour information in the damaged parts can be faithfully recovered from measurements of absorption spectra in the invisible region (IR and UV) and from the full spectral data of the well preserved parts of the image. We use the mathematical framework of sparse matrix recovery: The singular value thresholding (SVT) algorithm by Cai, Candès and Shen, and the iteratively re-weighted least squares minimization (IRLS) by Daubechies, DeVore, Fornasier and Güntürk. In addition to these two algorithms, which are iterative in nature, we propose a third, non-iterative, method (block completion, BC), which can be applied in the situation when the missing elements of a low-rank matrix constitute a block (submatrix); this is always true in our application. We shortly introduce the SVT and IRWLSM algorithms, perform a simple analysis of the BC method and, finally, demonstrate the performance of these three methods on a sample fresco.