Nonlinear Projection Recovery in Digital Inpainting for Color Image Restoration

This paper concerns with nonuniform sampling and interpolation methods combined with variational models for the solution of a generalized color image inpainting problem and the restoration of digital signals. In particular, we discuss the problem of reconstructing a digital signal/image from very few, sparse, and complete information and from a substantially incomplete information, which will be assumed as the result of a nonlinear distortion. Differently from well known inpainting applications for the recovery of gray images, the proposed techniques apply to color images embedding blanks where only gray level information is given. As a typical and inspiring example, we illustrate the concrete problem of the color restoration of a destroyed art fresco from its few known fragments and some gray picture taken prior to the damage. Numerical implementations are included together with several examples and numerical results to illustrate the proposed method. The numerical experience suggests furthermore that a particular system of coupled Hamilton-Jacobi equations is well-posed. Massimo Fornasier received his Ph.D. degree in Computational Mathematics on February 2003 at the University of Padova, Italy. Within the European network RTN HASSIP (Harmonic Analysis and Statistics for Signal and Image Processing) HPRN-CT-2002-00285, he cooperated as PostDoc with NuHAG (the Numerical Harmonic Analysis Group), Faculty of Mathematics of the University of Vienna, Austria and the AG Numerical/Wavelet-Analysis Group of the Department of Mathematics and Computer Science of the Philipps-University in Marburg, Germany (2003). Since June 2003 he is research assistant at the Department of Mathematical Methods and Models for the Applied Science at the University of Rome “La Sapienza”. Since May 2004 he is Individual Marie Curie Fellow (project FTFDORF-FP6-501018) at NuHAG. His research interests include applied harmonic analysis with particular emphasis on time-frequency analysis and decompositions for applications in signal and image processing. Since 1998, he developed with Domenico Toniolo the Mantegna Project (http://www.pd.infn.it/~labmante/) at the University of Padova and the local laboratory for image processing and applications in art restoration. Recently he has focused his attention on adaptive and dynamical schemes for the numerical solution of (pseudo) differential equations and inverse problems in digital signal processing.