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.