Filling-in by joint interpolation of vector fields and gray levels

A variational approach for filling-in regions of missing data in digital images is introduced. The approach is based on joint interpolation of the image gray levels and gradient/isophotes directions, smoothly extending in an automatic fashion the isophote lines into the holes of missing data. This interpolation is computed by solving the variational problem via its gradient descent flow, which leads to a set of coupled second order partial differential equations, one for the gray-levels and one for the gradient orientations. The process underlying this approach can be considered as an interpretation of the Gestaltist's principle of good continuation. No limitations are imposed on the topology of the holes, and all regions of missing data can be simultaneously processed, even if they are surrounded by completely different structures. Applications of this technique include the restoration of old photographs and removal of superimposed text like dates, subtitles, or publicity. Examples of these applications are given. We conclude the paper with a number of theoretical results on the proposed variational approach and its corresponding gradient descent flow.