Inpainting and Visual Interpolation

Inpainting is the artistic synonym for image interpolation, initially circulated among museum restoration artists who manually retouch degraded ancient paintings. Image and visual interpolation have broad application in vision cognitive science, digital technology, and telecommunication. Shannon's Interpolation Theorem (also known as the Sampling Theorem) fails for generic image interpolation because (a) images are intrinsically band-unlimited, and even non-smooth, and (b) the available samples are often degraded (by noise or blurring), or are irregularly distributed and have complex topology on 2-D image planes. Other well-known tools such as polynomial interpolants, splines, and wavelets share the same problems. In this talk, we present our recent efforts in developing generic inpainting models that combine in spirit the Bayesian framework (or Helmholtz principle in vision) and the geometry of edges and level sets. Our models are either variational or based on nonlinear geometric PDEs, among which two have been based on the celebrated Mumford and Shah's free boundary image model and Rudin and Osher's BV (bounded variation) image model. We discuss the modeling processes, their vision foundation, analysis of the models, algorithms based on numerical PDEs, and several important applications.