MRA contextual-recovery extension of smooth functions on manifolds

In a recent paper, the first author introduced an MRA (multi-resolution or multi-level approximation) approach to extend an earlier work of Chan and Shen on image inpainting, from isotropic diffusion to anisotropic diffusion and from bi-harmonic extension to multi-level lagged anisotropic diffusion extension. The objective of the present paper is to extend and generalize this work to nonstationary smooth function extension to meet the goal of inpainting missing image features, while matching the existing image content without apparent visual artifact. Our result is formulated as an MRA contextual-recovery extension for the completion of smooth functions on manifolds by deriving an error formula, from which sharp error estimates can be derived. A novel estimate for the biharmonic operator derived in this paper is a formulation of the error bound in terms the volume, as opposed to the diameter, of the image hole.