Pseudodifferential Inpainting: The Missing Link Between PDE- and RBF-Based Interpolation

Abstract

Inpainting with partial differential equations (PDEs) has been used successfully to reconstruct missing parts of an image, even for sparse data. On the other hand, sparse data interpolation is a rich field of its own with different methods such as scattered data interpolation with radial basis functions (RBFs). The goal of this paper is to establish connections between inpainting with linear shift- and rotation-invariant differential operators and interpolation with radial basis functions. The bridge between these two worlds is built by generalising inpainting methods to handle pseudodifferential operators and by considering their Green’s functions. In this way, we find novel relations of various multiquadrics to pseudodifferential operators. Moreover, we show that many popular radial basis functions are related to processes from the diffusion and scale-space literature. We present a single numerical algorithm for all methods. It combines conjugate gradients with pseudodifferential operator evaluations in the Fourier domain. Our experiments show that the linear PDE- and the RBF-based communities have developed concepts of comparable quality.