Connecting Image Inpainting with Denoising in the Homogeneous Diffusion Setting

TL;DR

This paper establishes theoretical links between image inpainting and denoising within homogeneous diffusion, demonstrating their equivalence in 1D and extending insights to 2D, highlighting the importance of data adaptivity.

Contribution

It derives explicit relations between inpainting and diffusion filtering in 1D and empirically extends these results to 2D, emphasizing data adaptivity's potential.

Findings

Equivalence between DbI and diffusion filtering in 1D

Empirical extension of theory to 2D

Data adaptivity can match operator adaptivity

Abstract

While local methods for image denoising and inpainting may use similar concepts, their connections have hardly been investigated so far. The goal of this work is to establish links between the two by focusing on the most foundational scenario on both sides: the homogeneous diffusion setting. To this end, we study a denoising by inpainting (DbI) framework: It averages multiple inpainting results from different noisy subsets. We derive equivalence results between DbI on shifted regular grids and homogeneous diffusion filtering in 1D via an explicit relation between the density and the diffusion time. We also provide an empirical extension to the 2-D case. We present experiments that confirm our theory and suggest that it can also be generalized to diffusions with non-homogeneous data or non-homogeneous diffusivities. More generally, our work demonstrates that the hardly explored idea of data adaptivity deserves more attention: It can be as powerful as some popular models with operator adaptivity.