Connecting image inpainting with denoising in the homogeneous diffusion setting

• Published in: Advances in continuous and discrete models Vol. 2025; no. 1; p. 74
• Main Authors: Gaa, Daniel, Chizhov, Vassillen, Peter, Pascal, Weickert, Joachim, Adam, Robin Dirk
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 28.03.2025; Springer Nature B.V
• Subjects: Analysis; Difference and Functional Equations; Diffusion; Functional Analysis; Mathematics; Mathematics and Statistics; Noise reduction; Ordinary Differential Equations; Partial Differential Equations; 68U10; Inpainting; Partial differential equations; 94A08; Diffusion; Sampling; 65D18; Denoising
• ISSN: 2731-4235; 1687-1839; 2731-4235; 1687-1847
• DOI: 10.1186/s13662-025-03935-7

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 2D case. We present experiments that confirm our theory and suggest that it can also be generalized to diffusions with nonhomogeneous data or nonhomogeneous 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.