Existence of solution for a coupled diffusion PDE system for various noise reduction

Abstract

In this work, we introduce a new model based on a high-order, non-linear PDE system for image denoising, which is controlled by a function $h$ that detects the type of noise. The proposed model generalizes and improves upon the coupled PDE system of Jain et al. (2019), allowing for the handling of various noise types, such as Gaussian, Speckle, and Salt & Pepper’s noise. Our model is based on a diffusion tensor that corrects the anisotropic, coherent diffusion property of the Weickert operator near tiny edges with a high diffusion order. This configuration exhibits flexibility in the diffusion speed, allowing for efficient smoothing near flat areas without changing directions along the edges or across them. We perform a rigorous analysis of the existence and uniqueness of the weak solution of the proposed coupled PDE system in a suitable functional framework, using the Schauder fixed-point theorem. Finally, we present representative numerical results to demonstrate the effectiveness of our model against various noise types, by comparing the obtained results with those of some competitive models.