Multiscale method for image denoising using nonlinear diffusion process: local denoising and spectral multiscale basis functions

We consider image denoising using a nonlinear diffusion process, where we solve unsteady partial differential equations with nonlinear coefficients. The noised image is given as an initial condition, and nonlinear coefficients are used to preserve the main image features. In this paper, we present a multiscale method for the resulting nonlinear parabolic equation in order to construct an efficient solver. To both filter out noise and preserve essential image features during the denoising process, we utilize a time-dependent nonlinear diffusion model known as Perona-Malik. Here, the noised image is fed as an initial condition and the denoised image is stimulated with given parameters. We numerically implement this model by constructing a discrete system for a given image resolution using a finite volume method and employing an implicit time approximation scheme to avoid time-step restriction. However, the resulting discrete system size is proportional to the number of pixels which leads to computationally expensive numerical algorithms for high-resolution images. In order to reduce the size of the system and construct efficient computational algorithms, we construct a coarse-resolution representation of the system using the Generalized Multiscale Finite Element Method (GMsFEM). We incorporate local noise reduction in the coarsening process to construct an efficient algorithm with fewer denoising iterations. We propose a computational approach with two main ingredients: (1) performing local image denoising in each local domain of basis support; and (2) constructing spectral multiscale basis functions to construct a coarse resolution representation by a Galerkin coupling. We present numerical results for several test images to demonstrate the effectiveness of the proposed multiscale approach with local denoising and local spectral representation.