Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2409.15952

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

Maria Vasilyeva·Aleksei Krasnikov·Kelum Gajamannage·Mehrube Mehrubeoglu
View original source ↗Read source document ↗Export metadata ↓

Abstract

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. 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. 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 multiscale basis functions to construct a coarse resolution representation by a Galerkin coupling. We present numerical results for several classic and high-resolution image datasets to demonstrate the effectiveness of the proposed multiscale approach with local denoising and local multiscale representation.

Explore related subjects

math.NAcs.NA

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Maria Vasilyeva, Aleksei Krasnikov, Kelum Gajamannage, Mehrube Mehrubeoglu. 2025-05-13. Multiscale method for image denoising using nonlinear diffusion process: local denoising and spectral multiscale basis functions. https://arxiv.org/abs/2409.15952

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Fully spectral scheme for the linear BGK equation on the whole space

In this article, we design a fully spectral method in both space and velocity for a linear inhomogeneous kinetic equation with mass, momentum and energy conservation. We focus on the linear BGK equation with a confinement potential $Φ$, even if the method could be applied to different collision operators. It is based upon the projection on Hermite polynomials in velocity and orthonormal polynomials with respect to the weight $e^{-$Φ$}$ in space. The potential $Φ$ is assumed to be a polynomial. It is, to the author's knowledge, the first scheme which preserves hypocoercive behavior in addition to the conservation laws. These different properties are illustrated numerically on both quadratic and double well potential.

math.NA↗
arXiv2026

Inverse inequalities for kernel-based approximation on bounded domains and Riemannian manifolds

This paper establishes inverse inequalities for kernel-based approximation spaces defined on bounded Lipschitz domains in $\mathbb{R}^d$ and compact Riemannian manifolds. While inverse inequalities are well-studied for polynomial spaces, their extension to kernel-based trial spaces poses significant challenges. For bounded Lipschitz domains, we extend prior Bernstein inequalities, which only apply to a limited range of Sobolev orders, to the full range of lower and upper orders, and derive Nikolskii inequalities that bound $L_\infty$ norms by $L_2$ norms. For compact Riemannian manifolds, we focus on restricted kernels, which are defined as the restriction of positive definite kernels from the ambient Euclidean space to the manifold, and prove their counterparts.

math.NA↗
arXiv2026

Error Estimates for Hyperbolic Scaling Limits of Linear Kinetic Models on Networks

This paper studies linear discrete kinetic models on networks and their asymptotic behavior in the small Knudsen number limit. For coupling conditions at an n-edge junction under a symmetric formulation, we introduce a change of variables that reformulates the system into n independent initial-boundary value problems. The asymptotic expansions are then constructed and rigorously justified by deriving an error estimate based on the energy method.

math.NA↗