[Paper Review] An efficient feature-preserving PDE algorithm for image denoising based on a spatial-fractional anisotropic diffusion equation

Maoyuan Xu, Xiaoping Xie|arXiv (Cornell University)|Jan 5, 2021

Image and Signal Denoising Methods41 references4 citations

TL;DR

This paper proposes a novel feature-preserving image denoising algorithm based on a two-sided spatial-fractional anisotropic diffusion equation using Grümwald-Letnikov fractional derivatives. By leveraging two-sided derivatives to better capture local self-similarity and applying the Short Memory Principle for computational efficiency, the method achieves superior balance between noise removal and edge/textural feature preservation, outperforming existing integer-order and one-sided fractional PDE methods in PSNR and SSIM metrics.

ABSTRACT

How to effectively remove the noise while preserving the image structure features is a challenging issue in the field of image denoising. In recent years, fractional PDE based methods have attracted more and more research efforts due to the ability to balance the noise removal and the preservation of image edges and textures. Among the existing fractional PDE algorithms, there are only a few using spatial fractional order derivatives, and all the fractional derivatives involved are one-sided derivatives. In this paper, an efficient feature-preserving fractional PDE algorithm is proposed for image denoising based on a nonlinear spatial-fractional anisotropic diffusion equation. Two-sided Grumwald-Letnikov fractional derivatives were used in the PDE model which are suitable to depict the local self-similarity of images. The Short Memory Principle is employed to simplify the approximation scheme. Experimental results show that the proposed method is of a satisfactory performance, i.e. it keeps a remarkable balance between noise removal and feature preserving, and has an extremely high structural retention property.

Motivation & Objective

• To address the challenge of balancing noise removal and structural feature preservation in image denoising.
• To overcome limitations of one-sided fractional derivatives in capturing full neighborhood information.
• To improve upon the staircase effect and speckle artifacts common in integer-order PDE-based denoising.
• To develop an efficient numerical scheme using the Short Memory Principle for practical implementation.
• To demonstrate superior performance in preserving image structure and texture while removing noise.

Proposed method

• Formulates a nonlinear spatial-fractional anisotropic diffusion equation using two-sided Grümwald-Letnikov fractional derivatives of orders α ∈ (1.25, 1.75) and β ∈ (1, 2).
• Employs a diffusion coefficient g(|∇βu|) that adapts to local image gradients, enabling edge-preserving smoothing.
• Applies the Short Memory Principle to reduce computational cost in approximating the fractional derivatives.
• Uses a semi-implicit finite difference scheme for time discretization to ensure stability and efficiency.
• Implements the PDE model in a level set-like framework to handle complex image structures.
• Solves the resulting system iteratively to evolve the noisy image toward a denoised state.

Experimental results

Research questions

• RQ1 Can two-sided fractional derivatives improve image denoising performance compared to one-sided derivatives?
• RQ2 Does the proposed fractional PDE model better preserve image edges and textures than classical integer-order PDE models?
• RQ3 To what extent does the Short Memory Principle maintain accuracy while reducing computational cost in fractional PDE-based denoising?
• RQ4 How does the proposed method compare in PSNR and SSIM to state-of-the-art denoising techniques under varying noise levels?
• RQ5 Can the model effectively mitigate both the staircase effect and speckle artifacts in denoised images?

Key findings

• The proposed method achieves the highest PSNR and SSIM values across all test images, including Lenna, Barbara, Baboon, and Pepper, under additive Gaussian white noise with δ = 25.
• For the Baboon image with δ = 25, the method achieves 23.84 dB PSNR and 0.6750 SSIM, significantly outperforming all competing methods.
• On the Lenna image with δ = 25, the method reaches 29.98 dB PSNR and 0.8105 SSIM, the best among all compared models.
• The method demonstrates superior structural retention, as evidenced by consistently higher SSIM values, particularly on textured and edge-rich images like Barbara and Baboon.
• Compared to TV-Stokes-based methods in [21,23], the proposed method yields higher PSNR and SSIM across all test images, confirming its enhanced performance.
• The use of two-sided fractional derivatives enables better modeling of local self-similarity, leading to improved preservation of fine textures and edges.