A novel Perona-Malik model driven by fractional regularized diffusion mechanism with caputo derivative applied to magnetic resonance image processing

Abstract

This research paper addresses a new class of nonlinear fractional parabolic equations that emerge in the realm of image processing. The proposed models are characterized by Caputo fractional time derivatives combined with nonlinear fractional gradient diffusion. Our proposed model is designed to address the challenge of balancing effective noise removal with fine feature preservation in medical images. Firstly, we investigate the theoretical solvability and numerical application of these models. We demonstrate two significant results related to the existence and uniqueness of weak solutions. The first result applies to the scenario where the diffusion coefficient is determined by a symmetric matrix and is independent of the solution. In this case, we establish the existence and uniqueness of the solution using the Galerkin method. The second result concerns the situation where the diffusion coefficient incorporates a nonlinear fractional regularized Perona-Malik mechanism. To prove the existence of a weak solution in this context, we primarily use Schauder’s fixed point theorem, supplemented with new technical estimates. Moreover, we validate our theoretical findings through numerical experiments conducted on MRI images corrupted with Gaussian noise. The performance of the proposed model is compared against classical and state-of-the-art denoising methods. Visual comparisons highlight the model’s ability to preserve fine anatomical details. Additionally, quantitative evaluations using PSNR and SSIM demonstrate superior performance over conventional models. The results confirm that the model achieves enhanced denoising while preserving critical structural details, making it a robust and efficient tool for challenging medical imaging scenarios.