Local Adaptiveness of Mixed Higher Order Partial Differential Equations and ItsApplication in Image Denoising

Background: Image denoising methods based on partial differential equations have attracted much attention due to their "infinite" local adaptation capabilities, high flexibility, and strong mathematical theoretical support. Methods: This paper proposes a mixed higher order partial differential equation denoising model for the step effect caused by the second-order denoising model and the edge blur caused by the fourth-order denoising model. The model combines the second-order and fourth-order terms based on the relationship between the variational energy minimization and the partial differential equations. The fourth-order term is used to remove noise in the uniform area of the image to avoid the step effect, and the second-order term is used at the edge to avoid boundary blur. Results: Theoretical analysis and numerical experiment results show that the proposed model has weak solutions and can effectively avoid the step effect and maintain the edge. Conclusion: The image denoising results of the model are better than those of other improved denoising models in subjective effect, and objective evaluation indicators, such as SNR, PSNR, and MSSIM.