Due to the strong ability of restoring textures and details in images, nonlocal equations have attracted an extensive interest for image denoising. However, the lack of regularity causes residual noise in the restored images. In this paper, we propose and study an evolution equation consisting of the weighted local and the weighted nonlocal $p$-Laplacian equations. The existence and uniqueness of solutions for the proposed equation are proven under the assumption that the weights vanish in sets of positive measure. We also show that solutions of the proposed equation converge to the solution of the usual $p$-Laplacian equation if the kernel is rescaled appropriately. Comparisons with local and nonlocal diffusion equations for removing Gaussian noise in images are presented.

由于具有很强的恢复图像纹理和细节的能力，非局部方程引起了图像去噪的广泛兴趣。然而，缺乏规律性会导致恢复图像中存在残留噪声。在本文中，我们提出并研究了由加权局部和加权非局部构成的演化方程$磷$-拉普拉斯方程。在权重在正测度集中消失的假设下，证明了所提出方程解的存在性和唯一性。我们还表明，所提出方程的解收敛到通常的解$磷$-Laplacian 方程，如果内核被适当地重新缩放。比较了用于去除图像中高斯噪声的局部和非局部扩散方程。