In this work, we propose a non-linear hyperbolic-parabolic coupled Partial Differential Equation (PDE) based model for image despeckling. Here, a separate equation is used to calculate the edge variable, which improves the quality of edge information in the despeckled images. The existence of the weak solution of the present system is achieved via Schauder fixed point theorem. We used a generalized weighted average finite-difference scheme and the Gauss-Seidel iterative technique to solve the coupled system. Numerical studies are reported to show the effectiveness of the proposed approach with respect to standard PDE-based and nonlocal methods available in the literature. Numerical experiments are performed over gray-level images degraded by artificial speckle noise. Additionally, we investigate the noise removal efficiency of the proposed algorithm when applied to real synthetic aperture radar (SAR) and Ultrasound images. Overall, our study confirms that in most cases, the present model performs better than the other PDE-based models and shows competitive performance with the nonlocal technique. To the best of our knowledge, the proposed despeckling approach is the first work that utilizes the advantage of the non-linear coupled hyperbolic-parabolic PDEs for image despeckling.

In this work, a gray level indicator-based nonlinear telegraph diffusion model is presented for image despeckling. Most of the researchers focus only on diffusion equation-based filter for multiplicative noise removal process. The proposed technique uses the benefit of the combined effect of diffusion equation as well as the wave equation. The wave nature of the system preserves the high oscillatory and texture patterns in an image. In this model, the diffusion coefficient depends not only on the image gradient but also on the gray level of the image, which controls the diffusion process better than only gradient-based diffusion approaches. Moreover, we prove the well-posedness of the present system using the Schauder fixed point theorem. Furthermore, we show the superiority of the proposed method over three recently developed methods on a set of gray level test images corrupted by speckle noise and check the noise removal capability of the present technique over some real SAR images corrupted by speckle noise with different noise levels.