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Fourth-order partial differential equations (PDEs) for noise removal are able to provide a good trade-off between noise removal and edge preservation, and can avoid blocky effects often caused by second-order PDE. In this study, the authors propose a fourth-order telegraph-diffusion equation (TDE) for noise removal. In the authors method, a domain-based fourth-order PDE is proposed, which takes advantage of statistic characteristics of isolated speckles in the Laplace domain to segment the image domain into two domains: speckle domain and non-speckle domain. Then, depending on the domain type, they adopt different conductance coefficients in the proposed fourth-order PDE. The proposed method inherits the advantage of fourth-order PDE which is able to avoid the blocky effects widely seen in images processed by second-order PDE. Furthermore, a TDE processing scheme is derived from previously proposed domain-based fourth-order PDE by adding second time derivative, which results in better edge preservation, whereas yielding better improvement in signal-to-noise ratio and low noise sensitivity. Experimental results show the effectiveness of the proposed method.