Summary: To address the problems of edge blurring and detail preservation in filtering, a novel high-dimensional filtering using a gradient minimized Mumford-Shah model is proposed, which uses the minimization of \(L_0\) and \(L_1\) regularization terms to achieve edge-preserving and texture-smoothing. For 2D images, the \(L_0\) norm describes the number of non-smooth pixels in the image, which is minimized to obtain the local flat region, that is, to make the filtered output as smooth as possible in the Mumford-Shah model. The \(L_1\) norm (total variation term) describes the length of all level-sets in the image, which is minimized to control the sharpness of the edges, that is, the length constraint in the Mumford-Shah model. Due to the robustness of the Mumford-Shah model to edge-preserving and texture-smoothing, a sound component separation can be obtained in high-dimensional signal decomposition. In the experiments, it is demonstrated that the proposed high-dimensional filter can achieve both edge-preserving and texture-smoothing. The characteristic is helpful for obtaining a perfect structure-texture separation and optimizing the result in some specific visual applications.