To reduce ringing and preserve edges, smoothness must be adaptive in image restoration. We improve on the Laplacian operator and propose an anisotropic regularizing operator. The variables are substitutes for the constants in the Laplacian operator. The anisotropic regularizing operator can control adaptively the direction and amount of smoothing in image restoration. Although the anisotropism idea is closely related to edge-preserving regularization, the anisotropic regularizing operator has different regularization terms and simpler conditions. The iterative equations of anisotropic regularizing operators have unified form. By imposing some constraints, iterative equations can become equations of image restoration with ringing reduction and edge-preserving regularization. The method for linearizing the anisotropic regularization term is provided.