The paper demonstrates potential advantages of using implicit finite differencing and filtering schemes for fast, accurate, and reliable differentiating and filltering of multidimensional signals defined on regular grids. In particular, applications to image enhancement and Gaussian image debluring are considered. The theoretical contribution of the paper is threefold. The first adapts the Fourier-Pade-Galerkin approximations approach for constructing compact implicit finite difference schemes with desirable spectral resolution properties. The second establishes a link between implicit and explicit finite differences used for gradient estimation. Finally, the third one consists of introducing new implicit finite difference schemes with good spectral resolution properties.