A method for optimal linear super-resolution image restoration

In this paper, we propose a super-resolution (pixel grid refinement) method for digital images. It is based on the linear filtering of a zero-padded discrete signal. We introduce a continuous-discrete observation model to create a reconstruction system. The proposed observation model is typical of real-world imaging systems - an initially continuous signal first undergoes linear (dynamic) distortions and then is subjected to sampling and the effect of additive noise. The proposed method is optimal in the sense of mean square recovery error minimization. In the theoretical part of the article, a general scheme of the linear super-resolution of the signal is presented and expressions for the impulse and frequency responses of the optimal reconstruction system are derived. An expression for the error of such restoration is also derived. For the sake of brevity, the entire description is presented for one-dimensional signals, but the obtained results can easily be generalized for the case of two-dimensional images. The experimental section of the paper is devoted to the analysis of the super-resolution reconstruction error depending on the parameters of the observation model. The significant superiority of the proposed method in terms of the reconstruction error is demonstrated in comparison with linear interpolation, which is usually used to refine the grid of image pixels.