A unified approach to optimal image interpolation problems based on linear partial differential equation models

A unified approach to optimal image interpolation problems, based on separable linear partial differential equation models for the image signal processes, is presented. This approach provides a constructive procedure for finding explicit and closed-form optimal solutions to the image interpolation problems under investigation where the type of interpolation can be either spatial or time-spatial. The main idea in reconstructing the unknown image from a finite set of sampled data so that a mean-square error is minimized is first to express the solution in terms of the reproducing kernel of a related Hilbert space, and then to construct this reproducing kernel using the fundamental solution of an induced linear partial differential equation, or the Green's function of the corresponding self-adjoint operator. It is proved in this paper that in most cases, closed-form fundamental solutions (or Green's functions) for the corresponding linear partial differential operators can be found in the general image reconstruction problem described by a first- or second-order linear partial differential operator. All of these operators have been classified in this paper, and an efficient method for obtaining their corresponding closed-form fundamental solutions (or Green's functions) is also presented. Finally, a computer simulation is described to demonstrate the procedure of reconstructing an image using the proposed method.