Optimal image reconstruction based on general PDE models

The problem of model-based optimal reconstruction of an image from its samples is studied. It is assumed that the image signal is described by a linear PDE (partial differential equation) model from which a sequence of output frames is available. These frames need not have rectangular boundaries or be uniformly sampled. Under the criterion that the image to be reconstructed is the one that is created by an unknown input signal with minimum energy, the authors obtain a unique optimal solution by interpolating the image samples with a recently formulated generalized spline called the PDL_g spline. It is possible to show that such a reconstruction corresponds to a minimum-mean-squares estimate of the image, given its samples. The authors also derive an algorithm for finding the optimal solution in an explicit and closed form. They include some computer simulation results to show the quality of the images obtained with the method.