Image resolution measure with applications to restoration and zoom

Traditionally, discrete images are assumed to be sampled on a square grid and from a special kind of band-limited continuous image, namely one whose Fourier spectrum is contained within the rectangular "reciprocal cell" associated with the sampling grid. With such a simplistic model, resolution is just given by the distance between sample points. Whereas this model matches to some extent the characteristics of traditional acquisition systems, it doesn't explain aliasing problems, and it is no longer valid for certain modern ones, where the sensors may show a heavily anisotropic transfer function, and may be located on a non-square (in most cases hexagonal) grid. In this work we first summarize the generalizations of Fourier theory and of Shannon's sampling theorem, that are needed for such acquisition devices. Then we explore its consequences: (i) A new way of measuring the effective resolution of an image acquisition system; (ii) A more accurate way of restoring the original image which is represented by the samples. We show on a series of synthetic and real images, how the proposed methods make a better use of the information present in the samples, since they may drastically reduce the amount of aliasing with respect to traditional methods. Finally we show how in combination with Total Variation minimization, the proposed methods can be used to extrapolate the Fourier spectrum in a reasonable manner, visually increasing image resolution.