Band-limited 2-D interpolation using NUFFT

Abstract

There exists a class of algorithms for fast approximate evaluation of trigonometric sums arising from extension of discrete Fourier transforms (DFT and IDFT) to irregular sample locations. These are referred to in the literature as “Non-uniform Fast Fourier Transforms” or “NUFFT”. These allow Fourier (trigonometric) interpolation to be done, for a given set of complex DFT coefficients, to a high degree of uniform approximation, in order O(N2) flops. This facilitates the theoretically pleasing prospect of using a Discrete Fourier Series to interpolate image data from a regularly sampled grid to intermediate irregular sample locations. Brute-force evaluation at N2 spatial locations, of a discrete 2-D Fourier series consisting of N2 modes scales as O(N4) and is prohibitive for image arrays on the order N ≈ 103 and above.. There are three variants of the trigonometric sums to be evaluated by such algorithms. The “Type-2” variant is that in which we sum a set of regular locations in the discrete Fourier domain, to evaluate the inverse transform at an irregular set of locations in the image domain. In other words, it is a method that allows us to use Fourier interpolation, with a relatively large number of DFT modes and a relatively large number of interpolation sites.