Azimuthal jittered sampling of bandlimited functions in the two-dimensional Fourier transform and the Hankel transform domains

Abstract

Stark’s sampling theorems in polar coordinates have important significance in the field of medical imaging. However, the direct use of Stark’s results may mislead as in most real-world scenarios the jitter occurs in azimuth due to uncertainty of sampling at the medical imaging system transmitter end. On the basis of the classical Stark’s interpolation formulae, this paper addresses the image reconstruction problem when the sample data are collected on nonuniform polar radius and polar azimuth lattices, where the sample points are normalized zeros of Bessel functions in radius while random jittered ones in azimuth that are clustered around uniform points with a given probability distribution. By using the existing basis functions found in Stark’s formulae, it follows a biased estimate of the original angularly periodic images that are bandlimited both to the highest frequency and in the Fourier transform or the Hankel transform domain. With the newly designed angular generating functions, there are a kind of unbiased estimators whose variance approaches zero as the jitter becomes less and less pronounced.