Reconstruction and Sampling Constraints for Spiral Data

This paper addresses the problem of reconstructing a circularly band-limited two-dimensional function from its samples on a general spiral contour. The spiral data are represented in terms of evenly spaced samples from a nonlinear two-dimensional transformation of the Cartesian coordinates. This transformation enables us to use unified Fourier reconstruction sampling principles [10] to obtain an accurate reconstruction based on a set of constraints imposed on the spiral parameters. The results are then utilized to develop efficient sampling strategies for the linear class of spirals, the spiral of Archimedes. Sampling efficiency for spiral data, analogous to the sampling efficiencies of hexagonally and rectangularly sampled data, is defined. A sampling scheme on a spiral is introduced that possesses a uniform sampling efficiency comparable to the sampling efficiency of rectangularly sampled data.