Abstract

A survey of 2-D nonuniform sampling theorems is given in [1] and [2]. This correspondence is a generalization of Stark’s theorem in polar coordinates [3] based on. the work of Butzer and Hinsen [4]. Stark [3] extended the Bessel interpolation to 2-D signals in polar coordinates (r,θ), where the nonuniform samples are derived from uniform samples in θ direction and zero-crossings of the Bessel function J_n(.) in the r direction, as shown in Figure 1. Butzer et al [4] extended the Lagrange expansion for 2-D signals. They showed that if the set of nonuniform samples lie on parallel straight lines (where the lines are nonuniformly spaced as shown in Figure 2), Lagrange interpolation is possible provided that the parallel lines and the set of nonuniform samples on each line satisfy one of the sufficient conditions for a 1-D set of uniqueness. In this correspondence, we prove that Lagrange interpolation is possible in polar coordinates under a more general setup than that of [3]; i.e., the nonuniform samples are on nonuniformly spaced concentric circles (Figure 3), or alternatively, the nonuniform samples lie on radial lines (Figure 4) and the rotational distances of radial lines are nonuniformly spaced; the nonuniform samples need not be limited to zeros of Bessel functions. The interpolation formulas for the cases depicted in Figures 3 and 4 are given in the next section.

© 1989 Optical Society of America