Gerchberg's extrapolation algorithm in two dimensions

Abstract

Gerchberg's 1-D iterative extrapolation algorithm for bandlimited signals is generalized to two dimensions in two distinct ways. One generalization requires knowledge of the entire spectral pupil of the bandlimited image. The second requires only knowledge of two 1-D intervals formed by the vertical and horizontal projections of the pupil. For real bandlimited images of the low-pass type, this corresponds to knowing only the maximum x and y spatial frequencies of the image. The utilization of information of the known portion of the image in the extrapolation process is discussed for both algorithms. The second algorithm, reformulated discretely, is placed in closed form.