Two-dimensional image reconstruction from Fourier coefficients computed directly from zero crossings

Two-dimensional image reconstruction using Fourier coefficients that are computed directly from the sampled representation of zero crossings is demonstrated. A two-dimensional image of dimensions N(x) × N(y) is interpreted as a set of N(y) independent x-space lines (in gray-scale format) that are arranged uniquely along the y direction. Each line has N(x) elements. Reconstruction is achieved first by computing the entire set of N(y) one-dimensional Fourier transforms from the measured zero crossings using Newton's formula. Each N(y)th line spectra has N(x) Fourier coefficients. The inverse Fourier transform is then applied to each of the line spectra to obtain a set of N(y) reconstructed x-space lines. The reconstructed image is obtained by arranging the reconstructed lines properly along they direction.