Abstract — The problem of reconstructing an image from irregular frequency samples arises in synthetic aperture radar (SAR), magnetic resonance imaging (MRI), limited angle tomography, and 2-D filter design. Since there is no 2-D Lagrange interpolation formula, this problem is usually solved using an iterative algorithm, such as POCS (Projection Onto Convex Sets), or CG (Conjugate Gradient) applied to a linear system of equations with the image pixels as unknowns. However, these require many iterations, and each iteration requires a non-uniform forward 2-D Discrete Fourier Transform (DFT). We present a non-iterative algorithm for the reconstruction of an (M × M) image from a sufficient number of arbitrary samples of its (N × N) 2-D DFT, where N>> M. The algorithm requires only a single sparse (N × N) 2-D DFT, followed by two roughly (M × M) 2-D DFTs. Precomputation for a given configuration of irregular (N × N) 2-D DFT samples is also required. Small and large examples illustrate the algorithm
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