Closed-form reconstruction of images from irregular 2-D discrete Fourier samples using the Good-Thomas FFT

The problem of reconstructing an image from irregular (not on a cartesian grid) samples of its 2-D DTFT arises in synthetic aperture radar (SAR) and magnetic resonance imaging (MRI), in which the 2-D DTFT is known only on part of a polar raster. It also arises in limited angle tomography, in which the 2-D DTFT is known in a bowtie region. Reconstruction requires either nearest-neighbor interpolation or solution of a large linear system of equations, both of which are computationally intensive and can lead to errors due to poor conditioning of the problem. An explicit formula does not seem to exist, since there is no 2-D Lagrange interpolation formula. This paper uses the Good-Thomas FFT to unwrap the 2-D problem into a 1-D problem, to which the 1-D Lagrange interpolation formula can be applied, either directly or recursively (the latter is much faster). This approach results in a sufficient condition to ensure a unique reconstruction