  We introduce a method to recover a continuous domain representation of a
piecewise constant two-dimensional image from few low-pass Fourier samples.
Assuming the edge set of the image is localized to the zero set of a
trigonometric polynomial, we show the Fourier coefficients of the partial
derivatives of the image satisfy a linear annihilation relation. We present
necessary and sufficient conditions for unique recovery of the image from
finite low-pass Fourier samples using the annihilation relation. We also
propose a practical two-stage recovery algorithm which is robust to
model-mismatch and noise. In the first stage we estimate a continuous domain
representation of the edge set of the image. In the second stage we perform an
extrapolation in Fourier domain by a least squares two-dimensional linear
prediction, which recovers the exact Fourier coefficients of the underlying
image. We demonstrate our algorithm on the super-resolution recovery of MRI
phantoms and real MRI data from low-pass Fourier samples, which shows benefits
over standard approaches for single-image super-resolution MRI.
