  We introduce a Prony-like method to recover a continuous domain 2-D piecewise
smooth image from few of its Fourier samples. Assuming the discontinuity set of
the image is localized to the zero level-set of a trigonometric polynomial, we
show the Fourier transform coefficients of partial derivatives of the signal
satisfy an annihilation relation. We present necessary and sufficient
conditions for unique recovery of piecewise constant images using the above
annihilation relation. We pose the recovery of the Fourier coefficients of the
signal from the measurements as a convex matrix completion algorithm, which
relies on the lifting of the Fourier data to a structured low-rank matrix; this
approach jointly estimates the signal and the annihilating filter. Finally, we
demonstrate our algorithm on the recovery of MRI phantoms from few
low-resolution Fourier samples.
