  We introduce a structured low rank matrix completion algorithm to recover a
series of images from their under-sampled measurements, where the signal along
the parameter dimension at every pixel is described by a linear combination of
exponentials. We exploit the exponential behavior of the signal at every pixel,
along with the spatial smoothness of the exponential parameters to derive an
annihilation relation in the Fourier domain. This relation translates to a
low-rank property on a structured matrix constructed from the Fourier samples.
We enforce the low rank property of the structured matrix as a regularization
prior to recover the images. Since the direct use of current low rank matrix
recovery schemes to this problem is associated with high computational
complexity and memory demand, we adopt an iterative re-weighted least squares
(IRLS) algorithm, which facilitates the exploitation of the convolutional
structure of the matrix. Novel approximations involving two dimensional Fast
Fourier Transforms (FFT) are introduced to drastically reduce the memory demand
and computational complexity, which facilitates the extension of structured low
rank methods to large scale three dimensional problems. We demonstrate our
algorithm in the MR parameter mapping setting and show improvement over the
state-of-the-art methods.
