  We present an algorithm for resampling a function from its values on a
non-Cartesian grid onto a Cartesian grid. This problem arises in many
applications such as MRI, CT, radio astronomy and geophysics. Our algorithm,
termed SParse Uniform ReSampling (SPURS), employs methods from modern sampling
theory to achieve a small approximation error while maintaining low
computational cost. The given non-Cartesian samples are projected onto a
selected intermediate subspace, spanned by integer translations of a compactly
supported kernel function. This produces a sparse system of equations
describing the relation between the nonuniformly spaced samples and a vector of
coefficients representing the projection of the signal onto the chosen
subspace. This sparse system of equations can be solved efficiently using
available sparse equation solvers. The result is then projected onto the
subspace in which the sampled signal is known to reside. The second projection
is implemented efficiently using a digital linear shift invariant (LSI) filter
and produces uniformly spaced values of the signal on a Cartesian grid. The
method can be iterated to improve the reconstruction results. We then apply
SPURS to reconstruction of MRI data from nonuniformly spaced k-space samples.
Simulations demonstrate that SPURS outperforms other reconstruction methods
while maintaining a similar computational complexity over a range of sampling
densities and trajectories as well as various input SNR levels.
