We present a sampling theory for a class of binary images with finite rate of
innovation (FRI). Every image in our model is the restriction of
\mathds{1}_{\{p\leq0\}} to the image plane, where \mathds{1} denotes the
indicator function and p is some real bivariate polynomial. This particularly
means that the boundaries in the image form a subset of an algebraic curve with
the implicit polynomial p. We show that the image parameters --i.e., the
polynomial coefficients-- satisfy a set of linear annihilation equations with
the coefficients being the image moments. The inherent sensitivity of the
moments to noise makes the reconstruction process numerically unstable and
narrows the choice of the sampling kernels to polynomial reproducing kernels.
As a remedy to these problems, we replace conventional moments with more stable
\emph{generalized moments} that are adjusted to the given sampling kernel. The
benefits are threefold: (1) it relaxes the requirements on the sampling
kernels, (2) produces annihilation equations that are robust at numerical
precision, and (3) extends the results to images with unbounded boundaries. We
further reduce the sensitivity of the reconstruction process to noise by taking
into account the sign of the polynomial at certain points, and sequentially
enforcing measurement consistency. We consider various numerical experiments to
demonstrate the performance of our algorithm in reconstructing binary images,
including low to moderate noise levels and a range of realistic sampling
kernels.Comment: 12 pages, 14 figure