  We introduce a continuous domain framework for the recovery of a planar curve
from a few samples. We model the curve as the zero level set of a trigonometric
polynomial. We show that the exponential feature maps of the points on the
curve lie on a low-dimensional subspace. We show that the null-space vector of
the feature matrix can be used to uniquely identify the curve, given a
sufficient number of samples. The worst-case theoretical guarantees show that
the number of samples required for unique recovery depends on the bandwidth of
the underlying trigonometric polynomial, which is a measure of the complexity
of the curve. We introduce an iterative algorithm that relies on the low-rank
property of the feature maps to recover the curves when the samples are noisy
or when the true bandwidth of the curve is unknown. We also demonstrate the
preliminary utility of the proposed curve representation in the context of
image segmentation.
