  When determining the parameters of a parametric planar shape based on a
single low-resolution image, common estimation paradigms lead to inaccurate
parameter estimates. The reason behind poor estimation results is that standard
estimation frameworks fail to model the image formation process at a
sufficiently detailed level of analysis. We propose a new method for estimating
the parameters of a planar elliptic shape based on a single photon-limited,
low-resolution image. Our technique incorporates the effects of several
elements - point-spread function, discretisation step, quantisation step, and
photon noise - into a single cohesive and manageable statistical model. While
we concentrate on the particular task of estimating the parameters of elliptic
shapes, our ideas and methods have a much broader scope and can be used to
address the problem of estimating the parameters of an arbitrary parametrically
representable planar shape. Comprehensive experimental results on simulated and
real imagery demonstrate that our approach yields parameter estimates with
unprecedented accuracy. Furthermore, our method supplies a parameter covariance
matrix as a measure of uncertainty for the estimated parameters, as well as a
planar confidence region as a means for visualising the parameter uncertainty.
The mathematical model developed in this paper may prove useful in a variety of
disciplines which operate with imagery at the limits of resolution.
