We present an asymptotic treatment of errors involved
in point-based image registration where control point (CP)
localization is subject to heteroscedastic noise; a suitable model
for image registration in fluorescence microscopy. Assuming an
affine transform, CPs are used to solve a multivariate regression
problem. With measurement errors existing for both sets of CPs
this is an errors-in-variable problem and linear least squares
is inappropriate; the correct method being generalized least
squares. To allow for point dependent errors the equivalence of a
generalized maximum likelihood and heteroscedastic generalized
least squares model is achieved allowing previously published
asymptotic results to be extended to image registration. For a
particularly useful model of heteroscedastic noise where covariance
matrices are scalar multiples of a known matrix (including
the case where covariance matrices are multiples of the identity)
we provide closed form solutions to estimators and derive their
distribution. We consider the target registration error (TRE) and
define a new measure called the localization registration error
(LRE) believed to be useful, especially in microscopy registration
experiments. Assuming Gaussianity of the CP localization errors,
it is shown that the asymptotic distribution for the TRE and LRE
are themselves Gaussian and the parameterized distributions are
derived. Results are successfully applied to registration in single
molecule microscopy to derive the key dependence of the TRE and
LRE variance on the number of CPs and their associated photon
counts. Simulations show asymptotic results are robust for low
CP numbers and non-Gaussianity. The method presented here is
shown to outperform GLS on real imaging data.