  A method for estimating the axis of reflectional symmetry of an image
$f(x,y)$ on the unit disc $D=\{(x,y):x^2+y^2\leq1\}$ is proposed, given that
noisy data of $f(x,y)$ are observed on a discrete grid of edge width $\Delta$.
Our estimation procedure is based on minimizing over $\beta\in[0,\pi)$ the
$L_2$ distance between empirical versions of $f$ and $\tau_{\beta}f$, the image
of $f$ after reflection at the axis along $(\cos\beta,\sin\beta)$. Here, $f$
and $\tau_{\beta}f$ are estimated using truncated radial series of the Zernike
type. The inherent symmetry properties of the Zernike functions result in a
particularly simple estimation procedure for $\beta$. It is shown that the
estimate $\hat{\beta}$ converges at the parametric rate $\Delta^{-1}$ for
images $f$ of bounded variation. Further, we establish asymptotic normality of
$\hat{\beta}$ if $f$ is Lipschitz continuous. The method is applied to
calibrating the point spread function (PSF) for the deconvolution of images
from confocal microscopy. For various reasons the PSF characterizing the
problem may not be rotationally invariant but rather only reflection symmetric
with respect to two orthogonal axes. For an image of a bead acquired by a
confocal laser scanning microscope (Leica TCS), these axes are estimated and
corresponding confidence intervals are constructed. They turn out to be close
to the coordinate axes of the imaging device. As cause for deviation from
rotational invariance, this indicates some slight misalignment of the optical
system or anisotropy of the immersion medium rather than some irregular shape
of the bead. In an extensive simulation study, we show that using a symmetrized
version of the observed PSF significantly improves the subsequent
reconstruction process of the target image.
