Testing for Image Symmetries—With Application to Confocal Microscopy

Statistical tests are introduced for checking whether an image function
f
(
x
,
y
) defined on the unit disc
D
={(
x
,
y
):
x

²
+
y

²
les 1} is invariant under certain symmetry transformations of
D
, given that discrete and noisy data are observed. Invariance under reflections or under rotations by rational angles is considered, as well as rotational invariance. These symmetry relations can be naturally expressed as restrictions for the Zernike moments of
f
(
x
,
y
). Therefore, the test statistics are based on the
L

₂
distance between Zernike series estimates of the image function itself and its version obtained after applying the symmetry transformation. The asymptotic distribution of the test statistics under both the hypothesis of symmetry as well as under fixed alternatives is derived. Furthermore, the quality of the asymptotic approximations via simulation studies is investigated. The usefulness of our theory is verified by examining an important problem in confocal microscopy, i.e., possible imprecise alignments in the optical path of the microscope are investigated. For optical systems with rotational symmetry, the theoretical point-spread function (PSF) is reflection symmetric with respect to two orthogonal axes, and rotationally invariant if the detector plane matches the optical plane of the microscope. The tests are used to investigate whether the required symmetries can indeed be detected in the empirical PSF.