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) : x2 + y2 . 1} is invariant under certain symmetry transformations of D, given that discrete and noisy data are observed. We consider invariance under reflections or under rotations by rational angles, as well as joint invariance under both reflections and rotations. Furthermore, we propose a test for rotational invariance of f(x, y), i.e., for checking whether f(x, y), after transformation to polar coordinates, only depends on the radius and not on the angle. These symmetry relations can be naturally expressed as restrictions for the Zernike moments of the image function f(x, y), i.e., the Fourier coefficients with respect to the Zernike orthogonal basis. Therefore, our test statistics are based on checking whether the estimated Zernike coefficients approximately satisfy those restrictions. This is carried out by forming the L2 distance between the image function and its transformed version obtained by some symmetry transformation. We derive the asymptotic distribution of the test statistics under both the hypothesis of symmetry as well as under fixed alternatives. Furthermore, we investigate the quality of the asymptotic approximations via simulation studies. The usefulness our theory is verified by examining an important problem in confocal microscopy, i.e., we investigate possible imprecise alignments in the optical path of the microscope. 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. We use our tests to investigate whether the required symmetries can indeed be detected in the empirical PSF.