Statistical assessment of image symmetries

This paper formulates the problem of assessing reflection and rotations symmetries of an image function observed in the presence of noise. Rigorous statistical tests are developed for testing image invariance under reflections or under rotations through rational angles, as well as joint invariance under both reflections and rotations. The symmetry relations are expressed as restrictions for Fourier coefficients with respect to a class of radial orthogonal functions. Therefore, our test statistics are based on checking whether the estimated radial coefficients approximately satisfy those restrictions. We derive the asymptotic distribution of the test statistics under both the hypothesis of symmetry and under fixed alternatives. The tests require novel statistical analysis based on the theory of quadratic forms of random variables. Furthermore, the problem of estimating the parameters defining a given class of symmetry is also tackled. This is resolved by using the theory of semiparametric inference.