Statistical Assessment of Signal and Image Symmetries

This paper formulates the problem of assessing the reflection symmetry of a function f observed in the presence of noise. We consider both univariate and bivariate characteristics representing signal and image functions. First the problem of estimating a parameter defining the reflection symmetry is examined. This is followed by the question of testing the given symmetry type. The estimation/detection procedure is based on minimizing the L-2-distance between empirical versions of f and its reflected version. For univariate functions this distance is estimated by the Fourier series type estimate. In the bivariate case we utilize a class of radial series represented by the Zernike functions. It is shown that the symmetry parameter can be recovered with the parametric optimal rate for all functions f of bounded variation.