Density‐free test for symmetry verification in images

Presence of symmetry is utilized in multiple machine vision systems to help achieve their goals. In numerous scenarios, this goal is to verify that certain symmetry is indeed exhibited by an image. However, we find that there is a shortage of methods for symmetry verification that would be capable of asserting an arbitrary reflectional or rotational symmetry. Using symmetry detectors to merely perform symmetry verification is improvident and not justified. We thus propose a novel statistical test for symmetry verification that fulfills the requirement of versatility. The proposed test is based on the principle that if an image is invariant to some hypothesized transformation, then an averaged image, obtained by averaging pixel intensities of an input image and of its transformed copy, looks exactly the same as an input image. Adopting the viewpoint that images are visual messages that convey some information allows us to expect that the amount of information in both images is the same. On the contrary, an incorrectly chosen transformation shall result in the information content being different. Based on this equality, we construct the test statistic and show that, when samples are large, the test statistic is asymptotically normally distributed. Finally, to verify the validity of the proposed principle and the performance of the method, a meticulous experimental study was performed on a large set of images. The results of this study confirmed the postulated ability of the method to verify an arbitrary symmetry and are demonstrated with several examples.