Pixel isotropy test based on directional perimeters

Abstract

In this paper we consider the so-called directional perimeters of a thresholded gray-level image. These geometrical quantities are built by considering separately the horizontal and vertical contributions of the pixel. We explicitly compute the first two moments of the directional perimeter under the hypothesis of an underlying discrete Gaussian stationary random field. We establish a central limit theorem (CLT), as the number of pixels goes to infinity, for the joint directional perimeters at various levels under a weak summability condition of the covariance function. By using the CLT previously established, we construct a consistent pixel isotropy test, based on the ratio of the directional perimeters. Our theoretical study is completed by extensive numerical illustrations based on simulated data. Finally, we apply our method to detect pixel anisotropy in calcaneus X-ray images.