Local isotropy test based on the oriented perimeter of digitalized images

Mariem Abaach (1) , Hermine Biermé (2) , Elena Di Bernardino (3) , Anne Estrade (1)

1 MAP5 - UMR 8145 - Mathématiques Appliquées Paris 5
2 IDP - Institut Denis Poisson
3 LJAD - Laboratoire Jean Alexandre Dieudonné

In this paper we consider the oriented perimeter of a thresholded grey-level image. This geometrical quantity is built by considering separately the horizontal and vertical contributions of the pixel. We explicitly compute the first two moments of the socalled oriented perimeter under the hypothesis of an underlying discrete Gaussian stationary field. We establish a Central Limit Theorem, as the number of pixels goes to infinity, for the joint oriented perimeters at various levels under a summability condition of the covariance function. By using the CLT previously established, we then construct a consistent local isotropy test, based on the ratio of the oriented perimeters. Our theoretical study is completed by extensive numerical illustrations based on simulated data. Finally, we apply our method to detect local isotropy in calcaneus X-ray images.

Keywords: Central Limit Theorem; Discrete geometry; Excursion sets; Gaussian fields; Geometric statistics; Medical imaging

Domains: Mathematics [math]

Mariem Abaach, Hermine Biermé, Elena Di Bernardino, Anne Estrade. Local isotropy test based on the oriented perimeter of digitalized images. 2023. ⟨hal-04037394⟩