In the present article we study the average of Lipschitz-Killing (LK) curvatures of the excursion set of a stationary isotropic Gaussian field X on ℝ2${ℝ}^2$${ℝ}^2$. The novelty is that the field can be nonstandard, that is, with unknown mean and variance, which is more realistic from an applied viewpoint. To cope with the unknown location and scale parameters of X, we introduce novel fundamental quantities called effective level and effective spectral moment. We propose unbiased and asymptotically normal estimators of these parameters. From these asymptotic results, we build a test to determine if two images of excursion sets can be compared. This test is applied on both synthesized and real mammograms. Meanwhile, we establish the consistency of the empirical variance estimators of the third LK curvature under a weak condition on the correlation function of X.

中文翻译：

---

使用 Lipschitz-Killing 曲率统计具有未知位置和尺度的高斯随机场

在本文中，我们研究了静止各向同性高斯场X的偏移集的 Lipschitz-Killing (LK) 曲率的平均值ℝ2${ℝ}^2$${ℝ}^2$. 新颖之处在于该领域可以是非标准的，即具有未知的均值和方差，从应用的角度来看这更现实。为应对X的未知位置和尺度参数，我们引入了称为有效能级和有效谱矩的新基本量。我们提出了这些参数的无偏和渐近正态估计量。根据这些渐近结果，我们构建了一个测试，以确定是否可以比较两个偏移集图像。该测试适用于合成和真实的乳房 X 线照片。同时，我们在X的相关函数上建立了弱条件下第三LK曲率的经验方差估计量的一致性。