Minkowski functionals quantify the morphology of smooth random fields. They are widely used to probe statistical properties of cosmological fields. Analytic formulas for ensemble expectations of Minkowski functionals are well known for Gaussian and mildly non-Gaussian fields. In this paper, we extend the formulas to composite fields which are sums of two fields and explicitly derive the expressions for the sum of uncorrelated mildly non-Gaussian and Gaussian fields. These formulas are applicable to observed data which is usually a sum of the true signal and one or more secondary fields that can be either noise, or some residual contaminating signal. Our formulas provide explicit quantification of the effect of the secondary field on the morphology and statistical nature of the true signal. As examples, we apply the formulas to determine how the presence of Gaussian noise can bias the morphological properties and statistical nature of Gaussian and non-Gaussian cosmic microwave background temperature maps.

中文翻译：

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复合平滑随机场的闵可夫斯基泛函

闵可夫斯基泛函量化平滑随机场的形态。它们被广泛用于探测宇宙学领域的统计特性。闵可夫斯基泛函的系综期望的解析公式在高斯场和轻度非高斯场中是众所周知的。在本文中，我们将公式扩展到两个场之和的复合场，并明确推导了不相关的轻度非高斯场和高斯场之和的表达式。这些公式适用于观测数据，这些数据通常是真实信号和一个或多个次级场的总和，这些次级场可以是噪声，也可以是一些残留的污染信号。我们的公式提供了二次场对真实信号的形态和统计性质的影响的明确量化。作为示例，我们应用公式来确定高斯噪声的存在如何使高斯和非高斯宇宙微波背景温度图的形态特性和统计性质产生偏差。