The halfspace depth can be seen as a mapping that to a finite Borel measure $\mu$ on the Euclidean space ${\mathbb{R}}^d$ assigns its depth, being a function ${\mathbb{R}}^d\to [0,\infty ):x\mapsto D\left(x;\mu \right)$. The depth of $\mu$ quantifies how much centrally positioned a point $x$ is with respect to $\mu$. This function is intended to serve as generalization of the quantile function to multivariate spaces. We consider the problem of finding the inverse mapping to the halfspace depth: knowing only the function $x\mapsto D\left(x;\mu \right)$, our objective is to reconstruct the measure $\mu$. We focus on $\mu$ atomic with finitely many atoms, and present a simple method for the reconstruction of the position and the weights of all atoms of $\mu$, from its depth only. As a consequence, (i) we recover generalizations of several related results known from the literature, with substantially simplified proofs, and (ii) design a novel reconstruction procedure that is numerically more stable, and considerably faster than the known algorithms. Our analysis presents a comprehensive treatment of the halfspace depth of those measures whose depths attain finitely many different values.

半空间深度可以看作是对有限Borel测度的映射 $\mu$ 在欧几里得空间上 ${\mathbb{[R}}^d$ 分配其深度，作为一个函数 ${\mathbb{[R}}^d\to [0，\infty ）：X\mapsto d\left(X;\mu \right)$。深度$\mu$ 量化一个点的中心位置 $X$ 是关于 $\mu$。该函数旨在用作将分位数函数推广到多元空间。我们考虑找到与半空间深度的逆映射的问题：仅知道函数$X\mapsto d\left(X;\mu \right)$，我们的目标是重建措施 $\mu$。我们专注于$\mu$ 具有有限多个原子的原子，并提出了一种简单的方法来重建原子中所有原子的位置和权重 $\mu$，仅从深度来看。结果，（i）我们用实质上简化的证明恢复了文献中已知的几个相关结果的概括，并且（ii）设计了一种新的重建程序，该程序在数值上比已知算法更稳定，并且速度明显更快。我们的分析提供了对那些深度有限地达到许多不同值的度量的半空间深度的综合处理。