Improving the Cavalieri estimator under non-equidistant sampling and dropouts

Abstract

Motivated by the stereological problem of volume estimation from parallel section profiles, the so-called Newton-Cotes integral estimators based on random sampling nodes are analyzed. These estimators generalize the classical Cavalieri estimator and its variant for non-equidistant sampling nodes, the generalized Cavalieri estimator, and have typically a substantially smaller variance than the latter. The present paper focuses on the following points in relation to Newton-Cotes estimators: the treatment of dropouts, the construction of variance estimators, and, finally, their application in volume estimation of convex bodies.Dropouts are eliminated points in the initial stationary point process of sampling nodes, modeled by independent thinning. Among other things, exact representations of the variance are given in terms of the thinning probability and increments of the initial points under two practically relevant sampling models. The paper presents a general estimation procedure for the variance of Newton-Cotes estimators based on the sampling nodes in a bounded interval. Finally, the findings are illustrated in an application of volume estimation for three-dimensional convex bodies with sufficiently smooth boundaries.

Improving Cavalieri volume estimation based on non‐equidistant planar sections: The trapezoidal estimator

“…In contrast, the trapezoidal estimator clearly shows a Zitterbewegung in the variance plot for the 1-oriented object, since the first derivative of this area function has two discontinuities (at the boundary of the support of the area function). Moreover and most importantly, we see that the trapezoidal estimator decreases as 𝑇 4 with decreasing 𝑇, whereas the Cavalieri estimator decreases approximately as 𝑇 3 .…”

“…Cavalieri estimation is thus solving the problem of estimating the integral $$Q=\int _{\mathbb {R}} f(x) dx$$ from finitely many values of f at random sampling points. This mathematical formulation of the problem is used in the papers 1–3, which we will repeatedly refer to.…”

“…Secondly, with the notation used in Ref. 3, we need estimates of certain moments denoted $$\gamma _{i,j}$$. The quantity $$T\gamma _{i,j}$$ represents the expected j th power of the typical distance between an observed section and its i th neighbour.…”

“…The quantity of interest is the volume 𝑄 of some nonrandom and bounded object 𝑌 ⊆ ℝ 3 . Traditionally, this quantity is estimated from parallel planar sections placed systematically with a fixed distance 𝑇 apart: With {𝑆 𝑘 } 𝑁 𝑘=0 denoting the cross sections of 𝑌 and {Area(𝑆 𝑘 )} 𝑁 𝑘=0 their associated known areas, the Cavalieri volume estimator of 𝑄 is…”

“…We describe them informally and refer to Definitions 1 and 2 in Ref. 3 for a mathematically rigorous exposition. We emphasize that no model assumptions are needed to compute the trapezoidal estimator $$\hat{Q}_1$$, and similarly its variance can be estimated without such assumptions (see Definition 1).…”