A limitation of the estimation of intrinsic volumes via pixel configuration counts (Q2874620)

Let \(V_k(C)\) denote the \(k\)-th intrinsic volume of a convex set \(C \subset \mathbb{R}^d\) with \(0 \leq k \leq d\). In computer applications \(C\) can usually only be represented by a pixel approximation. Thus, it is natural to ask whether intrinsic volumes of the original body can be obtained from its pixel approximation.NEWLINENEWLINEThe paper under review studies algorithms based on linear combinations of \(2^d\)-pixel configuration counts, which are computationally efficient estimators of intrinsic volumes. The coefficients of this linear combination are called weights. The main results of the paper state lower bounds for the worst case asymptotic bias over for all possible weights and their corresponding estimators \(\hat{V}_k\) for \(V_k\), for \(1 \leq k \leq d-2\). In particular, the results show that estimators which approximate intrinsic volumes other than volume or surface area by linear combinations of pixel configuration counts have the severe shortcoming that the estimated value may be very far from the true value (even if the resolution of the pixel image goes to 0).