A Limitation of the Estimation of Intrinsic Volumes via Pixel Configuration Counts

Abstract

It is often helpful to compute the intrinsic volumes of a set of which only a pixel image is observed. A computational efficient approach, which is suggested by several authors and used in practice, is to approximate the intrinsic volumes by a linear functional of the pixel configuration histogram. Here we want to examine, whether there is an optimal way of choosing this linear functional, where we will use a quite natural optimality criterion that has already been applied successfully for the estimation of the surface area. We will see that for intrinsic volumes other than volume or surface area this optimality criterion cannot be used, since estimators which ignore the data and return constant values are optimal w.r.t. this criterion. This shows that one has to be very careful, when intrinsic volumes are approximated by a linear functional of the pixel configuration histogram.

Local Digital Estimators of Intrinsic Volumes for Boolean Models and in the Design-Based Setting

“…For this, we need some stronger boundary conditions on X. For instance, Jürgen Kampf has shown in a yet unpublished paper (see [5]) that without the isotropy of the lattice, there are no local estimators for V 0 that are asymptotically unbiased for all polyconvex sets. On the other hand, it is well known that there exists a local algorithm for V 0 , which is asymptotically unbiased on the class of so-called r-regular sets; see, for example, the discussion in [15].…”

Valuations in Image Analysis

“…This is part of a more general phenomenon. Even when the underlying set is a convex polytope, the following was proved in [23], see also [8] when k = 2:…”

Local Digital Algorithms Applied to Boolean Models

“…Local algorithms have been studied theoretically in the design‐based setting where Z is a deterministic set and is stationary random. With the exception of volume and Euler characteristic, results show that they are almost always biased (Ziegel & Kiderlen, ; Kampf, ; Svane, ), even asymptotically when the resolution goes to infinity.…”