## A limitation of the estimation of intrinsic volumes via pixel configuration counts

• 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.

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Author: Jürgen Kampf urn:nbn:de:hbz:386-kluedo-32736 Report in Wirtschaftsmathematik (WIMA Report) (144) Preprint English 2012/09/28 2012 Technische Universität Kaiserslautern 2012/10/01 19 Fachbereich Mathematik 0 Informatik, Informationswissenschaft, allgemeine Werke / 00 Informatik, Wissen, Systeme / 006 Spezielle Computerverfahren 5 Naturwissenschaften und Mathematik / 51 Mathematik / 516 Geometrie 5 Naturwissenschaften und Mathematik / 51 Mathematik / 519 Wahrscheinlichkeiten, angewandte Mathematik 52-XX CONVEX AND DISCRETE GEOMETRY / 52Cxx Discrete geometry / 52C07 Lattices and convex bodies in n dimensions [See also 11H06, 11H31, 11P21] 62-XX STATISTICS / 62Hxx Multivariate analysis [See also 60Exx] / 62H35 Image analysis 65-XX NUMERICAL ANALYSIS / 65Dxx Numerical approximation and computational geometry (primarily algorithms) (For theory, see 41-XX and 68Uxx) / 65D18 Computer graphics, image analysis, and computational geometry [See also 51N05, 68U05] Standard gemäß KLUEDO-Leitlinien vom 10.09.2012

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