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 (permanent link):urn:nbn:de:hbz:386-kluedo-32736
Serie (Series number):Report in Wirtschaftsmathematik (WIMA Report) (144)
Document Type:Preprint
Language of publication:English
Publication Date:2012/09/28
Year of Publication:2012
Publishing Institute:Technische Universität Kaiserslautern
Number of page:19
Faculties / Organisational entities:Fachbereich Mathematik
DDC-Cassification:006 Spezielle Computerverfahren
516 Geometrie
519 Wahrscheinlichkeiten, angewandte Mathematik
MSC-Classification (mathematics):52C07 Lattices and convex bodies in n dimensions [See also 11H06, 11H31, 11P21]
62H35 Image analysis
65D18 Computer graphics, image analysis, and computational geometry [See also 51N05, 68U05]

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