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A LIMITATION OF THE ESTIMATION OF INTRINSIC VOLUMES VIA PIXEL CONFIGURATION COUNTS

Part of: Discrete geometry Multivariate analysis Numerical approximation and computational geometry

Published online by Cambridge University Press:  28 March 2014

Jürgen Kampf
Jürgen Kampf*
Affiliation:
FB Mathematik, TU Kaiserslautern, Postbox 3049, 67653 Kaiserslautern,Germany email [email protected]
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Abstract

It is often helpful to compute the intrinsic volumes of a set of which only a pixel image is observed. A computationally efficient approach, which is suggested by several authors and used in practice, is to approximate the intrinsic volumes by linear combinations of the pixel configuration counts. However, we will show that when this approach is used for the computation of an intrinsic volume other than volume or surface area, an asymptotic error of 100% of the correct value cannot be avoided. As a consequence we derive that estimators which ignore the data and return constant values are optimal with respect to a natural criterion which has already been applied successfully for the estimation of the surface area.

MSC classification

Secondary: 52C07: Lattices and convex bodies in $n$ dimensions 62H35: Image analysis 65D18: Computer graphics, image analysis, and computational geometry
Type
Research Article
Information
Mathematika , Volume 60 , Issue 2 , July 2014 , pp. 485 - 511
Copyright
Copyright © University College London 2014 

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