Estimation of surface area and surface area measure of three-dimensional sets from digitizations |
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Authors: | Johanna Ziegel Markus Kiderlen |
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Affiliation: | 1. Department of Mathematics, ETH Zurich, 8092 Zurich, Switzerland;2. Department of Mathematical Sciences, University of Aarhus, Ny Munkegade Build. 1530, DK-8000 Aarhus C, Denmark |
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Abstract: | A local method for estimating surface area and surface area measure of three-dimensional objects from discrete binary images is presented. A weight is assigned to each 2 × 2 × 2 configuration of voxels and the total surface area of an object is given by summation of the local area contributions. The method is based on an exact asymptotic result that holds for increasing resolution of the digitization. It states that the number of occurrences of a 2 × 2 × 2 configuration is asymptotically proportional to an integral of its “h-function” with respect to the surface area measure of the object. We find explicit representations for these h-functions. Analyzing them in detail, we determine weights that lead to an asymptotic worst case error for surface area estimation of less than 4%. We show that this worst case error is the best possible. Exploiting the local nature of the asymptotic result, we also establish two parametric estimators for the surface area measure. The latter allow to quantify anisotropy of the object under consideration. Simulation studies illustrate the validity of the estimation procedure also for finite, but sufficiently high resolution. |
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Keywords: | Surface area Surface area measure Anisotropy 3D binary image Configuration Gauss digitization Local method Rose of normal directions |
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