'l0tl1 lnternatiunal Wurkshap, IWEIH 200-4
`Auckland, New Zealand, December 2004
`Pmceedinégs
`
`EN BIo3—o "
`
`

`
`Reinhard Klette Jovisa Zunic (Eds.)
`
`Combinatorial
`
`Image Analysis
`
`10th International Workshop, IWCIA 2004
`Auckland, New Zealand, December 1-3, 2004
`
`Proceedings
`
`@ Springer
`
`Iibraryresearch staff@\«siIeyrein.com
`
`FRESENIUS KABI 1034-0002
`
`

`
`Volume Editors
`
`Reinhard Klette
`
`University of Auckland
`Tamaki Campus, CITR
`Glen Innes, Morrin Road, Building 731, Auckland 1005, New Zealand
`E-mail: r.klette@cs.auckland.ac.nz
`
`Jovisa Zunié
`Exeter University
`Computer Science Department
`Harrison Building, Exeter EX4 4QF, U.K.
`E-mail: J.Z.unic@ex.ac.uk
`
`Library of Congress Control Number: 2004115523
`
`CR Subject Classification (1998): 1.4, L5, 1.3.5, F.2.2, G.2.1, G.1.6
`
`ISSN 0302-9743
`
`ISBN 3-540-23942-1 Springer Berlin Heidelberg New York
`
`This work is subject to copyright. All rights are reserved, whether the whole or part of the material is
`concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting,
`reproduction on rnicrofilrns or in any other way, and storage in data banks. Duplication of this publication
`or parts thereof is permitted only under the provisions of the German Copyright Law of September 9. 1965,
`in its current version, a11d permission for use must always be obtained from Springer. Violations are liable
`to prosecution under the German Copyright Law.
`Springer is a part of Springer Science+Busi11ess Media
`springero11line.co1n
`© Springer-Verlag Berlin Heidelberg 2004
`Printed in Germany
`
`Typesetting: Camera-ready by author, data conversion by Scientific Publishing Services, Chennai, India
`Printed on acid-free paper
`SPIN: 11360971
`06/3142
`5 4 3 2 1 0
`
`Iibraryresearch staff@\«siIeyrein.com
`
`FRESENIUS KABI 1034-0003
`
`

`
`Table of Contents
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`Discrete Tomography
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`Binary Matrices Under the Microscope: A Tomographical Problem
`Andrea Froséné,
`]VIa'u7"z'ce Nzirat .
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`On the Reconstruction of Crystals Through Discrete Tomography
`KJ. Batenb-urg, W]. Palenstijn .
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`1
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`23
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`Binary Tomography by lterating Linear Programs from Noisy
`Projections
`Stefan l-/Veber, Thomas Schxiile, Joachim Hornegger,
`Christoph Schnorr .
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`38
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`Combinatorics and Computational lVIodels
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`Hexagonal Pattern Languages
`K.S. Dersanambrika, K. Krithévasan, C.
`K. G. Sub'raman7Lan .
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`]VIa7"t2'.n— Vida,
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`52
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`A Combinatorial Transparent Surface Modeling from Polarization
`Images
`Jvfohamad Ivan Fanany, Ifiichi Kobayashri, Itsruo Kumazawa .
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`65
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`77
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`Integral Trees: Subtree Depth and Diameter
`Walter G. Kropatsch, Yll Haschévnusa, Zygmunt Pizlo .
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`Supercover of Non—square and Non—cubic Grids
`Tr'o'u.ng K/ée/a Lmh, Atsu.shz' Irmlya, Robzn Strand, Gun/élla Bor_gefor.s'
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`88
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`Calculating Distance With Neighborhood Sequences in the Hexagonal
`Grid
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`Benedek Nagy .
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`98
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`On Correcting the Unevenness of Angle Distributions Arising from
`Integer Ratios Lying in Restricted Portions of the Farey Plane
`Imants Svalbe,
`/lndrerzu Kingston .
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`Iibraryresearch staff@\«siIeyrein.com
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`FRESENIUS KABI 1034-0004
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`

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`Vlll
`
`Table of Contents
`
`Combinatorial Algorithms
`
`Equivalence Between Regular n—G'—Maps and n—Surfaces
`Sylvie Alayrangues, Xavier Daragon, Jacque3—Oliver Lacliaud,
`Pascal Lienliardl .
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`Z—Tilings of Polyoininoes and Standard Basis
`Olivier B0(li.n.i., Bertrand Noiwel
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`Curve Tracking by Hypothesis Propagation and V<:>ting—Based
`Verification
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`151
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`Kaz/aliiko Kawamoto, Kaoru Hirota .
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`3D Topological Thinning by Identifying Non—siinple Voxels
`Gisela Klette,
`lVIian. Pan.
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`Convex Hulls in a 3-Dimensional Space
`Vladimir Kovaleizsky, H6'fL7"il€ Schulz .
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`197
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`A Near—Linear Time Algorithm for Binarization of Fingerprint Images
`Using Distance Transforrn
`Xuefeng Liang, Arijit Bislina, Tetsuo Asano .
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`Combinatorial Mathematics
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`On Recognizable Infinite Array Languages
`S.
`Jnanasekaranl, VB. Dare .
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`On the Number of Digitizations of a Disc Depending on Its Position
`M’ar'lin N. H/arlgley, Joiiisa Zunié .
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`On the Language of Standard Discrete Planes and Surfaces
`Damien. Jamel
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`248
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`Characterization of Bijective Discretized Rotations
`Bertrand N0-Lwel, Eric Remila .
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`Digital Topology
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`Magnification in Digital Topology
`Akira Nakam/ara .
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`CurVes7 Hypersurfaces7 and Good Pairs of Adjacency Relations
`Valentin.
`Brimkoir, Reinhard Klette .
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`250
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`276
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`FRESENIUS KABI 1034-0005
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`

`
`Table of Contents
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`IX
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`A Maximum Set of (26,6)—Connected Digital Surfaces
`J.C. C2?/"7}a, A. De Jl/I7’/guel,
`,[')0/rim?/gaez, AH. F’/"an/cés,
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`Simple Points and Generic Axiomatized Digital Surface—Structures
`Sebastien. Fourey .
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`318
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`334
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`Minimal Non—sirnple Sets in 4—Di1nensional Binary Images with
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`T. Yang Kong, Chy7J—J0’u. Ga/a .
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`Jordan Surfaces in Discrete Antimatroid Topologies
`Ralph Kopperman, John. L. Pfaltz .
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`How to Find a Khalimsky—Continuous Approximation of a Real—\/Valued
`Function
`Erllc ll/feliwz .
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`351
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`Digital Geometry
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`Algorithms i11 Digital Ceometry Based on Cellular Topology
`V. Kovalezwkry .
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`An Efficient Euclidean Distance Transform
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`366
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`Donald G. Bailey .
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`TWo—Dimensional Discrete Morphing
`Isameddme Bouykhmss, Serge llflguet, Laure Taugne .
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`42],
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`A Comparison of Property Estimators in Stereology and Digital
`Geometry
`Yu/man Huang, Re/mh/ard Klette .
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`Thinning by Curvature Flow
`Atuslhi. Imlya, lllasahlko Salto, Klwamu Nakamura .
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`Convex Functions on Discrete Sets
`Chrlster O. Kiselman .
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`Discrete Surface Segmentation into Discrete Planes
`Isabelle S2?v7}gn077,, 14’l07"e77,t,[')u/pant,
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`

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`X
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`Table of Contents
`
`Approximation of Digital Sets by Curves and Surfaces
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`Sketch—Based Shape Retrieval Using Length and Curvature of 2D
`Digital Contours
`Abdolah Chalechale, Golshah Yaghdy, Pmshan Premamtne .
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`474
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`Surface Smoothing for Enhancement of 3D Data Using Curvature—Based
`Adaptive Regularization
`Hyunjong Ki, Jeongho Shin, Junghoon Jung, Seongwon Lee,
`Joonki Paik .
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`488
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`502
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`512
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`522
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`Minimum—Length Polygon of a Simple Cube/—CurVe in 3D Space
`Fajie Li, Reinhard Klette .
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`Corner Detection and Curve Partitioning Using Arc—Chord Distance
`Jbfajed Jbfarji, Reinhard Klette, Pepe S-iy .
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`Shape Preserving Sampling and Reconstruction of Grayscale Images
`Pee’/" Stelldingev" .
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`Algebraic Approaches
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`534
`
`Comparison of Nonparametric Transformations and Bit Vector
`Matching for Stereo Correlation
`Bogusiaw Cyganek .
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`Exact Optimization of Discrete Constrained Total Variation
`Minimization Problems
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`548
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`Jéréme Darbon, Jllarc S-igelle .
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`Tensor Algebra: A Combinatorial Approach to the Projective Geometry
`of Figures
`David N.R. ]llcKi.nn0n, Brian C. Lovell
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`558
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`558
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`Junction and Corner Detection Through the Extraction and Analysis
`of Line Segments
`Cfirisiiawi Perwas.s'
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`leometric Algebra for Pose Estimation and Surface Morphing in
`Human Motion Estimation
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`Bode Hosanna/in, Rein/Lard Klette .
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`Iibraryresearch staff@\«siIeyrein.com
`
`FRESENIUS KABI 1034-0007
`
`

`
`Fuzzy Image Analysis
`
`Table of Contents
`
`XI
`
`A Study on Supervised Classification of Remote Sensing Satellite Image
`by Bayesian Algorithm Using Average Fuzzy lntracluster Distance
`Youn_g—J00'rL Jean, Jae—G'a'r’k Chm’, Jm—Il Kim .
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`Tree Species Recognition with Fuzzy Texture Parameters
`Ralf Raul/cc, Norbert Haala .
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`Image Segmentation
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`597
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`607
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`Fast Segmentation of High—Resolution Satellite Images Using VVatershed
`'I‘ransform Combined with an Efficient Region Merging Approach
`Qriuyxiao Chen, C-'hen.gh.u Zhoru, Jiancheng Lruo, D0n.gp2'.n.g ]VI2'.ng .
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`621
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`Joint Non—rigid Motion Estimation and Segmentation
`B07"-is Flach, Radém Sam .
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`631
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`Field Segmentation of Soccer Video
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`639
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`Sequential Probabilistic
`Images
`Karveh. Kcmgarloo, Ehscmollah Kabiv" .
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`Adaptive Local Binarization Method for Recognition of Vehicle
`License Plates
`
`Byeong Rae Lee, Kyw/zg.s()o Park, Hymzchul Kcmg, Haksoo Kim,
`Chung/cyue Kim .
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`646
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`Blur Identification and Image Restoration Based on Evolutionary
`Multiple Object Segmentation for Digital Auto—focusing
`Jeongho Sh/e'n, Sun,gh,yu'rL Hwang, Kimcm Kim,
`.]/my0'ung Kang,
`Seong-won Lee, Joonki Pa-Zk .
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`Performance Evaluation of Binarizations of Scanned Insect Footprints
`Yrmng W.
`ll/00 .
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`656
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`669
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`l\/Iatching and Recognition
`
`2D Shape Recognition Using Discrete VVaVelet Descriptor Under
`Similitude Transform
`
`K7’/mc/7,677/g Kit/7,, E[—/2/adzi Za/Lza/7, .
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`679
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`Iibraryresearch staff@\.«.=*iIeyrein.com
`
`FRESENIUS KABI 1034-0008
`
`

`
`Xll
`
`Table of Contents
`
`VVhich Stereo Matching Algorithm for Accurate 3D Face Creation?
`P/L. Leclemq, J. Lziu, A. VV00dward, P. Delmas .
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`690
`
`Video Cataloging System for Real—Tirne Scene Change Detection of
`Video
`
`l/Vanjoo Lee, Hyoki Kim, Hyunchul Kcmg, Jmsung Lee,
`Yonglsryu Kiln/, Seals:/we Jean/
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`Automatic Face Recognition by Support Vector Machines
`Huaqing Li, Shcmyu. ll/any, Feihu
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`Practical Region—Based Matching for Stereo Vision
`Brian. ]l1cKi7rm0n, Jacky Baltes .
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`705
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`715
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`726
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`739
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`Video Program Clustering Indexing Based 011 Face Recognition Hybrid
`Model of Hidden Markov Model and Support Vector Machine
`Yuehua l/Van, Shim/mg J2’, Y2? Xie, Xian Zhang, P62:/j'un Xie .
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`Texture Feature Extraction and Selection for Classification of Images
`in a Sequence
`K/vi/rz. Wm, Sung Bmlk,
`
`Sung A/Ln, Sang Kim, Yang J0 .. .
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`.
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`750
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`Author Index .
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`759
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`Iibraryresearch staff@\«siIeyrein.com
`
`FRESENIUS KABI 1034-0009
`
`

`
`A Comparison of Property Estimators in
`
`Stereology and Digital Geometry
`
`Yuman Huang and Reinhard Klette
`
`CITR, University of Auckland, Tamaki Campus, Building 731,
`Auckland, New Zealand
`
`Abstract. VVe consider selected geometric properties of 2D or 3D sets,
`given in form of binary digital pictures, and discuss their estimation.
`The properties examined are perimeter and area ir1 2D, and surface area
`and volume in 3D. We evaluate common estimators in stereology and
`O
`0
`di ‘ital eometr
`accordincr to their multi )robe or multicrrid conver ‘ence
`
`properties, and precision and efficiency of estimations.
`
`1
`
`Introduction
`
`digital geometry are mainly oriented towards property
`Both stereology as well
`estimations. Stereologists estimate geometric properties based on stochastic ge-
`ometry and probability theory
`Key intentions are to ensure isotropic, uniform
`and random (IUR) object— probe interactions to ensure the unbias of estimations.
`The statistical behavior of property estimators is also a subject in digital geom—
`etry. But it seems that issues of algorithmic efficiency and multigrid convergence
`became more dominant in digital geometry.
`Both disciplines attempt to solve the same problem, and sometimes they fol-
`low the same principles, and in other cases they apply totally different methods.
`In this paper, a few property estimators of stereology and digital geometry are
`comparatively evaluated, especially according to their multiprobe or multigrid
`convergence behavior, precision and efficiency of estimations.
`From the theoretical point of view, one opportunity for comparison is to study
`how testing probe and resolution affect the accuracy of estimations. VVe define
`multiprobe convergence in stereology analogously to multigrid convergence in
`digital geometry. Both definitions can be generalized to cover not only the es-
`timation of a single property such as length, but also of arbitrary geometric
`properties, including area in 2D, and surface area and volume in 3D. Measure
`ments follow stereological formulas, which connect the measurements obtained
`using different probes with the sought properties. VVe define multiprobe conver-
`gence in stereology analogously to multigrid convergence in digital geometry.
`
`Definition 1. Let Q be the object of interest for estimation, and assume -we
`have a way of obtaining discrete testing probes about Consider’ an estimation
`method E which takes testing probes as input and estimates a geometric property
`of Q. Let XE(Tn,) be the value estimated by
`with input Tn, where Tn
`
`R. Klette and J. Zunic (Eds): lVVClA 2004, LNCS 3322, pp. 42l,—431, 2004.
`CC)Springer—Verlag Berlin Heidelberg 2004
`
`Iibraryresearch staff@\«sileyrein.ccm
`
`FRESENIUS KABI 1034-0010
`
`

`
`422
`
`Y. Huang and R. Klette
`
`contains exactly n > 0 testing probes. The method E is said to be multiprobe
`convergent
`it conuerges as the number n of testing probes tends to
`{i.e., lirnnaoo XE(T,L) ::::::: c. c E R2), and it converges to the true ualue e.,
`c : X (Q)
`
`lines. planes, dis-
`Testing probes are measured at. along or Within points.
`ectors and so forth. The geometric property X might be the length (e.g.,
`perimeter). area. surface area. or volume. and we write L, A, S. or V for these.
`respectively. (Note that the true value X (Q) is typically unknown in applica-
`tions.) The method E can be defined by one of the stereological formulas.
`The grid resolution h of a digital picture is an integer specifying the number of
`grid points per unit. There are different models of digitizing sets Q for analyzing
`them based on digital pictures. Grid squares in 2D or grid cubes in 3D have
`grid points as their centers. The Gauss digitization G is the union of the grid
`squares (or grid cubes) Whose center points are in Q. The study of multigrid
`convergence is a common approach in digital geometry
`However. We recall
`the definition:
`
`Definition 2. Let Q be the object of interest for estimation, and assume we haue
`a way of digitizing Q into a 2D or 3D picture Ph using grid resolution h. Consider
`an estimation method E which takes digital pictures as input and estimates a
`geometric property
`of
`Let XE(Ph) be the ualue estimated by E with
`input Ph. The method E
`said to be multigrid convergent ijjt it converges as the
`number h of grid resolution tends to infinity
`limnaoo XE(Ph) : c, c E R2),
`and it converges to the true /value
`c :
`
`For simplicity We assume that our 2D digital pictures Ph, are of size hr x h.
`digitizing always the same unit square of the real plane. Analogously, We have
`h x h x h in the 3D case. An estimation method in digital geometry is typically
`defined by calculating approximative geometric elements (e. straight segments.
`surface patches, surface normals) which can be used for subsequent property
`calculations.
`
`2 Estimators
`
`Perimeter and surface area estimators in stereology and in digital geometry are
`totally different by applied method. ln stereology, for example a line testing probe
`(a set of parallel straight lines i11 2D or 3D space) is used. and the perimeter
`or surface area of an object is approximated by applying stereological formulas
`with the count of <:>bject—line intersections.
`ln digital geometry. estimators for measuring perimeter or surface area can be
`classified as being either local or global (see. e.g..
`[1, 4,
`[10] suggested using
`a fuzzy approach for perimeter and area estimations in gray level pictures. The
`chosen perimeter estimator for our evaluation (for binary pictures) is global and
`based 011 border approximations by subsequent digital straight segments (DSS)
`of maximum length.
`
`Iibraryresearch staff@\«sileyrein.com
`
`FRESENIUS KABI 1034-0011
`
`

`
`A Comparison of Property Estimators in Stereology and Digital Geometry
`
`423
`
`The surface area can be estimated for “reasonable small” 3D objects by us-
`ing time—eflicient local polyhedrizations such as Weighted local configurations
`For higher picture resolutions it is recommended to apply multigrid convergent
`techniques. For example.
`illustrates that local polyhedrizations such as based
`on marching cubes are not multigrid convergent. Global polyhedrization tech-
`niques e.g.. using digital planar segments) or surface—normal based methods
`can be applied to ensure multigrid convergence. Because of the space constraint.
`none of these surface area estimators are covered in this paper. The 2D area and
`volume estimators of stereology and those of digital geometry follow the same
`point—cou11t principle. In stereology. a point probe
`of systematic or random
`points) is placed in 2D or 3D space. the points Within the object are counted.
`and the stereology formulas are applied to estimate the area or volume of an
`object of interest.
`In digital geometry. a common estimator of area is pixel counting. whereas
`that of volume is voxel counting. These two estimators are also a special case
`of point counting in stereology Where a set of pixel or voxel centers is used
`as the point probe. Because the resolution directly relates to the number of
`pixels or voxels in the digital picture. it will consequently affect the precision of
`estimations if the stereological formulas are applied.
`
`2.1
`
`Perimeter and Surface Area
`
`Line intersection count method (LICM) is a common stereology estimator for 2D
`perimeter and 3D surface area. This estimator involves four steps: generate a line
`probe. find object border in a digital picture. count object—probe intersections.
`and apply stereological formulas to obtain an estimation of the property.
`The line probes i11 2D or 3D may be a set of straight or circular lines (see.
`e.g..
`8. 9]). or made up of cycloid arcs. VVe use a set of straight lines as the
`testing probe of both perimeter and surface area estimations.
`In this
`There are many possible Ways of defining a digital line (see. e.g..
`paper. a 2D or 3D straight line in digital pictures is represented as a sequence of
`pixels or voxels Whose coordinates are the closest integer values to the real ones.
`There exist different options for defining borders of objects in 2D or 3D pictures.
`In this paper. in 2D the 8—border of Q is used to observe intersections between
`Q and the line probe. Whereas in 3D the 26—border of Q is used (for definitions
`of 8—border or 26—border see. for example.
`Isotropic. uniform. and random
`(IU object—probe intersections are required for obtaining unbiased estimation
`results. Note that any digital line (which is an 8—curve) intersects an 8—border
`(Which is a 4—curve) if it “passes through”.
`
`Definition 3. A pixel (vessel) visited along a digital line is an intersection iff it
`is (m. 8—bo'rde7“ pizrel [26—b0rde7" Uoxel) of the object Q and successor [along the
`line) is a 7ri07ri—b07"de7" pixel (E05660.
`
`Since the objects of interest are general (not specified as a certain type). they
`may not be IUR in 2D or 3D space. To ensure IUR object—probe intersections.
`either the object or the probe must be IUR. or the combination of both must
`
`Iibraryresearch staff@\«sileyrein.com
`
`FRESENIUS KABI 1034-0012
`
`

`
`424
`
`Y. Huang and R. Klette
`
`;« EC:~¢nsUvvl*3w\s»vxrmoI:i31»Vvs'ué'f.9
`
`"
`
`
`
`Fig. 1. Screen shots of pivot lines of the LICW estimator using line probes of 16 (left)
`and 32 (right) directions
`
`In the experiments we attempt to generate an IUR straight
`be isotropic
`line probe for the stereology estimator (LICM) to produce unbiased results for
`perimeter and surface area measurements.
`
`Perimeter. The perimeter estimators examined are the stereology estimator
`(LlCl\/l) and the digital geometry estimator based on maximum length digital
`straight segments
`The stereology estimator corresponds to the stereological formula
`
`which is basically the calculation of perimeter density (length per unit area) of
`a 2D object. P is the number of intersections between line probe and (border
`of the) object Q. LT is the total length of the testing line probe, and PL is the
`point count density (intersections per unit length). As a. corollary of this7 the
`perimeter of an object can be estimated by multiplying 15,4 by the total area AT
`that the line probe occupies:
`
`£:LA‘AL
`
`7? P
`
`, 2 . LT .
`
`AT
`
`At the beginning of our experiments we consider how the estimation precision
`is influenced by an increase in the number of directions n of line probes. The
`direction of line probes are selected by dividing 180° equally by //7.. For instance,
`if there are 4 directions required then lines of slopes 007 4507 9007 and 135° are
`generated. Figure 1 illustrates the lines for 16 or 32 directions. Note that only
`one “pivot line” per direction is shown in this figure.
`To avoid errors or a bias caused by direction or position of line probes. every
`pivot line of one direction is shifted along the IL’ and y axes by just one pixel.
`Assume that the pivot line which is incident with (07 0) (i.e., the left lower corner
`of a picture) intersects the frontier of the unit square [0, 1] X [0, 1] again at point
`p7 with an Euclidean distance L between (07 0) and p. The total length of a line
`probe in direction 0*‘, 45° or 90“ is HL», and equal to
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`A Comparison of Property Estimators in Stereology and Digital Geometry
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`
`3h — h/tana],
`2
`,J
`
`where oz is the smaller angle between 0°‘ and 45° defined by the pivot line in our
`unit square.
`However. a stereological bias [8] ca11 not be avoided in this case due to the
`preselection of start position and direction of line probes for perimeter estima—
`tion of non—lUR objects. Therefore. we also include an estimator which uses lines
`at random directions. generated using a system function mnd{) (Which is sup-
`posed to generate uniformly distributed numbers in a given interval). Although
`a random number generator is used. the generated line probes are not necessarily
`isotropic in 2D space. (An improved IU R direction generator is left for future
`research.) For the digital geometry estimator DSS. We start at the clockwise
`loWer—leftmost object pixel and segment a path of pixels into subsequent DSSs
`of maximum length. Debled—Rennesson and Reveilles suggested an algorithm in
`[2] for 8—curves. and earlier Kovalevsky suggested one in [6] for 4—curves. In our
`evaluation. both methods have been used. but we only report on the use of the
`second algorithm (as implemented for the experiments reported in
`in this
`paper.
`
`The DSS estimator is multigrid convergent. Whereas the multiprobe conver-
`gent behavior of the LICM estimator depends on the used line probe. The digital
`geometry estimator DSS is time—eflicient because it traces only borders of ob-
`jects. and used one of the linear on—line DSS algorithms.
`The time—efliciency of the stereology estimator LlCl\/l depends on the number
`of line pixels involved. since it checks every pixel in a line probe to see Whether
`there is an intersection. In both implementations of LICM (i.e.. n directions by
`equally dividing i800. and random directions generated by the system function
`mnd()). every pivot line into one direction is translated along 27- and y—axes at
`pixel distance (in horizontal or vertical direction We have a total of h lines; in any
`other direction We have a total of 2}; — 1 lines). which results into multiple tests
`(intersection?) at all pixels just by considering them along different lines. In case
`of pictures of “simple objects” We can improve the efiiciency by checking along
`borders only instead of along all lines. However. normally We can not assume
`that for pictures in applications.
`
`‘We tested speed and multiprobe convergence of the stereology
`Surface Area.
`estimator LICM for surface area measurements. using the stereological formula
`
`where IL, is the density of intersections of objects with the line probe. and this is
`equal to the result of the line intersection count PL divided by the total length
`LT of the line probe.
`Consequently. the surface area of Q can be estimated by multiplying its
`surface density (obtained from the previous relationship) by the total volume of
`the testing space VT (i.e.. hg. which is the occupied volume of a 3D picture).
`§ 1111 SV -
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`Y. Huang and R. Klette
`
`Similar to the stereology estimator of perimeter measurement. a way of cre-
`ating an IUR straight line probe in 3D is required for unbiased surface area
`estimations. The LICM estimator may not be multiprobe convergent because of
`the used line probes. The efficiency of the estimator depends on the total number
`of line voxels.
`
`2.2 Area and Volume
`
`The stereology estimator for both 2D area (3D volume) measurements counts
`21)
`points which are within the object of interest. When the point probes
`used are the centers of all pixels (voxels) of a given picture. then this estimator
`coincides with the method used in digital geometry.
`Basic stereological formulas (see. eg..
`such
`
`are applied for 2D area estimation. The area of an object A can be estimated
`by multiplying the number of incident points P by the area Ap per point. In
`other words. the area occupied by all incident points approximates the area of
`the measured object. The area Ap per point is the total area AT of the picture
`(i.e.. width A3: times height Ay) divided by the total number PT of points of
`probe T.
`The principle of Bonaventura Cavalieri (1598-1647) is suggested for estimat-
`ing the volume of an object Q in 3D space in stereology books such as
`8.
`of
`[8] combines the Cavalieri principle with the point count method. A
`parallel 2D planes is used as test probe. where a set of points is placed into
`each plane to obtain the intersected area of the plane with the object. If there
`are m planes used in the process. the volume of the object V can be estimated
`
`by multiplying the distance 9 between the planes by the sum of the intersected
`areas A1.A2.A3. .
`. ..Am of all planes. V : <9 - (A1 + A-3 +14,-3 + — ~ Am).
`Russ and Dehofl use a set of 3D points as the test probe in
`to estimate the
`volume fraction VV from calculating a point fraction Pp. the ratio of incident
`points P (with the object) to the population of points IJT. VV : Pp : FLT. As
`a corollary of this. the volume V of the object is equal to the product of the
`volume fraction and the total volume occupied by the testing probe VT (i.e.. I23
`in an h x h X h picture). which can be calculated by multiplying the number of
`incident points by the volume per point Vp.
`
`V VV - VT £ - h3 P ~ Vp
`PT
`
`The results of both methods are identical when all the 2D planes are coplanar
`at equal distances. the set of points chosen in every plane is uniform. and the
`interspacing between these 2D points equals to the distance between planes.
`For area and volume estimations. since the estimators in both fields follow the
`same point count principles. the choice of poi11t probes will definitely influence
`the performance of the algorithm. lf a point probe is randomly picked from
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`427
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`A circle with radius DA has
`perimeter £3.52-Pi are area
`0 met.
`
`The lunula has perimeter
`C|.8Pi and area
`a 3va444as=~a+u 194916
`
`
`
`
`7 mg ,m.; Qbjegg has
`pew-eler 2392896 and
`ma 0»33‘i5*4‘*97847»
`
`5% square rotated 45 cegrees A square rotated 22.5 degrees
`The yin part oi yenyarg
`object has p-arimeler see: with width 033 has perimeter with wimh C3! 6 has perimeter
`and 8'98 0»08F‘iy
`2.4 and area C133.
`2,4 am: area 0.36.
`
`Fig. 2. Six pictures suggested as test data in
`
`regular 2D or 3D grid points, and the chosen number of points is less than the
`total number of pixels or voxels in the picture, then the method is (trivially)
`more time—eflicient than the digital geometry approach.
`Area and volume estimators are “very precise”, suggesting a fast multiprobe
`convergence
`experiments i11 Section
`they have been widely used i11 re-
`search and commercial fields. The digital geometry estimators (i.e., considering
`all pixels or voxels) are known [5] to be multigrid convergent
`for particular
`types of convex sets).
`
`3 Evaluation
`
`We tested 2D perimeter and area estimators by using six objects shown in Fig-
`ure 2. For the area of the lunula We used the formula I)
`- 7" — a.(r — h) for the
`area of the “removed” segment of the disk, with arc length b : 27m" - o/360 (us-
`ing integral part at : 139 of the estimate 04 : 1390253698 .
`.
`segment height
`h : 0.26, and a : 2 - x/0.1404. ln
`of 3D objects,