Table Of ContentLecture Notes in Computer Science 7346
CommencedPublicationin1973
FoundingandFormerSeriesEditors:
GerhardGoos,JurisHartmanis,andJanvanLeeuwen
EditorialBoard
DavidHutchison
LancasterUniversity,UK
TakeoKanade
CarnegieMellonUniversity,Pittsburgh,PA,USA
JosefKittler
UniversityofSurrey,Guildford,UK
JonM.Kleinberg
CornellUniversity,Ithaca,NY,USA
AlfredKobsa
UniversityofCalifornia,Irvine,CA,USA
FriedemannMattern
ETHZurich,Switzerland
JohnC.Mitchell
StanfordUniversity,CA,USA
MoniNaor
WeizmannInstituteofScience,Rehovot,Israel
OscarNierstrasz
UniversityofBern,Switzerland
C.PanduRangan
IndianInstituteofTechnology,Madras,India
BernhardSteffen
TUDortmundUniversity,Germany
MadhuSudan
MicrosoftResearch,Cambridge,MA,USA
DemetriTerzopoulos
UniversityofCalifornia,LosAngeles,CA,USA
DougTygar
UniversityofCalifornia,Berkeley,CA,USA
GerhardWeikum
MaxPlanckInstituteforInformatics,Saarbruecken,Germany
Ullrich Köthe Annick Montanvert
Pierre Soille (Eds.)
Applications of
Discrete Geometry and
Mathematical Morphology
First International Workshop, WADGMM 2010
Istanbul, Turkey, August 22, 2010
Revised Selected Papers
1 3
VolumeEditors
UllrichKöthe
UniversityofHeidelberg
HeidelbergCollaboratoryforImageProcessing
SpeyererStrasse6,69115Heidelberg,Germany
E-mail:[email protected]
AnnickMontanvert
GIPSA-lab
961,ruedelaHouilleBlanche
38402SaintMartind’Hèrescedex,France
E-mail:[email protected]
PierreSoille
EuropeanCommission
JointResearchCentre
ViaE.Fermi,2749
21027Ispra(Va),Italy
E-mail:[email protected]
ISSN0302-9743 e-ISSN1611-3349
ISBN978-3-642-32312-6 e-ISBN978-3-642-32313-3
DOI10.1007/978-3-642-32313-3
SpringerHeidelbergDordrechtLondonNewYork
LibraryofCongressControlNumber:2012943171
CRSubjectClassification(1998):I.4,I.2.10,I.3.5,I.5,H.2-3,J.3
LNCSSublibrary:SL6–ImageProcessing,ComputerVision,PatternRecognition,
andGraphics
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Preface
Discrete geometry and mathematical morphology play essential roles in image
analysis,computergraphics,patternrecognition,shapemodeling,andcomputer
vision because they acknowledge, from the outset, the inherently discrete na-
ture of the data to be processed and thus provide theoretical sound, yet com-
putationally efficient frameworks for digital image analysis in two and higher
dimensional spaces. Important recent results include methods for the represen-
tation and analysis of topological maps, fast algorithms for three-dimensional
skeletons, topological watersheds, connected filters, and hierarchical image seg-
mentation, as well as application-specific ones in areas such as remote sensing,
medical imaging, and document analysis.
ThesuccessfulDGCIconferenceseries(“DiscreteGeometryforComputerIm-
agery”)hasbecome the mainforumfor expertsin the field ofdiscretegeometry.
However,nocorrespondingforumfortheexchangeofideasbetweenexpertsand
potential users existed to date. The same applies to mathematical morphology
where the main conference ISMM (“International Symposium on Mathematical
Morphology”)is mainly targeted at experts in a similar way.
The First Workshop on Applications of Discrete Geometry and Mathemat-
ical Morphology (WADGMM 2010) was held at the International Conference
on Pattern Recognition (ICPR) 2010 in Istanbul in order to close this gap. It
was specifically designed to promote interchange and collaboration between ex-
pertsindiscretegeometry/mathematicalmorphologyandpotentialusersofthese
methodsfromotherfieldsofimageanalysisandpatternrecognition.Itcomprised
four invited review talks by leading researchers in the field and 18 poster pre-
sentationsofnew researchresultsthathadbeenselectedamong25submissions.
This volume presents greatly enhanced and thoroughly reviewed versions of se-
lected contributions that nicely reflect the diversity of discrete geometry and
mathematicalmorphologyapplicationsand offer,as we hope,a varietyof useful
tools for the image analysis practitioner.
The workshop was organized by the Technical Committee 18 on Discrete
Geometry of the InternationalAssociationfor PatternRecognition (IAPR). We
would like to take the opportunity to thank IAPR for their continuing sup-
port of our activities. We are also very grateful to the Organizing and Program
Committees, who did a great job in making WADGMM 2010 a success. Last
but not least, many thanks go to the authors and to the invited speakers (Peer
Stelldinger, David Coeurjolly, Jacques-Olivier Lachaud, Laurent Najman, and
Pierre Soille) who kindly accepted our invitation to present their work at the
workshopand in this extended volume.
June 2012 Ullrich K¨othe
Annick Montanvert
Pierre Soille
Organization
Scientific Committee
Ullrich K¨othe University of Heidelberg, Germany
Annick Montanvert GIPSA-Lab Grenoble, France
Pierre Soille JRC Ispra, Italy
Organizing Committee
Joost Batenburg University of Antwerp, Belgium
Guillaume Damiand LIRIS Lyon, France
Georgios Ouzounis JRC Ispra, Italy
Yukiko Kenmochi CNRS Marne-la-Vall´ee,France
Program Committee
Isabelle Bloch ENST, Paris, France
Gunilla Borgefors CBA, Uppsala, Sweden
Srecko Brlek LaCIM, Monteal, Canada
Jacopo Grazzini Los Alamos National Lab, USA
Atsushi Imiya Chiba University, Japan
Ingela Nystr¨om CBA, Uppsala, Sweden
Jos Roerdink University Groningen, The Netherlands
Christian Ronse University of Strasbourg,France
Philippe Salembier UPC, Barcelona,Spain
Gabriella Sanniti di Baja Istituto di Cibernetica, Naples, Italy
Peer Stelldinger University of Hamburg, Germany
Robin Strand CBA, Uppsala, Sweden
Akihiko Sugimoto NII, Tokyo, Japan
Peter Veelaert HogeschoolGent, Belgium
Michael Wilkinson University of Groningen, The Netherlands
Table of Contents
Connect the Dots: The Reconstruction of Region Boundaries from
Contour Sampling Points ......................................... 1
Peer Stelldinger
Digital Shape Analysis with Maximal Segments...................... 14
Jacques-Olivier Lachaud
Discrete Curvature Estimation Methods for Triangulated Surfaces...... 28
Mohammed Mostefa Mesmoudi, Leila De Floriani, and Paola Magillo
On Morphological Hierarchical Representations for Image Processing
and Spatial Data Clustering....................................... 43
Pierre Soille and Laurent Najman
Radial Moment Invariants for Attribute Filtering in 3D ............... 68
Fred N. Kiwanuka and Michael H.F. Wilkinson
Volumetric Analysis of Digital Objects Using Distance Transformation:
Performance Issues and Extensions................................. 82
David Coeurjolly
Geometric Analysis of 3D Electron Microscopy Data.................. 93
Ullrich K¨othe, Bj¨orn Andres, Thorben Kr¨oger, and Fred Hamprecht
Machine Learning as a PreprocessingPhase in Discrete Tomography.... 109
Miha´ly Gara, Tama´s S´amuel Tasi, and P´eter Bala´zs
Fast Planarity Estimation and Region Growing on GPU .............. 125
Micha¨el Heyvaert and Peter Veelaert
Writing Reusable Digital Topology Algorithms in a Generic Image
Processing Framework............................................ 140
Roland Levillain, Thierry G´eraud, and Laurent Najman
A New Image-Mining Technique for Automation of Parkinson’sDisease
Research........................................................ 154
Igor Gurevich, Artem Myagkov, and Vera Yashina
Author Index.................................................. 169
Connect the Dots: The Reconstruction of Region
Boundaries from Contour Sampling Points
Peer Stelldinger
International Computer ScienceInstitute(ICSI)
Berkeley, USA
[email protected]
Abstract. Twodimensional contourreconstruction from aset ofpoints
isaverycommonproblemnotonlyincomputervision.I.e.ingraphthe-
ory onemay ask for theminimal spanningtree ortheshortest Hamilto-
niangraph.Inpsychologythequestionarisesunderwhichcircumstances
people are able to recognize certain contours given only a few points.
In the context of discrete geometry, there exist a lot of algorithms for
2D contour reconstruction from sampling points. Here a commonly ad-
dressedproblemistodefineanalgorithmforwhichitcanbeprovedthat
the reconstuction result resembles the original contour if this has been
sampled according to certain density criteria. Most of these algorithms
can not properly deal with background noise like humans can do. This
paper gives an overview of the most important algorithms for contour
reconstructionandshowsthatarelativelynewalgorithm,called‘cleaned
refinement reduction’ is the most robust one with regard to significant
background noise and even shows a reconstruction ability being similar
to theone of a child at theage of 4.
1 Perceptually Meaningful Shape Reconstruction from
Point Sets
Twodimensionalcontourreconstructionfromasetofpointsisaveryoldproblem
not only in computer vision. I.e. in graph theory one may ask for the miniml
spanning tree or the shortest Hamiltonian graph. In Gestalt psychology the
question arises under which circumstances people are able to recognize certain
contours given only a few points, see e.g. [5, 18]. Also in computer science, the
probem of reconstructing some contours from a given 2D point set has a long
history. Algorithms for connecting points have early been proposed for specific
taskslikefinding theEuclideanminimumspanningtree(EMST)orthe shortest
round tour (TSP).
While these problems are easy to define (although not always easy to solve),
it is more complicated to define what kind of graph drawings are perceptually
meaningful.
The human ability to reconstruct curves given some set of points is remark-
able.E.g.considerthe widely known‘connectthe dots’drawings,whereonehas
to draw a picture by connecting some givennumbereddots in the correctorder.
U.K¨othe,A.Montanvert,andP.Soille(Eds.):WADGMM2010,LNCS7346,pp.1–13,2012.
(cid:2)c Springer-VerlagBerlinHeidelberg2012
2 P. Stelldinger
Such games often have the goal to practice the number reading ability of chil-
dren.Neverthelessinmostcasesitisquite obviousforahumantofindthe right
way to connect the points without looking at the numbers at all. One example
is givenin fig. 1, a drawingsolvedby a 4 years oldboy who had not yet learned
toreadthe numbers.Ifthe pointsareappropratelyalignedonesimply ‘sees’the
correct solution - even if no further information is given.
a) b)
c) d)
Fig.1. a) A Connect the Dots drawing of a 4 year old boy. Note that the points are
correctly connectedalthoughtheboywasnotabletoread thenumbersatthisage.b)
A point set with random background noise, c) the result when the boy was asked to
connect the dots, d) theoutput of the cleaned refinement reduction algorithm.
Humans can easily find meaningful structures in point sets even if they do
not know if the solution has to be open or closed, connected or disconnected,
branchedornot, orif they haveto use all givenpoints oronly a subsetofthem.
All these decisions can be made purely by looking at the points themselves.
E.g. being asked to ‘connect the dots’ shown in fig. 1b (without any further
Connect theDots: The Reconstruction of Region Boundaries 3
instructions) the same 4 year old boy easily found the house which had been
hidden in the data, see fig. 1c. Note, that he autonomously decided to use more
than one simple line and to connect only a subset of the points1.
Asecondexampleisshowninfig.2:Givenanoisypointseta),Eventhemost
common curve reconstruction algorithms fail, as shown for the crust algorithm
[3]inc)whilehumanscaneasilyfindagoodsolution.Whenaskingsomeonewho
isnotanexpertinthetopichowonecanfindasolutionasgivenind),hereplied
‘Itis obvious,can’tyousee it?’ After askingfora moredetaileddescription(i.e.
an algorithm) on how to connect the edges he replied ‘You just have to connect
each point on both sides with its nearest neighbor’. b) shows the graph which
onegetsbyconnectingeachpointwithitstwonearestneighborsintheDelaunay
graph (see below or a definition). This illustrates the hidden complexity of this
problem.
a) b)
c) d)
Fig.2. Boundary reconstruction of a point set: a) point set, b) two nearest Delaunay
neighbors, c) crust algorithm [3], d) perceivingly correct reconstruction
Since human observers can easily extract perceptually meaningful structures
in point sets, researchers tried to understand how structures in point patterns
are perceived and tried to find algorithms which are able to do the same [13].
Well-known graph structures have originally been introduced in this context,
e.g. the relative neighbourhood graph was proposed in [17] as a graph being
perceptually more meaningful than the Euclidean minimum spanning tree and
the Delaunay triangulation. Similar structures like the Gabriel graph and other
proximity graphs have also been discussed [10].
1 When being asked why he did not usethe otherpoints, hereplied ‘these are stars’.
4 P. Stelldinger
Definition 1. Given a point set S ∈ R2, the Voronoi diagram is the partition
of R2 into regions such that each point s ∈ S is accociated with the region of
all points in R2 for which s is the nearest of all point in S. These regions are
called Voronoi regions. The Delaunay triangulation (DT) is the unique straight
line graph which one gets by connecting any two points of S with an edge if
their Voronoi regions are adjacent to each other. The Gabrielgraph(GG) is the
uniquestraight linegraph which onegets byconnectinganytwopoints of S if the
smallest circle going through them does not enclose or touch any other point of
S. The relativeneighborhoodgraph(RNG) is the unique straight line graph one
gets by connecting any two points of S if the intersection of the two circles being
centered in one of the two points and going through the other does not enclose
or tough any other point in S. Moreover the Euclidean minimal spanning tree
(EMST) is the tree of smallest overall length connecting exactly the points of S.
It is well known that the EMST is a subgraph of the RNG, the RNG is a
subgraph of the GG and the GG is a subgraph of the DT. It is a common
concepttorestrictthesearchofagoodreconstructiontoedgesbeingpartofthe
Delaunaytriangulation.Basicallyallalgorithmsbeingdescribedinthefollowing
use only Delaunay edges.
O’Rourke et al. define the so-called minimal spanning Voronoi tree without
proving its existence and approximate it by a heuristic algorithm which shows
somekindofanaturalbehaviourinaperceptualsense[12]. However,since they
basically try to find a simple closed polygon connecting all points and being
minimal in some sense, their result can be seen as a simple heuristic for solving
the travelling salesmen problem (TSP). Indeed, the examples presented in [12]
show visually pleasing polygons which are all also optimal in the sense of the
TSP.In[1]ithasbeenproventhattheTSPcansuccessfullybeappliedtorecon-
structingasufficiently dense sampledsimple closedcurvebyusingapolynomial
time algorithm. Unfortunately such an approach is obviously restricted to the
task of finding one simple closed curve connecting all given points.
2 Provably Correct Shape Reconstruction from Point
Sets
With the definition ofalpha-shapes,Edelsbrunnerintroducedascale-dependent
concept for reconstructing not only thin structures but also planar regions in
a perceptual meaningful way given an unordered set of points [9]. The idea is
that the alignment of the points themselves determines the intrinsic local di-
mension of the reconstruction. While originally been introduced for defining
perceptually meaningful shapes for a point set, the sound underlying theory
led to the derivation of several mathematically justified approaches for shape
reconstruction. E.g. alpha-shapes have been used by Bernardini and Bajaj for
reconstructingsufficientlydensesampledsmoothboundariesofshapeswithcor-
rectnessguarantees[6].Togetherwiththe reconstructionalgorithmproposedby
Attali [4], this was the beginning of a paradigm shift from perceptual justifica-
tion to the reconstruction of object boundaries from a set of points originating
