HindawiPublishingCorporation
InternationalJournalofBiomedicalImaging
Volume2011, ArticleID516942, 13pages
doi:10. 1155/2011/516942
ResearchArticle
3D WindingNumber:TheoryandApplicationtoMedicalImaging
AlessandroBecciu,
1  AndreaFuster, MarkPottek, BartvandenHeuvel,
1 2 1
andHansvanAssen
1
BartterHaarRo meny,
1
1
2
DepartmentofBiomedicalEngineering, EindhovenUniversityofTechnology, 5600 MBEindhoven, TheNetherlands
DepartmentofBiology, KaiserslauternUniversityofTechnology, 67653Kaiserslautern, Germany
CorrespondenceshouldbeaddressedtoAndreaFuster,a.fuster@tue. nl
Received2June2010; Revised20September2010; Accepted1 November2010
AcademicEditor:YangboYe
Copyright(c)2011AlessandroBecciuetal. Thisis anopen access article distributed under theCreative Co m monsAttribution
License, whichpermitsunrestricteduse, distribution,andreproductioninanymedium, providedtheoriginalworkisproperly
cited.
Wedevelopanewformulation, mathe maticallyelegant,todetectcriticalpointsof3Dscalarimages.Itisbasedonatopological
nu mber, whichisthegeneralizationtothreedimensionsofthe2Dwindingnumber. Weillustrateourmethodbyconsideringthree
dif f erentbiomedicalapplications, namely, detectionandcountingofovarianfolliclesandneuronalcellsandestimationofcardiac
motionfromtaggedMRi mages. Qualitativeandquantitativeevaluationemphasizesthereliabilityoftheresults.
1.Introduction  fromthesamepit/peak. Thesemethodshavebeenextensively
employed for 2-di mensional applications. In recent years,
therehasbeenastrongincreaseofcomputationalpower, and
3Dscalarimagesarebecomingthestandarddataofinves-
tigation, especially in medical imaging. Three-dimensional
critical point techniquesallowfora more realistic analysis
ofhumanorganbehavior. Forexample,trackingalgorithms
applied on a 2-di mensional heart i mage sequence retrieve
onlyin-planecontractionsandrotationsofthecardiacwalls
butmissthethrough-planeco mponents. Thethrough-plane
co mponentsareinsteadretrievedwith3-dimensionaloptic
fl owapproaches.Inthispaper, weshowanapplicationwhere
thepresentedcriticalpointdetectionalgorithmisembedded
inafeaturepoint-basedmotionestimationtechnique.
Inthispaper, weworkwithatopologicalnumber(from
ho motopy theory) that can locate critical points of scalar
imagesinanarbitrarynumberofdi mensions.Intwodi men-
sions, it reducesto the so-called winding nu mber and has
beenstudiedindetailin[13–15].Inphysics,andinmodern
cosmologyinparticular,thewindingnumberappearsinthe
context of topological defects such as monopoles, cosmic
strings, and do main walls (see, e. g., [16] and references
therein). We consider this topological nu mber in three
di mensionsandrefertoitas3Dwindingnu mber. Properties
ofthisapproacharesigni cant. f i
Critical points are very helpful for dif f erent purposes and
applications in co mputer vision as key points, land mark
points, anchor points, and others. In segmentation, for
example, critical points have been used to characterize
deformingareas ofthebrain[1]ortoenhance ridgesand
valleys in MR i mages [2]. In i mage matching, mappings
between the considered i mages are computed based on
their critical points [3, 4]. I mage matching has been also
performedthroughtheso-calledtoppoints,criticalpointsfor
whichthedeterminantoftheHessianmatrixisequaltozero
[5, 6], orthroughthepopularHarrispoints[7]andtheSIFT
keypointdetector[8]. Criticalpointshavealsobeenusedin
motionestimationalgorithms,wheretheoptic ow eldis f l f i
generatedfromasparsesetofvelocitiesassociatedtomulti-
scaleanchorpoints[9, 10].
Critical pointdetectionisanestablishedresearch  eld. f i
Blom[11], for example, classi es critical points by count f i -
ing the sign changes between the analyzed pixels and its
neighbors in a hexagonal grid. Nackman [12] de nes the f i
image topology in terms of slope districts. The ridge and
valley lines are described as the ascending and descending
slopes co ming from saddle points. The dales and hills are
identi edasdistrictswhoselinesofslopeconvergeto/come f i