Topology preserving warping of binary images: Application to atlas-based skull segmentation

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Topology preserving warping of binary images:

Application to atlas-based skull segmentation

Sylvain Faisan, Nicolas Passat, Vincent Noblet, Renée Chabrier, Christophe Meyer

To cite this version:

Sylvain Faisan, Nicolas Passat, Vincent Noblet, Renée Chabrier, Christophe Meyer. Topology pre-

serving warping of binary images: Application to atlas-based skull segmentation. Medical Image

Computing and Computer-Assisted Intervention (MICCAI), 2008, New York, United States. pp.211-

218, �10.1007/978-3-540-85988-8_26�. �hal-01695012�

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Appliation to atlas-based skull segmentation

SylvainFaisan

1

,NiolasPassat

1

,VinentNoblet

1

,RenéeChabrier

2

,and

ChristopheMeyer

3

1

LSIIT-UMRCNRS7005, StrasbourgIUniversity,Frane

2

LINC-UMRCNRS7191,StrasbourgIUniversity,Frane

3

UniversityHospitalofBesançon,Frane

Abstrat. Lots ofworkshave beenreentlyarriedout inthe eldof

non-rigidregistration toensuretheestimationof one-to-onemappings.

However, warping a binaryimagewith suhtransformationsmay alter

itsdisrete topologial properties if ommon resampling strategies are

onsidered. This paper proposes an original method for warping a bi-

naryimage aordingto someontinuousandbijetive mapping,while

preservingitsdisretetopologialproperties.Resultsobtainedintheon-

textofatlas-basedsegmentationhighlight theinterestofthe approah.

Indeed,themethodhasbeensuessfullyappliedtothesegmentationof

skullstruturesfromadatabaseof15^CT-sans,^providing^both^geomet-

riallyandtopologiallysatisfatoryresults.

1 Introdution

Imagewarpingis theproess ofapplying somegeometri transformationto an

image. Given an image M ând â ôntinuous deformation eld h^, ^the ^goal îs

to ompute the warped image S^, ^so ^that ^for ^eah ^voxel v^, S(v) = M(h(v))^.

Sine h(v)^does^not^neessarily^orrespond^with ^grid^point, ^someinterpolation tehniquesarerequiredtoevaluateM(h(v))^.

Althoughseveralimageinterpolationtehniques(linear,ubi)[1℄havebeen

proposed forgrey-levelimages, nospei attentionhas been paid to thease

ofbinarydata.Commoninterpolationtehniques,exeptthenearestneighbour

interpolation, donotguaranteetheresampled imageS ^to^remain ^a^binary^im-

age.Toirumventthis limitation,it is possibleto use athresholdingas post-

proessingofinterpolationtogetabinaryimage.Unfortunately,warpingadis-

rete image aordingto a ontinuous and bijetive ( i.e. topology-preserving)

deformationeld withthese ommon interpolation tehniques may fail in pre-

serving itsdisrete topologialproperties. Quitesurprisingly,manyworkshave

beendevotedtodevelopregistrationmethodsprovidingdeformationeldswhih

preservetheontinuoustopology,whilethetopologypreservationofdisreteob-

jetsdeformed bysuh elds hasnotyet beenonsidered.Based ontheseon-

siderations,weproposeanalgorithmforwarpingabinaryimageaordingtoa

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ontextofsegmentation,wheretopologypreservationisaruialissue.Theseg-

mentationmethods dealingwith thisonstraintare oftenbasedontheonept

of 3^-D ^simple ^points ^[2℄ ^{( i.e.} ^points ^whose âddition ôr ^removal^from â ^binary

objetdoesnot alter its topology),whih an be loally haraterised in on-

stant time, leadingto fast algorithms. The basiideaof the proposed method

is to modifythe initialimage in ahomotopy-preservingfashion byadding and

removingsimplepointsuntilonvergingtoasolutionthatisasloseaspossible

totheontinuouswarpedimage.

The paper is organised as follows. In Setion 2, the proposed method is

desribed. In Setion 3,results in the ontext of atlas-based segmentation are

presented.ConlusionsandperspetivesareprovidedinSetion 4.

2 Method

ThemethodofwarpingabinaryimageM âording^to âôntinuousând^bije-

tivedeformationeldhân ^be^statedâs^the^followingônstrainedoptimisation problem:

Sˆ=arg min

S∼Md(S, M, h), ⁽¹⁾

where S îsâ^binary îmageônstrained^to^betopologiallyequivalentto M ând d(S, M, h)â^distane ^betweenS ând ^theôntinuous^warpedîmage M(h)^. ^This

paperpresentsa method to taklethis problem. We rstintroduein Se. 2.1

the distane d(S, M, h)^. ^Then, ^we ^explain ⁱⁿ ^Se. ^2.2 ^how ^to ^onstrain S ^to

betopologiallyequivalentto M ^during ^the optimisationproess. Finally, the optimisationstrategyisdetailedinSe.2.3.Aglobaloverviewofthemethodis

givenin Alg.1.

2.1 Costfuntion

Sine M ând S âre ônstrained ^to ^have ^the ^same ^topology, ^there îs â ône-to-

one relation between onneted omponents (CCs) of M ând ^the ônes ôf S^.

These CCs an be bakground CCs (BCCs) or objet CCs (OCCs), eah CC

orrespondingtoadistintlabel.WedeneN(v, S, M)^as ^the^CCⁱⁿ M ^whih

orresponds to theCC in S ^that ^enloses^voxelv^. ^The^distane d(S, M, h) ^be-

tweenS ând ^theôntinuous^warpedîmageM(h)îs ônsidered^hereafterâs ^the

ostfuntion,andisomputedasfollows:

d(S, M, h) =X

v∈S

ρ(v, S, M, h), ^with ρ(v, S, M, h) = min

v^′∈N(v,S,M)kv^′−h(v)k ,

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where ρ(v, S, M, h) îs ^the ^distane ^between h(v) ând ^the ^CC ôf M ^whih îs

assoiatedto theCCthat enlosesvⁱⁿ S^.^To^larify^the^idea,^the^omputation

of the ost funtion is illustrated in a 2^-D ^ase ⁱⁿ ^Fig. ^1. ρ(v, S, M, h) ^an ^be

eiently evaluated by omputing the hamfer distane map of the CC of M

(the oneassoiatedtotheCCthatenlosesv ⁱⁿS⁾ând^byêvaluatingîts^value

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v₁ v₂ v₃

v₄ v₅ v₆ M S

first object connected component

second object connected component

background connected component

a

b c

Fig.1.Illustrationoftheomputationofρ(v, S, M, h)^for ⁶^pixels.M îsômposedôf

twoOCCsand ofoneBCC. ρ(v, S, M, h)îs êqual^to0^for v2^,v3^,ând v5 ^sineh(v2)^, h(v3)^,ând h(v5) ^belong ^to ^the^same ^CCâs v2^,v3^, ândv5^, respetively. However, v6

belongsto theseond OCCwhereash(v⁶) ^belongs^to^the ^BCC^so ^thatρ(v⁶, S, M, h)

isequaltob^,^namely,^the^distane^betweenh(v6)ând^the^seondÔCC⁽ⁱⁿM^).În^the

sameway,ρ(v¹, S, M, h) =a^andρ(v⁴, S, M, h) =c^.

at position h(v)^. ^Notie ^that ^the ^hamfer ^distane ^map ân ^beômputed ône

timeforeah CC.

2.2 Topology handling

Atthe beginning ofthe method, S îs initializedto M ândîs ^then ^modied^by

iterativeremoval/additionofsimplepoints.Thelabelofasimplepointishanged

ifit dereasesthe ostfuntion.Whenhangingthelabel ofv^, îtîs âssoiated

unambiguouslytoauniqueCC,sineitisasimplepoint.TodeterminethisCC,

twoimagesrepresentingthelabelsofS âreûsed. ^Theyâre ûpdated^during ^the

wholeproesswithS^.

Theremoval/additionofsimplepointsanbenotappropriatedwhenavoxel

hastobetranslated.Thetranslationanbeinterpretedintermsofanaddition

(resp.aremoval)followedbyaremoval(resp.anaddition)ofsimplepoints.How-

ever,theostfuntion isestimatedaftereahlabelmodiation.Consequently,

therstmodiationmayinreasetheostfuntion,leadingtorefusethisoper-

ation, whereasbothmodiationsmaydereasetheostfuntion.That iswhy

the onept of topology-preservingtranslation is dened. This notionis inter-

pretedhereas thesimultaneous modiationofthestatusofaouple(v, v^′)^of

adjaentvoxelssuhthatS(v) = 1−S(v^′)^.^To^guarantee^topologypreservation, itissuienttohekthatv ^(resp.v^′⁾îs^simple^forS ândv^′ ^(resp.v⁾îs^simple

in S^′ ôbtained^from S âfter ^the^modiationôfv ^(resp.v^′^).^The translationat voxelv îs ^performedîfît âtually^redues ^theôst ^funtion.Îf^the ^pointv ân

betranslatedin dierentways,thetranslationwhihminimisesatbesttheost

funtion ishosen.

2.3 Optimisationstrategy

Thepurposeoftheoptimisationstrategyistoreahtheminimalvalueoftheost

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aspossible-geometriallysimilartotheontinuousdeformedimageM◦h^.^The

seletion of simple points to remove/add requires a list L ^whih ^ontains ^all

simplepointsof S ^presentingâ^positiveôst. ^Theôstîs^dened âs^the^benet

to hange thelabelat simplepoint v^. ^More^preisely, ^the^modiation^ofS(v)

enablestodereasetheostfuntion fromtheostc(v, S, M, h)^:

c(v, S, M, h) =d(S, M, h)−d(S^′, M, h) =ρ(v, S, M, h)−ρ(v, S^′, M, h), ⁽³⁾

whereS^′ îs^theîmageôbtained^fromS ^by^modifying^the^valueât v^.

During the dynamial sheme, when modifying a simple point v ⁱⁿ S ^to

obtain a new image S^′^, ^there îs ^no ^need ^to ^reompute ^the ^whole ^list L ^sine (i)c(v^′, S, M, h) =c(v^′, S^′, M, h)^forâll^voxelsv^′6=v^,ând (ii)^simple^property

ofpointsanonlybemodiedinthe26-neighbourhoodofv^. Consequently,the algorithmproeedsasfollowsuntilLîsêmpty.^The^pointôf^highestôst,^denoted v0^, îs ^removed ^from ^the ^list. ^The ^label ôf S ât v0 îs ^then ^modied. ^This ^may

hangethesimplepointswhihareinthe26-neighbourhoodofv0^:^points^whih

werenotsimple(resp.simple)andwhihbeomesimple(resp.non-simple)must

beaddediftheyhaveapositiveost(resp.removed)in(resp.from)L^.

When ρ(v, S, M, h) = 0^, ^the ^voxel v ^belongs ^to ^the ôrret ^CC. Â ôst ρ(v, S, M, h) > 0 ân ^result ^from ^the ^fat ^that h(v) îs ât ^the înterfae ôf ôb-

jets (h ^being â ôntinuous êld) ôr ^from ^topologial onstraints. However, it mayalsoresultfromtheonvergeneofthemethodtoaloalminimum.Todeal

with thisissue, wehekforallvoxelsv ^verifyingρ(v, S, M, h)>0 îfîtîs ^pos-

sible to translatev ^to ^redue ^the^ost ^funtion ^without ^topology^modiation.

Itmayhappenthat atranslationgeneratesnewsimplepointsintheneighbour-

hoodoftheinvolvedpoints,enablingtokeepdeformingtheurrentimageS ^by

lassial simplepointmodiation.

Inorder to avoid onvergene onto loal minima (resulting from geometri-

al or topologialonstraints) whih an appearwith large displaements, the

deformationisperformedinasmooth waybyonsideringN+ 1intermediate deformationeldsomputedfromh^,^namelyh⁽⁰⁾^,h⁽¹⁾^,. . .^,h^(N⁾^suh^that:

h⁽⁰⁾=Id, h^(N)=h (i)

∀j∈[0, N−1], ∀v∈S, kh^(j+1)(v)−h^(j)(v)k<1 (ii) ⁽⁴⁾

Constraint (ii) ^provides ^a ^lower ^bound ^for N^: N ≥ maxv∈Skh(v)−vk ^. ^The

deformationeldsh⁽ⁱ⁾ ⁽0< i < N⁾^are^nally ^dened^by:

∀v∈S, h⁽ⁱ⁾(v) =v+ i

N(h(v)−v). ⁽⁵⁾

Theoptimisationsheme just desribedis ahievedbyonsidering sequentially

h⁽¹⁾^,h⁽²⁾^,. . .^,h^(N)^.^Theâlgorithm îs^nally^desribedⁱⁿÂlg.^1.

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Input:M ^(binaryîmage^to^warpâording^toh^),h(transformationeld) Output:S^(warped^binaryîmage)

S=M

(h⁽ⁱ⁾)^Ni=1=transformationeldsobtainedfromh

forh^∗⁼h⁽¹⁾^toh⁽^N⁾^do

L=^list^of^simple^points^with^positive^ostc(v, S, M, h)ⁱⁿS

repeat

whileL 6=∅^do

v=^point^of^highest^ostⁱⁿL

S(v) = 1−S(v)^/*^swith^the^label^ofvⁱⁿS^*/

UpdateL^onsidering^the^new^status^of^pointsⁱⁿ^theneighbourhoodofv

endwhile

forallvoxelsv^verifyingρ(v, S, M, h)>0^and^whih^an^be^translated^withv^′ ^by^reduing

theostfuntion do

(S(v), S(v^′)) = (1−S(v),1−S(v^′))^/*^perform^thetranslation*/

UpdateL^onsidering^the^new^status^of^pointsⁱⁿ^theneighbourhoodofv, v^′

endfor

untilL=∅

endfor

3 Appliation: skull segmentation from CT san data

3.1 Experiments

Oneimportant appliationof theproposed approah onernsatlas-based seg-

mentation [3℄. Suh methods rely on a binary model M ^of ^the ^strutures ^of

interestwhih hasbeenobtainedfrom thesegmentationof animage R^.^When

searhing the struturesof interest in anew imageI^, ^the ^rst^step ^onsistsⁱⁿ

estimatingadeformationeldh^byregisteringRôntoI^.^The^struturesôfînter-

est in I^, ^denoted S^,âre ^thenôbtained^bytransformingM âording^to h^.^The

binary imagetopology-preservingdeformationisalso ofgreat interestsinewe

guaranteethat M ândS ^have^the^same^topology.În^the^sequel, ^the^method îs

proposedforthesegmentationofskull struturesfromCT-sandata.

Theproposed strategyonsistsin deforming apre-proessedskull template

assoiatedtoarefereneCTimageR^.^This^template^models^the^parts^of^the^skull

whihhavetobesegmentedinageometriallyandtopologiallyorretfashion.

Inpartiular,itisomposedofoneonnetedomponent,andhasnoavitybut

tenholesorrespondingtotheforamenmagnum,thezygomatiarhes,et.(see

Fig.2). Thistemplate isatually thebinaryimage M ^whih ^has^to ^be^warped

aordingtoa3^-D deformationeldh^estimated^byregisteringR^onto^the^CT

imageI ^to^be^segmented.

3.2 Results

Theeienyofthemethodisnotevaluatedintermsofsegmentationauray

sineitlargelydependsonthepreisionoftheestimateddeformationeld.The

goalof thisexperiment isto validatethe proposed method (P M⁾ ^and^to ^show

the benet of the approah with omparison to other interpolation methods,

namely, thenearestneighbour interpolation(M1^), ^and^the ^linearinterpolation

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Right:templatevisualisedwithitstopologialskeleton.Ithastobenotiedthat this

isapartial template:struturessuhasthevertebrae,for example,are notmodelled

sinetheirsegmentationisnotrequiredhere.

followed by a thresholding with a threshold of 0.5 (M2^). ^These ^methods ^are

omparedfrom twopointsofview,atopologialoneandageometrialone.

From a topologial point of view, the average number of OCCs (see b0 ⁱⁿ

Tab.1), ofholes (b1^),ând ôf âvities⁽b2^=numberôf ^BCCs −1⁾âre ômputed

for the15transported segmentationmaps obtainedforeah method. Thepro-

posedmethod istheonlyone guaranteeingtopologypreservation: thetopology

is strongly altered by methods M1 ând M2 ^(leading ^to ônneted ômponent

splitting,holeandavitygenerations,et.). Forexample,methodM1^generates

on averagea number of 150 undesired avities in the segmentation result. To

illustrate this point, Fig. 3 presents a typial result for the proposed method

(right)andfortheM2 ^method^(left).

Fromageometrialpointofview,thesegmentationmapsobtainedwiththe

proposed approah (Fig. 3,right)issatisfatory sinesimilar to the resultob-

erroneous holes

Fig.3.SegmentationresultsobtainedwithmethodM2^(left)^and^the^proposed^method

(right).Thetopologyhasbeenalteredintheleftimage,butpreservedintherightone.

Notethat thesurfaes are visuallynoisy sinethe real disreteresults are visualised

herewithoutmeshgeneration.

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P1 P2 P3 P4 P5 d dmax b0 b1 b2

M1 61.87 30.43 7.39 0.31 0.00 3.70.10⁻³ 0.87 1.33±0.59 109±51.0 150±71.0 M2 62.18 37.82 0.00 0.00 0.00 3.44.10⁻³ 0.50 1.06±0.25 24.8±8.00 7.20±2.80 P M 62.19 37.77 0.04 10⁻³ 10⁻⁴ 3.44.10⁻³ 1.06 1.00±0.00 10.0±0.00 0.00±0.00

Table 1. Comparisonof theproposed method(P M⁾^from ^a^geometrial⁽Pi^,d^,^and dmax)andfromatopologial(bi)pointofviewwiththenearestneighbourinterpolation

(M1)andthelinearinterpolationwiththresholding(M2).Pi^:^ratio^(%)^of^pointsv^for

whih ρ(v, S, M, h) îs ⁱⁿ](i−1), i].25.10⁻² ^mm ^(this^ratio îs ômputed^without ôn-

sideringvoxelsforwhihρ(v, S, M, h)îsêqual^to^0);d^:^mean^distaneôfρ(v, S, M, h)^; dmax^:^maximal^valueôfρ(v, S, M, h)^for^the¹⁵âses;b0^(resp.b1^,b2^):^numberôfôbjet

onnetedomponents(resp.holes,avities).

tainedwithM2^(theM1^andM2^methods^provide^by^onstrutiongeometrially orret results). To provide quantitativeomparison, the following strategy is

used. As M îs ômposed ôf ônly ône ÔCC ând ône ^BCC, ^the ômputation ôf ρ(v, S, M, h) îs ^possible ^for êah ^method ^(to ômpute ρ(v, S, M, h) ^for ^the M1

and M2 âpproahes,^voxels,^for ^whih S(v) = 1^, âre âssoiated^with ^the ÔCC,

and the others orrespond to the BCC). We observe rstly that the ratios of

points forwhih ρ(v, S, M, h) = 0 ^are ^idential^for ^the^three ^methods ^(98.6%).

ResultsresumedinTab.1providetheratiosoftheotherpoints(namely1.4%)

intermsofdistane.Theinspetionoftheresultsshowsthatthethreemethods

arerelativelysimilar.Notethat themaximal distaneisalittlebit higher(but

still low)for theproposedmethod. More preisely,on the15onsidered ases,

5segmentationmaps haveauniquevoxelv ^whose^valueρ(v, S, M, h)^is^lightly

greaterthanone (the maximalvalueenounteredin the15asesforρ^is^1.06).

This is due to the fat that the topology preservation indues here geometri

onstraints.Finally,wean onludethat thegeometryis similarfor thethree

methods, butonlytheproposedapproahanorretlyhandle topology.Ithas

tobenotiedthatthealgorithmhasalsobeentestedwithobjetsomposedof

severalCCs.Resultsaresatisfatorybut notpresentedin thispaper.

3.3 Disussion

Weannotiethatmedialimagesegmentationmethods[46℄basedontopology-

preservingdeformationofabinarymodel 1

havebeenproposedinthelastyears.

Theygenerally relyonsimplealgorithmiproessesandhypotheses:(i)^they^use

monotoni transformations whih either removeor add simple points from/to

amodelM ^neessarilysurrounding/surroundedbyS^,(ii)^suh^models^are^pro-

posedforstrutureshavinganon-omplextopologyortopologiallysimplied,

and (iii) ^only ^simpledeformationfuntions (based ongrey-levelvaluesor dis- tanemaps)areonsidered.Thedeformationstrategyproposedhereleadstoa

1

Somemethodsarebasedonlabelmodels,unfortunatelywithseveralapproximations

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tant improvements: themethods an use non-trivial topologial models whih

evolvein anon-monotoni andtopology-preservingfashionunder theguidane

ofomplex deformationfuntions.

4 Conlusion

A new method for warping a binary image in a disrete topology preserving

fashionaordingtoaontinuoustopologypreservingdeformationeldhasbeen

proposed.Furtherworks will onsistin extendingthis methodto label images.

Suh extensionwill requireto developasound theoretialframeworkfortopo-

logialmodellinganddeformationofsuhimages.

Theproposedmethodhasbeensuessfullyappliedtomedialimagesegmen-

tation. Anotherperspetiveis toonsider theproposed framework fordevising

strategieswhose behaviourmayevolve during thedeformationproess, forex-

ample by performing in parallel segmentation and registration, as proposed in

[9℄.

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