Multi Channel MRI Segmentation With Graph Cuts Using Spectral Gradient And Multidimensional Gaussian Mixture Model

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Multi Channel MRI Segmentation With Graph Cuts

Using Spectral Gradient And Multidimensional

Gaussian Mixture Model

Jérémy Lecoeur, Jean-Christophe Ferré, D. Louis Collins, Sean Patrick

Morrissey, Christian Barillot

To cite this version:

Jérémy Lecoeur, Jean-Christophe Ferré, D. Louis Collins, Sean Patrick Morrissey, Christian Barillot.

Multi Channel MRI Segmentation With Graph Cuts Using Spectral Gradient And Multidimensional

Gaussian Mixture Model. SPIE - Medical Imaging, Feb 2009, Orlando, United States.

pp.72593X1-72593X11, �10.1117/12.811108�. �inria-00354050�

Multi Channel MRI Segmentation With Graph Cuts Using

Spectral Gradient And Multidimensional Gaussian Mixture

Model

Jérémy Le oeur

∗†‡

Jean-Christophe Ferré

§∗†‡

D. Louis Collins

¶

Sean Morrissey

k∗†‡

Christian Barillot

∗†‡

ABSTRACT

A new segmentation framework is presentedtaking advantageof multimodal image signature of the dierent braintissues(healthyand/or pathologi al). This isa hievedbymergingthreedierentmodalitiesofgray-level MRI sequen esintoasingleRGB-likeMRI, hen e reatingaunique3-dimensionalsignatureforea h tissueby utilising the omplementaryinformationofea hMRIsequen e.

Usingthe s ale-spa espe tralgradientoperator, we anobtainaspatial gradientrobust to intensity inho-mogeneity. Eventhoughitisbasedonpsy ho-visual olortheory, it anbeverye ientlyappliedto theRGB oloredimages. Moreover,itisnotinuen edbythe hannelassigmentofea hMRI.

Itsoptimisationbythegraph utsparadigmprovidesapowerfulanda uratetooltosegmenteitherhealthy orpathologi altissuesin ashorttime(averagetimeaboutninetyse ondsforabrain-tissues lassi ation).

Asitisasemi-automati method, werunexperimentstoquantifytheamountofseedsneededtoperforma orre tsegmentation(di esimilaritys oreabove0.85). DependingonthedierentsetsofMRI sequen esused, thisamountofseeds(expressedasarelativenumberinpour entageofthenumberofvoxelsofthegroundtruth) isbetween6to16%.

We tested this algorithm on brainweb for validation purpose (healthy tissue lassi ation and MS lesions segmentation)andalsoon lini aldatafortumoursandMSlesionsde te tion andtissues lassi ation.

1. INTRODUCTION

Takingadvantageofthevariousproto olsthata quiremultiplemodalityimagesisa urrentissue(typi allyT1, T2,PD,DTI orFlair sequen es in MRneuroimaging). Thedata are be oming more andmore multi- hannel dataandtheiruniqueand omplementaryinformationshouldbemergedtogetherbeforesegmentationtogetrid ofthein onsisten iesone anen ounterwhensegmentingea hmodalityseparately. Today,reliableregistration methods, using dierent resolution and time, are available, nevertheless, a simple, robust, fast and reliable segmentationapproa hstilldoesnotexist forsu hkindofproblem.

Multi hannel segmentation usually relies on lustering or lassi ation. Inthe urrentwork,wepropose a new and original s ale-spa e approa h for segmenting organsand tissues from multidimensional images. We propose ate hniquethat anperform ajoint segmentation of three MRI volumes at a time. Theaim of this

∗

INRIA,VisAGeSUnit/Proje t,IRISA,Rennes,Fran e

†

UniversityofRennesI,CNRSIRISA,Rennes,Fran e

‡

INSERM,U746Unit/Proje t,IRISARennes,Fran e

§

DepartmentofNeuroradiology,Pont haillouUniversityHospital,Rennes,Fran e

¶

MontrealNeurologi alInstitute,M GillUniversity,Montreal,Canada

evolution.

As theintensity distributionofthe interestingtissuesfollowsaGaussian lawin ea hmodality, bymerging three volumes into asingle  olor MRI - ea h volume be oming a olor hannel - the olordistribution, thus reated, follows also a multidimensional Gaussian law. Ea h tissue being hara terized by a

3

-dimensional signature, dis riminating ea h tissuefrom oneanother iseasier. The main idea, presented here,to segmenta volumeistouseas ale-spa e olorinvariantedgedete tor-

i.e.

thespe tralgradient-inagraph utframework.

Among allthe various energy minimizationte hniques for segmentation, Greig et al. 1

proposed amethod basedonpartioningagraphbyaminimum ut/maximumowalgorithminheritedfromFord.

2

Then,Boykov etal.

3,4

enhan edthisapproa hbyimprovingits omputationale ien y,theirmethodisnowreferredasGraph Cut.

Inthefollowingse tions,wewillrstpresentthespe tralgradientandhowit anbeembeddedingraph uts optimization. Then, wewillshowvalidationandresultsonsynheti and lini aldatasu hasMultiple S lerosis lesionsandbraintumors. Finally,wewill dis ussonthe ontributionsandfuture improvementsto bemade.

2.METHODS

The framework we've designed is as follow : from three grey-level MRI sequen es, we build a olor MRI by assigningea h red,green orblue hannel to asequen e. Then we omputethespe tralgradientand useit in agraph utframework whi h requires seeds asinput. In theend of this framework,weobtainthe segmented stru tures(

e.g.

brain,MSlesions,tumors). Figure1summarizesthisframework.

Figure1.Ourframework

As it anbeseenon thegureabove, thegraph ut takestwodierent inputs: the Spe tralGradient- a boundary term - and the sour e and sink, a regional term whi h onstitutesthe intera tivepart of the algo-rithm. Inthefollowingsub-se tions,wewillexplainthegeneralframeworkoftheGraphCut,themathemati al observationsusedto omputethesegmentationand, nally,howwe've ombinedthemalltoa hieveourgoal. 2.1 Introdu tion to Graph Cut

A ording to the s heme by Boykovet al., 5,6

the segmentation problemati is des ribed by a dire tionalow graph

G = hV, Ei

whi h representstheimage. Thenode set isdened by twoparti ularnodes alled terminal nodes- also known as sour e and sink - whi h respe tivelyrepresentsthe lass obje t and ba kground. The other nodes orrespond to the 3D volume voxels and the dire ted weighted edges onne ting the nodes somehowen ode thesimilaritybetweenthe two onsideredvoxels. This method issemi-automati asthe user needtosele ttwosetsofvoxelsoftheimage,theobje tset

v

o

ontainingvoxelsoftheobje tandtheba kground set

v

b

ontaininvoxelsoftheba kground.

Let the set

P

ontain all the voxels

p

of the image, the set

N

be all the pair

{p, q}

of the neighboring elementsof

P

and

V = (V

1

, V

2

, ..., V

|P|

)

beabinaryve torwhere ea h

V

p

anbeone ofthetwolabelsobje t

orba kground. Therefore,theve tor

V

denesasegmentation. Theenergywewanttominimizebythegraph uthastheformgivenby:

E(V) = α ·

X

p∈P

R

p

(V

p

) +

X

{p,q}∈N

Vp 6=Vq

B

{p,q}

(1)

Theterm

Rp(·)

, ommonlyreferredastheregionalterm,expresseshowthevoxel

p

tsintogivenmodelsof theobje tandba kground. Theterm

B

{p,q}

,knownastheboundaryterm,ree tsthesimilarityofthevoxels

p

and

q

. Hen e,itislargewhen

p

and

q

aresimilarand losetozerowhentheyareverydierent. The oe ient

α

isusedto adjusttheimportan eoftheregionandboundaryterms.

Theedge onne tingapairofneighboringwoxelis alled

n

-linkandits ost anbebasedonvariousmetri s su h aslo alintensity gradient,Lapal ianzero- rosing,gradientdire tion orother riteriawith therestri tion thatthis ost annotbenegative. It'stheborderterm

B

{p,q}

ofequation1. Thesimplestimplementationofthe weight ostof anedgebetweenneighboringwoxels

p

and

q

isthen

w

{p,q}

= K · exp(−

|I

p

−I

q

|

σ

2

)

where

I

p

and

I

q

aretheintensitiesatvoxel

p

and

q

. Thisweightfun tionfor esthesegmentationboundariesatpla eswithhigh intensitygradient.

All the

v

o

nodesare onne tedto thesour e andallba kgroundseeds(

v

b

)are onne tedto thesink,those twosetsoflinksare alled

t

-link(terminal links). Those

t

-linksaretheregionalterm

R

p

(V

p

)

ofequation1.The simplestimplementationinvolvesinnite osttoall

t

-linksbetweentheseedsandtheterminals. Inmoreadvan ed work,the ostis basedonhowthe intensityof voxel

p

ts into given intensity models(

e.g

histograms)of the obje tandba kground,hen egivingapie eofregionalinformation.

The graphis now ompletely dened and the segmentation ontouris drawn by nding the minimum ut ofthisgraph. A utin agraphis asubsetofedge that dividesthegraphinto twoparts : thesour e-and the sink-part,hen eseparatingtheobje tfrom theba kgroundin abinarysegmentation ; the ostof a utbeing thesumofthe ostofitsedges. Theminimum utisthusthe utwiththeminimum ostand anbe omputed in polynomialtime using max-owalgorithm,

2

push-relabel te hni 7

or the now lassi alBoykov-Kolmogorov method.

8

2.2 ImageObservation : Maths At Work

Asexplainedabove,weneedtwokindsofinformationto ompleteourtask: someontheregionwhi hweobtain from multivariatemixture model and someonthe border whi h isprovided bythe spe tralgradient. Wewill presentinthefollowingpartsthosetwo on epts.

2.2.1 MultivariateGaussian Mixture Model : The Basi s

Toexploit the omplementarypie eofinformation ontainedin threeregisteredMRIsfrom dierentsequen es, we use a multivariate Gaussian mixture model. For ea h point labeled as aparti ular lass

c

, we onsider a 3- omponentsve tor

Ψ

omposed with the intensities of ea h MRI at this point. We then omputethe mean ve tor

Ψ

c

and the ovarian ematrix

Σ

c

. The lass-membership probability ofa voxel

v

is omputed with the multivariatenormaldistributionformula:

P

(Ψ

v

|c) = exp −

1

2

(Ψ

v

− Ψ

c

)

T

_{· Σ}

−1

c

· (Ψ

v

− Ψ

c

)

(2)

Aswe onsideratwo(orthree) lassesobje t(dependingonwhatwewanttoa hieve),theseedsofea h lass are dierentiated from the beginning and the sour e-membershipprobability of avoxel

v

is then thehightest ofthetwo(or three) lass-membershipprobabilities. As thesink isalwaysasingle lass, thesink-membership

Inorder to use the olorof tissus asan higly dis riminative de ision riterion, we need to build an invariant olor-edgedete tor. Wepropose to usethespe tralgradient,rstintrodu edby Geusebroeaket al.,

9

whi his basedonthepsy ho-visual olortheoryandonKoenderink'sGaussianderivative olormodel.

10

Color an be interpretedby spe tralintensity (

e

)that falls onto theretina ; this intensity depends on the spe tralree tionfun tion

r

ofthesurfa ematerialbut alsoonthelightspe trum

l

-whi hisafun tionofthe wavelength-fallingontoit. Inaddition,theshading

s

takesagreatpartinthisseenintensitywhereasitisonly position-dependent. Tosumitup,we anwritethisequation:

e(x, y, z, λ) = r(x, y, z, λ) · l(λ) · s(x, y, z)

(3)

As weseek olorsinvariants, weneed to get ridof

l(.)

and

s(.)

sin e

r

isthe onlytrue olor,whi h does notdepend onillumination onditions. This anbea hievedquiteeasily followingsthese steps: rst, wetake thederivativewithrespe tto

λ

andwenormalizetheexpression,thusleadingto:

1

e(x, y, z, λ)

∂e(x, y, z, λ)

∂λ

=

l

λ

l

+

r

λ

r

(4)

Then, asimpledierentiation to thespatial variable (

x

,

y

or

z

)makesanexpression whi h suits allof our onstraints:

∂(

1

e(x,y,z,λ)

∂e(x,y,z,λ)

∂λ

)

∂x

= 0 ⇐⇒

e

· e

xλ

− e

x

· e

λ

e

2

= 0

(5)

Hen e,wenowhavean expressionwhi h isfully expressedby spatialand spe tralderivativeof

e

, whi his theobservablespatio-spe tralintensitydistribution

Geusebroeketal. 11

haveproventhatthese terms anbeverywellapproximatedbysimplymultiplyingthe RGBvalues(seenasa olumnve tor)bytwomatri es:





e

e

λ

e

λλ





=





−0.019

0.048

0.011

0.019

0

−0.016

0.047

−0.052

0





|

{z

}

XY Z to e

·





0.621

0.133 0.194

0.297

0.563 0.049

−0.009 0.027 1.105





|

{z

}

RGB to XY Z

·





R

G

B





(6)

TherstmatrixtransformstheRGBvaluestotheCIE 1964XYZbasisfor olorimetryandthese ondone givesthebest lineartransformfrom the XYZvaluesto theGaussian olor model. These twomatri es anbe mergedina

3 × 3

matrix

M

that hara terisesthetransformationfrom RGBvaluesto

e

anditsderivatives.

Figure2.Fromlefttoright: ColorMRI(fromT1,T2andFlairsequen es),spe tralintensity

e

,rstorderderivativeof spe tralintensity

e

λ

andse ondorderderivativeofspe tralintensity

e

λλ

.

ε

=

1

e

·

∂e

∂λ

=

e

λ

e

(7)

Asstated in Geusebroek'swork, 11

yellow-blue transitions anbefoundwiththe rstorder gradient,whi h magnitudeis:

Γ =

q

(∂

x

ε)

2

+ (∂

y

ε)

2

+ (∂

z

ε)

2

(8)

These ondordergradientdete tsthepurple-greentransitions. Itsmagnitude anbe omputedasfollows:

Υ =

q

(∂

x,λ

ε)

2

+ (∂

y,λ

ε)

2

+ (∂

z,λ

ε)

2

=

q

(∂

x

ε

λ

)

2

+ (∂

y

ε

λ

)

2

+ (∂

z

ε

λ

)

2

(9) with:

ε

λ

=

∂ε

∂λ

=

e

· e

λλ

− e

2

λ

e

2

(10)

Finally,thedete tionofall oloredges anbeperformedwith:

ℵ =

p

Γ

2

_{+ Υ}

2

=

q

(∂

x

ε)

2

+ (∂

y

ε)

2

+ (∂

z

ε)

2

+ (∂

x

ε

λ

)

2

+ (∂

y

ε

λ

)

2

+ (∂

z

ε

λ

)

2

(11)

Figure3.Left:ColorMRI-Right: Spe tralgradient.

2.3 Introdu ing Multivariate Mixture Model And Spe tral Gradient in the Graph Cut Paradigm

Let'sgetba ktotheenergyformulationoftheGraphCut:

E(V) = α ·

X

p∈P

R

p

(V

p

) +

X

{p,q}∈N

Vp 6=Vq

B

{p,q}

(12)

Toformthegraph,weneedto omputethe

t

-links. Two asesaretobe onsidered: rst,theweight

W

so

of the

t

-linkinvolvingthesour e nodeis:

W

so

=











0

if

p

∈ B

∞

if

p

∈ O

α

· R

p

(B)

elsewhere (13)

Similar,theweight

W

si

ofthe

t

-linkinvolvingthesink nodeis omputedasfollows:

W

si

=











∞

if

p

∈ B

0

if

p

∈ O

α

· R

p

(O)

elsewhere (14)

A ordingtothes hemebyBoykovetal., 5,6

the

t

-linkweightsofavoxel

p

arethenegativelog-likelihoods:

R

p

(B) = − ln P (Ψ

p

|B)

and

R

p

(O) = − ln P (Ψ

p

|O)

(15)

To omputethe

n

-links,weuseanad-ho fun tion :

B

{p,q}

∝ exp



−

(ε(p) − ε(q))

2

_{+ (ε}

λ

(p) − ε

λ

(q))

2

2σ

2



·

1

dist(p, q)

(16)

where

ε

and

ε

λ

arethequantitiesdenedinequations7and10. Changingthe hannelsassignmentonly hange the spe tral gradient intensity but not the lo ation of the border, so, in this graph ut s heme, the hannel assigmentdoesnotimpa tthesegmentation.

In order to deal with the quite huge data sets we have whi h are very resour e onsuming, we needed a hierar hi al s heme that allows us to ompute the graph ut in a de ent time. Consequently, we hoose to implement the graph ut as amultiresolution algorithm. This method, rst developed by Lombaert et al.,

12 is inspired by the multilevel graph partition te hnique

13

and the narrow band from the level sets. 14

It rst omputesthegraph utonthe oarsestlevelandthenin thesu essivehigherlevelbut onlyonanarrowband derived from theminimal ut found at the previous oarserlevel. This pyramidal approa h with aGaussian de imationhasproventoberobust,evenwithhighdowns alingfa tor.

3.EXPERIMENTS AND RESULTS

First,we willpresentthevalidation experimentsrun onsyntheti data,thus permittingus a omparison with methodsfromtheliterrature. Then,wewillshowtheresultsobtainedwithrealdatawithgroundtruthfroman expert.

3.1 Validation on syntheti data

Thevalidationof thea ura yof oursegmentationmethod wasperformedon syntheti phantombyusing the BrainWebdata.

15

Webuiltthe olorMRIfrom simulatedT1,T2 andPD sequen es. All theimagesbelongto thesamesubje tandare onstitutedof217sli esof181x181isometri 1mmvoxelswith3%noise(relativeto thebrightesttissueintheimages).

Inorderto lassify orre tlythedierent lassesoftissues,wefollowedahierar hi alsegmentations heme. ArstGraphCut withsour eseedsmixing allthetissuesandsink seedsontheba kgroundgivesus thebrain mask; then inside this mask, we perform onegraph ut per tissue with only its own sour e seeds - the sink seedsbeingtheba kgroundseeds andtheothertissuesseeds. Thewhole omputation timeisbetween50to 80 se ondsonalaptop(Dual oreat2.16Ghzand 2GBofRAMrunningLinux).

We tested ourmethod on dierent non-uniformity levels and ompared the obtained segmentation to the groundtruth available,using the Di eSimilarity Coe ient with ve omponents being onsidered : erebro-spinaluid,whitematter,greymatter,thewholebrainandthelateralventri les. Allthetests wereperformed ondatawith3%ofnoise,relativetothebrightesttissue.

Thefollowingtableshowsthatthea ura yofthesegmentationisnotsigni antlyinuen edbybiasasthe DSCisquitesimilarforallnon-uniformityvalues. Moreover,thesegmentationismoreorlessequallya urate

NonUniformity

0% 20% 40%

CerebroSpinalFluid 0.892 0.891 0.892 GreyMatter 0.932 0.924 0.927 White matter 0.961 0.954 0.958 Wholebrain 0.983 0.981 0.982

Ventri les 0.946 0.944 0.944

Thoseresultsarebetterorsimilartothosefoundintheliteratureonsimilardata,su hasAshburneretal. 16 withDSCrunningfrom

0.932

forgreymatterto

0.978

forthewholebrain;theLOCUSapproa hbyS herreret al.

17

s ores

0.83

forCSF,

0.94

forgreymatterand

0.96

forwhitematter, orBri q etal. 18

withaDSC around

0.96

forea h lassoftissu.

An important data in the evaluation of this tool is the amount of seeds needed to a hieve a orre t seg-mentationonpathologi altissue. Weranexperimentsinorder toquantifythesimilaritybetweentheobtained segmentationand thegroundtruth. Asinputtoouralgorithm, weusedthegroundtruthrandomlyde imated.

Inordertoassessthisalgorithm, weusedsyntheti datafromBrainWeb 15

withsimulatedmultiple s lerosis lesions. Resultsofthis testarepresentedongure5.

Figure 5.Di e Similarity Coe ient versus relative numberof seeds(in per entage of the ground truth). Blue line : DSCobtainedafterrunningouralgorithm. Bla k line: DSC frominitialisationseedsonly. RedLine: dieren eofthe pre edingtwo urves,showingtheeen yoftheproposedmethod

As explained byZijdenbos, 19

a DSC s oreabove

0.7

is generally onsidered asverygood, espe ially when thesegmentedstru tures aresmall. Here,thisthresholdisrea hedwhentheinputseeds arearound

5

% ofthe

algorithm onmoderate MSlesions. Fortheoptimalvalue(10% of relativenumberofseed), our DSCis 0.93 whenVan-Leemput s ores0.80( al ulated byFreifeld in

21

),Freifeld0.77,Rousseauonly0.63andBri q0.79. 3.2 Quanti ation of the a ura y on various sequen es

Usually simulated data don't address the same issues than lini al data. That's why it is important to run experiments onthistype ofdata whi h rarelyallowstheuse ofautomati orgeneri appro hes. As weaimto usethis new toolin a lini al ontext, weevaluated the resultson dierent lini aldata sets. Those sets are theonesthataremostlikelytobeusedfordiagnosispurpose. Three ombinationsofsequen eswere onsidered relevantwith ourgoal: T1-weighted,T2-weightedandFlair(whi hwill bereferredasTTF), T1-w,T2-wand PD(whi hwillbereferredasTTP)andT1-w,T1-winje tedwithGadoliniumandFlair(whi hwillbereferred asTGF).Thedata oversalargerangeoflesionloadand lini algrades(fromRRto SP).

The following table summarizesthe dierent parametersof the olorMRIs webuilt in order to assess our tool:

Typeof Numberof Number of Sizeof Sizeof MRI Subje ts sli es sli es voxels

TTF

6

138

256 × 256

isometri

1

mm

TTP

8

217

181 × 181

isometri

1

mm

TGF

5

160

256 × 256

isometri

1

mm

We run similar experimentsthan those for BrainWeb validation with an addition : wealso used a ground truthde imatedbysu essiveerosionsasinput,hen esimulatingthe lassi albehavioroftheuserswhi hwould preferablytakeseeds inthe enterof thebiggerlesionsand forgetsmallerlesions.

Figure6.Di eSimilarityCoe ientversusrelativenumberofseeds(inper entageofthegroundtruth). Redlines: TTF MRIs,greenlines: TTP MRIs,bluelines : TGFMRIs,bla kline : DSC frominputdataalone ;solidline: randomly de imatedgroundtruthasinput,dashdottedline: groundtruthde imatedbysu essiveerosionsasinput.

Figure 6presentsthe resultsof this experiment. One annoti e that the behavior of the urvesare quite similar,however,theTTFsetsseemtobemoresuitableforsegmentingMSlesions,withanaverage

0.972

DSC s orefor anumberofseeds around

6

%of thegroundtruth; TTPbeingse ond bestwith a

0.777

averageDSC s oreforthesamerelativenumberofseedsandTGFonlys oring

0.593

.

Theotherimportantfa t showedbythisexperimentisthat arandom de imationperformsbetterthanthe su essiveerosionsas theDSC is lowerbyabout10%when theerosionsare used. We aninterpretthat from twohypothesis : thismay ome fromkeepingsmalllesionsintheinitialstage intherandomde imation (those

Redlines: TTFMRIs,greenlines: TTP MRIs,bluelines: TGFMRIs;solidline: randomly de imatedgroundtruth asinput,dashdottedline: groundtruthde imatedbysu essive erosionsas input. Thevariationisshownas anerror barontheoptimalpointofea h urveandisabout

±2

%forTTF,

±1.5

%forTTPand

±1

%fotTGF.

fromtheseedshasalowervarian ethantherealoneasweonlykeepthe entersoflesionswhoseintensitiesare notfullyrepresentativeofthenormalintensitydistributionofalllesions.

Figure7showsthedieren ebetweentheDSCs oreoftheinitialportionofthegroundtruthretainedfrom initialisation alone and the DSC s ore obtained after running our algorithm from the same initialisation. It somehow omputes the enhan ementgiven by our tool with respe t to the DSC. From top to bottom of the urvesin gure7, themaximum enhan ementis obtainedrespe tivelyfor

6

,

8

,

11

,

14

,

16

and

18

% ofrelative numberof seeds. This givesouroptimal rangeforinitialisation onstraintswith respe tto theexpe tedlesion load.

Resultsandseedsofthealgorithmareshownongure8forTTFsequen es.

Figure8.ResultsonTTFimages. Upperrowfromlefttoright: T1-w,T2-wandFlairSequen e. Lowerleft:initialisation fromrandomly de imatedground truth. Lower enter : initialisationfrom ground truthde imatedby erosion. Lower

weretobeabletosegmenta uratelytheperi-tumoraledemaandthetumoritself. TheMRsequen esusedfor thispurpose areT1 and T1with Gadoliniuminje tion images omposed of 182sli es of256x 256isometri 1 mmvoxelsand aninterpolatedFLAIRimage(original size704x704x34). As inthepre eding appli ations, the rstpass gives us the brain mask, then we apply the graph ut for ea h lass of interest (white matter, grey matter, edemaand tumor). On thefollowinggure, we ansee that the tumor (in blue) and edema(in dark grey)regionsarewellsegmented. This resultwasobtainedin roughly90se onds. Visualassesmenthavebeen performedbyanexpert.

Figure9.Oursegmentation(right),T1image(middle)andFlairimage(left).

4.CONCLUSION AND FUTURE WORK

In this paper, we have introdu ed the spe tral gradient in the eld of 3D medi al images. This new, fast and robust multiple images segmentation framework allows us to pro ess multi hannel data from s ale-spa e derivatives.

Itsoptimization by ahierar hi algraph utshas provento be a uratewith veryee tiveresults. It om-binesregion-basedintensity pro essingand ontour-baseds ale-spa eapproa hin anewway. With su h little omputationaltime,itallowstheusertointera tively orre ttheresultsfrom averylimitedinputeort.

We'vebuiltourworkonananalogybetweenRGB hannelsandmulti-modalMRIs. Thismethodmaynotbe optimal,espe iallythetransformmatrix

M

(

cf

eq.6). Tondtheoptimalmatrix,we ouldlearnitsparameters fromasupervisedpro edure.

We propose here asemi-automati but supervisingthe intensity model of lesions ould give an automati initialisation. Su hmodels ouldbegivenbytheSTREMalgorithmfromAit-Alietal.

23

Thenextstepsofthisworkwillbetorunmoreadvan edvalidationstudiesandparti ularlyonlongitudinal datasetsinordertobeable,forinstan e,toquantifytheevolutionofmultiples lerosislesionsandtostudythe inuen eoftheparameterofthe

M

matrixparameters.

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