A probabilistic framework for point-based shape modeling in medical image analysis

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A probabilistic framework for point-based shape

modeling in medical image analysis

Heike Hufnagel

To cite this version:

Heike Hufnagel. A probabilistic framework for point-based shape modeling in medical image analysis.

Medical Imaging. universität Lübeck, 2010. English. �tel-00844717�

der Universität zu Lübe k

Direktor: Prof. Dr. rer. nat. habil. Heinz Handels

A Probabilisti Framework

for Point-Based Shape Modeling

in Medi al Image Analysis

Inauguraldissertation

zur

Erlangung der Doktorwürde

der Universität zu Lübe k

 Aus der Te hnis h-Naturwissens haftli hen Fakultät 

Vorgelegt von

HEIKE HUFNAGEL

aus Lüneburg

Christian-Förster-Straÿe 12

20253Hamburg

Deuts hland

Email: hufnagelimi.uni-luebe k.de

Inauguraldissertation zur Erlangung der Doktorwürde

der Te hnis h-Naturwissens haftli hen Fakultät

der Universität zu Lübe k

Prüfungsvorsitzender: Prof. Dr. rer.nat. Thorsten M.Buzug

Erstberi hterstatter: Prof. Dr. rer.nat. habil.Heinz Handels

Zweitberi hterstatter: Prof. Dr.rer. nat. habil.Bernd Fis her

To Emmi and Evi

"Die gefährli hste aller Weltans hauungen ist die Weltans hauung der Leute,

wel he dieWelt nie anges hauthaben."

(The mostdangerous ofall world-viewsistheone ofpeoplewhohavenever viewed

theworld.)

A knowledgments

Before starting this thesis I did not know what I would get myself into, and

when Inallyrealized it and itwas too late, Iwasvery glad to dis overthatI did

not haveto walkthis path alone.

To begin with, I would like to thank my dire tor and do toral advisor Heinz

Handelsfor oeringmethe opportunity andthe work environment for myresear h

at theIMI, and I also thank him greatly for his trust in my apabilities, his good

advi eand onstru tive dis ussions.

Ithankmydire torNi holasAya hefromINRIA forkindly integratingmeinto his

team,for supporting mywork and givingmedire tion.

My deep gratitude goes to my advisors, they gave me inspiration, did not avoid

heated dis ussions and taught me a lot about omputer s ien e and the world of

resear h in general: Xavier Penne who guided my exploration of the fas inating

realms of mathemati s and Jan Ehrhardt who managed the pre arious balan e

between supervisoranddear friend.

Ialsowantto takethis opportunityto thankBerndFis herwhohasalways

a om-panied mywork from afarfor hisvaluable omments. I thank Tobias Heimann for

enthusiasti dis ussionsabout shape andhis kind ooperation.

Fromall myheartIthankEnder Konukogluand AndreaMartiniwhostillbelieved

inmeintimeswhenIdidnot anymoreand whoIalways ouldrelyonfors ienti

and emotional support.Mylife wouldbe alot more onfused withoutthem.

With Floren e Billet and Jean-Mar Peyrat I walked the same path throughout,

and I annot thank them enough for their empathi ompanionship whi h made

everything somu heasier.

In both my teams I was lu ky to meet great assistants: I thank Isabelle Strobant

and Renate Re he for e iently simplifyingadministrative matters and for having

an open earfor allkind of problems.

Ithank allmy olleagues fromtheAs lepiosteam:open doorsand theQueen's leg

alwaysinvitedfruitfuls ienti dis ussionsbutalso-maybeevenmoreimportantly

-valuableso ialand ulturalex hange inan internationalenvironment.Iespe ially

thank Marius Linguraru, TomVer auteren, Mauri io Reyes-Aguirre, Olivier Clatz,

and JimenaCostafor ties beyonda ademi issues.

Profoundly I thank my olleagues at the IMI who be ame my friends. Apart

from the s ienti support and en ouragement they oered, they were the reason

I always liked going to work even in di ult too- lose-to-deadline or

hidden-program-bug times: Ithank René Werner for letting mebathein hisserenity,Alex

S hmidt-Ri hbergfor for ing hisprogramming skills upon meand forhis kindness,

Nils Forkert for philosophi al ( igarette) breaks even in the middle of the night,

Dennis Säring for down-to-earth words at the right time, and Matthias Färberfor

lightening my view on things. For his patien e and willingness to help me with

te hni al omputer mattersI thankMartin Riemer.

I thank Ri ardo Martinez for sharing my life and making it more

Ithank the GermanA ademi Ex hange Servi e(DAAD) and theGerman

This thesis enters on the development of a point-based statisti al shape model

relyingon orresponden eprobabilitiesinasoundmathemati alframework.Further

fo us lies on the integration of the model into a segmentation method where a

novelapproa h istaken by ombiningan expli itlyrepresentedshape priorwithan

impli itly representedsegmentation ontour.

Inmedi alimageanalysis,thenotionofshapeisre ognizedasanimportant

fea-tureto distinguishandanalyseanatomi al stru tures. Themodeling ofshape

reali-zedbythe on eptofstatisti alshapemodels onstitutesapowerfultooltofa ilitate

the solutions to analysis, segmentation and re onstru tion problems. A statisti al

shape model tries to optimally represent a set of segmented shape observations of

anygivenorganviaameanshapeandavariabilitymodel.Afundamental hallenge

in doing statisti s on shapes lies in thedetermination of orresponden es between

the shape observations. The prevailing assumption of one-to-one point

orrespon-den es seems arguable due to un ertainties of the shape surfa e representations as

well asthe general di ulty ofpinpointing exa t orresponden es.

In this thesis, the following solution to the point orresponden e problem is

derived:Forallpointpairs,a orresponden eprobabilityis omputedwhi hamounts

to representing the shapesurfa es byMixtures of Gaussians. Thisapproa h allows

to formulate the model omputation in a generative framework where the shape

observationsareinterpretedasrandomlygeneratedbythemodel.Basedonthat,the

omputationofthemodelisthentreatedasanoptimizationproblem.Analgorithm

is proposedto optimize for model parameters and observation parameters through

asinglemaximumaposteriori riterion whi hleadsto amathemati ally soundand

uniedframework.

The method is evaluated and validated in a series of experiments on syntheti

and real data. To do so, adequate performan e measures and metri s are dened

based on whi h the quality of the new model is ompared to the qualities of a

lassi al point-based model and of an established surfa e-based model that both

relyon one-to-one orresponden es.

A segmentation algorithm isdeveloped whi hemploys theapriorishape

know-ledgeinherent inthestatisti alshapemodelto onstrain thesegmentation ontour

to probable shapes. An impli it segmentation s heme is hosen instead of an

ex-pli itone,whi h isbene ial regardingtopologi al exibilityandimplementational

issues.Themathemati allysoundprobabilisti shapemodelenablesthe hallenging

integration of an expli it shape prior into an impli it segmentation s heme in an

elegant formulation. A maximuma posteriori estimation isdeveloped ofa level set

fun tionwhose zerolevelsetbestseparatestheorganfromtheba kground undera

shape onstraintintrodu ed bythemodel.Thisleads toanenergy fun tionalwhi h

isminimized withrespe t tothelevelset usingan Euler-Lagrangian equation.

Sin- e both the model and the impli itly dened ontour are well suited to represent

multi-obje t shapes, an extension of the algorithm to multi-obje t segmentation

Ein probabilistis hes Framework

für punktbasierte Formmodellierung

in der medizinis hen Bildanalyse

Die vorliegende Doktorarbeit konzentriert si h auf die Entwi klung eines auf

Kor-respondenzwahrs heinli hkeiten beruhenden punktbasierten statistis hen F

ormmo-dellsineinem mathematis h fundiertenund ges hlossenenFramework.Ein weiterer

S hwerpunkt liegt in der Integration des entwi kelten Modells in eine

Segmentie-rungsmethode.HierwirdeinneuartigerAnsatzvorgestellt,inwel hemexplizit

de-niertes Formwissen mit einer implizit deniertenSegmentierungskontur kombiniert

wird.

Indermedizinis henBildanalysegiltderBegriderFormalswi htigesMerkmal

für dieErkennungunddieAnalyse anatomis her Stukturen.Die Formmodellierung

mittelsdesKonzeptesderstatistis henFormmodelleverkörperteinmä htigesW

erk-zeug, wel hes zu Lösungenfür Analyse-, Segmentierungs- und

Rekonstruktionspro-blemebeiträgt.Einstatistis hesFormmodellversu ht,einenSatzvonsegmentierten

Formbeoba htungeneines gegebenen Organsoptimaldur heine mittlereFormund

einVariabilitätsmodell zu repräsentieren. Einegroÿe Herausforderung für jegli hen

statistis hen Ansatz stellt hierbei die Bestimmung von Korrespondenzen zwis hen

den Formen dar. Die übli he Annahme von 1-zu-1 Korrespondenzen ist

problema-tis haufgrundderUnsi herheitendieGenauigkeitderSegmentierungbetreendals

au h aufgrundderallgemeinen S hwierigkeit, exakte Korrespondenzen zu

lokalisie-ren.

IndieserArbeitwirdalsLösungfürdasPunkt-Korrespondenzproblemeine

Kor-respondenzwahrs heinli hkeit für alle Punktepaare bere hnet. Dies bedeutet, daÿ

dieFormoberä hendur hGauÿ's heMis hverteilungenrepräsentiertwerden.Diese

Herangehensweise erlaubt eine Formulierung derModellbere hnung ineinem

gene-rativen Rahmen, indem dieBeoba htungen alszufällig dur h dasModell

generier-teSti hproben interpretiert werden. Daraufaufbauend wirddie Modellbere hnung

als Optimierungsproblem behandelt. Es wird ein Algorithmus zur Bere hnung der

Modell-undBeoba htungsparameter ineinemeinzigenMaximum-A-Posteriori

Kri-terium vorges hlagen. Dies führt zu einem mathematis h fundierten und

ges hlos-senenFramework.

Die Methode wirdin einer Experimentserie an synthetis hen und realen Daten

evaluiert und validiert. Dafür werden adäquate Leistungsmaÿe und Metriken

de-niert, anhand derer die Qualität desneuen Modells mit den Qualitäten eines

klas-sis hen punktbasierten Modellsund eines etabliertenoberä henbasierten Modells,

diebeideauf 1-zu-1 Korrespondenzenberuhen,vergli hen wird.

Es wird ein Segmentierungsalgorithmus entwi kelt, wel her das im Modell

ent-halteneVorwissenüberdieFormeneinsetzt,umdieSegmentierungskonturauf

Seg-Implementierungsfragen Vorteile aufweist. Das mathematis h fundierte

probabili-stis he Formmodell ermögli ht auf elegante Weise die anspru hsvolle Integrierung

vonexplizitrepräsentiertemVorwissenüberdieFormineinenimpliziten

Segmentie-rungansatz. Es wird eine Maximum-A-Posteriori S hätzung einer Levelsetfunktion

so formuliert, daÿ das zugehörige Zero-Levelset das zu segmentierende Organ

un-terEinbeziehung derFormbes hränkung,diedur h dasModellgegebenist,optimal

vomHintergrund trennt.DiesführtzueinemEnergiefunktional,wel hesunter

Nut-zung der Euler-Lagrange-Glei hung in Ri htung der Levelsetfunktion dierenziert

wird.DasowohldasModellalsau hderSegmentierungsansatzgutgeeignetsindfür

die Bes hreibung von Formen, die aus mehreren Objekten bestehen, wird eine

Er-weiterungdesAlgorithmus zueiner Multi-Objekt-Segmentierung entwi kelt undin

die glei he probabilistis he Formulierung integriert.Der Segmentierungalgorithmus

1 Introdu tion 1

1.1 Motivation. . . 1

1.2 Obje tives . . . 2

1.3 Stru tureof Manus ript . . . 3

1.4 List ofPubli ations . . . 6

2 Current Methods in Statisti al Shape Analysis 9 2.1 Shape Modeling inMedi al Imaging . . . 9

2.1.1 ShapeAnalysis . . . 9

2.1.2 Doing Statisti son Shapes. . . 11

2.2 TheCorresponden eProblem . . . 12

2.2.1 IterativeClosest Point Algorithm . . . 13

2.2.2 Spheri alHarmoni Des ription . . . 15

2.3 Computationof Statisti al Shape Models . . . 17

2.3.1 A tive Shape Models . . . 17

2.3.2 SSMBased on MinimumDes riptionLength . . . 18

2.4 Segmentation Using Shape Priors . . . 21

2.4.1 Deformable Models . . . 22

2.4.2 Expli itlyRepresentedShape Priors . . . 24

2.4.3 Impli itlyRepresentedShapePriors . . . 25

2.5 Dis ussion . . . 26

3 A Generative Gaussian Mixture Statisti al Shape Model 27 3.1 Motivation. . . 27

3.2 Expe tationMaximization - ICPAlgorithm . . . 29

3.2.1 Algorithm . . . 29

3.2.2 Generalizationto Ane Transformation . . . 32

3.2.3 EM-ICPMulti-S aling . . . 33

3.3 TheUnied Framework . . . 37

3.3.1 TheGenerative Model . . . 37

3.3.2 Optimizationof Parameters througha Single MAPCriterion 38 3.4 Computationof theObservation Parameters . . . 42

3.4.1 Transformation . . . 42

3.4.2 DeformationCoe ients . . . 44

3.5 Computationof the Model Parameters . . . 45

3.5.1 Mean Shape . . . 45

3.5.2 StandardDeviation. . . 45

3.5.3 VariationModes . . . 46

3.6 Pra ti al Aspe ts . . . 49

3.7 Extension of theCriterionfor Non-Convex Stru tures. . . 50

3.7.1 Integrationof Normals . . . 51

3.7.2 Estimating Normalsfor Unstru tured Point Clouds . . . 52

3.8 Dis ussion . . . 52 4 Evaluation of the GGM-SSM 55 4.1 Performan eMeasures . . . 55 4.1.1 Assessing SSMQuality . . . 55 4.1.2 Distan e Measures . . . 58 4.2 Comparison to anICP-SSM . . . 59 4.2.1 Syntheti Data . . . 59

4.2.2 Brain Stru ture MR:Putamen . . . 66

4.3 Comparison to ICP-SSMand MDL-SSM . . . 69

4.4 Unsupervised Classi ation . . . 74

4.5 Dis ussion . . . 75

5 Using the GGM-SSM as a Prior for Segmentation 79 5.1 Initialization . . . 80

5.1.1 Distribution Modelsfor Prior IntensityKnowledge . . . 80

5.1.2 Initial Pla ement Problem . . . 81

5.2 The GGM-SSMinImpli it Fun tion Segmentation . . . 82

5.2.1 Segmentation Using LevelSets . . . 83

5.2.2 MAP Estimationon the Level Sets . . . 85

5.2.3 Derivationof the Energy Fun tional . . . 87

5.2.4 Optimization oftheEnergy Fun tional . . . 90

5.3 Evaluationon KidneyCT Images . . . 91

5.3.1 Segmentation Experiment . . . 93

5.3.2 The Roleof theParameters . . . 96

5.4 Multiple Shape Class Segmentation . . . 97

5.4.1 Development of theAlgorithm . . . 98

5.4.2 ExperimentalEvaluationon HipJoint CTs . . . 101

5.5 Dis ussion . . . 108 6 Con lusion 111 6.1 Contributions . . . 111 6.1.1 ModelComputation . . . 111 6.1.2 Segmentation . . . 113 6.2 Perspe tives . . . 115 6.2.1 Parameters . . . 115 6.2.2 Appli ation . . . 115 6.2.3 Related Work . . . 116

A Mathemati al Ba kground 119

A.1 Mathemati alPrepositions . . . 119

A.2 TheICPasa spe i aseof theEM-ICP . . . 121

A.3 Mathemati alDerivations Chapter 3 . . . 121

A.4 Mathemati alDerivations Chapter 5 . . . 124

A.4.1 Divergen e Cal ulus . . . 124

A.4.2 Helpful Derivations . . . 125

Abbreviations and A ronyms 126

Introdu tion Contents 1.1 Motivation . . . 1 1.2 Obje tives . . . 2 1.3 Stru ture of Manus ript . . . 3 1.4 ListofPubli ations . . . 6 1.1 Motivation

Sin ethe dis overyof X-rays in1895, manydierent imagingte hniqueshavebeen

developedwhi hgainvisuala esstotheinteriorofa losedbodywithoutopeningit.

Nowadays,these te hniquesarewidelyusedinhealth- are and biomedi alresear h

and onstitute a substantial part of the lini al pra ti e. In order to fa ilitate the

interpretationofthegeneratedbodyimages,amultitudeofmedi alimageanalysing

methodshasbeen realizedwhi hsupportthephysi iansintheeldsofdiagnosti s,

surgi al planning and image guided surgery as well as medi al resear h. With the

progress of image a quisition te hniques, the modeling of anatomi al stru tures in

3Doreven4Dhasbe omeanimportant omponent inmedi alimage omputingas

thesemodelsoeranadditionalperspe tiveforthesurgeonsandareusedfor

model-based analysis, segmentation and lassi ation problems. A popular approa h for

shape modeling is onstituted bystatisti almethods whi haim to represent an

or-gan bystatisti alshapemodels. Asopposedto asingle3Dmodel or anatlas ofan

organwhi hareonly(typi al) shapeexamples, astatisti alshapemodelrepresents

aset ontainingsegmentedorgansbyameanshapeandavariabilitymodel. Hen e,

statisti alshapemodels in orporatea priorishapeknowledgedrawnfrommany

or-gan examples. Espe ially for segmentation problems, theappli ation of statisti al

shape models hasbeen proven to bevery su essfulfor a wide rangeof anatomi al

stru turesinCT, MRandultrasound images.

Theideaofdoingstatisti sonshapesrstleadstotheproblemofdistin tlydening

the on eptofa shape. A wellknowndenitionproposedbythemathemati ianD.

G. Kendallin1984 reads asfollows: "Shape isall thegeometri alinformation that

remains whenlo ation,s ale and rotational ee ts arelteredout from an obje t"

[Kendall 1984℄. However, when dealing withanatomi al stru tures, amore exible

denitionisneededwhi h alsore ognizesdeformable obje tsbasedontheirshapes.

Therefore,atleastee tslikeexionandshearinghavetobeintegrated. Thismeans

The hara teristi s of a statisti al shape model essentially depend on the

repre-sentation of the shape surfa e. Basi ally, a surfa e an be seen as a boundary

whi h separatesgeometri al regionsin3D. Mostly,itisrepresented expli itlyusing

meshes or point louds or impli itly basedon distan e fun tions. Inorder to

om-pute a surfa e representation for a binaryobje t, a sampling of the isosurfa e has

to be performed. The sampling isa ru ial stepwhi h - together with theimaging

te hnique -determines the detailedness oftheresulting surfa e model.

A fundamental problem for the omputation of statisti al shape models is the

de-terminationof orresponden esbetweentheobservations. Inordertoquantitatively

analyse shape dieren es, a method is needed to lo ate a orresponding point

lo- ation on one shape for a given point lo ation on another shape. Obviously, the

solution tothis problemalways dependsonthe shaperepresentation. Most urrent

methods relyonsurfa e-basedrepresentationsandworkwithone-to-one

orrespon-den es. Theydo not onsiderthe un ertainties neitherof thesegmentations nor of

thesampling output nor ofthe registration results. Moreover, even for theutopian

ase ofperfe t segmentationand ontinuoussurfa erepresentation, orresponden e

determination isnevernon-ambiguous butfor reprodu ible prominentlandmark

lo- ations.

The motivation of this thesis is to develop an alternative statisti al shape model

whi h takes into a ount the un ertainties of the whole s ene and to investigate

methods of applying this model for automati segmentation. Most urrent

algo-rithms ompute the mean shape and variability model on a step-by-step basis.

Therefore, a spe i goal of this thesis is to realize the model omputation in a

sound mathemati al framework.

1.2 Obje tives

Following the motivation phrased inthe previous se tion, we argue thatwhen

seg-mentinganatomi alstru turesinnoisyimage data,thesampledsurfa epointsonly

representprobablesurfa e lo ationsandnot ne essarilytheexa t"true"shape

sur-fa e. Besides, the hoi e ofsampling method signi antly inuen es thestatisti al

analysis ofthe shapes. For instan e,when the samebinary obje tissampledtwi e

with dierent resolutions, the resulting surfa e representations will not be

identi- al whi h makesthedetermination ofexa t orresponden es impossible. Moreover,

even for theoreti ally perfe tly ontinuous surfa es, a unique and reprodu ible

de-termination of orresponden es is an open problem. It even be omes impossible if

one of the surfa es features a shape detail that the other one la ks. For an

illus-tration, imagine a re onstru ted head of the sphinx ontaining a nose, and then

imagine the hallengeof determininga orrespondingpoint for thetipof thatnose

on the original sphinxhead. It isdesirable to expli itly model theun ertainties of

the s ene. In order to ome up witha realisti modeling ofa surfa e basedon the

sampledpoints,the goalisto investigatethepossibilitiesofrepresentingtheshapes

ina probabilisti framework where ea h sampledsurfa e point is drawn from a3D

basedonasetofsegmentedorganshapesforwhi hthebeststatisti alshape model

mustbe omputed. In orderto developa theoreti alfoundation ofthealgorithm it

might be of interest to adoptan alternative view on theproblem ofmodel

ompu-tation. Thefo usofthis thesisliesonthe development of astatisti alshape model

based on orresponden e probabilities ina sound mathemati al framework and its

appli ation inmedi al image segmentation.

Thesedemands leadmainly to thefollowing threeobje tives:

•

Development of a probabilisti framework to ompute a generative statisti al shape model based on orresponden e probabilities: The

rstproblemta kledisthe omputationofagenerativestatisti alshapemodel

thatoptimally representstheshapesina trainingdataset. Theaim isto

de-sign apoint-based parametri model whi hallows themodeling of variability

for ea h point. This might help physi ians to physi ally interprete the

de-formations. The fo us lies on the development of a generative probabilisti

framework whi h in ludes all variables needed to des ribe the s ene.

Ad-ditionally, the framework has to integrate a solution to the orresponden e

problem.

•

Development ofa deformablemodel segmentation ina probabilisti framework: A major problemin medi alimage pro essingis theautomati

segmentation of anatomi al stru tures. Therefore, the se ond problem to be

dealt withis the integration ofthegenerative statisti al shape modelinto an

automati segmentation s heme. The obje tive isto develop asound

mathe-mati al formulationwhi h is based on thesame probabilisti assumptions as

the framework for the omputation of the statisti al shape model. It is

in-tended to develop a segmentation algorithm whi h enables the segmentation

ofobje ts withnon-spheri al topology aswell asmultiple-obje t shapes.

•

Evaluation and validation with respe t toexisting methods: Amain advantage of working withpoint-based shape representation isthe simpli ity

of the resulting modelwithrespe tto its power. Ontheotherhand,

surfa e-basedmodelsgenerallyfeaturebetterqualitymeasuresthanpoint-based

mod-els. However,thequalityofthesurfa einformationtheyusedependsonimage

qualityandonthesegmentationmethod(whi hisoftenbasedonpointsdrawn

byexperts). Inordertopla ethenewmethodinthestate-of-the-art, itis

ru- ialtoevaluatethequalityoftheprobabilisti modelin omparisonwithother

statisti al shape models, investigate appli ations like lassi ation methods

and expose advantages and limits ofthe new model. Se ondly, an evaluation

of the segmentation method on dierent real data segmentation problems is

needed in order to identify the strengths of the method with respe t to the

state-of-the-art.

1.3 Stru ture of Manus ript

analysis. Chapter 3, 4 and 5 ontain the main ontributions regarding the

development and appli ation of a new statisti al shape model and a new level set

segmentation method relying on the model. Chapter 6 on ludes the manus ript.

In the following,a ondensedsummaryis givenfor ea h hapter.

In Chapter 2 the ba kground information needed about urrent methods in

statisti al shape analysis is summarized. It begins with a des ription of the

state-of-the-art regarding the use and types of statisti al shape models. Then

the point orresponden e problem is overed in detail before dierent methods

forthe omputationofstatisti alshapemodelsandtheirappli ationsarepresented.

In Chapter 3 an approa h to the problem of designing a generative

statisti- al shape model is developed [Hufnagel2007b , Hufnagel 2008b ℄. First, a solution

to the point orresponden e problem is derived by representing the shapes by

Mixtures ofGaussians. Following that,a soundanduniedframeworkisdeveloped

for the omputation of model parameters and observation parameters as well as

nuisan e parameters, and a maximum a posteriori estimation is formulated whi h

leads to a global riterion. Expli it formulas are derived for its optimization with

respe t to all parameters. Finally, pra ti al aspe ts of the implementation and

adaptions ofthe algorithm for spe ial ases aredis ussed.

In Chapter 4 an evaluation and validation of the generative Gaussian

Mix-ture statisti al shape model as developed in this thesis is performed. First, the

hoi e of performan e measures is established. Then, the performan e of the new

statisti al shape model is ompared to the performan e of a lassi al point-based

statisti al shape model based on the iterative losest points registration and the

prin ipal omponent analysis [Hufnagel2009a ℄. Furthermore, the performan e

of the new statisti al shape model in omparison with a surfa e-based statisti al

shape model whi h is omputed by the minimum-des ription-length approa h is

evaluated. The evaluation is done on syntheti and real data. Dierent examples

overing onvexand non- onvex aswell asspheri and non-spheri shape dataare

hosen.

In Chapter 5 an automati segmentation algorithm is developed whi h

em-ploys the a priori shape knowledge inherent in the new statisti al shape model.

After explaining the benets of employing a non-parametri segmentation ontour

instead of a parametri one, the problem of integrating an expli itly represented

statisti al shape model into an impli it segmentation s heme is ta kled. To our

knowledge, very few works onsidered that option. The problem is solved by

developing a novel maximum a posteriori estimation of the segmentation ontour

whi h is optimized based on the image information as well as on the statisti al

shape model information. Here, the respe tive steps whi h nally leadto a sound

probabilisti segmentation s heme are explained elaborately. It is demonstrated

in detail how to optimally exploit theimage information to guide the evolution of

probabilities insteadofone-to-one orresponden es,themodeling andsegmentation

of non-spheri and multi-obje t stru tures is possible. Consequently, an extension

of the algorithm to multi-obje t segmentation is developed whi h is integrated in

the same framework by adapting the orresponden e riterion. Experiments are

designed and ondu ted in order to validate the segmentation method on kidney

data and on hip joint data. Finally, the results are riti ally dis ussed, and the

advantages and limits of this segmentation method are revealed. Part of this

hapteris publishedin[Hufnagel2009 ℄.

In Chapter 6 the ontributions of this thesis are dis ussed and perspe tives

for futurework aregiven.

Appendix A ontains the mathemati al ba kground and detailed

1.4 List of Publi ations

This thesis is a monograph whi h ontains unpublished material. It is however

largely basedon thefollowing international publi ations:

Generative Gaussian Mixture Statisti al Shape Model (GGM-SSM)

[Hufnagel 2007b℄: H. Hufnagel, X. Penne , J. Ehrhardt, H. Handels, and

N. Aya he. Shape Analysis Using a Point-Based Statisti al Shape Model Built on

Corresponden e Probabilities. In Pro eedings of the Medi al Imaging Computing

and Computer AssistedIntervention (MICCAI)2007, volume 4791 ofLNCS,pages

959-967, 2007.

[Hufnagel 2007a℄: H. Hufnagel, X. Penne , J. Ehrhardt, H. Handels, and

N.Aya he. Point-BasedStatisti alShapeModels withProbabilisti Corresponden es

and AneEM-ICP. Bildverarbeitung für dieMedizin 2007,Springer Verlag,pages

434-438, 2007.

[Hufnagel 2008b℄: H. Hufnagel, X. Penne , J. Ehrhardt, N. Aya he, and

H. Handels. Generation of a Statisti al Shape Model withProbabilisti Point

Cor-responden es andEM-ICP.International Journal forComputer AssistedRadiology

and Surgery(IJCARS)vol. 2,no. 5,pages265-273, Mar h2008.

[Hufnagel 2008 ℄: H. Hufnagel, X. Penne , J. Ehrhardt, H. Handels, and

N. Aya he. A Global Criterion for the Computation of Statisti al Shape Model

Parameters Based on Corresponden e Probabilities. Bildverarbeitung für die

Medizin 2008, Springer Verlag,pages277-282, 2008.

Evaluation of the GGM-SSM

[Hufnagel 2008a℄: H. Hufnagel, X. Penne , J. Ehrhardt, N. Aya he, and

H.Handels. Comparisonof Statisti al Shape Models Builton Corresponden e

Prob-abilities and One-to-One Corresponden es. InPro eedingsof theSPIE Symposium

onMedi alImaging2008,vol. 6914ofSPIEConferen eSeries,pages4T1-4T8,2008.

[Hufnagel 2009a℄: H. Hufnagel, J. Ehrhardt, X. Penne , N. Aya he, and

Heinz Handels. Computation of a Probabilisti Statisti al Shape Model in a

Maximum-a-posteriori Framework. Methods of Information in Medi ine, vol. 48,

no. 4,pages 314-319,2009.

Segmentation Using the GGM-SSM

[Hufnagel 2009b℄: H. Hufnagel, J. Ehrhardt, X. Penne , and H. Handels.

Appli ation of a Probabilisti Statisti al Shape Model to Automati Segmentation.

Engi-[Hufnagel 2009 ℄: H. Hufnagel, J. Ehrhardt, X. Penne , A. S hmidt-Ri hberg,

and H. Handels. Level Set Segmentation Using a Point-Based Statisti al Shape

Model Relying on Corresponden e Probabilities. In Pro eedings of the MICCAI

Workshop Probabilisti Models for Medi al Image Analysis (PMMIA)2009, pages

34-44, 2009.

[Hufnagel 2010℄: H. Hufnagel, J. Ehrhardt, X. Penne , A. S hmidt-Ri hberg,

and H. Handels. Coupled Level Set Segmentation Using a Point-Based Statisti al

Shape Model Relying on Corresponden e Probabilities. In Pro eedings of theSPIE

Current Methods in Statisti al

Shape Analysis

Contents

2.1 Shape Modeling in Medi al Imaging . . . 9

2.2 The Corresponden eProblem . . . 12

2.3 Computationof Statisti al Shape Models . . . 17

2.4 SegmentationUsing Shape Priors . . . 21

2.5 Dis ussion . . . 26

The extra tionofinformation out of 2Dor 3Dimages oftenrelies onthe

dete -tion, re ognition and interpretationof shapesand shape variabilities. This dire tly

involvesthe (mathemati al)representation ofshapesaswellasmethodsto a ount

forandmeasurethemorphologi aldieren es. Eventhoughin lini alroutineshape

analysisisfrequently donebyviewingtheimagesalone, thereisawiderangeof

ap-pli ationswhereautomati almethodswithformalizedmetri sareneededforoverall

datainterpretationandshapestatisti s. This hapterisdedi atedtothedes ription

ofthesemethodsandisdividedasfollows: First,theimportan e ofshapemodeling

in medi al image analysis is outlined and the on ept of statisti al shape models

and their representations are dis ussed in se tion 2.1. Following that, we expand

on the fundamental problem of determining orresponden es between shapes and

on several methods of solution (se tion 2.2) whi h dire tly leads us to dis uss the

asso iated statisti alshape models in se tion2.3. Se tion 2.4explores the benets

of statisti alshape models for medi al image segmentation and dis ussesexpli itly

and impli itlyrepresentedshape priors.

2.1 Shape Modeling in Medi al Imaging

Shapemodelsareusedforawide rangeofmedi alimagingproblemslike

segmenta-tion, re onstru tion or shape analysis. In this se tion, a ondensed overviewabout

thedomainofshapeanalysiste hniquesinnowadays medi alresear hisgiven

(se -tion2.1.1)andthenthesubje tofdoingstatisti sondierentshaperepresentations

isintrodu ed (se tion2.1.2).

2.1.1 Shape Analysis

under-is to nd information based on the shape deformation or shape dieren es whi h

eventually helpinthediagnosti s, espe iallyin the neuroimaging ommunity. The

modeling of shape and themeasuring of morphologi al hanges inshape instan es

is also of great interest for the dis rimination between healthy and pathologi al

anatomi al stru tures. An intuitive approa h for dete ting shape dieren e is the

measurement of theglobal shape volume, however, this feature isoften not

signi- ant withrespe ttothe studieddisease. Thishasbeen shownfor examplebyGerig

etal.[Gerig 2001 ℄basedonthedete tionofgroupdieren esinhippo ampalshapes

in s hizophrenia. Resultsof higher signi an e are oftenobtained byperforming a

lo alshape analysis. A wide rangeof approa hes existsintheliterature whi h an

beroughly ategorized a ordingto the(shape)features hosento dothestatisti s

on. Inthefollowing,an overviewof developmentsinthateldis given bymeans of

exemplarily sele ted publi ations.

Earlymethodsproposedtoanalyseand ompare thetransformationelds obtained

whenregisteringanorgantoatemplate,whi hisusede.g.intheworkofDavatzikos

etal.[Davatzikos1996 ℄whoanalysethemorphologyofthe orpus allosum. A

sim-ilar idea is applied in the work of Boisvert et al. [Boisvert 2008 ℄ who model spine

shapedeformation bya ve tor of rigid transformations. First eorts in

mathemat-i ally apturing morphology hanges by doing statisti s on anatomi al landmarks

wereundertaken byF.L.Bookstein[Bookstein1986 , Bookstein 1991 ℄. The on ept

of statisti al shape analysis based on landmarks and pseudo-landmarks was taken

on byDrydenand Mardia [Dryden 1993 ℄ for thedete tion of genderrelated

dier-en esinmonkey raniaandbyBookstein[Bookstein1997 ℄forthedete tionofbrain

dieren es in s hizophrenia patients. In both approa hes, the shape variations are

measured based on Pro rustes or Riemannian distan es. Another shape analysis

method is based on a medial shape des ription to model lo al and global hanges

as e.g. used by Styner et al. [Styner 2003b ℄ who analyse the hippo ampus shape

of s hizophrenia patients. In several works the shapes are represented by distan e

fun tions whose feature ve tors are used as input for a learning algorithm, e.g. in

the work of Golland etal. [Golland 2001 ℄ who ompute a lassierfor healthy and

pathologi alhippo ampal shapesins hizophrenia or intheworkof Kodipakaetal.

[Kodipaka 2007 ℄ whose Kernel Fisher dis riminant distinguishes between ontrols

and epilepti sbyanalysing the shapeofthetemporal lobeor inthework ofTsaiet

al. [Tsai 2005 ℄ who proposean EM formulation to automati ally label lungshapes

representedbylevelsetfun tionsto belongtothe normal or theemphysema shape

lass. IntheworkofPeteretal.[Peter 2006a ℄,shapesarerepresentedbyaGaussian

MixtureModelonthelandmarks,andtheshapedieren es(hereof orpus allosum

shapes)aremeasuredusing geodesi distan es undertheFisher-Raometri .

Naturally,all of theseapproa heshave their strengths and weaknesses. The hoi e

of feature suited as inputfor the statisti alanalysis depends on therepresentation

of the shapes aswell ason the demands of the appli ation. The work done inthe

framework of this thesis on entrates on the ategory of shape analysis based on

point representations sin e statisti s on points are easily interpretable and have a

physi al signi an e. The general on ept however is not ne essarily onned to

2.1.2 Doing Statisti s on Shapes

Commonly, a shape lass an be des ribed by one typi al shape example of the

respe tiveorgan. However, thisapproa hisneitherspe i nor mathemati ally

a - urate. In orderto reliablydes ribea shape lass, we need to statisti allyevaluate

theshapesofasmanyobservationsoftheorganaspossible. Thisisusually donein

four steps: First, atraining data setwhi h ontains segmented observations of the

respe tiveorganhasto beprovided. Next,the observations have to be aligned ina

ommon referen eframeinorderto eliminateposevariations. Then, ameanshape

whi h optimally represents all aligned observations an be omputed. Finally, a

variabilitymodela ountingfortheshapedieren esisdetermined. Thevariability

ontains information about how mu h and in whi h way the mean shape an be

deformed whilestill representing a plausibleanatomi al stru ture.

In the state-of-the-art, shape models ontaining a mean shape and a variability

modelarereferredtoasstatisti alshape models(SSMs). Themethods

implement-ing the alignment as well as the statisti al methods used for the omputation of

meanshapeandvariabilitymodeldependontherepresentationoftheobservations.

An intuitive and widely-used method is to ompute SSMs on observations

repre-sentedby(triangulated) pointswhi haredistributedoverthesurfa eoftheshapes.

These so- alled point distribution models (PDMs) are either based on anatomi al

landmarks[Huysmans2005 ℄,onpseudo-landmarksthatarestrategi allydistributed

overthe observations' surfa es(e.g.[Frangi 2001,Rajamani 2004 ℄) or on points

re- onstru ted from impli it surfa es (e.g. [Kohlberger 2009 ℄) or on a ombination of

these. Point-based shape samples represented by a number of

N

points in 3D are

usually des ribed by a shape ve tor

S

k

∈ R

3×N

ontaining the point oordinates.

Thealignment to a ommon referen e frameis oftenperformedbya mesh-to-mesh

registration over the shape ve tors. The statisti evaluation then uses the aligned

shape ve torsasinputfor omputation of meanshape andvariability model.

For thesesteps, anotionof orresponden ehastobedened. A ommon approa h

isto assumeand determine one-to-onepoint orresponden esoverall observations.

In that ase, the oordinates of orresponding points are sorted in orresponding

entry positions in the shape ve tors sothat for all shape pairs

S

k

and

S

l

the i-th element

S

k

(i)

orresponds to

S

l

(i)

for all

i = 1, ..., 3N

. The omputation of the meanshape isthenstraightforward with

M =

¯

1

n

P

n

k=1

S

k

fora numberof

n

obser-vations. The subsequent omputation of variation modes is usually a omplished

by a prin ipal omponent analysis (PCA)on all observations and themean shape.

Thevariationmodes

∈ R

3N

arepairwiseorthogonalandspantheshapespa eofthe

SSM. Mathemati ally, the representation ofa random shape

M

inthe shapespa e

spannedbythe variationmodes anbeformulated usinga linearmodel:

M = ¯

M + P b

where the matrix

R ∈ R

N ×N

′

with

0 < N

′

≤ N

ontains the variation modes in

its rows and the ve tor

b ∈ R

N

ontains the oe ients whi h ontrol the extent

of deformation. The variation modes are ranked a ording to their varian e. For

The employment of the PCA is not onned to point representations but an be

employed to otherappli ationswhere theshapepropertiesareen oded ina feature

ve tor. Early methods in lude the representation of shapes by spheri al

harmon-i s (SPHARM) whi h parameterize the surfa e by a mapping on the unit sphere

[Bre hbühler 1995 , Székely 1996 ℄ or by Fourier surfa es whi h employ an ellipti

Fourier de omposition of the boundary and use the Fourier oe ients as feature

ve tors[Staib 1996 ,Floreby1998 ℄. Thestatisti sarethusdone inparameterspa e.

Re ently,therepresentation ofSSMs inimpli itframeworkshasbe ome ofinterest

as level setbased segmentation is explored more deeply. Here, the observations in

the training data set are often represented by signed distan e maps. The

align-mentoftheobservationsandthe subsequentstatisti sarethendonedire tlyonthe

distan e maps whi h are used as feature ve tors with individual voxels being the

ve tor omponents. The variabilitymodels an simplybe omputed by a prin ipal

omponent analysis [Leventon 2000a ℄ or by using more hallenging methods whi h

for example a ount for lo al variations as well [Rousson2002 ℄. Another strategy

represents the surfa es bymedial models whi h onsist of a enterline and ve tors

stret hing from there to the organ surfa e [Pizer1999, Styner2001 ℄. Here,

orre-sponden e between shapes aredened on themedial manifold. For omputing the

variabilityofmanifold-valueddata,aprin ipalgeodesi analysisisintrodu edwhi h

is adire tgeneralization of prin ipal omponent analysis.

It has to be kept in mind that the PCA is done under the assumption that the

shape ve torsare samples of a random variable under a normal distribution. This

is only the ase if the shape dieren es in the training data set are normally

dis-tributed whi h is di ult to establish. Another theoreti al problem o urs as the

dimensionsoftheshaperepresentation nearlyalwaysex eedthenumberofavailabe

samples. Besides,the PCAisoptimalinaleast-squaresenseandthereforesensitive

to outliers and lastly,all shapes have to be represented byfeature ve torsof equal

lengths. Nevertheless, the employment of the PCA for SSM omputation hasbeen

proven to ome to a eptable results and is su essfully applied in pra ti e. An

alternativefornon-normallydistributeddataisoeredbytheso- alledindependent

omponentanalysis(ICA)[Hyvärinen 2001 ℄. TheICAde orrelatesthe omponents

by maximizing their statisti al independen e. Another interesting approa h is to

do a prin ipal fa tor analysis(PFA) whi h leads to variation modes thatare more

easily interpretable in medi al sense [Ballester2005, Reyes2009 ℄. However, these

methods have the disadvantage that the variation modes annot be ranked easily

whi h posesa problemfor dimensionalityredu tion.

2.2 The Corresponden e Problem

A fundamentalproblemwhen omputing statisti alshapemodelsisthe

determina-tion of orresponden es between the observations in thetraining data set.

Mathe-mati ally,this problemdoesnothavea uniquesolutionanddependsheavilyonthe

denitionof 'shape'aswell ason itsrepresentation. Forshapesrepresentedas

between the ngers, and by then adding a xed number of equidistant landmarks

betweenthese. Inthatway,the orresponden esfromonelabeledshapetothenext

equallylabeledoneisstraightforwardanduniquelydened. In3D,however, a

man-ualdetermination of orresponden eis hardlyfeasible asitis verytime- onsuming

ingeneral. Inparti ular, the pinpointing of exa t orresponden es without relying

on lear anatomi al landmarks on 3D surfa es is an impossible task. In order to

automatizethe dete tionoflandmarks,severalmethodsextra tshapefeaturessu h

as high surfa e urvatures (e.g. [Benayoun 1994 ℄). Mostly however, automati

de-termination of orresponden es is done by performing a registration of model and

observation. Obviously, the solutions to the orresponden e problem highly

de-pend on the shape representations. For meshes, a straightforward approa h is to

ompute a similarity transformation found byleast-square point distan e

minimiz-ers. For non-linear registration,often spline-baseddeformations areused. Another

approa h is the mat hing of an atlas or template mesh to all observations in the

trainingdataset. Thewarped mesheshavetoberelaxedinorderto tthe

observa-tions. This an be donefor examplebyusingaMarkovrandom eldregularization

as proposed by Paulsen and Hilger [Paulsen 2003 ℄ or by employing a spring-mass

modelbasedonthesurfa epointsetandthe onne tingedgesasrealizedbyLorenz

andKrahnstöver[Lorenz 2000 ℄. Amethodforvolumetri representationsisto

om-pute a volumetri atlas withmanually added surfa e landmarks and then register

the atlas to volumetri ally represented observations. The warped landmarks then

determine the orresponden es.

In this se tion, two popular methods for orresponden e determinations are

de-s ribed basedondierent shape representations whi h playa roleintheremainder

of this thesis: First, the lassi al Iterative Closest Points (ICP) registration

algo-rithm that nds one-to-one orresponden es between two unstru tured point sets

isexplained. Then, an alternative approa h to orresponden e determinationusing

spheri al harmoni s surfa es parameterization is presented. Here, the

orrespon-den esare omputed byaregistrationbetweentheparameterizationsoftheshapes.

As an example for methods whi h solve the orresponden e problem in a

group-wiseoptimization approa h togetherwith the SSM omputation the maximum

de-s ription length (MDL) approa h is summarized in se tion 2.3. A omprehensive

omparison of dierent solutions to the orresponden e problem an be found in

[Styner 2003 ℄.

2.2.1 Iterative Closest Point Algorithm

The Iterative Closest Point algorithm is an e ient method used for registration

of 2D and 3Dshapes asrst shownand elaborately explained 1992 in[Besl 1992℄.

The ICPregistration is an interesting approa h asit an be used for dierent

rep-resentationsof geometri data like point sets,triangle sets, andimpli it or expli it

surfa es. Itisappliedto registrationproblems wherethepoint orresponden es are

not known in advan e. The ICP algorithm oers many re ognized advantages as

it does not need prepro essing or lo al feature extra tions in normal appli ations,

is given.

Let

S

beasetof

N

s

points

s

i

whi hdes ribetheobservationand

M

beasetof

N

m

points

m

j

whi hdes ribethemodel. TheICPalgorithmwillmat hea hobservation point

s

i

withone ofthemodelpoints. Basedon thosemat hes,a transformation

T

issoughtwhi hregisterstheobservationwiththemodel. The losestpointoperator

CP

isdened asadistan e metri

CP (s

i

, M ) = min

m

j

∈M

km

j

− s

i

k.

Weuse

m

i

j

= CP (s

i

, M )

where

m

i

j

is the losestpoint in

M

to a given s ene point

s

i

. TheICPalgorithm omputing

T

is implementedasfollows: 1.

T

(0)

_{= T}

k

is hosenasinitial estimate ofthetransformation

T

.

2. Repeatfor

k

iterations or until onvergen e:

•

Compute the losest point

m

i

j

∈ M

in the model for ea h observation point

s

i

∈ S

. The olle tion ofresulting point pairs

(s

i

, m

i

j

)

is alledset of orresponden es

C

with

C

_k−1

= ∪

N

s

i=1

{s

i

, CP (T

k−1

⋆ s

i

, M )}.

•

Compute

T

k

that minimizes the mean square error between all point

pairs in

C

.

For arigid registration,the appli ation of

T

to

S

lookslike this

T ⋆ s

i

= Rs

i

+ t

∀i

withtherotationmatrix

R ∈ R

3x3

andthetranslationve tor

t ∈ R

3

. The

minimiza-tion of the error between all point pairs in

C

is omputed using theLeast Squares

riterion:

T

= argmin

T

1

N

s

N

s

X

i=1

km

i

_j

− T ⋆ s

i

k

2

= argmin

R,t

1

N

s

N

s

X

i=1

km

i

_j

− Rs

i

− tk

2

.

The ICP algorithm onverges always monotoni ally to the nearest lo al minimum

where nearest ismeant inthesense of amean-square distan emetri .

As main disadvantage itmust be notedthat theICPis sus eptible to gross

statis-ti al outliers. Several approa hes deal with this problem by e.g. proposing robust

estimators [Zhang1994 , Masuda1996 ℄. Moreover, as any method minimizing a

non- onvex ostfun tion,the ICPla ksrobustnesswithrespe ttotheinitial

trans-formation be ause of lo al minima. This problem has been ta kled by the work

m

j

s

1

s

s

s

s

2

3

4

5

?

?

?

Figure 2.1: A orresponden e problem: One shape features two bumps, the other

onlyone. How an we determine orresponden es between the two?

Overall,theICPalgorithmanditsderivativesworkwellforalotofregistration

prob-lems. However, the determination of one-to-one orresponden es between

unstru -tured point sets is di ult when e.g. one shape features a ertain stru ture detail

andthe otherone doesnot,seegure2.1. Moreover,intheabsen eof(anatomi al)

landmarks, the determination of orresponden e depends heavily on the sampling

of theshape. To over ome this problem, the Expe tation Maximization - Iterative

ClosestPoints(EM-ICP) algorithm introdu es orresponden e probabilities instead

of exa t orresponden es. This on ept isexploredin se tion3.2.

2.2.2 Spheri al Harmoni Des ription

The use of spheri al harmoni s for statisti al shape modeling was introdu ed by

Bre hbühler et al. in 1995 [Bre hbühler 1995 ℄ in order to approximate one-to-one

orrespondingpointsondierent entities ontainingin lusionsandprotrusions. As

opposedto theuseofatorusparameterspa eusingFourier des riptorsasproposed

in [Staib 1992 ℄, the SPHARM surfa e des ription maps the observation surfa es

into a spheri al two- oordinate spa e, so it an only be onsidered for shapes with

spheri al topology whi h applies for most anatomi al stru tures. If the mapping

in ludesanoptimizationofthedistributionofnodesonthesphere, orresponden es

an thenbeestablisheddire tly bytheparametri des ription.

Surfa eobje tswithspheri altopology anbeparameterizedbytwopolarvariables,

thelongitude

θ = [0, ..., 2π]

and thelatitude

φ = [0, ..., π]

. Two verti es have to be

sele ted asthe poles forthis pro ess. Thelatitude shouldgrow smoothlyfrom

0

at

thenorth poleto

π

at the south pole. The longitudeon the other hand isa y li

parameter. Letx,yandz denoteCartesianobje tspa e oordinates. Thefun tion

whi h spe iesthe mapping of the oordinatesfrom theunitsphere on thesurfa e

isspe ied with

v(θ, φ) =





x(θ, φ)

y(θ, φ)

z(θ, φ)





.

B-splines or wavelets. The SPHARM algorithm makes use of spheri al harmoni s

asthey oerthe advantageof hierar hi al shape representation whi hnally

fa ili-tatesthe orresponden edetermination[Bre hbühler 1995 ℄. Typi ally,thefollowing

trun ated seriesexpansion isused:

v(θ, φ) =

R

X

r=0

r

X

−r

c

m

_r

Y

_r

m

(θ, φ)

where

Y

m

r

denotesthefun tionofdegree

r

andorder

m

with

Y

m

r

: [0, 2π]×[0, π] → C

.

A omplete denition an be found in e.g. [Bronstein2000℄. The shape des riptor

oe ients

c

m

r

are 3D ve torswith omponents

(x, y, z)

. Formally,the oe ients are omputedbythe inner produ toffun tion

v

and thebasisfun tion

c

m

_r

=

Z

π

0

Z

2π

0

v(θ, φ)Y

_r

m

(θ, φ)dφ sin θdθ.

(2.1)

Eventually,ea h shape surfa e

S

k

isuniquely des ribed bya set ofdes riptor oef- ients

C

k

= c

m

k,r

.

Due to the hierar hi al shape representation, inpra ti e thelevel of shape details

whi h are modeled depends on the maximal degree

R

in the spheri al harmoni s.

The parameterization fordegree

1

mapsthesurfa e to an ellipsoid. In orderto

de-termine shapepoint orresponden es byparameterizationto asphere,themapping

between surfa e and spheremust be bije tivewhi h isdes ribedinthis ase by





x

y

z





=





sin θ cos φ

sin θ sin φ

cos θ





.

Furthermoreitmustbe ontinuoussothatneighbouringpointsontheshapesurfa e

aremapped to neighbouring lo ationsonthesphere. The mappingfun tion should

be topology-preserving, and distortions whi h inevitably appear when mapping a

surfa e fa et to a spheri al square should be minimal. This is done by solving the

surfa e parameterizationasa onstrainedoptimization problemwithrespe ttothe

optimal oordinates for allsurfa e points[Bre hbühler 1995 ℄. Another problem

o - urs as the oe ients obtained by approximating equation (2.1) depend on the

rotation of the surfa e in spa e. Thus, for the determination of orresponden es

between dierent shape observations, a rotation of all observations to a anoni al

position in parameter spa e is needed. This an be done using the spheri al

har-moni s of degree

1

byrotating the parameter spa e so that the north pole (where

θ = 0

) is positioned at one end of the shortest main axis of the ellipsoid, and the

point where the Greenwi h meridian (

φ = 0

) rosses theequator (where

θ = π/2

)

is positioned at one end ofthelongestmain axis.

Thestatisti sontheshapesarenowdonebyevaluationoftheshapedes riptors. The

mean shape thenis des ribed bythespheri al harmoni s using the setof averaged

shapedes riptor oe ients

C =

¯

1

N

P

N

k

C

k

andthe prin ipal omponentanalysisis done usingthe ovarian ematrix

1

N −1

P

While the SPHARM parameterization is apable to smoothly represent highlevels

of shape details, it suers from the fa t that for shapes featuring rotational

sym-metryinthe spheri alharmoni s ofdegree

1

the mappingto the anoni alposition

inparameter spa e isnot unique. Therefore,the orresponden edetermination for

su h shapes be omes inappropriate as shown in a study on e.g. femoral heads by

Styneret al.[Styner 2003 ℄.

2.3 Computation of Statisti al Shape Models

In order to ompute a SSM, a su iently large training data set with segmented

organ observations is needed. Obviously,thetraining data set should only ontain

shapes onformingtotheshape lasswhi hismodeled, thatis, foraSSMofnormal

organ variability, only healthy patient data is permitted. Ea h observation has to

be segmented a urately. This is mostly done manually or semi-automati ally by

medi al experts who delineate the organ ontours sli e by sli e in medi al images.

Someorgans anbesegmentedalsoin3Dunderthesupportofautomati te hniques

like volume growing of thresholding. For binary segmentation, the onversion to

a surfa e representation is typi ally performed by the Mar hing Cubes algorithm

[Lorensen 1987℄. The rst step is ommonly the alignment of the observation in

a referen e oordinate system. Then, a mean shape and a variability model are

omputed su h asto optimally represent the shapesinthetraining dataset. Here,

thea uratedete tionof orresponden ebetweentheshapesplaysanimportantrole

regarding the quality of the nal SSM. The resulting SSM produ es new plausible

shapes or represents unknown shape observations of the same organ in dierent

patientsor underdierent onditions.

In this hapter, the omputation of two widely-used point distribution models is

summarized: Se tion2.3.1des ribesthe lassi alA tiveShapeModels(ASM)while

se tion2.3.2 presentsamethodto build ASMsusing gradient des ent optimization

of themaximumdes ription length.

2.3.1 A tive Shape Models

With the introdu tion of the 'A tive Contour Models' (ASMs) or 'Snakes' in 1988

by Kass et al. rst attempts were made to integrate a priori knowledge into the

segmentation pro ess by for ing the segmentation ontour to omply to a ertain

amount ofsmoothness[Kass1988 ℄. Thete hnique makesuseof aniterative energy

minimizationwhere only lo alshape onstraintsareapplied. Cooteset al. adopted

an iterative approa h but instead of applying a simple snake ontour, they

devel-oped a point distribution model or 'A tive Shape Model' to in orporate a priori

knowledge about the shape [Cootes 1992 , Cootes1995 ℄. When applying the ASM

to segmentation,they useglobal shape onstraints.

Let us des ribe the

N

observations

S

k

inthe training data set by meshes onsist-ing of

n

k

points

s

ki

∈ R

3

. Furthermore, let us assume that

n

k

= n ∀k

and that the points with the same index

i

orrespond. The set of observations an then be

transformation

T

k

. Foranexampleseegure2.2(a). Ifthealignmentisomitted,the variationinsizeand posearein ludedinthe nalvariabilitymodel. Thepoints

m

¯

i

ofthemeanshape

M

¯

arethen omputedbyaveragingoverallaligned orresponding

observation points

m

¯

i

=

1

N

P

N

k=1

T

k

⋆ s

ki

.

For an illustration see gure 2.2(b). In order to ompute the variabilitymodel, a prin ipal omponents analysis (PCA) is

performed. Under the assumption of dealing with normallydistributed data

sam-ples,thePCA determinesa lineartransformation whi h transformsthedatainto a

oordinate systemwhere the axes (=eigenve tors) lie inthe same dire tion asthe

greatest orrelations in the data. By transforming the data into the new

oordi-nate system, the orrelations of the original data set be ome un orrelated. Thus,

the new axes lie in the dire tions of the greatest varian e of the transformed data

set. Hen e,the dataisrepresentedinasystemwhere itssimilaritiesanddieren es

an be seen learly whi h makes the PCA a well-suited tool in the des ription of

shape variability. The

N

a tual eigenve tors

v

p

and asso iated eigenvalues

λ

p

are omputed by e.g. doing a diagonalisation on the ovarian e matrix with elements

cov

ij

=

P

N

k=1

(s

ki

− ¯

m

i

)(s

kj

− ¯

m

j

)

T

N −1

,so

v

p

∈ R

3n

whi hamountstoone3Deigenve tor

v

ip

permean shape point

m

¯

i

,see gure 2.2( ). A plausible new instan e of the shape lass annowbemodeledby

M = ¯

M +

N

X

p=1

ω

p

v

p

(2.2)

where

ω

p

∈ R

are the deformation oe ients whi h are typi ally onstrained to

ω

p

≤ 3λ

p

inorderto onlygenerate plausible shapes. Furthermore,ashapeanalysis an be donebyinterpreting the deformationsa ordingto theeigenmodeswiththe

greatest eigenvalue (seegure2.2(d,e,f)).

InordertobetteradapttheASMtosegmentation,Cootesetal.proposedtheA tive

Appearan e Models (AAMs) whi h in orporate a priori knowledge not only about

the shape but also about mean and variation of the image intensities (appearan e

or texture). This prin iple an be adapted in a simplied manner to all point

distributionmodelsgiventhattheoriginalimagedataisstillavailable. Basi ally,the

grey valueappearan esaroundea hpoint

s

ki

inthe trainingdatasetareevaluated bysamplingthe pixelinformation oneithersideofthe ontour innormaldire tion.

Then a lo al statisti al appearan e model is onstru ted with mean prole and

asso iated variability. During the image sear h along the normal, the quality of

the urrent prole around the model points is assessed with respe t to the lo al

appearan e model.

2.3.2 SSM Based on Minimum Des ription Length

While theSPHARM modelaswellastheASMdetermine orresponden es

individ-ually forea h observation,othermethodspropose toassign orresponden esa ross

all observations at the same time. This approa h is driven by the idea that the

best orresponden es are those whi h lead to the optimal SSM given the training

indi-a) d)

b) e)

) f)

Figure 2.2: ASM example. a) Aligned observations of a training data set. Ea h of

the 5 observationsisrepresented by 10pointsin2Danddepi ted inanother olour.

b) Mean shape point loud depi ted by red dots. ) axesof rst eigenmode depi ted

for ea h of the orresponding points. d) Mean shape

M

¯

of point distribution model.

e,f)Mean shape deformed a ording torst eigenmode

¯

M − 3λv

1

and

¯

are found. The rst to introdu e this approa h were Kot he et al. who use the

determinant of the ovarian e matrix as obje tive fun tion for the omputation of

2DSSMs[Kot he1998 ℄. Byminimizingthe determinantofthe ovarian e matrix,

they expli itly favor ompa t models whi h means low eigenvalues and few

eigen-ve tors. Davies et al. take up on that idea but propose another obje tive fun tion

in order to nd a sound theoreti al foundation as well as to ensure onvergen e

[Davies2002 ℄. Theirkeyprin iple isto favour thesimplest solutionout of all

sat-isfying ones (following the prin iple of O am's razor). Furthermore, they dene

the model quality over three parameters, the ompa tness, the generalization

abil-ityand the spe i ity. Amodel ismore ompa t than another ifit odesthe same

variability information in less omponents. A great generalization ability means

that the model is able to des ribe unknown possible instan es of the shape lass.

A spe i model only represents valid instan es of theshape lass. Themethod of

Davies etal. introdu es the appli ation of theminimumdes ription length (MDL)

asmeasure forthe simpli ityof theSSM. Underthe MDLapproa h,thenalSSM

optimally balan es omplexity and the quality of t between model and

observa-tions. Originally, theMDL isa on ept usedininformation theory for theoptimal

odingof messages. While theMDL framework ismathemati ally soundand leads

toverygoodresults[Davies2002a,Styner 2003b ℄,theobje tivefun tionis omplex

and omputationally expensive. Several approa hes have been proposed to redu e

the omplexity. Heimannet al. employ a simplied MDL ost fun tion introdu ed

in [Thodberg2003 ℄ and use a gradient des ent optimization to minimize it. They

an show that their approa h is faster and less likely to onverge to lo al minima

than previousapproa hes[Heimann 2005℄. In this se tion, theprin ipal on ept of

their algorithm isexplained and themesh parameterization aswell asthe optimal

determination of orresponden es usedin their framework are outlined. The

algo-rithm is onstrained to SSMs oforgans withspheri al topology.

The ostfun tion

F

whi h isbased onthe MDL of theresulting SSMisdened as

F =

n

X

p=1

L

p

with

L

p

=

 1 + log(λ

p

/c

cut

)

for

λ

p

≥ c

cut

λ

p

/c

cut

for

λ

p

< c

cut

(2.3)

where

λ

p

denotes the squareroot of the eigenvalues of the ovarian e matrix. The parameter

c

cut

isa uto onstantwhi hdes ribestheexpe tednoiseinthetraining data.

Regarding the mesh parameterization, a mapping of allsurfa es to theunit sphere

isperformed. Themapping hasto assignforevery point onthesurfa eofthemesh

a unique position on the sphere. The problem of mesh parameterization is that of

mapping apie ewise linearsurfa e withadis reterepresentation onto a ontinuous

spheri al surfa e. In ontrast to Davies et al. who use initial diusion mapping,

Heimann etal. reate a onformal mapping thatfo useson preservingangles. The

fun tion

L

maps ea h point

s

i

of the surfa e

S

to the unit sphere whi h results in a spheri al parameterization of

S

. Themapping fun tion isdened as

L : S → R

3

with

|L(s

i

)| = 1

for all points

s

i

. The initialization is done bymapping ea h

s

i

to

map-who propose a variational method whi h an nd a unique mapping between any

two genus zero manifolds [Gu 2003 ℄. Basi ally, two steps are exe uted: First, a

bary entri mapping isperformedwhi h positions ea h point

s

i

at the enter ofits neighbouringpoints. Next,a onformalmappingisobtainedbytakinginto a ount

theangles between edgesof the mesh for theparameterization. The mathemati al

proofof orre tnessof this approa h isgiven in[Gotsman2003 ℄.

Afterobtaininga onformalmapping

L

k

forea hsurfa eobservation

S

k

, orrespon-den es a ross the training data set are determined by mapping a set of spheri al

oordinates to ea h

S

k

. Subsequently, the optimal orresponden es and therefore theoptimalpositionsof allpointson thesurfa eshave tobedetermined. Todoso,

Heimannetal. hoosetomodifytheindividualparameterizations

L

k

forallsurfa es: In short, the orresponding landmarks of all observations are leared of the mean

and then stored in a matrix

B

′

. By employing a singular value de omposition to

B =

√

1

n−1

B

′

,the eigenve tors and eigenvalues

λ

p

for the systemof orresponding landmarks an be omputed. Thismeans that the

λ

p

inthe ost fun tion in equa-tion (2.3) an be expressed independen e of the singular values of

B

. Eventually,

the ostfun tionisminimizedwithrespe ttotheelementsof

B

bysolving

∂F

∂b

ij

= 0

.

Thisderivationleadsto a hange fortheindividuallandmarkpositionsasshownin

[Eri sson2003 ℄ as it yields a 3D gradient for every landmark. In order to onvert

thegradientsinto optimal kernel movements

(△θ, △φ)

,

∂F

∂(△θ,△φ)

is omputed by

∂F

∂(△θ, △φ)

=

∂F

∂b

ij

∂b

ij

∂(△θ, △φ)

where the surfa e gradients

∂b

ij

∂(△θ,△φ)

areestimated bynite dieren es.

It has to be taken into a ount that when moving one landmark, the adja ent

landmarks should be ae ted in a similar manner depending on their loseness.

Therefore,a trun ated Gaussian fun tion isdened with

c(x, σ) =

(

exp(

−x

_2σ

2

2

−

−(3σ)

2

2σ

2

)

for

x < 3σ

0

for

x ≥ 3σ

where

x

denotes the distan e between the spe i landmark and the enter of

the kernel and

σ

ontrols the size of the kernel. If a point at position

x

is

moved by

(△θ, △φ

), all other points are ae ted by

c(x, σ)(△θ, △φ)

. This

re-parameterization is done iteratively over all landmarks and all observations. For

a detailed derivation of this algorithm as well as an evaluation please refer to

[Heimann 2005 ,Heimann2007 ℄.

Note thatthis approa h only makessense for mesh representations of surfa esbut

not for point loud representations.

2.4 Segmentation Using Shape Priors

Thegoalofasegmentationpro essisthepartitioningofanimageintoregionswhi h