and evaluation metrics Blood vessel segmentation algorithms —Review of methods, datasets Computer Methods and Programs in Biomedicine

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Computer Methods and Programs in Biomedicine

journalhomepage:www.elsevier.com/locate/cmpb

Blood vessel segmentation algorithms — Review of methods, datasets and evaluation metrics

Sara Moccia

^a,b

, Elena De Momi

^a

, Sara El Hadji

^a,∗

, Leonardo S. Mattos

^b

aDepartment of Electronics, Information and Bioengineering, Politecnico di Milano, Milan, Italy

bDepartment of Advanced Robotics, Istituto Italiano di Tecnologia, Genoa, Italy

a rt i c l e i n f o

Article history:

Received 3 November 2017 Revised 23 December 2017 Accepted 2 February 2018

Keywords:

Blood vessels Medical imaging Review Segmentation

a b s t r a c t

Background: Blood vesselsegmentationisatopicofhighinterestinmedicalimageanalysissince the analysis ofvesselsis crucialfor diagnosis, treatmentplanningand execution, and evaluationofclini- caloutcomes indifferentfields, includinglaryngology,neurosurgery and ophthalmology.Automaticor semi-automaticvesselsegmentationcansupportcliniciansinperformingthesetasks.Differentmedical imagingtechniquesarecurrentlyusedinclinicalpracticeandanappropriatechoiceofthesegmentation algorithmismandatorytodealwiththeadoptedimagingtechniquecharacteristics(e.g.resolution,noise andvesselcontrast).

Objective: Thispaperaimsatreviewingthemostrecentandinnovativebloodvesselsegmentationalgo- rithms.Amongthealgorithmsandapproachesconsidered,wedeeplyinvestigatedthemostnovelblood vesselsegmentationincludingmachinelearning,deformablemodel,andtracking-basedapproaches.

Methods: Thispaperanalyzesmorethan100articlesfocusedonbloodvesselsegmentationmethods.For eachanalyzed approach, summary tablesarepresented reporting imaging technique used,anatomical regionandperformancemeasuresemployed.Benefitsanddisadvantagesofeachmethodarehighlighted.

Discussion: Despitetheconstantprogressandeffortsaddressedinthefield,severalissuesstillneedto beovercome.A relevantlimitationconsists inthesegmentationofpathologicalvessels. Unfortunately, notconsistentresearchefforthasbeenaddressedtothisissueyet.Researchisneededsincesomeofthe mainassumptionsmadeforhealthyvessels(suchaslinearityandcircularcross-section)donot holdin pathologicaltissues,whichontheotherhandrequirenewvesselmodelformulations.Moreover,image intensitydrops,noiseand low contraststill representan importantobstacleforthe achievement ofa high-qualityenhancement.Thisisparticularlytrueforopticalimaging,wheretheimagequalityisusu- allylowerintermsofnoiseandcontrastwithrespecttomagneticresonanceandcomputertomography angiography.

Conclusion: Nosinglesegmentationapproachissuitableforallthedifferentanatomicalregionorimaging modalities,thusthe primary goalofthisreviewwas to providean upto datesource ofinformation aboutthestateoftheartofthevesselsegmentationalgorithmssothatthemostsuitablemethodscan bechosenaccordingtothespecifictask.

© 2018ElsevierB.V.Allrightsreserved.

1. Introduction

Bloodvesselanalysisplaysafundamentalroleindifferentclin- ical fields, such as laryngology, oncology [1], ophthalmology [2], andneurosurgery[3–6],bothfordiagnosis,treatmentplanningand execution,andfortreatmentoutcomeevaluationandfollowup.

Theimportanceofvesselanalysisissupportedbytheconstant introductioninclinicalpracticeofnewmedicaltechnologiesaimed

∗ Corresponding author.

E-mail addresses: sara.moccia@polimi.it , sara.moccia@iit.it (S. Moccia), sara.elhadji@polimi.it (S. El Hadji).

at enhancing the visualization of vessels, as endoscopy in Nar- rowBandImaging (NBI)[7] andcone beamComputed Tomogra- phy(CT)3DDigitalSubtractionAngiography(DSA)[8].Atthesame time, standard techniques, such as Magnetic Resonance Angiog- raphy (MRA) and Computed Tomography Angiography (CTA), are constantlyimprovedtoenhancevasculartreevisualization[9–11]. Manualsegmentation of blood vessels is an expensive proce- dure in terms of time and lacking intra- and inter-operator re- peatabilityandreproducibility.Ontheotherhand,semi-automatic orautomaticvesselsegmentationmethodsrequireatleastoneex- pertclinician to segment orto evaluatethe segmentation results obtained. In addition, support for the development and evalua- https://doi.org/10.1016/j.cmpb.2018.02.001

0169-2607/© 2018 Elsevier B.V. All rights reserved.

Nomenclature

2D Bidimensional 3D Tridimensional

3DRA TridimensionalRotationAngiography Acc Accuracy

AUROC Area Under the Receiver Operating Characteristic Curve

AUPRC AreaUnderthePrecision–RecallCurve CNN ConvolutionalNeuralNetwork CRF ConditionalRandomField CT ComputedTomography

CTA ComputedTomographyAngiography DSA DigitalSubtractionAngiography DSC DiceSimilarityCoefficient EM Expectation-Maximization

FCCRF FullyConnectedMarkovRandomField FMM FastMarchingMethod

FN FalseNegative FP FalsePositive FPrate FalsePositiverate

FCN FullyConvolutionalNetworks GF GaborFilter

GPU GraphicProcessorUnit GS GoldStandard GVF GradientVectorFlow H Hessianmatrix HD Hausdorff distance IR Infrared

κ

^Cohen’s

κ

^coefficient

LNND LatticeNeuralNetworkwithDendritic LS LevelSet

MCC MatthewsCorrelationCoefficient M Metrictensorforminimumcostpath MCP MinimumCostPath

MF MatchedFilter

MHTT MultipleHypothesisTemplateTracking MIP MaximumIntensityProjection MRA MagneticResonanceAngiography MRF MarkovRandomField

MRI MagneticResonanceImaging NBI NarrowBandImaging NPV NegativePredictiveValue OCT OpticalCoherenceTomography OF Overlapuntilfirsterror OOF OptimalOrientedFlux

OT Overlapwiththeclinicallyrelevantpartoftheves- sel

OV Overlap

PBT ProbabilisticBoostingTree PF ParticleFiltering

PPV PositivePredictiveValue PSO ParticleSwarmOptimization RACAL RAdius-basedClusteringALgorithm RANSAC RANdomSAmpleConsensus RF RandomForest

ROC ReceiverOperatingCharacteristic Se Sensitivity

Sp Specificity

STAPLE SimultaneousTruth AndPerformanceLevelEstima- tion

SVM SupportVectorMachine TN TrueNegative

TP TruePositive US Ultrasound

tion of such algorithms is still poor as publicly available image datasetswithassociatedGoldStandard(GS)segmentationarecur- rently limitedto specific anatomical regions, such as retina [12]. However, automaticorsemi-automaticblood vesselsegmentation couldassistcliniciansand,therefore,aretopicsofgreatinterestin medicalresearch, asdemonstratedby thehighamount ofpapers annuallypublishedinthisfield.Indeed,anextensiveliteratureal- readyexistsonvesselsegmentationandinthepastyearsdifferent reviews on vessel segmentation algorithms have been published, such as [12–19]. However, due tothe strong developmentin the field,updated reviewsarerequiredtoanalyzeandsummarizethe actualstateoftheart.

Thisreviewaimsatanalyzingawidespectrumofthemostre- centand innovative vessel segmentation techniquesfound in the literature, reportingonstate of theartapproaches basedon ma- chinelearning(Section5),deformablemodel(Section6)andtrack- ing methods (Section 7). Moreover, it reports on the most com- monlyadoptedmetrics fortheevaluationofsegmentation results (Section3)andidentifiestheavailabletestingdatasets(Section4).

Thegoalofthisreviewistoprovidecomprehensiveinformation forthe understanding of existing vessel segmentation algorithms bysummarizingtheir advantagesandlimitations.Eachsegmenta- tionapproachisfirstanalyzedinthegeneralcontextofimageseg- mentationandtheninthespecificcontextofvesselsegmentation.

Foreach segmentation category, papersare discussed, illustrating their benefits and potential disadvantages. In addition, summary tablesreportingperformancemeasuresarepresentedforeachcat- egory.Thepaperconcludeswithadiscussiononfuturedirections andopenissuesinthefieldofvesselsegmentation.

Asummary ofthe papersanalyzed inthis review considering yearofpublication,anatomicalregionandimagingtechniqueisre- ported inTable 1. In addition, Fig. 1 highlights the categoriesof vesselsegmentationalgorithms analyzedinthefollowingsections ofthispaper.

2. Algorithmworkflow

AsshowninFig.1,invesselsegmentationalgorithmstheinput image first undergoes apre-processing step,whichtypically con- cernsnoisesuppression,datanormalization,contrastenhancement and conversion of color image to grayscale image. Since differ- entimagingmodalities produceimagescharacterized bydifferent resolution,noiseandcontrast,differentpre-processingtechniques havetobe employed. Anexhaustivereview onpre-processingal- gorithmsispresentedin[122].

Thecoreofthevesselsegmentationworkflowconcernstheseg- mentation process,which can beclassified infourdifferentcate- gories:

• Vesselenhancement

• Machinelearning

• Deformablemodels

• Tracking

Through vessel enhancement approaches, the quality of ves- sel perception is improved, e.g. by increasing the vessel contrast withrespecttobackgroundandother non-informative structures.

A strong and established literature on vessel enhancement ap- proachesalreadyexists.Examplesincludematchedfiltering[123], vesselness-basedapproaches[124],Wavelet[67]anddiffusionfil- tering[125]. Dueto the extensiveliterature on the enhancement methods andthe wideness ofthissubject,in thisreviewwe will notdealwithit.Acompletereviewonthetopiccanbefounde.g.

in[12].

The vessel enhancement can be followed by a thresholding steptodirectly obtainthevesselbinarymask. Nonetheless,mod- ern methods employ the enhanced vasculature as a preliminary

Table 1

Vessel segmentation categorization. MRA: Magnetic Resonance Angiography, CT: Computed Tomography, CT:A Computed To- mography Angiography, CFP: Color Fundus Photography, OCT: Optical Coherence Tomography, US: Ultrasound, FA: Fluorescein Angiography, DSA: Digital Subtraction Angiography, 3DRA: 3D Rotational Angiography.

Method Year Anatomical region Imaging technique Image processing method

Feng et al. [20] 2010 Brain MRA Unsupervised machine learning

Hassouna et al. [21] 2006 Brain MRA ( Section 5.1 )

Oliveira et al. [22] 2011 Liver CT

Goceri et al. [23] 2017 Liver MRI

Bruyninckx et al. [24] 2010 Liver CT

Bruyninckx et al. [25] 2009 Lung CT

Asad et al. [26] 2017 Retina CFP

Mapayi et al. [27] 2015 Retina CFP

Sreejini et al. [28] 2015 Retina CFP Cinsdikici et al. [29] 2009 Retina CFP Al-Rawi et al. [30] 2007 Retina CFP

Hanaoka et al. [31] 2015 Brain MRA Supervised machine learning

Sironi et al. [32] 2014 Brain Microscopy ( Section 5.2 ) Merkow et al. [33] 2016 Cardiovascular and Lung CT and MRI

Sankaran et al. [34] 2016 Coronary CTA

Schaap et al. [35] 2011 Coronary CTA

Zheng et al. [36] 2011 Coronary CT

Nekovei et al. [37] 1995 Coronary CT Smistad et al. [38] 2016 Femoral region, Carotid US

Chu et al. [39] 2016 Liver X-ray fluoroscopic

Orlando et al. [40] 2017 Retina CFP

Dasgupta et al. [41] 2017 Retina CFP

Mo et al. [42] 2017 Retina CFP

Lahiri et al. [43] 2017 Retina CFP

Annunziata et al. [44] 2016 Retina Microscopy

Fu et al. [45] 2016 Retina CFP

Luo et al. [46] 2016 Retina CFP

Liskowski et al. [47] 2016 Retina CFP

Li et al. [48] 2016 Retina CFP

Javidi et al. [49] 2016 Retina CFP

Maninis et al. [50] 2016 Retina CFP Prentasvic et al. [51] 2016 Retina CT

Wu et al. [52] 2016 Retina CFP

Annunziata et al. [53] 2015 Retina Microscopy Annunziata et al. [54] 2015 Retina Microscopy

Vega et al. [55] 2015 Retina CFP

Wang et al. [56] 2015 Retina CFP

Fraz et al. [57] 2014 Retina CFP

Ganin et al. [58] 2014 Retina CFP

Orlando et al. [59] 2014 Retina CFP

Becker et al. [60] 2013 Retina CFP

Rodrigues et al. [61] 2013 Retina OCT

Fraz et al. [62] 2012 Retina CFP

Zhang et al. [63] 2012 Retina CFP

Marin et al. [64] 2011 Retina CFP

Lupascu et al. [65] 2010 Retina CFP

Salem et al. [66] 2007 Retina CFP

Soares et al. [67] 2006 Retina CFP

Staal et al. [68] 2004 Retina CFP

Lee et al. [69] 2015 Aorta & mesenteric artery CTA Edge-based deformable models

Valencia et al. [70] 2007 Artery MRA ( Section 6.1 )

Law et al. [71] 2009 Brain & Coronary MRA & CTA

Moreno et al. [72] 2013 Coronary CTA

Wang et al. [73] 2012 Coronary CTA

Cheng et al. [74] 2015 Carotid,Coronary Liver, & Lung

Zhu et al. [75] 2009 Lung CTA

Zhang et al. [76] 2015 Retina CFP

Patwardhan et al. [77] 2012 – US

Klepaczko et al. [78] 2016 Brain MRA Region-based deformable models Tian et al. [79] 2014 Abdomen, Brain, CT, DSA ( Section 6.2 )

Heart, Lung & Retina Infrared, US & MRA

Law et al. [80] 2007 Brain MRA

Wang et al. [81] 2009 Carotid US

Liang et al. [82] 2015 Liver Microscopy

Zhao et al. [83] 2015 Retina CFP & FA

Zhao et al. [84] 2015 Retina CFP

Wang et al. [85] 2015 Retina CFP

Xiao et al. [86] 2013 Retina CFP

( continued on next page )

Table 1 ( continued )

Method Year Anatomical region Imaging technique Image processing method

Law et al. [87] 2006 Retina CFP

Robben et al. [88] 2016 Brain MRA Tracking approaches

Rempfler et al. [89] 2015 Brain MRA ( Section 7 )

Yureidini et al. [90] 2012 Brain 3DRA

Cetin et al. [91] 2015 Brain MRA

Coronary CTA

Cetin et al. [92] 2013 Brain MRA

Coronary CTA

Shim et al. [93] 2006 Brain CTA

Cherry et al. [94] 2015 Colon CTA

Shin et al. [95] 2016 Coronary FA

Carrillo et al. [96] 2007 Carotid, aorto-iliac MRA Coronary, pulmonary arteries CTA Amir-Khalili et al. [97] 2015 Carotid US

Benmansour et al. [98] 2011 Carotid CTA

Biesdorf et al. [99] 2015 Coronary CTA

Lugauer et al. [100] 2014 Coronary CTA

Tang et al. [101] 2012 Coronary MR

Wang et al. [102] 2012 Coronary CTA

Friman et al. [103] 2010 Coronary & CTA Liver

Li et al. [104] 2009 Coronary CTA

Wink et al. [105] 2002 Coronary MRA

Zeng et al. [106] 2017 Liver CTA

Bauer et al. [107] 2010 Liver CT

Amir-Khalili et al. [108] 2015 Kidney Endoscopy images Amir-Khalili et al. [105] 2002 Kidney Endoscopic video

Chen et al. [109] 2016 Retina CFP

Chen et al. [110] 2014 Retina CFP

Bhuiyan et al. [111] 2013 Retina CFP

Liao et al. [112] 2013 Retina CFP

Rouchdy et al. [113] 2013 Retina CFP

Stuhmer et al. [114] 2013 Retina CFP

Turetken et al. [115] 2013 Retina Microscopy

Liao et al. [116] 2012 Retina CFP

Kaul et al. [117] 2012 Retina CFP

Delibasis et al. [118] 2010 Retina CFP

Breitenreicher et al. [119] 2013 — —

Benmansour et al. [120] 2009 — —

Wink et al. [121] 2004 — X-ray

Fig. 1. Vessel segmentation workflow. The analyzed vessel segmentation approaches are presented highlighting vessel enhancement approaches, topic of this paper. A pre-processing step is usually performed, which concerns noise suppression, data normalization, contrast enhancement, and conversion of color image to grayscale image.

Post-processing can be performed to refine the segmentation result. Dotted lines show possible influences between segmentation algorithms.

Table 2

Contingency table for vessel segmentation.

Gold Standard segmentation Vessel Non-vessel

Algorithm Vessel TP FP

segmentation Non-vessel FN TN

Table 3

Performance measures for vessel segmentation algorithms.

Index Description

Accuracy ( Acc ) ^TP+_n^TN Sensitivity ( Se ) _{T P+}^{T P}_{F N} Specificity ( Sp ) _{T N+}^{T N}_{F P} False Positive rate ( FP rate ) 1 −Sp Positive Predictive Value ( PPV ) _{T P+}^{T P}_{F P} Negative Predictive Value ( NPV ) _{T N+}^{T N}_{F N}

AUROC Area Under the Receiver

Operating Characteristic curve Matthews Correlation Eq. (1)

Coefficient ( MCC )

Cohen’s κ^{coefficient (} κ^{) Eq. (2)} Dice Similarity Coefficient ( DSC ) Eq. (3) Hausdorff Distance HD Eq. (4)

Connectivity Eq. (5)

Area Eq. (6)

Length Eq. (7)

Overlap ( OV ) Eq. (8) Overlap until first error ( OF ) Eq. (9) Overlap with clinically relevant Eq. (10) part of the vessel ( OT )

step for more sophisticated segmentation algorithms. In particu- lar,theenhancedvasculaturecanbeusedtoextractfeaturestobe classifiedwith machinelearningalgorithms (Section 5),to define forces that constraint vessel model deformation for deformable model-basedsegmentation(Section6),ortoguidevasculartrack- ingthroughenhancedvasculatureintensityorgradient-basedcon- straints(Section7),asexplainedindepthinthisreview.

Apost-processingstepmayalsobeemployed,e.g.toreconnect vascularsegmentsorremovetoosmallsegmentedareas,whichof- tencorrespondtoimageartifactsornoise.

3. Evaluationmetrics

Segmentationperformanceiscommonlyevaluatedwithrespect to GSmanual segmentation performedby an expertclinician. To attenuate intra-subject variability when performing the manual segmentation,andobtainatruthfulGS,acombinationofsegmen- tations by multiple experts is usually employed. Different strate- gieshavebeenproposedtocombinethesegmentations:forexam- ple, a votingrule, oftenused inpractice, selectsas GSall voxels wherethemajorityofexpertsagreethestructuretobesegmented ispresent[126].However,suchapproachdoesnotallowforincor- poratingaprioriinformationofthestructurebeingsegmentedor estimatingthepresenceofanimperfectorlimitedreferencestan- dard.

To solve this issue, the Simultaneous Truth And Performance Level Estimation (STAPLE) hasbeen introduced in[127]. The ap- proach takes a collection of segmentations and computessimul- taneously aprobabilistic estimate ofthetruesegmentation anda measure oftheperformancelevel representedbyeach segmenta- tionusinganExpectation-Maximization(EM)algorithm.

When evaluating the performance of segmentation algorithms withrespect toGS, a contingencytable (Table2) withTruePosi- tive(TP),TrueNegative(TN),FalseNegative(FN),andFalsePositive (FP) iscommonlyused, wherepositive andnegativereferto pix-

elsbelongingtovesselsandbackgroundasinaccordwiththeGS segmentation,respectively.

Segmentation performance measures are summarized in Table3.Accuracy(Acc),Sensitivity(Se),andSpecificity(Sp)arethe mostfrequentlyadoptedmeasures,whereAccistheproportionof trueresults,bothTPandTN,amongthetotalnumberofexamined cases(n).Se,also referredasTP rate, measures theproportionof positives,bothTPandFN,thatarecorrectlyidentified.Spmeasures the proportion of negatives, both TN and FP, that are correctly identified. Although a high Se reflects the desirable algorithm inclinationto detectvessels, ahighSewithlow Spindicatesthat the segmentation includes many pixels that do not belong to vessels,i.e.highFP.Consequently,analgorithmthatprovideshigh Seandlow Spisacceptable ifthe post-processingstep isable to removepossibleFP.

Despite the fact that Acc, Se and Sp are the most frequently adoptedperformancemetrics,otherderivedmetricsarealsooften employed.ExamplesincludeFPrate,whichisequalto1−Sp,Pos- itivePredictiveValue (PPV), whichis theproportionofTPamong TP+FP, and Negative Predictive Value (NPV), which is the ratio betweenTNandTN+FN.PPVgivesanestimationofhowlikelyit isthatapixelbelongstoa vesselgiventhat thealgorithmclassi- fies itas positive.NPV corresponds to the likelihood that a pixel doesnotbelongtoavessel,giventhatthealgorithmclassifiesitas negative.

ReceiverOperatingCharacteristic(ROC) curve,whichillustrates theperformanceofabinaryclassifiersystemasitsdiscrimination thresholdisvaried,isalsooftenreported.TheareaundertheROC (AUROC)isusedasametric,indicatingtheprobabilitythataclas- sifierwillrankarandomlychosen positiveinstancehigherthana randomlychosen negativeone.AUROC assumesvalue 1foraper- fectclassifier.DifferentalgorithmsfortheAUROCestimationarere- portedin theliterature [128]. Precision-recall curve can be used, too.PrecisioncorrespondstoPPV,whilerecalltoSe.Theprecision- recallcurve compares TP withFN andFP,excluding TN,which is lessrelevant forthe vessel segmentation performance evaluation sincetheproportionofTP(vessels)andTN(background)ishighly skewed.Alsointhiscase,theareaundertheprecision-recallcurve (AUPRC)canbeexploited.

Anothermetric that can be used is the Matthews Correlation Coefficient(MCC)[129]:

MCC=

(

^TP∗TN

)

−

(

^FP∗TN

)

((

^TP+FP

)(

^TP+FN

)(

^TN+FP

)(

^TN+FN

)

⁽¹⁾

SomeauthorsalsoreporttheCohen’s

κ

^{coefficient[130],whichis}

ameasuresofinter-rateragreement:

κ

=Acc−pe

1−pe

(2)

where pe is the hypothetical probability of chance agreement, equaltotheprobabilityofGStogeneratepositivestimestheprob- abilityofthealgorithmtogeneratepositives.Cohen’s

κ

^coefficient

isconsidered a robustmetric sinceit takesinto accountalso the agreementbetweenalgorithmandGSoccurringbychance.

Spatialoverlappingindexescanbe used,too.Themostusedis theDice Similarity Coefficient(DSC) [131], whichis computedas theratio ofthe numberof elements(card) in the intersectionof twoclustersAandBbythemeanlabelimage,whereAandBindi- catethesegmentedvesselsanditscorrespondingGS,respectively:

DSC= card

(

^A∩B

)

card(A)+card(B) 2

= 2TP

FP+FN+2TP (3)

DSCisalsoknownasF1score.

Hausdorff distance is another overlapping index, which mea- sureshowfartheGSsegmentationandthesegmentedimageare

fromeachother:

HD=max

(

^supa∈Ain f_b∈Bd

(

^a,b

)

^,supb∈Bin f_a∈Ad

(

^a,b

) )

⁽⁴⁾

wheresup represents the supremum,inf the infimumandd isa chosenmetric,e.g.absolutevaluedistance.

All metrics described above are based on the pixel-to-pixel comparison between the segmented image and the GS, without considering that vessel pixels are part of a connected vascular structurewithspecific features, such asarea andlength. Forthis reason,theuseofthreeadditionalmetricfunctionsissuggestedin [132]:

Connecti

v

ity=1−min

(

¹,

|

^card

(

^A

)

−card

(

^B

) |

card

(

^A

) )

⁽⁵⁾

Area=card

(( δ

r1

(

^A

)

∩B

)

∪

( δ

r1

(

^B

)

∩A

)))

card

(

^A∪B

)

⁽⁶⁾

Length=card

( ψ (

^A

)

∩

δ

^r2

(

^B

))

∪

( δ

^r2

(

^A

)

∩

ψ (

^B

)))

card

( ψ (

^A

)

∪

ψ (

^B

))

⁽⁷⁾

where

δ

^r1 and

δ

^r2 are morphologicaldilatations obtainedusinga discofradiusr₁andr₂,respectively,and

ψ

^{isahomotopicskele-}

tonization[133].The Connectivitytermpenalizesfragmentedseg- mentation.TheAreafactormeasuresthedegreeofoverlappingbe- tweenA andB, beinglesssensitive toslight differencesbetween thesegmentation andtheGSifcompared tothe DSC,dueto the introduced dilatation.The Length factor evaluates theconsistency betweenthesegmentedandtheGSvessellength.

Another class ofevaluationmetrics proposed in [134]aims at quantifyingtheperformance ofsegmentation algorithmsinterms ofpoint-to-pointcorrespondencebetweentheGSvesselcenterline andthe computedcenterline. The point correspondence is com- monly computedwith the mean shift algorithm [135] and three differentcenterlineoverlapmeasuresarederived.Theoverlap(OV) measures theability to trackthe overall vessel annotatedby the observersanditisdefinedasfollows:

OV= TPMov+TPRov

TPMov+TPRov+FNov+FPov (8) whereTPR_ovreferstopointsoftheGScenterlinewhosedistanceto thecorrespondentpointson theevaluated centerline islessthan thelocalvesselradius.Pointsforwhichthisdistanceishigherthan theradiusare markedasFN_ov.Pointsonthe evaluatedcenterline aremarkedasTPMov ifthereis atleastone pointon theGSata distancelessthantheradius,otherwisetheyaremarkedasFPov.

The overlapuntilfirst error(OF) istheratioof thenumberof TPRovbeforethefirsterror(TPR_of)andtheoverallnumberofrefer- encepoints(NR):

OF= TPRo f

NR (9)

ThefirsterrorreferstothefirstpointoftheGScenterlinethat is atadistancehigherthantheradiusfromthecorrespondentpoint ontheevaluatedcenterline.

Overlapwiththeclinicallyrelevantpartofthevessel(OT)pro- videsan estimationoftheability ofthemethod tosegment ves- selsegmentsthatareconsideredclinicallyrelevant,e.g.haveadi- ameterequalorlargerthen 1.5mm.Inthiscase, thepoint(p_end) closesttotheendofthereferenceGSwitharadiuslargerthanor equalto0.75mmisdeterminedandOTiscomputedas:

OT= TPMot+TPRot

TPMot+TPRot+FNot+FPot

(10)

whereTPMot,TPRot,FNot,FPot arecomputedasTPMov,TPRov,FNov, FPov butconsideringonly pointsbetweenp_end andthebeginning oftheGScenterline.

4. Evaluationdatasets

Phantoms presenting meaningful features of interest with re- spect to the vascular tree(e.g. intensity profile, thickness, tortu- osity)are oftenconsidered forthe evaluationofsegmentational- gorithms. Phantomsareeasy tocontrolandmodifywiththegoal of understanding how and to which degree the algorithm per- formance dependson parametersettings. Moreover, by providing a GS, phantoms allow simple algorithm validation and training, whichisnot alwaysaneasytaskforrealclinicalimagessincethe correspondentGSmaynotbeavailable[146].Consideringtheben- efitofusingphantoms,methods todevelop realisticdigitalphan- tomshavebeenpresented,e.g.in[78]forintracranial arterialtree ontime-of-flightMRI.

Althoughphantomshaveanimportantroleinquantifyingalgo- rithmperformance,theydonotalwaysfullyreflectclinicalimages, e.g.duetothehighinter-patientvariability.Toovercomethisissue, a number of publicly available databases with associatedGS has beenpublishedin thelast few years.Publiclyavailable databases encourage consistent andfaircomparison of vesselsegmentation algorithms. However, thispositivetrend still concernsonly a few anatomical regions. A list of publicly available databases is pre- sentedinTable4.

5. Machinelearning

There are two main classes of machine learning approaches:

unsupervisedandsupervised.Theformerfindsamodelabletode- scribehiddenarrangementofinput image-derivedfeatures,with- out anyprior knowledgeor supervision,whilethe latterlearnsa datamodelfromasetofalreadylabeledfeatures,asexplainedin Sections5.1and5.2,respectively.Sinceunsupervisedlearningdoes not requireGSsegmentation, itisusefulforcaseswherepublicly available GS datasets are not available aswell as forexploratory dataanalysis.Ontheother hand,supervisedlearningrequiresGS segmentation to train the learningmodel.The training computa- tional cost varies depending on the adopted supervised learning approach.However, duringtestingphases, thecomputational cost isusuallynegligible.

5.1. Unsupervised

Unsupervisedlearningapproachesrepresentparticularfeatures onthebaseofthestatisticaldistributionoftheoverallinputdata.

The absence of available GS for supervised training justifies the employmentofsuchmethods,atthecostofsegmentation perfor- mance usually less satisfying withrespect to the supervised ap- proachones. A typical unsupervised learningalgorithmworkflow isshowninFig.2.

In[21],stochasticmodelingisusedtosegmentcerebrovascular structures fromtime offlightMRA. The pixelintensityhistogram is described by two major classes: vessel andbackground. Back- groundclassis approximatedby twoGaussiansandone Rayleigh distribution,while thevessel class isapproximatedby one Gaus- sian.EMalgorithm[147]isemployedtoautomaticallyestimatethe GaussiansandRayleighdistributionparameters.Spatialconstraints areincludedthroughMarkovRandomField(MRF)modeling[148], privilegingconnectedsetsofdata.Thus,MRFareparticularlyuse- fulwhenhighnoiselevelispresentintheimages.

In [23], k-means clusteringis usedfor rough liver vessel seg- mentation.Furtheriterativerefinementstepsbasedonmorpholog- icaloperationsareappliedtorefinethesegmentation.Thismethod reliesonautomatic k-meansandmorphologicaloperatorparame- ter selection. Thus, thealgorithm can adapt todifferent pixelin- tensitydistributionsintheimage.

Table 4

Publicly available databases with associated Gold Standard segmentation. CFP: Color Fundus Pho- tography, CTA: Computed Tomography Angiography, CT: Computed Tomography, FA: Fluorescein Angiography.

Name Anatomical region Number of images/volumes

STARE [136] Retina 20 CFP

DRIVE [68] 40 CFP

ARIA [137] 143 CFP

CHASE [138] 28 CFP

HRF [139] 45 CFP

IMAGERET [140] 219 CFP

MESSIDOR 1200 CFP

( http://messidor.crihan.fr )

REVIEW [141] 16 CFP

ROC [142] 100 CFP

VICAVR 58 CFP

( http://www.varpa.es )

VAMPIRE 8 FA

( http://vampire.computing . dundee.ac.uk)

CASDQEF [143] Coronary 48 CTA

ROTTERDAM [144] 20 CTA

VESSEL12 [145] Lung 20 CT

3D-IRCADb Liver 22 CT

( http://www.ircad.fr/research/3dircadb/ )

OSMSC Cardiovascular and 93 MRA and CT

( http://www.vascularmodel.com ) Lung

Vascular Synthesizer [146] 3D Synthetic data 120

Fig. 2. Unsupervised learning approaches build segmentation models based on un- labeled image features, such as local intensity and gradient. During model tuning, the goodness of the model is evaluated and the model is tuned according to a min- imization function that aims at finding the best separation between the vascular and background classes. Usually, such function is defined upon metrics such as Eu- clidean or probabilistic distance.

In [22], liver vessel segmentation is performed with region- growing inCT images.Apixel isincorporated inthegrowing re- gion if its intensity falls in a predefined range. The range ex- trema are defined by approximating the image histogram with three Gaussians, through Gaussian Mixture Model (GMM) [149]. This method isrelevant forseveral imagingtechniques, asfar as the pixel intensity distribution is nearly Gaussian shaped. Portal andhepaticveinsaresubsequentlyseparatedaccordingtogeomet- ric features asdimension and connectivity. In [20], a similar ap- proach is used to segment the brain vascular pattern. Maximum IntensityProjection(MIP)versionoftheCTisusedtoenhancethe vascular structures andonly two classes are considered for ves- selandbackgroundinthehistogramapproximation,resultingina lowerGMMcomputationalcost.

Fuzzy C-means segmentation of retinal blood vessels is em- ployed in [27]. To face non-uniform illumination and contrast, phase-congruency [150] is first performed, which preserves fea- tures within-phase frequency components, such asedges, while suppressing the others. Consequently, accurate segmentation can beperformedalsoinpresenceofintensitydropsandvaryingillu- minationlevelsintheimage.

In[28],Particle Swarm Optimization(PSO) isusedto segment retinalvessels.PSOisusedtoiterativelyfindtheoptimalmatched filter(MF) [123] parameters. The MF locallyexploits thecorrela- tion betweenlocal image areas and filterkernel that reproduces the blood vessel architecture in terms of width and orientation.

During the PSOiterative process, the AUROC ofthe MF response isusedasfitnessfunctionforthePSO.TwoMFsareemployed to separately enhancesmall andthick vessels.Similarly, optimalMF parametersareretrievedusinggeneticalgorithmin[30].However, PSOhastheshortcomingofeasilyfallingintolocaloptima,influ- encingthesegmentationperformance.

In [24,25] ant colony optimization, a population-based meta- heuristicusedtofindapproximatesolutionstooptimizationprob- lems,isusedtosegmentboth lungandlivervessels. Themethod connectsvesselbifurcationbycostpathalgorithmandusestheant colonyoptimizationmethodtoretrievetheoptimalvesseltreebe- tweenallpossiblepaths.Asimilarapproachisexploitedin[26]to segmentretinalvessels.Antcolonysegmentationresultsarecom- binedwithMFonesin[29]toimprovetheretinalvesselsegmen- tationaccuracy.

A summary of the analyzed unsupervised approaches is pre- sentedinTable5.

5.2.Supervised

Supervisedlearningforvessel segmentation infersa rulefrom labeledtrainingcouples,oneforeachpixel,whichconsistofanin- putvector offeatures (suchaspixelintensity,MF response,etc.), andan outputvalue, whichstateswhetherthepixelbelongsto a vesselornot accordingtoa GS.The workflow ofatypical super- visedapproachisshowninFig.3.

From the first attempts of using machine learning for vessel segmentation (including [37,67,68]), severalalgorithms have been

Table 5

Summary of unsupervised blood vessel segmentation algorithms (for performance indexes refer to Table 3 ). CT: Computed Tomography, MRA: Magnetic Resonance Angiography.

Method Testing dataset Synthetic data Segmentation performance measure

Feng et al. [20] 136 MR slices No Visual

Hassouna et al. [21] MRA Yes Visual

Goceri et al. [23] 14 MRI No DSC, HD

Bruyninckx et al. [24] 5 CT images (3D-IRCADb-01) No DSC ( http://www.ircad.fr/research/3dircadb/ )

Bruyninckx et al. [25] 1 CT volume No Euclidean distance

Oliveira et al. [22] 15 CT volumes No Visual

Al-Rawi et al. [30] 20 images (DRIVE database [68] ) No AUROC = 0 . 96 Asad et al. [26] 20 images (DRIVE database [68] ) No Se = 0 . 75 Cinsdikici [29] 20 images (DRIVE database [68] ) No AUROC = 0 . 94 Mapayi et al. [27] 20 images (STARE database [136] ) No Acc = 0 . 93

20 images (DRIVE database [68] ) Acc = 0 . 94

Sreejini et al. [28] 20 images (STARE [136] ) No Acc = 0 . 95 , Se = 0 . 72 , Sp = 0 . 97 20 images (DRIVE database [68] ) Acc = 0 . 96 , Se = 0 . 71 , Sp = 0 . 99

Fig. 3. Supervised approach workflow. During the training phase, image features (e.g. intensity, gradient, color) are extracted from the training images. A machine learning model is trained with such features and the corresponding labels, taken from the Gold Standard segmentation. Once the model is trained, it can be applied to a new, unseen, testing image to obtain the vessel segmentation.

publishedfollowingacontinuousprogressofresearchonthetopic.

Sofar,supervisedlearninghasbeenmainlyappliedtoretinalim- ages, since different labeled databases for training are publicly available(Table4).

In[61],retinalvasculatureissegmentedfromOpticalCoherence Tomography(OCT)images.Asetof2Dfundusreferenceimagesare computedfromthe3DOCTvolumeandusedasinputtoaSupport VectorMachine(SVM)withtheGaussiankernel.Theeffectiveness ofthemethoddependsonthechoice oftheselectedSVM kernel aswell asonthe tuning ofits parameters. Thisapproach is able tosegmentbothhealthy andpathologicalretinalvessels. Asa re- sult,thestudyofdiseaseprogressionisoneofthemajorfieldsof applicationofthismethod.

In [31]a geometricalfeatureset isdefinedtoclassify cerebral vesselmorphology inMRA. Thefeaturesarespecificallychosento accountalsoformorphologicallyabnormallesions,makingtheal- gorithmsuitableforsegmentedpathologicalstructures.Vesselseg- mentation is obtained via region-growing and the binary vessel maskisapproximatedbyagraph,whosenodesbelongtotheves- seltree.Foreachnode,the3Dhistogramofshortestpathlengths

betweentheconsiderednodeandtheothersiscomputedandused asfeaturevector.SVMisusedtoclassifyvasculartreemorphology inhealthyorpathologicalcondition.

Fully-Connected Conditional Random Field (FCCRF) is used in [40,59] to segment retinal vessels in color fundus photography.

FCCRF mapsthe image intoa fully connectedgraph structure,in which everypixel (graph node)is influencedby the others.Each pixel isrepresentedby a set offeatures extractedthrough vessel enhancementapproaches,suchasgradientmagnitude,andMFre- sponse.StructuredSVM[153]areemployedtolearntheFCCRFpa- rameters. The fully-connected framework leadsto a more robust segmentation withrespect totheclassificationperformedconsid- eringeachvesselpixel asan isolatedpoint,orasinfluencedby a restrictedneighborhood(asfortraditionalCRF).

RAdius-based Clustering ALgorithm (RACAL), introduced in [154],isusedin[66]tosegmentretinalvasculatureincolorfundus photography. RACALis usedto clusterpixels, through adistance- basedprinciple,inthefeaturespacebuiltconsideringgreenchan- nelintensity,gradientmagnitudeandmaximumimageHessian(H) eigenvalue.Theassignment ofeachcluster toeitherthevesselor backgroundclass is madeaccordingto a trainingprocedure, em- ployingasGSathresholdedversionofthevesselness:

V_Salem=max_σ

( λ

²

( σ ))

φ

std

(

^e1

)

⁽¹¹⁾

where e₁ is the H eigenvector associated to the smallest eigen- value,

λ

2 isthebiggestHeigenvalue,and

φ

std thestandard devi- ationoftheorientationofe₁ computedwithdifferentscale value (

σ

^).e1 inclinationisconstantforlongitudinalvessels thusprovid- ing a high vesselness value. Sparse codingis used in [49,63] for retinal vessel enhancement andsegmentation. Sparse coding ap- proximatestheimage intensityby asparse linearcombinationof itemsfroman overcompletedictionarybuiltfromtrainingimages patches.

In [36], Probabilistic Boosting Tree (PBT), is used to segment coronaryarteriesinCTimages.Geometricfeatures,whichdescribe thepositionofan imagevoxelinaheart-orientedcoordinatesys- tem, are used.In addition, image steerablefeatures are included in the boosting phase, which takes into account image intensity andgradientinformation.Asimilarapproachisexploitedin[57]to segmentretinalvessels.Inthiscase, thefeaturevector consistsof Gabor Filter (GF) [67] and Gaussian filter outputs. A similar ap- proachisusedin[62].

In [65], the feature-based AdaBoost classifier is used to seg- mentretinal vessels.Numerousfeatures areused, suchasMF,GF andGaussianderivatives.However,themostinformativeones,ac- cordingtotheauthors’analysis,arethesecondorderderivativeof Gaussian,multiscaleMFusingaGaussianvesselprofile,andStaal’s

ridges[68].WithrespecttoSVM,AdaBoostreliesontheconstruc- tionofanaccurateclassification modelfromalinearcombination ofweakclassifiers,makingiteasierandfastertotrain.AdaBoostis alsousedin[32,35]toextractvesselcenterline byusingconvolu- tionalfilterandintensity-basedfeaturevector.

Randomdecisionforest (RF) isusedin[53] tosegment highly tortuousorirregularstructures.AmodifiedGaussian-likebankfil- ter is designedto detectbended tubular structures andthe filter outputsareusedasfeaturesfortheclassification.Whencompared to SVM, RF usually has comparable performance with a lower training computational cost. Similarly, RF regressors are used in [34] for the estimation of vessel diameter in coronary artery from CTA. The rationale is evaluating the presence and the degree of stenosis using downstream and upstream properties of coronary treevasculatureasfeaturesfortheregression.

InspiredbyTuandBai[155],contextfiltersandappearancefil- tersareusedin[54].OptimalOrientedFlux(OOF)[156]isusedto exploit vesselappearanceinformation.OOFfindsthe optimalaxis on which image gradients are projectedin orderto compute the imagegradientflux.K-meansareusedtolearninanunsupervised way a bank of context filters fromthe OOF-filtered image. RF is usedto classifyafeature vectormadeofOOF-outputandcontext filteroutput. Asimilar approach isused in[44] usingasappear- ance filter theridge detectorfilter definedin[53].The main ad- vantage ofOOF is its robustnessagainst the disturbance induced bycloselylocatedadjacentobjects.

Neural networks are used in [64] to segment retinal vessels.

Gray-level-based andmoment invariants-based features are used totrainthenetwork,whichisdefinedasamultilayerfeedforward networkwiththreehiddenlayers.Thevesselmask isobtainedby thresholding thesigmoidoutput. Afurtherimprovementisintro- duced in [55], where intensity-based and moment-invariant fea- tures areused to segmentthe retinalvasculature throughLattice NeuralNetworkwithDendritic processing(LNND).As amatterof fact,LNNDarchitecturedoesnotrequiretosetthenumberofhid- denlayersinthenetwork,allowing fora simplenetworktraining andconsequentlyforareductionofthecomputationalcost.

In thepast years,ConvolutionalNeuralNetworks (CNNs)have becomestronglypopular.ACNNisafeed-forwardartificialneural networkinwhichtheconnectivitybetweenitsneuronsisinspired bytheorganizationofthehumanvisualcortex.Thebuildingblocks ofaCNNareconvolutionalandfullyconnectedlayers.Theconvo- lutional layerparameters consist ofa set offilters, whosevalues arelearnedduringtheCNNtraining.Fullyconnectedlayersrepre- sentthehigh-levelreasoningblockintheCNN.Neuronsinafully connected layer have connections to all activations in the previ- ouslayer.CNN-basedvascularsegmentationworkflowisshownin Fig.4.

CNNs have been exploited in [157] to extract esophageal mi- crovesselfeaturesfromNBImicroscopy.Theextractedfeaturesare then classifiedwithSVM.Similarly,in [56],CNNs areusedto ex- tracthierarchicalfeaturesfromretinalcolorfundusimages,which are then classified with ensemble RF. In [58], the feature vector extractedwithCNNsiscomparedwithadictionaryfeaturevector that referstoseveralvascularpatterns.Thenearest featurevector extracted from the dictionary, according to the nearest neighbor algorithm,iselectedasoutputvascularpattern. Theseapproaches areparticularlyusefulforsmalldatasets.Indeed,whenthedataset variabilityissmall,machinelearningapproaches,suchasSVMand RF,arebettersuitedforachievingpixelclassification.

On the other side, CNNs are trained to directly obtain vascu- larsegmentationin[51]forretinalvesselsegmentationinOCTan- giography, in [38] for carotid segmentation in ultrasoundimages andin [47]forretinal segmentation incolor fundusphotography images.Specifically,theCNNfullyconnectedlayerisusedtoclas- sify eachpixelintheimageasbelongingtovesselorbackground.

Fig. 4. Convolutional Neural Networks (CNN) for vascular segmentation have been used in two ways: (i) the CNN convolutional layers are used to automatically ex- tract image features, which are then classified with standard supervised learning approaches; (ii) CNN are directly used to obtain the vascular segmentation by em- ploying fully connected layers.

Thisapproach leadsto a fastCNN training, asit alreadyembeds theclassificationstep.However,alargedatasetisrequiredforpre- ventingfully-connected CNN overfitting. Indeed,as alreadyhigh- lighted,ifthedatasetissmallitisrecommendedtouseCNNonly toextractfeatures.

In[47] theuse ofimage pre-processingforretinal vessel seg- mentationwithCNNisalsoinvestigated.Imagesarepre-processed with different methods, such as global contrast normalization, zero-phase whitening, data augmentation using geometric trans- formations and gamma corrections. Authors report an increment insegmentationperformance.

Cross-modalitylearningisusedin[48]tosegmentretinalves- sels.Themappingfunctionbetweentheretinalimageandtheves- selmapislearnedthroughadeepneuralnetwork.

A unified framework of retinal image analysis that provides bothretinalvesselandopticdiscsegmentationisproposedin[50]. ACNNisdesignedtosegmentbothretinalvesselandopticdiscin singleforwardpass.

CNNs andConditional Random Field (CRF)are combined into an integrated deepnetwork called DeepVessel in [45] for retinal vessel segmentation. CRF helps modeling the long-range interac- tionsbetweenpixelsandincreasesthesegmentationperformance.

Asimilarapproachisexploitedin[46].CRFinclusionallows good segmentationperformancealsoinpresenceofintensitydropsand noise.

In[39]CNNisusedtoproducerobustvesselsegmentationand trackinginX-ray imagesequences. The trackingexploits a multi- dimensionalassignmentproblem,whichissolvedwithrank-1ten- sor approximation. Similarly, a deep CNN is trained forestimat- inglocalretinal vesselprobability viaprincipalcomponentanaly- sisandnearestneighborsearchin[52].Theresultingvesselmapis exploitedto extracttheentireconnectedtreewithaprobabilistic trackingapproach.

A studyon the effectiveness of gradient boosting fortraining CNNsisproposedin[60].Themainbenefitofthisapproachisthat bothfeaturesandtheclassifierthat usesthemarelearnedsimul- taneously,resultinginafasterprocedurethatdoesnotrequireany parametertuning.

Further developments are presented in [41,42], where Fully ConvolutionalNetworks(FCN)are usedtosegmentretinalvessels incolorfundusphotographyimages.Thefullyconnectedlayersare replacedbydeconvolutionallayersallowingtoobtainafasterand

Fig. 5. Fully Convolutional Neural networks (FCN) for vascular segmentation replace fully connected layers with one or more deconvolutional layers, making the seg- mentation faster.

moreprecisevessellocalizationwithrespectto approachesbased onfullyconnectedlayerclassification.FCN-basedvascularsegmen- tationworkflowisshowninFig.5.

In [33], theFCN approachis extendedto perform3D vascular segmentationincardiovascularandpulmonaryvesselsinMRIand CTvolumes,respectively.

In[43],apreliminaryattemptofusingadversariallearningfor vessel segmentation in retinal color fundus photography images isexploited. Inadversarialnetwork setup,one networkgenerates candidatesegmentationsandone evaluatesthem.Withrespectto standard network learning, adversarial learninghas the potential ofimprovingthesegmentation outcome, loweringthenumber of wronglyclassifiedpixels[158].

A summary of the supervised approaches analyzed above is presentedinTable6.

6. Deformablemodel

Deformable models consider curves or surfaces (S), defined within the image domain, that can move anddeform underthe influence ofinternal (F_int) and external (Fext) forces.The former aredesignedtokeep S smoothduringthe deformationwhilethe latterattract S toward the vessel boundary. Since S initialization isrequiredto start thedeformation process, a robustdeformable modelshould be insensitive to the initial position, as well as in generalto noise.Recent efforts indeformablemodel formulation focusoneasilyincorporatinginthemodelformulationbothimage- guideddeformationconstraintsanda prioriclinicalknowledge of vesselgeometry. Thisclass ofalgorithms appears suitable toface the segmentation of vessels with complex architecture and high shape andsize variability, both in pathological andphysiological context.However,the requiredcomputationalcost ingeneralstill representsalimitforrealtimeapplications.

Deformable model approaches can be divided in edge-based andregion-based,whicharehereafterindepthanalyzed.

6.1.Edge-based

According to the representation of S, edge-based deformable modelscanbeclassifiedinparametricorgeometricmodels[159].

Fig. 6. In parametric deformable model approaches, the segmentation is obtained by evolving a parametrized curve ( S ) according to external ( F ext) and internal forces ( F int). F extis formulated according to image-dependent features, such as intensity or gradient. F intdeals with constraint imposed to the curve evolution, such as cur- vature and perimeter.

6.1.1. Parametric

Parametricdeformable models, whose diffusion ismainly due totheworkofKassetal.[160],representS ina parametricform.

Thedeformablemodelproblemcanbeformulatedas:

c

∂

^S

∂

^{t =Fint}

⁽

^S

⁾

^+Fext

⁽

^S

⁾

⁽¹²⁾

beingcadampingcoefficient.F_int consistsoftwomaincontribu- tions:

F_int=

∂

∂

^s

⁽

^p

∂

^S

∂

^s

⁾

⁻

∂

²

∂

^s2

⁽

^q

∂

^2S

∂

^s2

⁾

⁽¹³⁾

withs∈[0, 1], andpand q beingthe weighting parameters that controltheSelasticityandresistancetobending,respectively.The Fexttermvariesaccordingtothemethod.Thegeneralworkflowof parametricdeformablemodelsisshowninFig.6.

Withrespect togeometric models, themain limitationofthis classofalgorithms,isthedifficultyinadaptingtochangingvessel topology,dueto the parametrizationof S.Nonetheless, the para- metricframeworkiseasytoformulateandallowsfastconvergence, whichisasuitablepropertytolowercomputationalcosts.

In[74],aB-snakeactivecontour[161]isemployedtosegment retinal vessels. Fext consists of (i) a Gradient Vector Flow (GVF) term[162],whichdescribeshowthegradientvectorsofanimage- derived edge-map diffuses inside the image domain, and (ii) a forcecontributionsthatimposeconstrainsontheSevolution,such asvessel cross-section shape,position andsize.The good perfor- mance,achievedwithbothlow contrastedandthinvessels,dete- rioratesinpresenceofpathology, i.e.whentheassumptionsmade onthevesselgeometryarenotanymorevalid.

In[69],anactivecontourstrategycoupledwithKalmanfiltering isemployedtosegmentthevasculatureinCTA.Theactivecontour provides the vessel segmentation in the first CTslice, employing image intensity-andgradient-basedF_ext,whiletheKalmanfilter- ing isusedtotrackthevessel acrossother CTslices.TheKalman tracking-basedapproachprovidesautomatic contourinitialization, reducing the computational cost withrespect to methods based solely on deformable models. Similarly, in [77], a single spatial Kalman-filtertrackerkeepstrackofthevesselcenter-linein3Dul- trasound.Thevesselboundariesarethenestimatedbygrowingan areaweightedactivecontouroutwardfromthecenterline.

In[76], active contours are usedto segment retinalvessels. A roughvesseledgemapiscomputedbythresholdingthevesselness measure definedin [124]. The vesseledges are then used to ini- tializetwoactivecontoursS₋andS₊.Theactive contoursdeform accordingtointensity-basedFext.Afurtherconstraintisintroduced intheF_ext formulation,toavoidtheintersectionofS₋andS₊,by controlling their relative distance. The vesselness measure allows