Research Article Automatic Segmentation and Measurement of …downloads.hindawi.com/journals/cmmm/2013/260410.pdf · 2019-07-31 · Research Article Automatic Segmentation and Measurement

Hindawi Publishing CorporationComputational and Mathematical Methods in MedicineVolume 2013 Article ID 260410 16 pageshttpdxdoiorg1011552013260410

Research ArticleAutomatic Segmentation and Measurement of Vasculature inRetinal Fundus Images Using Probabilistic Formulation

Yi Yin Mouloud Adel and Salah Bourennane

Institut Fresnel Ecole Centrale de Marseille Aix-Marseille Universite Domaine Universitaire de Saint-Jerome 13397Marseille France

Correspondence should be addressed to Yi Yin yiyininriafr

Received 2 August 2013 Accepted 21 October 2013

Academic Editor Seiya Imoto

Copyright copy 2013 Yi Yin et al This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The automatic analysis of retinal blood vessels plays an important role in the computer-aided diagnosis In this paper we introducea probabilistic tracking-based method for automatic vessel segmentation in retinal images We take into account vessel edgedetection on the whole retinal image and handle different vessel structures During the tracking process a Bayesian method withmaximum a posteriori (MAP) as criterion is used to detect vessel edge points Experimental evaluations of the tracking algorithmare performed on real retinal images from three publicly available databases STARE (Hoover et al 2000) DRIVE (Staal et al2004) and REVIEW (Al-Diri et al 2008 and 2009) We got high accuracy in vessel segmentation width measurements and vesselstructure identification The sensitivity and specificity on STARE are 07248 and 09666 respectively On DRIVE the sensitivity is06522 and the specificity is up to 09710

1 Introduction

Automatic vessel segmentation in medical images is a veryimportant task in many clinical investigations In ophthal-mology the early diagnosis of several pathologies such asarterial hypertension arteriosclerosis diabetic retinopathycardiovascular disease and stroke [1 2] could be achieved byanalyzing changes in blood vessel patterns such as tortuositybifurcation and variation of vessel width on retinal images

Early detection and characterization of retinal bloodvessels are needed for a better and effective treatment ofdiseases Hence computer-aided detection and analysis ofretinal images could help doctors allowing them to usea quantitative tool for a better diagnosis especially whenanalyzing a huge amount of retinal images in screeningprograms

Many methods for blood vessel detection on retinalimages have been reported in the literature [3ndash5] Thesetechniques can be roughly classified into pixel-basedmethods[6ndash14] model-based methods [15ndash21] and tracking-basedapproaches [22ndash29] respectively

Pixel-based approaches consist in convolving the imagewith a spatial filter and then assigning each pixel to back-ground or vessel region according to the result of a second

processing step such as thresholding or morphological oper-ation Chaudhuri et al [8] used 2D Gaussian kernels with12 orientations retaining the maximum response Hoover etal [6] improved this technique by computing local featuresto assign regions to vessel or background A multithresholdscheme was used by Jiang and Mojon [9] whereas Sofka andStewart [10] presented a multiscale matched filter Zana andKlein [11] usedmorphological filter combined with curvatureevaluation for retinal segmentation Neimeijer et al [12] useda classification scheme based on a simple feature computa-tion Gabor wavelet transform with a Bayesian classificationis performed by Soares et al [13] Staal et al [7] applied asupervised classification based on features computed near thecenterline The same scheme was used by Ricci and Perfetti[14] but with a modified line operator to train a supervisedpixel classifier

Model-based approaches use parametric models toextract vessels Al-Diri et al [30] extracted blood vesselsegment profiles and measured vessel width using a para-metric active contour based on two contours coupled byspring models Active contour model was applied by Kozerkeet al [15] to automatically segment vessels A level setgeometric-based regularization approach is given by Gooyaet al [16] Vasilevskiy and Siddiqi [17] developed FLUX for

2 Computational and Mathematical Methods in Medicine

narrow elongated vessel segmentation and Law and Chung[18] improved this technique for spherical flux computationCoronary arterial trees are reconstructed using ellipticalparametric cross-section model by Kayikcioglu and Mitra[19] Hessian-based method is proposed by Sato et al [20] tocharacterize stenosis in coronary angiograms whereas Wanget al [21] used Hermite multiresolution model for retinalvessel analysis A multiresolution approach based on a scale-space analysis is presented by Martınez-Perez et al [31] inwhich the width the size and orientation of retinal vesselsare obtained

Tracking-based approaches are based on local techniqueswhere either the centerline or vessel edges or both areextracted Starting from seed points these methods progressalong vessels by iterative prediction and parameter estima-tion Many methods including several 2D and 3D medicalimaging modalities have been published [4] An advantageof tracking methods is the guaranteed connectedness ofvessel segment whereas in pixels-processing-basedmethodsconnectedness is not guaranteed A whole vessel tree canbe tracked by these methods without examining the vastmajority of the image Starting from the optic disc Toliasand Panas [22] developed a fuzzy tracking model of 1Dvessel profile Their method however did not manage tohandle branch points Starting from seed points Can et al[23] used an iterative tracking algorithm updating at eachstep the position and the orientation of vessel points Zouet al [24] developed a recursive tracking technique guidedby accurate vessel direction and robust a priori knowledgeand termination criteria Compared to others this approachcould extract automatically most of the vessels with accuratevessel characteristics

Among tracking methods few probabilistic approacheshave been reported in the literature [32ndash34] The main ideasof a probabilistic trackingmethod have been presented in ourprevious work [35 36] which needed some improvementslike modeling blood vessels more accurately handling differ-ent vessel configurations and evaluating it on large databaseimages

In this paper a novel fully automatic tracking basedmethod using probabilistic formulation is introduced Firstseed points located inside vessels are obtained from theimage From these points an iterative tracking algorithmdetects simultaneously edges diameter width centerline andorientation of the whole vascular tree A branch predictionscheme to handle bifurcation and crossing is also developedThe major novelty lies in defining the likelihood of thehypothesis of edge points and the a prioriprobability based onGibbs formulation This new approach uses Gaussian modelto approximate vessel sectional intensity profiles identifiesbifurcation and crossing structures using gradient analy-sis tracks centerline and local vessel edges geometry andimproves the detection results as shown in the experimentsand discussion section A probabilistic segmentation schemeis associated with the maximum a posteriori (MAP) as acriterion to estimate local vessel edges

In the following Section 2 gives a general descriptionof the proposed method Section 3 is devoted to the expla-nations of the tracking algorithm Bayesian segmentation

is described in Section 4 Finally experiments on STAREDRIVE and REVIEW databases are presented and discussedin Section 5

2 General Description of the Method

In this paper we propose a tracking-based method to detectretinal vascular trees First a number of seed points areselected automatically on the retinal image which provideinitial parameters for the tracking algorithm The trackingprocess starts then from each of the seed points and detectsvessel edge points iteratively until end conditions are satisfiedFinally when all the seed points are processed the wholevascular tree is detected and the proposed algorithm stops

A tracking process from one seed point is described indetail as follows

21 Initialization Initial parameters which are obtainedfroma seed point include initial vessel center point and vesseldirection The blood vessel is then detected by the trackingalgorithm from the initial center point along the initial vesseldirection

22 Iteration At a given step a dynamic search windowis defined based on local vessel parameters including vesselcenter point direction and diameter We adopt a statisticalsampling method to select new candidate edge points onthe search window A Bayesian method with the MAP as acriterion is used to pick out new vessel edge points from thesecandidate points New vessel parameters are then updated forthe next iteration In addition during the tracking process abranch prediction scheme is applied at each iteration to detectvessel branches which exist in the local search area

23 End There are three end conditions for the proposedtracking process Under each of these conditions the trackingof current vessel stops The three end conditions are asfollows

(i) Current vesselrsquos end is considered to be found whenthe vesselrsquos diameter is less than one pixel or thecontrast between the vessel and background is low

(ii) Current vessel encounters a blood vessel which isalready detected by a tracking process starting fromanother seed point

(iii) Vessel branches are found All initial information ofthe branches is obtained by the algorithm and thetracking of these vessel branches starts The branchwith the biggest diameter is handled first

3 Tracking Algorithm

31 Selection of Seed Points In retinal image blood vesselsare not all connected because of the imaging conditions anddifferent eye diseasesTherefore the tracking algorithm startsfrom several initial points selected on the whole image andthe tracking results are combined to get the final detectedvascular tree In this study we use an automatic two-step

Computational and Mathematical Methods in Medicine 3

method based on grid line analysis to pick out initial seedpoints The first step searching over grid lines is similar tothe procedure used byCan et al [23]The second step consistsin filtering the image obtained in the previous step by the 2DGaussian filter and getting the validated seed points

First a set of grid lines is drawn over the retinal imageas shown in Figure 1(b) Local intensity minima on each gridline (horizontal or vertical) are detected (Figure 1(c)) andconsidered as candidate seed points Among the detectedlocal minima some are related to noise or other eye tissuessuch as fovea So rejection of the false candidate seed pointsis needed in order to avoid unnecessary tracking To testthe validity of the candidates we use a set of 2D Gaussiandirectional matched filters which were first proposed byChaudhuri et al [8]There are 12 different orientations for theGaussian kernels which are spaced in 15∘ from each otherWeconvolve the 12 oriented filters with the given retinal imagefor the selected candidate points If the highest response of12 directional filters for a candidate seed point is above alocal adaptive threshold it is validated as a seed point andthe corresponding filter direction is regarded parallel to localvessel direction In our study the local adaptive threshold isdefined based on local intensity distribution For a candidateseed point the local adaptive threshold is computed as

119879seed = 120583seed + 120572120590seed (1)

where 120583seed and 120590seed are the mean grey level and standarddeviation of the neighborhood of the candidate seed pointrespectively After many experiments the value of parameter120572 is fixed to 12 and the size of the neighborhood is fixed to61 times 61 pixels around each candidate seed point

This method is fast and effective because it considers onlythe pixels on the grid lines and validates seed points basedon the 2D Gaussian filter All the verified seed points areused for the initialization of the tracking algorithm Eachseed point is considered as local vessel center point andinitiates the tracking algorithm twice once in its related filterdirection and the other along the opposite direction If aseed point is out of vessels the tracking algorithm stopsafter one or two iterations according to the end conditionsAs shown in Figure 1(c) there is at least one seed point oneach of the blood vessels Some examples of seed points areshown in Figure 2 including locations and correspondingfilter directions Figure 2 is a subimage of Figure 1(c) markedby blue box

32 Statistical Sampling The proposed tracking algorithmstarts from each of the seed points and detects vessel edgepoints iteratively During the tracking process new vesseledge points are detected based on a statistical samplingscheme As shown in Figure 3 at a given step a line segmentis obtained perpendicular to the tracking direction This linesegment is regarded as a linear searchwindow which restrictsthe possible locations of new vessel edge points The width ofthe linear window is adaptive to local vessel diameter in orderto cover the potential positions of new edge points

At iteration 119896 (when 119896 = 0) the initial vessel centerpoint 119874

0is an initial seed point 119874

0may be not exactly

in the middle of local vessel However this deviation willbe corrected after two or three iterations The initial vesseldirection

0is along or opposite to the related filter direction

of the initial seed point A line segment is drawn centeredon 1198740and perpendicular to

0 which is regarded as the

initial linear window 1198710 The width of 119871

0is set bigger than

the widest blood vessel in the retinal image Two initial vesseledge points

0and

0are detected on 119871

0around the seed

point by the proposed method Initial vessel diameter is 1198890=

|00|

When 119896 ge 1 vessel parameters at the previous iteration119896minus1 are known including vessel edge points

119896minus1 119896minus1

centerpoint 119874

119896minus1 direction

119896minus1(119896minus1 = 1) and diameter 119889

119896minus1

In order to get the linear window 119871119896 we first extrapolate the

vessel center point119874119896minus1

to a point119874119896119904 which is 119904 pixels away

along 119896minus1

The extrapolated point 119874119896119904

is described as

997888997888997888997888997888997888997888rarr119874119896minus1119874119896119904= 119904119896minus1 (2)

Thevessel direction at119874119896119904is obtained by performing gradient

analysis of this point (see Section 33) and is denoted by119896119904 A line segment is then drawn centered on 119874

119896119904and

perpendicular to 119896119904

as shown in Figure 3This line segmentis regarded as the linear search window 119871

119896at iteration 119896 The

width of the linear window is set twice the diameter of localblood vessel 119889

119896minus1

We select119873119871119896

points which are numbered from 1 to119873119871119896

on 119871119896(119896 ge 0) The interval between two adjacent points

is fixed to 1 pixel 119873119871119896

selected points are considered ascandidate edge points New edge points

119896and

119896are chosen

from these candidates by a Bayesian method (see Section 4)Then new vessel parameters at iteration 119896 are updatedaccordingly The new center point 119874

119896is the middle point

of [119896 119896] vessel direction

119896is defined as the direction at

point119874119896by gradient analysis (see Section 33) and new vessel

diameter is 119889119896= |119896119896|

33 Gradient Analysis In this study the vessel direction ata point is assumed to be parallel to the nearest blood vesselThis direction can be estimated based on the local gradientinformation Let the gradient vector at point (119894 119895) in the imagehas the polar representation The modulus 119866

119894119895and the angle

120579119894119895are obtained based on the classic Sobel mask Since the

projection of a vector is maximized in the direction parallelto it the dominant gradient direction at (119894

119900 119895119900) is assumed

as the optimum direction in which the projections of thegradient vectors in the neighborhood119872 times 119873 around (119894

119900 119895119900)

aremaximized In practice the size of neighborhood used forcalculating the gradient direction is fixed to 5 times 5 If the angleof the dominant gradient direction at (119894

119900 119895119900) is denoted by

120579119892 the projection function of the gradients is expressed as

follows [41]

119865 (120579119892) = sum

(119894119895)isin119872lowast119873

1198662

119894119895

cos2 (120579119894119895minus 120579119892) (3)


(a) (b) (c)

Figure 1 Selection of seed points on a retinal image (a) retinal image (b) grid lines and candidate seed points (c) validated initial seedpoints

Figure 2 Examples of seed points

Figure 3 Linear search window and are the vessel edge points119874 is the center point is the vessel direction and 119896 is the indexof the iteration 119871

119896

is the linear search window at iteration 119896 Blackpoints on 119871

119896

show the possible locations of new edge points

119865(120579119892) is maximized by calculating the value of 120579

119892when

making its derivative equal to zeroThe angle of the dominantdirection at (119894

119900 119895119900) is

120579119892=1

2arctan(

sum(119894119895)isin119872lowast119873

1198662

119894119895

sin (2120579119894119895)

sum(119894119895)isin119872lowast119873

1198662

119894119895

cos (2120579119894119895)

) (4)

On a grayscale image the gradient vector points to the direc-tion of the greatest change of the grey level Therefore thedominant gradient direction at point (119894

119900 119895119900) is perpendicular

to the nearest blood vessel on the retinal image The angle ofthe vessel direction at (119894

119900 119895119900) can be estimated as

120579119894119900119895119900= 120579119892plusmn120587

2 (5)

The sign before 1205872 is determined by local vessel parametersas discussed later Finally the vessel direction at (119894

119900 119895119900) is

described as a unit vector by 120579119894119900119895119900

119894119900119895119900= (cos (120579

119894119900119895119900) sin (120579

119894119900119895119900)) (6)

As mentioned in the previous section the deployment oflinear search window and update of new vessel direction areboth accomplished based on the gradient analysis of specificpoints At iteration 119896 in order to get the linear searchwindowwe need to calculate the vessel direction at the extrapolatedpoint 119874

119896119904 We compare the dominant gradient direction

119892

at 119874119896119904

and the local vessel direction 119896minus1

(see Figure 4) Theangle difference between these two directions is

Δ120579 = 120579119892119874119896119904

minus 120579119863119896minus1

(7)

where 120579119892119874119896119904

is the angle of the dominant gradient directionat 119874119896119904 which can be calculated by (4) According to (5) the

angle of the vessel direction at 119874119896119904

is

120579119874119896119904=

120579119892119874119896119904

minus120587

2if 0 lt Δ120579 le 120587

120579119892119874119896119904

+120587

2if minus 120587 lt Δ120579 le 0

(8)

Finally the vessel direction at 119874119896119904

is described as a unitvector

119896119904= (cos(120579

119874119896119904) sin(120579

119874119896119904)) The new vessel direction

119896is defined as the vessel direction at the new center point

119874119896 which is calculated similarly to the direction at point119874

119896119904

34 Detection of Branches The vascular tree in retinal imagehas complex structuresThe linear search windowmentionedin Section 32workswell when tracking along a vessel withoutbranches However it is not suitable when encounteringvessel branches Therefore we consider using a semicirclesearch window (see Figure 7(a)) instead of the linear onewhen branches are predicted


Figure 4 Calculation of the vessel direction at the extrapolatedpoint 119874

119896119904

Figure 5 Different search windows during the tracking processBlack points are the candidate edge points and the white ones arethe detected vessel edge points and center points The blue arrowspoint to the vessel direction at the edge point or center point

The choice of different search windows depends on abranch prediction schemeWe have found that a single vesselwithout branches has two parallel edge lines Thus at a givenstep of the tracking process we compare the directions attwo local vessel edge points (see Figure 4) If the differencebetween two direction angles exceeds a fixed threshold 119879anglewe predict that current blood vessel will bifurcate into newvessel branches If not current vessel will be linear

At iteration 119896 vessel directions at two previous edgepoints

119896minus1and

119896minus1are obtained according to (6) Direction

angles of the edge points are denoted by 120579119880119896minus1

and 120579119881119896minus1

respectively The angle difference is calculated as

Δ120579119906V =

10038161003816100381610038161003816120579119880119896minus1

minus 120579119881119896minus1

10038161003816100381610038161003816 (9)

In this study the threshold 119879angle is chosen as 5∘ WhenΔ120579119906V lt 119879angle a linear search window is used Otherwise

when Δ120579119906V ge 119879angle a semicircle search window is used

instead of the linear one Figure 5 shows the different searchwindows during the tracking process However two vesseledge lines can be sometimes unparalleled due to the noise oreye diseases rather than vessel branches In these situationssemicircle search window is still used The proposed methodcan handle this situation and identify if branches really existor not

Background

Background

Vessel

Lk

Ukminus1

Vkminus1

Mi

Ukminus4 Ukminus3 Ukminus2

Vkminus2Vkminus3

Vkminus4

Mm1

Mm2

Figure 6 Normal configuration on a linear search window 1198721198981

and1198721198982

are selected candidate points on linear search window 119871119896

The two selected points give an assumption of local blood vessel anddefine a normal configuration

4 Bayesian Method for Vessel Segmentation

41 Configuration Model Vessel edge points are detectediteratively by a Bayesian method based on the proposedstatistical sampling scheme In order to choose new edgepoints among the candidate points on the dynamic searchwindow we define configurationmodels by a set of candidatepoints to describe the possible local vesselrsquos structures Inthis study vessel structures are categorized into three typesnormal bifurcation and crossing Normal case is regardedas the situation in which only a single vessel exists in thecurrent search area The case of bifurcation means thatone single vessel bifurcates into two branches A crossingcase is described when one vessel overlaps another In theproposed method there are three types of configurationmodels according to the different vessel structures

At a given step if a linear search window is used whenlocal blood vessel is predicted to be linear without brancheswe define normal configurations only As shown in Figure 6119871119896is the linear search window at iteration 119896 We select

two candidate points1198721198981

and1198721198982 which are the 119898th

1

and119898

th2

points on 119871119896(1 lt 119898

1lt 1198982lt 119873119871119896) and assume

them as new vessel edge points The two selected pointsdivide all the candidate points into three parts two partsbelonging to the background and one part belonging to thevessel Candidate points between the two selected pointsare assumed to belong to the vessel and others belong tothe background So the 119894th candidate 119872

119894(119894 isin [1119873

119871119896]) is

assumed to belong to the blood vessel if 119894 isin [1198981 1198982] or

to the background if 119894 isin [11198981[⋃]119898

2 119873119871119896] Two candidate

points selected on the linear search window can be definedas a normal configuration There are119873

119871119896(119873119871119896minus 1)2 normal

configurations at iteration 119896 which is the number of 2combinations from the set of119873

119871119896candidates on 119871

119896

Otherwise when vessel branches are predicted a semi-circle search window is used Three types of configurations(normal bifurcation and crossing) are defined In thissituation the normal configuration is defined to describethe possible local linear blood vessel which is mispredicted


as vessel branches by the branch prediction scheme Thebifurcation and crossing configurations are used to describethe possible new vessel branches At iteration 119896 if a semicirclesearch window 119862

119896is used three types of configurations are

illustrated in Figure 7 A normal configuration is definedby two selected candidate points on 119862

119896 similarly to the

definition on the linear search window For a bifurcationconfiguration four candidate points are selected to describethe edge points of two possible branches Six points areneeded for a crossing configuration Two of the six ones areassumed to be the new edge points of the same vessel whilethe other four points are considered as the edge points ofanother vessel which is over or under the current one

42 Probability of Configurations The configuration modelsare defined by a set of candidate edge points on the dynamicsearch window At a given step the proposed methodattempts to find a configuration which best matches currentvessel structure among all the possible configurations Thebest configuration at a given step is obtained by a maximuma posteriori estimation Local vessel structure and new vesseledge points are then detected based on this configuration

At a given step by computing the position of each of the119873 candidate edge points on the search window (line segmentor semicircle) and assigning the grey level value of the nearestpixel to that point we obtain the observed discrete intensityprofile 119884 = 119910

119894 119894 = 1 119873 The posteriori distribution of a

configuration 120594 is described as 119875(120594 | 119884) According to Bayesrsquorule

119875 (120594 | 119884) =119875 (119884 | 120594) 119875 (120594)

119875 (119884) (10)

where 119875(119884 | 120594) is the conditional probability of 119884 for agiven configuration 120594 and 119875(120594) is the a priori probabilityof 120594 When 119873 candidate points are selected the observedintensity profile 119884 is fixed and has no relationship with theconfiguration So119875(119884) does not depend on the configurationand will be disregarded Based on the Maximum a posteriori(MAP) criterion the best configuration is obtained as

120594 = argmax120594

119875 (120594 | 119884) = argmax120594

119875 (119884 | 120594) 119875 (120594) (11)

421 Configuration on Linear Search Window In order toexplain (11) in detail we first discuss the normal configura-tions on the linear search window As shown in Figure 6 119871

119896is

the linear search window at iteration 119896The observed discreteintensity profile on 119871

119896is 119884119871119896= 119910119894 119894 = 1 119873

119871119896 A normal

configuration which is defined by two selected candidatepoints119872

1198981and119872

1198982 is also shown in this figure Assuming

that the discrete grey levels on 119871119896are independent the

likelihood function of this normal configuration is computedas

119875 (119884119871119896| 120594) =

119873119871119896

prod

119894=1

119875 (119910119894| 120594) (12)

where 119875 (119910119894| 120594) is a conditional probability model which is

defined to describe the variability of the 119894th candidate point

on the search window belonging either to the backgroundor to the blood vessel As shown in Figure 6 all candidatepoints on 119871

119896are divided by119872

1198981and119872

1198982into three parts

which belong to the background vessel and backgroundrespectively so

119875 (119884119871119896| 120594) =

1198981minus1

prod

119894=1

119875 (119910119894| 119887)

1198982

prod

1198981

119875 (119910119894| V)

119873119871119896

prod

1198982+1

119875 (119910119894| 119887)

(13)

where 119887 and V denote background and vessel respectivelyIn Bayesian framework we assume that 119883

119871119896= 119909119894 119894 =

1 2 119873119871119896 is the true intensity profile associated with the

119873119871119896

candidate points on 119871119896 In this study we assume that the

retinal image is only affected by additive noise 120585 so

119884119871119896= 119883119871119896+ 120585 (14)

During the tracking process the local background is assumedto have a constant intensity Vesselrsquos sectional intensity profilecan be approximated by a Gaussian curve [8 40] The greylevel of a point 119872 in local vessel cross-section as shown inFigure 8 is computed as

119866 (119872) =

(119868119888minus 119868119887) exp(minus 119897

2

21205902

) + 119868119887

if 119872 isin vessel

119868119887

if 119872isin background(15)

where 119868119888is the grey level of local vesselrsquos center 119868

119887is the grey

level of local background 119897 is the distance between point119872 and the straight line which passes through vesselrsquos centeralong local vessel direction and 120590 defines the spread of theintensity profile and is set to the value of half of local vesselrsquosradius A selected vessel cross-section and its observed andestimated intensity profiles are show in Figure 9 For thenormal configuration shown in Figure 6 the true grey levelof the 119894th candidate point119872

119894on 119871119896 is estimated as

119909119894=

119909119894(V) = (119868

119888119896minus 119868119887119896) exp(minus

119897119894

2

2120590120594

2

) + 119868119887119896

if 119894 isin [1198981 1198982]

119909119894(119887) = 119868

119887if 119894 isin [1119898

1)⋃(119898

2 119873119871119896]

(16)

where 119868119888119896

is the grey level of local vessel center and 119868119887119896

is the mean grey level of local background areas 119897119894is the

distance between 119872119894and the straight line which passes

through themiddle point of [11987211989811198721198982] and is perpendicular

to 997888997888997888997888997888997888997888rarr11987211989811198721198982 The spread parameter 120590

120594= (12)|119872

11989811198721198982|

At iteration 119896 the local intensity parameters 119868119888119896

and 119868119887119896

are calculated on three moving regions one inside and twoothers outside the local blood vessel as shown in Figure 10These three regions are selected by the algorithm in localresearch area based on local vessel edge points

119896minus1and

119896minus1

The size of these three regions is adaptive to local vesseldiameter 119868

119888119896is the grey level of local vessel center point119874

119896minus1

119868119887119896

is the mean grey level of two regions outside local bloodvessel


(a) Normal (b) Bifurcation

(c) Crossing

Figure 7 Semicircle search window and three types of configurations 119862119896

is the semicircle search window at iteration 119896 and points markedby small circles are selected to define different types of configurations

Background

Vessel

Background

2120590

M

l

Figure 8 Illustration of (15) The dotted line is the vesselrsquos centerline The pink point is local vessel center point while the blue one ispoint119872

The observed grey level profile on 119871119896can be expressed as

119910119894= 119909119894+ 120585

= 119909119894(V) + 120585V if 119894 isin [119898

1 1198982]

119909119894(119887) + 120585

119887if 119894 isin [1119898

1)⋃(119898

2 119873119871119896]

(17)

where 120585V and 120585119887 are the Gaussian noise in local blood vesseland background respectively

120585V sim 119873(0 1205902

V)

120585119887sim 119873(0 120590

2

119887

)

(18)

The statistical parameters 120590V and 120590119887are also computed by

threemoving regions as shown in Figure 10 120590V is estimated asthe standard deviation of grey levels in the vessel region and120590119887is the standard deviation of grey levels in two background

regionsThen conditional probability model of the proposednormal configuration is

119875 (119910119894| V) =

1

radic2120587120590V

exp(minus(119910119894minus 119909119894(V))2

21205902

V

)

119875 (119910119894| 119887) =

1

radic2120587120590119887

exp(minus(119910119894minus 119909119894(119887))2

21205902

119887

)

(19)

and the likelihood function can be calculated by (13)In our method the a priori probability is based on the

Gibbs formulation [42]

119875 (120594) =1

119885exp (minus120582119880 (120594)) (20)


Background Vessel Background

2120590

Ml

Ib

Ic

(a)

minus10 minus8 minus6 minus4 minus2 0 2 4 6 8 10160

170

180

190

200

210

220

Distance in pixels from vessel center line

Observed profileEstimated profile

Gre

y le

vel

(b)

Figure 9 Vesselrsquos sectional intensity profile (a) Gaussian intensity model (b)The observed and estimated profiles of the vessel cross-sectionmarked by a black line on the retinal image

Lk

Ukminus1

Vkminus1


Vkminus2Vkminus3

Vkminus4

E1

E2

Mm1

tm1

Mm2

tm2

Figure 10 Estimation of vessel edge lines and local statisticparameters

where 119885 is a normalization parameter and 120582 is a regulariza-tion term In our study 119885 was fixed to 1 As 119885 was set tobe a constant it has no influence on the computation of themaximum of (11) After many experiments 120582 was set to 001119880(120594) is the energy function of a given configuration Gibbsformulation aims at linking a configuration with an energyfunction to penalize high energetic configurations

At a given step when a linear search window is usedthe local blood vessel is assumed to be linear The localvessel edges can be estimated as two straight lines and newedge points are supposed to be aligned on the local vesseledge lines At iteration 119896 (119896 lt 5) the vessel edge lines 119864

1

and 1198642are estimated as two straight lines along the local

direction 119896minus1

and passing through two edge points 119896minus1

and 119896minus1

respectively When 119896 ge 5 1198641and 119864

2are defined as

the least square straight lines obtained by four pairs of edgepoints detected in the previous iterations (see Figure 10)Considering the proposed normal configuration of Figure 6we compute the distance between 119872

1198981and 119864

1and

the distance between 1198721198982

and 1198642 which are denoted

by 1199051198981

and 1199051198982 respectively (see Figure 10) The energy

function can be defined as119880(120594) = 11990521198981

+ 1199052

1198982

[42] The a prioriprobability has the expression

119875 (120594) =1

119885exp (minus120582 (1199052

1198981

+ 1199052

1198982

)) (21)

At iteration 119896 the likelihood functions and the a prioriprobabilities of all the 1198622

119873119871119896

normal configurations on 119871119896are

computed similarly to the proposed normal configurationof Figure 6 The best configuration at iteration 119896 is obtainedby (11) The two candidate points used to define the bestconfiguration are regarded as new vessel edge points

422 Configurations on Semicircle Search Window In theother situation when a semicircle search window is used ata given step three types of configurations are defined Asshown in Figure 7 119862

119896is the semicircle search window at

iteration 119896 The observed discrete intensity profile on 119862119896is

denoted by 119884119862119896= 119910119894 119894 = 1 119873

119862119896 The probabilities of all

the configurations on 119862119896are computed

For normal configurations on 119862119896 the likelihood function

is computed similarly using (13) An example of bifurcationconfiguration is shown in Figure 7(b) It has four selectedpoints which are the 119901th

1

119901th2

119901th3

and 119901th4

candidate pointson 119862119896 Its likelihood function can be computed as

119875 (119884119862119896| 120594) =

1199011minus1

prod

119894=1

119875 (119910119894| 119887)

1199012

prod

1199011

119875 (119910119894| V)1199013minus1

prod

1199012+1

119875 (119910119894| 119887)

1199014

prod

1199013

119875 (119910119894| V)

119873119862119896

prod

1199014+1

119875 (119910119894| 119887)

(22)

The sectional intensity distribution of the two assumed vesselbranches is also approximated by the Gaussian shaped curve


The true discrete intensity profile on 119862119896is obtained similarly

to (16) according to the given bifurcation configuration Theconditional probability models 119875 (119910

119894| 119887) and 119875 (119910

119894| V) in

(22) are calculated similarly to (19) In addition the likelihoodfunction of a crossing configuration is obtained based on itssix selected points

The a priori model which is defined by the previousdetected vessel edge points (see (21)) has no influenceon bifurcation or crossing configurations Therefore thea priori probability of a configuration on the semicirclesearch window no matter normal bifurcation or crossing isdisregarded According to (11) the best configuration on 119862

119896

is obtained as

120594 = argmax120594

119875 (119884119862119896| 120594) (23)

If 120594 is a normal configuration the local blood vessel isconsidered to be linear and two candidate points used todefine 120594 are regarded as new vessel edge points at iteration119896 So the proposed method solves the problem that locallinear blood vessel is mispredicted to bifurcate into vesselbranches If 120594 is a bifurcation configuration two vesselbranches are detected and the four selected candidate pointsare considered as the initial edge points of two branches If 120594is a crossing configuration current vessel encounters anotherone The new detected blood vessel is regarded as two vesselbranches starting from the crossingThe third and the fourthselected points of the six ones are regarded as new edge pointsof current blood vessel The other four points are consideredas the initial edge points of two branches starting from thecrossing

5 Experiments and Discussion

The performance of our method was evaluated on threepublicly available databases STARE [6] DRIVE [7] andREVIEW [30 43] There are three initial parameters in ouralgorithm the distance between grid lines in the selectionof seed points the look-ahead distance 119904 (see (2)) and thelength of the initial linear window 119871

0 In our experiments

the distance between grid lines is fixed to 30 pixels 119904 isset to 3 pixels and 119871

0is set to 10 pixels after many trials

These parameter values can give the best results includingcomputational time when testing on the three databases

The STARE database contains 20 retinal images whichare captured by the TopCon TRV-50 fundus camera at a 35∘field of view (FOV) 10 of the images are from healthy ocularfundus and the other 10 are from unhealthy ones All theimages are segmented manually by two independent special-ists The proposed method is tested with the segmentation ofthe first observer as ground truth

TheDRIVE database contains 40 retinal images which arecaptured by theCanonCR5 camera at 45∘ FOVThe 40 imageswere divided into a training set and a test set each of whichcontains 20 images They had been manually segmented bythree observers trained by an ophthalmologist The imagesin the training set were segmented once while images in thetest set were segmented twice resulting in sets A and B Our

method is tested on the test set using the segmentations of setA as ground truth

The REVIEW database consists of four image sets whichinclude 16 images with 193 vessel segments demonstrating avariety of pathologies and vessel types The high resolutionimage set (HRIS) represents different sever grades of diabeticretinopathy The vascular disease image set (VDIS) containsa range of normal and diseased retina including diabetic andarteriosclerotic retinopathies The central light reflex imageset (CLRIS) represents early atherosclerotic changes with anexaggerated vascular light reflexThe images of the kick pointimage set (KPIS) are taken from clean large vessel segmentsThese four image sets contain 5066manually marked profilesassessed by three independent expertsTheperformance of analgorithm can be compared with manual measurement withaccuracy up to 025 of a pixel

51 Segmentation Performance We used STARE and DRIVEdatabases to evaluate the segmentation performance of theproposed method In practice our algorithm is developedin Matlab (version 760324) environment Some functionsare programmed using C language in order to save com-putation time The average running time of our trackingmethod on one image is 96min on STARE and 63min onDRIVE Our runtimes are higher than the ones obtainedusing nontracking vessel segmentation For example theprocessing time of Mendoncarsquos method [1] is 3min for aSTARE image and 25min for a DRIVE image MendoncaandCampilho [1] used a pixel-based global approach and alsoimplemented their algorithm usingMatlab In order to assessthe proposed method and compare it with the other state-of-art methods we use three widely known performancemeasures the detection accuracy sensitivity and specificityThe accuracy is defined as the ratio of the total number ofcorrectly classified pixels to the number of pixels in the FOV(field of view) The sensitivity is defined as the ratio of thenumber of correctly classified vessel pixels to the numberof total vessel pixels in the ground truth The specificity isdefined as the ratio of the number of correctly classifiednonvessel pixels to the number of total nonvessel pixels insideFOV in the ground truth

First we discuss the segmentation performance of theproposed method on 20 retinal images from the STAREdatabase For example we discuss the test result of ourmethod on a retinal image (im0077) from the STAREdatabase as shown in Figure 11(a) The segmentation resultof the proposed method and the ground truth are shown inFigures 11(b) and 11(c) respectively In order to obtain thefinal detected blood vessels we first obtain vessel edge linesby linking the detected vessel edge points Then the pixelsbetween two vessel edge lines are considered to belong tothe blood vessel By comparing with the ground truth wecan see that the proposed method is able to detect most ofthe blood vessels The detection sensitivity specificity andaccuracy of the proposed method on im0077 are 0802009600 and 09426 respectively Besides we selected fourregions on im0077 as shown in Figure 11(a) which presentfour types of vessel structures line curve bifurcation and


(d)(g)

(e) (f)

(a) (b) (c)

(d) (e) (f) (g)

Figure 11 Tests on different vessel structures (a) Retinal image (im0077) from the STARE database Four selected regions show differentvessel structures such as line curve bifurcation and crossing (b) Segmentation result by the proposed method (c) Ground truth (d)ndash(g)Subimages obtained from (a) and the related dynamic search window and detected vessel edge points

crossing The enlarged regions are shown in Figures 11(d)ndash11(g) respectively In the second rowof Figures 11(d)ndash11(g) wecan see that the linear search window is used when trackingfor linear blood vessels while the semicircle search windowis used to detect vessel branches in the bifurcation or crossingstructures The candidate points on each search window aremarked by block dots while detected vessel edge points aremarked by small circles

Threemethods Hooverrsquos [6] Soaresrsquo [13] andMendoncarsquos[1]methods are used for comparison purposes on the STAREdatabase Hoover et al [6] proposed a filter-based methodand made their segmentation results on the STARE databasepublic on their website Soaresrsquo [13] method is based onthe 2D Gabor wavelet transform and its test results onthe STARE database are presented on the public website(httpretinalsourceforgenet) Mendonca and Campilho[1] used morphological operators for vessel segmentationTable 1 lists the segmentation performance of different meth-ods on the STARE database As mentioned in the beginningof this section segmentation results of the first observer areused for the ground truth In Table 1 the manual detectionresults of the second observer are considered as a refer-ence used for comparison The performance measures ofHooverrsquos [6] and Soaresrsquo [13]methods are calculated using thesegmented images from their public websites respectivelyThe performance of Mendoncarsquos method is obtained from

Table 1 Evaluation results of different segmentation methods onSTARE database

Method Sensitivity Specificity AccuracyThe 2nd observer 08949 09390 09354Hoover et al [6] 06751 09567 09267Soares et al [13] 07165 09748 09480Mendonca and Campilho [1] 06996 09730 09440Proposed 07248 09666 09412

the original paper [1] The experimental results show thatour method has a higher sensitivity (07248) than the otherthree methods The specificity and accuracy of the proposedmethod are 09666 and 09412 respectively which are bothhigher than those of the second observer and Hooverrsquosmethod [6]

As mentioned above the STARE database includes 10normal and 10 abnormal retinal images In Table 2 we showthe performance measures of different methods in the twocases on STARE database In normal cases the proposedmethod outperforms Hooverrsquos method [6] and has a similarperformance to Mendoncarsquos method [1] In abnormal casesthe proposed method is better than the other three methodsThe sensitivity of the proposedmethod is up to 07034 which


(a) (b) (c)

(d) (e)

Figure 12 Results on a normal retinal image from the STARE database (a) retinal image (im0255) (b) Hoover et al [6] (c) Soares et al [13](d) the proposed method (e) ground truth

(a) (b) (c)

(d) (e)

Figure 13 Results on an abnormal retinal image from the STARE database (a) retinal image (im0044) (b) Hoover et al [6] (c) Soares et al[13] (d) the proposed method (e) ground truth


(a) (b)

(c) (d)

(e) (f)

Figure 14 Results on a retinal image from DRIVE database (a) retinal image (01 test) (b) ground truth (c) Martınez-Perez et al [31] (d)Chaudhuri et al [8] (e) Jiang and Mojon [9] (f) the proposed method

is much higher than that of the others To make the com-parison fairer we show the segmentation results of differentmethods on two retinal images from the STAREdatabase onenormal im0255 and one abnormal im0044 (see Figures 12 and13) The ground truth image is the segmentation result of thefirst observer Result images of Hooverrsquos [6] and Soaresrsquo [13]methods are downloaded directly from their public website(httpretinalsourceforgenet) In the normal case as shownin Figure 12 the proposed method detected more detailsthan the others especially in the area near the fovea In the

abnormal case as shown in Figure 13 our method detectedmost of the vascular tree while Hooverrsquos method lost somethin blood vessels and themain blood vessels in Soaresrsquo resultare discontinued

Secondly we discuss the segmentation performanceon DRIVE database Several popular vessel segmentationmethods are considered when testing on the DRIVEdatabase for comparison purposes as shown in Table 3The performance measures of the compared methodsare obtained from the website of DRIVE database


(a) (b)

(c) (d)

Figure 15 Test results of the proposed method on the REVIEW database Images (a)ndash(d) show the vessel segments obtained from HRISVDIS CLRIS and KPIS respectively

Proposed methodGround truth

Vessel diameter (pixels)

Freq

uenc

y

1000

800

600

400

200

02520151050

Figure 16 Distribution of vessel diameters on the REVIEWdatabase

(httpwwwisiuunlResearchDatabasesDRIVE) We cansee that the proposed method has higher specificity andaccuracy than those of the others The sensitivity is a bit

Table 2 Evaluation results on normal and abnormal retinal imagesfrom STARE database

Method Sensitivity Specificity AccuracyNormal cases

The 2nd observer 09646 09236 09283Hoover et al [6] 06766 09662 09324Soares et al [13] 07554 09812 09542Mendonca and Campilho [1] 07258 09791 09492Proposed 07463 09664 09405

Abnormal casesThe 2nd observer 08252 09544 09425Hoover et al [6] 06736 09472 09211Soares et al [13] 06869 09682 09416Mendonca and Campilho [1] 06733 09669 09388Proposed 07034 09668 09420

inferior to some state-of-the-art methods Because ourmethod seems to be effective in abnormal cases as shown inTable 2 most of the images are normal in DRIVE database

Figure 14 shows the segmentation results of four differentmethods on the retinal image 01 test from the DRIVE


Table 3 Evaluation results of different segmentation methods onDRIVE database

Method Sensitivity Specificity AccuracyThe 2nd observer 07761 09725 09473Al-Diri et al [30] 07282 09551 09258Jiang and Mojon [9] 06478 09625 09212Martınez-Perez et al [31] 07086 09496 09181Chaudhuri et al [8] 02716 09794 08773Proposed 06522 09710 09267

database As shown in Figure 14(a) 01 test is a normal retinalimage without retinopathy The ground truth as shown inFigure 14(b) is manual segmentation result Figures 14(c) and14(d) show the original classification results of the Martnez-Perezrsquos [31] and Chaudhurirsquos [8] algorithms respectivelyFor these two methods the final segmentation results areobtained from the original results by thresholding at a certainvalue Jiang andMojon [9] use the adaptive local thresholdingand the segmentation result is shown in Figure 14(e) Thereare disconnected main blood vessels and missed thin bloodvessels in Jiangrsquos results The segmentation result of theproposed method is shown in Figure 14(f)

52 Width Measurement Performance To assess the diame-termeasurement performance we have used the four datasets(HRIS VDIS CLRIS and KPIS) of the REVIEW databaseIn REVIEW database the profiles marked by the observersprovide the locations of vessel edge points for the vesselsegments inHRIS VDIS andCLRIS Local vessel parametersare calculated based on these marked vessel edge points Ifa pair of local marked vessel edge points are denoted by(1199091 1199101) and (119909

2 1199102) the local vessel center point is ((119909

1+

1199092)2 (119910

1+ 1199102)2) diameter isradic(119909

1minus 1199092)2

+ (1199101minus 1199102)2 and

direction is (1205872) + arctan((1199102minus 1199101)(1199092minus 1199091)) For the

dataset KPIS the manually marked profiles supply the vesselcenter point diameter and direction directly without givingthe vessel edge points When testing our method on theimages of REVIEW database the first profile of each selectedvessel segment is used for the initialization The trackingprocess is stopped when the search area exceeds the selectedvessel segment Figure 15 shows examples of vessel segmentsobtained from the four datasets respectivelyThe edge pointsdetected by the proposed method are marked by black starswhile the ground truth points given by the database aremarked by white dots Because the database does not givethe observed edge points in KPIS there are no ground truthpoints in Figure 15(d)

It is known that the actual vessel diameters in theREVIEW database are obtained by observers In order tocompare the actual diameters with the diameters detectedby the proposed method we analyze the distribution of thevessel diameters For this purpose the distribution of thediameters is obtained by two steps rounding the valuesof vessel diameters to integers and then computing the

frequency of the diameters The distribution curves of theactual and the detected vessel diameters on the REVIEWdatabase are shown in Figure 16The actual distribution curveis marked by stars while the detected distribution is markedby circles From the distribution curves we can see that mostof the blood vessels have diameters between 3 and 10 pixelsThe largest blood vessel does not exceed 23 pixels in widthThe proposed method has a high accuracy in vessel widthmeasurement

We have also compared the performance of the proposedmethod with the half height full width (HHFW) [38]Gregsonrsquos [37] 1D Gaussian [39] 2D Gaussian [40] andAl-Dirirsquos [30] methods by presenting the success rate andmean width When an algorithm fails to detect the vesselwidth at a given point (eg does not converge) the currentwidth measurement is considered meaningless The successrate of an algorithm is defined as the ratio between thenumber of meaningful measurements and the number oftotal measurements Table 4 shows the performance of thedifferentmethods by presenting the success rate and themeanwidth For each dataset the success rate is shown in the firstcolumn and the mean width is shown in the second columnThe results of three observers are obtained from the databaseFor each dataset the mean result of the three observers isused as ground truth The results of the compared methodsare obtained from the original paper of Al-Diri We can seefrom the success rates that the HHFW 1D Gaussian 2DGaussian and Al-Dirirsquos methods failed to detect some vesselsegments on datasets HRIS VDIS and CLRIS In particularthe success rate of the HHFW is down to 0 on the CLRIS ForGregsonrsquos and the proposedmethods the success rate is 100on all of the four datasets However the proposedmethod hashigher accuracy in the mean width than Gregsonrsquos methodIn general the proposed method outperforms all the otheralgorithms especially on the CLRIS dataset On the HRISandVDIS datasets ourmethod is slightly inferior to Al-Dirirsquosmethod which however did not use all its measurementresults

6 Conclusions

In this paper we have introduced an automatic trackingmethod for vessel detection in retinal images The trackingalgorithm is able to segment themain vascular tree and obtainlocal vessel parameters such as vessel edge points directionand the vessel width along the retinal vessels The proposedmethod starts from a number of seed points selected all overthe image During the tracking process a dynamic searchwindow is used to get the information of local grey levelsrsquostatistics Different types of configurations are defined tocombine the intensity distribution and the geometrical struc-ture of local blood vessel Vesselrsquos edge points are detectedby a Bayesian method with the MAP criterion Experimentson three publicly available databases show that the proposedmethod detects blood vessels effectively and outperformssome classical vessel detection methods We compared ourraw results with the other techniques Some postprocessingwill be done later in order to improve the detection accuracy


Table 4 Performance of different methods on REVIEW database

HRIS VDIS CLRIS KPISMethod Mean Mean Mean MeanObserver 100 435 100 885 100 138 100 752Gregson et al [37] 100 764 100 1007 100 128 100 729HHFW [38] 883 497 784 794 0 mdash 963 6471D-G [39] 996 381 999 578 986 63 100 4952D-G [40] 989 418 772 659 267 70 100 587Al-Diri et al [30] 997 463 996 88 930 157 100 656Proposed 100 481 100 785 100 1339 100 671

Besides the improvements of vessel sectional intensitymodeland the reconstruction method of the detected vascular treeare both necessary work in the future

Conflict of Interests

There is no conflict of interests in the present study

References

[1] A M Mendonca and A Campilho ldquoSegmentation of retinalblood vessels by combining the detection of centerlines andmorphological reconstructionrdquo IEEE Transactions on MedicalImaging vol 25 no 9 pp 1200ndash1213 2006

[2] B FangW Hsu andM L Lee ldquoOn the detection of retinal ves-sels in fundus imagesrdquo 2003 httphdlhandlenet172113675

[3] C Kirbas and F Quek ldquoA review of vessel extraction techniquesand algorithmsrdquoACMComputing Surveys vol 36 no 2 pp 81ndash121 2004

[4] D Lesage E D Angelini I Bloch and G Funka-Lea ldquoA reviewof 3D vessel lumen segmentation techniques models featuresand extraction schemesrdquoMedical Image Analysis vol 13 no 6pp 819ndash845 2009

[5] M M Fraz P Remagnino A Hoppe et al ldquoBlood ves-sel segmentation methodologies in retinal imagesmdasha surveyrdquoComputer Methods and Programs in Biomedicine vol 108 pp407ndash433 2012

[6] A Hoover V Kouznetsova andM Goldbaum ldquoLocating bloodvessels in retinal images by piecewise threshold probing of amatched filter responserdquo IEEETransactions onMedical Imagingvol 19 no 3 pp 203ndash210 2000

[7] J Staal M D Abramoff M Niemeijer M A Viergever andB van Ginneken ldquoRidge-based vessel segmentation in colorimages of the retinardquo IEEETransactions onMedical Imaging vol23 no 4 pp 501ndash509 2004

[8] S Chaudhuri S Chatterjee N Katz M Nelson and MGoldbaum ldquoDetection of blood vessels in retinal images usingtwo-dimensionalmatched filtersrdquo IEEETransactions onMedicalImaging vol 8 no 3 pp 263ndash269 1989

[9] X Jiang and D Mojon ldquoAdaptive local thresholding byverification-based multithreshold probing with application tovessel detection in retinal imagesrdquo IEEE Transactions on PatternAnalysis andMachine Intelligence vol 25 no 1 pp 131ndash137 2003

[10] M Sofka and C V Stewart ldquoRetinal vessel centerline extractionusing multiscale matched filters confidence and edge mea-suresrdquo IEEE Transactions onMedical Imaging vol 25 no 12 pp1531ndash1546 2006

[11] F Zana and J C Klein ldquoSegmentation of vessel-like patternsusing mathematical morphology and curvature evaluationrdquoIEEE Transactions on Image Processing vol 10 no 7 pp 1010ndash1019 2001

[12] M Niemeijer J Staal B van Ginneken M Loog and M DAbramoff ldquoComparative study of retinal vessel segmentationmethods on a new publicly available databaserdquo in MedicalImaging 2004 Image Processing vol 5370 of Proceedings of SPIEpp 648ndash656 San Diego Calif USA February 2004

[13] J V B Soares J J G Leandro R M Cesar Jr H F Jelinekand M J Cree ldquoRetinal vessel segmentation using the 2-DGabor wavelet and supervised classificationrdquo IEEE Transactionson Medical Imaging vol 25 no 9 pp 1214ndash1222 2006

[14] E Ricci and R Perfetti ldquoRetinal blood vessel segmentationusing line operators and support vector classificationrdquo IEEETransactions on Medical Imaging vol 26 no 10 pp 1357ndash13652007

[15] S Kozerke R Botnar S Oyre M B Scheidegger E MPedersen andP Boesiger ldquoAutomatic vessel segmentation usingactive contours in cine phase contrast flow measurementsrdquoJournal of Magnetic Resonance Imaging vol 10 pp 41ndash51 1999

[16] A Gooya H Liao K Matsumiya K Masamune Y Masutaniand T Dohi ldquoA variationalmethod for geometric regularizationof vascular segmentation inmedical imagesrdquo IEEE Transactionson Image Processing vol 17 no 8 pp 1295ndash1312 2008

[17] A Vasilevskiy and K Siddiqi ldquoFlux maximizing geometricflowsrdquo IEEE Transactions on Pattern Analysis and MachineIntelligence vol 24 no 12 pp 1565ndash1578 2002

[18] M W K Law and A C S Chung ldquoEfficient implementationfor spherical flux computation and its application to vascularsegmentationrdquo IEEE Transactions on Image Processing vol 18no 3 pp 596ndash612 2009

[19] T Kayikcioglu and S Mitra ldquoA new method for estimatingdimensions and 3-d reconstruction of coronary arterial treesfrom biplane angiogramsrdquo in Proceedings of 6th Annual IEEESymposium on Computer-Based Medical Systems pp 153ndash158Jun 1993

[20] Y Sato T Araki M Hanayama H Naito and S TamuraldquoA viewpoint determination system for stenosis diagnosis andquantification in coronary angiographie image acquisitionrdquoIEEE Transactions on Medical Imaging vol 17 no 1 pp 121ndash1371998

[21] L Wang A Bhalerao and R Wilson ldquoAnalysis of retinalvasculature using a multiresolution hermite modelrdquo IEEETransactions onMedical Imaging vol 26 no 2 pp 137ndash152 2007

[22] Y A Tolias and S M Panas ldquoA fuzzy vessel tracking algorithmfor retinal images based on fuzzy clusteringrdquo IEEE Transactionson Medical Imaging vol 17 no 2 pp 263ndash273 1998


[23] A H Can H Shen J N Turner H L Tanenbaum and BRoysam ldquoRapid automated tracing and feature extraction fromretinal fundus images using direct exploratory algorithmsrdquoIEEE Transactions on Information Technology in Biomedicinevol 3 no 2 pp 125ndash138 1999

[24] P Zou P Chan and P Rockett ldquoA model-based consecutivescanline tracking method for extracting vascular networksfrom 2-D digital subtraction angiogramsrdquo IEEE Transactions onMedical Imaging vol 28 no 2 pp 241ndash249 2009

[25] O Chutatape L Zheng and S Krishnan ldquoRetinal blood vesseldetection and tracking bymatched gaussian and kalman filtersrdquoin Proceedings of the 20th Annual International Conference of theIEEE Engineering in Medicine and Biology Society (EMBS rsquo98)pp 3144ndash3149 1998

[26] E Grisan A Pesce A Giani M Foracchia and A RuggerildquoA new tracking system for the robust extraction of retinalvessel structurerdquo in Proceedings of the 26th Annual InternationalConference of the IEEE Engineering in Medicine and BiologySociety (EMBC rsquo04) pp 1620ndash1623 San Francisco Calif USASeptember 2004

[27] J Ng S T Clay S A Barman et al ldquoMaximum likelihoodestimation of vessel parameters from scale space analysisrdquoImage and Vision Computing vol 28 no 1 pp 55ndash63 2010

[28] T Yedidya and R Hartley ldquoTracking of blood vessels in retinalimages using kalman filterrdquo in Proceedings of the Digital ImageComputing Techniques and Applications (DICTA rsquo08) pp 52ndash58 Canberra Australia December 2008

[29] M Vlachos and E Dermatas ldquoMulti-scale retinal vessel seg-mentation using line trackingrdquo Computerized Medical Imagingand Graphics vol 34 no 3 pp 213ndash227 2010

[30] B Al-Diri A Hunter andD Steel ldquoAn active contourmodel forsegmenting and measuring retinal vesselsrdquo IEEE Transactionson Medical Imaging vol 28 no 9 pp 1488ndash1497 2009

[31] M E Martınez-Perez A D Hughes A V Stanton S AThom A A Bharath and K H Parker ldquoRetinal blood vesselsegmentation by means of scale-space analysis and regiongrowingrdquo in Proceedings of the Medical Image Computing andComputer-Assisted Intervention (MICCAI rsquo99) pp 90ndash97 1999

[32] S Zhou W Chen Z Zhang and J Yang ldquoAutomatic segmen-tation of coronary angiograms based on probabilistic trackingrdquoin Proceedings of the 3rd International Conference on Bioinfor-matics and Biomedical Engineering (ICBBE rsquo09) pp 1ndash4 BeijingChina June 2009

[33] C Florin N Paragios and J Williams ldquoParticle filters aquasi-monte carlo solution for segmentation of coronariesrdquo inProceedings Medical Image Computing and Computer-AssistedIntervention (MICCAI rsquo05) pp 246ndash253 2005

[34] K A Al-Kofahi S Lasek D H Szarowski et al ldquoRapidautomated three-dimensional tracing of neurons from confocalimage stacksrdquo IEEE Transactions on Information Technology inBiomedicine vol 6 no 2 pp 171ndash187 2002

[35] M Adel A Moussaoui M Rasigni S Bourennane and LHamami ldquoStatistical-based tracking technique for linear struc-tures detection application to vessel segmentation in medicalimagesrdquo IEEE Signal Processing Letters vol 17 pp 555ndash558 2010

[36] Y Yin M Adel and S Bourennane ldquoRetinal vessel segmenta-tion using a probabilistic trackingmethodrdquoPattern Recognitionvol 45 no 4 pp 1235ndash1244 2012

[37] P H Gregson Z Shen R C Scott and V Kozousek ldquoAuto-mated grading of venous beadingrdquo Computers and BiomedicalResearch vol 28 no 4 pp 291ndash304 1995

[38] O Brinchmann-Hansen and H Heier ldquoTheoretical relationsbetween light streak characteristics and optical properties ofretinal vesselsrdquoActaOphthalmologica vol 64 no 179 pp 33ndash371986

[39] L ZhouM S Rzeszotarski L J Singerman and J M ChokreffldquoDetection and quantification of retinopathy using digitalangiogramsrdquo IEEE Transactions on Medical Imaging vol 13 no4 pp 619ndash626 1994

[40] J Lowell A Hunter D Steel A Basu R Ryder and R LKennedy ldquoMeasurement of retinal vessel widths from fundusimages based on 2-D modelingrdquo IEEE Transactions on MedicalImaging vol 23 no 10 pp 1196ndash1204 2004

[41] A R Rao and R C Jain ldquoComputerized flow field analysisoriented texture fieldsrdquo IEEE Transactions on Pattern Analysisand Machine Intelligence vol 14 pp 693ndash709 1992

[42] M V Ibanez and A Simo ldquoBayesian detection of the fovea ineye fundus angiographiesrdquo Pattern Recognition Letters vol 20no 2 pp 229ndash240 1999

[43] B Al-Diri A Hunter D Steel M Habib T Hudaib and SBerry ldquoReviewmdasha reference data set for retinal vessel profilesrdquoin Proceedings of the 30th Annual International Conference of theIEEE Engineering in Medicine and Biology Society (EMBS rsquo08)pp 2262ndash2265 Vancouver Canada August 2008

Submit your manuscripts athttpwwwhindawicom

Stem CellsInternational

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


MEDIATORSINFLAMMATION

of


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OncologyJournal of



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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Evidence-Based Complementary and Alternative Medicine

Volume 2014Hindawi Publishing Corporationhttpwwwhindawicom


narrow elongated vessel segmentation and Law and Chung[18] improved this technique for spherical flux computationCoronary arterial trees are reconstructed using ellipticalparametric cross-section model by Kayikcioglu and Mitra[19] Hessian-based method is proposed by Sato et al [20] tocharacterize stenosis in coronary angiograms whereas Wanget al [21] used Hermite multiresolution model for retinalvessel analysis A multiresolution approach based on a scale-space analysis is presented by Martınez-Perez et al [31] inwhich the width the size and orientation of retinal vesselsare obtained

Tracking-based approaches are based on local techniqueswhere either the centerline or vessel edges or both areextracted Starting from seed points these methods progressalong vessels by iterative prediction and parameter estima-tion Many methods including several 2D and 3D medicalimaging modalities have been published [4] An advantageof tracking methods is the guaranteed connectedness ofvessel segment whereas in pixels-processing-basedmethodsconnectedness is not guaranteed A whole vessel tree canbe tracked by these methods without examining the vastmajority of the image Starting from the optic disc Toliasand Panas [22] developed a fuzzy tracking model of 1Dvessel profile Their method however did not manage tohandle branch points Starting from seed points Can et al[23] used an iterative tracking algorithm updating at eachstep the position and the orientation of vessel points Zouet al [24] developed a recursive tracking technique guidedby accurate vessel direction and robust a priori knowledgeand termination criteria Compared to others this approachcould extract automatically most of the vessels with accuratevessel characteristics

Among tracking methods few probabilistic approacheshave been reported in the literature [32ndash34] The main ideasof a probabilistic trackingmethod have been presented in ourprevious work [35 36] which needed some improvementslike modeling blood vessels more accurately handling differ-ent vessel configurations and evaluating it on large databaseimages

In this paper a novel fully automatic tracking basedmethod using probabilistic formulation is introduced Firstseed points located inside vessels are obtained from theimage From these points an iterative tracking algorithmdetects simultaneously edges diameter width centerline andorientation of the whole vascular tree A branch predictionscheme to handle bifurcation and crossing is also developedThe major novelty lies in defining the likelihood of thehypothesis of edge points and the a prioriprobability based onGibbs formulation This new approach uses Gaussian modelto approximate vessel sectional intensity profiles identifiesbifurcation and crossing structures using gradient analy-sis tracks centerline and local vessel edges geometry andimproves the detection results as shown in the experimentsand discussion section A probabilistic segmentation schemeis associated with the maximum a posteriori (MAP) as acriterion to estimate local vessel edges

In the following Section 2 gives a general descriptionof the proposed method Section 3 is devoted to the expla-nations of the tracking algorithm Bayesian segmentation

is described in Section 4 Finally experiments on STAREDRIVE and REVIEW databases are presented and discussedin Section 5

2 General Description of the Method

In this paper we propose a tracking-based method to detectretinal vascular trees First a number of seed points areselected automatically on the retinal image which provideinitial parameters for the tracking algorithm The trackingprocess starts then from each of the seed points and detectsvessel edge points iteratively until end conditions are satisfiedFinally when all the seed points are processed the wholevascular tree is detected and the proposed algorithm stops

A tracking process from one seed point is described indetail as follows

21 Initialization Initial parameters which are obtainedfroma seed point include initial vessel center point and vesseldirection The blood vessel is then detected by the trackingalgorithm from the initial center point along the initial vesseldirection

22 Iteration At a given step a dynamic search windowis defined based on local vessel parameters including vesselcenter point direction and diameter We adopt a statisticalsampling method to select new candidate edge points onthe search window A Bayesian method with the MAP as acriterion is used to pick out new vessel edge points from thesecandidate points New vessel parameters are then updated forthe next iteration In addition during the tracking process abranch prediction scheme is applied at each iteration to detectvessel branches which exist in the local search area

23 End There are three end conditions for the proposedtracking process Under each of these conditions the trackingof current vessel stops The three end conditions are asfollows

(i) Current vesselrsquos end is considered to be found whenthe vesselrsquos diameter is less than one pixel or thecontrast between the vessel and background is low

(ii) Current vessel encounters a blood vessel which isalready detected by a tracking process starting fromanother seed point

(iii) Vessel branches are found All initial information ofthe branches is obtained by the algorithm and thetracking of these vessel branches starts The branchwith the biggest diameter is handled first

3 Tracking Algorithm

31 Selection of Seed Points In retinal image blood vesselsare not all connected because of the imaging conditions anddifferent eye diseasesTherefore the tracking algorithm startsfrom several initial points selected on the whole image andthe tracking results are combined to get the final detectedvascular tree In this study we use an automatic two-step




119879seed = 120583seed + 120572120590seed (1)






0may be not exactly






0is set bigger than


0and


0around the seed


|00|



centerpoint 119874



119896minus1




along 119896minus1


is described as

997888997888997888997888997888997888997888rarr119874119896minus1119874119896119904= 119904119896minus1 (2)



119896119904and





119896minus1

We select119873119871119896



is fixed to 1 pixel 119873119871119896


119896and

119896are chosen






diameter is 119889119896= |119896119896|





119900 119895119900) is assumed


119900 119895119900)


119900 119895119900) is denoted by


follows [41]

119865 (120579119892) = sum

(119894119895)isin119872lowast119873

1198662

119894119895

cos2 (120579119894119895minus 120579119892) (3)


(a) (b) (c)




119896


119896



119892when


119900 119895119900) is

120579119892=1

2arctan(

sum(119894119895)isin119872lowast119873

1198662

119894119895

sin (2120579119894119895)

sum(119894119895)isin119872lowast119873

1198662

119894119895

cos (2120579119894119895)

) (4)





120579119894119900119895119900= 120579119892plusmn120587

2 (5)


119900 119895119900) is


119894119900119895119900= (cos (120579

119894119900119895119900) sin (120579

119894119900119895119900)) (6)



119892

at 119874119896119904



Δ120579 = 120579119892119874119896119904

minus 120579119863119896minus1

(7)

where 120579119892119874119896119904



is

120579119874119896119904=

120579119892119874119896119904

minus120587

2if 0 lt Δ120579 le 120587

120579119892119874119896119904

+120587

2if minus 120587 lt Δ120579 le 0

(8)



119896119904= (cos(120579

119874119896119904) sin(120579




119896119904




119896119904




119896minus1and



and 120579119881119896minus1


Δ120579119906V =

10038161003816100381610038161003816120579119880119896minus1

minus 120579119881119896minus1

10038161003816100381610038161003816 (9)




Background

Background

Vessel

Lk

Ukminus1

Vkminus1

Mi


Vkminus2Vkminus3

Vkminus4

Mm1

Mm2


and1198721198982








1

and119898

th2

points on 119871119896(1 lt 119898

1lt 1198982lt 119873119871119896) and assume


119894(119894 isin [1119873

119871119896]) is



2 119873119871119896] Two candidate


119871119896(119873119871119896minus 1)2 normal


119871119896candidates on 119871

119896












119875 (120594 | 119884) =119875 (119884 | 120594) 119875 (120594)

119875 (119884) (10)


120594 = argmax120594

119875 (120594 | 119884) = argmax120594

119875 (119884 | 120594) 119875 (120594) (11)


119896is


119896is 119884119871119896= 119910119894 119894 = 1 119873

119871119896 A normal


1198981and119872




119875 (119884119871119896| 120594) =

119873119871119896

prod

119894=1

119875 (119910119894| 120594) (12)





1198981and119872



119875 (119884119871119896| 120594) =

1198981minus1

prod

119894=1

119875 (119910119894| 119887)

1198982

prod

1198981

119875 (119910119894| V)

119873119871119896

prod

1198982+1

119875 (119910119894| 119887)

(13)


119871119896= 119909119894 119894 =


119873119871119896



119884119871119896= 119883119871119896+ 120585 (14)


119866 (119872) =

(119868119888minus 119868119887) exp(minus 119897

2

21205902

) + 119868119887


119868119887



119887is the grey



119909119894=

119909119894(V) = (119868

119888119896minus 119868119887119896) exp(minus

119897119894

2

2120590120594

2

) + 119868119887119896

if 119894 isin [1198981 1198982]

119909119894(119887) = 119868

119887if 119894 isin [1119898

1)⋃(119898

2 119873119871119896]

(16)

where 119868119888119896






120594= (12)|119872

11989811198721198982|


and 119868119887119896


119896minus1and

119896minus1



119896minus1

119868119887119896




(c) Crossing



Background

Vessel

Background

2120590

M

l



119910119894= 119909119894+ 120585

= 119909119894(V) + 120585V if 119894 isin [119898

1 1198982]

119909119894(119887) + 120585

119887if 119894 isin [1119898

1)⋃(119898

2 119873119871119896]

(17)


120585V sim 119873(0 1205902

V)

120585119887sim 119873(0 120590

2

119887

)

(18)




119875 (119910119894| V) =

1

radic2120587120590V

exp(minus(119910119894minus 119909119894(V))2

21205902

V

)

119875 (119910119894| 119887) =

1

radic2120587120590119887

exp(minus(119910119894minus 119909119894(119887))2

21205902

119887

)

(19)



119875 (120594) =1

119885exp (minus120582119880 (120594)) (20)



2120590

Ml

Ib

Ic

(a)


170

180

190

200

210

220



Gre

y le

vel

(b)


Lk

Ukminus1

Vkminus1


Vkminus2Vkminus3

Vkminus4

E1

E2

Mm1

tm1

Mm2

tm2




1




and 119896minus1


2are defined as


1198981and 119864

1and



by 1199051198981



+ 1199052

1198982


119875 (120594) =1

119885exp (minus120582 (1199052

1198981

+ 1199052

1198982

)) (21)


119873119871119896






denoted by 119884119862119896= 119910119894 119894 = 1 119873





1

119901th2

119901th3

and 119901th4


119875 (119884119862119896| 120594) =

1199011minus1

prod

119894=1

119875 (119910119894| 119887)

1199012

prod

1199011

119875 (119910119894| V)1199013minus1

prod

1199012+1

119875 (119910119894| 119887)

1199014

prod

1199013

119875 (119910119894| V)

119873119862119896

prod

1199014+1

119875 (119910119894| 119887)

(22)





119894| 119887) and 119875 (119910

119894| V) in



119896

is obtained as

120594 = argmax120594

119875 (119884119862119896| 120594) (23)















(d)(g)

(e) (f)

(a) (b) (c)

(d) (e) (f) (g)









(a) (b) (c)

(d) (e)


(a) (b) (c)

(d) (e)



(a) (b)

(c) (d)

(e) (f)






(a) (b)

(c) (d)




Freq

uenc

y

1000

800

600

400

200

02520151050















1+

1199092)2 (119910


1minus 1199092)2

+ (1199101minus 1199102)2 and






6 Conclusions








References


















































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of



















119879seed = 120583seed + 120572120590seed (1)






0may be not exactly






0is set bigger than


0and


0around the seed


|00|



centerpoint 119874



119896minus1




along 119896minus1


is described as

997888997888997888997888997888997888997888rarr119874119896minus1119874119896119904= 119904119896minus1 (2)



119896119904and





119896minus1

We select119873119871119896



is fixed to 1 pixel 119873119871119896


119896and

119896are chosen






diameter is 119889119896= |119896119896|





119900 119895119900) is assumed


119900 119895119900)


119900 119895119900) is denoted by


follows [41]

119865 (120579119892) = sum

(119894119895)isin119872lowast119873

1198662

119894119895

cos2 (120579119894119895minus 120579119892) (3)


(a) (b) (c)




119896


119896



119892when


119900 119895119900) is

120579119892=1

2arctan(

sum(119894119895)isin119872lowast119873

1198662

119894119895

sin (2120579119894119895)

sum(119894119895)isin119872lowast119873

1198662

119894119895

cos (2120579119894119895)

) (4)





120579119894119900119895119900= 120579119892plusmn120587

2 (5)


119900 119895119900) is


119894119900119895119900= (cos (120579

119894119900119895119900) sin (120579

119894119900119895119900)) (6)



119892

at 119874119896119904



Δ120579 = 120579119892119874119896119904

minus 120579119863119896minus1

(7)

where 120579119892119874119896119904



is

120579119874119896119904=

120579119892119874119896119904

minus120587

2if 0 lt Δ120579 le 120587

120579119892119874119896119904

+120587

2if minus 120587 lt Δ120579 le 0

(8)



119896119904= (cos(120579

119874119896119904) sin(120579




119896119904




119896119904




119896minus1and



and 120579119881119896minus1


Δ120579119906V =

10038161003816100381610038161003816120579119880119896minus1

minus 120579119881119896minus1

10038161003816100381610038161003816 (9)




Background

Background

Vessel

Lk

Ukminus1

Vkminus1

Mi


Vkminus2Vkminus3

Vkminus4

Mm1

Mm2


and1198721198982








1

and119898

th2

points on 119871119896(1 lt 119898

1lt 1198982lt 119873119871119896) and assume


119894(119894 isin [1119873

119871119896]) is



2 119873119871119896] Two candidate


119871119896(119873119871119896minus 1)2 normal


119871119896candidates on 119871

119896












119875 (120594 | 119884) =119875 (119884 | 120594) 119875 (120594)

119875 (119884) (10)


120594 = argmax120594

119875 (120594 | 119884) = argmax120594

119875 (119884 | 120594) 119875 (120594) (11)


119896is


119896is 119884119871119896= 119910119894 119894 = 1 119873

119871119896 A normal


1198981and119872




119875 (119884119871119896| 120594) =

119873119871119896

prod

119894=1

119875 (119910119894| 120594) (12)





1198981and119872



119875 (119884119871119896| 120594) =

1198981minus1

prod

119894=1

119875 (119910119894| 119887)

1198982

prod

1198981

119875 (119910119894| V)

119873119871119896

prod

1198982+1

119875 (119910119894| 119887)

(13)


119871119896= 119909119894 119894 =


119873119871119896



119884119871119896= 119883119871119896+ 120585 (14)


119866 (119872) =

(119868119888minus 119868119887) exp(minus 119897

2

21205902

) + 119868119887


119868119887



119887is the grey



119909119894=

119909119894(V) = (119868

119888119896minus 119868119887119896) exp(minus

119897119894

2

2120590120594

2

) + 119868119887119896

if 119894 isin [1198981 1198982]

119909119894(119887) = 119868

119887if 119894 isin [1119898

1)⋃(119898

2 119873119871119896]

(16)

where 119868119888119896






120594= (12)|119872

11989811198721198982|


and 119868119887119896


119896minus1and

119896minus1



119896minus1

119868119887119896




(c) Crossing



Background

Vessel

Background

2120590

M

l



119910119894= 119909119894+ 120585

= 119909119894(V) + 120585V if 119894 isin [119898

1 1198982]

119909119894(119887) + 120585

119887if 119894 isin [1119898

1)⋃(119898

2 119873119871119896]

(17)


120585V sim 119873(0 1205902

V)

120585119887sim 119873(0 120590

2

119887

)

(18)




119875 (119910119894| V) =

1

radic2120587120590V

exp(minus(119910119894minus 119909119894(V))2

21205902

V

)

119875 (119910119894| 119887) =

1

radic2120587120590119887

exp(minus(119910119894minus 119909119894(119887))2

21205902

119887

)

(19)



119875 (120594) =1

119885exp (minus120582119880 (120594)) (20)



2120590

Ml

Ib

Ic

(a)


170

180

190

200

210

220



Gre

y le

vel

(b)


Lk

Ukminus1

Vkminus1


Vkminus2Vkminus3

Vkminus4

E1

E2

Mm1

tm1

Mm2

tm2




1




and 119896minus1


2are defined as


1198981and 119864

1and



by 1199051198981



+ 1199052

1198982


119875 (120594) =1

119885exp (minus120582 (1199052

1198981

+ 1199052

1198982

)) (21)


119873119871119896






denoted by 119884119862119896= 119910119894 119894 = 1 119873





1

119901th2

119901th3

and 119901th4


119875 (119884119862119896| 120594) =

1199011minus1

prod

119894=1

119875 (119910119894| 119887)

1199012

prod

1199011

119875 (119910119894| V)1199013minus1

prod

1199012+1

119875 (119910119894| 119887)

1199014

prod

1199013

119875 (119910119894| V)

119873119862119896

prod

1199014+1

119875 (119910119894| 119887)

(22)





119894| 119887) and 119875 (119910

119894| V) in



119896

is obtained as

120594 = argmax120594

119875 (119884119862119896| 120594) (23)















(d)(g)

(e) (f)

(a) (b) (c)

(d) (e) (f) (g)









(a) (b) (c)

(d) (e)


(a) (b) (c)

(d) (e)



(a) (b)

(c) (d)

(e) (f)






(a) (b)

(c) (d)




Freq

uenc

y

1000

800

600

400

200

02520151050















1+

1199092)2 (119910


1minus 1199092)2

+ (1199101minus 1199102)2 and






6 Conclusions








References


















































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of

















(a) (b) (c)




119896


119896



119892when


119900 119895119900) is

120579119892=1

2arctan(

sum(119894119895)isin119872lowast119873

1198662

119894119895

sin (2120579119894119895)

sum(119894119895)isin119872lowast119873

1198662

119894119895

cos (2120579119894119895)

) (4)





120579119894119900119895119900= 120579119892plusmn120587

2 (5)


119900 119895119900) is


119894119900119895119900= (cos (120579

119894119900119895119900) sin (120579

119894119900119895119900)) (6)



119892

at 119874119896119904



Δ120579 = 120579119892119874119896119904

minus 120579119863119896minus1

(7)

where 120579119892119874119896119904



is

120579119874119896119904=

120579119892119874119896119904

minus120587

2if 0 lt Δ120579 le 120587

120579119892119874119896119904

+120587

2if minus 120587 lt Δ120579 le 0

(8)



119896119904= (cos(120579

119874119896119904) sin(120579




119896119904




119896119904




119896minus1and



and 120579119881119896minus1


Δ120579119906V =

10038161003816100381610038161003816120579119880119896minus1

minus 120579119881119896minus1

10038161003816100381610038161003816 (9)




Background

Background

Vessel

Lk

Ukminus1

Vkminus1

Mi


Vkminus2Vkminus3

Vkminus4

Mm1

Mm2


and1198721198982








1

and119898

th2

points on 119871119896(1 lt 119898

1lt 1198982lt 119873119871119896) and assume


119894(119894 isin [1119873

119871119896]) is



2 119873119871119896] Two candidate


119871119896(119873119871119896minus 1)2 normal


119871119896candidates on 119871

119896












119875 (120594 | 119884) =119875 (119884 | 120594) 119875 (120594)

119875 (119884) (10)


120594 = argmax120594

119875 (120594 | 119884) = argmax120594

119875 (119884 | 120594) 119875 (120594) (11)


119896is


119896is 119884119871119896= 119910119894 119894 = 1 119873

119871119896 A normal


1198981and119872




119875 (119884119871119896| 120594) =

119873119871119896

prod

119894=1

119875 (119910119894| 120594) (12)





1198981and119872



119875 (119884119871119896| 120594) =

1198981minus1

prod

119894=1

119875 (119910119894| 119887)

1198982

prod

1198981

119875 (119910119894| V)

119873119871119896

prod

1198982+1

119875 (119910119894| 119887)

(13)


119871119896= 119909119894 119894 =


119873119871119896



119884119871119896= 119883119871119896+ 120585 (14)


119866 (119872) =

(119868119888minus 119868119887) exp(minus 119897

2

21205902

) + 119868119887


119868119887



119887is the grey



119909119894=

119909119894(V) = (119868

119888119896minus 119868119887119896) exp(minus

119897119894

2

2120590120594

2

) + 119868119887119896

if 119894 isin [1198981 1198982]

119909119894(119887) = 119868

119887if 119894 isin [1119898

1)⋃(119898

2 119873119871119896]

(16)

where 119868119888119896






120594= (12)|119872

11989811198721198982|


and 119868119887119896


119896minus1and

119896minus1



119896minus1

119868119887119896




(c) Crossing



Background

Vessel

Background

2120590

M

l



119910119894= 119909119894+ 120585

= 119909119894(V) + 120585V if 119894 isin [119898

1 1198982]

119909119894(119887) + 120585

119887if 119894 isin [1119898

1)⋃(119898

2 119873119871119896]

(17)


120585V sim 119873(0 1205902

V)

120585119887sim 119873(0 120590

2

119887

)

(18)




119875 (119910119894| V) =

1

radic2120587120590V

exp(minus(119910119894minus 119909119894(V))2

21205902

V

)

119875 (119910119894| 119887) =

1

radic2120587120590119887

exp(minus(119910119894minus 119909119894(119887))2

21205902

119887

)

(19)



119875 (120594) =1

119885exp (minus120582119880 (120594)) (20)



2120590

Ml

Ib

Ic

(a)


170

180

190

200

210

220



Gre

y le

vel

(b)


Lk

Ukminus1

Vkminus1


Vkminus2Vkminus3

Vkminus4

E1

E2

Mm1

tm1

Mm2

tm2




1




and 119896minus1


2are defined as


1198981and 119864

1and



by 1199051198981



+ 1199052

1198982


119875 (120594) =1

119885exp (minus120582 (1199052

1198981

+ 1199052

1198982

)) (21)


119873119871119896






denoted by 119884119862119896= 119910119894 119894 = 1 119873





1

119901th2

119901th3

and 119901th4


119875 (119884119862119896| 120594) =

1199011minus1

prod

119894=1

119875 (119910119894| 119887)

1199012

prod

1199011

119875 (119910119894| V)1199013minus1

prod

1199012+1

119875 (119910119894| 119887)

1199014

prod

1199013

119875 (119910119894| V)

119873119862119896

prod

1199014+1

119875 (119910119894| 119887)

(22)





119894| 119887) and 119875 (119910

119894| V) in



119896

is obtained as

120594 = argmax120594

119875 (119884119862119896| 120594) (23)















(d)(g)

(e) (f)

(a) (b) (c)

(d) (e) (f) (g)









(a) (b) (c)

(d) (e)


(a) (b) (c)

(d) (e)



(a) (b)

(c) (d)

(e) (f)






(a) (b)

(c) (d)




Freq

uenc

y

1000

800

600

400

200

02520151050















1+

1199092)2 (119910


1minus 1199092)2

+ (1199101minus 1199102)2 and






6 Conclusions








References


















































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of


















119896119904




119896minus1and



and 120579119881119896minus1


Δ120579119906V =

10038161003816100381610038161003816120579119880119896minus1

minus 120579119881119896minus1

10038161003816100381610038161003816 (9)




Background

Background

Vessel

Lk

Ukminus1

Vkminus1

Mi


Vkminus2Vkminus3

Vkminus4

Mm1

Mm2


and1198721198982








1

and119898

th2

points on 119871119896(1 lt 119898

1lt 1198982lt 119873119871119896) and assume


119894(119894 isin [1119873

119871119896]) is



2 119873119871119896] Two candidate


119871119896(119873119871119896minus 1)2 normal


119871119896candidates on 119871

119896












119875 (120594 | 119884) =119875 (119884 | 120594) 119875 (120594)

119875 (119884) (10)


120594 = argmax120594

119875 (120594 | 119884) = argmax120594

119875 (119884 | 120594) 119875 (120594) (11)


119896is


119896is 119884119871119896= 119910119894 119894 = 1 119873

119871119896 A normal


1198981and119872




119875 (119884119871119896| 120594) =

119873119871119896

prod

119894=1

119875 (119910119894| 120594) (12)





1198981and119872



119875 (119884119871119896| 120594) =

1198981minus1

prod

119894=1

119875 (119910119894| 119887)

1198982

prod

1198981

119875 (119910119894| V)

119873119871119896

prod

1198982+1

119875 (119910119894| 119887)

(13)


119871119896= 119909119894 119894 =


119873119871119896



119884119871119896= 119883119871119896+ 120585 (14)


119866 (119872) =

(119868119888minus 119868119887) exp(minus 119897

2

21205902

) + 119868119887


119868119887



119887is the grey



119909119894=

119909119894(V) = (119868

119888119896minus 119868119887119896) exp(minus

119897119894

2

2120590120594

2

) + 119868119887119896

if 119894 isin [1198981 1198982]

119909119894(119887) = 119868

119887if 119894 isin [1119898

1)⋃(119898

2 119873119871119896]

(16)

where 119868119888119896






120594= (12)|119872

11989811198721198982|


and 119868119887119896


119896minus1and

119896minus1



119896minus1

119868119887119896




(c) Crossing



Background

Vessel

Background

2120590

M

l



119910119894= 119909119894+ 120585

= 119909119894(V) + 120585V if 119894 isin [119898

1 1198982]

119909119894(119887) + 120585

119887if 119894 isin [1119898

1)⋃(119898

2 119873119871119896]

(17)


120585V sim 119873(0 1205902

V)

120585119887sim 119873(0 120590

2

119887

)

(18)




119875 (119910119894| V) =

1

radic2120587120590V

exp(minus(119910119894minus 119909119894(V))2

21205902

V

)

119875 (119910119894| 119887) =

1

radic2120587120590119887

exp(minus(119910119894minus 119909119894(119887))2

21205902

119887

)

(19)



119875 (120594) =1

119885exp (minus120582119880 (120594)) (20)



2120590

Ml

Ib

Ic

(a)


170

180

190

200

210

220



Gre

y le

vel

(b)


Lk

Ukminus1

Vkminus1


Vkminus2Vkminus3

Vkminus4

E1

E2

Mm1

tm1

Mm2

tm2




1




and 119896minus1


2are defined as


1198981and 119864

1and



by 1199051198981



+ 1199052

1198982


119875 (120594) =1

119885exp (minus120582 (1199052

1198981

+ 1199052

1198982

)) (21)


119873119871119896






denoted by 119884119862119896= 119910119894 119894 = 1 119873





1

119901th2

119901th3

and 119901th4


119875 (119884119862119896| 120594) =

1199011minus1

prod

119894=1

119875 (119910119894| 119887)

1199012

prod

1199011

119875 (119910119894| V)1199013minus1

prod

1199012+1

119875 (119910119894| 119887)

1199014

prod

1199013

119875 (119910119894| V)

119873119862119896

prod

1199014+1

119875 (119910119894| 119887)

(22)





119894| 119887) and 119875 (119910

119894| V) in



119896

is obtained as

120594 = argmax120594

119875 (119884119862119896| 120594) (23)















(d)(g)

(e) (f)

(a) (b) (c)

(d) (e) (f) (g)









(a) (b) (c)

(d) (e)


(a) (b) (c)

(d) (e)



(a) (b)

(c) (d)

(e) (f)






(a) (b)

(c) (d)




Freq

uenc

y

1000

800

600

400

200

02520151050















1+

1199092)2 (119910


1minus 1199092)2

+ (1199101minus 1199102)2 and






6 Conclusions








References


















































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of


























119875 (120594 | 119884) =119875 (119884 | 120594) 119875 (120594)

119875 (119884) (10)


120594 = argmax120594

119875 (120594 | 119884) = argmax120594

119875 (119884 | 120594) 119875 (120594) (11)


119896is


119896is 119884119871119896= 119910119894 119894 = 1 119873

119871119896 A normal


1198981and119872




119875 (119884119871119896| 120594) =

119873119871119896

prod

119894=1

119875 (119910119894| 120594) (12)





1198981and119872



119875 (119884119871119896| 120594) =

1198981minus1

prod

119894=1

119875 (119910119894| 119887)

1198982

prod

1198981

119875 (119910119894| V)

119873119871119896

prod

1198982+1

119875 (119910119894| 119887)

(13)


119871119896= 119909119894 119894 =


119873119871119896



119884119871119896= 119883119871119896+ 120585 (14)


119866 (119872) =

(119868119888minus 119868119887) exp(minus 119897

2

21205902

) + 119868119887


119868119887



119887is the grey



119909119894=

119909119894(V) = (119868

119888119896minus 119868119887119896) exp(minus

119897119894

2

2120590120594

2

) + 119868119887119896

if 119894 isin [1198981 1198982]

119909119894(119887) = 119868

119887if 119894 isin [1119898

1)⋃(119898

2 119873119871119896]

(16)

where 119868119888119896






120594= (12)|119872

11989811198721198982|


and 119868119887119896


119896minus1and

119896minus1



119896minus1

119868119887119896




(c) Crossing



Background

Vessel

Background

2120590

M

l



119910119894= 119909119894+ 120585

= 119909119894(V) + 120585V if 119894 isin [119898

1 1198982]

119909119894(119887) + 120585

119887if 119894 isin [1119898

1)⋃(119898

2 119873119871119896]

(17)


120585V sim 119873(0 1205902

V)

120585119887sim 119873(0 120590

2

119887

)

(18)




119875 (119910119894| V) =

1

radic2120587120590V

exp(minus(119910119894minus 119909119894(V))2

21205902

V

)

119875 (119910119894| 119887) =

1

radic2120587120590119887

exp(minus(119910119894minus 119909119894(119887))2

21205902

119887

)

(19)



119875 (120594) =1

119885exp (minus120582119880 (120594)) (20)



2120590

Ml

Ib

Ic

(a)


170

180

190

200

210

220



Gre

y le

vel

(b)


Lk

Ukminus1

Vkminus1


Vkminus2Vkminus3

Vkminus4

E1

E2

Mm1

tm1

Mm2

tm2




1




and 119896minus1


2are defined as


1198981and 119864

1and



by 1199051198981



+ 1199052

1198982


119875 (120594) =1

119885exp (minus120582 (1199052

1198981

+ 1199052

1198982

)) (21)


119873119871119896






denoted by 119884119862119896= 119910119894 119894 = 1 119873





1

119901th2

119901th3

and 119901th4


119875 (119884119862119896| 120594) =

1199011minus1

prod

119894=1

119875 (119910119894| 119887)

1199012

prod

1199011

119875 (119910119894| V)1199013minus1

prod

1199012+1

119875 (119910119894| 119887)

1199014

prod

1199013

119875 (119910119894| V)

119873119862119896

prod

1199014+1

119875 (119910119894| 119887)

(22)





119894| 119887) and 119875 (119910

119894| V) in



119896

is obtained as

120594 = argmax120594

119875 (119884119862119896| 120594) (23)















(d)(g)

(e) (f)

(a) (b) (c)

(d) (e) (f) (g)









(a) (b) (c)

(d) (e)


(a) (b) (c)

(d) (e)



(a) (b)

(c) (d)

(e) (f)






(a) (b)

(c) (d)




Freq

uenc

y

1000

800

600

400

200

02520151050















1+

1199092)2 (119910


1minus 1199092)2

+ (1199101minus 1199102)2 and






6 Conclusions








References


















































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of


















(c) Crossing



Background

Vessel

Background

2120590

M

l



119910119894= 119909119894+ 120585

= 119909119894(V) + 120585V if 119894 isin [119898

1 1198982]

119909119894(119887) + 120585

119887if 119894 isin [1119898

1)⋃(119898

2 119873119871119896]

(17)


120585V sim 119873(0 1205902

V)

120585119887sim 119873(0 120590

2

119887

)

(18)




119875 (119910119894| V) =

1

radic2120587120590V

exp(minus(119910119894minus 119909119894(V))2

21205902

V

)

119875 (119910119894| 119887) =

1

radic2120587120590119887

exp(minus(119910119894minus 119909119894(119887))2

21205902

119887

)

(19)



119875 (120594) =1

119885exp (minus120582119880 (120594)) (20)



2120590

Ml

Ib

Ic

(a)


170

180

190

200

210

220



Gre

y le

vel

(b)


Lk

Ukminus1

Vkminus1


Vkminus2Vkminus3

Vkminus4

E1

E2

Mm1

tm1

Mm2

tm2




1




and 119896minus1


2are defined as


1198981and 119864

1and



by 1199051198981



+ 1199052

1198982


119875 (120594) =1

119885exp (minus120582 (1199052

1198981

+ 1199052

1198982

)) (21)


119873119871119896






denoted by 119884119862119896= 119910119894 119894 = 1 119873





1

119901th2

119901th3

and 119901th4


119875 (119884119862119896| 120594) =

1199011minus1

prod

119894=1

119875 (119910119894| 119887)

1199012

prod

1199011

119875 (119910119894| V)1199013minus1

prod

1199012+1

119875 (119910119894| 119887)

1199014

prod

1199013

119875 (119910119894| V)

119873119862119896

prod

1199014+1

119875 (119910119894| 119887)

(22)





119894| 119887) and 119875 (119910

119894| V) in



119896

is obtained as

120594 = argmax120594

119875 (119884119862119896| 120594) (23)















(d)(g)

(e) (f)

(a) (b) (c)

(d) (e) (f) (g)









(a) (b) (c)

(d) (e)


(a) (b) (c)

(d) (e)



(a) (b)

(c) (d)

(e) (f)






(a) (b)

(c) (d)




Freq

uenc

y

1000

800

600

400

200

02520151050















1+

1199092)2 (119910


1minus 1199092)2

+ (1199101minus 1199102)2 and






6 Conclusions








References


















































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of


















2120590

Ml

Ib

Ic

(a)


170

180

190

200

210

220



Gre

y le

vel

(b)


Lk

Ukminus1

Vkminus1


Vkminus2Vkminus3

Vkminus4

E1

E2

Mm1

tm1

Mm2

tm2




1




and 119896minus1


2are defined as


1198981and 119864

1and



by 1199051198981



+ 1199052

1198982


119875 (120594) =1

119885exp (minus120582 (1199052

1198981

+ 1199052

1198982

)) (21)


119873119871119896






denoted by 119884119862119896= 119910119894 119894 = 1 119873





1

119901th2

119901th3

and 119901th4


119875 (119884119862119896| 120594) =

1199011minus1

prod

119894=1

119875 (119910119894| 119887)

1199012

prod

1199011

119875 (119910119894| V)1199013minus1

prod

1199012+1

119875 (119910119894| 119887)

1199014

prod

1199013

119875 (119910119894| V)

119873119862119896

prod

1199014+1

119875 (119910119894| 119887)

(22)





119894| 119887) and 119875 (119910

119894| V) in



119896

is obtained as

120594 = argmax120594

119875 (119884119862119896| 120594) (23)















(d)(g)

(e) (f)

(a) (b) (c)

(d) (e) (f) (g)









(a) (b) (c)

(d) (e)


(a) (b) (c)

(d) (e)



(a) (b)

(c) (d)

(e) (f)






(a) (b)

(c) (d)




Freq

uenc

y

1000

800

600

400

200

02520151050















1+

1199092)2 (119910


1minus 1199092)2

+ (1199101minus 1199102)2 and






6 Conclusions








References


















































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of



















119894| 119887) and 119875 (119910

119894| V) in



119896

is obtained as

120594 = argmax120594

119875 (119884119862119896| 120594) (23)















(d)(g)

(e) (f)

(a) (b) (c)

(d) (e) (f) (g)









(a) (b) (c)

(d) (e)


(a) (b) (c)

(d) (e)



(a) (b)

(c) (d)

(e) (f)






(a) (b)

(c) (d)




Freq

uenc

y

1000

800

600

400

200

02520151050















1+

1199092)2 (119910


1minus 1199092)2

+ (1199101minus 1199102)2 and






6 Conclusions








References


















































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of

















(d)(g)

(e) (f)

(a) (b) (c)

(d) (e) (f) (g)









(a) (b) (c)

(d) (e)


(a) (b) (c)

(d) (e)



(a) (b)

(c) (d)

(e) (f)






(a) (b)

(c) (d)




Freq

uenc

y

1000

800

600

400

200

02520151050















1+

1199092)2 (119910


1minus 1199092)2

+ (1199101minus 1199102)2 and






6 Conclusions








References


















































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of

















(a) (b) (c)

(d) (e)


(a) (b) (c)

(d) (e)



(a) (b)

(c) (d)

(e) (f)






(a) (b)

(c) (d)




Freq

uenc

y

1000

800

600

400

200

02520151050















1+

1199092)2 (119910


1minus 1199092)2

+ (1199101minus 1199102)2 and






6 Conclusions








References


















































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of

















(a) (b)

(c) (d)

(e) (f)






(a) (b)

(c) (d)




Freq

uenc

y

1000

800

600

400

200

02520151050















1+

1199092)2 (119910


1minus 1199092)2

+ (1199101minus 1199102)2 and






6 Conclusions








References


















































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of

















(a) (b)

(c) (d)




Freq

uenc

y

1000

800

600

400

200

02520151050















1+

1199092)2 (119910


1minus 1199092)2

+ (1199101minus 1199102)2 and






6 Conclusions








References


















































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of






















1+

1199092)2 (119910


1minus 1199092)2

+ (1199101minus 1199102)2 and






6 Conclusions








References


















































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of






















References


















































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of











































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of





















of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of
















Documents

Research Article Automatic Segmentation and Measurement of …downloads.hindawi.com/journals/cmmm/2013/260410.pdf · 2019-07-31 · Research Article Automatic Segmentation and Measurement