a nonparametric shape prior constrained active contour model for

Research ArticleA Nonparametric Shape Prior Constrained Active ContourModel for Segmentation of Coronaries in CTA Images

Yin Wang12 and Han Jiang1

1 State Key Laboratory of Virtual Reality Technology and Systems Beihang University Beijing 100191 China2 College of Astronautics Nanjing University of Aeronautics and Astronautics Nanjing 210016 China

Corrspondence should be addressed to Yin Wang yinwangeenuaaeducn

Received 3 January 2014 Accepted 5 March 2014 Published 1 April 2014

Academic Editor Emil Alexov

Copyright copy 2014 Y Wang and H Jiang This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

We present a nonparametric shape constrained algorithm for segmentation of coronary arteries in computed tomography imageswithin the framework of active contours An adaptive scale selection scheme based on the global histogram information ofthe image data is employed to determine the appropriate window size for each point on the active contour which improvesthe performance of the active contour model in the low contrast local image regions The possible leakage which cannot beidentified by using intensity features alone is reduced through the application of the proposed shape constraint where the shape ofcircular sampled intensity profile is used to evaluate the likelihood of current segmentation being considered vascular structuresExperiments on both synthetic and clinical datasets have demonstrated the efficiency and robustness of the proposed methodTheresults on clinical datasets have shown that the proposed approach is capable of extracting more detailed coronary vessels withsubvoxel accuracy

1 Introduction

Reliable and correct vessel segmentation is a crucial andfundamental step towards the designing of computer-aidedsystems in diagnosis of vascular diseases Despite intensiveamounts of research efforts dedicated to the developmentof semi- or fully automated algorithms for extraction ofvascular structures in medical images the task of definingcorrect boundaries of vascular structures is still challengingbecause of the complex geometrical structure of vessels andpathological lesions that could introduce undesired artifactsBesides images features may change dramatically in terms ofimage modalities and anatomical applications

Existing methods can be generally classified into two cat-egories namely tracking based algorithms and deformablemodel based approaches respectively Vessel segmentationusing tracking based techniques generally begins with anumber of starting points also known as seeds and thenthe neighboring pixels are added to the foreground objectsin terms of predefined criteria Among them minimum pathbased methods [1ndash4] are particularly popular in the context

of vessel segmentationWink and his coworkers [5] proposedthe application of the filter response fromvessel enhancementalgorithms [6ndash8] as the minimum path energy for extractionthe vessels in MRA images The work was further adoptedin [9] to improve the performance of their method in thepresence of varying vessel diameters and junctions by usingthe scale factor of vessel enhancement filters as an additionalparameter However the vesselness measurement derivedfrom second order derivative information is obtained basedon single branch vesselmodelThus it could lead to erroneoussegmentation in case where such assumption cannot be holdfor example in the proximal areas of aneurysms stenosisand bifurcations Based on local intensity profiles of vesselsKaftan et al [10] proposed a ldquomedialnessrdquo vesselness mea-surement as the minimum path energy which improves theperformance of the proposed algorithm with respect to localintensity variationsThemethodwas further improved in [11]where the authors proposed calculating the medialness met-ric only in selected pixels in terms of current segmentationand thus the efficiency of their algorithm has been greatly

Hindawi Publishing CorporationComputational and Mathematical Methods in MedicineVolume 2014 Article ID 302805 11 pageshttpdxdoiorg1011552014302805

2 Computational and Mathematical Methods in Medicine

increased Hua and Yezzi [12] proposed modeling vascularstructures as a succession of spheres to simultaneously detectboth the vessel centerline and associated boundaries In theirmethod the regional statistics are only derived from theboundaries of the spheres thus improving the robustness ofthe segmentation respect to image noise In the same researchline Antiga et al [13] proposed a surface domain methodby finding the locus of centers of maximal spheres inscribedinto the tubular structures usingVoronoi diagram Florin andhis colleagues [14] modeled vessels as a series of ellipticalcross-sections and detected the vessels based on particlefilters Compared to the previously discussed deterministicapproaches their algorithm is developed based on stochasticfilters which provides additional statistical properties aboutthe resulting segmentation Following their seminal workSchaap et al [15] applied a succession of short tubes to modelthe vessel segments which greatly enhances the robustnessand accuracy of particle filter based methods

Active contour models [16] known as snakes representone of the most popular model based methods in medicalimages processing communities The algorithm in generalis performed by iteratively deforming a contoursurface untilthe associated contour energy is minimized The activecontour energy is usually determined by its shape and image-driven features such as image edges After the pioneeredwork in [17 18] a great deal of researches have been dedicatedto develop varieties of image features to drive the activecontour to the desired boundaries Yang et al [19] proposedincorporating active contour segmentation into a Bayesianprobabilistic framework where the image-driven energy intheir method is redefined by posterior probabilities As acontrast to early works using image edges to formulate image-based energies region based image features relying on globalinformation derived from image regions are more robust tonoisy and inhomogeneous gradients A well-known regionbased active contour model was reported by Chan and Vese(known as CV model) [20] who proposed a simplifiedsolution toMumford-Shah functionalminimization problembased on the framework of level sets Following their seminarwork Rousson et al proposed more robustly model regionalstatistics by using both mean and variance information [21]Lecellier et al improve the accuracy and robustness of themodeled statistics by using the entire family of exponentialfunctions [22] Localized active contours [23ndash25] have beenintroduced to improve the performance of the active contoursin the presence of varying brightness across the imageLankton and Tannenbaum [26] proposed the use of differentmetrics to measure the similar intensity distributions derivedfrom image regions Despite the benefits introduced bymeasuring the regional statistics locally the selection of theappropriate scale poses new difficulties in these models Ifthe scale is selected to be small the intensity density wouldbe estimated based on a small number of pixels this wouldincrease the capability of the active contour to identify weakedges but it also makes the active contour be sensitive toimage noise and initialization On the other hand when atoo large window size is used the intensity distribution iscalculated based on ldquoglobalrdquo information which cannot takethe advantage of localized techniques

The leakage problem is often encountered in medicalimage segmentation when only intensity features are utilizedsince boundaries between different objects cannot be alwaysdefined by intensities Shape priors [27 28] generally definedbased on a training set have been introduced as an addi-tional hard constraint to improve the segmentation whenthe objects to be segmented exhibit similar appearances tothe ones present in the training sets Reliable shape priorshowever are in general difficult to determine in practice dueto the high interpatient variability of the vessel geometriesand the limited availability of training datasets In additionparameterization of shape features from training set anddetermination of their correspondences are also challengingTo address these problems Nain et al [29] incorporate a softshape prior into the framework of active contours whichallows the shape information to be directly derived from thecurrent segmentation In their work the shape descriptor wasdefined as a ball structure centered on each point along theactive contour and the likelihood of current segmentationto the vascular structures is determined by measuring thepercentage of pixels belonging to both the ball and the objectThe leakage can be therefore detected when output of theshape descriptor is high By the introduction of the shapedescriptor the potential leakage would be penalized duringcontour evolution Yet their proposed shape descriptor isunable to discriminate vessel bifurcations from leakage areaswhich leads to undesired gaps in the vicinity of these regions

The contributions of the present work are twofold Firstlywe propose an active contour energy functional for ves-sel segmentation with an adaptive scale selection schemewhich allows the localized kernel varying in an automatedfashion based on prior knowledge regarding the histogramdistribution of the CTA images The proposed adaptive scaleselection scheme improves the performance of the localizedactive contour model for weak edges Secondly we proposecoupling a nonparametric shape constraint into the activecontour framework which prevents the contour leak intoneighboring regions and improves the segmentation resultsFollowed by this introduction we provide great details aboutthe proposed method in Section 2 This is followed by thepresentation of the experimental results on both syntheticand clinical datasets demonstrating the efficiency of theproposed approach The conclusions of this work and futureresearch are presented in the final section

2 Materials and Methods

Let Ωx denote a local image with a radius 119903 centered at x onthe active contour 119862(x) As depicted in Figure 1 the localizedimage Ωx can be partitioned into two subregions by theactive contour If we assume that the intensity distributionwithin these image regions can be approximated by a two-parameter Gaussian function the probability of a pixel beingclassified as belonging to the region Ω

119894can be calculated as

follows

119875119894= 119875 (119868 (y) | y isin Ω

119894cap Ωx)

=1

radic2120587120590119894

exp(minus(120583119894minus 119868 (y))2

21205902

119894

)

(1)

Computational and Mathematical Methods in Medicine 3

(a)

Ω1

Ω2

(b)

Figure 1 Synthetic image illustrates the active contour model based on localised regional statistics (a) A randomly selected centre point isshown in yellow from the active contour red circle defines a localised image (b)The zoomed-in image of circular neighbourhood defined in(a) Note that we neglect the pixels outside of the local window

(a)

Intensities

Nor

mal

ized

wei

ghts

0 1000 2000 3000 40000

1

2

3

4

Fitted histogramOriginal histogram

times10minus3

(b)

Figure 2The histogram distribution of CTA image (a) An axial slice of the CTA image randomly picked from the datasets (b)The intensitydistribution (blue) and the fitted mixture model (red) of the CTA image

where Ω119894| 119894 = 1 2 represent the regions inside and outside

the contour respectively 119868(y) denotes the image intensityvalues at position y 120583

119894and 120590

119894are the mean and the variance

derived from region Ω119894 respectively It should be noted that

we use x and y as two independent spatial variables in thispaper

21 Adaptive Scale Selection Scheme As discussed in previ-ous section the localized active contours using a constantwindow size may lead to erroneous segmentation whenlocal image contrast is weak To remedy this problem wepresent an adaptive scale section scheme based on the globalhistogram distribution of the input image which allows forthe determination of the appropriate scale for each point

along the active contour in an automated fashionWhen localimage contrast is low we use a relatively smaller window sizeto increase the sensitivity of the active contour in detectingweak edges on the other hand a larger localized image wouldbe considered when image contrast is relatively higher whichimproves the robustness of the active contour to image noise

Figure 2 shows a typical histogram distribution of acontrast enhanced CTA image data and three major peakswhich correspond to the air in the lungs blood filled regionsand soft tissues can be observed from the histogram Toparameterize the histogram a mixture of Gaussian modelis used and the model parameters that is the mean andvariance for each Gaussian function are determined bythe application of Expectation Maximization (EM) methodCoronary arteries are vessels filled with blood thus we


assume that the intensities distribution derived within thesevessels should be similar to the ones obtained from otherblood-filled regions Based on this assumption we define thewindow size for each point on the active contour as follows

119903 (x) = min scale +max scale sdot exp (120591 sdot 119863 (x)) (2)

where 119903(x) denotes the radius of the local window centeredat x on the active contour and 120591 is a tuning parameterdetermining the decay speed of the exponential functionmin scale and max scale are constants that can be definedbased on prior knowledge about the maximum size of thevessel to be segmented 119863

119861represents the Bhattacharyya

distance between the intensity distribution derived from theinterior regions of the active contour within a local imageand the blood filled regions According to (2) it favors alarge window size when the intensity distribution of currentsegmentation is similar to the one extracted from blood filledregionsOn the other hand if the intensity statistics of currentsegmentation deviate from the preassumed distributionindicating that the localized image lacks contrast a smallwindow size would be selected Since we utilize Gaussianmodel to approximate regional statistics in this paper theBhattacharyya distancemeasurement can be computed usingthe following simplified form

119863119861(119901 119902) =

1

4ln(1

4(1205902

1

1205902

119887

+1205902

119887

1205902

1

+ 2)) +1

4((1205831minus 120583119887)

1205902

1+ 1205902

119887

)

(3)

where 1205831 1205901denote the mean and variance calculated from

the interior areas of the active contour within the local imagewhile 120583

119887 120590119887are the mean and variance of the blood filled

regions estimated from the Gaussian mixture model

22 Nonparametric Shape Constraints Image segmentationbased on intensity information alone is not able to alwaysproduce desired results since the boundaries between objectscannot be always characterized by image intensities In thispaper we propose incorporation of a nonparametric shapeconstraint into active contour framework which prevents thecontour deviated from being tubular structures Motivatedby the work reported by Qian and his coworkers [30] weuse the intensity features calculated on a sphericalcircularimage profile to measure the likelihood of a pixel belongingto the vascular structures As shown in Figure 3 the intensitydistribution along the circular sampled profile has limitedpeaks for vascular structure On the other hand it can beobserved that the probability density function sampled fromthe circular image profile for nonvascular structures is moredispersed (see Figure 4) Based on this observation we definethe nonparametric shape descriptor as

V (x) = 2119867 (x)max (119867 (x))

minus 1 x isin inside (119862 (x)) (4)

119867(x) = int ℎ (x 119877 120579) sdot log ℎ (x 119903 120579) 119889120579 (5)

where 119867(x) denotes the entropy of the profile ℎ(x 119903 120579)sampled on a circular image profile centered at x with

radial 119877 and 120579 indicates the orientation of each samplingpoint on the image profile It should be noticed that theradius of the circular sampling profile is not necessarilyequal to the local window size 119903 119867(x) is the entropy ameasurement of the randomness of the intensity profile Theentropy will be large in magnitude if an intensity profileis evenly distributed and on the other hand the value of119867(x) approaches 0 when the sampled intensity is sparselydistributed (totally random) The maximum value of 119867(x)can be found when the sampled profile is equally distributedfor all of the directions V(x) defines the likelihood of a voxelbeing considered vascular structure based on the shape ofcurrent segmentation which is normalised between minus1 and1 Since entropy is a computational consuming calculationin this paper we only perform the vesselness measurementwithin the interior region of the active contour Accordingto (4) if the shape of current segmentation deviates frombeing vascular that is V(x) rarr 1 for pixels inside the activecontour the shape based active contour energy would serveas an erosion term leading to inwards movement of theactive contour On the other hand the shape based energycomponent would expand the active contour outwards forvoxels with negative values of V(x)

23 Active Contour Energy Let 119862(x) denote the activecontour representing the boundaries of the object to besegmented Given an image 119868(y) and the shape of currentsegmentation V(y) the posterior probability of a pixel beingclassified as belonging to the object can be calculated as

119875 (y isin Ω119894cap Ωx | 119868 (y) V (y))

=119875 (119868 (y) V (y) | y isin Ω

119894cap Ωx) 119875 (y isin Ω119894 cap Ωx)

119875 (119868 (y))

(6)

where 119875(y isin Ω119894cap Ωx) denotes the prior probability of

the a voxel being classified to region Ω119894among all the

possible partitions in the local image Ωx Assuming thatthe probabilities of a pixel being assigned to any partitionare equal we ignore this term to simplify our analysis119875(119868(y)) represents the probability of the gray level values119868(y) which can be neglected since it is independent of thesegmentation of the image 119875(119868(y) V(y)) denotes the jointprobability density distribution of the gray level value 119868(y)and the shapemeasurement V(y) If the gray level distributionof the image and the shape of the contour are assumed tobe independent of each other given the class of the pixelthe posterior probability about the intensity values and shapemeasurement for each voxel can be defined as

119875 (119868 (y) V (y) | y isin Ω119894cap Ωx)

= 119875 (119868 (y) y isin Ω119894cap Ωx) 119875 (V (y) y isin Ω119894 cap Ωx)

(7)

Theprior probability of119875(119868(y) | y isin Ω119894capΩx) has been already

defined in (1) Since the shape descriptor is only defined


120579

(a)

Inte

nsity

val

ues

0 50 100 150 200 250 300 350

50

100

150

200

250

300

0

120579 (deg)

(b)

Figure 3 Synthetic image shows the intensity distribution along a circular sampled image profile for vascular structure (a) The syntheticvessel image for straight and Y-shaped segments is depicted (b)The intensity distribution along the circular sampled profile shown in (a) forstraight vessel segment (blue) and vessel bifurcation (red) respectively

(a)

Inte

nsity

val

ues

0 50 100 150 200 250 300 3500

50

100

150

200

250

120579 (deg)

(b)

Figure 4 Synthetic vessel image illustrates the intensity distribution (shown in (b)) sampled along a circular profile (red circle in (a)) centeredwithin the nonvascular structure

within the interior region of the active contour that is y isin Ω1

thus (7) can be written as

119875 (119868 (y) V (y) | y isin Ω119894cap Ωx)

=

119875 (119868 (y) | y isin Ω119894cap Ωx)

times 119875 (V (y) | y isin Ω119894cap Ωx) y isin Ω

1

119875 (119868 (y) | y isin Ω119894cap Ωx) y isin Ω

2

(8)

Maximizing the posterior probability in (6) is equivalent tominimizing its negative logarithm Hence for each given

point x belonging to the contour 119862(x) the image-basedenergy 119864x can be defined as

119864x = minus2

sum

119894=1

intΩ119894capΩx

log119875 (y isin Ω119894cap Ωx | 119868 (y) V (y)) 119889y

= minus

2

sum

119894=1

logintΩ119894capΩx

119875 (119868 (y) V (y) | y isin Ω119894cap Ωx) 119889y

= minus

2

sum

119894=1

[intΩ119894capΩx

log119875 (119868 (y) | y isin Ω119894cap Ωx) 119889y]

minus 120582intΩ1capΩx

log119875 (V (y) | y isin Ω1cap Ωx) 119889y

(9)


The prior probabilities that is 119875(119868(y) | y isin Ω119894cap Ωx)

and 119875(V(y) | y isin Ω1cap Ωx) were defined in (1) and

(4) respectively 120582 is a weighting term compromising thecontribution between the intensity based energy and theshape constraint

Let 120601 be the signed distance function (ie the level setfunction) representing the active contour andwe assume thatit takes positive values in the interior of the contour and isnegative for regions outside of the contour the active contourenergy can be formulated as

119864 = minusint1198671015840(120601 (x))

times

2

sum

119894=1

intΩ119894capΩx

[log119875 (119868 (y) | y isin Ω119894cap Ωx)]

times 119872119894(120601 (y)) 119889x

+ 120582intΩ1capΩx

log119875 (V (y) | y isin Ω1cap Ωx)

times 119872119894(120601 (y)) 119889y 119889x

+ 120583int1003816100381610038161003816nabla119867 (120601 (x))

1003816100381610038161003816 119889119909

(10)

where 1198721(120601(y)) = 119867(120601) and 119872

2(120601(y)) = 1 minus 119867(120601)119867(sdot)

and1198671015840(sdot) denotes the Heaviside and its first order derivativerespectively The first two terms in the right hand side of(10) are the negative logarithm posterior probability definedin (7) and the minimums occur when the active contour islocated on desired boundaries of the vessel while no leakagesare detected The last term is a smoothness regulator whichpenalizes jagged edges by using the total length of the result-ing boundaries The constant 120583 controls the contribution ofthis smoothness term in the entire active contour functionalThe associated Euler-Lagrange equation is defined as

120597120601

120597119905= 120575 (120601)(120583 div(

nabla120601

1003816100381610038161003816nabla1206011003816100381610038161003816

) + log1199011

1199012


V (x) 119889y)

(11)

where

1199011= intΩ1capΩx

1

radic21205871205901

exp(minus(1205831(x) minus 119868(y))2

21205902

1(x)

) 119889y


1

radic21205871205902

exp(minus(1205832 (x) minus 119868 (y))

2

21205902

2(x)

) 119889y

(12)

1199011and 119901

2represent the prior probability density distribution

of the object and background respectively 120575(sdot) denotes theDirac delta function The width of the localized kernel 119903 isupdated according to (2) at each iteration

3 Experimental Results and Analysis

In this section the proposed method is applied to both2D synthetic and 3D real clinical images to demonstrate its

efficiency We firstly compare the proposed approach witha localized regional statistics based active contours modelreported by Li et al [24] to analyze the benefits introducedby the application of the adaptive scale scheme and shapeconstraint in terms of 2D synthetic images The tuningparameters of the proposed techniquewere empirically deter-mined and fixed throughout this experiment In particularwe chose the smoothness weight 120583 at 02 and the shapebased energy factor V was set to 04 The localized windowsize 119903 is initialized as the maximum size of the vessel tobe segmented The width of the sampled circular profile isset to twice of the maximum size of the vessel of interestin order to ensure correct shape measurement for vesselbifurcations and the number of sampled directions is 36to balance its efficiency and robustness The minimum andmaximum scales for localized window are set to 4 and 10voxels respectively

Three metrics that is true positive (TP) false positive(FP) and overlappingmeasurement (OM) [31] are employedto quantify the performance of the resulting segmentationproduced by various methods and their definitions aredefined as follows

TP =119873119861cap 119873119877

119873119877

FP =119873119861minus 119873119861cap 119873119877

119873119877

OM = 2 sdot119873119861cap 119873119877

119873119861+ 119873119877

(13)

where the ground truth data119873119877is defined as a binary image

with pixels labelled to one representing the object and zerofor others and119873

119861represents pixels identified as the object by

the vessel segmentation algorithmsFigure 5(a) illustrates a 2D synthetic vessel image with

additive Gaussian noise and the active contour is initializedas a circle In Figures 5(b) and 5(d) we show the segmentationresults obtained by utilizing localized CV model in [24]and the proposed method respectively Figure 5(c) is theproposed shape measurement calculated within the interiorareas of the synthetic vessel where bright pixels indicatingsuch high possibilities considered as belonging to the vesselwhile the dark ones imply that those pixels are more likely tobe classified as nonvascular structures

The quantitative analysis of the resulting segmentationsin terms of the three metrics defined in (13) is presented inFigure 6 In general it can be observed that the localizedCV model is unable to extract the corrected vessel segmentand leak into the proximal blob-like structure which resultsin high FP rate and relatively smaller OM values On theother hand the proposed technique combining intensity andshape information greatly improves the segmentation resultsin terms of FP and OM metrics In addition by comparingFigures 5(b) and 5(d) we can find that the proposed methodextracts more distal vessel segment than localized CVmodeldue to the application of the adaptive scale scheme

Twelve coronary CT volumes were acquired from ourcalibrators themean size of the images is 512times512times282withan average in-plane resolution of 040mmtimes040mm and themean voxel size in the 119911-axis is 041mm The ground truthdata were defined through manual delineation by trained


(a) (b)

(c) (d)

Figure 5 Segmentation results on 2D binary synthetic image (a) Contour initialization shown as red circle (b) results obtained usinglocalized Chan-Vese method (c) vesselness measurement calculated for each pixel within the vascular structure and (d) illustration of thesegmentation results from the proposed algorithm

0

01

02

03

04

05

06

07

08

09

1

TP FP OM

0858 0893

0688

0001

0674

Localized CV model Proposed method

0943

Figure 6 Comparison of the 2D synthetic image segmentationresults obtained through the application of localized CV model andthe proposed method

biomedical student using interactive software developed inour lab In this paper three major branches of the coronaryartery (ie the left and right coronary artery plus anotherlargest branch among all of the coronary segments) are eval-uated Hausdorff distance a measurement of the overlappingbetween surfaces is utilized to evaluate the accuracy of thesesegmentation algorithms

119889119867 (XY) = maxsup

119909isinXinf119910isinY119889 (119909 119910) sup

119910isinYinf119909isinX119889 (119909 119910) (14)

where X and Y are the vertices of the mesh surfaces ofthe arteries corresponding to the segmentation results andthe ground truth respectively and 119889(119909 119910) measures theEuclidean distance between points 119909 and 119910 belonging tovertices X and Y The marching cube algorithm is usedto construct the surface representation of the coronaryarteries from the binary volume obtained from the segmen-tationmanual delineation

In Figures 7 and 8 we present the segmentation resultsin terms of 3D surface image and 2D axial images of thecoronary arteries respectively The quantification resultsare given in Figure 9 The tuning parameters of both ofthe two techniques were empirically determined and fixedthroughout the experiments Both the proposed algorithmand localized CV model were implemented in visual studioC++ 2008 on a standard desktop PC and the average execu-tion time was around 90 seconds for the proposed methodto process a CTA dataset On the other hand localized CVmodel takes roughly 70 s to perform the same task

As illustrated in Figure 9 the overall performance of theproposed method in terms of TP rate and OM metric isbetter than localized CV model indicating the extraction ofmore details of the coronary arteries Meanwhile the valueof FP measurement of the proposed approach is higher thanthe localized CV algorithm which implies that the proposedmethod tends to oversegment the vessels This observationis also demonstrated in Figure 8 where we illustrate theresulting segmentation on 2D axial image as contours Inthese images the red contours denote manually delineatedground truth and the segmentations obtained from theproposed model and localized CV method are shown asblue and black curves respectively As the proposed method


(a) (b)

Figure 7 3D surface data illustrates the resulting segmentation (red surfaces) obtained from the proposed method (a) and localized CVmodel (b) respectively Blue surface is obtained through manual delineation

(a) (b)

(c) (d)

Figure 8 Comparison of the segmentation results in 2D axial images (a) and (b) show the resulting segmentation on 2D cross-sectionalimages randomly taken from dataset number 5 while (c) and (d) are the resulting segmentation illustrated on 2D axial image picked fromdataset number 9The red contour represents themanually delineated reference standards and the segmentation obtained from the proposedmethod and localizedmodel are shown as blue and black contours respectively Note that these images are upsampled by a factor of 5 in orderto increase the in-plane resolution


0

10

20

30

40

50

60

70

80

90

100TP

rate

Localized CV modelProposed method

Datasets1 2 3 4 5 6 7 8 9 10 11 12

0

(a)


Datasets

FP ra

te

0

5

15

20

25

30

35

40

45

1 2 3 4 5 6 7 8 9 10 11 12

(b)


1 2 3 4 5 6 7 8 9 10 11 12Datasets

0

01

02

03

04

05

06

07

08

09

OM

met

ric

(c)


1 2 3 4 5 6 7 8 9 10 11 12Datasets

0

02

04

06

08

14H

ausd

orff

dist

ance

10

12

(d)

Figure 9 Bar plots depict the performance of the proposed technique and localized CV model in terms of TP rate (a) FP metric (b) OMrate (c) and Hausdorff distance (d) on extraction of the coronary arteries in clinic CTA images

employs an adaptive scale selection scheme it increases thecapability of the proposedmethod in defining relatively weakedge indicating higher OM and TP rates In addition theproposed method outperforms localized CV model in termsof Hausdorff distance metric implying that the proposedmethod is able to more precisely extract the coronary arterieswith subvoxel accuracy

4 Conclusion and Future Work

Precise and robust segmentation of vascular structures playsa vital role in clinical tasks such as stenosis grading andfunctional analysis of blood circulations In this paper wepresented novel algorithm for the segmentation of coronary

arteries in 3D CTA images based on the framework ofactive contours Compared with conventional local regionalstatistics based models the proposed method enables theextraction of more detailed vessels by the introduction ofan adaptive scale selection scheme The leakage problemwhich cannot be identified by using intensity informationalone was solved by using a nonparametric shape constraintExperimental results on both synthetic and real clinicaldatasets demonstrated the efficiency and robustness of theproposed model in defining the correct boundaries of vascu-lar structures In terms of future research we aim to identifyand segment the soft plaques within the coronary arterieswhich is amore challenging task due to the lack of appropriateintensities features


Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work is funded by the Opening Foundation of StateKey Laboratory of Virtual Reality Technology and Systems ofChina under Grant no BUAA-VR-13KF-09

References

[1] L D Cohen and R Kimmel ldquoGlobal minimum for active con-tour models a minimal path approachrdquo International Journal ofComputer Vision vol 24 no 1 pp 57ndash78 1997

[2] M Wan F Dachille and A Kaufman ldquoDistance-field basedskeletons for virtual navigationrdquo in Proceedings of the Confer-ence on Visualization pp 239ndash245 October 2001

[3] M Wan Z Liang Q Ke L Hong I Bitter and A KaufmanldquoAutomatic centerline extraction for virtual colonoscopyrdquo IEEETransactions on Medical Imaging vol 21 no 12 pp 1450ndash14602002

[4] Y Samara M Fiebich A H Dachman J K Kuniyoshi K Doiand K R Hoffmann ldquoAutomated calculation of the centerlineof the human colon on CT imagesrdquo Academic Radiology vol 6no 6 pp 352ndash359 1999

[5] O Wink A F Frangi B Verdonck M A Viergever andW J Niessen ldquo3D MRA coronary axis determination using aminimumcost path approachrdquoMagnetic Resonance inMedicinevol 47 no 6 pp 1169ndash1175 2002

[6] C Lorenz I C Carlsen TM Buzug C Fassnacht and JWeeseldquoMulti-scale line segmentation with automatic estimation ofwidth contrast and tangential direction in 2D and 3D medicalimagesrdquo in CVRMed-MRCAS rsquo97 vol 1205 of Lecture Notes inComputer Science pp 223ndash242 Springer Berlin Germany 1997

[7] A F Frangi W J Niessen P J Nederkoorn J Bakker W PMali and M A Viergever ldquoQuantitative analysis of vascularmorphology from 3D MR angiograms in vitro and in vivoresultsrdquo Magnetic Resonance in Medicine vol 45 pp 311ndash3222001

[8] A F Frangi W J Niessen K L Vincken and M A ViergeverldquoMultiscale vessel enhancement filteringrdquo in Proceedings of theMedical Image Computing and Computer-Assisted Intervention(MICCAI rsquo98) pp 130ndash137 Berlin Germany 1998

[9] O Wink W J Niessen and M A Viergever ldquoMultiscale vesseltrackingrdquo IEEE Transactions on Medical Imaging vol 23 no 1pp 130ndash133 2004

[10] J N Kaftan H Tek and T Aach ldquoA two stage approach for fullyautomatic segmentation of venous vascular structures in liverCT imagesrdquo inMedical imaging 2009 Image Processing J P WPluim and B M Dawant Eds vol 7259 of Proceedings of SPIE2009

[11] M A Gulsun and H Tek ldquoRobust vessel tree modelingrdquo inMedical image computing and Computer-Assisted InterventionmdashMICCAI rsquo08 vol 5241 of Lecture Notes in Computer Science pp602ndash611 Springer Berlin Germany 2008

[12] L Hua and A Yezzi ldquoVessels as 4-D curves global minimal 4-D paths to extract 3-D tubular surfaces and centerlinesrdquo IEEETransactions on Medical Imaging vol 26 no 9 pp 1213ndash12232007

[13] L Antiga B Ene-iordache and A Remuzzi ldquoCenterline com-putation and geometric analysis of branching tubular surfaceswith application to blood vessel modelingrdquo in Proceedings ofthe 11th International Conference in Central Europe on ComputerGraphics Visualization and Computer Vision pp 11ndash18 PlzenmdashBory Czech Republic 2004

[14] C Florin N Paragios and J Williams ldquoParticle filters a quasi-Monte-Carlo-solution for segmentation of coronariesrdquoMedicalImage Computing and Computer-Assisted Intervention vol 8pp 246ndash253 2005

[15] M Schaap R Manniesing I Smal T van Walsum A D vanLugt and W Niessen ldquoBayesian tracking of tubular structuresand its application to carotid arteries in CTArdquo Medical ImageComputing and Computer-Assisted InterventionmdashMICCAI rsquo07Springer Berlin Germany vol 4792 no 2 pp 562ndash570 2007

[16] MKass AWitkin andD Terzopoulos ldquoSnakes active contourmodelsrdquo International Journal of Computer Vision vol 1 no 4pp 321ndash331 1988

[17] V Caselles R Kimmel and G Sapiro ldquoGeodesic Active Con-toursrdquo International Journal of Computer Vision vol 22 no 1pp 61ndash79 1997

[18] S Kichenassamy A Kumar P Olver A Tannenbaum and AYezzi ldquoGradient flows and geometric active contour modelsrdquoin Proceedings of the 5th International Conference on ComputerVision pp 810ndash815 Boston Mass USA June 1995

[19] Y Yang A Tannenbaum D Giddens and A Stillman ldquoAuto-matic segmentation of coronary arteries using bayesian drivenimplicit surfacesrdquo in Proceedings of the 4th IEEE InternationalSymposium on Biomedical Imaging pp 189ndash192 Arlington VaUSA April 2007

[20] T F Chan and L A Vese ldquoActive contours without edgesrdquo IEEETransactions on Image Processing vol 10 no 2 pp 266ndash2772001

[21] M Rousson C Lenglet and R Deriche ldquoLevel set and regionbased surface propagation for diffusion tensor MRI segmenta-tionrdquo inComputer Vision andMathematical Methods inMedicaland Biomedical Image Analysis vol 3117 of Lecture Notes inComputer Science pp 123ndash134 Springer BerlinGermany 2004

[22] F Lecellier J Fadili S Jehan-Besson G Aubert M Revenuand E Saloux ldquoRegion-based active contours with exponentialfamily observationsrdquo Journal of Mathematical Imaging andVision vol 36 no 1 pp 28ndash45 2010

[23] M W K Law and A C S Chung ldquoWeighted local variance-based edge detection and its application to vascular segmenta-tion in magnetic resonance angiographyrdquo IEEE Transactions onMedical Imaging vol 26 no 9 pp 1224ndash1241 2007

[24] C Li C Y Kao J CGore andZDing ldquoImplicit active contoursdriven by local binary fitting energyrdquo in Proceedings of the IEEEComputer Society Conference on Computer Vision and PatternRecognition (CVPR rsquo07) pp 1ndash7 Minneapolis Minn USA June2007

[25] C Li C Y Kao J C Gore and Z Ding ldquoMinimization ofregion-scalable fitting energy for image segmentationrdquo IEEETransactions on Image Processing vol 17 no 10 pp 1940ndash19492008

[26] S Lankton and A Tannenbaum ldquoLocalizing region-basedactive contoursrdquo IEEE Transactions on Image Processing vol 17no 11 pp 2029ndash2039 2008

[27] Y Chen H D Tagare S Thiruvenkadam et al ldquoUsing priorshapes in geometric active contours in a variational frameworkrdquoInternational Journal of Computer Vision vol 50 no 3 pp 315ndash328 2002


[28] A Tsai A Yezzi Jr and A S Willsky ldquoCurve evolutionimplementation of the Mumford-Shah functional for imagesegmentation denoising interpolation and magnificationrdquoIEEE Transactions on Image Processing vol 10 no 8 pp 1169ndash1186 2001

[29] D Nain A Yezzi and G Turk ldquoVessel segmentation usinga shape driven flowrdquo in Proceedings of the Medical ImageComputing and Computer-Assisted Intervention (MICCAI rsquo04)pp 51ndash59 September 2004

[30] X Qian M P Brennan D P Dione et al ldquoA non-parametricvessel detection method for complex vascular structuresrdquoMed-ical Image Analysis vol 13 no 1 pp 49ndash61 2009

[31] A P Zijdenbos BMDawant R AMargolin andAC PalmerldquoMorphometric analysis of white matter lesions in MR imagesmethod and validationrdquo IEEE Transactions onMedical Imagingvol 13 no 4 pp 716ndash724 1994


increased Hua and Yezzi [12] proposed modeling vascularstructures as a succession of spheres to simultaneously detectboth the vessel centerline and associated boundaries In theirmethod the regional statistics are only derived from theboundaries of the spheres thus improving the robustness ofthe segmentation respect to image noise In the same researchline Antiga et al [13] proposed a surface domain methodby finding the locus of centers of maximal spheres inscribedinto the tubular structures usingVoronoi diagram Florin andhis colleagues [14] modeled vessels as a series of ellipticalcross-sections and detected the vessels based on particlefilters Compared to the previously discussed deterministicapproaches their algorithm is developed based on stochasticfilters which provides additional statistical properties aboutthe resulting segmentation Following their seminal workSchaap et al [15] applied a succession of short tubes to modelthe vessel segments which greatly enhances the robustnessand accuracy of particle filter based methods

Active contour models [16] known as snakes representone of the most popular model based methods in medicalimages processing communities The algorithm in generalis performed by iteratively deforming a contoursurface untilthe associated contour energy is minimized The activecontour energy is usually determined by its shape and image-driven features such as image edges After the pioneeredwork in [17 18] a great deal of researches have been dedicatedto develop varieties of image features to drive the activecontour to the desired boundaries Yang et al [19] proposedincorporating active contour segmentation into a Bayesianprobabilistic framework where the image-driven energy intheir method is redefined by posterior probabilities As acontrast to early works using image edges to formulate image-based energies region based image features relying on globalinformation derived from image regions are more robust tonoisy and inhomogeneous gradients A well-known regionbased active contour model was reported by Chan and Vese(known as CV model) [20] who proposed a simplifiedsolution toMumford-Shah functionalminimization problembased on the framework of level sets Following their seminarwork Rousson et al proposed more robustly model regionalstatistics by using both mean and variance information [21]Lecellier et al improve the accuracy and robustness of themodeled statistics by using the entire family of exponentialfunctions [22] Localized active contours [23ndash25] have beenintroduced to improve the performance of the active contoursin the presence of varying brightness across the imageLankton and Tannenbaum [26] proposed the use of differentmetrics to measure the similar intensity distributions derivedfrom image regions Despite the benefits introduced bymeasuring the regional statistics locally the selection of theappropriate scale poses new difficulties in these models Ifthe scale is selected to be small the intensity density wouldbe estimated based on a small number of pixels this wouldincrease the capability of the active contour to identify weakedges but it also makes the active contour be sensitive toimage noise and initialization On the other hand when atoo large window size is used the intensity distribution iscalculated based on ldquoglobalrdquo information which cannot takethe advantage of localized techniques

The leakage problem is often encountered in medicalimage segmentation when only intensity features are utilizedsince boundaries between different objects cannot be alwaysdefined by intensities Shape priors [27 28] generally definedbased on a training set have been introduced as an addi-tional hard constraint to improve the segmentation whenthe objects to be segmented exhibit similar appearances tothe ones present in the training sets Reliable shape priorshowever are in general difficult to determine in practice dueto the high interpatient variability of the vessel geometriesand the limited availability of training datasets In additionparameterization of shape features from training set anddetermination of their correspondences are also challengingTo address these problems Nain et al [29] incorporate a softshape prior into the framework of active contours whichallows the shape information to be directly derived from thecurrent segmentation In their work the shape descriptor wasdefined as a ball structure centered on each point along theactive contour and the likelihood of current segmentationto the vascular structures is determined by measuring thepercentage of pixels belonging to both the ball and the objectThe leakage can be therefore detected when output of theshape descriptor is high By the introduction of the shapedescriptor the potential leakage would be penalized duringcontour evolution Yet their proposed shape descriptor isunable to discriminate vessel bifurcations from leakage areaswhich leads to undesired gaps in the vicinity of these regions

The contributions of the present work are twofold Firstlywe propose an active contour energy functional for ves-sel segmentation with an adaptive scale selection schemewhich allows the localized kernel varying in an automatedfashion based on prior knowledge regarding the histogramdistribution of the CTA images The proposed adaptive scaleselection scheme improves the performance of the localizedactive contour model for weak edges Secondly we proposecoupling a nonparametric shape constraint into the activecontour framework which prevents the contour leak intoneighboring regions and improves the segmentation resultsFollowed by this introduction we provide great details aboutthe proposed method in Section 2 This is followed by thepresentation of the experimental results on both syntheticand clinical datasets demonstrating the efficiency of theproposed approach The conclusions of this work and futureresearch are presented in the final section

2 Materials and Methods

Let Ωx denote a local image with a radius 119903 centered at x onthe active contour 119862(x) As depicted in Figure 1 the localizedimage Ωx can be partitioned into two subregions by theactive contour If we assume that the intensity distributionwithin these image regions can be approximated by a two-parameter Gaussian function the probability of a pixel beingclassified as belonging to the region Ω

119894can be calculated as

follows

119875119894= 119875 (119868 (y) | y isin Ω

119894cap Ωx)

=1

radic2120587120590119894

exp(minus(120583119894minus 119868 (y))2

21205902

119894

)

(1)


(a)

Ω1

Ω2

(b)


(a)

Intensities

Nor

mal

ized

wei

ghts

0 1000 2000 3000 40000

1

2

3

4


times10minus3

(b)




119894and 120590













119863119861(119901 119902) =

1

4ln(1

4(1205902

1

1205902

119887

+1205902

119887

1205902

1

+ 2)) +1

4((1205831minus 120583119887)

1205902

1+ 1205902

119887

)

(3)






V (x) = 2119867 (x)max (119867 (x))






119875 (y isin Ω119894cap Ωx | 119868 (y) V (y))

=119875 (119868 (y) V (y) | y isin Ω


119875 (119868 (y))

(6)




119875 (119868 (y) V (y) | y isin Ω119894cap Ωx)


(7)




120579

(a)

Inte

nsity

val

ues

0 50 100 150 200 250 300 350

50

100

150

200

250

300

0

120579 (deg)

(b)


(a)

Inte

nsity

val

ues

0 50 100 150 200 250 300 3500

50

100

150

200

250

120579 (deg)

(b)




119875 (119868 (y) V (y) | y isin Ω119894cap Ωx)

=

119875 (119868 (y) | y isin Ω119894cap Ωx)


1


2

(8)



119864x = minus2

sum

119894=1

intΩ119894capΩx


= minus

2

sum

119894=1



= minus

2

sum

119894=1

[intΩ119894capΩx




(9)






119864 = minusint1198671015840(120601 (x))

times

2

sum

119894=1

intΩ119894capΩx


times 119872119894(120601 (y)) 119889x



times 119872119894(120601 (y)) 119889y 119889x

+ 120583int1003816100381610038161003816nabla119867 (120601 (x))

1003816100381610038161003816 119889119909

(10)

where 1198721(120601(y)) = 119867(120601) and 119872

2(120601(y)) = 1 minus 119867(120601)119867(sdot)


120597120601

120597119905= 120575 (120601)(120583 div(

nabla120601

1003816100381610038161003816nabla1206011003816100381610038161003816

) + log1199011

1199012


V (x) 119889y)

(11)

where


1

radic21205871205901


21205902

1(x)

) 119889y


1

radic21205871205902


2

21205902

2(x)

) 119889y

(12)

1199011and 119901







TP =119873119861cap 119873119877

119873119877

FP =119873119861minus 119873119861cap 119873119877

119873119877

OM = 2 sdot119873119861cap 119873119877

119873119861+ 119873119877

(13)









(a) (b)

(c) (d)


0

01

02

03

04

05

06

07

08

09

1

TP FP OM

0858 0893

0688

0001

0674


0943



119889119867 (XY) = maxsup


119910isinYinf119909isinX119889 (119909 119910) (14)





(a) (b)


(a) (b)

(c) (d)



0

10

20

30

40

50

60

70

80

90

100TP

rate


Datasets1 2 3 4 5 6 7 8 9 10 11 12

0

(a)


Datasets

FP ra

te

0

5

15

20

25

30

35

40

45

1 2 3 4 5 6 7 8 9 10 11 12

(b)


1 2 3 4 5 6 7 8 9 10 11 12Datasets

0

01

02

03

04

05

06

07

08

09

OM

met

ric

(c)


1 2 3 4 5 6 7 8 9 10 11 12Datasets

0

02

04

06

08

14H

ausd

orff

dist

ance

10

12

(d)









Acknowledgment


References


































(a)

Ω1

Ω2

(b)


(a)

Intensities

Nor

mal

ized

wei

ghts

0 1000 2000 3000 40000

1

2

3

4


times10minus3

(b)




119894and 120590













119863119861(119901 119902) =

1

4ln(1

4(1205902

1

1205902

119887

+1205902

119887

1205902

1

+ 2)) +1

4((1205831minus 120583119887)

1205902

1+ 1205902

119887

)

(3)






V (x) = 2119867 (x)max (119867 (x))






119875 (y isin Ω119894cap Ωx | 119868 (y) V (y))

=119875 (119868 (y) V (y) | y isin Ω


119875 (119868 (y))

(6)




119875 (119868 (y) V (y) | y isin Ω119894cap Ωx)


(7)




120579

(a)

Inte

nsity

val

ues

0 50 100 150 200 250 300 350

50

100

150

200

250

300

0

120579 (deg)

(b)


(a)

Inte

nsity

val

ues

0 50 100 150 200 250 300 3500

50

100

150

200

250

120579 (deg)

(b)




119875 (119868 (y) V (y) | y isin Ω119894cap Ωx)

=

119875 (119868 (y) | y isin Ω119894cap Ωx)


1


2

(8)



119864x = minus2

sum

119894=1

intΩ119894capΩx


= minus

2

sum

119894=1



= minus

2

sum

119894=1

[intΩ119894capΩx




(9)






119864 = minusint1198671015840(120601 (x))

times

2

sum

119894=1

intΩ119894capΩx


times 119872119894(120601 (y)) 119889x



times 119872119894(120601 (y)) 119889y 119889x

+ 120583int1003816100381610038161003816nabla119867 (120601 (x))

1003816100381610038161003816 119889119909

(10)

where 1198721(120601(y)) = 119867(120601) and 119872

2(120601(y)) = 1 minus 119867(120601)119867(sdot)


120597120601

120597119905= 120575 (120601)(120583 div(

nabla120601

1003816100381610038161003816nabla1206011003816100381610038161003816

) + log1199011

1199012


V (x) 119889y)

(11)

where


1

radic21205871205901


21205902

1(x)

) 119889y


1

radic21205871205902


2

21205902

2(x)

) 119889y

(12)

1199011and 119901







TP =119873119861cap 119873119877

119873119877

FP =119873119861minus 119873119861cap 119873119877

119873119877

OM = 2 sdot119873119861cap 119873119877

119873119861+ 119873119877

(13)









(a) (b)

(c) (d)


0

01

02

03

04

05

06

07

08

09

1

TP FP OM

0858 0893

0688

0001

0674


0943



119889119867 (XY) = maxsup


119910isinYinf119909isinX119889 (119909 119910) (14)





(a) (b)


(a) (b)

(c) (d)



0

10

20

30

40

50

60

70

80

90

100TP

rate


Datasets1 2 3 4 5 6 7 8 9 10 11 12

0

(a)


Datasets

FP ra

te

0

5

15

20

25

30

35

40

45

1 2 3 4 5 6 7 8 9 10 11 12

(b)


1 2 3 4 5 6 7 8 9 10 11 12Datasets

0

01

02

03

04

05

06

07

08

09

OM

met

ric

(c)


1 2 3 4 5 6 7 8 9 10 11 12Datasets

0

02

04

06

08

14H

ausd

orff

dist

ance

10

12

(d)









Acknowledgment


References







































119863119861(119901 119902) =

1

4ln(1

4(1205902

1

1205902

119887

+1205902

119887

1205902

1

+ 2)) +1

4((1205831minus 120583119887)

1205902

1+ 1205902

119887

)

(3)






V (x) = 2119867 (x)max (119867 (x))






119875 (y isin Ω119894cap Ωx | 119868 (y) V (y))

=119875 (119868 (y) V (y) | y isin Ω


119875 (119868 (y))

(6)




119875 (119868 (y) V (y) | y isin Ω119894cap Ωx)


(7)




120579

(a)

Inte

nsity

val

ues

0 50 100 150 200 250 300 350

50

100

150

200

250

300

0

120579 (deg)

(b)


(a)

Inte

nsity

val

ues

0 50 100 150 200 250 300 3500

50

100

150

200

250

120579 (deg)

(b)




119875 (119868 (y) V (y) | y isin Ω119894cap Ωx)

=

119875 (119868 (y) | y isin Ω119894cap Ωx)


1


2

(8)



119864x = minus2

sum

119894=1

intΩ119894capΩx


= minus

2

sum

119894=1



= minus

2

sum

119894=1

[intΩ119894capΩx




(9)






119864 = minusint1198671015840(120601 (x))

times

2

sum

119894=1

intΩ119894capΩx


times 119872119894(120601 (y)) 119889x



times 119872119894(120601 (y)) 119889y 119889x

+ 120583int1003816100381610038161003816nabla119867 (120601 (x))

1003816100381610038161003816 119889119909

(10)

where 1198721(120601(y)) = 119867(120601) and 119872

2(120601(y)) = 1 minus 119867(120601)119867(sdot)


120597120601

120597119905= 120575 (120601)(120583 div(

nabla120601

1003816100381610038161003816nabla1206011003816100381610038161003816

) + log1199011

1199012


V (x) 119889y)

(11)

where


1

radic21205871205901


21205902

1(x)

) 119889y


1

radic21205871205902


2

21205902

2(x)

) 119889y

(12)

1199011and 119901







TP =119873119861cap 119873119877

119873119877

FP =119873119861minus 119873119861cap 119873119877

119873119877

OM = 2 sdot119873119861cap 119873119877

119873119861+ 119873119877

(13)









(a) (b)

(c) (d)


0

01

02

03

04

05

06

07

08

09

1

TP FP OM

0858 0893

0688

0001

0674


0943



119889119867 (XY) = maxsup


119910isinYinf119909isinX119889 (119909 119910) (14)





(a) (b)


(a) (b)

(c) (d)



0

10

20

30

40

50

60

70

80

90

100TP

rate


Datasets1 2 3 4 5 6 7 8 9 10 11 12

0

(a)


Datasets

FP ra

te

0

5

15

20

25

30

35

40

45

1 2 3 4 5 6 7 8 9 10 11 12

(b)


1 2 3 4 5 6 7 8 9 10 11 12Datasets

0

01

02

03

04

05

06

07

08

09

OM

met

ric

(c)


1 2 3 4 5 6 7 8 9 10 11 12Datasets

0

02

04

06

08

14H

ausd

orff

dist

ance

10

12

(d)









Acknowledgment


References


































120579

(a)

Inte

nsity

val

ues

0 50 100 150 200 250 300 350

50

100

150

200

250

300

0

120579 (deg)

(b)


(a)

Inte

nsity

val

ues

0 50 100 150 200 250 300 3500

50

100

150

200

250

120579 (deg)

(b)




119875 (119868 (y) V (y) | y isin Ω119894cap Ωx)

=

119875 (119868 (y) | y isin Ω119894cap Ωx)


1


2

(8)



119864x = minus2

sum

119894=1

intΩ119894capΩx


= minus

2

sum

119894=1



= minus

2

sum

119894=1

[intΩ119894capΩx




(9)






119864 = minusint1198671015840(120601 (x))

times

2

sum

119894=1

intΩ119894capΩx


times 119872119894(120601 (y)) 119889x



times 119872119894(120601 (y)) 119889y 119889x

+ 120583int1003816100381610038161003816nabla119867 (120601 (x))

1003816100381610038161003816 119889119909

(10)

where 1198721(120601(y)) = 119867(120601) and 119872

2(120601(y)) = 1 minus 119867(120601)119867(sdot)


120597120601

120597119905= 120575 (120601)(120583 div(

nabla120601

1003816100381610038161003816nabla1206011003816100381610038161003816

) + log1199011

1199012


V (x) 119889y)

(11)

where


1

radic21205871205901


21205902

1(x)

) 119889y


1

radic21205871205902


2

21205902

2(x)

) 119889y

(12)

1199011and 119901







TP =119873119861cap 119873119877

119873119877

FP =119873119861minus 119873119861cap 119873119877

119873119877

OM = 2 sdot119873119861cap 119873119877

119873119861+ 119873119877

(13)









(a) (b)

(c) (d)


0

01

02

03

04

05

06

07

08

09

1

TP FP OM

0858 0893

0688

0001

0674


0943



119889119867 (XY) = maxsup


119910isinYinf119909isinX119889 (119909 119910) (14)





(a) (b)


(a) (b)

(c) (d)



0

10

20

30

40

50

60

70

80

90

100TP

rate


Datasets1 2 3 4 5 6 7 8 9 10 11 12

0

(a)


Datasets

FP ra

te

0

5

15

20

25

30

35

40

45

1 2 3 4 5 6 7 8 9 10 11 12

(b)


1 2 3 4 5 6 7 8 9 10 11 12Datasets

0

01

02

03

04

05

06

07

08

09

OM

met

ric

(c)


1 2 3 4 5 6 7 8 9 10 11 12Datasets

0

02

04

06

08

14H

ausd

orff

dist

ance

10

12

(d)









Acknowledgment


References






































119864 = minusint1198671015840(120601 (x))

times

2

sum

119894=1

intΩ119894capΩx


times 119872119894(120601 (y)) 119889x



times 119872119894(120601 (y)) 119889y 119889x

+ 120583int1003816100381610038161003816nabla119867 (120601 (x))

1003816100381610038161003816 119889119909

(10)

where 1198721(120601(y)) = 119867(120601) and 119872

2(120601(y)) = 1 minus 119867(120601)119867(sdot)


120597120601

120597119905= 120575 (120601)(120583 div(

nabla120601

1003816100381610038161003816nabla1206011003816100381610038161003816

) + log1199011

1199012


V (x) 119889y)

(11)

where


1

radic21205871205901


21205902

1(x)

) 119889y


1

radic21205871205902


2

21205902

2(x)

) 119889y

(12)

1199011and 119901







TP =119873119861cap 119873119877

119873119877

FP =119873119861minus 119873119861cap 119873119877

119873119877

OM = 2 sdot119873119861cap 119873119877

119873119861+ 119873119877

(13)









(a) (b)

(c) (d)


0

01

02

03

04

05

06

07

08

09

1

TP FP OM

0858 0893

0688

0001

0674


0943



119889119867 (XY) = maxsup


119910isinYinf119909isinX119889 (119909 119910) (14)





(a) (b)


(a) (b)

(c) (d)



0

10

20

30

40

50

60

70

80

90

100TP

rate


Datasets1 2 3 4 5 6 7 8 9 10 11 12

0

(a)


Datasets

FP ra

te

0

5

15

20

25

30

35

40

45

1 2 3 4 5 6 7 8 9 10 11 12

(b)


1 2 3 4 5 6 7 8 9 10 11 12Datasets

0

01

02

03

04

05

06

07

08

09

OM

met

ric

(c)


1 2 3 4 5 6 7 8 9 10 11 12Datasets

0

02

04

06

08

14H

ausd

orff

dist

ance

10

12

(d)









Acknowledgment


References


































(a) (b)

(c) (d)


0

01

02

03

04

05

06

07

08

09

1

TP FP OM

0858 0893

0688

0001

0674


0943



119889119867 (XY) = maxsup


119910isinYinf119909isinX119889 (119909 119910) (14)





(a) (b)


(a) (b)

(c) (d)



0

10

20

30

40

50

60

70

80

90

100TP

rate


Datasets1 2 3 4 5 6 7 8 9 10 11 12

0

(a)


Datasets

FP ra

te

0

5

15

20

25

30

35

40

45

1 2 3 4 5 6 7 8 9 10 11 12

(b)


1 2 3 4 5 6 7 8 9 10 11 12Datasets

0

01

02

03

04

05

06

07

08

09

OM

met

ric

(c)


1 2 3 4 5 6 7 8 9 10 11 12Datasets

0

02

04

06

08

14H

ausd

orff

dist

ance

10

12

(d)









Acknowledgment


References


































(a) (b)


(a) (b)

(c) (d)



0

10

20

30

40

50

60

70

80

90

100TP

rate


Datasets1 2 3 4 5 6 7 8 9 10 11 12

0

(a)


Datasets

FP ra

te

0

5

15

20

25

30

35

40

45

1 2 3 4 5 6 7 8 9 10 11 12

(b)


1 2 3 4 5 6 7 8 9 10 11 12Datasets

0

01

02

03

04

05

06

07

08

09

OM

met

ric

(c)


1 2 3 4 5 6 7 8 9 10 11 12Datasets

0

02

04

06

08

14H

ausd

orff

dist

ance

10

12

(d)









Acknowledgment


References


































0

10

20

30

40

50

60

70

80

90

100TP

rate


Datasets1 2 3 4 5 6 7 8 9 10 11 12

0

(a)


Datasets

FP ra

te

0

5

15

20

25

30

35

40

45

1 2 3 4 5 6 7 8 9 10 11 12

(b)


1 2 3 4 5 6 7 8 9 10 11 12Datasets

0

01

02

03

04

05

06

07

08

09

OM

met

ric

(c)


1 2 3 4 5 6 7 8 9 10 11 12Datasets

0

02

04

06

08

14H

ausd

orff

dist

ance

10

12

(d)









Acknowledgment


References




































Acknowledgment


References

































a nonparametric shape prior constrained active contour model for

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