Vasculature Segmentation in MRA Images Using Gradient Compensated Geodesic Active Contours

J Sign Process Syst (2009) 54:171–181DOI 10.1007/s11265-008-0216-4

Vasculature Segmentation in MRA Images Using GradientCompensated Geodesic Active Contours

Dornoosh Zonoobi · Ashraf A. Kassim · Weijia Shen

Received: 31 December 2007 / Revised: 16 March 2008 / Accepted: 14 April 2008 / Published online: 30 May 2008© 2008 Springer Science + Business Media, LLC. Manufactured in The United States

Abstract Precise segmentation of vasculature fromthree-dimensional (3D) magnetic resonance angiogra-phy (MRA) images is playing an important role inimage-guided neurosurgery, pre-operation planningand clinical analysis. Active Contour based evolutionalgorithms are being widely applied to MRA data sets,however existing approaches exhibit some difficultiesin extracting tiny parts of the vessels. Our objective isto develop an automated segmentation scheme to ac-curately extract vasculature of the brain, especially tinyvessels. Inspired by the intrinsic properties of MRA, wehave proposed a scheme called the gradient compen-sated geodesic active contours (GCGAC), which com-pensates for low gradients near edges of thin vessels.The GCGAC, which is implemented based on level set,has been tested on both synthetic volumetric image andreal 3D MRA images. Our experiments show that theintroduced gradient compensation can facilitate moreaccurate segmentation of tiny blood vessels.

Keywords Vasculature segmentation · MRA images ·Active contours

1 Introduction

Segmentation of structures is an important goal inthree-dimensional (3D) medical image processing.Magnetic resonance angiography (MRA) is a nonin-vasive medical imaging technology that produces 3D

D. Zonoobi · A. A. Kassim (B) · W. ShenNational University of Singapore, Singapore, Singaporee-mail: [email protected]

images of vessels. Accurate extraction of 3D vascularstructures from MRA images has become increasinglyimportant for detecting and aiding the treatment ofheart disorders, stroke, and blood vessel diseases [3].

A variety of methods have been developed forsegmenting vessels within MRA data. These can bebroadly divided into two categories: skeleton-based andnonskeleton-based methods [5]. Skeleton-based tech-niques are those indirect methods that segment andreconstruct vessels by first calculating the centerlinesand cross sections of vessels [2, 5].

Contrary to these methods, nonskeleton-based tech-niques are those that segment vessels in 3D directly.In this category, active contours based methods havereceived considerable attention and success [5]. Sinceits introduction by Kass et al. [7] as a parametric ac-tive contours model or snake, numerous models havebeen proposed and used successfully for medical imagesegmentation. However, it is difficult to employ para-meteric models to effectively extract whole vessel treesin MRA images, as the models would be required tochange topology during evolution [5]. Another class ofactive contours models are geometric ones which arebased on level set theory [8, 9]. These methods aregoverned by curvature dependent speeds, also calledlevel set stoppers and are able to freely adapt intocomplex topologies of objects. Therefore, they havebeen proposed and applied widely on MRA image seg-mentation. In the original model proposed by Casellesand Malladi [8, 10], the governing force is based onthe image gradient, however, this stopping term is notrobust enough and hence may not stop the bleedingor leaking of the boundaries. Furthermore, the pullingback feature is also not strong enough [10]. To solvethese problems, Kichenassamy et al. [12] and Yezzi

172 D. Zonoobi et al.

et al. [13], introduced an extra stopping term, alsocalled the pull back term, to make the front returnto its position once it crosses the object boundary.However, these methods may have some difficulty inextracting tiny vessels from 3D images as they sufferfrom boundary leakage for complex structures [10].Siddiqi et al. [10, 14, 15] introduced area minimizationwhich functions as an additional stopping force. Thisterm provides an additional attraction force when theevolving surface is in the vicinity of an edge. Although,this new term improves the speed of evolution , it isnot sufficiently robust for segmenting complex shapesin brain vasculature [2].

One of the major challenges in segmentation ofMRA images is the accurate extraction of thin vessels.As they exhibit much variability in shape and sizesand usually have a low intensity contrast with the sur-rounding tissues. To deal with this problem, Pingkunet al. [2, 3] proposed a new capillary geodesic activecontours (CGAC) which is modeled based on the phys-ical phenomenon of capillary action. The incorporatedcapillary force adapts the evolution surface into thinbranches of blood vessels and obtains more accuratesegmentation results [2, 3]. Nevertheless, the CGACdoes not make use of any priori knowledge of theMRA image properties for the propagation and thespeed term is not specialized for MRA images. Thus,it is not surprising that it may exhibit some difficulty inextracting very tiny vessels from 3D MRA images ofbrain vasculature.

Specialized methods for particular applications canoften achieve better performance compared to generalones by incorporating some prior knowledge which actlike regularizers in the segmentation process. In thispaper, we present an image segmentation algorithmfor automatically extracting the whole vasculature from3D angiography by incorporating a priori knowledgeof MRA images of brain vasculature into the geodesicactive contours methods and making improvements totheir speed term. Tests show that our proposed algo-rithm has great potential for segmenting tiny vesselsunder low contrast conditions. In our work, to differ-entiate the vessels, we call those with thickness of lessthan 6 voxels as tiny vessels while those that are thickerbut less than 12 voxels as thin vessels.

2 Method

2.1 Level Set Method

Level set methods are numerical techniques for analyz-ing and computing interface motions that have been

used in geodesic active contours models [1, 9]. Sinceimplicit representation is used, the level set methodallows for topological changes to occur without any ad-ditional computational complexity [2]. Therefore theycan be applied with no prior assumption about theobject topology. When solving problems, the level setmethod increases the dimensionality of the problem.Let φ : �2 → � be the signed distance function to curveC. If C is evolving according to:

−→Ct = F

−→N (1)

it can be shown that the level set approach will followcontour C using the following level set equation:

�t + F|∇�| = 0 (2)

where F is a speed function. F is also known as thestopping force and it is used in the level set approachto stop the deformation process. The accuracy of thesegmentation process depends upon when and wherethe propagating curve needs to stop [10]. Therefore,the speed force should approximate closely to zeronear the target boundary as the propagating surfaceneeds to stop close to the vicinity of the target shape [1].Different types of stopping forces have been proposedsuch as those based on image gradients [1], edgestrength [13], area minimization [14] and capillary ac-tion [2, 3]. In all of these stopping forces, the key featureis the intensity gradient of the objective image.

2.2 Capillary Geodesic Active Contours

The capillary geodesic active contours (CGAC) [1]method is modeled based on the physical phenomenonof capillary action. In this method, the evolution is doneusing the following stopping force:

∂�

∂t= g(κ + V0)|∇�| + ∇g · ∇�

+ (κ̂2 + λ

) |∇�| f(1 − cos2 θ

). (3)

where

cos θ = −→N · ∇g

|∇g| .

The third term is the novel term which comes from thecapillary action. The gradient map of the image g(x),is a monotonically decreasing function of the imagegradient and is the only image dependant term in theCGAC stopping force. Its role is to help the evolvingcontour, reside on the desired object boundaries.

g(x) = 1

1 + |∇(Gσ (x) ∗ I(x))| (4)

Vasculature segmentation in MRA images 173

or

g (|∇ I|) = e|ξ∇(Gσ (x)∗I(x))| (5)

and κ is the curvature of the evolving contour.

2.3 Thickness vs. Brightness

In MRA images, vessels appear brighter than the back-ground. However, the range of the intensity values ofblood vessels in MRA images are not restricted to asmall interval but are spread out widely. Figure 1ashows the Maximum Intensity Projection (MIP) of aMRA data set. It can be observed that thick vesselsappear much brighter than the thinner ones. Thus,thick vessels are easily distinguishable while thin vesselsappear to not differ much from the surrounding tissues.

From the gradient magnitude of a MIP image shownin Fig. 1b, it is evident that thick vessels have highgradient values while those of thin vessels are muchlower.

Within the blood vessels, the intensity depends verymuch on the blood flow [16]. The velocity of blood flowin thick vessels is higher than in thin vessels. Therefore,the thicker the vessel, the more will be the flow andhence the higher the intensity and gradient values.Also, thin vessels are affected by the partial volumeeffect, which further reduces their intensity [6].

The boundary gradient vs. the thickness of vessels isplotted for a sample data set in Fig. 2. Clearly, thereis an almost exponential relationship between vesselthickness and the gradient on their boundaries whichis approximated as follows:

|∇ I(x)| ∼= MaxGrad ∗ (1 − e(−λ∗Th(x))

)(6)

(a) (b)Figure 1 (a) MIP of a MRA data set and (b) its gradientmagnitude.

Figure 2 The boundary gradient vs. the thickness of the vesselsfor a sample data set.

where x is any point on the vessels boundaries, ∇ I(x)

is the gradient at x, MaxGrad is the maximum of theimage gradient values, Th(x) is the thickness of thevessel at point x and λ is a small constant dependingon the image properties.

It can be seen from Fig. 3 that the intensity gradientinitially increases sharply with the thickness and levelsoff beyond a certain thickness. In MRA images, fora fixed repetition time (TR), if the velocity exceedsa certain value, the intensity due to blood flow willremain constant [11].

2.4 Gradient Compensated Geodesic Active Contours(GCGAC)

As discussed in Section 2.2, the gradient map (g(|∇ I|))of the image is the only image-dependant component ofthe stopping forces and its role is to stop the contour’sevolution near the boundaries of the desired objects.The gradient map is a static function of the image

Figure 3 Vessel gradient vs. its thickness.


gradient (Eqs. 4, 5) i.e., its values do not change asthe contour deforms. However, it can be seen fromFig. 1b, that the gradient values are much lower for thethin vessels compared to the thick ones, thus makingit difficult to place the contour on the boundaries ofthin vessels. To address this problem, we propose adynamic gradient map that is not only a function ofthe image gradient but also a function of time andcontour location. As the evolving contour gets smaller,it is very likely that it is approaching the thinner vesselswhose gradient values may not be high enough to at-tract the contour. Therefore, it is desirable to make theevolving contour more sensitive to the image gradientsas it shrinks. To achieve this, a novel gradient mapfunction is defined such that it produces reasonablevalues even if the image gradient is not high enough,provided that the contour thickness is also small. Toreach this goal, an equalizer function is defined (Eq. 7),which is a function of the contour’s thickness and itsrole is to strengthen the image gradient values whenthe contour’s thickness is small (i.e., when the contouris most probably in the vicinity of the thin vessels).This equalizer function is called gradient compensatorfunction, M(x, t), and is defined as follows:

M(x, t) = α ∗ (e(−λ∗Th(x,t))) (7)

where x is any point on the contour, Th(x) is theestimated thickness of the surface at that point, λ isthe same parameter as in Eq. 6 and α is a constant.From Fig. 4, it can be seen that the value of thisfunction varies between 0 and α for very thick andextremely tiny contours, respectively. The same λ as inEq. 6 has been chosen so that the rate of the changeof M(x, t) vs. Th(x) is the same as |∇ I(x)| vs. Th(x)

Figure 4 M(Th(x)) vs. estimation of thickness at point x.

in Eq. 6. This ensures that M(x, t) increases as thegradient values decrease at the thin vessel boundaries.Function M(x, t) is incorporated into the gradient mapequation by replacing the term |∇ I|(x) in Eq. 5, with(1 + M(x, t))|∇ I|(x), as follows:

g(x, t) = e|ξ∗∇(Gσ (x)∗(1+M(x,t))∇ I(x))|

= e|ξ∗∇(Gσ (x)∗(1+α∗e(−λ∗Th(x,t)))∇ I(x))| (8)

Gradient compensator function compensates for theinsufficient amount of |∇ I| on the thin vessels’ bound-aries. M(x, t) is nearly zero for thick vessels, hence,(1 + M(x, t)) ∗ |∇ I| � |∇ I|. However, for thin vessels,M(x, t) > 0 and is proportional to the gradient differ-ence of that vessel and a thick one. Thus, (1 + M(x, t)) ∗|∇ I| > |∇ I|. α, in Eq. 7, is a constant which functions asa trade-off between this compensator term and otherforces in the evolution process. Its value can be chosenin the range of 0 and Max(|∇ I|)

Min(|∇ I|) − 1, where the upper limitis chosen in a way as to strike a balance between thegradients of thin and thick vessels. Choosing a large α,the effect of makeup function will be more prominentand more thin vessels will be extracted (Fig. 7). How-ever, the approach may become more sensitive to noise.Thus, as in any other segmentation approach, there is aneed to keep a balance between these two issues.

The evolution equations for the Gradient Compen-sated Geodesic Active Contour (GCGAC) is obtainedby replacing the g(x) with g(x, t) (Eq. 8) in the CGACevolution equation (Eq. 3) as follows:

∂�

∂t= g(κ + V0, t)|∇�| + ∇g(t) · ∇�

+(κ̂2 + λ

)|∇�| f(1 − cos2 θ

). (9)

In the GCGAC, we need to estimate the thicknessof the evolving contour. For this purpose the normalvector at each point x is first obtained as follows:

−→N = ∇φ

|∇φ| (10)

Since blood vessels in MRA images appear brighterthan the background, the directions of normals in theimage could be known. Assuming that the contour isalmost tubular at each point, the thickness of point x(Th(x)) could be estimated as the number of voxelsalong the normal direction within the contour, as shownin Fig. 5. This computation is not costly, since thenormal of the surface has been calculated in other partsof the level set algorithm.


Figure 5 Estimation of the local contour thickness.

2.4.1 Implementation

The GCGAC described in this paper has been codedfor 3D segmentation based on the insight segmentationand registration toolKit (ITK; http://www.itk.org/). Theresults of the algorithm have been visualized based onthe visualization toolKit (VTK; http://www.vtk.org/).

Figure 6 shows the major components of theGCGAC which is used in the segmentation task. Aninitial segmentation estimate is generated by simplythresholding the 3D MRA image with thin tubularstructures. Before applying the algorithm, the image issmoothened by using a small isotropic Gaussian filter,since the level set algorithm inherently requires somesmoothness of gradients. The result is used to generatean initial signed distance function �0, which has neg-ative values inside objects and positive values on theoutside. As thick vessels appear quite bright in MRAimages, we are able to extract the thick parts of thevasculature in the early stage of the algorithm. A 3-Darray keeps track of the evolving contour thicknessright from the initial contour and is used in the gradientcompensator function. The level set function � is theniteratively updated according to

�n+1 = �n + ∇�n∇t, (11)

where ∇� is calculated using Eq. 9. For the evolv-ing contour, convergence is achieved when volumet-ric change is very small over some iterations. TheNarrow band level set method [17] has been used for

Figure 6 An overview of the gradient compensated geodesicactive contours algorithm (GCGAC).

the implementation, in order to restrict most computa-tions to a thin band of active voxels immediately sur-rounding the interface. Furthermore, the level set map� is periodically reinitialized to be a signed distancefunction. It means that, the zero level set c is extractedfrom �, and then the value at each point is set to be itsdistance to C.

(a) (b)

(c) (d) (e)Figure 7 Illustration of the effects on varying α in Eq. 8 on (a) asynthetic 3D vessel whose cross section along the axis is shown in(b). Segmented vessel when (c) α = 0, (d) α = 1.2, and (e) α=3.2.

http://www.itk.org/

http://www.vtk.org/


(a) (b)Figure 8 MIP of the two segmented MRA data sets.

3 Results and Discussion

The proposed algorithm has been applied on both syn-thetic data and MRA data sets. First, the effectivenessof the GCGAC is studied using a vessel-like computermodel. Secondly, the vasculature tree from three dif-ferent sets of real MRA data is extracted, using thealgorithm described in Section 2.4.1 and the resultsare compared with both that ones of the capillarygeodesic active contours (CGAC) algorithm and thereference manual segmentations performed by medicalradiologists.

3.1 Results Based on Synthetic Data

Figure 7 shows the application of the our proposedapproach on a computer model to illustrate the effectsof the gradient compensator function. As seen fromFig. 7a, b the intensity of the voxels in the model de-creases as the thickness of the vessel decreases. Startingfrom the initialization, evolution progresses accordingto the level set speed function in Eq. 3 and our pro-posed gradient map in Eq. 8. In this experiment, all the

(a) The proposed algorithm results (b) Manually segmented resultsFigure 9 Results of the first data set.

(a) The proposed algorithm results (b) Manually segmented results

Figure 10 Results of the second data set.

parameters are fixed except for α (in the compensatorfunction) which varies between 0 and Max(|∇ I|)

Min(|∇ I|) − 1. Theevolution results with different parameter settings aftera number of iterations are visualized in Fig. 7c, d, e.When α is set to zero, our algorithm evolves exactlylike the CGAC and is not able to extract the thin anddarker portions of the vessel. Other extracted resultswith different values of α in Fig. 7 shows that thecompensator function can facilitate the evolution of thecontour in tiny portions of vessels.

When segmenting actual MRA vasculature data sets,we have observed that α > 3, makes our approach toosensitive to the noise. On the other hand, choosingα < 2 would also weaken our gradient compensatorfunction which results in the smaller amount of seg-mented vessels. An α of 2.7 gives reasonable resultsand this value has been used for all the subsequentsegmentations.

3.2 Segmentation Results of 3D MRA Images

The algorithm has been also applied on different setsof real 3D time of flight MRA (TOF-MRA) images.These images and their manual segmentations by clin-icians were provided by the Department of Diagnostic

(a) (b)Figure 11 (a) MIP of a MRA data set. (b) Its segmented regionof interest (ROI).


Figure 12 Results for MRAdata set 1.

(a) Proposed algorithm results (b) Corresponding CGAC results

Radiology at the National University Hospital. Each ofthe these data sets contains approximately 120 512×512slices at spacing of 0.43×0.43×1.2 and 16 bits of graytone resolution. The following sections present the per-formance and effectiveness of our approach as com-pared with other methods.

3.2.1 Comparison with Manually Segmented Results

Two sets of real MRA data are segmented using ourproposed algorithm and the extracted vessels are com-pared with manual segmentation results obtained byclinicians. The MIP of these two data sets are shown


Figure 13 Results for MRAdata set 2.


in Fig. 8. The initial segmentation is obtained by thresh-olding the raw data set. Figures 9, 10 show clinically(manual) segmented results and the correspondingGCGAC segmentation ones.

The proposed algorithm is able to extract much ofthe vascular tree that appears in the clinically seg-mented images. In addition, it can be seen that a fair

amount of tiny vessels have been successfully seg-mented using our proposed algorithm.

3.2.2 Comparison with CGAC

Three MRA data sets have been segmented usingour proposed method. Figure 11a shows the MIP of


Figure 14 Results For MRAdata set 3.


one of these MRA data sets where vessels within theregion of interest (Fig. 11b) are enhanced for bettervisualization.

We have compared the segmentation results of ourproposed method with the state-of-the-art capillary ac-tive contours algorithm [3]. Identical parameter settingsin the evolution equations are used for both methodsexcept for the gradient compensator term, α , which is

specific to the our proposed method and determines thecontribution of this proposed force. The segmentationresults of the CGAC and our method are visualized inFigs. 12, 13, 14 from different points of view. From theseresults, it is evident that while the CGAC is able toextract some thin vasculature, it is not able to segmentthe tiny vessels while our proposed method, on theother hand, is capable able to do so.


4 Concluding Remarks

A novel gradient compensated geodesic active contours(GCGAC) has been proposed and developed to ex-tract vasculature from MRA images. The performanceof this algorithm has been validated by experimentalresults on both synthetic images and medical MRAimages and better results have been obtained com-pared with other state-of-the-art MRA segmentationalgorithms.

Acknowledgements We wish to thank Prof. Wang Shih-Chang,Dr. Yeh Ing Berne and other clinicians from the Departmentof Diagnostic Radiology at National University Hospital ofSingapore, specially prof. Wang Shih-Chang and Dr. Yeh IngBerne, for their kind help and support.

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Dornoosh Zonoobi received her B.Sc. degree from Shiraz Uni-versity, Iran in 2005. She obtained her M.Eng. degree in Com-puter Engineering from the National University of Singapore(NUS) in 2008. Since then she is working as a researcher withthe Department of Electrical and Computer Engineering at NUS.Her research interests include medical image analysis, signalprocessing, and machine learning.


Ashraf A. Kassim received his B.Eng. (First Class Honors)and M.Eng. degrees in Electrical Engineering from the NationalUniversity of Singapore (NUS) in 1985 and 1987, respectively.From 1986 to 1988, he worked on machine vision systems at TexasInstruments. He went on to obtain his Ph.D. degree in Electricaland Computer Engineering from Carnegie Mellon University,Pittsburgh, in 1993. Since 1993, he has been with the Electricaland Computer Engineering Department at NUS, where he iscurrently an Associate Professor and Vice Dean of EngineeringFaculty. Dr Kassim’s research interests include image analysis,machine vision, video/image processing and compression.

Weijia Shen received his B.Eng. degree from Xi’an JiaotongUniversity, Xi’an, China in 2003. Since 2003, he has been a Ph.D.student in Electrical and Computer Engineering Department atNational University of Singapore. His research interests includeimage processing, machine vision, and machine learning.

Vasculature Segmentation in MRA Images Using Gradient Compensated Geodesic Active Contours

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