research article coronary arteries segmentation based on...

Research ArticleCoronary Arteries Segmentation Based on the 3D DiscreteWavelet Transform and 3D Neutrosophic Transform

Shuo-Tsung Chen123 Tzung-Dau Wang4 Wen-Jeng Lee5 Tsai-Wei Huang6

Pei-Kai Hung1 Cheng-Yu Wei78 Chung-Ming Chen1 and Woon-Man Kung79

1 Institute of Biomedical Engineering National Taiwan University Taipei 10617 Taiwan2Department of Applied Mathematics Tunghai University Taichung 40704 Taiwan3Sustainability Research Center Tunghai University Taichung 40704 Taiwan4Cardiovascular Center and Division of Cardiology Department of Internal MedicineNational Taiwan University Hospital Taipei 10048 Taiwan5Department of Medical Imaging National Taiwan University Hospital Taipei 10048 Taiwan6Department of Nursing College of Medicine and Nursing Hungkuang University Taichung 43302 Taiwan7Department of Exercise and Health Promotion College of Education Chinese Culture University Taipei 11114 Taiwan8Department of Neurology Chang Bing Show Chwan Memorial Hospital Changhua 50544 Taiwan9Department of Neurosurgery Lo-Hsu Foundation Lotung Poh-Ai Hospital Luodong Yilan 26546 Taiwan

Correspondence should be addressed to Chung-Ming Chen chungntuedutw andWoon-ManKung nskungwmyahoocomtw

Received 19 July 2014 Accepted 11 October 2014

Academic Editor Kuo-Sheng Hung

Copyright copy 2015 Shuo-Tsung Chen et al 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

Purpose Most applications in the field of medical image processing require precise estimation To improve the accuracy ofsegmentation this study aimed to propose a novel segmentation method for coronary arteries to allow for the automatic andaccurate detection of coronary pathologiesMethodsThe proposed segmentationmethod included 2 parts First 3D region growingwas applied to give the initial segmentation of coronary arteries Next the location of vessel information HHH subband coefficientsof the 3DDWT was detected by the proposed vessel-texture discrimination algorithm Based on the initial segmentation 3DDWTintegrated with the 3D neutrosophic transformation could accurately detect the coronary arteries Results Each subbranch of thesegmented coronary arteries was segmented correctly by the proposed method The obtained results are compared with thoseground truth values obtained from the commercial software from GE Healthcare and the level-set method proposed by Yanget al 2007 Results indicate that the proposed method is better in terms of efficiency analyzed Conclusion Based on the initialsegmentation of coronary arteries obtained from 3D region growing one-level 3D DWT and 3D neutrosophic transformation canbe applied to detect coronary pathologies accurately

1 Introduction

Efficient and automatic image segmentation methods areuseful for the isolation and visualization of vessels in com-puted tomographic angiography (CTA) There are manyproposed methods for the segmentation of vessels [1ndash14] Avessel filter [1] can be used to enhance tubular structurehowever it cannot address the problem of the image forceand veins which can lead to a narrowed or broken seg-mentation of vessels Parametric shape models [2ndash5] do notdirectly allow for the detection of topological changes and

they usually obtain a seriously narrowed segmentation inthe neighborhood of a branch point in the vessel Level-setapproaches [6ndash13] are computationally expensive They alsosuffer from leakage at places where the intensity gradientsof the edges are relatively weak and are very sensitive tothe placement of the initial contour of the propagatingfront Metz et al [14] used the minimum cost path ofthe specified start and end points in vessel to detect thecoronary arteries centerlineThis is not an automaticmethoddetecting branches is difficult Friman [15] proposedmultiplehypothesis template tracking which follows the direction of

Hindawi Publishing CorporationBioMed Research InternationalVolume 2015 Article ID 798303 9 pageshttpdxdoiorg1011552015798303

2 BioMed Research International

centerline obtained in advance However it is difficult todetect small branches and vessels by using this method

In this study we propose a newmethod for automaticallyand correctly segmenting coronary arteries from CTA datasets In image preprocessing we detected the aorta auto-matically by using methods proposed in the literature [516] The proposed coronary arteries segmentation method issummarized as follows First of all we automatically obtainedthe seed point of a 3D region growing by the differencebetween the two adjacent slices due to the small changesof the aorta between two adjacent slices Next 3D regiongrowing was applied to initially search for the probablelocation of coronary arteries which was then dilated by 3voxels Based on the dilation of the probable location wedetected the coronary arteries accurately by applying the 3Ddiscrete wavelet transformation (DWT) and 3Dneutrosophictransformation to the CTA volume The location of vesselinformation in HHH subband coefficients was detectedby the proposed vessel-texture discrimination algorithmAccordingly HHH subband coefficients were used whichwere characterized and classified by 120572-means operation and119870-means clustering Finally the proposed method was testedon several CTA data sets and the experimental resultsindicated that the proposedmethod had a good performance

The rest of this study is organized as follows Section 2reviews some preliminaries and Section 3 uses 3D regiongrowing and 3DDWT to propose a newmethod for segment-ing coronary arteries Section 4 contains the experiments anddiscussion and the conclusions are drawn in Section 5

2 Preliminaries

In this section we will briefly introduce the concepts of DWTand provide an overview of some fundamental mathematicalconcepts that are used in this study

21 Region Growing Region growing is a simple well-developed region-based image segmentation technique [17]It postulates that neighboring voxels within the same regionhave similar intensity values and is also classified as a voxel-based image segmentation method since it involves theselection of initial seed points In other words this methodof segmentation examines neighboring voxels of initial seedpoints and determines whether neighboring voxels should beadded to the region Consequentially the general concept ofregion growing is to group voxels with the same or similarintensities to one region according to the given seed pointsand a homogeneity criterion

22 Discrete Wavelet Transform Wavelet transform isobtained by a single prototype function 120595(119909) which isregulated with scaling and shift parameters To construct120595(119909) a scaling function 120593(119909) is determined The discretenormalized scaling and wavelet basis functions are definedas

120593119894119899 (

119905) = 21198942

120593 (2119894119905 minus 119899)

120595119894119899 (

119905) = 21198942

120595 (2119894119905 minus 119899)

(1)

where 119894 and 119899 are the dilation and translation parametersOrthogonal wavelet basis functions not only provide a simplemethod to calculate coefficient expansion but also span 119871

2(R)

in signal processing As a result signal 119878(119905) isin 1198712(R) can

be expressed as a series expansion of orthogonal scalingfunctions and wavelets More specifically

119878 (119905) = sum

ℓ

1198881198950(ℓ) 1205931198950 119896

(119905) + sum

119896

infin

sum

119895=1198950

119889119895 (

119896) 120595119895119896 (119905) (2)

where 119888119895(ℓ) = int

R119878(119905)120593119895ℓ

(119905)119889119905 and 119889119895(119896) = int

R119878(119905)120595119895119896

(119905)119889119905

are the low-pass and high-pass coefficients respectively 1198950is

an integer to define an interval on which 119878(119905) is a piecewiseconstant The two-scale equations for scaling and waveletbasis function are given as follows

120593 (119905) = radic2 sum

119898isinZ

ℎ119898120593 (2119905 minus 119898) (3)

120595 (119905) = radic2 sum

119898isinZ

119892119898120593 (2119905 minus 119898) (4)

where 119892119898

= (minus1)119898ℎ1minus119898

The coefficient ℎ119898in (3) has to meet

several conditions for the set of the wavelet basis functionto be unique and orthonormal and have a certain degree ofregularity

The coefficients ℎ119898and 119892

119898play a very crucial role in a

given DWT Performing the wavelet transformation does notrequire the explicit forms of 120593(119905) and 120595(119905) but only dependson ℎ119898and 119892

119898 The final output of the wavelet decomposition

includes a set of 119895-level wavelet coefficients One method toimplement DWT is to use a filter bank that provides perfectreconstruction DWT involves local analysis of frequency inspace and time domains and it provides multiscale imagedetails step by step If the scale becomes smaller every partbecomes more accurate and ultimately all imaging detailscan be focalized accurately If DWT is applied to a volumeit will produce the highest-frequency middle-frequency andlowest-frequency parts Figure 1 shows the results of applying3D DWT to a volume which includes eight parts LLLLLH LHL LHH HLL HLH HHL and HHH The lowest-frequency and highest-frequency parts are LLL and HHHrespectively [16 17]

23 119870-Means Clustering 119870-means clustering is a method ofcluster analysis which aims to partition 119899 observations into 119896

clusters in which each observation belongs to the cluster withthe nearest mean Given a set of observations (119909

1 1199092 119909

119899)

where each observation is a 119889-dimensional real vector 119896-means clustering aims to partition the 119899 observations into119896 (119896 le 119899) sets 119904

1 1199042 119904

119896 so as to minimize the within-

cluster sum of squares

arg min119896

sum

119894=1

sum

119909119895isin119904119894

10038171003817100381710038171003817119909119895minus 120583119894

10038171003817100381710038171003817

2

(5)

where 120583119894is the mean of points in 119904

119894

BioMed Research International 3

Rows

Columns

Slices

Original volume

LxLyLz

LxLyHz

LxHyLz

LxHyHz

HxLyLz

HxLyHz

HxHyLz

HxHyHz

LA darr

LA darr

LA darr

LA darr

LA darr

LA darr

LA darr

HA darr

HA darr

HA darr

HA darr

HA darr

HA darr

HA darr

Figure 1 The structure of applying 3D DWT to a volume

3 The Proposed Segmentation Method

In order to segment coronary arteries accurately from CTAdata sets 3D region growing was initially applied to searchfor the probable location of the coronary arteries Next weused the 3D DWT and 3D neutrosophic transformation toaccurately detect the coronary arteries

31 Initial Segmentation of Coronary Arteries This sectiondiscusses the initial segmentation of coronary arteries using3D region growing In image preprocessing we found theaorta automatically by using methods proposed in the liter-ature [5 16] The selection of the seed point of 3D regiongrowing was initially made to check which slice beganthe information of the coronary arteries Due to the smallchanges in aorta area between the two adjacent slices weautomatically obtained the seed points by the differencebetween the two adjacent slices

As shown in Figure 2 we use the difference between thetwo adjacent slices (a) and (b) to automatically obtain theseed points bounded by the blue line in (c) which indicatesthe boundary of coronary arteriesThe boundary of coronaryarteries denotes the high-frequency subband 119889

119895(119896) in (2)

when comparing vessel lumen and backgroundSince coronary arteries do not exhibit abrupt intensity

changes along their centerline [4] a rough tubular mask ofcoronary arteries can be easily constructed by 3D region

growing We chose a 26-connected neighborhood for ouradjacent pixel relationship and then the 3D region growingmethod was applied with a set of prespecified seed voxel(s)and grown from these seeds by merging neighboring voxelswhose properties were most similar to the premerged regionThe homogeneity criterion was defined as the differencebetween the intensity of the candidate voxel and the averageintensity of the premerged region The selection of the seedpoint was initially intended to check which slice began theinformation of the coronary arteries Next the homogeneitycriterion was applied to group voxels with the same or similarintensities into one region If the homogeneity criterion wassatisfied the candidate voxel was merged with the premergedregion The process was repeated until no more voxelswere assigned to the region and then the number of allmerged voxels was calculated In order to avoid leakagethe total number of merged voxels was limited to 12000otherwise 3D region growing was restarted by automaticallyusing an improved homogeneity criterion Finally the initialsegmentation of the coronary arteries in a volume wascompleted

32 Vessel-Texture Discrimination According to Parsevalrsquostheorem the energy in a signal 119878(119905) is given as follows [16 17]

int |119878(119905)|2119889119905 =

infin

sum

119897=minusinfin

|119888(119897)|2+

infin

sum

119895=0

infin

sum

119896=minusinfin

10038161003816100381610038161003816119889119895(119896)

10038161003816100381610038161003816

2

(6)


This equation implies that the energy of a signal is the sum-mation of low-frequency and high-frequency coefficientsDWT is a good analytic tool for image texture analysis orline-based patterns [18ndash21] Since a vessel is a type of 3D line-based pattern inCT volume we used 119897

2-normof thesewavelet

coefficients to find the energy of line-based patterns 1198972-norm

was defined as

119864 (119862) = 1198622=

119901

sum

119894=1

1003816100381610038161003816119888119894

1003816100381610038161003816

2 (7)

where the vector 119862 = [119888119894]1times119901

was the wavelet coefficients ofa frequency channelThe searching algorithm is summarizedas follows

Algorithm 1 (1) Transform a given vessel volume into fre-quency channels by a specified number of decompositionlevels We usually set the number to one in the first search

(2) Use (7) to calculate the average 1198972-norm of each

channel and maximum of 119864(119862) for the vessel volume(3) If the maximum of 119864(119862)was significantly greater than

another channelrsquos 119864(119862) the search was stopped Otherwisethe number of decomposition levels was increased followedby a repeat of step 1

By using the above algorithm we observed that the mostsignificant information of the vessel texture often appearedin the high frequency channels Thus we used the subbandHHH to detect vessels in this study

33 Accurate Detection of Coronary Arteries The initialsegmentation of coronary arteries in a volumewas completedby 3D region growing as described in Section 31 Sinceregion growing is a simple region-based image segmentationmethod it was only used to search for the initial locationof the coronary arteries We then accurately detected thecoronary arteries by applying DWT to each slice in thevolume as described in this subsection

First the initial location of the coronary arteries obtainedfrom 3D region growing was dilated by 3 voxels Nextwe used the Haar wavelet bases in (1) to transform thehost images into the orthogonal DWT domain by one-leveldecomposition Only HHH subbands were employed forfurther processes because most of the information on thecoronary arteries and boundarieswere in theHHHsubbandsWe calculated the mean energy using coefficients of HHHsubbands in a local window 119908 as follows

HHH (119894 119895 119896) =

1

119908 times 119908 times 119908

119903

sum

119897=119894

119904

sum

119898=119895

119905

sum

119899=119896

HHH (119897 119898 119899) (8)

where

119903 = round(119894 +

119908

2

)

119904 = round(119895 +

119908

2

)

119905 = round(119896 +

119908

2

)

(9)

Next the subbands HHH were characterized by 3 member-ship sets 119879 119865 and 119880 Consider

119879HHH (119894 119895 119896) =

HHH (119894 119895 119896) minusHHHmin

HHHmax minusHHHmin

119865HHH (119894 119895 119896) = 1 minus 119879HHH (119894 119895 119896)

119880HHH (119894 119895 119896) =

120575 (119894 119895 119896) minus 120575min120575max minus 120575min

(10)

where

HHHmin = min HHH (119894 119895 119896)

HHHmax = max HHH (119894 119895 119896)

120575 (119894 119895 119896) =

10038161003816100381610038161003816HHH (119894 119895 119896) minusHHH (119894 119895 119896)

10038161003816100381610038161003816

120575min = min 120575 (119894 119895 119896)

120575max = max 120575 (119894 119895 119896)

(11)

That is a pixel could be represented as a neutrosophic domain119875(119905 119891 119906) which means the pixel is 119905 true 119891 false and 119906uncertain where 119905 varies in 119879 119891 varies in 119865 and 119906 varies in119880 In order to reduce the uncertainty 119906 120572-means operationwas employed as follows

119879 (120572) =

119879 if 119880 lt 120572

119879120572 if 119880 ge 120572

119865 (120572) =

119865 if 119880 lt 120572

119865120572 if 119880 ge 120572

119880120572(119894 119895 119896) =

120575119879(119894 119895 119896) minus 120575

119879min

120575119879max

minus 120575119879min

(12)

where the parameter 120572 is a positive number and

119879120572(119894 119895 119896) =

1

119908 times 119908 times 119908

119903

sum

119897=119894

119904

sum

119898=119895

119905

sum

119899=119896

119879 (119897 119898 119899)

119865120572(119894 119895 119896) =

1

119908 times 119908 times 119908

119903

sum

119897=119894

119904

sum

119898=119895

119905

sum

119899=119896

119865 (119897 119898 119899)

120575119879(119894 119895 119896) =

1003816100381610038161003816100381610038161003816

119879 (119894 119895 119896) minus 119879 (119894 119895 119896)

1003816100381610038161003816100381610038161003816

119879 (119894 119895 119896) =

1

119908 times 119908 times 119908

119903

sum

119897=119894

119904

sum

119898=119895

119905

sum

119899=119896

119879 (119897 119898 119899)

(13)

The inverse DWT was then applied to obtain a new volumewhich possessed the true subset Finally we applied119870-meansclustering (119870 = 3) in (2) to differentiate vessel lumen vesselboundary (true subset) and background The true subset 119879was retained respectively


(a) (b)

(c)

Figure 2 The 3D region growing seed points in (c) were automatically obtained by using the difference between the two areas bounded bythe blue lines in adjacent slices (a) and (b)

4 Experiments and Discussion

To test the proposed method CTA volumes obtained froma CT system were segmented for coronary arteries The slicethickness was 0625mm and the volume was 512lowast512lowast(sdot) indifferent data sets The window size 119908 was set to 3 whichwas enough to capture the local texture characteristics Theparameter 120572 was set to 02 We tested 20 data sets most ofwhich were segmented successfully except for a few smallbranches that were lost in 2 of the data sets due to the localfailure in region growing To evaluate the performance ofour segmented coronary arteries we compared our resultswith that obtained from the ground truth values obtainedfrom the commercial software from GE Healthcare and thelevel-set method We used the overlapping metric (OM) andHausdorff distance (119889

119867) to analyze the efficiency of each

method

41 The Segmenting Efficiency of 2D Imaging Figure 3(b)shows the results of 6 slices in 1 CTA volume obtainedusing the proposed method The areas bounded by the red

line are coronary arteries The segmenting efficiency wascomparedwith themanually delineated ground truth data119873

119877

in Figure 3(a) by using OM which was defined as

OM = 2(

119873119879cap 119873119877

119873119879+ 119873119877

) (14)

where 119873119879indicates the pixelsvoxels of the segmented coro-

nary arteries The OM was close to 1 when the segmenta-tion was well matched to the reference ground truth andapproached zero when the results had no similarity to thereference

In the 6 slices in Figure 3(b) the segmentation resultsshowed that the proposed method detected coronary arteriesaccurately As shown in Table 1 the average OM of theproposed method was 096

42The Segmenting Efficiency on a 3DVolume Thefirst focusof the comparisonwas the correctness of the 4main branchesthe right coronary artery (RCA) the left anterior descendingartery (LAD) the circumflex (CRX) and the first diagonal


(a) Ground truth data (b) The segmented results

Figure 3 Comparison of the 2D segmentation (b) with respect to the ground truth data (a)

artery (DA) Figures 4 and 5 show the detected coronaryarteries in the CTA volume obtained from the GE Health-care and the proposed method and the proposed methodsegmented these 4 main branches correctly compared to the4 main branches in Figure 5

Another test of performance is the correctness of theremaining branches Due to the multiresolution of the DWTeach subbranch in Figures 3 and 5 was correctly segmentedFurthermore the coronary arteries obtained from the pro-posed method were much better than those obtained from


Figure 4The 3D coronary arteries manually edited by a radiologistusing an AW workstation (GE Healthcare Wisconsin USA)

200

150

100

100100

50

50

50150

150

Figure 5 The 3D coronary arteries in the CTA volume obtainedfrom the proposed method

Table 1 Comparison of the 2D image segmentation results

Yang et al [11] method Proposed methodMean OM 068 096

the level-set method which had several leakages as seen inFigure 6

In addition to the OM the difference between thesegmented vessel surface and themanually delineated groundtruth datawasmeasured by theHausdorffdistance [22]whichwas defined as follows

119889119867 (119883 119884) = maxsup

119909isin119883

inf119910isin119884

119889 (119909 119910) sup119910isin119884

inf119909isin119883

119889 (119909 119910) (15)

where 119883 and 119884 are the vertices of the mesh surfaces of thearteries corresponding to the segmentation results and theground truth and 119889(119909 119910) measure the Euclidean distancebetween points 119909 and 119910 belonging to vertices 119883 and 119884

150

150150

200

200

250

100

100

100

50

50

50

3D coronary tree

Figure 6 The 3D coronary arteries in the CTA volume obtainedfrom the level-set approach

Table 2 Comparison of the 3D segmentation results

Yang et al [11] method Proposed methodMean OM 060 092Mean 119889

119867(119883 119884) 108 068

Table 3 Comparison of the cross-sectional diameter

Ground truth Yang et al [11] method Proposed method47 44 48

Table 2 lists the mean OM and mean Hausdorff distance forthe proposed method and the method of Yang et al [11] Theresults show that the proposed method was much better thanthat of Yang et al [11] in terms of both OM and Hausdorffdistance

43 DiameterMeasurement In this subsection we computedthe diameters of segmented coronary arteries using theproposed method and the method of Yang et al Manyefficient algorithms have been proposed to extract the tubecenterline We applied the algorithm proposed by Lee etal [23] to extract the centerline of the segmented coronaryarteries Using these extracted centerlines we obtained thecross sections of the segmented coronary arteries as shownin Figure 7 By computing the area 119860 of each cross sectionthe diameter 119903 was estimated as follows

119903 asymp radic119860

120587

(16)

Table 3 shows the estimated diameter of the same crosssection for the proposed method and the method of Yang etal [11] The diameter of the proposed method was closer tothe diameter in ground truth data at the same cross sectionthan to that obtained from the method of Yang et al

44 Experimental Environment and Execution Time Theproposed method was implemented in MATLAB (R2011a)on a standard specification PC with a 32GHz CPU and12GB RAM The average execution time was 58 seconds to


(a) (b)

Figure 7 (a) The cross section of the segmented coronary arteries (b) Magnified view of (a)

extract the entire coronary tree compared to approximately47 seconds for themethod by Yang et al for the same process

5 Conclusions

Accurate extraction of coronary arteries is important to assessartery lesions in clinical practice In this study we proposea novel method to segment coronary arteries automaticallyBased on the initial segmentation obtained from 3D regiongrowing one-level 3D DWT and 3D neutrosophic transfor-mation were applied to detect coronary arteries accuratelyThe location of vessel information in HHH subband coef-ficients of DWT was successfully detected by the proposedvessel-texture discrimination algorithm Accordingly theHHH subband coefficients were used and characterized andclassified by 3D neutrosophic transformation and 119870-meansclustering The experimental results verify the efficiency ofthe proposed method

Conflict of Interests

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

Acknowledgments

This study was supported by the National Taiwan UniversityHospital and theNational ScienceCouncil Taiwan under theNSC Grant NSC 98-2221-E-002-098-MY3

References

[1] A F Frangi W J Niessen K L Vincken and M A ViergeverldquoMultiscale vessel enhancement filteringrdquo in Medical ImageComputing andComputer-Assisted InterventationmdashMICCAIrsquo98vol 1496 of Lecture Notes in Computer Science pp 130ndash137Springer Berlin Germany 1998

[2] P J Yim J J Cebral R Mullick H B Marcos and P L ChoykeldquoVessel surface reconstruction with a tubular deformablemodelrdquo IEEE Transactions on Medical Imaging vol 20 no 12pp 1411ndash1421 2001

[3] D Nain A Yezzi and G Turk ldquoVessel segmentation using adriven flow shaperdquo in Proceeding of International Conferenceon Medical Image Computing Computer-Assisted Intervention-MICCAI 2004 vol 3216 of Lecture Notes in Computer Sciencepp 51ndash59 2004

[4] H Tek Y Zheng M A Gulsun and G Funka-Lea ldquoAnautomatic system for segmenting coronary arteries from CTArdquoin Proceedings of the MICCAI Workshop on Computing andVisualization for Intravascular Imaging (MICCAI-CVII rsquo11) pp47ndash54 2011

[5] A Hennemuth T Boskamp D Fritz et al ldquoOne-click coronarytree segmentation in CT angiographic imagesrdquo InternationalCongress Series vol 1281 pp 317ndash321 2005

[6] Y Yang A Tannenbaum and D Giddens ldquoKnowledge-based3D segmentation and reconstruction of coronary arteries usingCT imagesrdquo in Proceedings of the 26th Annual InternationalConference of the IEEE Engineering in Medicine and BiologySociety (EMBC rsquo04) vol 1 pp 1664ndash1666 San Francisco CalifUSA September 2004

[7] H Shikata G McLennan E A Hoffman and M SonkaldquoSegmentation of pulmonary vascular trees from thoracic 3DCT imagesrdquo International Journal of Biomedical Imaging vol2009 Article ID 636240 11 pages 2009

[8] T Brox and J Weickert ldquoLevel set segmentation with multipleregionsrdquo IEEE Transactions on Image Processing vol 15 no 10pp 3213ndash3218 2006

[9] W K Pratt Digital Image Processing John Wiley amp Sons LosAltos Calif USA 4th edition 2007

[10] L Wang L He A Mishra and C Li ldquoActive contours drivenby local Gaussian distribution fitting energyrdquo Signal Processingvol 89 no 12 pp 2435ndash2447 2009

[11] 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 From Nano to Macro (ISBIrsquo07) pp 189ndash192 April 2007


[12] Y Wang and P Liatsis An Automatic Method for Segmentationof Coronary Arteries X-Ray in Coronary CT Imaging IEEEComputer Society Developments in E-Systems Engineering2010

[13] Y Wang and P Liatsis ldquoAutomatic segmentation of coronaryarteries inCT imaging in the presence of kissing vessel artifactsrdquoIEEE Transactions on Information Technology in Biomedicinevol 16 no 4 pp 782ndash788 2012

[14] C T Metz M Schaap A C Weustink N R Mollet T VanWalsum and W J Niessen ldquoCoronary centerline extractionfrom CT coronary angiography images using a minimum costpath approachrdquo Medical Physics vol 36 no 12 pp 5568ndash55792009

[15] O Friman M Hindennach C Kuhnel and H-O PeitgenldquoMultiple hypothesis template tracking of small 3D vesselstructuresrdquo Medical Image Analysis vol 2010 no 14 pp 160ndash171 2010

[16] S C Saur C Kuhnel T Boskamp G Szekely and P C CattinldquoAutomatic ascending aorta detection in CTA datasetsrdquo inBildverarbeitung fur die Medizin 2008 pp 323ndash327 SpringerBerlin Germany 2008

[17] T Pavlidis and Y-T Liow ldquoIntegrating region growing and edgedetectionrdquo IEEE Transactions on Pattern Analysis and MachineIntelligence vol 12 no 3 pp 225ndash233 1990

[18] M Unser ldquoLocal linear transforms for texture measurementsrdquoSignal Processing vol 11 no 1 pp 61ndash79 1986

[19] T Chang and C C J Kuo ldquoTexture analysis and classificationwith tree-structured wavelet transformrdquo IEEE Transactions onImage Processing vol 2 no 4 pp 429ndash441 1993

[20] M Unser ldquoTexture classification and segmentation usingwavelet framesrdquo IEEE Transactions on Image Processing vol 4no 11 pp 1549ndash1560 1995

[21] Z-Z Wang and J-H Yong ldquoTexture analysis and classificationwith linear regressionmodel based on wavelet transformrdquo IEEETransactions on Image Processing vol 17 no 8 pp 1421ndash14302008

[22] R T Rockefellar and R J-BWetsVariational Analysis vol 317Springer New York NY USA 1988

[23] T C Lee R L Kashyap and C N Chu ldquoBuilding skeletonmodels via 3-D medial surface axis thinning algorithmsrdquoGraphical Models and Image Processing vol 56 no 6 pp 462ndash478 1994

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Volume 2014Hindawi Publishing Corporationhttpwwwhindawicom


centerline obtained in advance However it is difficult todetect small branches and vessels by using this method

In this study we propose a newmethod for automaticallyand correctly segmenting coronary arteries from CTA datasets In image preprocessing we detected the aorta auto-matically by using methods proposed in the literature [516] The proposed coronary arteries segmentation method issummarized as follows First of all we automatically obtainedthe seed point of a 3D region growing by the differencebetween the two adjacent slices due to the small changesof the aorta between two adjacent slices Next 3D regiongrowing was applied to initially search for the probablelocation of coronary arteries which was then dilated by 3voxels Based on the dilation of the probable location wedetected the coronary arteries accurately by applying the 3Ddiscrete wavelet transformation (DWT) and 3Dneutrosophictransformation to the CTA volume The location of vesselinformation in HHH subband coefficients was detectedby the proposed vessel-texture discrimination algorithmAccordingly HHH subband coefficients were used whichwere characterized and classified by 120572-means operation and119870-means clustering Finally the proposed method was testedon several CTA data sets and the experimental resultsindicated that the proposedmethod had a good performance

The rest of this study is organized as follows Section 2reviews some preliminaries and Section 3 uses 3D regiongrowing and 3DDWT to propose a newmethod for segment-ing coronary arteries Section 4 contains the experiments anddiscussion and the conclusions are drawn in Section 5

2 Preliminaries

In this section we will briefly introduce the concepts of DWTand provide an overview of some fundamental mathematicalconcepts that are used in this study

21 Region Growing Region growing is a simple well-developed region-based image segmentation technique [17]It postulates that neighboring voxels within the same regionhave similar intensity values and is also classified as a voxel-based image segmentation method since it involves theselection of initial seed points In other words this methodof segmentation examines neighboring voxels of initial seedpoints and determines whether neighboring voxels should beadded to the region Consequentially the general concept ofregion growing is to group voxels with the same or similarintensities to one region according to the given seed pointsand a homogeneity criterion

22 Discrete Wavelet Transform Wavelet transform isobtained by a single prototype function 120595(119909) which isregulated with scaling and shift parameters To construct120595(119909) a scaling function 120593(119909) is determined The discretenormalized scaling and wavelet basis functions are definedas

120593119894119899 (

119905) = 21198942

120593 (2119894119905 minus 119899)

120595119894119899 (

119905) = 21198942

120595 (2119894119905 minus 119899)

(1)

where 119894 and 119899 are the dilation and translation parametersOrthogonal wavelet basis functions not only provide a simplemethod to calculate coefficient expansion but also span 119871

2(R)

in signal processing As a result signal 119878(119905) isin 1198712(R) can

be expressed as a series expansion of orthogonal scalingfunctions and wavelets More specifically

119878 (119905) = sum

ℓ

1198881198950(ℓ) 1205931198950 119896

(119905) + sum

119896

infin

sum

119895=1198950

119889119895 (

119896) 120595119895119896 (119905) (2)

where 119888119895(ℓ) = int

R119878(119905)120593119895ℓ

(119905)119889119905 and 119889119895(119896) = int

R119878(119905)120595119895119896

(119905)119889119905

are the low-pass and high-pass coefficients respectively 1198950is

an integer to define an interval on which 119878(119905) is a piecewiseconstant The two-scale equations for scaling and waveletbasis function are given as follows

120593 (119905) = radic2 sum

119898isinZ

ℎ119898120593 (2119905 minus 119898) (3)

120595 (119905) = radic2 sum

119898isinZ

119892119898120593 (2119905 minus 119898) (4)

where 119892119898

= (minus1)119898ℎ1minus119898

The coefficient ℎ119898in (3) has to meet

several conditions for the set of the wavelet basis functionto be unique and orthonormal and have a certain degree ofregularity

The coefficients ℎ119898and 119892

119898play a very crucial role in a

given DWT Performing the wavelet transformation does notrequire the explicit forms of 120593(119905) and 120595(119905) but only dependson ℎ119898and 119892

119898 The final output of the wavelet decomposition

includes a set of 119895-level wavelet coefficients One method toimplement DWT is to use a filter bank that provides perfectreconstruction DWT involves local analysis of frequency inspace and time domains and it provides multiscale imagedetails step by step If the scale becomes smaller every partbecomes more accurate and ultimately all imaging detailscan be focalized accurately If DWT is applied to a volumeit will produce the highest-frequency middle-frequency andlowest-frequency parts Figure 1 shows the results of applying3D DWT to a volume which includes eight parts LLLLLH LHL LHH HLL HLH HHL and HHH The lowest-frequency and highest-frequency parts are LLL and HHHrespectively [16 17]

23 119870-Means Clustering 119870-means clustering is a method ofcluster analysis which aims to partition 119899 observations into 119896

clusters in which each observation belongs to the cluster withthe nearest mean Given a set of observations (119909

1 1199092 119909

119899)

where each observation is a 119889-dimensional real vector 119896-means clustering aims to partition the 119899 observations into119896 (119896 le 119899) sets 119904

1 1199042 119904

119896 so as to minimize the within-

cluster sum of squares

arg min119896

sum

119894=1

sum

119909119895isin119904119894

10038171003817100381710038171003817119909119895minus 120583119894

10038171003817100381710038171003817

2

(5)

where 120583119894is the mean of points in 119904

119894


Rows

Columns

Slices

Original volume

LxLyLz

LxLyHz

LxHyLz

LxHyHz

HxLyLz

HxLyHz

HxHyLz

HxHyHz

LA darr

LA darr

LA darr

LA darr

LA darr

LA darr

LA darr

HA darr

HA darr

HA darr

HA darr

HA darr

HA darr

HA darr






119895(119896) in (2)





int |119878(119905)|2119889119905 =

infin

sum

119897=minusinfin

|119888(119897)|2+

infin

sum

119895=0

infin

sum

119896=minusinfin

10038161003816100381610038161003816119889119895(119896)

10038161003816100381610038161003816

2

(6)





was defined as

119864 (119862) = 1198622=

119901

sum

119894=1

1003816100381610038161003816119888119894

1003816100381610038161003816

2 (7)










HHH (119894 119895 119896) =

1

119908 times 119908 times 119908

119903

sum

119897=119894

119904

sum

119898=119895

119905

sum

119899=119896

HHH (119897 119898 119899) (8)

where

119903 = round(119894 +

119908

2

)

119904 = round(119895 +

119908

2

)

119905 = round(119896 +

119908

2

)

(9)


119879HHH (119894 119895 119896) =

HHH (119894 119895 119896) minusHHHmin

HHHmax minusHHHmin

119865HHH (119894 119895 119896) = 1 minus 119879HHH (119894 119895 119896)

119880HHH (119894 119895 119896) =


(10)

where

HHHmin = min HHH (119894 119895 119896)

HHHmax = max HHH (119894 119895 119896)

120575 (119894 119895 119896) =

10038161003816100381610038161003816HHH (119894 119895 119896) minusHHH (119894 119895 119896)

10038161003816100381610038161003816

120575min = min 120575 (119894 119895 119896)

120575max = max 120575 (119894 119895 119896)

(11)


119879 (120572) =

119879 if 119880 lt 120572

119879120572 if 119880 ge 120572

119865 (120572) =

119865 if 119880 lt 120572

119865120572 if 119880 ge 120572

119880120572(119894 119895 119896) =

120575119879(119894 119895 119896) minus 120575

119879min

120575119879max

minus 120575119879min

(12)


119879120572(119894 119895 119896) =

1

119908 times 119908 times 119908

119903

sum

119897=119894

119904

sum

119898=119895

119905

sum

119899=119896

119879 (119897 119898 119899)

119865120572(119894 119895 119896) =

1

119908 times 119908 times 119908

119903

sum

119897=119894

119904

sum

119898=119895

119905

sum

119899=119896

119865 (119897 119898 119899)

120575119879(119894 119895 119896) =

1003816100381610038161003816100381610038161003816

119879 (119894 119895 119896) minus 119879 (119894 119895 119896)

1003816100381610038161003816100381610038161003816

119879 (119894 119895 119896) =

1

119908 times 119908 times 119908

119903

sum

119897=119894

119904

sum

119898=119895

119905

sum

119899=119896

119879 (119897 119898 119899)

(13)



(a) (b)

(c)





method



119877


OM = 2(

119873119879cap 119873119877

119873119879+ 119873119877

) (14)












200

150

100

100100

50

50

50150

150






119889119867 (119883 119884) = maxsup

119909isin119883

inf119910isin119884

119889 (119909 119910) sup119910isin119884

inf119909isin119883

119889 (119909 119910) (15)


150

150150

200

200

250

100

100

100

50

50

50

3D coronary tree




119867(119883 119884) 108 068






120587

(16)




(a) (b)



5 Conclusions




Acknowledgments


References






























of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of

















Rows

Columns

Slices

Original volume

LxLyLz

LxLyHz

LxHyLz

LxHyHz

HxLyLz

HxLyHz

HxHyLz

HxHyHz

LA darr

LA darr

LA darr

LA darr

LA darr

LA darr

LA darr

HA darr

HA darr

HA darr

HA darr

HA darr

HA darr

HA darr






119895(119896) in (2)





int |119878(119905)|2119889119905 =

infin

sum

119897=minusinfin

|119888(119897)|2+

infin

sum

119895=0

infin

sum

119896=minusinfin

10038161003816100381610038161003816119889119895(119896)

10038161003816100381610038161003816

2

(6)





was defined as

119864 (119862) = 1198622=

119901

sum

119894=1

1003816100381610038161003816119888119894

1003816100381610038161003816

2 (7)










HHH (119894 119895 119896) =

1

119908 times 119908 times 119908

119903

sum

119897=119894

119904

sum

119898=119895

119905

sum

119899=119896

HHH (119897 119898 119899) (8)

where

119903 = round(119894 +

119908

2

)

119904 = round(119895 +

119908

2

)

119905 = round(119896 +

119908

2

)

(9)


119879HHH (119894 119895 119896) =

HHH (119894 119895 119896) minusHHHmin

HHHmax minusHHHmin

119865HHH (119894 119895 119896) = 1 minus 119879HHH (119894 119895 119896)

119880HHH (119894 119895 119896) =


(10)

where

HHHmin = min HHH (119894 119895 119896)

HHHmax = max HHH (119894 119895 119896)

120575 (119894 119895 119896) =

10038161003816100381610038161003816HHH (119894 119895 119896) minusHHH (119894 119895 119896)

10038161003816100381610038161003816

120575min = min 120575 (119894 119895 119896)

120575max = max 120575 (119894 119895 119896)

(11)


119879 (120572) =

119879 if 119880 lt 120572

119879120572 if 119880 ge 120572

119865 (120572) =

119865 if 119880 lt 120572

119865120572 if 119880 ge 120572

119880120572(119894 119895 119896) =

120575119879(119894 119895 119896) minus 120575

119879min

120575119879max

minus 120575119879min

(12)


119879120572(119894 119895 119896) =

1

119908 times 119908 times 119908

119903

sum

119897=119894

119904

sum

119898=119895

119905

sum

119899=119896

119879 (119897 119898 119899)

119865120572(119894 119895 119896) =

1

119908 times 119908 times 119908

119903

sum

119897=119894

119904

sum

119898=119895

119905

sum

119899=119896

119865 (119897 119898 119899)

120575119879(119894 119895 119896) =

1003816100381610038161003816100381610038161003816

119879 (119894 119895 119896) minus 119879 (119894 119895 119896)

1003816100381610038161003816100381610038161003816

119879 (119894 119895 119896) =

1

119908 times 119908 times 119908

119903

sum

119897=119894

119904

sum

119898=119895

119905

sum

119899=119896

119879 (119897 119898 119899)

(13)



(a) (b)

(c)





method



119877


OM = 2(

119873119879cap 119873119877

119873119879+ 119873119877

) (14)












200

150

100

100100

50

50

50150

150






119889119867 (119883 119884) = maxsup

119909isin119883

inf119910isin119884

119889 (119909 119910) sup119910isin119884

inf119909isin119883

119889 (119909 119910) (15)


150

150150

200

200

250

100

100

100

50

50

50

3D coronary tree




119867(119883 119884) 108 068






120587

(16)




(a) (b)



5 Conclusions




Acknowledgments


References






























of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of




















was defined as

119864 (119862) = 1198622=

119901

sum

119894=1

1003816100381610038161003816119888119894

1003816100381610038161003816

2 (7)










HHH (119894 119895 119896) =

1

119908 times 119908 times 119908

119903

sum

119897=119894

119904

sum

119898=119895

119905

sum

119899=119896

HHH (119897 119898 119899) (8)

where

119903 = round(119894 +

119908

2

)

119904 = round(119895 +

119908

2

)

119905 = round(119896 +

119908

2

)

(9)


119879HHH (119894 119895 119896) =

HHH (119894 119895 119896) minusHHHmin

HHHmax minusHHHmin

119865HHH (119894 119895 119896) = 1 minus 119879HHH (119894 119895 119896)

119880HHH (119894 119895 119896) =


(10)

where

HHHmin = min HHH (119894 119895 119896)

HHHmax = max HHH (119894 119895 119896)

120575 (119894 119895 119896) =

10038161003816100381610038161003816HHH (119894 119895 119896) minusHHH (119894 119895 119896)

10038161003816100381610038161003816

120575min = min 120575 (119894 119895 119896)

120575max = max 120575 (119894 119895 119896)

(11)


119879 (120572) =

119879 if 119880 lt 120572

119879120572 if 119880 ge 120572

119865 (120572) =

119865 if 119880 lt 120572

119865120572 if 119880 ge 120572

119880120572(119894 119895 119896) =

120575119879(119894 119895 119896) minus 120575

119879min

120575119879max

minus 120575119879min

(12)


119879120572(119894 119895 119896) =

1

119908 times 119908 times 119908

119903

sum

119897=119894

119904

sum

119898=119895

119905

sum

119899=119896

119879 (119897 119898 119899)

119865120572(119894 119895 119896) =

1

119908 times 119908 times 119908

119903

sum

119897=119894

119904

sum

119898=119895

119905

sum

119899=119896

119865 (119897 119898 119899)

120575119879(119894 119895 119896) =

1003816100381610038161003816100381610038161003816

119879 (119894 119895 119896) minus 119879 (119894 119895 119896)

1003816100381610038161003816100381610038161003816

119879 (119894 119895 119896) =

1

119908 times 119908 times 119908

119903

sum

119897=119894

119904

sum

119898=119895

119905

sum

119899=119896

119879 (119897 119898 119899)

(13)



(a) (b)

(c)





method



119877


OM = 2(

119873119879cap 119873119877

119873119879+ 119873119877

) (14)












200

150

100

100100

50

50

50150

150






119889119867 (119883 119884) = maxsup

119909isin119883

inf119910isin119884

119889 (119909 119910) sup119910isin119884

inf119909isin119883

119889 (119909 119910) (15)


150

150150

200

200

250

100

100

100

50

50

50

3D coronary tree




119867(119883 119884) 108 068






120587

(16)




(a) (b)



5 Conclusions




Acknowledgments


References






























of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of

















(a) (b)

(c)





method



119877


OM = 2(

119873119879cap 119873119877

119873119879+ 119873119877

) (14)












200

150

100

100100

50

50

50150

150






119889119867 (119883 119884) = maxsup

119909isin119883

inf119910isin119884

119889 (119909 119910) sup119910isin119884

inf119909isin119883

119889 (119909 119910) (15)


150

150150

200

200

250

100

100

100

50

50

50

3D coronary tree




119867(119883 119884) 108 068






120587

(16)




(a) (b)



5 Conclusions




Acknowledgments


References






























of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of























200

150

100

100100

50

50

50150

150






119889119867 (119883 119884) = maxsup

119909isin119883

inf119910isin119884

119889 (119909 119910) sup119910isin119884

inf119909isin119883

119889 (119909 119910) (15)


150

150150

200

200

250

100

100

100

50

50

50

3D coronary tree




119867(119883 119884) 108 068






120587

(16)




(a) (b)



5 Conclusions




Acknowledgments


References






























of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of

















(a) (b)



5 Conclusions




Acknowledgments


References






























of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of


































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of
















research article coronary arteries segmentation based on...

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