Medical Image Segmentation of Blood Vessels Based on Clifford Algebra and Voronoi Diagram

Transcription

1 Medical Image Segmetatio of Blood Vessels Based o Clifford Algebra ad Vorooi Diagram Zhe Jie Huag, Guo Heg Huag*, Liag Lu Cheg College of Automatio, Guagdog Uiversity of Techology, Guagzhou510006, Guagdog, Chia. *Correspodig author. Tel: ; keviwogdh@126.com Mauscript submitted May 21, 2018; accepted Jue 5, doi: /jsw Abstract: First, a regio growig algorithm based o Clifford algebra is put forward to segmet blood vessels of abdomial aorta. Compared with the traditioal regio growig method, the proposed algorithm is easy to operate ad good at robustess. Furthermore, it is available to segmet both ormal ad abormal blood vessels of abdomial aorta: the seed poit ca keep o growig eve whe there are some isolated poits i regio of blood vessels. The key poit of this algorithm is that: the gray values of eight eighborhood poits are composed as uit feature vectors of curret seed poits, ad the Clifford algebra vector product represetatio will be applied. Secod, for the purpose to segmet thier global abdomial blood vessels rapidly, a ovel segmetatio algorithm of blood vessels based o 2D Vorooi diagram is put forward as a attempt. Experimets demostrate that it is potetial to segmet thier global abdomial blood vessels fast. I the future, this algorithm is potetial to segmet blood vessels i abdomial orgas. Key words: Medical image segmetatio, blood vessels, Clifford algebra, Vorooi diagram 1. Itroductio Recetly, virtual surgeo simulatio becomes more ad more importat to assist surgeo ad perform better therapy treatmet. Research o virtual surgeo simulatio geerally icludes extractio for the Regio of Iterest (ROI), 3D recostructio for medical images; surgeo simulatio ad so o. Therei, for the purpose to reduce the amout of data ad locate lesios, extractig the ROI is very importat, such as blood vessels, tumors ad other orgas segmetatio. There are may methods applied to segmet vessels from differet huma orgas. Segmetatio of vessels maily icludes segmetatio for liver vessels, retial vessels ad cholic vessels etc. May methods were proposed ito this field to segmet vessels of differet orgas. Therei, a algorithm based o a NN scheme for voxel classificatio was applied to segmet retial vessels [1]. However, it could oly be applied to 2D images but 3D image series. Eseault et al. proposed a fast ad fully automatic method based o geometric momets ad Graph Cut techique to segmet liver vessels [2]. However, this algorithm was ot effective to segmet thier vessels ad sesitive to iterpolatio of image slices. Bashar et al put forward a compoet-based structural measure amed Hole Area Idex (HAI) to isolate boe, soft-tissue structures ad segmet thier ad larger vessels [3]. However, pre-ehacemet was ecessary to icorporate some missig thier vessels, ad these ehacemets also ehaced udesired structures. Moreover, level set theory is oe of the most popular methods which are applied to vessels segmetatio. Therei, Hog et al put forward local biary fittig eergy i the hybrid level set method to extract local iformatio o MRA image more accurately, ad it could also be applied o 361 Volume 13, Number 6, Jue 2018

2 CT image [4]. Although it could solve the problem that traditioal hybrid level set could ot produce a satisfactory segmetatio for medical images with high itesity homogeeity, it was oly a solutio to segmet thier vessels. So far, oe of the above solutios could be successful to segmet vessels o low cotrast ad heterogeeous eviromets, especially for elimiatig boes aroud lager vessels. To our best kowledge, there are few methods so far which are applied for the global abdomial vessel extractio without the boes aroud vessels [3]. It is difficult to segmet vessels without boes. Therefore, we propose two ovel algorithms to segmet abdomial vessel. O oe had, 3D regio growig algorithm based o Clifford algebra vector product represetatio is put forward to segmet the blood vessels for abdomial aorta; o the other had, for the purpose to segmet thier global abdomial vessel without major boes ad orgas, aother ovel blood vessel segmetatio algorithm based o 2D Vorooi diagram is also put forward as a attempt. 2. Related Works 2.1. Clifford Algebra Let e0, e1..., e be a orthogoal basis i the liear space over R [5], the Clifford algebra A is a 2 2 associative algebra which is composed by e0, e1..., e, the e e, e e e ee e, e 1, i 1, 2,,. e e e e,1 i j. i j j i Elemets i 1 p 1 A are kow as the Clifford algebra, 0 0 i 0 0 i i i x e, x A, x have the form that 0 A A A A ( h,, h ), 1 h h,1 p, e e 1e 2 e, R. Obviously, A is 2-dimesioal associative p A h h hr A ad o-commutative algebra. If Clifford algebra x has the form x x0 xiei, x is called as Clifford vector. For is x A, Clifford mold of x is defied as xi i 1 x 2 ( ) If. x x e ad i i i 1 multiplicatio, i 1 i i x 2 ( A ) i 1. Especially, the mold of Clifford vector x y y e are two uit Clifford vector, i accordace with the rule of Clifford algebra xy ( x e )( y e ) x y ( x y x y ) e e (1) i i i i i i i j j i i j i 1 i 1 i 1 1 i, j Uder ormal circumstaces, the product of two -dimesioal Clifford vector is o loger a -dimesioal Clifford vector. I other words, the Clifford vector by the -dimesioal liear subspace Clifford multiplicatio is ot closed. x y is deoted by product of x ad y. 1 i, j ( x y x y ) e e i j j i i j, ad it is defied as the outer Theorem Suppose that q 1 ad q 2 are pure uit Clifford algebra, ad the compoets of q 1 ad q 2 are all o-egative. The q1 q2 q1q 2 q1 q Dimesio Vorooi Digram Dirichlet ad Vorooi firstly itroduced this cocept. This cocept was used to research the quadratic form: let quadratic matrix of data be as a ode, ad the scope of ode was reflected by the Euclidea orm. Later, this cocept was kow as Vorooi diagram [6]-[9]. Vorooi firstly studied this structure o the dual ature which was metioed by Descartes. He coected all adjacet odes through lies, ad got a triagular mesh fially. Subsequetly, Delauay studied the triagulatio of two-dimesioal poit set ad obtaied the same triagular mesh. The Delauay studied 362 Volume 13, Number 6, Jue 2018

3 two-dimesioal poit set triagulatio ad obtaied the same triagular mesh what Vorooi got before. Each edge of this triagle mesh was strictly limited what was give by Delauay: there is a roud that goes pass the edge of two edpoits, ad this circle does ot cotai the other poit. Vorooi ad Delauay s results proved the dual graph foud by Vorooi was the triagular mesh costructed by Delauay. Thereupo, the dual graph of Vorooi diagram was also amed as Delauay triagulatio. The priciple of 2D Vorooi diagram classificatio is that [10]: classify the pedig processed area i image by usig 2D, ad iterate util fid out the boudary of the image. I the case of 2D plae, plae is divided ito several areas refer to discrete poits of plae. Therei, there a discrete poit i each area which is a set that most early to the poit. It is defied as: For pi V, 1 2 d( p, x) ( p x ) ( p x ) 2 V { p, p, p}, the Vor( pi ) { x R d( x, pi ) d( x, p j), j i,1 j }. Therei, is the Euclidea distace betwee p p1, p2 ( ) ad x x, x. Besides, Delauay triagulatio ad Vorooi diagram is duality. There is a correspodig Delauay triagulatio to each give 2D Vorooi diagram. The specific relatioship is as follows: i 2D plae, the Vorooi vertex of each 2D Vorooi diagram correspods to a triagle i the Delauay triagulatio, ad this vertex is the circumceter of the triagle i the Delauay triagulatio; each edge i the 2D Vorooi diagram correspods to oe edge i the Delauay triagulatio, ad they will be vertical; each sub-structure for 2D Vorooi diagram correspods to a vertex i the Delauay triagulatio, ad this vertex is the Delauay triagulatio which correspods to the vertex of the sub-structure of Vorooi diagram. 1 2 (b) Fig. 1. 2D Vorooi digram: Delauay triagulatio of 16 poits; (b) correspodig Vorooi divisio. I additio, for image processig, the vertices of the 2D Vorooi diagram are the "skeleto" of a closed coected regio i the image plae. At preset, algorithms based o 2D Vorooi diagram eve 3D Vorooi diagram have bee applied i the field of image segmetatio, fluid simulatio ad so o. 3. Algorithms 3.1. Regio Growig Based o Clifford Algebra I order to go o growig i the blood vessels areas with isolated voxels whe we segmet the blood vessels, a ovel Regio Growig algorithm based o Clifford algebra is put forward to segmet more blood vessels of abdomial aorta. I accordace with Clifford algebra vector product represetatio theorem which was metioed at the previous sectio, if two uit vectors are equal whe ad oly whe their ier product value is 1; if two uit vectors are similar, their ier product will be closer to 1. I 3D data of CT image series, if two adjacet voxels are the same or similar, the ier product betwee the eight-dimesioal feature vectors composed by the gray values of their eight eighborhoods should be equal to 1 or ear to 1. Sum up, the procedure of Regio Growig algorithm based o Clifford algebra is: Step 1, iput k pieces of m cotiguous CT series images (3D volume data, size is m k), ad select the iitial seed poits i target areas. 363 Volume 13, Number 6, Jue 2018

4 Step 2, calculate the gray values of eight eighborhoods of curret seed poit, ad they will compose a eight-dimesioal vector ad be uitized. The the eight-dimesioal uit vector is defied as the eight-dimesioal uit feature vector of curret seed poit, it represets the feature of curret seed poit. The defiitio of eight eighborhoods i Regio Growig algorithm based o Clifford algreba is as Fig. 2. Fig. 2. The defiitio of eight eighborhoods of Regio Growig based o Clifford algebra. Step 3, calculate the uit eight-dimesioal feature vectors of six eighborhoods of curret seed poits, ad fid the eighborhoods whose ier products of feature vector with curret seed poit withi a give iterval which is close to 1 (i this paper, the give iterval is [0.995,1] ). The these eighborhoods will be icorporated ito the curret voxel regio of seed poit, ad become a ew seed poit. Iterate is over whe eighborhood which fulfills the rule caot be foud agai. The workflow chart of this algorithm is as Fig. 3. Etry Iput CT image series.; select iitial seed poits. Calculate Eight-dimesioal feature vectors of curret seed poits The eighborhoods will be the ew seed poits. Yes Is there eighborhood of curret seed poits? Is there ier product i give iterval? Yes, calculate Feature vectors of curret seed poits; feature vector ier product betwee seed poits ad their eighborhoods No No Iteratio is over; output the result. Exit Fig. 3. The flow chart of regio growig based o Clifford algebra. 364 Volume 13, Number 6, Jue 2018

5 Differet from traditioal Regio Growig algorithms which just cosider the gray value of the curret poits, Regio Growig algorithm based o Clifford algebra fully cosiders the feature of eighborhoods of curret poits Vessesl Segmetatio based o 2D Vorooi Digram It has bee lackig a feasible way to segmet the global thier abdomial blood vessels (thier blood vessels are defied as o-aortic vessels whose size are less tha or equal to the size of the third layer of abdomial aorta). I additio, the segmetatio of thier o-aortic abdomial blood vessels is very importat to surgeo of pacreas, splee ad other abdomial orgas. I CT series, it icludes three mai types of the thier abdomial blood vessels: horizotal vessels, diagoal vessels ad vertical vessels. The thier horizotal blood vessels would be rough thi lies i each CT slice; the thier vertical blood vessels would be thi poits i pre CT slice; the thier diagoal blood vessels would be regarded as the combiatio of thi lies ad thi poits i each CT slice. Therefore, at the premise of beig deoised, if thi lies ad poits i each slice ca be segmeted that meas it is possible to segmet the thier abdomial blood vessels. Combied with the Vorooi diagram theory which was metioed previously, a ovel segmetatio algorithm of blood vessels based o 2D Vorooi diagram is desiged specifically for segmetatio of the global thier abdomial blood vessels. The whole procedure will be: Step 1, import the origial CT series oe by oe, ad the Threshold segmetatio is applied as pre-segmetatio, the remaiig are all orgas, blood vessels ad boes. The, import the origial CT series oe by oe agai, ad the traditioal Regio Growig is applied to segmet abdomial aorta. Fially, subtract the results of Threshold segmetatio ad Regio Growig. The remaiig will be voxels of boes, orgas ad thier abdomial blood vessels. Step 2, cosider the edge voxels as samplig poits, ad the iput them as the iitial vertices of algorithm based o two-dimesioal Vorooi diagram. The, execute this algorithm ad the output result is the set of Vorooi vertices. For each closed iterval (ier vessels regio), there exist a cetral axis which coect to the vertices of each Vorooi diagram, ad it ca be see as "skeleto" of two-dimesioal blood vessel images. It is possible that a samplig poit is the itersectio of eclosed areas, so it may correspod to more tha oe axis. Defie the set of cetral axis which correspods to samplig poit P as Q, ad the defie the size of two-dimesioal image by usig the shortest distace from P to Q. Step 3, set the threshold radius, ad the samplig whose distace to cetral axis does ot exceed the threshold will be retaied. Durig actual operatio, the desity of samplig poits is high, so the samplig poit to the distace correspodig to two-dimesioal Vorooi vertex set ca be approximated as the distace betwee the samplig poit ad the cetral axis set. Whe the distace does ot exceed the radius is to keep the samplig poits. After this operatio, the samplig poit set is expressed as R. Step 4, calculate the Delauay triagulatio of R. The triagulatio impacts iside ad outside of the eclosed regio. As the result of a eclosed area iside is eeded, it is ecessary to remove the exteral triagle regio. Through observatio, the triagles i the triagulatio of outside regio, the shape mostly is the sleder-shaped obtuse triagle; however, more acute triagle is iside the eclosed regio, eve if there is a obtuse triagle, its circumcircle radius must ot exceed the radius. Therefore, to calculate the radius of all the circumcircle of the triagle, Threshold filterig, it ca remove the triagular area outside the eclosed area, ad results of the etire algorithms are obtaied fially. 365 Volume 13, Number 6, Jue 2018

6 (b) (c) (d) (e) Fig. 4. The whole process of segmetatio algorithm based o 2D Vorooi diagram: origial image; (b) iitial result of traditioal threshold segmetatio; (c) result of traditioal regio growig; (d) subtractig result betwee results of threshold segmetatio ad regio growig; (e) the fial result of the proposed segmetatio algorithm based o 2D Vorooi diagram. Etry Iput CT image. Threshold segmetatio Subtract ad obtai No abdomial vessels, boes ad orgas Threshold segmetatio Deaulay triagulatio is applied to R; output the result. Exit Threshold radius is give; samplig poits whose distace are ot exceed the threshold radius are retaied as the set R. Fig. 5.The flow chart of Vessel Segmetatio based o Vorooi diagram. I additio, because of the complexity of the eclosed area (several small holes i the iteral of large area), Deaulay triagulatio made mistakes o segmetig part of the backgroud. To solve this problem, sample the voxels iside the triagle by the color iformatio to determie whether it belogs to the foregroud. I egieerig, bitwise operatios are applied to combie with the fillig after Delauay triagulatio with the origial image. 4. Experimetal Results All experimets will be performed i Microsoft Widows7. I this thesis, all experimetal data for 366 Volume 13, Number 6, Jue 2018

7 Abdomial CT images were provided by the Pearl River Hospital from Souther Medical Uiversity, ad the size of each slice of images is The experimetal eviromet is: CPU is Itel Core i7-3517u 1.90GHz CPU, ad RAM is 4GB Regio Growig Based o Clifford Algebra Segmet S50 (365 slices) ad S70 (320 slices) these two sets of data with abdomial aorta respectively by regio growig based o cofidet iterval ad Clifford algebra, ad compare the results. First, segmet the abdomial aorta for S50 ad S70 by the two algorithms, ad the select four iitial seed poits. The iitial seed poits is distributed as Fig. 6. (b) Fig. 6. The distributio of iitial seed poits o S50 ad S70: the iitial seed poits o slice 183rd of S50; (b) the iitial seed poits o slice 161st of S70. The segmetatio results for abdomial aorta from S50 by usig traditioal regio growig based o cofidet iterval ad the proposed segmetatio algorithm based Clifford algebra are show by Fig. 7. (b) (c) (d) (e) (f) Fig. 7. The segmetatio result of abdomial aorta from two algorithms o S50: & (d) the origial images o slice 182d ad 205th; (b) & (e) results from regio growig based o cofidet iterval; (c) & (f) results from the proposed method based o Clifford algebra. 367 Volume 13, Number 6, Jue 2018

8 The segmetatio results for abdomial aorta from S70 by usig regio growig based o Cofidet iterval ad Clifford algebra are show by Fig. 8. (b) (c) (d) (e) (f) Fig. 8. The segmetatio result of abdomial aorta from three algorithms o S70: & (d) the origial images o slice 109th ad 199th; (b) & (e) results from regio growig based o cofidet iterval; (c) & (f) results from the proposed method based o Clifford algebra. The 3D recostructio results o S50 ad S70 which are obtaied after Regio Growig algorithms based o Cofidet iterval ad Clifford algebra are show respectively by Fig. 9 ad 10. (b) Fig. 9. The 3D recostructio results of two algorithms o S50: result from regio growig based o cofidet iterval; (b) result from the proposed method based o Clifford algebra. (b) Fig. 10. The 3D recostructio results of two algorithms o S70: result from regio growig based o cofidet iterval; (b) result from the proposed method based o Clifford algebra. 368 Volume 13, Number 6, Jue 2018

9 The, abdomial o-aorta from S50 with oly oe iitial seed poit will be segmeted by traditioal Regio Growig, Regio Growig based o Cofidet iterval ad Clifford algebra. The iitial seed poit ad results of segmetatio are show by Fig. 11. (b) (c) (d) (e) (f) (g) (h) (i) Fig. 11. Visual compariso o S50: seed poit of S50, slice 1st; (b) & (f) the origial images; (c) & (g) results from traditioal regio growig; (d) & (h) results from regio growig based o cofidet iterval; (e) & (i) results from the proposed method based o Clifford algebra. The 3D recostructio results o abdomial o-aorta from S50 which are extracted after Regio Growig algorithms based o Cofidet iterval ad Clifford algebra are show by Fig. 12. (b) (c) Fig. 12. The 3D recostructio result of three algorithms: result from traditioal regio growig; (b) result from regio growig based o cofidet iterval; (c) result from the proposed method based o Clifford algebra. Afterwards, performace aalysis of segmetatio of Regio Growig algorithm based o traditioal Cofidet iterval ad Clifford algebra o S50 (365 slices, four seed poits), S70 (320 slices, four seed poits) ad S50 (oe seed poit) will be show as the below tables. Table 1. Segmetatio Summary for S50 Part1 Method Rutime Aalysis Cofidet iterval s Fewer vessels were segmeted; boes ad orgas were ot segmeted. Clifford algebra s More vessels were segmeted; boes ad orgas were segmeted. 369 Volume 13, Number 6, Jue 2018

10 Table 2. Segmetatio Summary for S50 Part2 Method Rutime Aalysis Regio growig 68.34s Fewest vessels were segmeted. Cofidet iterval 99.02s Fewer vessels were segmeted. Clifford algebra s Most vessels were segmeted. Table 3. Segmetatio Summary for S70 Method Rutime Aalysis Cofidet iterval s Fewer vessels were segmeted. Clifford algebra s More vessels were segmeted, but part of liver was segmeted. 3D regio growig algorithm based o Clifford algebra vector product represetatio are proposed by this paper, ad experimetal performace data ca be see from the above tables: whe traditioal Regio Growig algorithm ad Regio Growig algorithm based o Cofidece iterval are applied to segmeted abdomial aorta, if it grows to the isolated voxels betwee the regios of vessels, it may ot keep o growig. However, it is ot oly that the curret voxels' gray values are cosidered by 3D regio growig algorithms based o Clifford algebra, but also the eighborhoods of the curret voxels are cosidered. Therefore, o oe had, if there is oly oe breakpoit i the vessel regios, the seed poits ca cotiue to grow ad segmet the blood vessels by these two methods; o the other had, comparig with algorithm based o Clifford algebra, though more layers of abdomial aorta are segmeted by the algorithm based o Clifford algebra tha traditioal Regio Growig, some of o-iterestig regios were be extracted such as other orgas or boes. The reaso is too may eighborhoods are cosidered to compose a feature vector of the seed poits to blur the differeces betwee poits. The seed poits would grow to o-iterestig regios of differet qualities Vessesl Segmetatio Based o 2D Vorooi Digram Compared with traditioal Threshold segmetatio, blood segmetatio algorithm based o 2D Vorooi diagram is applied to segmet thier abdomial blood vessels from first 80 slices of S50 ad S70. The segmetatio results are show by Fig. 13 ad 14. (b) (c) (d) (e) (f) Fig. 13. The segmetatio results of S50 i its first 80 slices: & (d) the origial images; (b) & (e) results from traditioal threshold segmetatio; (c) & (f) results from the proposed segmetatio based o 2D Vorooi diagram. 370 Volume 13, Number 6, Jue 2018

11 (b) (c) (d) (e) (f) Fig. 14. The segmetatio results of S70 i its first 80 slices: & (d) the origial images; (b) & (e) results from traditioal threshold segmetatio; (c) & (f) results from the proposed segmetatio based o 2D Vorooi diagram. The 3D recostructio of the segmetatio results of these algorithms are show by Fig. 15 ad 16. (b) Fig. 15. The recostructio of thier blood vessels of abdomial o-aorta o the first 80 slices of S50. (b) Fig. 16. The recostructio of thier blood vessels of abdomial o-aorta o the first 80 slices of S70. It ca be see from two previous figures: o oe had, more of thier blood vessels of abdomial 371 Volume 13, Number 6, Jue 2018

12 o-aorta excludig mostly boes ad soft tissue by algorithm based o 2D Vorooi diagram, but there will be more discrete poits i the segmetatio results; o the other had, Threshold segmetatio may ot oly segmet fewer thi blood vessels, but also o-iterested orgas. Compared with the 3D Regio Growig algorithm based o Clifford algebra, poorer segmetatio results are got by algorithm based o 2D Vorooi diagram. It eeds to select may iitial seed poits to segmet the global vessels that may waste a lot of time to compute. However, the thier blood vessels of abdomial o-aorta ca be segmeted rapidly by algorithm based o 2D Vorooi diagram. This method is potetial to be applied i cliical practice. Method Rutime Aalysis Threshold 14.25s 2D Vorooi diagram s Table 4. Segmetatio Summary for S50 Fewer thi vessels were segmeted. Orgas ad boes were segmeted. There was ot isolated voxel. More thi vessels were segmeted. Orgas ad boes were almost ot segmeted. But there were may isolated voxels. Table 5. Segmetatio Summary for S70 Algorithm Rutime Aalysis Threshold 18.94s Almost o thi vessel was segmeted. 2D Vorooi diagram s More thi vessels were segmeted. Orgas ad boes were almost ot segmeted. But there were may isolated voxels. A ovel segmetatio algorithm based o 2D Vorooi diagram is proposed to segmet more thi blood vessels of abdomial o-aorta excludig the boes ad soft tissue. However, there will be may discrete poits are segmeted by this algorithm. It is due to that this algorithm is a 2D algorithm that cosiders each slice oe by oe, but cosiders the relatioship betwee each slice. However, Vorooi diagram theory has bee promoted to 3D, thus blood segmetatio algorithm based o 3D Vorooi diagram is potetial to solve this problem i the future. 5. Coclusio Research o the segmetatio of the abdomial blood vessels is focused o 3D medical images. First, existig segmetatio methods applied to abdomial aorta segmetatio are ot good eough. Therefore, a ew algorithm based o hypercomplex-3d regio growig algorithm based o Clifford algebra is applied to segmet abdomial arterial vessels. Compared with traditioal regio growig, Hessia matrix ad geetic algorithm etc., this algorithm is easy to operate ad good at robustess. Therefore, it is available to segmet both ormal ad abormal blood vessels of abdomial aorta: the seed poit ca go o growig, eve whe there are some isolated poits i regio of blood vessels. However, compared with traditioal Regio Growig algorithms, this algorithm is ot fast eough i computig, thus the future work will be focused o algorithm optimizatio or GPU acceleratio. Secod, it has bee lackig a algorithm which ca segmet thier global abdomial blood vessels rapidly. A ovel segmetatio algorithm of blood vessels based o 2D Vorooi diagram is put forward as a attempt. Experimets demostrate that it is valid to segmet thier global abdomial blood vessels by usig this algorithm. However, the algorithm is still i the ifacy, ad has the followig shortcomigs: first, there are some isolated voxels i the segmetatio result. It is all because that segmetatio algorithm of blood vessels based o 2D Vorooi diagram is a 2D method, ad it does ot cosider the relatioship betwee the adjacet slices. Secod, the segmetatio result depeds partly o the pre-segmetatio result from Threshold ad Regio Growig. I future work, we will cotiue to study the blood vessels segmetatio algorithm based o 3D Vorooi diagram. The future work will be to cosider the relatioship betwee the adjacet slices ad remove discrete breakpoits. Besides, this algorithm will also be applied to segmet blood vessels i 372 Volume 13, Number 6, Jue 2018

13 abdomial orgas. Ackowledgmet This work was sposored by Natioal Nature Sciece Foudatio of Chia (grat umber ), Natioal Key R&D Pla of Chia (grat umber 2016YFC ). Refereces [1] Mari, D., Aquio, A., Gegudez-Arias, M. E., & Bravo, J. M. (2011). A ew supervised method for blood vessel segmetatio i retial images by usig gray-level ad momet ivariats-based features. IEEE Trasactios o Medical Imagig, 30, [2] Eseault, S., Lafo, C., & Dilleseger, J. L. (2010). Liver vessels segmetatio usig a hybrid geometrical momets/graph cuts method. IEEE Trasactios o Biomedical Egieerig, 57, [3] Bashar, M. K., Mori, K., & Kobayashi, T. J. (2010). Automatic segmetatio of abdomial blood vessels from cotrasted X-ray CT images. Proceedigs of the 2010 Iteratioal Coferece o Electrical ad Computer Egieerig (pp ). [4] Hog, Q., Li, Q., & Tia, J. (2010). Local hybrid level-set method for MRA image segmetatio. Proceedigs of the 2010 IEEE 10th Iteratioal Coferece o Computer ad Iformatio Techology. (pp ). [5] Gog, Y., Leog, I. T., & Qia, T. (2009). Two itegral operators i Clifford aalysis. Joural of Mathematical Aalysis ad Applicatios, 354, [6] Liu, W., & Li, X. A ew octoio method for edge detectioof color image. Preprited. [7] Okabe, A., Boots, B., & Sugihara, K. (1992). Spatial tessellatios: Cocepts ad applicatios of Vorooi diagrams: Joh Wiley & Sos, Ic. [8] Aurehammer, F. (1991). Vorooi diagrams ad mdash, a survey of a fudametal geometric data structure. ACM Comput. Surv., 23, [9] Reem, D. (2009). A algorithm for computig vorooi diagrams of geeral geerators i geeral ormed spaces. preseted at the Proceedigs of the 2009 Sixth Iteratioal Symposium o Vorooi Diagrams. [10] Vorooi_diagram. Retrieved from Zhejie Huag was bor i Guagdog provice, Chia, i He graduated i automatio from Guagdog Uiversity of Techology i He is a master s studet i school of automatio of Guagdog Uiversity of Techology. His curret research is about digital image processig, deep learig ad computer visio. Guoheg Huag was bor i Guagdog provice, Chia, i He received PhD degree i Macau Uiversity. He is a lecturer ad teaches i Guagdog Uiversity of Techology sice His research iterests iclude deep learig, computer visio ad machie leaig. Liaglu Cheg was bor i Hubei provice, Chia, i 1964, is a Prof. ad doctoral supervisor of Faculty of Automatio, Guagdog Uiversity of Techology. He received PhD degree i Huazhog Uiversity of Sciece ad Techology. His curret research is about computer visio, image data compressio etc. 373 Volume 13, Number 6, Jue 2018