An Efficient Method for Deformable Segmentation of 3D US Prostate Images

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1 An Effcent Method for Deformable Segmentaton of 3D US Prostate Images Yqang Zhan 1,2,3, Dnggang Shen 1,2 1 Sect. of Bomedcal Image Analyss, Dept. of Radology, Unversty of Pennsylvana, Phladelpha, PA dnggang.shen@uphs.upenn.edu 2 Center for Computer-Integrated Surgcal Systems and Technology, Johns Hopkns Unversty, Baltmore, MD 3 Dept. of Computer Scence, Johns Hopkns Unversty, Baltmore, MD yzhan@cs.jhu.edu Abstract. We prevously proposed a deformable model for automatc and accurate segmentaton of prostate boundary from 3D ultrasound (US) mages by matchng both prostate shapes and tssue textures n US mages[6]. Textures were characterzed by a Gabor flter bank and further classfed by support vector machnes (SVM), n order to dscrmnate the prostate boundary from the US mages. However, the step of tssue texture characterzaton and classfcaton s very slow, whch mpedes the future applcatons of the proposed approach n clnc applcatons. To overcome ths lmtaton, we frstly mplement t n a 3-level mult-resoluton framework, and then replace the step of SVMbased tssue classfcaton and boundary dentfcaton by a Zernke momentbased edge detector n both low and mddle resolutons, for fast capturng boundary nformaton. In the hgh resoluton, the step of SVM-based tssue classfcaton and boundary dentfcaton s stll kept for more accurate segmentaton. However, SVM s extremely slow for tssue classfcaton as t usually needs a large number of support vectors to construct a complcated separaton hypersurface, due to the hgh overlay of texture features of prostate and non-prostate tssues n US mages. To ncrease the effcency of SVM, a new SVM tranng method s desgned by effectvely reducng the number of support vectors. Expermental results show that the proposed method s 10 tmes faster than the prevous one, yet wthout losng any segmentaton accuracy. 1 Introducton Prostate cancer contnues to be the second-leadng cause of cancer death n Amercan men [1]. As transrectal ultrasound (TRUS) mages have been wdely used for the dagnoss and treatment of prostate cancer, the accurate segmentaton of the prostate from TRUS mages plays an mportant role n many clncal applcatons [1]. Accordngly, a number of automatc or sem-automatc segmentaton methods have been proposed. Ghane et.al. [2] and Hu et.al. [3] desgned 3D dscrete deformable models to sem-automatcally outlne the prostate boundares. Shao et al [4] proposed a level set method to detect the prostate n the 3D TRUS mages. Gong et al [5] provded a G.-Z. Yang and T. Jang (Eds.): MIAR 2004, LNCS 3150, pp , Sprnger-Verlag Berln Hedelberg 2004

2 104 Y. Zhan and D. Shen Bayesan segmentaton algorthm, based on deformable superellpses model, to segment 2D prostate contours. We prevously proposed a statstcal shape model to segment the prostate from 3D TRUS mages by matchng both prostate shapes and tssue textures n TRUS mages [6]. The effectveness of our prevous method s manly resulted from the jont use of two novel technques, () a Gabor flter bank used for 3D texture features extracton and () support vector machnes (SVM) used for texture-based tssue classfcaton. However, both of these two technques are computatonally very expensve, thereby mpedng the fast segmentaton of prostates from 3D TRUS mages. To overcome ths lmtaton, ths paper presents an effcent segmentaton approach, whch s mplemented n a 3-level mult-resoluton framework and s further speeded up by two technques respectvely desgned for dfferent resolutons. In both low and mddle resolutons, a Zernke moment-based edge detector s used to replace the step of SVM-based tssue classfcaton and boundary dentfcaton, for fast boundary detecton. In the hgh resoluton, a new SVM tranng method s desgned to mprove the effcency of SVMs by reducng the number of support vectors, whch are ntally requred to construct a very complcated separaton hypersurface for the classfcaton of the hghly confoundng prostate and non-prostate tssues n TRUS mages. By usng these technques, our approach for prostate segmentaton s hghly speeded up, yet wthout losng any segmentaton accuracy. 2 Methods Our prevous deformable shape model [6] uses both statstcal shape nformaton and mage texture nformaton to segment the prostate boundary. Its success n prostate segmentaton results from the texture analyss, whch dstngushes prostate and nonprostate tssues from nosy TRUS mages. However, the two technques employed n texture analyss,.e., a Gabor flter bank for texture characterzaton and SVMs for texture-based tssue classfcaton, are both computatonally very expensve. For example, t takes about 40 mnutes to segment a prostate from a 256x256x176 TRUS mage, usng SGI workstaton wth a 500MHz processor. Therefore, t s necessary to speed up the segmentaton approach. For fast segmentaton, we frstly formulate the segmentaton approach n a 3-level mult-resoluton framework, whch has been wdely used to ncrease the speed of the algorthms n the lterature [7,8]. For example, the orgnal TRUS mage s decomposed nto three mult-resoluton mages,.e. the mage of orgnal sze and the mages down-sampled by factors 2 and 4. The surface model s ntalzed at the lowest resoluton, and subsequently deforms to the prostate boundary. The segmentaton result n the lower resoluton s up-sampled to the next hgher resoluton, and used as ntalzaton of the deformable model n the hgher resoluton. These steps are terated untl the deformable model converges to the prostate boundary n the hghest resoluton. Besdes the mult-resoluton framework desgned above, two effectve methods,.e., Zernke moment-based edge detector and a new tranng method for generatng effcent SVMs, are partcularly desgned to speed up the segmentaton approach n

3 An Effcent Method for Deformable Segmentaton of 3D US Prostate Images 105 low resolutons and hgh resoluton, respectvely. The detals of these two technques are descrbed next. 2.1 Zernke Moment Based Edge Detecton As dscussed above, although a Gabor flter bank [9] s capable of extractng robust and rch texture features, t s computatonally very expensve due to the use of Gabor flters at multple scales and orentatons. Addtonally, as texture features are regonbased features, prostate tssues n the down-sampled TRUS mages usually have less dstngushed texture features, compared to those n the orgnal mages. Therefore, boundary nformaton, drectly computed from the ntenstes, s better than texture nformaton for gudng the deformable model n both low and mddle resolutons. Zernke moment-based edge detector has been proposed n [10]. It has three advantages n edge detecton. Frst, as Zernke moments are ntegral-based operators, t s nose tolerant, whch s especally mportant for detectng prostate boundares n the nosy TRUS mages. Second, as detaled next, ths edge detecton method provdes a more complete descrpton of the detected edges than the tradtonal edge detector, e.g. Canny edge detector. Thrd, as only three masks,.e., two real masks and one complex mask, are requred to get the edge features of each voxel, t s computatonally more effcent than the Gabor flter bank whch used 10 masks [6]. Zernke moment operator projects the mage data onto a set of complex polynomals, whch form a complete orthogonal set over the nteror of a unt crcle. For an mage f ( x, y), ts Zernke moment of order n and repetton m can be defned as: Z nm n + 1 = π 2 2 x + y 1 f ( x, y) V * nm ( ρ, θ ) dxdy (1) where V R ( ρ, θ ) = R ( ρ) e jmθ nm nm ( n m ) / 2 s n 2s n+ m n m nm ( ρ ) = [( 1) ( n s)! ρ ] [ s!( s)!( s)!], and (,θ ) s= ρ are the polar coordnates of (x,y). Consderng an deal step edge (c.f. Fg 1), ts mportant features nclude the step heght k, the background gray level h, the perpendcular dstance from the center of the crcular kernel l, and the angle of edge wth respect to the x-axs φ. All these features can be mathematcally represented by three low order Zernke moments (Z, Z, Z ) as: ϕ = tan [Im( Z11) Re( Z11)] (2a) l = Z 20 Z 11 (2b) h = k = 3Z l (2c) 2 3 / (1 ) 1 2 ( Z00 kπ / 2 + k sn ( l) + kl 1 l ) π (2d)

4 106 Y. Zhan and D. Shen jϕ where Z =, and Re(.) and Im(.) represent the real part and the magnary part 11 Z11e of a complex value, respectvely. Smlarly, Zernke moments can be used to measure the general edges n the 2D mage by usng eq. (2). z Coronal Plane l φ k k Cor φ Cor h Axal Plane x φ Ax k Ax y Fg. 1. A 2D deal step edge model. Fg. 2. Schematc explanaton of usng two 2D edge vectors to roughly reconstruct a 3D edge vector. Notably, rather than extendng the 2D Zernke moment to 3D, we smply apply two orthogonal 2D Zernke moment operators, whch respectvely le on the axal plane and the coronal plane (c.f. Fg 2), to get two sets of edge features for each voxel Ax v,.e., Ax Ax Ax Cor { h ( v), k ( v), l ( v), ϕ ( v)} and Cor Cor Cor { h ( v), k ( v), l ( v), ϕ ( v)}. As shown n Fg 2, two 2D edge vectors can be roughly consdered as two projectons (black dashed arrows) of a 3D edge vector (black sold arrow) n the axal and the coronal planes, respectvely. (Edge vector s a vector whose magntude and drecton represent the edge strength and the normal drecton of the edge, respectvely.) Thus, the 3D edge vector,.e., e(v), can be represented by two 2D edge vectors as follows: Ax Ax Ax Ax Cor T e ( v) = k ( v) [cos( ϕ ( v)),sn( ϕ ( v)),sn( ϕ ( v)) tan( ϕ ( v))] (3) In our prevous segmentaton approach [6], an energy functon s defned on each vertex P of the deformable surface model, and t s used to evaluate the matchng degree of the deformable model wth the prostate boundares n the TRUS mages. The energy functon conssts of two energy terms,.e., the external energy, whch drves the deformable model to the prostate boundary, and the nternal energy, whch preserves the geometrc regulaton of the model durng deformaton. By jontly mnmzng these two terms, the deformable model s able to converge to the prostate boundares. In ths study, the external energy s re-formulated such that the edge features captured by Zernke moments are employed to gude the deformable segmentaton, whle the nternal energy remans the same. Accordngly, for each vertex P, ts external energy s defned as: Ext E P = w E P + w E P + w E P (4) where E ( P ) E Str Int ( ) ( ) ( ) ( ) Str Str Dst Dst ( ) ( ) = < n( P ), e( v) > 1 E v N P v N P Ax Cor P = ( h v + h v ) 2 H P ( ) ( ) v N P v N P ( ) ( ) ( ) Int Int, Ax Cor ( P ) mn( l ( P ), l ( P )) Dst ( ) = and

5 An Effcent Method for Deformable Segmentaton of 3D US Prostate Images 107 There are three tems n Eq. (4). The frst tem denotes the ntegrated edge strength n the sphercal neghborhood of P,.e., N(P ). Notably, the edge strength s projected to the normal drecton of the deformable surface at P, n(p ), by the nner product < >. The second tem denotes the dstance from P to the boundary. The thrd tem requres that the deformable model converges only to the boundary wth the ntensty smlar to the learned average ntensty, H(P ), whch s captured for vertex P from a set of tranng samples. By jontly mnmzng these three tems, the deformable model s thus drven to the prostate boundares n both low and mddle resolutons of TRUS mages. 2.2 A Tranng Method for Increasng the Effcency of SVM Zernke moment-based edge detector s able to detect prostate boundares n the low and the mddle resolutons, however, t s not effectve n accurately delneatng pros- sub-surface. In the tate boundares n the hgh resoluton as the prostate boundares are usually blurred by speckle nose n the orgnal TRUS mages. Accordngly, we stll use the statstcal texture matchng method [6], whch conssts of texture characterzaton by a Gabor flter bank and texture-based tssue classfcaton by SVMs, for prostate segmentaton n the hgh resoluton stage of our mult-resoluton framework. In our method, a set of SVMs are employed for texture-based tssue classfcaton [6]. Each of them s attached to a sub-surface of the model surface and traned by the manually-labeled prostate and non-prostate samples around that testng stage, the nput of the SVM s a feature vector, whch conssts of Gabor features extracted from the neghborhood of a voxel, and the output denotes the lkelhood of the voxel belongng to the prostate. In ths way, the prostate tssues are dfferentated from the surroundng ones. However, snce the Gabor features of TRUS prostate mages vary greatly across the ndvduals and ther dstrbuton s hghly overlapped between prostate and non-prostate regons, the traned SVM usually has a huge number of support vectors. Ths s because () a large number of the support vectors, locatng at the margns, are requred to construct a hghly convoluted hypersurface, n order to separate two classes; () even the hghly convoluted separaton hypersurface has been con-structed, qute a lot of confoundng samples are stll msclassfed and thus selected as other support vectors, locatng beyond the margns. Notably, ths huge number of support vectors wll dramatcally ncrease the computatonal cost of the SVM. Therefore, t s necessary to desgn a tranng method to decrease the number of support vectors of the fnally traned SVM, by smplfyng the shape of the separaton hypersurface. The basc dea of ths tranng method s to selectvely exclude some tranng samples, thereby the remanng samples are possble to be separated by a less convoluted hypersurface. Snce the support vectors determne the shape of the separaton hypersurface, they are the best canddates to be excluded from the tranng set, n order to smplfy the shape of the separaton hypersurface. However, excludng dfferent sets of support vectors from the tranng set wll lead to dfferent smplfcatons of the separaton hypersurface. Fg 3 presents a schematc example n the 2-dmensonal feature space, where we assume support vectors exactly locatng on the margns. As shown n Fg 3(a), SVM traned by all the samples has 10

6 108 Y. Zhan and D. Shen support vectors, and the separaton hypersurface s convoluted. Respectve excluson of two support vectors, SV 1 and SV 2, denoted as gray crosses n Fg 3(a), wll lead to dfferent separaton hypersurfaces as shown n Fgs 3(b) and 3(c), respectvely. SVM n Fg 3(b) has only 7 support vectors, and ts hypersurface s less convoluted, after re-tranng SVM wth all samples except SV 1. Importantly, two addtonal samples, denoted as dashed crcle/cross, were prevously selected as support vectors n Fg 3(a), but they are no longer selected as support vectors n Fg 3(b). In contrast, SVM n Fg 3(c) stll has 9 support vectors, and the hypersurface s very smlar to that n Fg 3(a), even SV 2 has been excluded from the tranng set. Fg.3. Schematc explanaton of how to selectvely exclude the support vectors from the tranng set, n order to effectvely smplfy the separaton hypersurface. The sold and dashed curves denote the separaton hypersurfaces and ther margns, respectvely. The crcles and the crosses denote the postve and the negatve tranng samples, whch are dentcal n (a), (b) and (c). The tranng samples locatng on the margns are the support vectors. The reason of SVM n Fg 3(b) beng more effcent than that n Fg 3(c) s that the excluded support vectors SV 1 contrbutes more to the convoluton of the hypersurface. For each support vector, ts contrbuton to the convoluton of hypersurface can be approxmately defned as the generalzed curvature of ts projecton pont on the hypersurface. For example, for SV 1 and SV 2 n Fg 3(a), ther projecton ponts on the hypersurface are J 1 and J 2. The curvature of the hypersurface at pont J 1 s much larger than that at pont J 2, whch means the support vector SV 1 has more contrbuton to make the hypersurface convoluted. Therefore, t s more effectve to flatten the separaton hypersurface by excludng the support vectors, lke SV 1, wth ther projecton ponts havng the larger curvatures on the hypersurface. Accordngly, the new tranng method s desgned to have the followng four steps. Step 1. Use all the tranng samples to tran an ntal SVM, resultng n l 1 ntal In support vectors { SV, = 1,2,..., l1} and the correspondng decson functon d 1. Step 2. Exclude the support vectors, whose projectons on the hypersurface have the largest curvatures, from the tranng set: 2a. For each support vector SV In, fnd ts projecton on the hypersurface, P(SV In ), along the gradent of dstance functon d 1.

7 An Effcent Method for Deformable Segmentaton of 3D US Prostate Images 109 2b. For each support vector SV In, calculate the generalzed curvature of P(SV In ) on the hypersurface, c(sv In ). 2c. Sort SV In n the decrease order of c(sv In ), and exclude the top n percentage of support vectors from the tranng set. Step 3. Use the remanng samples to retran the SVM, resultng n l 2 support vec- Re tors { SV, = 1,2,..., l2} and the correspondng decson functon d 2. Notably, l 2 s usually less than l 1. Re Re Step 4. Use the l 2 pars of data ponts { SV, d 2 ( to fnally tran the SVRM SV )} (Support Vector Regresson Machne) [12], resultng n l 3 fnal support vectors Fl { SV, = 1,2, l3} and the correspondng decson functon d 3. Notably, l 3 s usually less than l 2. Usng ths four-step tranng algorthm, the effcency of the traned SVMs wll be hghly enhanced wth very lmted loss of classfcaton rate, whch wll be shown n the frst experment. Notably, as n the statstcal texture matchng method, the matchng degree of the deformable model wth the prostate boundares s defned n a nose tolerant fashon [6], a lttle loss of classfcaton,.e., a lttle number of ms-classfed voxels, wll not nfluence the segmentaton accuracy, whle the segmentaton speed s greatly ncreased. 3 Expermental Results The frst experment s presented to test the performance of the proposed tranng method n ncreasng the effcency of SVMs. We frstly select prostate and non- labeled TRUS mages samples from one prostate samples from sx manually mage are used as testng samples, whle samples from other fve mages are used as tranng samples. Each sample has 10 texture features, extracted by a Gabor flter bank [9]. We use our method to tran a seres of SVMs by excludng dfferent percentages of support vectors n Step 2c of our tranng method. The performances of these SVMs are measured by the number of support vectors fnally used and the number of correct classfcatons among 3621 testng samples. As shown n Fg 4(a), after excludng 50% of ntally selected support vectors, the fnally-traned SVM has 1330 support vectors, whch s only 48% of the support vectors (2748) ntally selected n the orgnal SVM; but ts classfcaton rate stll reaches 95.39%. Compared to 96.02% classfcaton rate acheved by orgnal SVM wth 2748 support vectors, the loss of classfcaton rate s relatvely trval. If we want to further reduce the computatonal cost, we can exclude 90% of ntally selected support vectors from the tranng set. Our fnally-traned SVM has only 825 support vectors, whch means the speed s trple, and t stll has 93.62% classfcaton rate. To further valdate the effect of our traned SVM n prostate segmentaton, the SVM wth 825 support vectors (denoted by the whte trangle n Fg 4(a)) s appled to a real TRUS mage for tssue classfcaton. As shown n Fgs 4(b1) and 4(b2), the result of our traned SVM s not nferor to that of the orgnal SVM wth 2748 support vectors (denoted by the whte square n Fg 4(a)), n terms of dfferentatng prostate tssues from the surroundng ones.

8 110 Y. Zhan and D. Shen In the second experment, the proposed segmentaton approach s appled to segment prostates from sx real 3D TRUS mages. A leave-one-out valdaton method s used,.e., each tme fve mages are used for tranng, and the remanng one s used for testng. The sze of 3D mages s 256x256x176, wth the spatal resoluton 0.306mm. Fg 5(a) shows the mult-resoluton deformaton procedure on one of the TRUS m-ages. The whte contours, labeled as LF, MF and HF, denote the fnally de-formed models n the low, mddle and hgh mages, respectvely. Notably, the models n both low and mddle resolutons are guded by the Zernke momentbased edge detector, whle the model n the hgh resoluton s guded by the statstcal texture matchng method. The algorthm-based segmentaton result s compared to the hand-labeled result n Fg 5(b). Moreover, Table 1 gves a quanttatve evaluaton of ths comparson to all the sx TRUS mages. From both vsual results and quanttatve analyss, we can conclude that our automated segmentaton method s able to segment the prostate from nosy TRUS mages. Importantly, usng a SGI workstaton wth 500MHz processor, the average runnng tme for segmentng a prostate s 4 mnutes, whch s 10 tmes faster than our prevous method [6]. Performance of Our Traned SVM vs Percentage of Intal Support Vectors Excluded from the Tranng Set 4000 Num of Correct Classfcaton (among 3621 testng samples) Num of Support Vectors (96.02%) 3454 (95.39%) 3390(93.62%) % 10% 20% 30% 40% 50% 60% 70% 80% 90% Percentage of Intal Support Vectors Excluded from the Tranng Set (a) (b1) Fg. 4. (a) The performance of the fnally-traned SVM changes wth the percentages of ntal support vectors excluded from the tranng set. (b) Comparsons of tssue classfcaton results usng (b1) the orgnal SVM wth 2748 support vectors and (b2) our traned SVM wth 825 support vectors. The tssue classfcaton results are shown only n an ellpsodal regon and mapped to 0~255 for the dsplay purpose. (b2)

9 An Effcent Method for Deformable Segmentaton of 3D US Prostate Images 111 Table 1. Comparson of the algorthm-based segmentaton and the hand-labeled segmentaton on sx real TRUS mages. Subjects Average Dstance Overlap Volume Volume (Voxels) Error (%) Error (%) Image Image Image Image Image Image Mean Stand. Devaton Concluson We have proposed an effcent segmentaton approach for fast segmentaton of prostates from 3D TRUS mages. Our segmentaton approach was formulated as a mult- and t was speeded up by two technques, respectvely deresoluton framework, sgned for dfferent resolutons. In both low and mddle resolutons, Zernke moment- based edge detector s used to replace the step of SVM-based tssue classfcaton and boundary dentfcaton, for fast capturng boundary nformaton for deformable segmentaton. In the hgh resoluton, a new tranng method has been desgned to ncrease the effcency of the fnally traned SVM for texture-based tssue classfcaton, thereby equally ncreasng the effcency of texture matchng step n deformable segmentaton procedure. Compared to our prevous segmentaton method [6], the proposed one s 10 tmes faster n segmentng 3D prostate from TRUS mages, yet wthout losng any segmentaton accuracy. Fg. 5. (a) A typcal mult-resoluton deformaton procedure. The contour denotes the model on a selected slce of the TRUS mage. The contour n the mage I s the ntalzed model n the low resoluton. The contours n the mages LF MF and HF denote the fnally deformed models n the low, mddle and hgh resoluton mages. (b) Vsual comparsons between algorthm-based and hand-labeled segmentaton results. The whte contours are the hand-labeled results, whle the dashed ones are the algorthm-based segmentaton results.

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