Recovering the 3D Shape and Respiratory Motion of the Rib using Chest X-Ray Image

Transcription

1 694 MEDICAL IMAGING ECHNOLOGY Vol.20 No.6 Noveber 2002 Recoverng he 3D Shape and Respraory Moon of he Rb usng Ches X-Ray Iage Myn Myn Sen *, Msuru Kozu *2, Yosho Yanaghara *3 and Hrosu Haa *3 Absrac -hs aper presens a new ehod for he reconsrucon of he 3D shape and respraory oon of he rb fro he 2D X-ray ages. Each rb has been represened as a curve and s conrol pons are esaed by he cubc B-splne curve eraon algorh. o recover he 3D oon and he conrol pons of he rbs usng he 2D conrol pons, he facor decoposon algorh for projecve srucure and oon has been developed. he easureen arx has been bul wh he correspondng conrol pons of each curve. By eans of he conrol pons, nsead of all pons of a curve, daa capacy and copuaonal es are reduced. Furherore, he precson of achng pars of a rb n ulple ages s proved. he esaon errors are no appeared only due o he 2D conrol pon approxaon bu also he copuaon of he 3D conrol pons. he experen wh X-ray ages has been perfored and he effecveness of he proposed echnque has been confred hrough he resuls wh accepable errors. hs approach can be appled o exane he ches expanson for nspecng he pulonary dseases. Key words: Respraory oon of he rb, B-splne curve conrol pons, easureen arx, sngular value decoposon, horax, ches X-ray ages Med Iag ech 20(6): , Inroducon Analyss of ches wall oon s an poran research feld n he respraory physology. Recenly, he sudes of he respraory oveen n he 3D confguraon of he ches wall have been pad subsanal aenon by researchers. he ches or breas shape and poson wll be changed hrough he rb oon on he occason of he breahng relave o he conracon of he breahng uscle. In any research works C ages produced by he hgh resoluon and fas C scanners are anly used o deec he ches wall oveen. here are soe relaed approaches [-7] o generae and recover he oon of he ches wall and rbs. Cala e al. [] appled four V caeras o deec he 86 hesphercal reflecve arkers arranged crcuferenally on he ches wall. Grooe e al. [2] also used an auoac oon analyzer, he ELIE syse, o nvesgae he ches wall oon durng dal breahng. wo V caeras are used n her syse o record he 36 arkers. Jordanoglou [3] recovered he oon vecor of a rb. Accordng o hs, he rb roaes around only one axs and s shape doesn change durng breahng. he spaal vecors a several pons along a rb are parallel o one anoher and hey have a consan drecon on successve breahng cycles. Park e al. [5] analyzed he 3D rb oon for C ages. In hs paper a new ehod o recover he shape and respraory oon of he rbs s proposed. Each rb s consd- * Kehanna Huan Info. Councaon Research Laboraory (CRL) [Hkarda2-2-2, Seka-cho, Soraku-gun, Kyoo, , JAPAN] e-al:yn@crl.go.jp *2 he Deparen of Radology, Shga Unversy of Medcal Scence, Sea gesunowa-cho, Osu-sh, Shga, , JAPAN *3 Dvson of Physcal Elecroncs and Inforacs Engneerng, Graduae School of Engneerng, Osaka Cy Unversy, Sugoo, Suyos-ku, Osaka , JAPAN. receve: Deceber 0, 200. accep: Sepeber 2, 2002.

2 Med Iag ech Vol. 20 N0.6 Noveber ered as a curve and s 3D shape and oon are recovered fro he 2D X-ray ages. he conrol pons of each curve are esaed by he cubc B-splne curve approxaon algorh. In hs algorh he eraon ehod based on he learnng algorh [8,9] s eployed for nzng he approxaon errors. A curve whch well resebles he orgnal curve can be reconsruced fro hese conrol pons. hen, he correspondng pars of conrol pons of each curve are nvesgaed hrough he ages. he easureen arx has been bul by gaherng he correspondng conrol pons of each curve of he X-ray age fraes. he shape and oon of a sngle objec can be recovered fro he age srea under orhography [7]. In our prevous work [9] he 3D shape of he objec has been reconsruced by he sngular value decoposon approach. In hs work we copue no only he 3D conrol pons bu also he oon of each rb. he 3D curve has been refored by hese conrol pons. Average oon paraeer of he rb s approxaed fro he oon of each curve. Moreover, 3D oon and he se of conrol pons of all curves can be copued fro he se of correspondng conrol pons of all curves n successve age fraes. I s possble o nspec soe pulonary dseases fro he ches expanson observed by hs approach. hs paper s organzed as follows. he srucure of horax and he B-splne conrol pons approxaon algorh are nroduced n Secon 2. he deal of he proposed ehod based on he facorzaon algorh s dscussed n Secon 3. he experen and resuls are descrbed n Secon 4. Fnally, he conclusons and dscusson are presened n Secon Rb and B-splne conrol pons ) Srucure of he horax In hs Secon he srucure of he horax wll be consdered. Fgure llusraes he odel of horax whch s fored by he welve pars of lef - rgh rbs conneced wh he spne (backbone). Fgure 2 shows an exaple of an X-ray age of a ale subjec. he poson of each rb wll change due o he breahng. Fgure 3 presens he oveen of Fg. Model of horax (back). Fg. 2 An X-ray age. Fg. 3 he oveen of a rb beween wo ages. Fg. 4 Roaon axs of rb.

3 696 Med Iag ech Vol. 20 N0.6 Noveber 2002 rb 2 beween wo ages. Each rb ouches he backbone n wo places, whch are he rb head (capu cosae) and process ransverses, and each rb roaes cenerng on he axs whch passes hese places [5]. hen, he roaonal axes of lef- rgh rb are llusraed by he cross secon of horax n fgure 4. 2) Process he general process of our ehod for recoverng he shape and oon of he rb s shown n fgure 5. he X-ray ages are obaned by an X-ray radology devce durng breahng. 3) Conrol pons of a rb Each rb can be represened as a curve. A se of conrol pons needs o esae for represenng he feaure of he curve. B-splne conrol pons esaon algorh s used n hs work. A B-splne curve P() of degree n can be defned n ers of he se of conrol pons Q ( 0,,., n) wh he paraeer ( 0 ), n = 0 = P ( ) = B n ( ) Q, () where B n () s called Bernsen polynoal. By assung n =3, he cubc B-splne curve P () can be reduced no a arx for fro Eq. () as: P ) = [ B ( ) B ( ) B ( ) B ( )]Q, (2) ( where Q = [ Q ] 0 Q Q 2 Q 3, B 0 ( ) = ( ), B ( ) = + +, B 2 ( ) = + + +, B 3( ) = (0 ). 0 ( ) 6 B, B ( ), B 2( ) and B 3( ) vary wh he paraeer ( 0 ). In case when = 0, he pon P of he cubc B-splne curve can be deduced fro Eq. (2). 2 P = Q - + Q + Q + ( =,.., ), (3) where s he nuber of copued conrol pons of he nal conrol pons on he curve. For nsance, he B-splne conrol pons for a free curve are shown n fgure 6. o exane he conrol pons Q concernng wh he Fg. 5 he schee of he proposed ehod. Fg. 6 B-splne conrol pons for a curve.

4 Med Iag ech Vol. 20 N0.6 Noveber assued pons P on a free curve, frs he locaon of he pons P on he curve us be esaed. I s clear ha he frs and las conrol pons of he curve are he end pons of he curve. he second and hrd conrol pons of he B-splne curve are assued o be locaed a he angen lnes ade by he end pons of he curve. he locaons of pon P 2 and P 3 are assued by he locaon of he ( N )h and ( N N )h 5 5 pons of he curve by he learnng algorh [8]. Here, N s he oal nuber of pons n he curve. A he k -h eraon of learnng he conrol pon Q ( =,.., ) s, 3 k k k Q ( = P Q + Q + ). (4) 2 2 o oban he accurae conrol pons of a curve, he dfference δ of he conrol pons Q fro ( k -)h o k -h eraon s copued,.e., k δ = Q Q. (5) k For any open curve, he se of Q j ( j = 0,.., ) whch conansq ( =,.., ), Q 0 = Q and Q = Q - s sad o be he se of robus B-splne conrol pons when δ s suffcenly sall,.e., ax δ ε. Le us suppose he value of he nu perssble error ε be equal o n hs work. We decded hs value eprcally. If ax δ > ε, Q j s replaced asq = δ + Q, Q 0 = Q and Q = Q - and exane agan for he dfferen δ fro he ( k ) h o he k -h eraon. If he error s no sall enough hen he new conrol pons are assued o be near he old one. 3. Recoverng he 3D rb oon and conrol pons based on he facor decoposon In hs Secon, 3D B-splne conrol pons of each rb are copued fro he 2D curve of he horax by he facorzaon approach. horax conans welve pars of lef-rgh rbs. he B-splne conrol pons of each rb are esaed fro he 2D X-ray ages acqured durng breahng. he 2 n easureen arx B ( =,2,..., k ) s generaed by rackng he conrol pons of he k -h B-splne curve hrough he successve ages. Le us consder he easureen arx under he full perspecve projecon odel. hen B can be expressed n arx for: X B = ( =,2,..., k ) (6) Y where X and Y are he x and y coponen of he conrol pons, respecvely. he regsered easureen arx B ~ s derved fro he arx B relave o he objec cener coordnae syse and facorzed no hree arces as follows: ~ B = U Σ V, (7) where he egen vecors of B ~ B ~ ~ ~ and B B are he coluns of U and V, and hey sasfy U U = V V = VV = I, (8) where Σ s a dagonal arx of posve decreasng sngular values and he nuber of nonzero sngular values s relaed o he rank of B ~, and I s he un arx. Due o he nosy and unached pon easureens, B ~ wll be a os rank 4 n real suaons. And he dagonal arx Σ wll also conan ore han 4 nonzero sngular values. herefore, we pay aenon only he frs four coluns of U and he frs four rows ofv, and denoe he U ~ andv ~, respecvely. Σ ~ s also creaed by he frs four dagonal eleens of Σ. hs consderaon s ade for

5 698 Med Iag ech Vol. 20 N0.6 Noveber 2002 sngle objec. U ~, Σ and V ~ are also changed relang o he rank. he regsered easureen arx B ~ can be expressed as follows ~ B = RS, (9) where R represens he caera roaon and S represens he 3D poson of he objec wh respec o he objec cener. For he bes possble approxaon, Eq. (7) can be reduced as ˆ ~ ~ V ~ B = U Σ. (0) Le us defne ˆ ~ /2 R = U ( ~ Σ ), () ˆ /2 (Σ ~ = ) V. S hen we can rewre Eq. (0) as Bˆ = Rˆ Sˆ, (2) where Rˆ and Ŝ are he lnear ransforaon of he roaon and shape arces R and S, respecvely. he roaon and shape are copued by usng any nverble 4 4 arx A, R = RA ˆ, (3) - S = ( A) Sˆ. (4) o oban he exac soluon, A s defned as concaenaon of wo blocks, [ ] A = A A. (5) he 4 3 arx A s relaed o he roaonal coponen and he 4 arx A s relaed o ranslaon. he arx A s consraned by he orhonoraly of axs vecors F and j F. We have he syse of equaons: ˆ A ˆj A A A ˆ ˆj =, =, ˆ ˆ A A j = 0. he arx A can be obaned fro he above syse of equaons. o deerne he arx A, he cenrod of he se of conrol pons has been consdered as, X N = = = ˆ ˆ P, (7) B RS [ RA RA ] Y N where P = ( P ) = 0 s he cenrod of he objec. I follows fro Eq. (7) ha N B = RA ˆ, (8) and he arx A s, A Rˆ = ( Rˆ) Rˆ B. (9) Once he arx A has been found, he roaon and srucure of each curve are recovered fro Eq. (3), Eq. (4) and Eq. (9). he roaon arx R conans oon and ranslaon as R =, (20) 0 (6)

6 Med Iag ech Vol. 20 N0.6 Noveber r r r where = r r r r r r and = [ ] 2 3. he oon of he rb can be descrbed by he roaon angle and drecon concernng wh he roaon axs. hree characersc values can be reduced fro by he egen value decoposon. One value wll be a real value and ohers wo are coplex. he egen vecor relang o he un value s represened for he drecon of he roaon axs. And he roaon angle s copued fro wo coplex values. 4. Experens and resuls he ches X-ray ages are obaned fro he healhy ale subjecs by he radology devce durng respraon. able suarzes he curren perforance. he cener pxel of he age s (256,256). he dsance beween a fl plae and he radology devce s fxed o 00 c. he caera odel can be consdered as a weak-perspecve odel by he rao of he objec deph and dsance fro he X-ray devce. A PC wh Penu III 750 MHz processor and 768 MB eory s used n hs experen. Malab verson 6. whch s he hgh perforance nuerc copuaon and vsualzaon sofware s eployed o analyze he daa. Soe processng seps of curve exracon have been perfored wh Adobe Phooshop 6.0J. Rb regons are exraced fro he X-ray age by processng he ages. Due o he naure of he rb, he sooh shape of curve can be obaned fro he cenerlne and upper edge lne of he rb. Rb s dffcul o exrac separaely and rb 0, rb and rb 2 can be seen clearly n soe ages. So he rbs 2 o 9 are only consdered o exrac he lower pars of her edges. he sulaon of he B-splne conrol pon esaon for a curve has been done. he sulaon resuls are shown n fgure 7. For each curve of rbs, he se of conrol pons ncludng four conrol pons s exaned by he cubc B-splne conrol pons fndng algorh. A B-splne curve and s conrol pons for rb 2 are shown n fgure 8. he accuracy of he conrol pon approxaon for he rbs s shown n fgure 9. he average errors for he curve deecon by B-splne conrol pons are shown n able 2. And he 3D oon and B-splne conrol pons for each curve are copued based on he sngular value able. Perforance of he subjecs and X-Ray devce Nuber of subjecs 0 Average age 42 Frae rae Average deph of he body Dsance beween a fl plae and he radology devce Up o 9 fraes durng 0.3sec 7 c 00 c (a) (b) Fg 7 Sulaon resuls:(a)orgnal curve and B-splne curve wh conrol pons, (b)errors concernng wh he eraon es.

7 700 Med Iag ech Vol. 20 N0.6 Noveber 2002 decoposon ehod. he 3D recovered shapes of rb2, rb 3, rb4, rb 5, rb 6 and rb 9 are shown n Fg. 0. he average copuaon es for recoverng he 3D oon and he drawng of B-splne curve for each curve are 0.2 sec. he ax-roaon angle of he rbs for each subjec (S) and average oon angles of he rbs are llusraed n Fg. (a) and (b), respecvely. he effecveness of our proposed ehod was confred fro he experenal resuls. (a) (b) Fg. 8 B-splne curves and conrol pons for rb 2:(a) Orgnal curve and reconsruced curve of rb 2, (b) he curves of rb 2 n successve ages. Fg. 9 X-ray age wh B-splne curves. Fg.0 Recovered 3D shapes of soe rbs. able 2 Average error for he curve deecon. Max-ean square errors () Average-ean square errors () Lef rbs Rgh rbs Average

8 Med Iag ech Vol. 20 N0.6 Noveber Conclusons A new ehod for deecng he 3D shape and oon of he rb s descrbed n hs paper. Frs, he se of conrol pons for each curve (rb) s beng exraced fro he dfferen ages durng breahng. Because our ehod uses only he conrol pons nsead of all pons of a curve, daa aoun and copuaonal es wll be drascally reduced. Even hrough ore han 60 pons are ncluded n a curve, only four conrol pons for each curve are needed for copuaon. Furherore, he accuracy of achng of a rb n ulple ages s proved. Nex, he 3D oon and 3D conrol pons of each curve are esaed based on he facor decoposon ehod. hen he 3D shape of each rb has been refored fro hese 3D conrol pons. he average oon of he rbs has been deerned. hs ehod does no use a hree-densonal C age bu he 2D X-ray ages. As for he paen, s desred o nvesgae he breahng by sandng up and s no essenal o pu h on he bed. I s an advanage o nvesgae he breahng. hs approach can be appled o exane he ches expanson for nspecng he pleural dseases such as chronc obsrucve pulonary dsease (COPD), uberculoss, ankylosng spondyls, and adul respraory dsress syndroe and so on. I s also possble o analyze he odelng of he respraory syse. Our nex work s o prove he precson of he oon and he shape of a rb wh he ore conrol pons for each hgher order splne curves. And we nend o reconsruc he whole shape of he rb cage based on he ul-objec facorzaon approach under a perspecve projecon odel. Moreover, we would lke o nvesgae he breahng rae of abnoral people for coparng wh hese resuls. Acknowledgeen he auhors express her hanks o Mr. Yoshk Arakawa, Chef of Iage Processng Secon, Kehanna Huan Info-Councaons Research Cener CRL for helpful suggesons and coens o hs work. Auhors also apprecae o he voluneer ebers who pered o acqure he X-rays age n Shga Prefecure Unversy hospal for hs experen. (a) Roaon angles of each subjec (b) Average roaon angles. Fg. Roaon angle of rbs.

9 702 Med Iag ech Vol. 20 N0.6 Noveber 2002 References [] Cala S.J., Kenyon C.M., Ferrgno G. e al.: Check wall and lung volue esaon by opcal reference oon analyss, J. of Appled Physology, Vol. 8, No. 6, pp , 996. [2] De Groue A., Waner M., Cheron G. e al.: Ches wall oon durng dal breahng, J. of Appled Physology, Vol. 83(5), pp , 997. [3] Jordanogulou J: Vecor analyss of rb oveen, Respraon Physology, Vol. 0, pp , 970. [4] Wlson.A., Rehder K., Krayer S. e al.: Geoery and respraory dsplaceen of he huan rbs, J. of Appled Physology, Vol. 62(5), pp , 987. [5] Park Y., Kaoka H., Sao Y. e al. : 3D Analyss of he Respraory Rb Moon for C Iages, IEICE DII, Vol. J83 -DII, No. 8, pp , Aug [6] Ferrgno G, Carneval P, Alver A e al.: hree-densonal Opcal Analyss of Ches Wall Moon. J Appl Physol Vol. 77, No.3, pp , 994. [7] oas C. and Kanade.: Shape and oon fro age sreas under orhography: A facorzaon ehod, In. J. Copuer Vson, 9(2), pp , 992. [8] Bhuyan M. A., Sen M. M. and Haa H.: On a Free Curve Machng Mehod Algorh for Fndng he Conrol Pons Bezer Curves and Slary beween Free Curves, Proc. of he IASED In. Conf. Sgnal and Iage Processng, SIP98, Lar Vegar, USA, pp , 998. [9] Sen M.M. and Haa H.: Recoverng he 3D B-splne Conrol pons of he Free Curve for Shape Reforng, IEICE ranslaons on Inforaon and Syses, Vol. E84-D, No. 8, pp , Aug. 200.