RECONSTRUCTION OF 3D TUBULAR STRUCTURES FROM CONE-BEAM PROJECTIONS

Transcription

1 RECONSTRUCTION OF 3D TUBULAR STRUCTURES FROM CONE-BEAM PROJECTIONS Ja L Dept. of Elect. and Comp. Engrg. Oakland Unversty Rochester, MI 48309, USA Laurent Cohen CEREMADE Unversty Pars - Dauphne Pars cede 16, France ABSTRACT A new method of reconstructng three-dmensonal (3D) tubular structures from cone-beam projectons s proposed n ths paper. Mnmal path method s appled to detect tubular structures n the cone-beam projectons. The etracted nformaton are fused to desgn a weght functon over the 3D volume so that mnmal path method can be appled agan to etract centerlnes of 3D tubular structures. The valdty of the proposed weght functon reles on the sparsty of 3D mages contanng tubular structures. The procedure of estmatng the radus of 3D tube from projectons s also descrbed. The proposed method requres nether cone-beam reconstructon, nor tree matchng, one of whch s usually necessary n the estng reconstructon methods. The method s robust to small motons between dfferent projectons. Inde Terms Tubular structures, cone-beam, mnmal path 1. INTRODUCTION Reconstructon of three dmensonal (3D) tubular structures from multple cone-beam projectons has mportant applcatons n medcal mage analyss. For eample, the etracton of 3D coronary arteres n the analyss of X-ray coronary angographes [1], and the ntracerebral vascular reconstructon n neurosurgeon practce [2], are both concerned wth reconstructng 3D blood vessels from conebeam projectons. Usng a dgtal flat panel, cone-beam projectons are usually obtaned n a rotatonal acquston mode wth a fed step n rotatonal angle. A few projectons can be selected from the whole sequence for the reconstructon task. For eample, n coronary arteres reconstructon, the projectons correspondng to the same cardac tme are selected such that a statc 3D reconstructon s possble. Moton estmaton can be performed after the statc reconstructons at dfferent cardac tme are obtaned. We are manly concerned wth statc reconstructon from a few projectons n ths paper, so the ssue of moton estmaton wll not be dscussed, although t s one of the man tasks n quanttatve coronary analyss. The estng reconstructon methods can be classfed nto two categores accordng to whether t requres 3D volume reconstructon or not. Methods n the frst category usually perform conebeam mage reconstructon, then apply 3D segmentaton to etract the tubular structures [3]. For general 3D cone-beam reconstructon, most common non-teratve algorthms are based on the backprojecton formula proposed by Feldkamp [4]. Although fltered back-projecton reconstructon s faster than teratve reconstructon, the resulted mages usually contan sever artfacts due to the over smplfed lne-ntegral model. Iteratve cone-beam reconstructons The author performed the work whle takng sabbatcal leave at CERE- MADE n the Unversty of Pars - Dauphne n clam better reconstructon qualty, but suffer from hgh computatonal cost. After 3D volume s reconstructed, tubular structures can be etracted usng mnmal path method [5]. L and Yezz proposed to represent 3D tubular structure as a 4D curve to ncorporate radus nformaton such that a global mnmzng 4D path can be a complete soluton to the reconstructon task [6]. The methods that do not requre 3D cone-beam reconstructon usually frst perform segmentaton n each projecton to etract centerlnes, then perform tree matchng over the trees from dfferent projectons to reconstruct 3D tubular structures. For eample, n [2], Bulltt et al addressed the specfc problem of reconstructon of 3D curves from a par of curves n the presence of error. In [1], Blondel et al performed multocular matchng to buld correspondences between the centerlnes of dfferent projectons. The approach developed by Jandt s group also falls nto ths category [7]. When processng projectons, each pel s assgned a rank to ndcate ts lkelhood of beng nsde a blood vessel. Then rankng functons of dfferent vewng angles are pared to form a weght functon of the volume that s used n 3D centerlne etracton. However, the ssue of radus estmaton s not addressed n Jandt s work. In ths paper, we present a reconstructon method, whch etracts both the centerlnes and rad of 3D tubular structures. The method does not requre cone-beam reconstructon, so t belongs to the second category dscussed above. centerlnes n each projecton are frst etracted usng mnmal path method. Then the dstance map correspondng to the dstance between pels and centerlnes s computed for each projecton. A 3D weght functon s obtaned by fusng the dstance maps of all the projectons. Fnally 3D centerlnes are reconstructed from the 3D weght functon va mnmal path method, and rad at dscrete centerlne ponts are estmated from the sze of tubular structures. Our contrbuton to the feld s the ntroducton of a new weght functon for mnmal path segmentaton, whch eplots the sparsty of tubular structure mages, as well as a low cost radus estmaton procedure. The remander of ths paper s organzed as follows. In Secton 2, mnmal path method and tubular structure detecton based on optmally orented flu are brefly revewed. Secton 3 detals the 3D weght functon desgn and radus estmaton. In Secton 4, the epermental results wth smulated coronary data sets are presented and dscussed. We conclude the paper n Secton MINIMAL PATHS AND TUBULAR STRUCTURE DETECTION 2.1. Mnmal Path Mnmal path, also called geodesc, s a path connectng a startng pont and an endng pont that mnmzes the total cost accumulated along the path. Let : [0, 1] Ω be a smooth curve, the cost

2 functonal can be regarded as a weghted length of and epressed as L() = Z 1 0 W((t)) (t) dt (1) where (t) s the dervatve of, and W s a weght functon defned on the doman Ω that can vary wth applcatons. Usng ths settng, the soluton to the mnmal path problem s a global mnmzer of the weghted length, = argmn L() (2) P( s, e) where s and e represent the startng and endng ponts of the path, and P( s, e) s the set of all the paths between s and e. In computer vson communty, mnmal path method has been appled to mage segmentaton, especally the centerlne etractons of tubular structures, such as roads and blood vessels. For ths knd of applcaton, weght functon s usually desgned such that the regon nsde the tubular structures has a small value relatve to the background or other objects. Therefore the optmzer prefers the path passng through tubular structures to the solutons that pass through background. To solve mnmal path problem, mnmal dstance map assocated wth the startng pont s must be computed. The mnmal dstance map U s () s a functon over the mage doman Ω, whose value equals the weghted length of the mnmal path connectng ponts s and,.e. U s () = mn L() (3) P( s,) where P( s, ) s the set contanng all the possble paths between s and, and Ω. U s () satsfes Ekonal equaton U s = W, whch can be solved by dfferent schemes after dscretzaton. An effcent non-teratve algorthm for solvng the equaton s fast marchng algorthm [8]. We have appled fast marchng algorthm to compute mnmal dstance map n both the and 3D centerlne etractons. Once the mnmal dstance map s computed, mnmal path can be obtaned by gradent descent startng from the endng pont e. Fg. 1 shows a cone-beam projecton and the centerlnes etracted from t. that the desred features can be represented by the mnmzng curve. In case of tubular structure segmentaton, the weght functon W should ncorporate the nformaton from tubular structure detecton. Such knd of detecton ams at detectng the presence of tubular structures as well as estmatng local orentaton and sze of the structure. The detectors that have been proposed nclude Hessan based flters and optmally orented flu (OOF). Hessan based flters have dffcultes when there are other objects adjacent to the tubular structure. To overcome ths drawback, Law and Chung ntroduced OOF, whch estmated the structure orentaton by fndng a projecton as on whch the projected gradent flu s mnmzed [9]. The advantage of OOF s that ts performance s not dsturbed by the adjacent objects. In [6], L and Yezz proposed to represent a tubular surface as the envelope of a famly of spheres wth contnuously changng center ponts and rad, and developed two dfferent 4D weght functons whch can ensure the sphere wth the desred radus has lower weght than those spheres wth naccurate rad. The centerlne etracton n our approach has adopted a verson of L and Yezz s method, whch allows smultaneous etracton of centerlne and radus. A drawback of the method s that t requres user nputs of start ponts and end ponts of tubular structures. The key pont detecton technque [10] can be ncorporated to remove the requrement of end ponts, but hasn t been mplemented by ths tme. 3. FROM CONE-BEAM PROJECTIONS TO 3D WEIGHT FUNCTION The frst step n our approach s to process each cone-beam projecton to etract centerlnes and estmate local thckness of the projected tubular structures. After that, 3D centerlne reconstructon and radus estmaton are done sequentally. As mentoned n Secton 1, our weght functon for 3D centerlne etracton s computed from centerlnes n projectons. So the focus of ths secton s to desgn a 3D weght functon usng the etracted nformaton. Table 1 lsts the notatons that wll be used n our dscusson. Fg. 2 llustrates the physcal meanngs of these notatons. Notatons S proj 3D d Table 1. Notatons Descrptons a pont n the 3D feld of vew radaton source locaton of the -th projecton the -th projecton operator centerlnes of the 3D tubular structure centerlnes n the -th projecton, equvalent to proj ( 3D ) Eucldean dstance map assocated wth, defned over the -th projecton (a) A cone-beam projecton (b) Etracted centerlnes Fg. 1. A cone-beam projecton and the assocated centerlnes Tubular Structure Detecton When applyng mnmal path method to solve mage segmentaton problem, the dffcult task s to desgn a weght functon W such 3.1. Weght Functon Desgn Mnmal path method reles on proper desgn of weght functon to obtan meanngful soluton. Based on the nformaton etracted from projectons, we are nterested n formng a 3D weght functon sutable for mnmal path method to etract centerlnes of 3D tubular structures. Due to the nature of mnmal path, the desred weght functon should have small values nsde the tubular structure, and lowest value on the centerlnes.

3 S S j 3D proj( ) S D 3D j A B j Fg. 2. Illustratons of notatons. For each projecton mage, we frst compute the dstance map assocated wth the etracted centerlnes. Usng the -th projecton as an eample, the map s defned as a pel s Eucldean dstance to,.e. d (proj ()) = mn y proj () y, (4) where proj represents the -th projecton operator, and s the centerlne etracted from the -th projecton. Based on the dstance functon d, we propose a weght functon W() = ma(d (proj ())), (5) for 3D centerlne etracton. We clam that W() has the followng propertes: 1., W() D, W() = The set X 0 = { : / 3D andw() = 0} s very small f the 3D volume contanng tubular structures s sparse. By sparse mage, we refer to mages where background pels/voels are domnant. For eample, n the X-ray coronary angography, the pels n vessel regons are about 5-10% of all the pels n a projecton. In terms of the 3D volume to be reconstructed, the voels n vessel regons are about % of the total number of voels. Proof 1. Ths property s nherted from the nature of the dstance functon d. 2. 3D, proj () W() = 0., d (proj ()) = 0 3. W() = 0 mples that proj (),. If / 3D, proj (), s an event that s unlkely to occur due to the sparsty of the mage, especally when the number of projectons used n reconstructon s larger than 2. For a pont / 3D, the event proj () belongs to occurs f and only f belongs to the surface formed by S and 3D. We can assume the probablty of ths event s less than a small constant c for all due to the sparsty of the volume. The value of c vares wth the level of sparsty. In Fg. 3. Estmaton of 3D tubular structure s radus. smulated data that we have tested, c s less than 0.1. It s also reasonable to assume that the event that proj () belongs to s ndependent to the event that proj j () belongs to j, for j. Let K be the number of projectons used n reconstructon, the probablty that proj (), or W() = 0 s less than c K. In other words, the sze of the set X 0 = { : / 3D andw() = 0} s less than N N y N z c K, where N, N y and N z are the sze of the three dmensons of the volume. We use Fg. 2 to llustrate the argument above. Fg. 2 shows two 3D ponts, 3D and / 3D. Because s n the lne formed by S j and, proj j ( ) equals proj j (), and belongs to j, whch mples d (proj j j ( )) = 0. However, due to the sparsty, s not n any lne formed by S and a pont n 3D. So proj ( ) does not belong to, whch mples d (proj ( )) > 0 and W( ) > 0. These propertes of W() can assure that t acheves mnmum value along 3D and larger values out of 3D, whch subsequently assures the accurate etracton of 3D centerlnes of the tubular structures n sparse mages. The estence of the set X 0 usually does not have an mpact to the 3D centerlne etracton because X 0 / 3D. As long as the startng pont provded by the user s n 3D, the ponts n X 0 wll not be etracted by the mnmal path solver. In real practce, a small postve constant ǫ s added to W() to prevent loops n the path Radus Estmaton We estmate the rad of 3D tubular structure from the rad etracted n projectons. Fg. 3 shows the general relatonshp between the thckness of the 3D tubular structure and the radus of ts projectons. S s the locaton of radaton source. For a pont n the 3D centerlne 3D, ts projecton s on the centerlne. A and B are two boundary ponts that form a segment passng through and perpendcular to. The three ponts, S, A and B, defne a plane that ntersects the 3D tube. Let the radus of the tube at be r. The dstance d(s, ) s usually much larger than the value of r. So lne SA and lne SB boundng the tube at are near parallel. Under the assumpton of parallelsm, the shortest dstance between the two lnes at, represented by D, can be regarded as the dameter of the tube. The value of D can be calculated from the locaton of S,, A and B. If the

4 (a) Projecton 1 (b) Projecton 2 (c) Projecton 3 (d) Projecton 4 Fg. 4. Cone-beam projectons used n reconstructon. We present the results of tubular structure reconstructon usng the proposed method n ths secton. The method has been tested over smulated data to quantfy the accuracy of the results. To generate the smulated data, blood vessels were constructed from bnary 3D coronary trees and embedded n a volume mage. Fg. 5 (a) shows the smulated coronary arteres. The volume mage was projected to produce a seres of cone-beam projectons of sze to smulate rotatonal angography. The projecton operator s a dstance drven operator. Four projectons, wth rotatonal angle at 0, 30, 60 and 90, have been selected as the nput to the reconstructon algorthm. They are shown n Fg. 4. After etractng the centerlnes n these projectons, the 3D weght functon s computed and fed to mnmal path method to etract 3D centerlnes. The result s shown n Fg. 5 (b). We quantfy the accuracy of mnmal path soluton usng a smple metrc. Let 3D = {p 1, p 2,..., p M} be the true centerlne and = {v 1, v 2,..., v K} be the mnmal path soluton. For each pont v, we fnd ts two nearest neghbors n 3D and compute the dstance between v and the lne formed by the two nearest neghbors. The resulted dstance s regarded as the error of at v. The error of s obtaned by takng average of ths error over the whole curve. It should be noted that curve smoothness, whch s also an mportant factor n fdelty quantfcaton, cannot be measured wth ths metrc. To test the robustness of the proposed method, random translatons have been added to the vertcal drecton of the four projectons to smulate the moton caused by mperfect synchronzaton n cardac tme. A standard Gaussan was sampled and multpled wth dfferent values of σ to generate dfferent levels of random motons. Table 2 shows a snapshot of such generated motons n vertcal drecton. Applyng the lsted motons, we got fve mnmal path solutons correspondng to σ = 0,0.5, 1, 2,4, separately. The errors of these solutons are recorded at the bottom of Table 2. Fg. 6 compares one of the 3D centerlnes etracted n the eperment of σ = 4 wth the ground truth. As ndcated n Table 2, the error of the mnmal path soluton equals pels when σ = 4. These results demonstrate that the proposed method s not very senstve to small motons n vertcal drecton. Table 2. Smulated dfferent levels of random motons and reconstructon errors n pels. (a) Smulated coronary arteres (b) Etracted centerlnes Motons n vertcal drecton (pel) σ = 0 σ = 0.5 σ = 1 σ = 2 σ = 4 Proj Proj Proj Proj Error Fg. 5. Reconstructon of smulated 3D coronary trees. assumpton of parallelsm doesn t hold, r can stll be estmated va a general procedure, whch nvolves a lttle bt more computaton. 4. RESULTS Fg. 6. One of the 3D centerlnes etracted when σ = 4. The blue dots represent the ground truth, whle the red lne s the mnmal path soluton. 5. CONCLUSION In ths paper, a new method for statc reconstructon of 3D tubular structures from cone-beam projectons has been presented. The sparsty of tubular structures s eploted to desgn a 3D weght functon

5 for the etracton of centerlnes by mnmal path method. The feasblty of the method has been evaluated over smulated data sets. The results show that the method can acheve farly good accuracy when the cone-beam projectons are dsturbed by small motons n the vertcal drecton. Current verson of the method requres both start ponts and end ponts of the tubular structure as nputs. In the future, a mnmal path method wth key pont detecton technque can be mplemented to remove the requrement for end ponts. 6. REFERENCES [1] Chrstophe Blondel, Grégore Malandan, Régs Vallant, and Ncholas Ayache, Reconstructon of coronary arteres from a sngle rotatonal -ray projecton sequence, IEEE Trans. on Medcal Imagng, vol. 25, no. 5, pp , 6. [2] E. Bulltt, A. Lu, and S. M. Pzer, Three-dmensonal reconstructon of curves from pars of projecton vews n the presence of error. I. Algorthms, Med. Phys., vol. 24, no. 11, pp , [3] Mehua L, Haquan Yang, and Hroyuk Kudo, An accurate teratve reconstructon algorthm for sparse objects: applcaton to 3D blood vessel reconstructon from a lmted number of projectons, Phys. Med. Bol., vol. 47, pp , 2. [4] L.A. Feldkamp, L.C. Davs, and J.W. Kress, Practcal conebeam algorthm, J. Opt. Soc. Am. A, vol. 1, no. 6, pp , [5] L.D. Cohen and T. Deschamps, Segmentaton of 3D tubular objects wth adaptve front propagaton and mnmal tree etracton for 3D medcal magng, Comput. Methods Bomech. Bomed. Eng., vol. 10, no. 4, pp , 7. [6] Hua L and Anthony Yezz, Vessels as 4-D curves: Global mnmal 4-D paths to etract 3-D tubular surfaces and centerlnes, IEEE Trans. on Medcal Imagng, vol. 26, no. 9, pp , 7. [7] Uwe Jandt, Drk Schäfer, Mchael Grass, and Volker Rasche, Automatc generaton of 3D coronary artery centerlnes usng rotatonal -ray angography, Med. Image Anal., vol. 13, pp , 9. [8] J. A. Sethan, A fast marchng level set method for monotoncally advancng fronts, Proc. of the Natonal Academy of Scences, vol. 93, no. 4, pp , [9] Ma W. Law and Albert C. Chung, Three dmensonal curvlnear structure detecton usng optmally orented flu, n ECCV 8: LNCS 5305, Berln Hedelberg, 8, pp , Sprnger-Verlag. [10] Fethallah Benmansour and Laurent D. Cohen, Fast object segmentaton by growng mnmal paths from a sngle pont on or 3D mages, Journal of Mathematcal Imagng and Vson, 8.