Algorithm for image reconstruction in multi-slice helical CT

Transcription

1 Algorihm for image reconsrucion in muli-slice helical CT Kasuyuki Taguchi a) and Hiroshi Aradae Medical Engineering Laboraory, Toshiba Corporaion, 1385 Shimoishigami, Oawara, Tochigi , Japan Received 7 February 1997; acceped for publicaion 14 January 1998 Effors are being made o develop a new ype of CT sysem ha can scan volumes over a large range wihin a shor ime wih hin slice images. One of he mos promising approaches is he combinaion of helical scanning wih muli-slice CT, which involves several deecor arrays sacked in he z direcion. However, he algorihm for image reconsrucion remains one of he bigges problems in muli-slice CT. Two helical inerpolaion mehods for single-slice CT, 36LI and 18LI, were used as saring poins and exended o muli-slice CT. The exended mehods, however, had a serious image qualiy problem due o he following hree reasons: 1 excessively close slice posiions of he complemenary and direc daa, resuling in a larger sampling inerval; 2 he exisence of several disconinuous changeovers in pairs of daa samples for inerpolaion; and 3 he exisence of cone angles. Therefore we have proposed a new algorihm o overcome he problem. I consiss of he following hree pars: 1 opimized sampling scan; 2 filer inerpolaion; and 3 fan-beam reconsrucion. Opimized sampling scan refers o a special ype of muli-slice helical scan developed o shif he slice posiion of complemenary daa and o acquire daa wih a much smaller sampling inerval in he z direcion. Filer inerpolaion refers o a filering process performed in he z direcion using several daa. The normal fan-beam reconsrucion echnique is used. The secion sensiiviy profile SSP and image qualiy for four-array muli-slice CT were invesigaed by compuer simulaions. Combinaions of hree ypes of opimized sampling scan and various filer widhs were used. The algorihm enables us o achieve accepable image qualiy and spaial resoluion a a scanning speed ha is abou hree imes faser han ha for single-slice CT. The noise characerisics show ha he proposed algorihm efficienly uilizes he daa colleced wih opimized sampling scan. The new algorihm allows suiable combinaions of scan and filer parameers o be seleced o mee he purpose of each examinaion American Associaion of Physiciss in Medicine. S Key words: muli-slice CT, helical scan, image reconsrucion, filer inerpolaion, opimized sampling scan I. INTRODUCTION Before he developmen of helical compued omography CT around 199, 1 5 advances in CT echnology focused mainly on faser scanning and on higher resoluion in he axial plane. Helical CT enabled us o obain volume daa in a single breah hold, and opened he door o he age of volumeric imaging. Helical scan needs inerpolaion in he longiudinal direcion z direcion before fan-beam reconsrucion. Wih he progress of helical scanning, various ypes of inerpolaion and reconsrucion mehods have been proposed. These include 36 linear inerpolaion LI, 18 linear inerpolaion 4,5 18LI, 18 linear exrapolaion 5 18LE, nonlinear inerpolaion, a general approach, 6 and a deconvoluion process. 7 Since higher-pich helical scanning degrades he secion sensiiviy profile SSP, we have o compromise beween scan speed and spaial resoluion in single-slice helical CT. There is, however, a need for faser scanning and higher resoluion in volume in various clinical fields. CT scanners are expeced o evolve ino volume scanners wih a small voxel size, large scanning range, and high speed. The use of a muli-slice deecor, consising of several deecor arrays sacked in he z direcion, is a promising approach o he developmen of volume scanners. I can also improve he ime resoluion of dynamic volume scans. The lack of a suiable algorihm for image reconsrucion is one of he bigges sumbling blocks in he developmen of muli-slice CT. The cone angle creaes problems even in he case of small angles. Several algorihms using cone-beam backprojecion have been proposed for helical scan. 8 1 Cone-beam backprojecion, however, requires complicaed calculaion relaive o fan-beam backprojecion. This is an imporan pracical problem hindering implemenaion. We have herefore developed a new algorihm for mulislice CT using fan-beam backprojecion. I consiss of following hree pars; 1 opimized sampling scan; 2 filer inerpolaion; and 3 fan-beam reconsrucion. Opimized sampling scan refers o a special ype of muli-slice helical scan echnique developed in order o shif he slice posiion of complemenary daa and o acquire daa wih a significanly smaller sampling inerval. Filer inerpolaion refers o a filering process performed in he z direcion using several daa samples in order o reduce he effec of disconinuous changeovers. Subsequenly, he normal fan-beam reconsrucion echnique is used. The proposed algorihm allows us o 55 Med. Phys. 25 4, April /98/25 4 /55/12/$ Am. Assoc. Phys. Med. 55

2 551 K. Taguchi and H. Aradae: Algorihm for image reconsrucion 551 FIG. 1. Geomery of hird-generaion single-slice and four-array muli-slice CT scanners. a Coordinae sysem; b side view of muli-slice CT; c side view of single-slice CT; d view along he z axis. achieve boh he image qualiy and he SSP levels sufficien for pracical use. In his paper we define he scan geomery and noaion in Sec. II. In Sec. III, we describe wo mehods for single-slice CT, ouline he exension of hese mehods for muli-slice CT, and discuss heir shorcomings. We hen propose a new algorihm in Sec. IV and use compuer simulaions in Sec. V o demonsrae ha his approach overcomes he problems discussed in Sec. III. The discussion and conclusions are presened, respecively, in Sec. VI and Sec. VII. II. DEFINITION OF SCANNER GEOMETRY AND PARAMETERS In his secion, we define he scanner geomery and esablish he noaion used in his paper. For he following discussion we focus our aenion on he hird generaion geomery for he case of four-array muli-slice CT and single-slice CT. The resuls can easily be exended o oher geomeries or o muli-slice CT wih a differen number of arrays. The geomery and coordinaes are shown in Fig. 1. The muli-slice deecor consiss of four arrays (N 4) sacked in he z direcion along a cylindrical surface. The view and channel angles are denoed by and, respecively. The maximum channel angle is m. The focus-o-cener disance is defined by fcd and he focus-o-deecor disance by fdd. BW refers o he beamwidh in he z direcion for one deecor array a he cener of roaion. BW for single-slice CT is defined in Fig. 1 c. The slice posiion of each array is defined as he cener of BW a he cener of roaion. The normalized helical pich P is given by P L BW, where L denoes he z incremen per roaion. III. SINGLE-SLICE CT AND EXTENSION TO MULTI- SLICE CT This secion oulines and exends 36LI and 18LI, which are helical inerpolaion mehods for single-slice CT. These wo mehods were chosen because of heir complemenariy. The characerisics of 36LI are adequae wih respec o boh noise and arifac eliminaion bu inadequae wih respec o he SSP while he converse is rue for 18LI. The use of he above wo algorihms for comparison will herefore be sufficien for evaluaion of he proposed algorihm. The helical pich was chosen as P 1 for single-slice CT and P 4 for muli-slice CT. P N was chosen because i is equivalen o he oal widh of he beam a he roaion cener and seems o be he simples condiion for he exension. The direc and complemenary daa are defined as pd(, ) and pc(, ), respecively, and zd( ) and zc(, ) are he corresponding slice posiions. Here, direc daa refer o daa obained a he focus posiion a he curren view angle. The daa obained a he opposie side are referred o as complemenary daa Fig. 2 a. For he slice z z, 36LI is described by Eq. 2 and 18LI by Eq. 3 : p(, ) w( ) pd( 2, ) (1 w( )) pd(, ), z zd w zd 2 zd ; 2 1

3 552 K. Taguchi and H. Aradae: Algorihm for image reconsrucion 552 FIG. 2. Diagram of helical scan in single-slice CT. a Direc daa and complemenary daa; b view diagram showing he slice posiion of all daa a whole channel angle a arbirary view angle ; c scan diagram showing he slice posiion of direc daa solid line and complemenary daa a dashed line in whole helical scan. The complemenary daa makes he daa sampling inerval in he z direcion small. p, w, pc, 1 w, pd,, w, z zd zc, zd, 3 pc, pd 2,, FIG. 3. Diagram of helical scan in muli-slice CT. a Direc daa four ses from one focus and he focus posiion of complemenary daa; b view diagram for a pich of 4 (P 4) showing he slice posiion of all daa a whole channel angle a arbirary view angle ; c scan diagram for a pich of 4 (P 4) showing he slice posiion of direc daa solid line and complemenary daa a dashed line in whole helical scan. The complemenary daa a coincides wih he direc daa of anoher array and does no provide sufficien benefi. zc, zd 2. Figure 2 shows he helical scan geomery. Figure 2 b is called he view diagram and Fig. 2 c is called he scan diagram. The view diagram shows he relaionship beween he channel angle and he slice posiion a an arbirary view angle, while he scan diagram shows he relaionship beween he view angle and he slice posiion for he whole helical scan. Noe ha he complemenary daa of only he cenral channel is drawn in he scan diagram. Since he disance beween he complemenary and direc daa is small, 18LI gives sharper SSP han 36LI. Figure 3 shows he helical scan geomery for muli-slice CT in he case of P 4. The direc and complemenary daa are idenified by he deecor array number (n 1,2,3,4). The direc and complemenary daa are hus defined as pd(,,n) and pc(,,n), respecively, and zd(,n) and zc(,,n) are he corresponding slice posiions. The exended 36 linear inerpolaion ex-36li is given by Eq. 4 and exended 18 linear inerpolaion ex-18li by Eq. 5 :

4 553 K. Taguchi and H. Aradae: Algorihm for image reconsrucion 553 FIG. 4. Diagram of helical scan wih he proposed opimized sampling scans. a Scan diagram for P 2.5; b scan diagram for P 3.5; c scan diagram for P 4.5 wih half-beam widh BW ; d view diagram for P 2.5. Boh he direc and complemenary daa make he daa sampling inerval in he z direcion significanly smaller han ha for a pich of 4 (P 4) Fig. 3. p, w pd,,n 1 1 w pd,,n w pd 2,,1 1 w pd,,n w z zd,n BW w 1 ; n N n N 4 p, w, ph,,j 1 1 w, ph,,j, z zh,,j w, zh,,j 1 zh,,j ; 5 where ph(,,j) and ph(,,j 1) are he lower and upper daa ses adjacen o he slice z z, respecively, and zh(,,j) and zh(,,j 1) are he corresponding slice posiions. Figure 3 and Eq. 5 indicae ha here are he following wo serious problems in muli-slice helical CT: 1 excessively close slice posiions for complemenary and direc daa, resuling in a larger sampling inerval; 2 several disconinuous changeovers in pairs of daa samples in he inerpolaed daa ses obained for one roaion. Because he slice posiion of he complemenary daa is exacly he same as ha of he direc daa of anoher array a he cenral channel ( ), he sampling disance beween wo daa poins used for inerpolaion is larger han ha for 18LI. In addiion, muli-slice CT needs several imes he number of changeovers required by single-slice CT; which degrades he SSP and image qualiy. IV. PROPOSED METHOD This secion inroduces a new algorihm which consiss of he following hree pars: A opimized sampling scan; B filer inerpolaion; and C fan-beam reconsrucion. A. Opimized sampling scan One problem in ex-18li is he slice posiion of he complemenary daa a he cenral channel. The mos effecive soluion is o change he helical pich in order o shif he complemenary daa. Figure 4 a, b, and c shows scan diagrams wih a pich of 2.5, 3.5, and 4.5, respecively (P 2.5,3.5,4.5). Complemenary daa of only he cenral channel is shown. Figure 4 d shows he view diagram for a pich of 2.5. Figure 4 a d shows ha he sampling inerval in he z direcion decreases significanly o abou one-fourh as compared o P 4. We call hese unique muli-slice helical scan modes opimized sampling scan. Noe ha he purpose of he opimized sampling scan is o shif he sampling posiion of he complemenary daa and o acquire daa wih a significanly smaller sampling inerval.

5 554 K. Taguchi and H. Aradae: Algorihm for image reconsrucion 554 FIG. 6. Examples of filer shape for filer inerpolaion. The recangular filer a is used in he compuer simulaions. FIG. 5. Concep and parameers for filer inerpolaion wih wo seps. Firs, he linear inerpolaion is processed using adjacen daa poins in order o obain muliple resampling daa ses. Then, he resampling daa ses are filered wihin he filer widh FW. The resampling inerval and posiions are defined by he filer widh FW and he number of resampling posiions (2I 1). B. Filer inerpolaion Anoher problem wih he exended mehod is he exisence of several disconinuous changeovers in pairs of daa samples in he inerpolaed daa ses obained for one roaion. The mos effecive mehod of eliminaing hese changeover effecs is o uilize more han wo daa ses for inerpolaion. Filer inerpolaion is a filering process ha is performed in he z direcion wih several daa ses and is used insead of helical inerpolaion. The concep is shown in Fig. 5. A widh is assumed in he z direcion and is defined as he filer widh FW. As a resul of he opimized sampling scan, several daa poins lie wihin FW. Filer inerpolaion can be divided ino he following wo seps as shown in Eq. 6 : 1 resampling by linear inerpolaion using adjacen daa poins; and 2 filering he resampled daa se. In he firs sep, he resampling posiions are deermined by he following wo facors: 1 he number of resampling poins, 2I 1; and 2 he filer widh, FW. Complemenary daa are used in resampling. 1 Sep 1: Resampling by linear inerpolaion for I i I pf,,i w,,j i ph,,j i 1 1 w,,j i ph,,j i, zf i zh,,j i w,,j i zh,,j i 1 zh,,j i, zf i z i z, z FW 2 I 1. 2 Sep 2: Filering p, I i I w i pf,,i I, 6 i I w i where ph(,,j(i)) and ph(,,j(i) 1) are he lower and upper daa ses adjacen o he resampling posiion z zf(i), respecively, and zh(,,j(i)) and zh(,,j(i) 1) are he corresponding slice posiions. z is he resampling inerval. The shape and widh of he filer are defined by w(i), I, and FW. They can be chosen freely for he expeced SSP, image qualiy, and noise. Some examples of filer shape are shown in Fig. 6: a smoohing; b edge enhancemen; and c modifying he shape of SSP. Filer inerpolaion, as given by Eq. 6, needs a large number of calculaions. Direc filering, which is a much faser implemenaion mehod, is oulined in he Appendix. Noe ha filer inerpolaion corresponds o ex-18li wih opimized sampling scan when FW mm. C. Fan-beam reconsrucion The normal fan-beam reconsrucion echnique, which uses filered backprojecion, is applied o obain slice images using he p(, ) daa of Eq. 6. V. COMPUTER SIMULATIONS This secion shows he resuls of compuer simulaions wih hree ypes of projecion daa: 1 coin phanom daa for he SSP; 2 random noise daa for he image noise; and 3 ball phanom daa for he image qualiy. A program was wrien o generae line inegrals hrough he phanoms. The TABLE I. Scan and filer parameers used in his paper: normalized helical pich, P; beamwidh, BW mm ; filer widh, FW mm ; and filer shape. The proposed mehods for muli-slice CT are denoed as 1 3, he exended mehods for muli-slice CT are denoed as 4, and he mehods for single-slice CT are denoed as 5. Scan parameers Filer parameers P BW FW Shape , 1., 2., 3., 4. Recangular , 1., 2., 3., 4. Recangular , 1., 2., 3., 4. Recangular

6 555 K. Taguchi and H. Aradae: Algorihm for image reconsrucion 555 FIG. 7. Secion sensiiviy profiles SSPs for a beam widh BW of2mm for single-slice CT and exended muli-slice CT; a 36LI (P 1), b 18LI (P 1), c ex-36li (P 4), and d ex-18li (P 4). The profiles of he exended muli-slice mehods are almos he same as ha of 36LI. FIG. 9. SSPs for he proposed mehod P 2.5 (BW 2 mm) for various filer widhs (FW,1,2,3 mm). The profiles for single-slice CT 36LI and 18LI wih P 1 are also shown. The profiles for he proposed mehod are similar o ha of 18LI when FW 1 mm and comparable o hose of 18LI and 36LI when FW 2 mm. focus, channel angle, and view angle are divided ino a number of micro-pars and summed in order o avoid digializaion errors. The values of all parameers are shown in Table I. We chose he following hree scan modes from a number of combinaions for discussion; 1 P 2.5, 2 P 3.5, and 3 P 4.5. The filer shape was fixed as recangular, and filer widh FW was varied from mm o 4 mm. The scan modes compared were 4 P 4, muli-slice CT, and 5 P 1, single-slice CT. The beamwidh BW and fcd were, respecively, fixed a 2 mm and a 6 mm in all he compuer simulaions. A. Secion sensiiviy profile SSP Coin phanoms were used o evaluae he SSP for various scan and filer parameers. The phanoms, which had a diameer of 2 mm and a hickness of.2 mm, were placed wihin he cenral slice (z mm) a four posiions along a circle FIG. 8. Variaion of a FWHM and b FWTM of he SSPs of he hree proposed mehods wih FW. BW is fixed a 2 mm. Proposed mehods: 1 P 2.5; 2 P 3.5; 3 P 4.5. These mehods allowed beer values han hose of he 36LI single-slice mehod (P 1) and he exended mehods ex-36li and ex-18li wih P 4 o be obained when FW 2 mm. FIG. 1. SSPs for he proposed scan modes P 2.5; P 3.5; P 4.5 wih FW 2 mm and BW 2 mm. The profiles for single-slice CT 36LI and 18LI wih P 1 are also shown. The profiles for he proposed mehods are comparable o hose for single-slice CT.

7 556 K. Taguchi and H. Aradae: Algorihm for image reconsrucion 556 FIG. 11. SSPs a five posiions for a beam widh BW of 2 mm for single-slice CT and exended muli-slice CT; a 36LI (P 1), b 18LI (P 1), c ex-36li (P 4), and d ex-18li (P 4). The profiles of he exended muli-slice mehods are almos he same as ha of 36LI. ( 2 mm) and a he cener of roaion. The (x,y,z) coordinaes of he ceners of he phanoms were mm, 1 mm,mm, 1 mm, mm, mm, mm, 1 mm, mm, 1 mm, mm, mm, and mm,mm,mm. Slice images were reconsruced wih a reconsrucion incremen of.1 mm, and he average CT value in he ROI in he images was defined as he response a he corresponding z posiion. These CT values were normalized agains he value a z mm. Figure 7 shows he SSPs a he cener of roaion for single-slice and exended muli-slice CT; 36LI and 18LI wih P 1, and ex-36li and ex-18li wih P 4. The profiles of ex-36li and ex-18li are broad and almos he same as ha of 36LI. Figure 8 a shows he full widh a half-maximum FWHM and Fig. 8 b shows he full widh a enhmaximum FWTM of SSPs a he cener of roaion for he proposed mehods wih various filer widhs FW ; 1 P 2.5, 2 P 3.5, and 3 P 4.5. The values for he singleslice mehods 36LI and 18LI wih P 1 and exended mehods ex-36li and ex-18li wih P 4 are also shown. The beamwidh, BW, is fixed a 2 mm. Noe ha he FWHM increases by less han 2% unil FW 2 mm for all he proposed mehods. Also, all he resuls lie beween 36LI and 18LI unil FW 2 mm. Figure 9 shows he SSPs of he proposed mehod wih P 2.5 and BW 2 mm for various filer widhs (FW,1,2,3, mm) a he cener of roaion. The profiles for single-slice CT 36LI and 18LI wih P 1 are also shown for comparison. I can be seen ha he profile broadens wih increase in FW. A FW of less han 1 mm will give a beer SSP han ha of 18LI, while an FW of 2 mm gives an inermediae SSP ha lies beween hose of 18LI and 36LI. Figure 1 shows he SSPs of he proposed mehods; 1 P 2.5, 2 P 3.5, 3 P 4.5; and of single-slice CT (P 1); 4 36LI and 5 18LI. FW is fixed a 2 mm. I can be seen ha all he proposed mehods give comparable profiles in spie of he hick FW. Figure 11 a d shows he SSPs a he five posiions for he following four cases: a 36LI (P 1); b 18LI (P 1); c ex-36li (P 4); and d ex-18li (P 4). Figure 12 a e shows he SSPs a he five posiions for he proposed mehods: a P 2.5 (FW mm); b P 2.5 (FW 2 mm); c P 2.5 (FW 4 mm); d P 3.5 (FW 2 mm), and e P 4.5 (FW 2 mm). I can be seen ha he spaial variaions of he SSP for he proposed mehods are comparable o hose for single-slice CT Fig. 11. The spaial variaions for P 2.5 are paricularly good. B. Image noise Random noise daa were used o evaluae image noise for various scan modes and filer widhs. The images were re-

8 557 K. Taguchi and H. Aradae: Algorihm for image reconsrucion 557 FIG. 12. SSPs a five posiions for he proposed mehods for various scan modes and filer widhs; a P 2.5 (FW mm), b P 2.5 (FW 2 mm), c P 2.5 (FW 4 mm), d P 3.5 (FW 2 mm), and e P 4.5 (FW 2 mm). The spaial variaions of he SSP for he proposed mehods are comparable o hose for single-slice CT Fig. 11. consruced and he sandard deviaion s.d. of he CT value in he ROI was measured a he same five posiions as he SSP. Figure 13 shows he s.d. s a he five posiions for he proposed mehods P 2.5, 3.5, and 4.5 wih FW, 2, and 4mm and single-slice CT 36LI and 18LI wih P 1. The s.d. s of he proposed mehods are similar when FW is mm, ha is, he noise characerisics are similar even hough he daa sampling inerval is differen. This resul indicaes ha only some daa samples are used in helical inerpolaion and he ohers are wased. The s.d. s, however, decrease wih he increase in FW, because he number of daa used in he filer inerpolaion increases. The raio of he s.d. s can be calculaed as he square roo of he number of daa used in he filer inerpolaion. For example, he s.d. s a FW 2mmand4mmare 62% and 46% of he s.d. a FW mm when P 2.5 in Fig. 13. According o Fig. 4 a, wo, five, and nine daa samples are, respecively, used in filer inerpolaion wih FW, 2, and 4 mm when P 2.5. The raio of s.d. s can be calculaed as he square roo of he number of daa used when he filer shape used is a recangle. The values of 2/5 and 2/9 are.63 and.47, respecively, and hus mach he simulaion resul. The higher he helical scanning pich, he less he number of daa sampled in he same filer widh. Thus a difference beween he s.d. s for he hree scan modes appears when he FW increases. Noe ha he s.d. of image noise for he proposed mehod

9 558 K. Taguchi and H. Aradae: Algorihm for image reconsrucion 558 FIG. 13. The sandard deviaions s.d. s of he image noise a five posiions for he proposed mehods P 2.5, 3.5, and 4.5 wih FW, 2, and 4 mm and single-slice CT 36LI and 18LI wih P 1. The s.d. s decrease wih he increase in FW. Noe ha he s.d. of image noise for he proposed mehod is smaller han ha for single-slice CT when FW is equivalen o BW. Thus he same noise level can be achieved using noisier projecion daa. is smaller han ha for single-slice CT when FW is equivalen o BW. Thus he same noise level can be achieved using noisier projecion daa. The spaial variaion of he image noise for our mehods was also smaller han ha for single-slice CT: 5% maximum P 2.5 wih FW, 2 mm and P 3.5 wih FW 2 mm agains 13% for 18LI and 8% for 36LI. C. Image qualiy FIG. 14. Reconsruced images of ball phanoms 2 mm, conras HU, background 1 HU. a 36LI; b 18LI; c Ex- 36LI; d Ex-18LI. BW is fixed a 2 mm. The slice posiion is 8 mm away from he ceners of he balls. The window level and widh are 1 and 8, respecively. The image qualiy is degraded in he exended mehods. FIG. 15. Reconsruced images of ball phanoms 2 mm, conras HU, background 1 HU by he proposed mehod for a filer widh of, 2, and 4 mm. BW is fixed a 2 mm. a P 2.5; b P 3.5; c P 4.5. The slice posiion is 8 mm away from he ceners of he balls. The window level and widh are 1 and 8, respecively. The image qualiy improves significanly wih he increase of FW. Ball phanoms were used o evaluae image qualiy wih various scan modes and filer widhs. Eigh balls ( 2 mm) were se a he cenral plane (z mm) along a circle ( 2 mm). The conrass of he ball and he background were HU and 1 HU, respecively. The scan parameers were he same as hose for he SSP. Slice images near he edges of he balls (z 8 mm) were reconsruced and compared wih each oher. The window level and window widh were fixed a 1 and 8, respecively. Images for single-slice and exended muli-slice CT 36LI, 18LI, ex-36li, and ex-18li are shown in Fig. 14. Severe arifacs appear for boh ex-18li and ex-36li. These arifacs will clearly be a serious problem in clinical use. Images obained by he hree proposed mehods are shown in Fig. 15; 1 P 2.5, 2 P 3.5, and 3 P 4.5 wih FW, 2, 4 mm. When FW mm, hese mehods correspond o ex-18li wih opimized sampling scan. I can be seen ha boh opimized sampling scan and filer inerpolaion improve he image qualiy, and ha he image qualiy improves markedly wih he increase of FW. Noe ha he image qualiy wih a pich of 2.5 or 3.5 is comparable o ha of 18LI when FW 2 mm cf. Fig. 14. As long as he arifac level is almos equal o ha of single-slice CT 36LI or 18LI, he resuls are accepable. For furher quaniaive analysis of he arifac level, he profiles across one ball phanom are shown in Fig. 16. The (x,y) coordinaes of he cener of he phanom was mm, 1 mm. The profiles in he horizonal direcion (x) are shown in a P 2.5 wih various filer widhs, b FW 2 mm wih various scan modes, and c single-slice 36LI and 18LI wih P 1 and exended muli-slice CT ex-36li and ex-18li wih P 4. The corresponding profiles in he verical direcion (y) are shown in Figs. 16 d f. Eiher an undershoo or an overshoo near he background indicaes an arifac. Filer inerpolaion improves

10 559 K. Taguchi and H. Aradae: Algorihm for image reconsrucion 559 FIG. 16. The profiles across one ball phanom. The (x,y) coordinaes of he cener of he phanom was mm, 1 mm. In he horizonal direcion (x); a P 2.5 wih various filer widhs, b FW 2 mm wih various scan modes, and c single-slice 36LI and 18LI wih P 1 and exended muli-slice CT ex-36li and ex-18li wih P 4. In he verical direcion (y); d P 2.5 wih various filer widhs, e FW 2 mm wih various scan modes, and f single-slice 36LI and 18LI wih P 1 and exended muli-slice CT ex-36li and ex-18li wih P 4. Filer inerpolaion improves boh he undershoo and he overshoo a and d. The arifac levels for he proposed mehods P 2.5 and 3.5 are comparable o ha for 18LI. boh when FW is 2 mm Fig. 16 a and d. I can also be seen ha he arifac levels wih he proposed mehods P 2.5 and 3.5 are comparable o ha wih 18LI. VI. DISCUSSION The compuer simulaions show ha he exended mehods ex-36li and ex-18li have severe problems as discussed in Sec. III. Boh he image qualiy and he SSP are seriously degraded. This is an inheren problem of wo-poin inerpolaion in muli-slice CT. The problem level was so severe ha a new mehod leading o drasic improvemen was needed. The compuer simulaions show ha he opimized sampling scan improves SSPs. This can be aribued o he significanly smaller sampling inerval. The simulaions also demonsrae ha filer inerpolaion produces beer image qualiy wih larger FW and sharper SSPs wih smaller FW. This is because he use of muliple daa sampled over a smaller inerval wihin FW can eliminae he effecs of disconinuous changeovers. In helical scan wih a pich of 2.5 or 3.5, boh he image qualiy and he SSPs are comparable o hose of single-slice CT when FW is equivalen o BW. When he helical pich is higher, boh he SSP and he image qualiy are degraded. Improvemen using half-backprojecion or oher mehods is required, especially for helical pich values higher han 4.5. In clinical pracice, we have wo choices wih muli-slice CT: 1 scanning faser over a greaer lengh wih equivalen image qualiy and SSP; or 2 obaining beer daa wihou parial volume effecs by summing several narrow beam

11 56 K. Taguchi and H. Aradae: Algorihm for image reconsrucion 56 widh daa ses. We can also change he spaial resoluion in he z direcion by adjusing he filer parameers, as we can do presenly in he ransaxial plane. In shor, we can choose a suiable combinaion of scan and filer parameers o mee he purpose of he examinaion. The scan parameers are he normalized helical pich ( P) and he beam widh for one deecor array a he cener of roaion BW. The filer parameers are he widh FW and he shape recangular, ec. of he filer in filer inerpolaion. Some algorihms for cone-beam backprojecion 8,9 have a paien dose problem, because one daa is seleced and he oher is hrown away in overlapped areas. Alhough here are overlapped areas in our proposed mehod, all he daa are efficienly uilized in filer inerpolaion. According o he resuls of our invesigaion on he image noise, he ube curren for our mehod can be reduced while keeping he oal paien dose and image noise a he same levels as hose for single-slice CT. This is advanageous in pracical sysems for clinical use. I should be noed here ha he proposed algorihm can also be applied o muli-slice CT wih a differen number of deecor arrays. VII. CONCLUSION We have presened a new reconsrucion algorihm for muli-slice helical CT based on he combinaion of opimized sampling scan, filer inerpolaion, and fan-beam reconsrucion. We have presened he resuls of compuer simulaions wih various scan and filer parameers for four-array mulislice CT. The algorihm enables us o achieve accepable image qualiy and spaial resoluion a a scanning speed ha is abou hree imes faser han ha for single-slice CT. The noise characerisics show ha he proposed algorihm efficienly uilizes he daa colleced wih opimized sampling scan. A suiable combinaion of scan and filer parameers can be chosen according o he purpose of he examinaion. ACKNOWLEDGMENTS The auhors would like o hank Dr. Ilan Zmora a Bio- Imaging Research, Inc. IL, USA for his helpful suggesions during he course of he sudy, anonymous reviewers for heir consrucive commens, and Dr. Takashi Ichihara for encouraging us o publish his paper. APPENDIX: DIRECT FILTERING METHOD FASTER FILTER INTERPOLATION TECHNIQUE In he Appendix we propose an implemenaion mehod for filer inerpolaion ha requires a shorer processing ime. This mehod is based on inegral calculus. The following explanaion assumes ha a recangular filer is used. Figure 17 shows he sampling daa poins in he z direcion a an arbirary view and channel angle. The number of daa poins used in his process is defined as J. The jh daa value is denoed by ph(j) and he corresponding slice posiion by zh(j). FW is he filer widh and z, zl, and zh are he z posiions of he cener, boom edge, and op edge of he filer, respecively. FIG. 17. Parameers for direc filering a faser filer inerpolaion. A. Calculaing he weighs for he daa Depending on he relaive z posiion, he corresponding equaion is seleced from among Eqs Weighs for all daa are calculaed using hese equaions. Noe ha he weighs are normalized agains FW. (1-1) Ouside case (j 1 orj) w j 1 zh j 1 zl u zh j 1 zh j du 1 zh j 1 zl FW 2 zh j 1 zh j 2 j 1, A1 w j 1 zh zh j 1 zh j zh j 1 d 1 zh zh j 1 FW 2 zh j zh j 1 2 j J. A2 (1-2) Inside case (j 2 orj 1) Variables and u for he case of j 2 are shown in Fig. A1: w j 1 zh j 1 zh j 1 zh j 1 zh j d u zh j zl 1 zh j zh j 1 du 1 FW zh j 1 zh j zh j zl 2 zh j zl 2 j 2 ; A3 2 zh j zh j 1

12 561 K. Taguchi and H. Aradae: Algorihm for image reconsrucion 561 w j 1 zh zh j 1 zh j 1 zh j d u zh j zh j 1 1 zh j zh j 1 du 1 FW zh j zh j 1 zh zh j 2 zh zh j 2 j J 1. A4 2 zh j 1 zh j (1-3) Inernal case (3 j J 2) Variables and s for he case of j 3 are shown in Fig. 17: w j 1 zh j zh j 1 zh j 1 zh j 1 zh j 1 zh j 1 2 FW zh j zh j 1 d s zh j 1 zh j ds 3 j J 2. B. Direc filering using he calculaed weighs A5 All he daa are assigned weighs and summed according o he following equaion in order o obain he daa z z: j p zc w j ph j. A6 j 1 When J 3 wih small FW, 1-2 is modified o (1-2) Inside case ( j 2 J 1) w j 1 zh zh j 1 zh j 1 zh j d u zh j zl 1 zh j zh j 1 du 1 1 zh zh j FW 2 2 zh j 1 zh j zh j zl 2 j 2 2 zh j zh j 1 zh zl FW. A7 When J 2 wih hin FW, his becomes a simple linear inerpolaion wih respec o z z. (1-4) Case of J 2 zh zh 1 w 2 1 zl zh 1 z zh 1 zh 2 zh 1, zh 2 zh 1 d w 1 1 w 2 zh 2 z zh 2 zh 1. A8 This mehod can easily be modified for a nonrecangular filer w(z). For example, Eqs. A5 and A6 will be modified o w j zh j zh j 1 zh j 1 zh j 1 ds; zh j zh j 1 w d s zh j 1 zh j w s A9 and p zc J j 1 w j ph j J. A1 w j j 1 a Elecronic mail: kaguchi@mel.nasu.oshiba.co.jp 1 I. Mori, Compuerized omographic apparaus uilizing a radiaion source, U.S. Paen No. 4,63, W. Kalender, W. Seissler, E. Kloz, and P. Vock, Spiral volumeric CT wih single-breah-hold echnique, coninuous ranspor, and coninuous scanner roaion, Radiology 176, Y. Toki, Principles of Helical Scanning, in Basic Principles and Clinical Applicaions of Helical Scan: Applicaions of Coninuous-Roaion CT, edied by K. Kimura and S. Koga Iryokagakusha, Tokyo, 1993, Chap. IV, Secion 1.1, pp Y. Toki, T. Rifu, H. Aradae, Y. Hirao, and N. Ohyama, New reconsrucion algorihm in helical-volume CT, Radiology 121P, C. Crawford and K. F. King, Compued omography scanning wih simulaneous paien ranslaion, Med. Phys. 17, J. Hsieh, A general approach o he reconsrucion of x-ray helical compued omography, Med. Phys. 23, H. Hu and Y. Shen, Helical reconsrucion algorihm wih userselecable secion profiles, Radiology 189P, H. Kudo and T. Saio, Three-dimensional helical-scan compued omography using cone-beam projecions, IEICE D-II J74-D-II, G. Wang, T. H. Lin, P. C. Cheng, and D. M. Shinozaki, A general cone-beam reconsrucion algorihm, IEEE Trans. Med. Imaging 12, S. Schaller and T. Flohr, A new approximae algorihm for image reconsrucion in cone-beam spiral CT a small cone-angles, Absrac book of IEEE Nucl. Sci. Symp. and Med. Imag. Conf., M1-3: