Research of Large Field of View Scan Mode for ...

V ol. 16　 N o . 1 　　　　　　　　　　　　 CHIN ESE JO U RN A L OF A ERO N A U TICS　　　　　　　　　　　　 February 2003

Research of Large Fiel d of View Scan Mode f or Industrial CT F U Jian, L U Hong -nian ( School of M echanical Engineering and A utomat ion, Beij ing University of A eronautics and A stronautics, Beij ing 　100083, China) Abstract: 　 F or industr ial computed tomog raphy systems, gener ation II scan mode has a lar ge field of v iew but time-consuming and g eneration III has a small field o f view but fast. In order t o realize the rapid ICT test of larg e objects , a scan mode based on generation III called larg e field of view scan was discussed and its reconstr uction alg orithm based on F BP was deduced. T he validity of the alg orithm w as v erified by the results of computer simulation and ex periments. A nalysis show ed that the effectiv e scan field o f view could be improv ed by more than 90% compar ed w ith that of gener ation III. Key words: 　industrial computed tomog raphy ; fan beam scan; filter back-pr ojection algor ithm; rebin alg or ithm; non-destr uctiv e testing 工 业 CT 大视场扫 描成像方 法研究. 傅健, 路宏年. 中国 航空学 报( 英文版) . 2003, 16( 1) : 5964. 摘　要: 工业 CT 二代扫描方式扫 描视场大, 但扫描 时间长; 三代扫描 方式扫描时 间短, 但扫描视 场小。为解决较大尺寸构件 ICT 快速检测问题, 讨论了一种基于三代扫 描的大视场扫描方 式, 推导 了它的 F BP 重构算法。计算机模拟和实验结果证明了该算法的正确性。分析表明, 其有效扫描视野 在三代的1. 9倍以上。 关键词: 计算机层析成像; 扇束扫描; FBP 算法; 重排算法; 无损检测 文章编号: 1000-9361( 2003) 01-0059-06　　　中图分类号: T P391　　　文献标识码: A

　 　Indust rial computed tomography ( ICT ) is a

needs less scan t ime than T R, large object s have to

novel non-contact ed measurement technolog y. Based on t he high -resolut ion , high -cont rast imag es

em ploy T R because of t he limits caused by the effective ang le of t he X-ray beam and t he lengt h of

generat ed by X-ray scan and reconst ruct ion, it can perm it an exact measurement of t he posit ion and

detectors. It results in a slice scan with no less t han 10min.

the dim ensions of the int ernal structure of m any industrial com ponent s and provide informat ion that

In order t o overcome t his diff icult y, a scan mode called large f ield of view ( LF OV ) based on

cannot currently be provided nondestructively by any ot her m eans[ 1] . T he measurement of the w all

RO is proposed and depict ed in Fig. 1( b) . Based

thickness of air engine blades is a good ex am ple. T he majority of scan m odes current ly adopted by ICT syst ems can be classified into tw o classes: generat ion II or T R and generat ion III or RO. T he

on the rebin reconst ruction alg orit hm of the f an beam , t he algorithm for L FOV is present ed . T he result s of com put er sim ulat ion and experim ent s verify t he validity of t his scan mode.

feat ure of T R is t hat the object s need to rot at e and

1　LF OV Scan M ode

transverse. T he feat ure of RO, depict ed in Fig. 1 ( a ) , is t hat t he objects only rotate . Alt hough RO

Fig . 1( c ) shows the scan principle . S 0 is t he X-ray source. D 1D 2 represent s t he linear array de-

Received dat e: 2002-06-27; R evision received dat e: 2002-12-16 A rt icle U RL: ht t p: / / ww w . hkx b. n et . cn/ cja/ 2003/ 01/ 0059/

FU Jian, L U Hong -nian

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tect or ( L DA) . is t he ef fective ang le of t he X-ray beam . D is t he dist ance from t he X-ray source to the detect ors. S 0D 1D 2 is the scan geometry of RO.

w here h ( x r ) is a filt er function. p ( x r , !) is t he project ion of t he parallel beam . T he formation of t he projection is depict ed in Fig. 2( b) . 2

RO can inspect the object s wit h t he radius no larger t han r 1 as long as the object s rotate 360° at t he point O1 . For LF OV, t he scan g eomet ry does not change and t he rotation cent er of t he table m oves to O 2. T he object s wit h t he radius no larger than r 2 can be t est ed w hen they rotate 360° . T he data collected by the actual detect or D 1D 2 can be convert ed int o the dat a collect ed by t he virt ual detector A B through rebin and interpolation. Wit hout changing t he posit ion and the length of t he detector D 1 D 2 and t he angle of the X -ray f an beam , t he L FOV scan mode can test the large object s that are usually inspect ed wit h t he det ector A B and the angle of t he X-ray beam as t w ice as that in RO.

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a( r , ) =

1

∫U ~p ( s , ∀) d∀ 2

e

1

( 2)

0

D + rsin( ∀- ) is a w eight ing f act or . D p ( s , ∀) is the project ion of the f an beam . p e ( s , ∀) D = p ( s, ∀) is t he w eight ed project ion. D 2 + s2 1 g ( s ) = 2 h( s) is a f ilt er f unction. ~ p e ( s, ∀) = p e ( s , ∀) * g ( s) is t he f iltered project ion . s1 = Drcos( ∀- ) is t he posit ion of the reconstructed D+ rsin( ∀- ) point . T he format ion of t he projection of t he f an w here U =

beam is described in F ig . 2( a ) . T he mission of t he reconstruct ion algorithm for the LF OV presented by t his paper is t o solve how t o reconstruct the object wit h t he radius r 2 using the detector D 1D 2 . T he convent ional f ilt er back project ion ( F BP ) alg orit hm f or t he f an beam described by Eq . ( 2) needs the complete f an beam . Obviously , L FOV can not acproject ions for 360° quire t he complet e fan beam projections f or 360° . So t his algorithm can not be used directly. T he incomplet e projections need to be rebinned or reorg anized. T his idea of rebin comes from t w o characterist ics of parallel beam projections implied in Fig. 2( b ) . One is that t he project ions f rom 0°t o 180° . Anot her are t he same as those from 180°t o 360° is that t he project ion does not change w hen t he project ing direct ion is converse.

F ig . 1　L F OV scan principle

2　Algorithm for L FOV T here exist reconst ruction form ulas f or t he parallel beam and fan beam

[ 2, 3]

. T he reconstructed

slice is represent ed by a( r, ) in polar coordinat es. a( r, ) =

( h( x ) * p ( x , !) ∫ r

r

x = rcos ( - !) r

) d!=

0

∫

g[ x r , !]

0

F ig . 2　T wo kinds o f scan m odes

T he principle of t he LF OV alg orit hm based on the above idea is to convert the incomplete project ions of t he f an beam collect ed by t he act ual de-

x = rcos( - !) r

d!

( 1)

tect or D 1D 2 into the equivalent complet e projec-

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Research of Large Field of V iew S can M ode for Indust rial CT

tions of t he fan beam collect ed by t he virt ual detectors A B . Fig . 3 is t he processing chart of t he L FOV algorithm present ed by this paper. According t o the arrang ement of det ector channels , there are tw o kinds of rebin algorithms convert ing fan beam project ions into parallel beam project ions: equivalent space and equivalent angle. T he alg orit hm f or an equivalent angle is described in Ref. [ 4] . T his part will f irst ly show the algorit hm f or the equivalent space. T hen an L FOV reconst ruction algorit hm is presented.

F ig . 3　R econstruction flow for L F OV

2. 1 　Rebin al gorithm for equival ent space fan beam

!= ∀+ #= ∀ + arct an( s/ D) , x r = scos#= sD /

D 2 + s2

Based on t he above equat ions, Fig . 5 depict s t he t ransform bet ween the t wo address spaces: ( s, ∀) and ( x r , !) . Every point of int ersection is the position of a ray . T he point of intersect ion of the solid lines is the address storing the f an beam projection. T he point of int ersect ion of t he dot lines is t he address st oring t he parallel beam project ion. T he point of intersect ion of t he dot lines is w hat t his paper needs t o calculat e.

Fig. 5　 T r ansform betw een space ( s, ∀) and ( x r , !)

From Fig. 5, one can know that it is possible t o get t he address storing t he parallel projections

T he essent ial to convert fan beam projections ( , p s ∀) into parallel beam projection p ( x r , !) is to

denot ed by the point of intersect ion of t he dot lines in virt ue of the int erpolat ion of the position of t he

construct a t ransform bet w een t w o address spaces:

point of intersect ion of the solid lines v ertically and horizont ally . x r , !, ∀ and s are discretized . p ( s , ∀)

( s, ∀) and ( x r , !) . Fig . 4 depict s t he geomet ry relat ionship betw een t he f an beam ray and parallel beam ray un-

and p ( x r , !) are represented by p ( i∃ s, j ∃∀) and p ( n∃ x r , m∃ !) . T hey are sim plified t o p ( i, j ) and p ( n, m ) . T here are int erpolat ion results as follow s ( linear int erpolation) p ( i, m) = ( 1 -

j

p ( n, m) = ( 1 -

i

j

) p ( i, j ) +

j

p ( i, j + 1)

) p ( i, m) +

i

p ( i + 1, m)

= ( m - arct an( i∃ s/ D) / ∃ ∀) Int [ m - arct an( i∃ s/ D) / ∃ ∀] =

n2d 2D 2/ ( D 2 - n2d 2 ) -

Int [

n2 d 2D 2 / ( D 2 - n2 d 2) ]

i

F ig. 4　 G eo met ry r elationship of parameters of equiv alent space fan beam ray

der a certain projection angle. T he relat ionships among these parameters of the f an beam and parallel beam im plied in Fig . 4 can be ex pressed as

Where Int [ ] is t he int eg er part . 　　From the above equat ions, t he parallel projections can be calculat ed based on t he fan beam project ions collected by the det ect or A D 2 . T his algorit hm can be also used to get the f an beam projections from parallel beam project ions conversely .

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2. 2　LFOV rebin reconstruction al gorithm Follow ing t he alg orithm illustrated in F ig . 3, it is necessary to t ransform t he project ions collected

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t he size of t he reconstruction imag e: 360× 360; t he number of view s: 360. T he equivalent ray beam angle is 13° . F ig . 7

by t he detect or t he D 1 D 2 to the detector the A D 2 . A m ethod is linear int erpolation . T he int erpolat ion

show s L FOV projections processing flow and reconst ruction image .

formulas as f ollow s can be deduced f rom the g eomet ry relationship bet ween D 1D 2 and A D 2.

Table 1　Geometry parameters of the model

t = s/ cos( / 2) p ( s, ∀) = p ( Int [ t] , ∀) + ( t - Int[ t ] )

( 3)

　　( p ( Int [ t ] + 1, ∀) - p ( Int [ t] , ∀) ) 　 　 T hen t he fan beam project ions collect ed by the det ect or A D 2 are rebinned int o parallel beam project ions using t he algorit hm described in 2. 1. T he follow ing step is t o mirror t he parallel beam projections f or 360°t o t he project ions collect-

ellipse 1 2 3 4 5 6 7 8 9 10 11 12

origin posit ion X Y 0 0 - 10 10 - 30 30 - 50 50 - 70 70 - 100 100

0 0 0 0 0 0 0 0 0 0 0 0

half of t he lengt h of axis long axis short axis 144 120 1 1 2 2 4 4 8 8 12 12

144 1. 5 1 1 2 2 4 4 8 8 12 12

at tenuat ion coef f icient 1500 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000

ed by t he detector BD 2 in virtue of t he sym met ry of the parallel beam t hat p ( x r , !) = p ( - x r , !± ). T he algorithm in 2. 1 is conversely used again to convert the parallel beam projections collect ed by the detect or BD 2 int o corresponding f an beam project ions. Finally t he equivalent and com plete fan beam project ions for 360°collected by t he det ect or A B can be created by connecting the fan beam projections collect ed by the det ect ors A D 2 and BD 2. Now the conventional algorit hm for f an beam in Eq. ( 2) can be used t o reconst ruct the image.

3　Com puter Sim ulation In order t o verif y t he L F OV scan mode and it s reconst ruct ion algorithm, a m odel show n in Fig. 6 is used. T able 1 g ives t he param et ers of t he model. T he simulat ion condit ions are ( unit : pixel) :

Fig . 6　 Simulation model

the distance from X-ray source to rotation cent er D : 800; the number of detect or channels: 360;

F ig . 7　 LF OV pr ojections processing flow and reco nstruct ion image ( a ) 180 × 373 or iginal fan beam pro jections; ( b) fan beam pro ject ions r esulting fr om Eq. ( 3) ; ( c) 180×360 equivalent par allel beam projectio ns for 360°resulting fr om t he alg orithm in 2. 1; ( d ) 180× 360 equiv alent parallel beam projections for 360°collected by detector BD 2 ; ( e) 180×360 equivalent fan beam pr ojections for 360°co llected by detector BD 2 r esulting fr om t he algo rithm in 2. 1; ( f ) 360× 360 fan beam pr ojections for 360°collected by detect or A B resulting from the connecting in 2. 2; ( g ) filter ed projections; ( h ) fan beam r econstr uctio n image

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Research of Large Field of V iew S can M ode for Indust rial CT

4　P erform ance Analysis T his part makes a research about t he precision of t he LF OV scan mode and its reconst ruct ion algorit hm. T he model in Fig . 6 is simulated and reconstructed respectively by fan beam RO, parallel

　 　Addit ionally , noise w it h t he st andard deviation 1% is added int o the project ions. T he signal noise rat io of the reconst ruct ion images is calculated t o evaluat e the anti-noise performance of t he algorit hms. T he results are illust rat ed in F ig . 9 and T able 3.

beam and L FOV. T he reconstruct ion images are compared w ith t he model. Regularized m ean square dist ance , reg ularized m ean absolute distance and t he worst distance are the estimation st andards [ 2]

of performance . T he result s are illustrated in Fig . 8 and T able 2.

Fig. 9　 Compar iso n in anti -noise per formance betw een par allel beam, fan beam and L FOV Table 3　SNR of reconstruction images

Fig. 8　 Comparison betw een the images reco nstruct ed by parallel beam, fan beam and L FO V

f an beam

LFOV

14. 13

14. 02

14. 82

　　 T he above result s show t hat t he L FOV scan mode and its reconstruction algorithm are right .

5　Experiments

Table 2　Image qual ity parameters mean square dist ance

mean absolut e dist ance

w orst distance

0. 0397

0. 0078

245. 2013

fan beam

0. 0407

0. 0081

299. 6062

LFOV

0. 0402

0. 0091

330. 8535

parallel beam

parallel beam

T he L FOV scan mode and it s alg orit hm are verified by t he experiments using ACT IS-450/ 600 ICT syst em of BIR Corp. and DR / ICT -320/ 400 ICT syst em of Beijing Universit y of Aeronat ics and

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FU Jian, L U Hong -nian

Astronautics w it h the blade of a jet engine as t he object. F ig . 10 is the reconst ruct ion image of a slice of the object .

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7　Conclusions When t he t able rotates 360°at the points O 1 and O2 respect ively, the project ions collected by t he det ect or D 1D 2 can be rebinned int o complet e fan beam projections f or 360°in virtue of anot her t echnology developed by the authors. T he result s of com put er simulation and ex perim ent s verify t he validit y of t he LF OV scan mode and it s reconstruct ion algorit hm . Because t here exist several int erpolat ion steps, the precision of t his algorithm is, to some ex tent , limit ed. T his kind of scan m ode is benef icial to t he rapid t est of large objects w hen the requirem ent of t he precision is medium.

Fig. 10　 LF OV imag e o f the blade of jet engine ( a) or iginal pr ojection ; ( b ) reconstr uction imag e

References [ 1] 　 E1441-00, St andard guide f or computed tomography ( CT ) imaging [ S] . A ST M , 2000.

6　Effective Scan View Analysis From F ig . 1 ( c ) , the maximum size of scan view f or RO is r 1 when the detect or D 1 D 2 remains it s length . U nder t he sam e conditions, t he max imum size of scan view for L FOV is r 2 . T he g eomet ry relationship implied in F ig . 1 can be ex pressed as

[ 2] 　 Herman G T . Im age reconst ruct ion f rom projection: t he fundament als of computed t omography [ M ] . N ew Y ork : A cadem ic Press , 1980. [ 3] 　 K ak A C , M alcolm S. Principles of computerized t omogr aphic imaging[ M ] . N ew Y ork: IEE E Press , 1999. [ 4] 　 庄天戈. CT 原理与 算法[ M ] . 上 海: 上 海交通大 学出版 社, 1992. Zhuang T G. C T principle and algorit hm [ M ] . Shanghai: Shanghai Jiao tong U niversit y Press, 1992. ( in Chinese)

Biography: r 2 = D/ ( 1 + 1/ ( 2 × r 1 / ( D - r 1 ) ) )

　 　 For ICT syst em s, the value of D lies f rom 950mm t o 1000m m. When D is 950mm and r 1 is 100mm, r 2 equals 190m m and r 2/ r 1 equals 1. 9. When D is 1000m m and r 1 is 100mm, r 2 equals 200mm and r 2/ r 1 equals 2. Obv iously , t he LF OV scan mode can im prove the ef fective scan field of view by more t han 90% compared w it h t hat of RO .

FU Jian 　Born in 1976, he r eceiv ed B . S . from Beijing U niversity o f Aer onautics and A stro nautics in 1999. He is studying fo r his doctor al deg ree in BU A A . Since 1999, he has done r esearch w ork on industrial digital r adiolog y and computed to mogr aphy . He has published some r elative papers in var ious per iodicals. T el: ( 010) 82317867, E -mail: fujian 706@ 263. net