Statistical Interior Tomography

Transcription

1 Statstcal Interor Tomography Qong Xu a, Hengyong Yu b,c, Xuanqn Mou a, Ge Wang c,d a Insttute of Image processng and Pattern recognton, X'an Jaotong Unversty, X'an, Shaanx 70049, P.R.Chna b Dept. of Radology, Dvson of Radologc Scences, Wake Forest Unversty Health Scences, Wnston-Salem, C 757, USA c Bomedcal Imagng Dvson, VT-WFU School of Bomedcal Engneerng and Scences, Wake Forest Unversty Health Scences, Wnston-Salem, C 757, USA d Bomedcal Imagng Dvson, VT-WFU School of Bomedcal Engneerng and Scences, Vrgna Tech., Blacksburg, VA 406, USA ABSTRACT The long-standng nteror problem has been recently revsted, leadng to promsng results on exact local reconstructon also referred to as nteror tomography. To date, there are two key computatonal ngredents of nteror tomography. The frst ngredent s nverson of the truncated Hlbert transform wth pror sub-regon knowledge. The second s compressed sensng (CS) assumng a pecewse constant or polynomal regon of nterest (ROI). Here we propose a statstcal approach for nteror tomography ncorporatng the aforementoned two ngredents as well. In our approach, projecton data follows the Posson model, and an mage s reconstructed n the maxmum a posteror (MAP) framework subject to other nteror tomography constrants ncludng known subregon and mnmzed total varaton (TV). A determnstc nteror reconstructon based on the nverson of the truncated Hlbert transform s used as the ntal mage for the statstcal nteror reconstructon. Ths algorthm has been extensvely evaluated n numercal and anmal studes n terms of major mage qualty ndces, radaton dose and machne tme. In partcular, our encouragng results from a low-contrast Shepp-Logan phantom and a real sheep scan demonstrate the feasblty and merts of our proposed statstcal nteror tomography approach. Keywords: Computed tomography (CT), nteror tomography, compressed sensng (CS), truncated Hlbert transform, maxmum a posteror (MAP) reconstructon. ITRODUCTIO Reconstructon from the truncated projecton data assocated wth lnes through the regon-of-nterest (ROI) s usually called nteror problem, whch has been studed for a long tme -3. The conventonal wsdom s that the nteror problem does not have a unque soluton. Recently, t has been reported that the nteror problem s solvable f some addtonal pror nformaton s avalable n advance. Ths new exactness-orented local reconstructon methodology s referred to as nteror tomography 4. To date, there are two key computatonal ngredents of nteror tomography. One ngredent s nverson of the truncated Hlbert transform () wth pror sub-regon knowledge 5-0, whch s based on the concept of the dfferentated back-projecton (DBP) -3. Its man dea s to use the analytc contnuaton technque to extend the known sub-regon to the whole ROI. In numercal mplementaton, frst chords/pi-lnes are defned passng through the known sub-regon, then the DBP along each lne s computed, fnally certan methods, such as projecton onto convex set (POCS) 5, 9, 3 and sngular value decomposton (SVD) 8, are appled to nvert the truncated Hlbert transform () to determne the -D mage on the lne. Another ngredent s compressed sensng (CS) assumng a pecewse constant or polynomal ROI 4-7. Its man dea s to defne an approprate sparsfyng transform and an assocated objectve functon, and then the mnmzaton of the objectve functon wll lead to the true mage n ROI. A commonly used sparsfyng transform s dscrete gradent transform (DGT) and the assocated objectve functon s the sum of DGT, whch s usually called total varaton (TV), and then the nteror ROI can be exactly reconstructed va a TV mnmzaton. Developments n X-Ray Tomography VII, edted by Stuart R. Stock, Proc. of SPIE Vol. 7804, 7804I 00 SPIE CCC code: X/0/$8 do: 0.7/ Proc. of SPIE Vol I-

2 The two aforementoned methods based on or CS s exactly for noseless data when the precse pror knowledge on a subregon n an ROI s known or the ROI s ndeed pecewse constant or polynomal 8. However, these methods dd not take nto account the statstcal nature of projecton data, and wll not work well n the case of low count data. In fact, the projecton measurements should be assumed to obey certan specfc statstcal dstrbuton 9-. Because the well-known statstcal teratve reconstructon (SIR) algorthm can accommodate the physcal models of data acquston protocols and demonstrate a better bas-varance performance, t s much more promsng than other reconstructon methods. In ths paper, we propose a statstcal nteror tomography approach to obtan a better performance of nteror tomography for practcal CT applcatons. In the next secton, we wll descrbe our algorthm scheme. In Secton 3, the expermental results for both smulated and real data are shown. Fnally, we wll dscuss the related ssues and conclude the paper n the last Secton 4.. METHODOLOGY In ths secton, a SIR algorthm wll be developed to solve the nteror problem. Frst, we wll revew the orgnal SIR dea. Then, we descrbe the CS method for regularzaton and POCS methods for ntalzaton. These ngredents wll be then ntegrated nto our statstcal nteror tomography scheme.. SIR algorthm For smplcty, we assume that the x-ray source s monochromatc and the measurements follow a Posson dstrbuton, where Posson b e, =, L, I, () p y ~ { } y s the measurement along the th projecton path wth be p beng the expected value, b the blank scan factor, p the lnear ntegral of lnear attenuaton coeffcent along the th projecton path and I s the number of x- ray paths. For SIR, the object s dscretzed as rectangular pxels, p can be dscretzed as follows, where J (,, ) [ ] l j j j= p = x y z dl a = Α, =, L, I, () l s the x-ray path, ( x, yz, ) the lnear attenuaton coeffcent of the materal at a 3-D locaton (,, ) x yz, the number of the pxels, A = { a j } the system matrx whch accounts for the system geometry, ( L ) and the symbol represents a transpose operator. For the normalzed ntersecton area between the pxel and the ray beam. th x-ray path and the J,, =, J th j pxel, a j can be calculated as the Because the measurements along dverse x-ray paths are ndependent of each other n a statstcal sense, the jont probablty dstrbuton of the data acquston process can be expressed as e y P y = =, (3) y I y = = y! I ( ) P( y ) and the correspondng log-lkelhood functon (gnorng the constant terms) can be wrtten as I I p ( y ) ln ( y ) ( ) [ Α ] [ Α] ( ) L = P = y p + be = y + be. (4) = = From the statstcal perspectve, the orgnal mage can be reconstructed by maxmzng a posteror (MAP) of functon P( y). Snce the natural logarthm s monotoncally ncreasng, the maxmzaton of a posteror P( y) Proc. of SPIE Vol I-

3 can be carred out by maxmzng ts logarthm. Accordng to the Bayesan rule P( y) P( y) = P( y ) P( ) mage reconstructon task s equvalent to the maxmzaton of the followng objectve functon { L( y ) + ln P( )}, the = arg max, (5) where ln P ( ) expresses the pror knowledge on the object. Because ( ) R ( ) and the objectve functon can be rewrtten as I [ ] ( y[ ] be Α Α + ) + R( ) ln P s a regularzaton term, we denote t as = arg mn. (6) = pˆ Applyng a second-order Taylor s expanson to functon ( ) p b = ln 9, Eq. (6) can be expressed as y g p = y p+ be around an estmated lne ntegral I y = arg mn ([ Α] pˆ ) + R( ). (7) = The regularzaton term R ( ) usually penalzes the dfference among neghborng pxels snce the ntenstes of adjacent pxels are normally smlar. A general form of the regularzaton term s J ( ) = kj ( j k ) R β ωϕ, (8) j= k Cj where β s an emprcal scalar to tradeoff the data fdelty and regularzaton term, ω kj s the weghts on dfferental components, C j s the neghborhood of the j th pxel, and ϕ s a potental functon that determnes the effect of the dfferental component.. CS method In the deal case of nose-free data, the nverson procedure usng the CS method can be expressed as follows mn Ψu, st.. Φu =b, (9) u where Φ s measurement matrx, b s the vector of measurements, u s the vector of pxels value, Ψu s a sparsfyng transform of u, and represents the l norm. Among all the exstng sparsfyng transforms, the dscrete gradent transform (DGT) s most commonly used. The sum of DGT for each pxel s usually called total th varaton (TV) of an mage. In a -D mage space, the lnear attenuaton coeffcent j of the j pxel can be redenoted n dual subscrpts as =, j = ( m ) W + n, m =,, L, H, n =,, L, W, (0) j m, n where W and H are respectvely the wdth and heght of the -D mage array, and mage can be expressed as, J = W H. Then, the TV of an Proc. of SPIE Vol I-3

4 TV H W ( ) dmn, =, and dmn, ( mn, m+, n) ( mn, mn, + ) m= n= = +. () R wth β Ψ. If we choose the DGT as sparsfyng transform of an mage, we arrve at a CS based SIR framework for mnmzng the followng objectve functon The CS method can be ncorporated nto the SIR framework n Eq. (7) by substtutng ( ) I y arg mn ([ ] pˆ ) TV ( ) β = Α +. () = There are varous ways to mnmze the above objectve functon. Whle Tang et al employed the Gauss-Sedel scheme, n ths paper we wll use an alternatve mnmzaton method n terms of soft-threshold flterng 3. Theren, for the CS based mage reconstructon expressed n Eq. (9), the data fdelty step wth the smultaneous algebrac reconstructon technque (SART) 4 and the TV mnmzaton step va soft-threshold flterng are performed n an alternatve manner. Smlarly, to mnmze the objectve functon Eq. (), we deal wth the log-lkelhood term I y arg mn ([ ] ˆ ) Α p and the TV term arg mn{ TV ( )} alternatvely. Wth the separable parabolod = surrogate subject to non-negatvty, each update for the log-lkelhood term s obtaned as follows = I n ( a ([ ] ˆ j y Α p )) n+ n = j j I J aj y aj = j= +. (3).3 Inverson of by POCS When there s a known subregon nsde an ROI, the nteror problem can be solved by the nverson of wth the POCS method. Frst, a set of chord/pi lnes are constructed, whch go through both the known and unknown regons n the ROI. The lnear attenuaton coeffcent ( x, yz, ) on such a chord/pi-lne L s re-denoted as f ( t ), where t s the -D coordnate along L. Let the support of f ( t ) on L be [ c, c ], the nterval of the ROI on L be ( c3, c 4), the nterval of known sub-regon on L be ( c5, c 6), and these constants satsfy c c3 c5 < c6 c4 c. as Second, the along each chord/pi-lne nsde the ROI s computed by the DBP method, whch can be expressed c ds g t PV f s H f t π t s () = ( ) = ( L )( ), t ( c3, c4) By the Trcom formula, f ( t ) can be recovered from ts Hlbert transform g( t ) where C () f c c c ( c t)( t c ) f () t = C + PV g( s) ( c s)( s c ) f π c = f t dt π s a known quantty (the projecton along the chord). c. (4) ds s t, (5) Proc. of SPIE Vol I-4

5 Thrd, along each chord s nverted by the POCS method. Because Eq. (5) can not be drectly used to nteror problem, the POCS method s commonly used to solve f from Eq. (4). It assumes that the -D functon f belongs to the ntersecton of J convex sets C, C, L, CJ. If the projecton operators onto these convex sets are denoted as k+ k P, P, L, PJ, POCS can be expressed as f = PP J J L Pf, where k ndcates the teraton tme. In other f t L words, the nteror problem s essentally to fnd ( ) ( ) { ( )( )() (), (, )} C = f L Hf t = g t t c c 3 4 { ( ) ( ) ( ), (, )} C = f L f t = f t t c c ( ) c C = f L f () t dt = Cf π c { ( ) ( ) 0, [, ]} C = f L f t t c c 4 { ( ) ( ), [, ]} C = f L f t f t c c 5 max n the ntersecton of the convex sets: where f0 () t and f max are the deal mage functon and ts upper bound, respectvely. More convex sets can be ntroduced f addtonal convex constrants are avalable..4 Scheme of statstcal nteror tomography As shown n Fgure., the proposed statstcal nteror tomography scheme combnes the -based and CS-based nteror reconstructon algorthms n a statstcal framework. Whle the result of the -based nteror reconstructon s used as the ntal guess, the CS-based nteror tomography s mplemented to mnmze the log-lkelhood term and the TV term alternatvely. -based nteror reconstructon by POCS Intal mage CS-based nteror reconstructon n statstcal framework arg mn = y arg mn TV ([ Α] pˆ ) { ( )} Fgure.. Scheme of the proposed statstcal nteror tomography Proc. of SPIE Vol I-5

6 3. EXPERIMETS AD RESULTS The proposed statstcal nteror tomography was evaluated n both numercal smulaton and practcal applcatons. 3. umercal Smulaton In the numercal smulatons, we used a low contrast -D Shepp-Logan phantom. A fan-beam geometry and an equspatal vrtual detector were assumed. The vrtual detector was centered at the system orgn and always perpendcular to the lne from the system orgn to the x-ray source. The dstance from the x-ray source to the system orgn was 57 cm and the detector ncluded 360 elements wth a total length of 0.8 cm. For a full scan, we equ-angularly collected 080 projectons wth 0 6 photons per detector element and 360 projectons wth photons, respectvely. The smulated statstcal nteror reconstructon conssts of two major steps: -based reconstructon and CSbased reconstructon. In the -based reconstructon, chords/pi- lnes were constructed along the horzontal drecton, and the mage values on a subregon of each chord were precsely known n advance. The ROI for the -based reconstructon was an nscrbed square nsde the FOV as llustrated n Fg.. The reconstructed mages covered an FOV of radus n a matrx. On each chord, the ROI covered 96 pxels wth a known sub-regon of 6 pxels. The DBP was carred out only n the square ROI. The maxmum teraton number was 500. In the CS-based reconstructon, the ROI was defned by the local scannng beam. An ordered subsets strategy was adopted to accelerate the TV mnmzaton based SIR. The maxmum teraton number was 0 wth 0 subsets Fgure. The D Shepp-Logan phantom wthn a dsplay wndow [0.9.]. We evaluated the results of the -based reconstructon (denoted by ), the CS-based statstcal reconstructon ntalzed by a zero mage (denoted by ) and the CS-based statstcal reconstructon ntalzed by the result of the (denoted by SIRCS-). Fgure. 3 showed the reconstructed mages by dfferent reconstructon schemes. It can be observed that the mages reconstructed by the are much nosy, especally n the condton of 360 projectons wth photons. The nose ntensty n the results reconstructed by other algorthms was lower and more stable. Moreover, the results reconstructed by had a bas but the SIRCS- acheved the best performance. Representatve profles along the lne a (see Fgure.) of the reconstructed mages were shown n Fgure. 4. The reconstructon accuracy of the was affected by the pxel poston. That s, the closer to the known subregon, the more accurate the result s, whch s consstent wth the theoretcal analyss on stablty of nteror tomography 5. Compared to the other two schemes, showed much stronger nose. The had weaker nose, but a larger bas. It may be due to several factors such as the teraton number, pxel sze, mage sparsfy, etc. However, SIRCS- seems always convergng to the truth. We selected a rectangular sub-regon b (see Fgure.) n the left ellpse wth the true attenuaton coeffcent The reconstructed results of the sub-regon b usng the three methods n the cases of 080 vews and 0 6 photons were evaluated n Table n terms of average error ε, maxmum error ε max and standard varaton σ. The standard varaton wth was the largest. The bas wth was the greatest. The performance of SIRCS- outperformed the other two methods consstently, beng the closest to the phantom mage. 0.9 Proc. of SPIE Vol I-6

7 Fgure. 3. Images reconstructed n the ROI by the,, and SIRCS- schemes. The dsplay wndow s [0.9.] for all the mages. 080 Vews 0 6 Photons 360 Vews Photons Phantom SIRCS Phantom SIRCS- f(t) f(t) Pxel Pxel Fgure. 4. Profle of the reconstructed mages along the lne a ndcated n Fg.. Whle the left s from 080 projectons and 0 6 photons, the rght s from 360 projectons and photons. Table. Comparsons of, and SIRCS- n terms of average error, maxmum error and standard devaton. Condtons Methods ε ε max σ Vews Photons SIRCS Proc. of SPIE Vol I-7

8 3. Real CT data To demonstrate the feasblty of the statstcal nteror reconstructon for practcal applcatons, we performed a CT scan of a lvng sheep, whch was approved by the Unversty of Iowa and Vrgna Tech IACUC commttees. The chest of the sheep was scanned n fan-beam geometry on a SIEMES 64-Slce CT scanner. The radus of the x-ray source scannng trajectory was 57 cm. Over a 360 range, 60 projectons were unformly collected. For each projecton, 67 detector elements were equ-angularly dstrbuted to defne a FOV of radus 5.05 cm. In our experments, two scans were performed wth a normal dose (00kVp, 50mAs) and a low dose (80kVp, 7mAs), respectvely. We frst reconstructed the entre lung cross-secton n a 5 5 matrx coverng a 9.06 cm 9.06 cm regon from the normal dose full-scan dataset. Then, a subregon of radus 6 pxels was selected n a trachea, where the attenuaton coeffcent was known to be zero. After that, a crcular regon of radus 60 pxels around the trachea was chosen as an ROI. Fnally, only the projecton data through the ROI were kept to smulate an nteror scan. In ths stuaton, the PI-lnes were constructed along all radal drectons from the center of the trachea. The maxmum teraton number of was fxed as 500. The maxmum teraton number of the SIR was 40 wth 40 subsets. Because the sheep s lve, the mages of the two scans at the normal and low dose levels were slghtly dfferent due to physologcal moton. Therefore, the reference mages were reconstructed from the correspondng full scan datasets, respectvely. To mprove the mage qualty, the reference mages were reconstructed usng the SIR method nstead of the commonly used flterng backprojecton (FBP) method. Fg. 6. Images n the ROI reconstructed by,, and SIRCS- from normal and lose dose datasets. Dsplay wndow s [-700HU 800HU] (HU = 0.08/mm). Fg. 6 showed the mages reconstructed by the aforementoned three reconstructon schemes wth dfferent dose levels. It can be observed that the mages reconstructed by the have a lower spatal resoluton. One reason s that the PI-lnes were constructed along radal drectons, and the nterpolaton was requred for a coordnate transformaton 5. The results reconstructed by the other two algorthms had a hgher spatal resoluton and also led to a bas as what we notced n the numercal smulatons. Typcal profles of the results were shown n Fg.7. It s notced that whle smoothened the mage, the other two methods reserved more detals. stll suffered from a substantal bas. Although the SIRCS- performed much better than both and methods, there were stll some resdual artfacts especally near the perpheral regon of the ROI. We are developng more sophstcated algorthms to suppress ths knd of artfacts. Because the deal mage s not known, t would be meanngless to compute the average error and standard devaton aganst the gold standard. Instead, we evaluated the results reconstructed wth the three methods from that reconstructed from the normal dose datasets n terms of spatal resoluton and an mage qualty assessment (IQA) ndex SSIM 6. The results were shown n Table. Whle the SSIM measures were computed n reference of the globally reconstructed mages, spatal resoluton was estmated across the nternal border of the trachea as the full-wdth-of-half-maxmum Proc. of SPIE Vol I-8

9 (FWHM) of the lne spread functon ftted nto the Gaussan form 7. Compared to the reference mages, SIRCS- had the best structural smlarty accordng to SSIM. Besdes, SIRCS- produced a hghest spatal resoluton. SIRCS- Zero dd not work as well as SIRCS- but t outperformed Reconstructed Value (HU) Reconstructed Value (HU) Vews and ormal Dose Ref SIRCS Pxel 60 Vews and Low Dose Reconstructed Value (HU) Vews and ormal Dose Ref SIRCS Pxel Ref SIRCS Pxel Pxel Fg. 7. Typcal profle of the reconstructed results. The upper and bottom rows are respectvely from normal and lose dose data. The left and rght columns are respectvely along the horzontal and vertcal central lnes of ROI. Reconstructed Value (HU) Vews and Low Dose Ref SIRCS- Table. Comparsons of the, and SIRCS- methods n terms of spatal resoluton and SSIM. Condtons Methods Resoluton (mm) SSIM Vews ormal Dose SIRCS DISCUSSIO AD COCLUSIO From the above expermental results, -based nteror tomography dd not have any sgnfcant bas but t s nosesenstve. Besdes, as demonstrated n Fg.4 the accuracy and robustness of ths algorthm become less further away from the known subregon. Snce the PI-lnes are constructed around a known subregon, they are usually not consstent wth Proc. of SPIE Vol I-9

10 the Cartesan grd, and the nvolved nterpolaton wll reduce the spatal resoluton as well. On the other hand, the proposed CS-based statstcal nteror tomography has dstnct merts n both nose and resoluton aspects. In the numercal smulaton, satsfactory results have been obtaned even n the case of 360 vews and photons. Usng the IQA ndex SSIM to assess the results from real CT datasets, the mages reconstructed usng ths method have produced a better structural smlarty n reference to the globally reconstructed counterpart. The spatal resoluton comparson has also llustrated that the proposed method has a better spatal resoluton. However, TV mnmzaton based nteror tomography assumes the smooth property of ROI mages, and produce suboptmal results when ths assumpton s sgnfcantly volated. In ths regard, the hgh order TV (HOT) mnmzaton approach would be a promsng tool 7. Addtonally, ths algorthm must be stopped after fntely many teratons. Hence, SIRCS wth a zero ntal mage usually leads to a based result. When the result s used as an ntal guess, SIRCS has been shown to arrve at a globally optmal result very relably. From another pont of vew, the based ntalzaton adds the Hlbert constrant, whch ncorporates the known subregon nto the fnal result. Therefore, the SIRCS- method not only has a strong ant-nose power and better structural detals but also t s capable of elmnatng any potental bas effectvely. In concluson, we have proposed a statstcal nteror tomography approach by combnng -based nteror tomography, CS-based nteror tomography and a statstcal reconstructon framework. Our smulaton and experments have shown that t s a powerful and useful tool for local CT reconstructon n practce applcaton. ACKOWLEDGEMET Ths work s partally supported by SFC (o ), the program of Chnese Mnstry of Educaton (o , CET ), SF/MRI program (CMMI-09397), and IH/IBIB grants (EB00667, EB0785). REFERECES [] atterer, F., [The mathematcs of computerzed tomography] Socety for Industral Mathematcs, (00). [] Maass, P., The nteror Radon transform, SIAM Journal on Appled Mathematcs, 5(3), (99). [3] Lous, A. K., and Reder, A., Incomplete data problems n X-ray computerzed tomography, umersche Mathematk, 56(4), (989). [4] Yu, H. Y., Ye, Y. B., and Wang, G., "Interor tomography: theory, algorthms and applcatons," Proc. SPIE. 7078, 70780F (008). [5] Ye, Y. B., Yu, H. Y., We, Y. C. et al., A general local reconstructon approach based on a truncated Hlbert transform, Internatonal Journal of Bomedcal Imagng, Artcle ID:63634, 8 (007). [6] Ye, Y. B., Yu, H. Y., and Wang, G., Exact nteror reconstructon wth cone-beam CT, Internatonal Journal of Bomedcal Imagng, Artcle ID:0693, 5 (007). [7] Ye, Y. B., Yu, H. Y., and Wang, G., Exact nteror reconstructon from truncated lmted-angle projecton data, Internatonal Journal of Bomedcal Imagng, Artcle ID:47989, 6 (008). [8] Yu, H. Y., Ye, Y. B., and Wang, G., Interor reconstructon usng the truncated Hlbert transform va sngular value decomposton, Journal of the X-Ray Scence and Technology, 6(4), 43-5 (008). [9] Kudo, H., Courdurer, M., oo, F. et al., Tny a pror knowledge solves the nteror problem n CT, Physcs n Medcne and Bology, 53, 07-3 (008). [0] Courdurer, M., oo, F., Defrse, M. et al., Solvng the nteror problem of computed tomography usng a pror knowledge, Inverse Problems, 4, (008). [] Gel'fand, I. M., and Graev, M. I., Crofton's functon and nverson formulas n real ntegral geometry, Functonal Analyss and Its Applcatons, 5(), -5 (99). [] oo, F., Clackdoyle, R., and Pack, J. D., A two-step Hlbert transform method for D mage reconstructon, Physcs n Medcne and Bology, 49, (004). [3] Defrse, M., oo, F., Clackdoyle, R. et al., Truncated Hlbert transform and mage reconstructon from lmted tomographc data, Inverse Problems,, (006). [4] Yu, H. Y., and Wang, G., Compressed sensng based nteror tomography, Physcs n Medcne and Bology, 54(9), (009). [5] Yu, H. Y., Yang, J. S., Jang, M. et al., Supplemental analyss on compressed sensng based nteror tomography, Physcs n Medcne and Bology, 54, (009). Proc. of SPIE Vol I-0

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