CT Image Reconstruction in a Low Dimensional Manifold

Transcription

1 CT Image Reconstructon n a Low Dmensonal Manfol Wenxang Cong 1, Ge Wang 1, Qngsong Yang 1, Jang Hseh 3, Ja L, Rongje La 1 Bomecal Imagng Center, Department of Bomecal Engneerng, Department of Mathematcal scences Rensselaer Polytechnc Insttute, Troy, Y GE Healthcare Technologes, Waukesha, WI Abstract: Regularzaton methos are commonly use n X-ray CT mage reconstructon. Dfferent regularzaton methos reflect the characterzaton of fferent pror knowlege of mages. In a recent work, a new regularzaton metho calle a low-mensonal manfol moel (LDMM) s nvestgate to characterze the low-mensonal patch manfol structure of natural mages, where the manfol mensonalty characterzes structural nformaton of an mage. In ths paper, we propose a CT mage reconstructon metho base on the pror knowlege of the low-mensonal manfol of CT mage. Usng the clncal raw projecton ata from GE clnc, we conuct comparsons for the CT mage reconstructon among the propose metho, the smultaneous algebrac reconstructon technque (SART) wth the total varaton (TV) regularzaton, an the fltere back projecton (FBP) metho. Results show that the propose metho can successfully recover structural etals of an magng object, an acheve hgher spatal an contrast resoluton of the reconstructe mage than counterparts of FBP an SART wth TV. Key Wors: X-ray compute tomography (CT), mage reconstructon, fltere backprojecton (FBP), smultaneous algebrac reconstructon technque (SART), total varaton (TV), low mensonal manfol moel (LDMM). 1. Introucton X-ray compute tomography (CT) s a major magng moalty n mecal, securty, an nustral applcatons. The fltere back-projecton (FBP) s an effcent an robust metho for x-ray CT mage reconstructon [1], but t generates strong mage nose an artfacts n the cases of low-ose or ncomplete atasets. Extensve efforts have been mae to mprove mage qualty for practcal purposes [-4]. Iteratve methos ncorporate pror nformaton of mages, an offer stnct avantages over the analytc methos n cases of nosy an few-vew ata. The statstcal teratve methos moel the statstcs of photons to mprove the reconstructe mage qualty from the low-ose acqustons [4, 5]. Recently, the compressve sensng (CS) approach [6, 7] s apple for the mage reconstructon from less measurements than that requre by the yqust-shannon samplng theorem. Base on the CS theory, mage reconstructon algorthms were evelope for varous problems of CT mage reconstructon for mprovng mage qualty an reucng raaton ose, such as total varaton (TV) regularzaton [3, 5], nonlocal mean (LM) [, 8], ctonary learnng (DL) [9], pror mage constrane compresse sensng (PICCS) [10], an pror rank an sparsty moel (PRISM)-base mage reconstructon [11]. TV s a typcal sparse transform for an mage, an s a popular regularzaton form for mage

2 reconstructon ue to ts ablty to preserve mage eges. However, t s effectve only for reconstructon of pecewse constant mages an woul over-smoothen texture regons, whch may sacrfce mportant etals. LM explots a hgh egree of reunancy of an mage for e-nosng [8]. The smlarty s erve from ntensty fferences between neghborng patches of pxels or voxels. A non-lnear flter can be use to reuce mage nose by upatng each pxel value wth a weghte average of ts neghbors accorng to the smlarty of nvolve patches. DL buls aaptve sparse representaton elements from a tranng set of mages, an utlzes oman knowlege at a eeper level [9]. The ctonary tens to capture local mage features effectvely an helps mage enosng an feature nference. However, the structural fferences between a true mage an tranng mages coul affect the mage reconstructon qualty. PICSS regularzes mage reconstructon wth a pror mage nstea of mage patches [10]. PRISM combnes sparsty an low rank expectatons of an mage. All these methos were reporte wth varous egrees of success but no perfect soluton exsts that s suffcently accurate an robust, an further mprovement n mage qualty remans a popular topc. The ea of the propose X-ray CT mage reconstructon moel s nspre by a recent metho calle the lowmensonal manfol moel (LDMM) [1, 13]. Usng the mage patches scusse n nonlocal methos [13], the LDMM nterprets mage patches as a pont clou sample n a low-mensonal manfol embee n a hgh mensonal ambent space, whch proves a new way of regularzaton by mnmzng the menson of the corresponng mage patch manfol. Ths can be explane as a natural extenson of the ea of low-rank regularzaton for lnear objects to ata wth more complcate structures. Moreover, the authors n [1] elegantly fn that the pont-wsely efne manfol menson can be compute as a Drchlet energy of the coornate functons on the manfol, whose corresponng bounary value problem can be further solve by a pont ntegral metho propose n [14]. The LDMM performs very well n mage mprntng an super-resoluton. In ths paper, the regularzaton metho base on LDMM s propose for CT mage reconstructon. The patch manfol of mages s generally a low mensonal structure, an yet accommoates rch structural nformaton [13]. Usng the Bregman teraton [15], the propose reconstructon moel can be teratvely solve by a sequence of soft thresholng, Posson equatons prove by the Laplace-Beltram operator over a pont clou usng the pont ntegral metho [1], an upatng the patch manfol structure by renewng the K-nearest neghborhoo. The rest of the paper s organze as follows. In secton, we prove a etale escrpton for the propose X-ray CT mage reconstructon moel base on LDMM. A umercal algorthm s also esgne base on Bregman teraton. In secton 3, we perform the mage reconstructon for the clncal raw projecton ata from GE Clnc usng the propose LDMM-base reconstructon metho. In aton, we also conuct reconstructon comparsons wth results obtane from FBP an SART wth TV. Our numercal results valate the effectveness of the propose metho. After that, we conclue the paper n the last secton.. Image Reconstructon Metho In ths secton, we frst revew the statstcal moel of x-ray CT magng. After that, we wll scuss the propose moel of CT mage reconstructon base on LDMM an ts numercal algorthm.

3 .1. Statstcal Moel for X-ray CT magng In x-ray CT magng, the number of x-ray photons recore by a etector element s a ranom varable, whch obeys a Posson strbuton [1]: y y p y exp y. (1) y! The expectaton value of x-ray photons along a path l from x-ray source to -th etector element obeys Beer- Lambert law:, () where b s the number of x-ray photons etecte by -th etector element n the blank scannng (wthout any object n the beam path), an s the lnear attenuaton coeffcent of the object. To mplement the numercal computaton, Eq. () can be scretze as, y b expa μ (3) where μ s a vector compose of pxel values on mage of lnear attenuaton coeffcents, an A s the weghtng coeffcents of the pxel values on -th beam path. Snce ata are nepenent between etectors, the lkelhoo functon for x-ray photons probablty strbuton on etectors s, 1 y y PY μ exp-y, (4) y! T where Y,,,. Accorng to the Bayesan rule: P P P P y1 y y μ Y Y Y μ μ, the mage reconstructon task can be mplemente by maxmzng a posteror (MAP) strbuton P μy [5, 16]. From the monotonc property of the natural logarthm, the mage reconstructon can be reuce to followng mnmzaton problem [5]: where R P μ arg mn y y log y R μ, (5) 1 μ ln μ s a regularzaton term expressng the pror knowlege about the attenuaton mage μ, an s the total number of x-ray beam paths. In the context, we propose to use the low-menson of an mage as pror knowlege to conuct mage reconstructon, whch s scusse n the next Secton. After nsertng Eq. (3) n Eq. (5), a secon-orer approxmaton s apple to smplfy the complcate optmzaton to a quaratc optmzaton: b μ argmn Aμ y R μ (6) 1

4 .. Image Reconstructon algorthm usng LDMM Fg.1. The patch manfol of a CT mage (left) an the corresponng menson functon of the patch manfol wth patch sze 16X16 (rght). Let I enote an mage contane m n pxels: I I, j 1 m,1 j n, an P I enotes a patch of mage I, whch s a sub-mage of I wth sze of s1 s, s,,, here, P I I j s s j s j j s s j s the central coornates of the patch. An mage s ecompose nto a set of patches. These patches can be overlappng or nonoverlappng. Let enotes all patch set such that the unon of the patch set covers the whole mage, for example 1, s 1, s 1,, m 1, s 1, s 1,, n s an nex set of the patch. pont set n n 1 1 R wth a menson of s1 s. P I samples a low mensonal manfol PI P I can be also seen as a MI embee R, whch s calle the patch manfol of I as shown n the left mage of Fg. 1. The patch manfol s low mensonal for many natural mages [13]. In fact, for X-ray CT mages, ths low-mensonal structure of the patch manfol s also true. As an example llustrate n the rght mage of Fg. 1, we construct a patch manfol of a CT mage usng patchng sze 16X16. Ths leas to a pont clou, whch pont-wse menson s colorcoe on the mage. Base on ths assumpton, one natural regularzaton term s efne as the menson of the patch manfol to seekng etal structure nformaton for the mage reconstructon. Ths metho recovers the CT mage such that the menson of ts patch manfol s as small as possble. Therefore, the optmzaton moel Eq. (6) s reformulate for the measurement ata felty an the manfol mensonal quantfcaton: μ arg mn b A μ y m μ (7) 1 where m μ enotes the mensonalty of the patch manfol μ of an mage μ. Wth fferental geometry, the mensonalty of the patch manfol can be calculate by the coornate functon [1], m M x μ (8) 1

5 where s the embeng coornate functon efne by, 1,,, 1 x x, x,, x M R. Combnng Eqs. (7) an (8), we obtan x x, for any mn M,.., M b A μ y s t P M μ, (9) μ 1 1 where M s a manfol, an P μ s the patch set. The optmzaton (9) can be solve by alternatng recton teraton. Gven an ntal mage μ, the manfols M s establshe. Then optmzaton (9) s mplemente to upate the mage μ. From the reconstructe mage μ, the manfol s further upate, an mage reconstructon s performe. Ths process s repeate untl convergence of teratve proceure. The computaton of manfol from an mage s rect an easy. Gven the manfol M, the optmzaton problem Eq. (9) can be solve to compute the coornate functons 1,,, Bregman teraton [15]. an upate the mage μ usng the splt 1,,, arg mn M μ -, 1 μ arg mn b A μ y P μ - P μ Q, b F 1 k1 k Q Q P P c P P μ Q a μ - μ, (10) In the Bregman teraton, Eq. (10b) can be reuce to a l mnmzaton, whch can be solve usng the conjugate graent (CG) metho, whch prouces the exact soluton after a fnte number of teratons. The most ffcult task s to solve the optmzaton (10a) because t contans fferental of coornate functons. Applyng the stanar varaton metho, Eq. (10a) s equvalent to solvng followng Laplace-Beltram equaton. Mu x x y u y v y 0, x M y u x 0, x M n where n s the out normal of M, an M s the bounary of M. Recently, the pont ntegral metho has been propose to solve Laplace-Beltram equaton over a pont clou [Ref]. The man ea of the pont ntegral metho s to apply followng ntegral approxmaton for the fferental term n Laplace-Beltram equaton: where R, t 1 u y Mu yrt x, yy u x u yrt x, yy Rt x, yy t n (1) M M M x y are kernel functons gven as follows, (11)

6 x y Rt x, y Ct exp, 4t (13) where C t s a normalzng factor. Usng the ntegral approxmaton (1), followng ntegral equaton can be obtane to approxmate the Laplace-Beltram equaton, ux uyrtx yy t Rtx yu x u y,, 0 (14) M The ntegral equaton (14) can be further scretze nto a matrx equaton over the pont set P μ usng some quarature rule [1]: y L+ W U = W V L = D -W (15) where t M, j Rt x, x j,, j 1,,,, W j, j 1,,, s the weght matrx, j D ag 1,,, s a agnoss matrx. Thus, the optmzaton (10a) can be solve base on the matrx equaton (15). The etale formulaton an alternatng mnmzaton steps for solvng Eq. (10) are escrbe n the flowchart for Algorthm 1. Algorthm 1 Intalze an ntal mage, Q0 0 an parameters an ; j1 1: Whle the current soluton s not converge o : Compute the weght matrx W w, j, L D W, D ag, w j j from the patch mage P n, an the matrces 3: Solve the lnear systems: L W U WV n n V P μ Q n1 4: Upate by solvng the problem: μ argmn b A y U P Q μ μ 1 5: Upate n Q : n1 n n1 Q Q U P μ n F 6: En Whle 3. Image reconstructon results In the secton, we test the propose LDM-base reconstructon moel wth real patent atasets obtane on a GE clncal CT scanner. In aton, we also compare our results wth those obtane by the conventonal fltere back projecton (FBP) metho an the smultaneous algebrac reconstructon technque (SART) wth a

7 total varaton (TV). All numercal computatons n ths secton are mplemente by MATLAB n a PC wth 16G RAM an.8ghz CPU Smulaton result: A realstc phantom aapte from a human CT slce s use to evaluate the propose algorthms. We use an computerassste tomography smulaton envronment (CatSm) [17], whch was evelope by GE Global Research Center, to smulate x-ray magng. CatSm ncorporates polychromatcty, realstc quantum an electronc nose moels, fnte focal spot sze an shape, fnte etector cell sze, an etector cross-talk for the smulaton of real x-ray magng. All acqustons are smulate wth polychromatc x-ray source operate at 10 kvp an 0.mSv ose for the low ose magng. The raus of the scannng Fg. The snogram from CatSm. trajectory s 54.1cm. 984 projectons are unformly acqure over a 360-egree angular range. For each projecton, 888 etector elements are equangular strbute. The phantom s scretze nto a matrx, an the snogram s forme by stackng all projectons of fferent vews, as shown n Fg.. We perform mage reconstructon respectvely usng the propose LDM-base mage reconstructon, the smultaneous algebrac reconstructon technque (SART) wth a total varaton (TV) an FBP metho. The comparsons show that the LDM-base mage reconstructon moel s better than the other reconstructon metho, as shown n Fg. 3. (a) (b) (c) () Fg. 3: Comparson of CT reconstructon results over lung tumor. (a) Groun truth CT mages, (b) the reconstructe mage usng the propose metho; (c) the reconstructe mage usng SART wth TV, an () the reconstructe mage usng a FBP metho. 3.. Expermental results In ths ata set, the scan s n a typcal helcal geometry. After approprate preprocessng, we obtane a set of 64-slce fan-beam snograms, as shown n Fg. 3. The raus of the scannng trajectory s 54.1cm. Over a 360- egree angular range 984 projectons are unformly acqure. For each projecton, 888 etector elements were equangular strbute. The fel of vew (FOV) s of a 5 cm raus. The mage matrx was of pxels. Then, the snogram s forme by stackng all projectons of fferent vews, as shown n Fg. 4.

8 From the snogram, we frst conuct mage reconstructon usng the propose metho. As the mage llustrate n Fg.5 (a), the propose LDMbase mage reconstructon well preserves structural nformaton especally texture features. In our metho, we choose the patch sze of 16x16 to form the patch manfol an regularzaton parameters are chosen as 0.5 an 0.. For comparson, the FBP metho an the SART wth TV regularzaton are apple as well to perform the mage reconstructon from same projecton ataset, whose results are showe n Fg.5 (b-c), respectvely. The comparsons show that the LDM-base mage reconstructon moel outperforms the other two reconstructon Fg. 4. The snogram from a clncal scanner. methos. The mage reconstructe va only SART teraton wth TV s blurry. SART wth TV s sutable to reconstruct smple structural mages. For complex mecal mages, SART-TV over-smoothens texture regons, resultng n the loss of etals. FBP keeps the structural nformaton but t makes the reconstructe mage nosy. (a) (b) (c) Fg. 5. Comparson of CT mage reconstructons from raw patent ata. (a) The reconstructe mage usng the LDM-base metho, (b) the reconstructe mage usng SART wth TV, an (c) the reconstructe mage usng FPB. 4. Dscussons an Concluson The major contrbuton n ths paper s to present an mage reconstructon metho ae by the regularzaton of a low mensonal manfol (LDM) moel. Ths metho promses substantally ncrease spatal an contrast resoluton. Our teratve algorthm also ncorporates pror knowlege, an account for photon statstcs at a low ose level. However, the computatonal cost of the propose LDM-base mage reconstructon metho s

9 hgher than the SART teratve methos. Major computatonal cost s matrx-vector multplcaton operatons n the teratve algorthm. Ths problem can be solve by parallel computaton on GPU computer because matrxvector multplcaton s hghly ata parallel computaton. The computatonal spee of the propose teratve metho can be mprove on a GPU workstaton. The comparson between the propose metho an several representatve methos has been performe to llustrate the merts of the LDMM-base reconstructon approach. The raw atasets from a clncal CT scanner have been use to evaluate the mage qualty. Results show that the regularzaton metho of low mensonal manfol s an effcent an robust mage reconstructon technque, an well preserves mage eges an structural etals of the reconstructe mage comparng to the FBP metho an the SART wth TV regularzaton. Ths LDM-base approach s very promsng for mecal magng an other applcatons. Acknowlegment: Ths work s partally supporte by the atonal Insttutes of Health Grant IH/IBIB R01 EB an U01 EB R. La s work s partally supporte by the atonal Scence Founaton SF DMS References [1] A. C. Kak, an M. Slaney, Prncples of computerze tomographc magng, Phlaelpha: Socety for Inustral an Apple Mathematcs, 001. [] S. Ha, an K. Mueller, Low ose CT mage restoraton usng a atabase of mage patches, Phys Me Bol, vol. 60, no., pp , Jan 1, 015. [3] E. Y. Sky, Y. Duchn, X. Pan, an C. Ullberg, A constrane, total-varaton mnmzaton algorthm for low-ntensty x-ray CT, Me Phys, vol. 38 Suppl 1, pp. S117, Jul, 011. [4] I. A. Elbakr, an J. A. Fessler, Statstcal mage reconstructon for polyenergetc X-ray compute tomography, IEEE Trans Me Imagng, vol. 1, no., pp , Feb, 00. [5] J. Tang, B. E. ett, an G. H. Chen, Performance comparson between total varaton (TV)-base compresse sensng an statstcal teratve reconstructon algorthms, Phys Me Bol, vol. 54, no. 19, pp , Oct 07, 009. [6] E. J. Canes, J. Romberg, an T. Tao, Robust uncertanty prncples: Exact sgnal reconstructon from hghly ncomplete frequency nformaton, IEEE Transactons on Informaton Theory, vol. 5, no., pp , Feb, 006. [7] E. J. Canes, J. K. Romberg, an T. Tao, Stable sgnal recovery from ncomplete an naccurate measurements, Communcatons on Pure an Apple Mathematcs, vol. 59, no. 8, pp , Aug, 006. [8] T. Brox, O. Klenschmt, an D. Cremers, Effcent nonlocal means for enosng of textural patterns, IEEE Trans Image Process, vol. 17, no. 7, pp , Jul, 008. [9] Q. Xu, H. Yu, X. Mou, L. Zhang, J. Hseh, an G. Wang, Low-ose X-ray CT reconstructon va ctonary learnng, IEEE Trans Me Imagng, vol. 31, no. 9, pp , Sep, 01. [10] G. H. Chen, J. Tang, an S. Leng, Pror mage constrane compresse sensng (PICCS): a metho to accurately reconstruct ynamc CT mages from hghly unersample projecton ata sets, Me Phys, vol. 35, no., pp , Feb, 008. [11] H. Gao, H. Yu, S. Osher, an G. Wang, Mult-energy CT base on a pror rank, ntensty an sparsty moel (PRISM), Inverse Probl, vol. 7, no. 11, ov 01, 011. [1] S. Osher, Z. Sh, an W. Zhu, Low mensonal manfol moel for mage processng, UCLA Tech. Rep., no. CAM 16-04, 016. [13] G. Peyre, Manfol moels for sgnals an mages, Computer Vson an Image Unerstanng, vol. 113, no., pp , Feb, 009. [14] Z. L, Z. Sh, an J. Sun, Pont ntegral metho for solvng posson-type equatons on manfols from pont clous wth convergence guarantees, arxv vol , 014.

10 [15] T. Golsten, an S. Osher, The Splt Bregman Metho for L1-Regularze Problems, Sam Journal on Imagng Scences, vol., no., pp , 009. [16] B. De Man, J. uyts, P. Dupont, G. Marchal, an P. Suetens, An teratve maxmum-lkelhoo polychromatc algorthm for CT, Ieee Transactons on Mecal Imagng, vol. 0, no. 10, pp , Oct, 001. [17] B. De Man, S. Basu,. Chanra, B. Dunham, P. Ec, M. Iatrou, S. McOlash, P. Sanath, C. Shaughnessy, an B. Tower, "CatSm: a new computer assste tomography smulaton envronment." pp. 6510G-6510G-8.