ne-sorce bsed X-ry Tomogrphy Deep Bhrhd Hengyong Y 4 Hong 3 Robert Plemmons 5 Ge Wng 4. Bomedcl mgng Dvson VT-WFU School o Bomedcl Engneerng & Scence We Forest Unversty Wnston-Slem C 757. Bomedcl Engneerng Deprtment We Forest Unversty School o edcne Wnston Slem C 757 3. School o Electrcl & Compter Engneerng Unversty o lhom ormn 7309 4. Bomedcl mgng Dvson VT-WFU School o Bomedcl Engneerng & Scence Vrgn Tech. Blcsbrg VA 406 5. Deprtments o themtcs nd Compter Scence We Forest Unversty Wnston-Slem C 709 Abstrct: Crrent compted tomogrphy (CT) scnners ncldng mcro-ct scnners tlze pont x-ry sorce. As we trget hgher nd hgher sptl resoltons the redced x-ry ocl spot sze lmts the temporl nd contrst resoltons chevble. To overcome ths lmtton n ths pper we propose to se lne-shped x-ry sorce so tht mny more photons cn be generted gven dt cqston ntervl. n reerence to the smltneos lgebrc reconstrcton technqe (SART) lgorthm or mge reconstrcton rom proecton dt generted by n x-ry pont sorce here we develop generlzed SART lgorthm or mge reconstrcton rom proecton dt generted by n x-ry lne sorce. r nmercl smlton reslts demonstrte the esblty o or novel lne-sorce-bsed x-ry CT pproch nd the proposed generlzed SART lgorthm. eywords: Compter tomogrphy (CT) mcro-ct smltneos lgebrc reconstrcton technqe (SART) generlzed SART pont sorce lne sorce.. ntrodcton Snce the rst compted tomogrphy (CT) scnner ws mde [] ll the commercl scnners hve been employng the x-ry sorce wth smll ocl spot whch cn be mthemtclly modeled s pont sorce. n mcro-ct nd even nno-ct pplctons the redced x-ry ocl spot sze hs become lmtng ctor to cheve desrble mge resolton n terms o sptl contrst nd temporl mesres. To ddress ths sse we propose to se lne-shped x-ry sorce so tht more photons cn be generted n gven dt cqston ntervl. n ths context the x-ry sorce cn be mthemtclly modeled s lne-segment. n pont x-ry sorce CT scnners the sptl resolton s lmted by the nte ocl-spot sze necessry to generte scent nmber o x-ry photons nd the temporl resolton s lmted by the tme necessry to cqre scent proecton dt over n nglr rnge.
n contrst to recently proposed sorce congrtons le the mltplexed [] nd mltple-sorce geometry [3] whch tlze mltple pont x-ry sorces to redce the cqston tme or technqe trets the entre lne segment s sngle x-ry sorce. Snce lne sorce covers wde nglr rnge per vew rrdton wth n ncresed nmber o photons s cheved long wth reltvely hgher coolng cpblty o the x- ry sorce. Thereore lne-shped sorce technqe cold be good cnddte to blnce mong sptl contrst nd temporl resolton. A lne-shped x-ry sorce cn be brcted sng eld emsson x-ry sorce technology. Feld emtters hve been sed s n electron sorces or long tme. The most sgncnt derence between the eld emsson x-ry tbe nd exstng tbes les n the eld emtter cthode. The cthode cn be mde wth n rry o mcro-mchned eld emsson tps. By dong so t s possble to obtn very shrp tps nd very close proxmty between the tps nd the gte electrode. Ths gretly redces the potentl derence between the tp nd the gte reqred to cheve the eld emsson. The rry my lso be very densely pced. As reslt even thogh the crrent tht cn be obtned rom sngle tp s smll the totl crrent tht cn be obtned rom n rry cn be mch lrger. Schwoebel et l. [4] reported tht they were ble to obtn crrent o 300mA by pcng 50000 tps nto crclr re o sqre mllmeter whch s eqvlent to 40A/cm. Usng the technqe s descrbed bove lne shped x-ry sorce cn be brcted. The wdth cn be mde s nrrow s 0.0mm (or less) nd the length o the sorce cn be mde n tens o centmeters or longer necessry. The orgnzton o ths pper s s the ollows. n the next secton we ormlte orwrd mgng model ssmng lne sorce. n the thrd secton we develop generlzed smltneos lgebrc reconstrcton technqe (SART) to enble lnesorce-bsed reconstrcton. n the orth secton we perorm nmercl tests to demonstrte the perormnce o or technqe. n the lst secton we dscss relevnt sses nd conclde the pper.. ne-shped X-ry mgng odel As shown n Fg. lner vrtl detector s ssmed n or lne-shped x-ry sorce cqston geometry. Vectors s ( v) nd t ( v) reer to the loctons o the lne-shped x-ry sorce nd the lne detector respectvely. The x-ry orwrd proecton model or lne sorce n terms o the nmber o photons rrvng t detector locton t cn be wrtten s ( x) dx x ry ( s t ) ( t) ( s ) e ds () where (s) s the orgnl nmber o photons emntng rom the sorce t pont s (t) s the nmber o photons rrvng t locton t on the detector x ( v) reers to poston long the x-ry pth connectng ponts s nd t (x) s lner ttenton coecent o pont x x nd s respectvely represent the D coordntes long xed ry pth nd the lne sorce nd the oter ntegrl s crred ot long the whole lne
sorce. The power o the exponentl term ( ( x) x ry ( s t) ttenton coecents long the x-ry connectng ponts s nd t. dx ) s the lne ntegrl o the the length o the lne s nntesmlly smll the x-ry ttenton model o pont sorce cn be obtned by removng the oter ntegrl n Eq. () s ( x) dx x ry ( s t ) ( t) ( s ) e () where s s the locton o the pont sorce nd hence cn ssme only one vle per vew. Applyng the logrthmc trnsormtons on both sdes o Eq. () we cn obtn lner eqton representng the lne ntegrl o the ttenton coecents long n x-ry pth s p ( ( t) t) log( ) ( x) dx. (3) x ry ( s ) ( s) t CT mge reconstrcton s typcl nverse problem o recoverng (x) rom the correspondng orwrd proecton models Eqs. () nd (3). Whle there re mny nlytc lgorthms or pont-sorce proecton dt modeled by Eq. (3) the non-lner ntre o Eq. () or the proecton dt o lne-shped x-ry sorce mes t dclt to obtn correspondng nlytcl method or the mge reconstrcton. However n tertve method cn be developed to cheve the reconstrcton s demonstrted n the next secton.. Generlzed SART lgorthm Althogh tertve methods hve not been employed by ny commercl CT scnners de to hgh compttonl costs ssocted wth them ther speror perormnce s well estblshed when the dt s ncomplete nosy nd dynmc. enwhle there s renewed nterest n tertve lgorthms de to the mprovement n compttonl cpbltes [5 7]. t s well nown tht smltneos lgebrc reconstrcton technqe (SART) [6 7] hs remned very powerl tool or tertve reconstrcton snce ts ntrodcton nd t hs been shown to converge to weghted lest sqres solton rom ny ntl gess [8]. oreover the SART method opertes n the whole rel spce whle the other poplr tertve lgorthm expectton mxmzton s dened only or nonnegtve spce lthogh t does preserve dt delty or non-negtve pxel nd voxel vles. Here we wll generlze the SART method or mge reconstrcton rom proecton dt o the proposed lne sorce model. 3. Pont Sorce SART Algorthm Becse proecton or the pont x-ry sorce s lner ntegrl o ttenton vles long the x-ry pth (Eq. 3) t cn be wrtten n dscrete orm s p x (4)
where p s the th proecton wth beng the nmber o proectons gven by the prodct o the nmber o vews nd the nmber o detector pxels s the nmber o smple ponts long the th ry pth re the ttenton coecents o the ponts on the th ry wth. Becse most o the ponts long the x-ry pths n Eq. (4) do not le on the dscrete mge grd t s necessry to obtn the ttenton vles or them rom the mge grd pxels v nterpolton. Assmng tht lner nterpolton s employed we cn rewrte Eq. (4) s p where w x w x re the ttenton vles o the mge pxels nmber o pxels n the mge (5) wth beng the w s the normlzed weght contrbton o on the mge grd to the proecton p s the length o the ntersecton between the th x-ry pth nd spport o the regon beng reconstrcted. et w x. Eq. (5) cn now be smpled s p. (6) Ths s eqvlent to lner system o eqtons gven by AF P (7) where A s mtrx F s vector nd P s vector wth A (8) T F ] (9) [ p p The SART lgorthm or solvng F rom P cn be wrtten n n tertve ormt s [7] ter ter + ter ( p F ) + () th where ter ndctes the terton nmber s the row o mtrx A T P [ p p ]. (0) ter nd p F s the estmted rom crrent mge th derence between the rel lne ntegrl nd the lne ntegrl ter F.
3. Generlzed ne Sorce SART Algorthm Followng the sme steps s or the pont sorce proecton or the lne-shped x- ry sorce bsed on Eq.() cn be wrtten n the dscrete orm s exp( x) s () th where s the proecton or the lne sorce wth beng the nmber o proectons gven by the prodct o the nmber o vews nd the nmber o detector pxels s the nmber o sorce ponts chosen or the dscretzton o the lne sorce s the orgnl nmber o photons n ll x-rys rom lne sorce pont s the nmber o smple ponts long the x-ry pth rom lne sorce pont to detector pxel correspondng to the proecton s the lner ttenton coecent o the ponts long the x-ry pth rom sorce pont to detector pxel correspondng to the proecton x s the smplng ntervl long the ry pth nd s s the smplng ntervl long the lne sorce. n the ollowng we ssme tht ll the x-rys rom lne sorce hve sme orgnl nmber o photons s. Althogh Eq. () s not lner eqton set we cn mody the SART lgorthm to solve Eq. (). Becse most o the ponts on the x-ry pths n Eq. () do not le on the dscrete mge grd gn t s necessry to obtn the ttenton vles or them rom the mge grd pxels. Assmng tht lner nterpolton s employed we rewrte Eq. () s where w x exp( ) s w x (3) re the ttenton vles o the mge pxels s the nmber o pxels n the mge w s the normlzed contrbton o pxel to the lne-ntegrl o the ttenton coecents long the x-ry rom lne sorce pont to the detector pxel correspondng to the proecton nd s the length o the ntersecton between the x-ry pth rom th sorce pont to the detector pxel correspondng to the proecton nd the spport o the regon beng reconstrcted. et the estmted th proecton dt nd or the rel or exp( d ) wth oe th proecton dt we hve oe beng or d log. (4) oe The rel proecton dt cn now be rewrtten n terms o the estmted proecton dt (Eq. (3)) s or exp( d ) exp( w whch cn be smpled s x) s (5)
or exp( ( w Agn let or x + d )) s. (6) w x Eq. (6) cn be rther smpled s exp( ( otce tht the sme + d )) s. (7) d s dded to the lne ntegrls ( ) o the ttenton coecents long ny x-rys rom ll the lne sorce ponts to the detector pxel correspondng to the lne sorce proecton. Consder hypothetcl stton n whch ll the ponts long the lne x-ry sorce serve s pont x-ry sorces nd let q denote the lne ntegrls o the x-rys rom these sorces contrbtng to the proecton. Althogh the lne ntegrls q re not nown we wll show n the ollowng tht t s not necessry to now them. ow we cn orm lner system o eqtons AF Q (8) U mtrx wth U F nd Q re nd where A s respectvely descrbed by U vectors
A (9) T ] [ F (0) T q q q q q q q q ] [ Q. () Along the rows o A nd Q there re two ndex vrbles nd. we combne them nto one ndex we obtn U U U U A ()
T Q [ q q q q U ]. (3) At rst loo t ppers tht we cn now se Eq. () only Q s nown s F U ter + ( q ) ter ter + U.. (4) otce tht ter q F s the derence between the rel lne ntegrl q nd the lne ter ter ntegrl F estmted rom the crrent terte vle F o the mge. Ths derence or prtclr proecton s gven by d obtned n Eq. (4) ths mng t nnecessry to now q. To be consstent wth ndexng let s lso cll d s d whle rememberng tht s combnton o ndexes nd nd tht d s the sme or ll the x-rys (ll ' s ) rom the lne-sorce or prtclr proecton. We cn now wrte Eq. (4) s U ter+ ter + U d. (5) Eq.(4) s the nl SART lgorthm or mge reconstrcton rom the proecton dt o lne-shped x-ry sorce. Becse t cn be ppled or mge reconstrcton rom ny non-lner proecton model we cll t generlzed SART lgorthm. V. mercl Smltons To vldte the esblty o lne-shped x-ry sorce nd demonstrte the merts o generlzed SART lgorthm we developed nmercl smltor. The thorx Phntom [9] ws sed n or smltons. A 60 cm vrtl lner detector ws ssmed nd ts detector element sze ws 0. cm. The center o the lne sorce ws t perpendclr dstnce ( D ) (see Fg. ) o 75 cm rom the so-center. The whole lne sorce ws rotted wth the center o the lne sorce trcng crcle o rds 75 cm. The sorce lengths o 3cm 5cm nd 8cm were employed wth proecton dt cqred or 60 eqnglr 7 vews. mber o photons sed per x-ry per sorce pont ws ssmed to be0. mercl smlton reslts re presented n Fgs. to 5. The sze o the mges presented s 3.65 4 cm wth 44 56 pxels. n Fg. mges re reconstrcted t derent nmber o tertons or lne sorce length o 3cm. Comprng Fgs b c nd d t cn be observed tht the ncresng the nmber o tertons mges wth shrper edges re obtned. Ths phenomenon s more obvos t the edges o the vertebr nd the lngs. mge reconstrcted ter 00 tertons or the x-ry sorce lengths o 3cm 5 cm nd 8 cm long wth the colmn nd the row proles re presented n Fgs. 3 4 nd 5 respectvely. For or smlton cses the best mge qlty s obtned wth sorce length o 3 cm nd reltvely poor mge qlty s obtned wth the sorce lengths o
3cm nd 8cm. The ncrese n sorce length reslts n ncresed blrrng especlly t the edges. Ths t becomes ncresngly dclt to reconstrct ne strctres whch n or cse re the rbs snce they hve shrp nd thn bondres. V. Dscssons nd Conclsons A pont x-ry sorce sed n commercl CT scnners lmts the temporl nd sptl resolton nd lso reslts n reqent hetng o the x-ry tbe. These lmttons my be overcome by sng the proposed lne-shped x-ry sorce bsed mge technqe. Reconstrctng mge rom lne-shped x-ry sorce s chllengng ts de to nonlner ntre o the resltng proecton dt. n ths rtcle we developed generlzed SART lgorthm to enble reconstrcton rom lne sorce. We beleve tht ths lgorthm cn be esly extended to more generl D x-ry sorce shpes nd even to D plnr x-ry sorces s well whch my hve pplctons n dynmc mgng. oreover the S-SART lgorthms [0] cn be smlrly moded to obtn generlzed S-SART lgorthm wth ster convergence. To the best o or nowledge ths s the rst pper ttemptng to solve the contrdcton between temporl nd sptl resoltons by non-pont sorce. Addtonl reserch eorts re necessry long ths drecton. n ths stdy we cqred proecton dt over n nglr rnge o π bt t wll be nterestng to nd ot or prctcl resons the mnmm nglr rnge necessry or reconstrcton whch my chnge or derent sorce lengths. t wll lso be exctng to see ner smplng o x-ry sorce drng nmercl smltons mproves sptl resolton. Althogh t hs not been theoretclly proved we beleve tht the generlzed SART s convergent to the rel mge vle s shown or SART n [8]. However detl stdy s beyond the scope o ths pper bt wll provde ddtonl nsghts nto the perormnce o the lgorthm. There re lso possbltes or enhncng or lgorthm or developng new lgorthms. n or vew the mn lmtton o or generlzed SART lgorthm s the blrrng o the edges nd lrge nmber o tertons reqred to reconstrct shrp mges. To ths end grdent or pror normton my be ncorported wthn or lgorthm or other lgorthms tlzng grdent normton cold be developed n the tre. Some exmples o pror normton nclde spport nd non-negtvty constrnts []. n ddton nvestgton o reglrzton pproches to llevte nose mgncton nd blrrng rtcts [] s n mportnt reserch drecton to prse n tre wor. Ths s esblty stdy nd there s not enogh normton to comment on sctter nd ts eects. Sctterng s rel concern n compted tomogrphy nd t wll become mportnt to stdy the mont o sctter nd resltng mge degrdton s the lneshped bsed x-ry sorce tomogrphy dvnces. However needed new lgorthms cold be developed to redce the eects o sctter. n conclson we proposed novel lne-shped x-ry sorce bsed CT mgng technqe nd correspondng reconstrcton method. The developed generlzed SART lgorthm enbles mge reconstrcton rom proecton dt o not only lne-shped x-
ry sorce bt lso more generlzed D nd D sorce. r nmercl smltons hve demonstrted the esblty nd merts o the proposed technqes nd lgorthms. Some nterestng tre reserch drectons were lso presented. Acnowledgements Ths wor ws prtlly spported by H/BB grnts (EB00667EB00487 nd EB00788). Reerences. Honseld G.. Compterzed Trnsverse Axl Scnnng (Tomogrphy).. Descrpton System (Reprnted From Brtsh-ornl--Rdology Vol 46 Pg 06-0 973). Brtsh ornl Rdology 995. 68(85): p. H66-H7.. Zhng. et l. ltplexng rdogrphy or ltr-st compted tomogrphy: A esblty stdy. edcl Physcs 007. 34(6): p. 57-57. 3. De n B. et l. lt-sorce nverse geometry CT: new system concept or x- ry compted tomogrphy. n edcl mgng 007: Physcs o edcl mgng. 007. Sn Dego CA USA: SPE. 4. Schwoebel P.R. C.A. Spndt nd C.E. Hollnd Hgh crrent hgh crrent densty eld emtter rry cthodes. ornl o Vcm Scence ∓ Technology B (croelectroncs nd nometer Strctres) 005. 3(): p. 69. 5. ng. nd G. Wng Convergence stdes on tertve lgorthms or mge reconstrcton. EEE Trnsctons n edcl mgng 003. (5): p. 569-579. 6. Andersen A.H. nd A.C. Smltneos Algebrc Reconstrcton Technqe (Srt) - A Speror mplementton The Art Algorthm. Ultrsonc mgng 984. 6(): p. 8-94. 7. A.C. Stnley Prncples o Compterzed Tomogrphc mgng.. 00: Socety o ndstrl nd Appled themtcs. 8. ng. nd G. Wng Convergence o the smltneos lgebrc reconstrcton technqe (SART). EEE Trnsctons n mge Processng 003. (8): p. 957-96. 9. Sorbelle. Thorx Phntom Avlble t: http://www.mp.nerlngen.de/phntoms/thorx/thorx.htm. [cted //007]. 0. Ge W. nd. ng rdered-sbset smltneos lgebrc reconstrcton technqes (S-SART). ornl o X-Ry Scence nd Technology 004. (3): p. 69.. Chen D. nd Plemmons R. onnegtvty Constrnts n mercl Anlyss. Proceedngs o the Symposm on the Brth o mercl Anlyss even Belgm World Scentc Press A. Bltheel nd R. Cools Eds. to pper 009.. Persons T. Hemler P. nd Plemmons R. 3D tertve Restorton o Tomosynthetc mges ntegrted Compttonl mgng Systems SA Techncl Dgest Seres (ptcl Socety o Amerc) 00.
s Vrtl Detector t v ne Sorce 0 φ D Fgre : ne x-ry sorce cqston geometry
b c d Fgre : Comprson o mges reconstrcted sng ne-sart lgorthm sng lne sorce o length 3 cm t derent terton nmbers. ) rgnl mge. Reconstrcted mges n ) b) nd c) re respectvely t 30 60 nd 00 tertons. Dsply wndow s [0.8.] b c d Fgre 3: ne SART reconstrcted mges ter 00 tertons sng lne sorce o length 3 cm t two derent dsply wndows wth proles long the dotted lne. ) Reconstrcted mge t dsply wndow o [0 ]. b) Reconstrcted mge t dsply wndow o [0.8.]. c) Prole o the row. d) Prole o the colmn.
b c d Fgre 4: ne SART reconstrcted mges ter 00 tertons sng lne sorce o length 5 cm t two derent dsply wndows wth proles long the dotted lne. ) Reconstrcted mge t dsply wndow o [0 ]. b) Reconstrcted mge t dsply wndow o [0.8.]. c) Prole o the row. d) Prole o the colmn.
b c d Fgre 5: ne SART reconstrcted mges ter 00 tertons sng lne sorce o length 8 cm t two derent dsply wndows wth proles long the dotted lne. ) Reconstrcted mge t dsply wndow o [0 ]. b) Reconstrcted mge t dsply wndow o [0.8.]. c) Prole o the row. d) Prole o the colmn.