Grangeat-Type Helical Half-Scan CT Algorithm for Reconstruction of a Short Object. Seung Wook Lee a, b and Ge Wang a. Department of Radiology

Transcription

1 Granga-Typ Hlical Half-Scan CT Algorihm for Rconsrucion of a Shor Objc Sung Wook L a, b and G Wang a a CT/Micro-CT Lab. Dparmn of Radiology Dparmn of Biomdical Enginring Univrsiy of Iowa Iowa Ciy, IA, USA b Dparmn of Nuclar and Quanum Enginring Kora Advancd Insiu of Scinc and Tchnology Dajon, 0-0, Souh Kora Corrsponding Addrss Sung Wook L and G Wang CT/Micro-CT Laboraory Dparmn of Radiology Univrsiy of Iowa Iowa ciy, IA, USA Tl -6-0 Fax swl@c.radiology.uiowa.du g@c.radiology.uiowa.du //00 : PM

2 ABSTRACT Currnly, con-bam CT and Micro-CT scannrs ar undr rapid dvlopmn for major biomdical applicaions. Half-scan conbam imag rconsrucion algorihms assum only par of a scanning urn, and ar advanagous in rms of mporal rsoluion and imag arifacs. Whil h xising half-scan con-bam algorihms ar in h Fldkamp framwork, w hav publishd a halfscan algorihm in h Granga framwork for a circular rajcory. In his papr, w xnd our prvious work o a hlical cas wihou daa runcaion. W modify h Granga's formula for uilizaion and simaion of Radon daa. Spcifically, w cagoriz ach characrisic poin in h Radon spac ino singly, doubly, riply sampld, and shadow rgions rspcivly. A smooh wighing sragy is dsignd o compnsa for daa for rdundancy and inconsisncy. In h hlical half-scan cas, h concps of projcd rajcoris and ransiion poins on mridian plans ar inroducd o guid h dsign of wighing funcions. Thn, h shadow rgion is rcovrd via linar inrpolaion afr smooh wighing. Th Shpp-Logan phanom is usd o vrify h corrcnss of h formulaion, and dmonsra h mris of h Granga-yp half-scan algorihm. Our Granga-yp hlical half-scan algorihm is no only valuabl for quaniaiv and/or dynamic biomdical applicaions of CT and micro-ct, bu also srv as an inrmdia sp owards solving h long objc problm. Ky words: half-scan, Granga-yp rconsrucion, con-bam CT, hlical CT, rconsrucion Running il: L and Wang: Hlical Half-scan Granga formula //00 : PM

3 Sung Wook L and G Wang: Hlical Half-Scan Granga 6 I. INTRODUCTION Sixn-slic hlical compurizd omography CT scannrs ar alrady commrcially availabl. C-arm sysms and svral micro-ct sysms ar also basd on con-bam acquisiion and rconsrucion -. Prooyps of h con-bam sysms wr rcnly rpord. Wih rapid incrmn in h numbr of dcor rows, h concp of con-bam CT or volumric CT will bcom mor and mor popular. Nw applicaions ar mad possibl by hs nw fas volumric imaging chnologis, such as for cardiac and lung xaminaions, CT angiography, and inrvnional procdurs 6,. In hos applicaions, high mporal rsoluion is on of h mos imporan rquirmns. 8 0 Half-scan chniqus wr dvlopd o improv h mporal rsoluion for axial and spiral CT imags 8,. Whil half-scan con-bam algorihms in h Fldkamp framwork hav rlaivly long hisory 0-, h half-scan conbam algorihms hav bn rcnly dvlopd in h Granga framwork for h circular scanning gomry by our Laboraory as wll by Noo and Huschr indpndnly,. Th diffrnc bwn hs rsuls is subsanial. Tha is, our work is in h rbinning framwork, whil h formulaion by Noo and Huschr is in h filrd backprojcion forma This rsarch is a naural xnsion of our half-scan algorihm from circular o hlical scanning gomry o solv h shor objc problm, which assums ha h objc is complly covrd by h X-ray con bam from any sourc posiion. Evn hough h gomry for daa runcaion along h axial dircion is h mos pracical, rsarch wih h shor objc gomry is no only valuabl on is own, such as for micro-ct imaging of sphrical sampls, bu also ssnial as an inrmdia sp oward solving h long objc problm,8. 0 This papr is organizd as follows. In Scion II.A h Granga algorihm is brifly rviwd for complnss. In II.B, h rbinning quaions for a hlical rajcory ar inroducd. In II.C, h hlical half-scan Granga formula is dscribd. In II.D, h wighing funcions ar drivd. In II.E, h inrpolaion mhod usd in his sudy is xplaind. In II.F, h implmnaion procdur is summarizd. In Scion III, h rsuls ar prsnd and discussd. Finally, In Scion VI, h papr is concludd. //00 : PM

4 Sung Wook L and G Wang: Hlical Half-Scan Granga II. MATERIALS AND METHODS A. GRANGEAT FRAMEWORK 6 A.. GRANGEAT'S DERIVATION For con-bam CT, i is insrumnal o connc con-bam daa o D Radon daa. Smih, Tuy and Granga indpndnly sablishd such conncions,,0. Granga s formulaion is gomrically araciv and bcoms popular. Hr, w rviw Granga's drivaion procss. I largly follows h noaions dfind in. In con-bam gomry as in Fig., h D Radon ransform a h characrisic poin C is dfind by 8 Rf n / / 0 f n, r, rdrd, 0 which mans h plan ingraion of h gray riangl wih on of h vrics bing h sourc poin S. Th plan is normal o h vcor n. Any poin on h plan is rprsnd wih h polar coordina r, on islf. A con-bam projcion is h lin ingraion from S o any poin A on h dcor plan and is mahmaically rprsnd by 0 Xf n f n, r, dr. As you can noic, hr is r in Eq., which prvns us from using h masurd con-bam projcion. To limina his, h firs drivaiv wih rspc o is applid o Eq. and w hav 6 Rf n / / 0 f n, r, rdrd. 8 To firs ordr, w hav h following rlaion a any poin on h plan, n, r, : d r cos d, whr is h angl bwn SO and SC. 0 By subsiuing d d, d d r cos w hav //00 : PM

5 Sung Wook L and G Wang: Hlical Half-Scan Granga Rf / f n, r, dr d. 6 cos / 0 Hr, r is canclld ou and h con bam projcion can b dircly uilizd as Rf / Xf n, d cos / 6 Th wo angular variabls, and can b changd o h variabls on h dcor plan, s and, wih h following gomrical rlaionships. s SO an 8 SC D an 8 Diffrniaing h wo yilds 0 SO ds d 0 cos d SA cos d Finally, w hav SO Rf n Xf s n, d cos s SA A.. CONE-BEAM DATA TO DERIVATIVE DATA IN THE RADON DOMAIN 6 Fig. is ransformd o Fig. o show h gomrical rlaionship bwn h dcor plan and h mridian plan in h Radon domain. Accordingly, Eq. can b rwrin as 8 Rf n Rf n cos s cos s SO X SA whr Xf s n,, n w D f Xf s n,, n d s n,, n d, is h dcor valu, which is dfind by h disanc s from h dcor O cnr O along h lin prpndicular o on h dcor plan D D D xnding an angl from h //00 : PM D C

6 Sung Wook L and G Wang: Hlical Half-Scan Granga SO x -axis, h disanc bwn h sourc and h dcor cnr, SA h disanc bwn h sourc and an D arbirary poin mridian plan A along, and h angl bwn SOD and SC. Givn a characrisic poin C on a in h Radon domain, h plan orhogonal o h vcor n is drmind. Thn, h M inrscion poins of h plan wih h sourc rajcory a can b found, and h dcor plans D spcifid, on which h lin ingraion ough o b prformd. L C dno h inrscion of h dcor plan D 6 D wih h ray ha coms from S and gos hrough C. Th posiion CD can b dscribd by a vcor sn D. To compu h drivaiv of Radon valu a C, h lin ingraion is prformd along, which is orhogonal o h 8 vcor sn D. Onc h firs drivaiv of Radon is calculad, h original funcion f can b rconsrucd wih h 0 following D Radon invrsion formula:, / x f d 8 Rf [ x n n ] sin d. / 0 B. REBINNING EQUATIONS FOR A HELICAL TRAJECTORY Th gomry for a hlical half-scan is shown in Fig., whr x, yz, is h rfrnc coordina sysm, y', z' dnos a mridian plan a from h x-axis, and h sourc vrx,, rangs from 0 o m. If h hlical 6 pich is dnod by h, h sourc rajcory can b paramrizd as a SO cos, SO sin, h z0. 8 To compu h drivaiv of h Radon valu a n, w nd firs calcula h lin ingraion poin C D on a dcor plan, hrough which a lin ingraion is prformd along h lin normal o OC D. Thr is a 0 gomrical rlaionship bwn h characrisic poin words, w can find s,, from,,.,, and h lin ingraion poin s,,. In ohr Th D Radon valu a a characrisic poin,, following quaion: is h ingraion of an objc on a plan saisfying h n x. 6 //00 : PM

7 Sung Wook L and G Wang: Hlical Half-Scan Granga Th inrscion of h plan wih h sourc rajcory is found by rplacing x wih a o solv n a whr n sin cos, sin sin, cos dnos h characrisic poin, and a R cos, R sin, h z h sourc posiion. Th quaion can b wrin as: 0 R cos sin cos R sin sin sin h z0 cos 0 R sin cos cos sin sin h z0 cos 8 6 Whil w can calcula h lin ingraion poin analyically in a circular rajcory cas, w can only do i numrically in a hlical rajcory cas. Onc w acquir, w hav lin ingraion poin, s on h dcor plan, D 8 whr. Th quaions ar xprssd as : co an sin, 0 h z0 cos sin s. 0 cos Thrfor, h Radon valu a h characrisic poin dfind by,, can b calculad by h ingraion along h lin rprsnd by, s, according o Eqs C. GRANGEAT-TYPE HELICAL HALF-SCAN FORMULA 6 Wih a circular half-scan, hr ar hr yps of rgions: shadow, singly and doubly sampld rgions, rspcivly. Wih a hlical half-scan, in addiion o hos hr yps of rgions, hr may b riply sampld rgions as wll. Thy ar schmaically illusrad in Fig.. Hnc, h Granga-yp hlical half-scan formula mus b in h following forma: 8 Rf n g g R f n, g whr g dno h wighing funcions, g is a group idnifir o b xplaind in dail lar, and //00 : PM

8 Sung Wook L and G Wang: Hlical Half-Scan Granga 6 SO f n Xf s,, d. cos s SA R g Th form is basically h sam as ha wih a circular half-scan rajcory bu hr wighing funcions and corrsponding Radon valus ar ndd for ach characrisic poin, whil w only nd wo wighing funcions in a circular half-scan cas. In Eq., h valu of h group idnifir g is drmind according o h following criria 6 g =, 0,,, 8, m. whr and ar angnial vrics and will b xplaind in dail in Sc D.. This mans ha h calculad 0 Radon daa mus blong o on of h hr groups dpnding on h saisfy. Rgarding h wighing funcions, hy mus g g. If w assum ha hr is no moion or daa inconsisncy during h scan, w can simply avrag Radon daa from diffrn vrics. In his cas, h wighing funcions bcom disconinuous. Howvr, in ral siuaions, moion ffcs and daa inconsisncy should b akn ino accoun. Thn, h disconinuous wighing funcions could 6 caus arifacs. Thrfor, h smooh wighing funcions ar ndd for saisfacory imag qualiy. Th nx scion will b dvod for his purpos. 8 D. WEIGHTING FUNCTIONS //00 : PM

9 Sung Wook L and G Wang: Hlical Half-Scan Granga Th concps of projcd rajcory and ransiion poins ar ndd bfor our wighing funcions ar dsignd. Hnc, w firs inroduc hm in D. and D.. Thn, w dsign h wighing funcions in D.. D.. PROJECTED TRAJECTORY ON A MERIDIAN PLANE AT Th projcd rajcory on a mridian plan a is usful for dsign of h wighing funcions. Svral projcd 6 rajcoris on diffrn mridian plans ar shown in Fig.. Th sourc rajcory of Eq. is firs ransformd 8 from x, y, z o x', y', z' coordina sysms, as shown in Fig.. Thrfor, for a givn mridian plan a projcd rajcory on his plan is dscribd as, h y SOsin. z h z0 0 Dashd lins in Fig. 6a rprsn h plans normal o h mridian plan. Th ingraion rsuls on h plans corrsponding o ach lin mus b qual o h Radon valu a,,. In h Granga framwork, h drivaiv of Radon daa is calculad from h dcor plans as dnod by h colord lins associad wih h inrscd vrx poins. Gomrically, hr can b maximally hr inrscion poins. For a givn,, h numbr of inrscing poins drmins h dgr of rdundancy and dpnds on. 6 D.. TRANSITION POINTS ON THE PROJECTED TRAJECTORY Evry projcd rajcory has wo nd poins xprssd by 8 y, z SOsin0,0 z0, 6 y, z SOsin m, h m z0. 0 For a givn on a mridian plan, hr may b plans normal o and angn o h projcd rajcory. Th angnial poins can b analyically spcifid by //00 : PM

10 Sung Wook L and G Wang: Hlical Half-Scan Granga 8 y, z SOsin, h z0, 8 y, z SOsin, h z0, whr, ar calculad in h following: A projcd rajcory on a mridian plan is drmind as y SOsin, 0 6 z h z 0. To xprss as a funcion of z, w obain 8 z z0 h, subsiu Eq. ino Eq. 0, and hav 0 z z0 y SOsin. h Thn, w compu h drivaiv and s i o h slop of h colord lin: dy z z SO cos 0 co dz h h. In ohr words, z hcos h SO co z0. Thn, w hav 6 cos h SO co. 6 //00 : PM

11 Sung Wook L and G Wang: Hlical Half-Scan Granga Th soluion of his quaion is maningful only whn 0 m, rsuling in up o wo soluions: and. I is assumd ha vrics in Subscion C. is grar han if i xiss. Rcall ha hs and ar usd for grouping In rms of h abov nd poins and angnial poins, w can find h ransiion poins as follows: 6 8, whr y, z sin, cos. As shown in Fig. 6a, w can idnify h yp of a rgion according o h 0 following cririon:, indicaing a doubly sampld rgion;, a singly sampld rgion;, a doubly sampld rgion. D.. SMOOTH WEIGHTING FUNCTIONS Wighing funcions ar dsignd in rfrnc o,,,, which ar funcions of and. Mahmaically spaking, hr ar up o wo possibiliis whn w hav nihr nor, which ar 6 and. Similarly, hr ar up o six cass whn w hav only ihr or. Furhrmor, hr ar up o wny four cass whn w hav boh and. Howvr, all h samns ar no maningful afr our cas-by- 8 cas inspcion. I is found ha hr xis only lvn cass acually. For xampl, in absnc of and, w always hav h cas of, and nvr hav h cas of. Similarly, i is impossibl o hav h 0 cass of and in absnc of. Thos lvn cass ar: i in absnc of and ; ii in absnc ; iii in absnc of ; iv in //00 : PM

12 Sung Wook L and G Wang: Hlical Half-Scan Granga 0 absnc of ; v in absnc of ; vi ; vii ; ix ; x ; xi. ; viii In addiion o our gomric analysis on projcd rajcoris on h mridian plan, w did xclusiv numrical simulaion o confirm ha hr ar only h abov lvn maningful cass. For all h,, w calculad h ransiion poins, sord h valus, and dcidd which on of h mahmaically possibl hiry wo 6 cass i blongs o. Onc a spcific cas was found, w s h flag for ha cas on. Afr his kind of numrical 8 vrificaion, w liminad h cass ha nvr happnd. Also, w rpad h simulaion wih rspc o rprsnaiv combinaions of imaging paramrs, including h sourc-o-origin disanc, hlical pich and con m angl. Finally, i was confirmd ha hr ar indd only h abov lvn cass ha ar fasibl. 0 For ach of hos maningful cass, smooh wighing funcions ar dsignd in rms of,,, 6. Th wo gnral rquirmns ar ha h sum of h wighs for ach characrisic poin b on, and h wigh profil along h dircion b coninuous. To furhr undrsand h dsigning procssing, som rprsnaiv illusraions ar considrd hlpful. Fig. 6b rprsns h cas whr hr is nihr nor for givn,, and only group xiss. Thrfor,, and and ar s o zro. Fig. 6c corrsponds o h cas whr hr is on angnial poin, and wo groups ar availabl. Th rajcory from o blongs o group, whil ha from o blongs o group. Th wighing funcion for group,, is dsignd o chang gradually from o 0 along h projcd rajcory from o, whil h wighing funcion for group,, 8 incrass from 0 o in h sam inrval, and says consan bwn and. is s o zro sinc hr is 0 no group. Of cours, h sum of h wighing funcions should b mad on. Fig. 6d illusras h cas whr hr ar wo angnial poins, and w hav group bwn and, group bwn and, and group bwn and. Th wighing funcion for h group,, should b on bwn o, and smoohly chang from o 0 along h projcd rajcory from o. Th wighing funcion for h group,, should smoohly chang from 0 o along h rajcory from o, and b 0 bwn and. Th //00 : PM

13 Sung Wook L and G Wang: Hlical Half-Scan Granga wighing funcion for h group,, is dsignd o smoohly incras from 0 a unil i rachs h middl poin bwn and, and dcras o 0 a. Som rprsnaiv disribuions of h wighing funcions ar includd in Fig.. Th wighing funcions ar formulad as follows, kyd o ach of h fasibl cass: i in absnc of and : 6, 0, 8 0. ii in absnc : 0 cos, 0, sin,, 0 iii in absnc of : 6 0, sin, //00 : PM

14 Sung Wook L and G Wang: Hlical Half-Scan Granga, cos, 0 iv in absnc of : sin,, 6 cos, 0, 8 0 v in absnc of : 0, cos, 0, sin, 0 vi 6 //00 : PM

15 Sung Wook L and G Wang: Hlical Half-Scan Granga 0, 0, sin, sin,, cos, 6 cos, 0, 8 0, 0 vii cos, 0, 0, sin,, 6 //00 : PM

16 Sung Wook L and G Wang: Hlical Half-Scan Granga cos, 0, 0, sin, viii, 6 cos, / 0, / 8 0, 0, 0 sin, / sin, / 0, 0, 0, / cos, / 6 //00 : PM

17 Sung Wook L and G Wang: Hlical Half-Scan Granga, ix 0, sin,,, 6,, 8, cos, 0 0, x 0, sin, / sin, / //00 : PM

18 Sung Wook L and G Wang: Hlical Half-Scan Granga 6 0,, cos, / 0, / 0,, 6 cos, / cos, / 8, xi 0, cos, / cos, /, 0, //00 : PM

19 Sung Wook L and G Wang: Hlical Half-Scan Granga 0, / cos, /, 0, sin, / 6 sin, / 0, 8 E. INTERPOLATION 0 As w did in h circular scanning cas,, in his sudy w coninu using h linar inrpolaion sragy o sima missing daa. Th boundaris of h shadow zon wr numrically drmind. Thn, h shadow zons wr linarly inrpolad along h dircion using h masurd boundary valus. To dmonsra his inrpolaion ida graphically, Fig. 8 shows h drivaivs of Radon daa wih no inrpolaion and wih linar inrpolaion, rspcivly. F. IMPLEMENTATION PROCEDURE 6 To summariz, h Granga-yp half-scan algorihm can b implmnd in h following sps: Spcify a characrisic poin,, whr h drivaiv of Radon daa can b calculad; 8 Calcula and ; //00 : PM

20 Sung Wook L and G Wang: Hlical Half-Scan Granga 8 Calcula,,, ; Calcula smooh wighing funcions,,,,,, and,, ; Drmin lin ingraion poins for h givn characrisic poin according o h rbinning Eqs. 8-0; 6 Calcula h drivaivs of Radon daa using Eqs., and sor hm according o hir group mmbrship as drmind by Eq. ; 6 Apply h wighing funcions o h drivaiv of Radon daa using Eq. ; 8 Rpa Sps - unil all h masurabl characrisic poins ar don; 8 Esima h Radon daa in h shadow zon using h linar inrpolaion mhod in his sudy; 0 Us h wo sag paralll-bam backprojcion algorihm as dfind by Eq. o rconsruc an imag volum. 0 No ha rsuls from sps - can b pr-calculad and sord for compuaional fficincy. Similarly, h rbinning cofficins from Sp can b found bforhand. 6 III. RESULTS AND DISCUSSIONS W dvlopd a sofwar simulaor in h IDL Languag Rsarch Sysms Inc., Bouldr, Colorado for Grangayp imag rconsrucion. In h implmnaion of h Granga-yp formula, h numrical diffrniaion was prformd wih a buil-in funcion basd on -poin Lagrangian inrpolaion. Th sourc-o-origin disanc was s o. Th numbr of dcors pr con-bam projcion was 6 by 8 6. Th siz of h D dcor plan was. by.. Th hlical pich was. Th full-con angl was abou 0 dgr. Th scan rang was from 0 o m. Th numbr of projcions was 0. Th numbr of mridian plans 0 was 80. Th numbrs of radial and angular sampls wr 6 and 60, rspcivly. Each rconsrucd imag volum had dimnsions of. by. by., and conaind 6 by 6 by 6 voxls. On migh us h sam numbr of projcions abov and blow ach and vry slic so ha all slics hav h sam imag qualiy. Our prfrrd approach uss an asymmric numbr of slics abov and blow h slic xcp for h z=0 slic. If w us h symmric half-scan hlical Fldkamp, which has symmric projcions for any z //00 : PM

21 Sung Wook L and G Wang: Hlical Half-Scan Granga 6 locaion, all slics would hav similar imag qualiy. Howvr, i mans ha ach slic is rconsrucd in a diffrn im window. Th primary purpos of our work is o dvlop half-scan algorihms in h Granga framwork wih suprior mporal characrisics mporal rsoluion and mporal consisncy; h lar rquirs ha a whol volum of inrs is rconsrucd wih projcions in h sam im window and lss imag arifacs which is achivd by appropria daa handling in h shadow zon. Th rason ha w us h asymmric projcions is o mainain his mporal consisncy. W can also us half-scan hlical Fldkamp mhods in h sam way. 8 0 Th D Shpp-Logan phanom was usd in h numrical simulaion as shown in Tabl. Fig. shows h drivaivs of Radon daa of h Shpp-Logan phanom, h wighing funcions for ach group, and h combind daa. Fig. 0 prsns ypical rconsrucd slics of h Shpp-Logan phanom. Wih h hlical half-scan Fldkamp mhod Fig. 0a and h hlical half-scan Granga and zro-padding mhod Fig. 0 b, h low innsiy drop was srious away from h cnr plan. Howvr, his yp of arifacs was ssnially liminad wih h hlical half-scan Granga and linar inrpolaion mhod Fig. 0c. IV. DISCUSSIONS AND CONCLUSION 6 In h wriing priod of our firs draf, i cam o our anion ha Noo and Huschr rcnly publishd a half-scan con-bam rconsrucion papr in an SPIE confrnc. Th similariy bwn our work and hir papr is ha 8 boh h groups considrd h half-scan con-bam rconsrucion using h Granga mhod. Howvr, hir work is in h filrd backprojcion framwork 6, whil ours is in h rbinning framwork. 0 6 W bliv ha boh halfscan algorihms ar complmnary. I sms ha daa filling mchanism is mor flxibl in our rbinning framwork. Noo and Huschr suggsd ha h paralll-bam approximaion of con-bam projcion daa b usd o sima missing daa, which is don in h spaial domain. This kind of spaial domain procssing is also allowd in our framwork. In addiion o h spaial domain approximaion, h Radon domain simaion, such as linar inrpolaion, splin inrpolaion and knowldg-basd inrpolaion, can b don in our framwork as wll. Howvr, in h filrd backprojcion framwork, ach fram of con-bam projcion daa can b procssd as soon as i is acquird, a dsirabl propry for pracical implmnaion. Clarly, a sysmaic comparison of h wo algorihms is worh of furhr invsigaion. //00 : PM

22 Sung Wook L and G Wang: Hlical Half-Scan Granga 0 Th hlical scanning gomry sudid in his papr is o solv h shor objc problm, which assums ha h objc is complly covrd by h X-ray con bam from any sourc posiion. Evn hough rsarch wih h shor objc gomry is valuabl on is own, such as for micro-ct imaging of sphrical sampls, h gomry for daa runcaion along h axial dircion is h mos pracical. Thrfor, h xnsion of our Granga-yp hlical half-scan CT work o h long objc cas is an imporan fuur opic. 6 In conclusion, w hav formulad a Granga-yp half-scan algorihm in h hlical scanning cas o solv 8 h shor objc problm. Th smooh half-scan wighing funcions hav bn dsignd o compnsa for daa rdundancy and inconsisnc. Numrical simulaion rsuls hav vrifid h corrcnss of our formulaion, and 0 dmonsrad h mris of our algorihm. W bliv ha h Granga-yp half-scan algorihm is promising for quaniaiv and dynamic biomdical applicaions of CT and micro-ct. ACKNOWLEDGEMENT W would lik o hank Dr. Dominic J. Huschr and Dr. Frdric Noo for mailing us hir SPIE papr. This work was suppord in par by h NIH gran R0 DC00 and EB0068. //00 : PM

23 L and Wang: Half-scan Granga formula REFERENCES S. W. L and G. Wang, "A Granga-yp half-scan algorihm for con-bam CT," Md. Phys. 0, D. W. Holdsworh, "Micro-CT in small animal and spcimn imaging," Trnds in Biochnology 0, S-S 00. G. Wang, "Micro-CT scannrs for biomdical applicaions: an ovrviw," Adv. Imaging 6, R. Ning, X. Tang, D. Conovr, and R. Yu, "Fla panl dcor-basd con bam compud omography wih a circl-plus-wo-arcs daa acquisiion orbi: Prliminary phanom sudy," Md. Phys. 0, K. Taguchi, "Tmporal rsoluion and h valuaion of candida algorihms for four-dimnsional CT," Md. Phys. 0, W. A. Kalndr, Compud omography: Fundamnals, Sysm Tchnology, Imag Qualiy, Applicaions Publicis MCD Vrlag, Munich, 000. G. Wang, C. R. Crawford, and W. A. Kalndr, "Mulirow dcor and con-bam spiral/hlical CT," IEEE Trans. Md. Imaging, D. L. Parkr, "Opimal shor scan convoluion rconsrucion for fanbam CT," Md. Phys., - 8. C. R. Crawford and K. F. King, "Compud omography scanning wih simulanous pain ranslaion," Md. Phys., G. T. Gullbrg and G. L. Zng, "A con-bam filrd backprojcion rconsrucion algorihm for cardiac singl phoon mission compud omography," IEEE Trans. Md. Imaging, -0. G. Wang, Y. Liu, T. H. Lin, and P. C. Chng, "Half-scan con-bam x-ray microomography formula," Scanning 6, 6-0. S. Zhao and G. Wang, "Fldkamp-yp con-bam omography in h wavl framwork," IEEE Trans. Md. Imaging, //00 : PM

24 Sung Wook L and G Wang: Hlical Half-Scan Granga Y. Liu, H. Liu, Y. Wang, and G. Wang, "Half-scan con-bam CT fluoroscopy wih mulipl x-ray sourcs," Md. Phys. 8, F. Noo and D. J. Huschr, "Imag rconsrucion from con-bam daa on a circular shor-scan," Proc. SPIE 68, 0-. P. Granga, "Mahmaical framwork of con bam D rconsrucion via h firs drivaiv of h Radon ransform," in Mahmaical Mhods in Tomography, Lcur nos in Mahmaics, did by G. T. Hrman, A. K. Louis, and F. Narr Springr Vrlag, Brlin,, M. Dfris and R. Clack, "A Con-bam rconsrucion algorihm using shif-varian filring and con-bam backprojcion," IEEE. Trans. Md. Imaging, 86-. Y. Wng, G. L. Zng, and G. T. Gullbrg, "A rconsrucion algorihm for hlical con-bam SPECT," IEEE Trans Nucl Sci 0, H. Kudo, F. Noo, and M. Dfris, "Con-bam filrd-backprojcion algorihm for runcad hlical daa," Phys. Md. Biol., B. D. Smih, "Imag rconsrucion from con-bam projcions: Ncssary and sufficin condiions and rconsrucion mhods," IEEE Trans Md Imaging MI-, H. K. Tuy, "An invrsion formula for con-bam rconsrucion," SIAM J. Appl. Mah., 6-8. C. Jacobson, Ph.D. dissraion Thsis, Linkoping Univrsiy, 6. S. R. Dans, Th Radon ransform and som of is applicaions Wily-Inrscinc, Nw York, 8. F. Narr, Th Mahmaics of Compurizd Tomography Sociy for Indusrial and Applid Mahmaics, Philadlphia, 86. S. W. L, G. Cho, and G. Wang, "Arifacs associad wih implmnaion of h Granga formula," Md. Phys., //00 : PM

25 Sung Wook L and G Wang: Hlical Half-Scan Granga F. Noo, M. Dfris, R. Clackdoyl, and H. Kudo, "Imag rconsrucion from fan-bam projcions on lss han a shor scan," Phys. Md. Biol., //00 : PM

26 L and Wang: Half-scan Granga formula TABLE CAPTIONS Tabl. Paramrs of h phanoms usd in our numrical simulaion. a, b, c dno h smi-axs of llipsoids, x, y, z h cnr coordinas of ach llipsoid, and h sam as in Fig.. Th acual dnsiy a a posiion is drmind by summing h dnsiis of h llipsoids covring ha poin. FIGURE CAPTIONS Figur. Granga's con bam gomry. Th con bam projcion and h firs drivaiv of Radon ransform ar linkd. Figur. Mridian and dcor plans. a Rlaionship bwn mridian and dcor plans, b mridian plan, and c dcor plan. Figur. Hlical Half-scan gomry. Figur. Classificaion of h Radon spac. a Shadow rgion, b singly sampld rgion, c doubly sampld rgion, and d riply sampld rgion. Figur. Projcd rajcoris on mridian plans of 0, h 0, and i 0. a 0, b 0, c 0, d 0, 0, f 0, g Figur 6. Transiion poins for dsign of h wighing funcions on mridian plans. a Paramrs of h projcd rajcory on a mridian plan, b cas i, c cas ii, and d cas viii. Figur. a-d Rgion map and wighing funcions for mridian plan a 0, -h 0, and i-l 0. ai Rgion map: h brighs is for riply sampld zons and h darks shadow zon. bfj Th firs wighing disribuion, cgk h scond wighing disribuion, and dhl h hird wighing disribuion. Figur 8. Firs drivaiv Radon daa of h D Shpp-Logan phanom afr daa filling. a zro padding, b linar inrpolaion. Figur. Firs drivaiv Radon daa of h D Shpp-Logan phanom a 0 for a group, b group, and c group. df Wighing funcions for ach group and g combind daa. //00 : PM

27 Sung Wook L and G Wang: Hlical Half-Scan Granga Figur 0. Rconsrucd imags of h D Shpp-Logan phanom. a Hlical half-scan Fldkamp, b Granga-yp hlical half-scan rconsrucion wih zro-padding, b Granga-yp hlical half-scan rconsrucion wih linar inrpolaion. Th sam projcion rang has bn usd o rconsruc h imag volum in h sam im window. Firs row: vrical slic a y=0.; scond row: vrical slic a x=-0.06; and hird row: ransvrs slic a z=0.. Th conras rang is [.00,.0]. //00 : PM

28 Phanom a b c x y z θ ϕ Dnsiy Shpp-Logan Tabl.

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