Centre of Rotation Determination Using Projection Data in X-ray Micro Computed Tomography

Transcription

1 Insttutonen för medcn och vård Avdelnngen för radofysk Hälsounverstetet Centre of Rotaton Determnaton Usng Proecton Data n X-ray Mcro Computed Tomography Brger Olander Department of Medcne and Care Rado Physcs Faculty of Health Scences

2 Seres: Report / Insttutonen för radolog, Unverstetet Lnköpng; 77 ISSN: -799 ISRN: LIU-RAD-R-77 Publshng year: 994 The Author(s)

3 Report 77 ISSN ISRN ULI-RAD-R 77 SE Centre of Rotaton Determnaton Usng Proecton Data n X-ray Mcro Computed Tomography Dept of Radaton Physcs Lnköpng Unversty, Sweden Brger Olander

4 - Centre of Rotaton Determnaton usng Proecton Data n X-ray Mcro Computed Tomography - CONTENTS CONTENTS...I CENTRE OF ROTATION DETERMINATION USING PROJECTION DATA IN X-RAY MICRO COMPUTED TOMOGRAPHY... INTRODUCTION... DEFINITIONS... CENTRE OF GRAVITY METHODS...5 THE sn METHOD...6 THE opp METHOD...7 TRANSLATED OPPOSITE PROJECTION METHODS...8 THE TOPln METHOD...9 THE TOPfft METHOD... BASELINE RESTORATION... RESULTS...5 CONCLUSIONS... REFERENCES... - Brger Olander, Dept of Radaton Physcs, Lnköpng Unversty, Sweden -

5 - Centre of Rotaton Determnaton usng Proecton Data n X-ray Mcro Computed Tomography - Centre of Rotaton Determnaton usng Proecton Data n X-ray Mcro Computed Tomography INTRODUCTION There are several methods avalable to determne the Centre Of Rotaton, COR, and algn detectors and X-ray focus to COR n X-ray computed tomography. Some methods use narrow rods/needles or specally made algnment obects or phantoms. In X-ray Mcro Computed Tomography (or Computerzed Mcro Tomography), µct (CMT), methods usng sample proecton data for COR measurements are preferred snce the replacement of algnment obects wth samples often dsplace translaton stages and make the algnment obsolete. To acheve an optmal mage qualty, precse postonng of COR to the detector and X-ray focus s essental. In µct ths can be accomplshed n an algnment procedure usng sample proecton data pror to scannng. Ths algnment procedure wll add examnaton tme and ncrease the dose to the sample. Therefore the algnment procedure should ncorporate as few proectons as possble and be nsenstve to nose. Some scannng equpment cannot be modfed to mplement such algnment procedure but actual COR can be determned from proecton data and used n reconstructon. Ths report ntroduces a new Translated Opposte Proecton, TOP, technque usng a par of opposte parallel proectons (8 apart). Two TOP methods are developed: TOPln usng lnear nterpolaton n the spatal doman and TOPfft n the frequency doman. The two TOP methods are compared to two Centre Of Gravty,, methods. The two methods are: sn, an enhancement of a method presented by Hogan et al [Hogan93] and opp, a smplfcaton of ths method possble wth a fxed COR. In ths report all proectons n one scan are assumed to have a fxed, COR, as n thrd (or hgher) generaton tomography or frst (and second generaton) tomography f the translaton stage errors s neglgble. Ths also means that the rotaton stage errors must be neglgble. The sn s the only method presented here capable of determne a COR for each proecton angle, thus allowng for a COR movng as a functon of proecton angle. The TOP methods normally gve better precson wth non-deal proecton data compared to the methods. Tests usng both smulated and scanned proecton data ndcate that the TOP methods gve hgher precson n presence of stochastc errors (nose) and system errors lke calbraton errors. A µct scan often takes a long tme and detector calbraton and X-ray ntensty profles mght vary wth tme gvng non-statonary system errors durng a sngle scan. If the system errors can be approxmated wth smple polynomal functons, a new Baselne Restoraton, BR, technque can be used together wth the TOP methods to reduce COR errors. - Brger Olander, Dept of Radaton Physcs, Lnköpng Unversty, Sweden -

6 - Centre of Rotaton Determnaton usng Proecton Data n X-ray Mcro Computed Tomography - DEFINITIONS All methods descrbed n ths report use parallel proecton data, Pθ ( t). The parallel proecton data s the Radon-transform of the obect attenuaton functon, f ( x, y) where Pθ ( t) = P( θ, t) = f ( x, y) δ( x cosθ + ysn θ t) dxdy δ( x ) s the Drac-functon. () Fan-beam proecton data needs to be rebnned to parallel proecton data f not D >> r max, where r max s the radus of the smallest crcle centred at the orgn subscrbng the obect and D s the orgn to source dstance. In µct the fan-beam geometry n fgure wth equdstant detectors n lne s the most common non-parallel geometry. The rebnnng procedure ncludes a twodmensonal nterpolaton technque, normally blnear nterpolaton, and uses the relatonshp between the co-ordnates ( β, s ) n fan-beam geometry (fgure ) and ( θ, t ) n parallel geometry (fgure ). t = s cos γ, θ = β + γ s D t =, θ = β + tan D + s s () D D R (s ) D R (s) B s=s s s= y detector D A f(x,y) x P (t ) detector P (t) t t= t=t y f(x,y) x D D source FIGURE. Fan-beam geometry. FIGURE. Parallel geometry. - Brger Olander, Dept of Radaton Physcs, Lnköpng Unversty, Sweden -

7 - Centre of Rotaton Determnaton usng Proecton Data n X-ray Mcro Computed Tomography - P t k t 3 c k N detector y P t t= t=t c t=t k c k 3 t f(x,y) x FIGURE 3. Proecton data samplng. The parallel data s a set of M proectons wth proecton angles θ, = M, each proecton wth N equdstant sampled proecton data at t = t, =... N. Let the angular ncrement be θ, the sample dstance t and the centre data ndex c. The dscrete sampled proecton data s denoted by P P, P ( t or P, t. θ, θ θ The term poston, υ R, wll be used here to specfy poston n terms of sample ndex. A sample wth ndex k at t = t k s at poston υ = k and poston υ = k + ε, < ε < s n between ndex k and k + at t = tk + ε t. We defne the algnment offset α as the postonal dsplacement of COR from centre data poston (see fgure 3). α = t c t (3) The Centre Of Rotaton, COR, wll have the poston υ COR = c α (4) As an example the quarter offset often used n X-ray CT corresponds to α = ±. 5. The Radon Transform from R needs to be defned for t R and θ [ π[ θ wth the followng perodcy property, and t s perodc n P( θ + π, t) = P( θ, t) (5) - Brger Olander, Dept of Radaton Physcs, Lnköpng Unversty, Sweden - 3

8 - Centre of Rotaton Determnaton usng Proecton Data n X-ray Mcro Computed Tomography - Another mportant property s that the generalsed Centre Of Gravty,, of f ( x, y) ( x, y ), x x f ( x, y) dx dy =, y = f ( x, y) dx dy y f ( x, y) dx dy f ( x, y) dx dy wth polar co-ordnates ( r, φ ), x = r cos φ, y = r snφ s transformed onto the one-dmensonal t = t P P ( θ, ) ( θ, ) t dt t dt of the proecton data at angle θ. That s t = r cos( θ φ ) (6) In the descrpton of the algnment methods the followng notatons wll be used: X θ, s data at proecton angle θ and ndex, X θ ( t ) X θ, α X θ α X θ, s measured data, s data translated α t, X θ ( t α t) s estmated translaton of measured data - Brger Olander, Dept of Radaton Physcs, Lnköpng Unversty, Sweden - 4

9 - Centre of Rotaton Determnaton usng Proecton Data n X-ray Mcro Computed Tomography - CENTRE OF GRAVITY METHODS The poston of proecton data at proecton angle θ s estmated usng υ ( θ ) N = = N = P( θ, t ) P( θ, t ) (7) Equaton (6) can be expressed n terms of poston υ for each proecton angle we get t = + υ t COR wth ρ r = and t υ ( θ ) = ρ cos( θ φ ) + υ (8) COR y = cos( - )+ COR r COR x + + COR ( FIGURE 4. Centre of Gravty methods prncple. - Brger Olander, Dept of Radaton Physcs, Lnköpng Unversty, Sweden - 5

10 - Centre of Rotaton Determnaton usng Proecton Data n X-ray Mcro Computed Tomography - THE sn METHOD A Mnmum Mean Square Error, MMSE, algorthm can be used to fnd ρ and υ COR for a fxed φ. Then the φ gvng mnmum Mean Square Error, MSE, s searched [Hogan93]. The MSE s calculated usng ( ) M MSE = υcor + ρ θ φ υ θ = cos( ) ( ) (9) The search of φ gvng MMSE results n computng a lot of cosne's and sums and solvng one second order lnear equaton system for each tested φ. If the ρ cos( θ φ ) -term s splt nto one COS- and one SIN-term, fndng MMSE can be reduced to solvng a thrd order lnear equaton system and no search for φ s necessary. Let where ρ cos( θ φ ) = ac + bs () a = ρ b = ρ c s = cos( θ ) = sn( θ ) cos( φ ) sn( φ ) Equaton (9) s rewrtten as M ( COR ( )) MSE = υ + a c + b s υ θ = () MMSE s found by settng each partal dervatve wth respect to a, b and υ COR to zero resultng n the lnear equaton system M M M c cs c a M M cs s s b M υ c s M = = = M = = = M = = COR = M = M = M = c υ s υ ( θ ) ( θ ) υ ( θ ) () The equaton system s solved usng Gaussan elmnaton. - Brger Olander, Dept of Radaton Physcs, Lnköpng Unversty, Sweden - 6

11 - Centre of Rotaton Determnaton usng Proecton Data n X-ray Mcro Computed Tomography - Ths method s called the sn method n ths report. The sn method needs several proecton angles to be accurate but proecton data from less than 8, ( θ M θ ) < π, s suffcent. An algnment offset, α, can be estmated for each proecton angle θ accordng to [Hogan93]. The other presented methods n ths report requre a constant algnment offset n all proectons (a fxed COR). THE opp METHOD The method can be consderably smplfed f proecton data from at least one par of opposte proecton angles θ and θ + π s avalable. Equaton (5) gves t ( θ + π ) = t ( θ). In terms of poston we get or υ ( θ + π ) υ = ( υ ( θ) υ ) υ COR COR COR υ ( θ) + υ ( θ + π ) = (3) Estmates of υ ( θ) and υ ( θ + π) are calculated usng equaton (7) and estmated υ COR s the mean value of the two postons. Fnally α s calculated usng equaton (4). Ths smplfed method s called the opp-method n ths report. - Brger Olander, Dept of Radaton Physcs, Lnköpng Unversty, Sweden - 7

12 - Centre of Rotaton Determnaton usng Proecton Data n X-ray Mcro Computed Tomography - TRANSLATED OPPOSITE PROJECTION METHODS The TOP methods need proecton data from at least one par of opposte proecton angles θ and θ + π. The dea s to estmate translated proecton data for the two opposte proectons α Pθ, = P( θ, t α t) α Pθ + π, = P( θ + π, t α t) (4) and a fxed α. Ths translaton wll algn the centre proecton data at ndex c to COR f α s correct. Equaton (5) gves P = P α α θ, c+ θ + π, c (5) P(,t ) translate t, shft c to P(,t c+ - t) c N -K K P( +,t ) translate t, P( +,t c- - t) shft c to and reverse c N -K K t t=t t=t c t=t N FIGURE 5. The prncple of the Translated Opposte Proecton methods (see text for explanaton). The measured proecton data can be modelled as the "true" data wth an addtonal error term P = P + ε θ, θ, θ, (6) - Brger Olander, Dept of Radaton Physcs, Lnköpng Unversty, Sweden - 8

13 - Centre of Rotaton Determnaton usng Proecton Data n X-ray Mcro Computed Tomography - The estmated Translated Opposte Proecton, TOP, dfference s α α α α α α α D = P P = P + ε P ε θ, θ, c+ θ + π, c θ, c+ θ, c+ θ + π, c θ + π, c (7) α Note that estmaton (nterpolaton) errors are ncluded n the ε -terms. θ, Equaton (5) gves α α α D θ, = ε θ, c ε θ π, c + + (8) The TOP methods fnd the algnment offset α gvng Mnmum Mean Square Dfference, MMSD, between the two estmated TOP s. The Mean Square Dfference, MSD, s calculated usng K α ( θ, ) MSD = D, K = max( c k, N c + k + ), k α < k +, k Z (9) = K The estmaton of the translated proecton data can be mplemented n both spatal doman and n Fourer doman. In spatal doman translated data s nterpolated from orgnal proecton data. The TOPln method uses lnear nterpolaton. Hgher order nterpolaton flters can be used but the lnear nterpolaton has one maor advantage besdes smple mplementaton and a moderate number of calculatons; the MSD s a second order polynomal of α n each nterval k α < k +. Thus the mnmum MSD n each such nterval s easly calculated, no search for mnmum MSD s needed. The TOPfft method calculates the Dscrete Fourer-transform of the proecton data usng FFT. The translaton s mplemented usng a phase-shft proportonal to α n the Fourer doman. The MSD s also calculated n the Fourer doman usng Parseval' s relaton, no nverse Fourer-transform s needed. The algorthm calculates the α gvng mnmum MSD. The TOPfft method has two maor advantages compared to the TOPln method: better precson because no nterpolaton s needed and the ablty to flter data pror to algnment wthout any extra convoluton or FFT and IFFT (nverse FFT) calculatons. The maor drawback s a hgher computatonal complexty compared to the TOPln method. The algnment offset α s searched n the algnment offset ntervalαmn α αmax. Ths nterval must be wde enough to nclude the correct α. A good way to get hgh precson wth a moderate computatonal effort s to make a coarse algnment usng the TOPln method and a wde nterval followed by a TOPfft algnment wth a narrow nterval. THE TOPln METHOD For α n the range k α < k +, the lnear nterpolaton s calculated as Pˆ Pˆ α θ, θ, = α Pˆ + ( α ) Pˆ, α = α k, k α < k +, k Z k θ, k = f < or > N k θ, k k () - Brger Olander, Dept of Radaton Physcs, Lnköpng Unversty, Sweden - 9

14 - Centre of Rotaton Determnaton usng Proecton Data n X-ray Mcro Computed Tomography - The estmated translated opposte proecton dfference s calculated accordng to equaton (7) and the MSD wll be ( θ, + ( ) θ, + θ + π, ( ) θ + π, ) K MSD = a P + a P a P a P = K K = max( c k, N c + k + ), ak = α k, k α < k +, k Z k k c k k c k k c k k c k () Ths MSD calculaton results n a polynomal of the form MSDk = c ak + c a k + c wth a c mnmum or maxmum for a =. If t s a mnmum for a k c k n the range a k < ths mnmum MSD k s stored. The stored mnma from all k n the algnment offset nterval αmn k ( αmax ) are compared and the a k resultng n the lowest mnmum MSD k gves the algnment offset α = k + a k. THE TOPfft METHOD The proecton data s ntally zero-padded and then shfted (crcular shft) to get the centre proecton data ndex, c, at orgn (the orgn s ndex n the FFT data array). Pθ,, = N Pθ, =, = N + L q θ, Pθ, + c, = L c = Pθ, + c L, = L c + L Zero paddng to L length Crcular shft () L must be chosen accordng to L max( L ), L = max( c k, N c + k + ) +, k α < k + k k for all k n the algnment offset nterval α translated proecton data. mn k ( α ) to avod crcular convoluton of the max The dscrete Fourer-transform of the zero-padded and shfted proecton data s calculated usng FFT accordng to π J / L Q = e q, J = L (3) θ, J L = θ, - Brger Olander, Dept of Radaton Physcs, Lnköpng Unversty, Sweden -

15 - Centre of Rotaton Determnaton usng Proecton Data n X-ray Mcro Computed Tomography - Let the dscrete spatal angular frequency be defned as ω J π J L, J = L = π ( J L) L, J = L L (4) πxx If a functon f ( x) has Fourer-transform F( X) = e f ( x) dx the translaton Xa g( x) = f ( x + a) n the spatal doman corresponds to a phase shft G( X) = e π F( X) n the Fourer doman. In the dscrete case a translaton of α, q α, corresponds to the phase shft α αω θ, J θ, J Q = e J Q. Together wth Parseval's theorem q L = L = Q and the knowledge L that the Fourer-transform of a Real functon s Hermtan we can wrte equaton (9) as L αω J αω J MSD = e Q, J e Qθ + π, J (5) L J= θ where Q s the complex conugate of Q. The TOPfft method calculates α gvng mnmum MSD MSD. Ths s done usng the Newton-Raphsson, NR, method to solve MSD = =. α There mght be several local mnma or maxma of MSD where the dervatve s zero. In order to fnd the global mnmum and a startng pont to the NR method we ntally calculate MSD k for allα k = α k n the algnment offset nterval αmn αk αmax where α s suffcently small to ensure that the NR method converges to the global mnmum. The α k gvng mnmum MSD k s used as startng pont n the MSD = calculatons. The frst and second order dervatve are calculated accordng to = L MSD 4 MSD = = ω J ( a J cos( αω J ) + b J sn( αω J )) α L J = L MSD 8 MSD = = + ω αω αω α L J = ( a sn( ) b cos( )) J J J J J (6) where,, Qθ, J Qθ + π, J = aj + bj aj bj R MSD The NR algorthm fnds the MSD root teratvely usng α n+ = α n repeated untl MSD α n+ α n s suffcently small. - Brger Olander, Dept of Radaton Physcs, Lnköpng Unversty, Sweden -

16 - Centre of Rotaton Determnaton usng Proecton Data n X-ray Mcro Computed Tomography - BASELINE RESTORATION In the X-ray CT applcatons proecton data s calculated usng a measured sgnal, I (absorbed X-ray energy n detector or number of detected photons), and a calbraton sgnal, I (ar scan), for each detector. I The proecton data s calculated as P = log for each detector and proecton. I If the detector senstvty s changed by a factor γ s or X-ray ntensty s changed by a factor γ snce the last calbraton the proecton data s based by a system error γ s Rθ, = log = logγ s + log γ γ Detector senstvty, γ s, fluctuatons mght orgnate from n temperature varaton or X-ray spectral varaton. The X-ray ntensty fluctuatons, γ, are manly a functon of proecton angle θ due to ntensty varatons wth tme. These fluctuatons can be compensated usng the edge detectors as reference ( I = I ) or usng the property that the ntegrated proecton data for any proecton angle s constant. Besde the ntensty varatons wth θ the X-ray ntensty as a functon of t (or detector poston) tends to vary wth tme. The γ s and γ fluctuatons as a functon of t are normally small but they mght cause sgnfcant COR determnaton errors. The best way to get small system errors n the algnment procedure s to make a separate algnment procedure usng sample proecton data pror to scannng. If a method s used whch rely on few proectons ths algnment scan has a low cost n tme and dose to the obect and the system errors are mnmsed. It s also possble to algn the COR to the X-ray focus and detector n an optmal way. Some reconstructon software also requres proecton data to be algned wth a specfc algnment offset, usually α =. If the scannng equpment cannot be modfed to mplement a separate algnment procedure the actual COR can be determned from scanned proecton data and used later n the reconstructon procedure (f supported by the reconstructon software). Another source of system errors n µct s detector non-lnearty. We can correct for detector non-lnearty f each ndvdual detector s characterstcs are known. These correctons can be dead-tme correctons for photon countng systems and lookup tables for semconductor detectors. These correctons are hard to get correct and often contrbute sgnfcantly to the system error. The TOP methods tend to be less senstve for these system errors than the methods. To further mprove algnment accuracy n presence of system errors an teratve Baselne Restoraton, BR, technque can be combned wth the TOP methods. Assume that an opposte proecton par has system errors R ( θ t ) and R ( t). θ + π A statonary system mples R ( t) = R + ( t). Let the proecton data error term consst of θ θ π - Brger Olander, Dept of Radaton Physcs, Lnköpng Unversty, Sweden -

17 . - Centre of Rotaton Determnaton usng Proecton Data n X-ray Mcro Computed Tomography - the system error and a nose term ε θ, = Rθ, + ηθ, = Rθ ( t ) + ηθ ( t ) (quantum, electronc and other types of random nose). If α s known we can wrte the estmated TOP dfference n equaton (8) as D R R α α α α α θ, = θ, c+ θ + π, c + ηθ, c+ ηθ + π, c (7) α The estmaton errors are ncluded n the η -terms. θ, P(,t ) P(,t c+ - t) translate t, shft c to c N -K K P( +,t ) translate t, P( +,t c- - t) shft c to and reverse c N -K K restore proectons subtract P(,t )- (t -c + t)/ proectons D, c N P( +,t )+ (t c- - t)/ c N K ( t) ft baselne K -K K FIGURE 6. Baselne Restoraton prncple (see text for explanaton). The BR-technque approxmates the TOP system error dfference, D = R ( t) R ( t), wth l l an l order polynomal β( t) = c t + c t + + c l l β( t) / s the baselne. θ θ θ+ π For a gven α the baselne s determned usng a Mnmum Mean Square Error, MMSE, ft to α D θ, K α l l MSE = Dθ, cl ( t ) cl ( t ) c, = K K = mn( c k, N c + k), k α < k +, k Z (8) - Brger Olander, Dept of Radaton Physcs, Lnköpng Unversty, Sweden - 3

18 where - Centre of Rotaton Determnaton usng Proecton Data n X-ray Mcro Computed Tomography - D = P P α α α θ, θ, c+ θ, c s calculated usng lnear nterpolaton accordng to equaton (). The MMSE calculatons lead to an l + order lnear equaton system solved usng Gaussan elmnaton. If the system error s statonary and odd t can be determned usng Rθ ( t) = Rθ + π ( t) = β( t) /. Even f the system error s varyng wth θ or not odd, β( t) / can be used to elmnate COR errors for each par opposte proectons. Proecton data s modfed pror to COR calculaton accordng to ~ Pθ, = Pθ, β( [ c + α] t) / ~ Pθ + π, = Pθ + π, + β( [ c α] t) / (9) The BR and algnment algorthm's are repeated teratvely untl the changes n α from one teraton to another s suffcently small α n α n α stop or a maxmum number of teratons s performed n = n stop. The BR algnment procedure s:. Set α to a ntal value α and teraton number, n, to. { Pθ,, Pθ + π, } orgnal set of measured opposte proecton data.. Calculate baselne restoraton usng α n and { Pθ,, Pθ + π, } { P ~ ~, θ P }, θ + π, n. n s the, The result s ~ ~. Calculate algnment usng modfed data, { Pθ,, Pθ + π, } = { Pθ,, Pθ + π, } result s α n. n n, The 3. If ( α n α n > α stop and n < n stop ) then ncrement teraton number, n = n + and repeat from, otherwse algnment s fnshed. Ths teratve technque usng BR and a TOP algnment method normally converges rapdly wth a relable algnment offset α n f the baselne polynomal s a good approxmaton to the opposte proecton system error dfference. The algnment methods are consderably more senstve to system errors and the BR stage often has so much mpact on the calculatons that α n don't converge at all or to a false value. - Brger Olander, Dept of Radaton Physcs, Lnköpng Unversty, Sweden - 4

19 - Centre of Rotaton Determnaton usng Proecton Data n X-ray Mcro Computed Tomography - RESULTS The four presented methods have been tested and compared usng both smulated proecton data (wth known COR and nose-level) and scanned proecton data. Fgures 7 and 8 llustrate one example usng smulated proecton data. The smulated data conssts of sets wth parallel proecton data of a flled tube wth some low and hgh densty obects of dfferent sze and shape. Each proecton data set has N = 55 samples per proecton and M = 8 proecton angles wth π θ = (64 opposte proecton data pars). To examne the senstvty to nose of the dfferent 8 methods, proecton data wth dfferent quantum nose levels are smulated - wthout nose, wth quantum nose levels I ar = 6 and I ar = 4 photons per detector n ar, respectvely. Fgure 7A, B and C show the frst proectons, θ =, of the proecton data usng the three dfferent quantum nose levels. Fgure 7D shows a reconstructon of the proecton data (no nose). To estmate the algnment errorσ α, the standard devaton of α, COR s estmated usng all the avalable 64 opposte proecton data pars wth the opp, TOPln and TOPfft methods. The sn method uses 64 proectons ( θ spacng) for each COR estmaton (3 tmes more data than the other methods) andσ α s calculated usng COR estmates from 9 overlappng proecton data subsets; =... 64, = up to = Fgure 8 shows the algnment results wth smulated algnment offset α sm = and α sm =. 5 (quarter offset) respectvely A) Frst proecton, no nose. B) Frst proecton, I ar = C) Frst proecton, I ar = 4. D) Reconstructed mage from data n A. FIGURE 7. Smulated proecton data. - Brger Olander, Dept of Radaton Physcs, Lnköpng Unversty, Sweden - 5

20 - Centre of Rotaton Determnaton usng Proecton Data n X-ray Mcro Computed Tomography opp TOPln TOPfft sn A) no nose,α sm =. B) no nose, α sm =. 5. opp TOPln TOPfft sn.. opp TOPln TOPfft sn C) I ar = 6,α sm =. D) I ar = 6, α sm = opp TOPln TOPfft sn.5 E) I ar = 4,α sm =. F) I ar = 4, α sm = opp TOPln TOPfft sn opp TOPln TOPfft sn FIGURE 8. Algnment results wth smulated proecton data, α sm = and α sm =. 5. Vertcal axs s calculated algnment offset, α, wth mnmum ( ), maxmum ( ), mean ( ) values and ± one standard devaton, σ α. It s clear that the methods shouldn t be compared to each other wth smulated data usng the lowest nose-levels snce the algnment errors are small and more due to smulaton errors, partal volume effect etc. than method errors. Wth α = and wthout nose all the opposte proecton methods (sn, TOPln and TOPfft) have zero error because all smulated opposte proecton pars are dentcal but reversed. Wth α =. 5 we have a small error even wthout nose usng the opposte proecton methods. The sn method has a small (σ ~.) error even wthout nose or wth low nose levels but ths error s approxmately the same for all α. The methods seems to be more senstve to nose then the TOP methods. Comparng the TOPln and TOPfft methods wth α = 5. (worst case) and hgher nose-levels clearly shows the mprovement of usng the FFT-method nstead of lnear nterpolaton. - Brger Olander, Dept of Radaton Physcs, Lnköpng Unversty, Sweden - 6

21 - Centre of Rotaton Determnaton usng Proecton Data n X-ray Mcro Computed Tomography - The second example uses a set of proecton data scanned wth the µct equpment at the department of Radaton Physcs. The scanned obect s a pencl and the rebnned proecton data s a set of data wth N = 3 samples per proecton, t = 5µm and M = 36 proecton π angles wth θ =. 6 opposte proecton data pars (opp, TOPln and TOPfft 36 methods) or set of 6 proectons ( π spacng, sn method) are used to estmate 6 algnment offset error. Correct algnment offset s α A) Frst proecton. B) Reconstructed mage. FIGURE 9. Scanned proecton data - pencl. In ths example usng real data wth rather hgh nose level (manly quantum nose), small but not neglgble detector calbraton and non-lnearty errors clearly show the mproved precson acheved usng the TOP methods compared to the methods. Usng the TOP methods wth ths data we can obtan the COR wth a standard devaton of approxmately µm (/5 of the ray wdth). The thrd example s a scan from a ndustral µct equpment wth bad calbraton manly due to varatons n X- opp TOPln TOPfft sn ray ntensty profle after several hours of scannng. The obect s a resoluton (both spatal and contrast) phantom made of steel wth both hgh and low densty contrast obects of dfferent sze. The rebnned proecton data set have N = 94 samples per proecton, t = 34µm and π M = 58 proecton angles wth θ = (from 64 fanbeam proectons wth β =. 7 ) FIGURE. Algnment result, scanned proecton data - pencl. α. 6. Vertcal axs s calculated algnment offset wth mnmum ( ), maxmum ( ), mean ( ) values and ± one standard devaton, σ α. Fgure shows the frst proecton and a reconstructon of the proecton data. - Brger Olander, Dept of Radaton Physcs, Lnköpng Unversty, Sweden - 7

22 - Centre of Rotaton Determnaton usng Proecton Data n X-ray Mcro Computed Tomography - Fgure llustrates the baselne restoraton usng ths proecton data and α = 85.. Ths example clearly shows the robustness of the TOP methods together wth baselne restoraton A) Frst proecton. B) Reconstructed mage. FIGURE. Scanned proecton data - resoluton phantom..5.5 Proecton Data,.5.5 Proecton Data, A) Translated proecton data. B) Translated and reversed proecton data Estmated TOP dfference Baselne, order= C) Estmated TOP dfference. D) Baselne of order l = 3..5 Estmated TOP dfference Zoomed.5 Estmated TOP dfference wth Baselne restoraton Zoomed E) Proecton data, and 8, zoomed. F) Same as E wth baselne restoraton. FIGURE. Scanned proecton data - resoluton phantom. Algnment usng baselne restoraton. - Brger Olander, Dept of Radaton Physcs, Lnköpng Unversty, Sweden - 8

23 - Centre of Rotaton Determnaton usng Proecton Data n X-ray Mcro Computed Tomography - Table lsts the algnment results usng the presented methods (no algnment errors are calculated snce we have only data from 8 ). We cannot use the methods wth ths data snce the estmated COR error s > 5. The TOP methods wthout baselne restoraton works better wth a COR error < but the bg mprovement comes wth baselne restoraton ( TOP methods only). The BR parameters used n ths example s baselne order l = 3, stop condtonα stop =. and max. number of teratons n stop =. Correct algnment offset s n the range α =... Baselne restoraton opp TOPln TOPfft sn no yes TABLE. Algnment results, scanned proecton data - resoluton phantom. - Brger Olander, Dept of Radaton Physcs, Lnköpng Unversty, Sweden - 9

24 - Centre of Rotaton Determnaton usng Proecton Data n X-ray Mcro Computed Tomography - CONCLUSIONS All the presented methods can be used to determne the Centre Of Rotaton, COR, from proecton data. The Centre Of Gravty,, method presented n [Hogan93] s mproved by a new way to obtan the φ gvng Mnmum Mean Square Error, the sn method. Usng the sn method an algnment offset, α, can be estmated for each proecton angle θ and parallel proecton data from less than 8 s suffcent to determne COR. The other presented methods need at least one par of parallel opposte proectons, separated exactly 8 and a fxed algnment offset for all proecton angles. A smplfed method, the opp method s also presented gvng an easly mplemented and smple way to determne COR usng proecton data. The methods are senstve to detector calbraton, detector non-lnearty, X-ray ntensty profle changes and other system-errors. The ntroduced Translated Opposte Proecton, TOP, methods mprove algnment precson n presence of system errors. The TOP-methods can also be used together wth a Baselne Restoraton, BR, technque to further mprove the algnment of data wth system errors. The TOPln method usng lnear nterpolaton to estmate the translated opposte proecton s easly mplemented and normally acheves suffcent algnment precson wth moderate computng efforts. The TOPfft method mproves algnment precson but s more complex to mplement, need more experence to tune (select proper parameters) and need more computng power. These methods and especally the TOP methods have been tested thoroughly and are used successfully on the µct equpment at the Department of Radaton Physcs and a few other stes workng wth µct. - Brger Olander, Dept of Radaton Physcs, Lnköpng Unversty, Sweden -

25 - Centre of Rotaton Determnaton usng Proecton Data n X-ray Mcro Computed Tomography - REFERENCES [Hogan93] John P. Hogan, Robert A. Gonsalves and Allen S. Kreger, Mcro Computed Tomography: Removal of Translaton Stage Backlash, IEEE Transactons on Nuclear Scence, vol. 4, No. 4, pp. 38-4, August Brger Olander, Dept of Radaton Physcs, Lnköpng Unversty, Sweden -