Acceleratng X-Ray data collecton usng Pyramd Beam ray castng geometres Amr Averbuch Guy Lfchtz Y. Shkolnsky 3 School of Computer Scence Department of Appled Mathematcs, School of Mathematcal Scences Tel Avv Unversty, Tel Avv 69978 3 Program of Appled Mathematcs, Department of Mathematcs Yale Unversty, New Haven, CT, USA Abstract Image reconstructon from ts projectons s a necessty n many applcatons such as medcal (CT), securty, nspecton and others. Ths paper extends the D Fan-beam method n [] to three dmensons. The algorthm, called Pyramd Beam (PB), s based on the parallel reconstructon algorthm n []. It allows fast capturng of the scanned data. In three dmensons, the reconstructons are based on the dscrete X-Ray transform []. The PB based algorthms have dfferent geometres where smplcty of the capturng geometres s emphaszed. The PB geometres are reordered to ft parallel projecton geometry. The underlyng dea n the paper s to use the algorthm n [] by portng the proposed PB geometres to ft the algorthm n []. The complexty of the algorthm s comparable wth the 3D FFT. The results show excellent reconstructon qualtes whle beng smple for practcal use. Introducton X-Ray magng s a crtcal component n many applcatons such as medcal scans (CT, MRI, PET), baggage scannng n arports, materal nspecton, cars tre nspecton, food nspecton, bology, electroncs and many more. In practce, emtters emanatng electromagnetc radaton and detectors, whch measure the radaton power arrved at them, are used n X-Ray tomography. The photons radaton, whch passes through the scanned object, decreases. From the radance at the detectors and from the emtter s radance, t s possble to reconstruct a 3D functon of the radance attenuaton. The attenuaton factor s unque for dfferent materals.
Wth the advance of technology and the rapd ncrease of computatonal power, 3D reconstructons become practcal. Usng 3D reconstructon methods together wth 3D vsualzaton, greatly mprove the analyss capabltes of the scanned results. In ths paper, we present several related methods to accelerate 3D X-Ray data acquston when only one emtter s used. These methods are based on the PB geometry. Its performance s compared wth the parallel beam geometry. The orgnal (source) mage s reconstructed by the applcaton of the nverse X-Ray algorthm ([]). All the proposed methods n the paper are based on careful postonng of multple detectors to enable smultaneous collecton of many rays that are emtted n all drectons by one emtter. Here s an herarchcal lst of the methods that are descrbed n the paper: Centered Pyramd Beam (CPB) method has a smple geometry. The emtter and the detectors are postoned accordng the structure that the nverse X-Ray transform [] dctates. It collects only porton of the actual X-Ray data. Thus, t s mpractcal. On the other hand, t demonstrates the capabltes of other practcal methods that are descrbed below. Boundary Algned emtter Pyramd Beam (BAPB) method collects all the requred data to perform an mage reconstructon. Sldng Boundary Algned emtter Pyramd Beam (SBAPB) method s a varaton of the BAPB method n whch the detectors are utlzed more effcently. Ths method reduces the number of detectors. It can be used wth all the X-Ray data acquston methods that are descrbed here. Mrrored Pyramd Beam (MPB) method collects only a porton of the requred data for the reconstructon. The rest of the data s collected by mrrorng the rays. Dstrbuted Pyramd Beam (DPB) s a dstrbuted algorthm that dvdes the boundng volume of the object to several sub-volumes, collects the X-Ray data and performs the reconstructon of each sub-volume separately. Then, all the reconstructons are concatenated to create the complete mage. The MPB and DPB methods requre that the emtter s located on planes nsde the boundng volume of the object. Therefore, they are applcable to process non-sold objects, to scan smultaneously several separated objects n dfferent X-Ray chambers, or to scan complex objects that are separated on the planes nsde the boundng volume where the emtter has to be located. Any type of object (sold and non-sold) can be scanned wth the BAPB method and there are no restrctons on ts structure. In secton 5., we show how to reduce the number of detectors to the mnmum dctated by [], by postonng them on movng boards. Ths dea s applcable to all the above methods.
In our mplementaton, the geometry n each axes was the same, each axes can have ts own geometry. The proposed PB ray castng topology speeds the 3D X-Ray data acquston by O(n ) factor n comparson to the parallel beam topology. The structure of the paper s as follows. Secton revews related works on fast nverson algorthms of the X-Ray Transform whle speedng data acquston usng pyramd beams or cone beams. In Secton 3, the X-Ray Transform and ts dscrete verson, whch appeared n [], are descrbed. Secton 4 demonstrates the parallel beam data acquston and reconstructon of X-Ray data produced by an analytc parallel beam X-Ray Transform. The pyramd beam projectons are defned n secton 5. It contans a descrpton of several acquston methods and how to convert from pyramd beam projecton data nto parallel projecton data. Related works Snce memory and tme complexty of the reconstructon algorthms grow polynomally wth the number of dmensons then any algorthmc acceleraton s crtcal. Acceleraton can come from speedng ether the acquston or the reconstructon or from both. Two man approaches are used to reconstruct 3D mages from X-Ray projectons. The frst approach reconstructs separately D slces of the mage and then concatenates the slces to form a 3D mage. Ths requres the mage to be statc to prevent regstraton problems. It also may generate dscontnutes n the reconstructed 3D mage. The second approach generalzes the D reconstructon algorthms to any number of dmensons. The frst approach for a 3D object segmentaton and reconstructon s used n [8, 9]. [] regsters the D slces and then reconstructs the 3D object. A technque, whch mproves the qualty of D slces and then uses the mproved slces to construct the 3D mage va mage processng methods, s descrbed n [, ]. In ths paper, we are nterested n acceleratng the acquston whle usng a fast 3D X-Ray reconstructon algorthm that s descrbed n []. Usually, fast 3D X-Ray reconstructon algorthms are based on the Fourer slce theorem. Some of these algorthms nterpolate the polar grd nto a Cartesan grd. The Fourer transform s senstve to nterpolaton and the reconstructed mage suffers from dstortons. The fltered back projecton based algorthms overcome ths problem but ther complexty s O(n 4 log n) where n s the mage resoluton n each axs. Accurate reconstructon that does not necesstate nterpolaton s descrbed n [] and t s based on the constructons n [3, 4]. Bresler et. al. [8] proposes herarchcal algorthm for applyng the back projecton of the 3D Radon transform. Ther algorthm s a natve 3D algorthm and does not rely on factorzaton of 3
the 3D Radon transform nto pars of D Radon transforms, whch makes the algorthm ndependent of the samplng geometry. The algorthm n [8] decomposes each projecton nto a sum of 8 back projectons each has n plane-ntegrals projectons onto n 3 /8 volumes. Each volume s one octant of the reconstructon. The algorthms are appled recursvely untl each octant s sze s one voxel. The complexty of the algorthm s O(n 3 log(n)). Another famly of reconstructon algorthms s the mult-level nverson algorthms. It dvdes the nput snogram to a number of subsnograms that uses ether exact or approxmate decomposton algorthms. The snograms are repeatedly subdvded untl they are represented by one voxel. Then, the nverse transformaton s appled to reconstruct the sub-volumes. The sub-volumes are aggregated to form the fnal volume. An exact method to decompose the snograms s descrbed n [3]. Ths paper also presents a fast algorthm whch approxmates the reconstructon. In addton, t also presents a method that combnes both. Maxmum lkelhood expectaton maxmzaton ([4]) s an teratve reconstructon method, n whch an ntal reconstructon s guessed, and then updated n order to mnmze the dfference between the projectons of the reconstructed mage and the measured projectons. In addton, [4] descrbes the cone beam data acquston method. An algorthm that decomposes the mage frequency doman to sub-bands and reconstructs the sub-bands on a down-sampled grd s gven n [7]. Cone beam projecton methods, whch s based on acceleratng the scanned data acquston by detectng multple rays emtted smultaneously from a sngle source, are gven n [4, 5, 6]. Ths paper proposes fast acquston algorthm whch s a varaton of the cone beam method. The projecton s assumed to be a collecton of rays that forms a pyramd. These rays are sampled smultaneously. The reconstructon algorthm, whch s descrbed n [], s algebracally accurate whle preservng the geometrc propertes of the contnuous transforms. It s also nvertble. 3 The X-Ray Transform The proposed fast data acquston methods n ths paper are based on the 3D X-Ray transform geometry that s descrbed n []. The 3D transform s outlned here. In the rest of paper, u denotes a unt vector. The X-Ray transform of a 3D functon f = f(x, y, z) s a collecton of all lne ntegrals of f over all the lnes n the 3D space. Defnton 3.. A lne l = p + t d, t R n the 3D space, s defned by ts drecton unt vector d R 3 and a pont p R 3 that the lne passes through. Defnton 3.. Drecton by angles. Two angles θ, φ R defne a lne drecton by a unt vector 4
d R 3 denoted d θ,φ, by rotatng the unt vector u x = (,, ) by φ around the Y axs, and then rotatng the resultng vector by θ around the Z axs. θ s also known as the vector headng and φ as the vector elevaton. A lne n R 3 wth drecton d θ,φ s denoted by l p θ,φ. Defnton 3.3. Drecton by a pont. A pont p dr R 3 defnes a lne drecton by a unt vector, denoted by d pdr, as the vector from that pont to the orgn (,, ). That s, d = wth d pdr s denoted by l p p dr. Defntons 3. and 3.3 are equvalent. p dr p dr. A lne n R3 Defnton 3.4. Lne ntegral. The lne ntegral of f(x, y, z) over the lne l p θ,φ, denoted by LIp θ,φ, s LI p θ,φ f = ( f p + t d ) θ,φ dt, θ, φ, t R, p R 3. From these defntons, the X-Ray transform of f(x, y, z), denoted by XRf, s the set { XRf = LI p θ,φ f θ, φ R, p R3}. In a smlar way, the lne ntegral of f(x, y, z) over the lne lp p dr, denoted by LI p p dr, s LI p p dr f = f (p + t d ) pdr dt, t R, p, p dr R 3. By usng these defntons, we get that the X-Ray transform of f(x, y, z), denoted by XRf, s the set XRf = { LI p p dr f p dr, p R 3}. Defnton 3.5. Parallel projecton. A parallel projecton of the X-Ray transform s a collecton of all the computed lne ntegrals that have the same drecton. These lnes are defned by a specfc drecton d θ,φ or d pdr by XR p θ,φ f or XRp p dr f, respectvely. where θ, φ R, p dr R 3 and an arbtrary p R 3. The projecton s denoted The Fourer Slce Theorem lnks between the parallel projectons XR p θ,φ f, θ, φ R, p R3, and the Fourer Transform. It establshes that the Fourer transform of a parallel projecton n drecton d θ,φ of a 3D functon f(x, y, z) s the Fourer transform of f(x, y, z) sampled on a hyper-space perpendcular to d θ,φ that passes through the orgn. Formally, XR p θ,φ f = f ( ) (ξ) where ξ dθ,φ s the hyper-space perpendcular to the vector d θ,φ, that passes through the orgn. In other words, the D Fourer transform of the parallel projecton XR p θ,φf equals to the 3D Fourer transform of f(x, y, z) sampled on ξ. Lemma 3.6. ([]) Assume I s a dscrete 3D mage wth resoluton n n each drecton. Then, the D FFT of a parallel projecton at some drecton d θ,φ, θ, φ R, s a D plane n the 3D FFT of the orgnal mage XR p x,α,β (k, l) = XR p y,α,β (k, l) = XR p z,α,β (k, l) = 5 Î ( αk βl, k, l) Î (k, αk βl, l) Î (k, l, αk βl)
where α, β are the slopes between the unt vector and the cartesan axes x, y, z, k, l = n,..., n and XR x XR y and XR z are explaned n secton 3.. Lemma 3.6 shows that n order to reconstruct the mage from parallel projectons, we have to apply the D Fourer transform to the projectons, re-organze them n the 3D space to get the Fourer Transform of the orgnal mage, and then the 3D Inverse Fourer Transform s appled to recover the orgnal mage. The Dscrete X-Ray Transform n [] provdes a O(n 3 log n) algorthm that reconstructs accurately the 3D mage. It s based on the reorganzaton of the Fourer Transforms of the projectons n the pseudo-polar grd as was explaned n [3, 4]. 3. The dscrete X-Ray Transform We assume that the data for the reconstructon s fnte and dscrete. The algorthm to compute the X-Ray transform of a 3D dscrete mage s gven n []. The nvertblty of the algorthm and ts valdty n representng dscrete mages are proven n detals there. Here s a descrpton how to dscretze the mage and the underlyng pseudo-polar grd (see [3, 4]). 3.. Dscretzaton of the X-Ray Transform data Followng are the defntons that descrbe the dscrete mage and the sets of ponts defnng the lnes drectons and ther translatons. We assume that the mage s a dscrete 3D functon f(x, y, z) that s defned as f = {f(x, y, z) n x, y, z n }. Accordng to [], the calculated projectons calculatons are separated nto three groups. Defnton 3.7. Man axs, denoted A, s X = A Y = Z = 3. Defnton 3.8. Secondary axes, denoted by S and S, are S Y = X otherwse S Y = 3 Z otherwse. The lnes n R 3 are also separated nto three subsets. Each subset s assocated wth a man axs X, Y or Z. At each subset of lnes, the absolute value of the angles between the projectons of the lnes on the planes XY, XZ and Y Z and the man axs A are smaller than 45. 6
Defnton 3.9. (Lnes dvson I) The three subsets of the lnes n R 3 are: L x = {l p θ,φ 45 θ < 45 or 35 θ < 5, 45 φ < 45 or 35 φ < 5, p R 3 } L y = {l p θ,φ 45 θ < 35 or 5 θ < 35, 45 φ < 45 or 35 φ < 5, p R 3 } L z = {l p θ,φ θ < 36, 45 φ < 35 or 5 φ < 35, p R 3 }. Loosely speakng, lnes that belong to L are closer to the man axs A than to any other axs. Ths dvson covers all the lnes n R 3 - see proof n []. Lemma 3.. Assume that two lnes pass through the same pont p R 3. The frst lne drecton s defned by φ, θ R. The second lne drecton s defned by φ + 8 and θ or by φ and θ + 8. Then, l p θ,φ lp θ,φ+8 l p θ+8, φ. From Lemma 3., defnton 3.9 becomes: Defnton 3.. (Lnes dvson II) The three subsets of the lnes n R 3 are: L x = {l p θ,φ 45 θ < 45, 45 φ < 45, p R 3 } L y = {l p θ,φ 45 θ < 35, 45 φ < 45, p R 3 } L z = {l p θ,φ θ < 36, 45 φ < 35, p R 3 }. For a spatally bounded functon f(x, y, z), the majorty of the lnes n R 3 do not ntersect the functon s boundng volume. As explaned n defnton 3.4, lne ntegral s defned by a drecton and a pont the lne passes through. The lmtatons on the drectons of the lnes, whch partcpate n the dscrete X-Ray transform, are defned n 3.. Lemma 3. determnes the mnmal set of ponts requred to defne lnes that produce non-trval lne ntegrals. Lemma 3.. Ponts of nterest ([]) Assume that each of the coordnates x, y, z of the functon f(x, y, z) are spatally bounded by the nterval [ n/, n/]. In addton, we restrct the drectons to the set defned n 3.. Then, the mnmal set of ponts, whch s requred to defne the non-trval lne ntegrals, ncludes ponts wth coordnate A = and the coordnates S and S are bounded by [ n, n]. The lnes pass through these ponts as was descrbed n defnton 3.4. Defnton 3.3. Dscrete set of ponts of nterest. A dscrete subset of ponts, whch were defned n Lemma 3., have the coordnates A = and S, S { n,..., n}. Ths set s denoted by P tr. The collecton of non-trval lne ntegrals over lnes from L, =,, 3, s denoted by XR x XR y and XR z, respectvely. For a pont p dr R 3, defnton 3.3 descrbes how to determne the lne drecton. In order to dscretze the lnes sets n defnton 3., a dscrete set of ponts s defned. 7
Defnton 3.4. Current lnes L are defned for each A by: L x = L = L y = L z = 3 Defnton 3.5. Dscrete set of drectons. ncludes ponts wth the coordnate A = n/ and S, S of the lne set L, =,, 3.. A dscrete set of ponts, denoted by P dr, whch { n/,..., n/}, defnes a dscrete subset For all the ponts n P tr, the coordnate A s equal to. Smlarly, for all the ponts n P dr, A s equal to -. Therefore, these ponts can be defned unquely by pars of values from the other two coordnates S and S. Defnton 3.6. Smplfed drectons and translatons sets. The ponts n the sets P tr and P dr can be defned unquely by the pars (s, t) and (k, l), respectvely, s, t, k, l R. k and s represent the coordnate S P dr, respectvely. whle l or t represent the coordnate S. These sets of pars are denoted by P tr and Accordng to [], the dscrete sets Ptr and Pdr, =,, 3, defne the exact set of lne ntegrals requred to arrange the data on the pseudo-polar grd. reconstructon method that was descrbed there. Ths enables to use the fast and accurate 4 Reconstructon from analytcal parallel projectons The valdty and the accuracy of the reconstructon algorthms were tested and verfed on the 3D Shepp-Logan mage whch was constructed analytcally. In ths secton, the reconstructon algorthm uses parallel projectons. In secton 5, the Pyramd-beam reconstructon s descrbed. Two methods are used to compute the projectons: Method : The parallel projectons arrays are computed analytcally. Method : The parallel projectons arrays are computed usng the dscrete X-Ray transform ([]). Table 4. dsplays the l error between the two methods that compute the projectons. The computatons that are marked by were done by a dstrbuted algorthm - see secton 6. 8
n l error l /n 3 error per voxel 4 5.9 8.8e-4 8 6.77 8.938e-4 6 66.434.83e-4 3 86.664 8.6e-5 64 494.57.36e-5 8 4896.59.485e-6 56 48.536.9e-7 Table 4.: The l error between the analytc and the dscrete computaton ([]) of the projectons of the Shepp-Logan mage. Table 4. shows that the computatonal error per voxel decreases as the resoluton of the mage ncreases. Fgure 4. dsplays the l error between projectons of the Shepp-Logan mage at dfferent drectons where n = 64. The mnmum and the maxmum of the l errors were.463 and 47.88, respectvely. Each pxel n the mage s the l of the dfference between the projectons computed at dfferent drectons accordng to defnton 3.3. The error per pxel s derved from the dfference between the analytc computaton and the exact X-Ray computaton from []. The mnmum and the maxmum error values show that the contrbuton of each lne ntegral to the l error s small as shown n table 4.. From Fg. 4., we see that when at least one of the secondary coordnates S and S of the drecton ponts p dr s close to zero, the error decreases. Ths s due to the fact that the ponts coordnates along the lne ntegral are proportonal to the tan of the angle. Therefore, the error from a lne ntegral computaton ncreases wth the angle. 3 3 3 4 4 4 5 5 5 6 6 6 3 4 5 6 3 4 5 6 3 4 5 6 (a) (b) (c) Fgure 4.: l error between projectons when the man axes s X (a), Y (b) and Z (c) 9
4. Parallel projecton geometry The dscrete parallel projectons n respect to a man axs A are retreved by restrctng the lne ntegrals from defnton 3.5 to the set of lnes defned by the ponts n P tr and P dr 3.6). For a pont p dr ntegrals whose drectons are defned by p dr. (see defnton P dr, the dscrete parallel projecton XR P tr f, =,, 3, contans lne, p dr For each pont p tr P tr, there s exactly one lne ntegral n the projecton that passes through the pont p tr. The mage s bounded n the nterval [, ] n each axs. The mage resoluton at each axs s n. Ths mples that the set of coordnates { n/,..., n/} s mapped to [, ]. The ponts n the set P tr, defned n 3.3, have the coordnates A = and S, S set P dr, defned n 3.5, have the coordnates A = and S, S {j/n j = n,..., n}. The ponts n the {j/n j = n/,..., n/}. In order to understand where the emtter and detector have to be placed, a specfc lne s analyzed. Defnton 4.. Generalzed pont descrpton. p G (u, v, w) s a pont where u, v, w R are the coordnates of A, S and S, respectvely. Defnton 4.. Generalzed planes. A plane, whch s defned by settng the man axs coordnate to a constant value A c, c R, s denoted by P (c). A lne that s defned by the translaton pont p tr(s, t) and by the drecton pont p G (,k,l) dr (k, l) passes through the pont p p G (, s, t). From defnton 3.3, the lne drecton s p G (,k,l). Therefore, ths lne ntersects the planes P () and P () at the ponts p G (, s + k, t + l) and p G (, s k, t l), respectvely, where s, t {j/n j = n,..., n} and, k, l {j/n j = n/,..., n/}. P dr All the lne ntegrals n the dscrete parallel projecton XR P tr f, where p, p dr dr s a specfc pont n and p tr, are all the ponts n P tr that have the same drecton. Each lne passes through a dfferent pont on the plane P (). Therefore, the lnes are parallel as ths method name suggests. For a specfc drecton defned by the pont p dr (k, l) P dr, the process, whch calculates the projecton XR P tr, p dr (k,l)f usng one emtter, s descrbed below. For each pont p tr(s, t) P tr, the emtter s placed at the pont p G (, s + k, t + l) and the detector s placed at the pont p G (, s k, t l), k, l {j/n j = n/,..., n/} and s, t {j/n j = n,..., n}. The emtter s postons are all n a square where coordnates A = and S, S are from the nterval [ 3, 3]. The detectors postons are the same as the emtter except A =. Ths geometry shows that the emtter and the detector are beng located on parallel planes P () and P (), respectvely. Lne ntegrals n the same parallel projecton cannot be calculated smultaneously. Therefore, only one detector s needed to compute ths projecton. Fgure 4. shows the lnes from L 3 (see defnton 3.4), whch are defned by p 3 tr(, ) and by dfferent ponts from P dr 3. Fgure 4.3 shows the lnes from L 3 whch are defned by p dr 3 (.5,.5) and by dfferent ponts from P 3 tr. In Fg. 4.3, the gray dashed lne s defned by the translaton p 3 tr(, ).
....4.4.4.6.6.6.8.8.8.5.5.5.5.5.5 (a) (b) (c)...4.4.6.6.8.8.5.5.5.5 (d) (e) Fgure 4.: Lnes defned by the pont p 3 tr(, ) and by dfferent p 3 dr P 3 dr. (a) Lne defned by p dr 3 (, ). (b) Lne defned by p 3 dr (.5,.5). (c) Lne defned by p 3 dr (.5,.5). (d) Lne defned by p dr (,.5). (e) All lnes drectons defned by the set P 3 dr....4.4.4.6.6.6.8.8.8.5.5.5.5.5.5 (a) (b) (c)...4.4.6.6.8.8.5.5.5.5.5 (d) (e) Fgure 4.3: Lnes defned by p dr 3 (.5,.5) and by dfferent p 3 tr P tr. 3 (a) Lne defned by p 3 tr(.,.5). (b) Lne defned by p 3 tr(.5,.5). (c) Lne defned by p 3 tr(.,.5). (d) Lne defned by p 3 tr(.5,.). (e) A subset of lnes from the parallel projecton XR P 3 tr 3, p 3 dr (.5,.5)f
Fgure 4.4 descrbes several subsets of lnes from parallel projectons at dfferent drectons.....4.4.4.6.6.6.8.8.8 (a) (b) (c) Fgure 4.4: Parallel projectons. (a) A subset of the projecton XR P tr 3 3, p dr 3 (.,.)f. (b) A subset of the projecton XR P tr 3 3, p dr 3 (.,.)f. (c) A subset of the projecton XR P 3 tr 3, p dr 3 (.5,.)f Accordng to Lemma 3. and the fact that f = f(x, y, z) s bounded n each drecton, t s easy to verfy that lne ntegrals over lnes wth translaton greater than. n one of the dmensons are equal to. Fgure 4.5 descrbes the parallel projecton XR P tr 3 3, p dr 3 (.,.)f. The bold lnes n Fg. 4.5 represent the boundng volume of f(x, y, z). The gray lnes begn n plane P () where the emtter s located and ends n the plane P () where the detector s located. The fgure shows that the emtter s and the detector s coordnates S and S, respectvely, are n the nterval [ 3, 3], as was mentoned n ths secton before. From defnton 3.3, p 3 tr P 3 tr are ponts on the plane P 3 (). Fgure 4.5 shows that the coordnates S and S of the ponts on the plane P 3 () are n the nterval [.,.]. It also shows that lnes, whch are defned by the ponts p 3 tr(s, t) where s or t equal. or., are tangent to the boundng volume. If s > or t > then the lnes wll not ntersect the boundng volume. These results are also true for projectons n drectons defned by p dr 3 (k, l) where k < or l <..8.6.4..8.6.4. 3 3 3 3.8...6.4.4.4.6.6..8.8. 4.4 5.6.8 3 3 5 4 3 3 Fgure 4.5: Dfferent vews of non-trval lnes that have the same drecton p dr 3 (.,.). The emtter s located on the plane P 3 () and the detector s located on the plane P 3 (). The nverse dscrete X-Ray transform ([]) reconstructs the mage from a set of parallel projectons. The sets Ptr and Pdr together defne all the lne ntegrals requred to reconstruct the mage.
Defnton 4.3. The nput to the nverse X-Ray transform. The nput for the nverse X-Ray transform s all the parallel projectons defned by the sets Ptr and Pdr. Ths set, denoted P P, s: { } P P = XR p tr f p,p dr P dr, p dr P dr, =,, 3. dr The parallel projecton XR p tr f s computed for each drecton defned by the pont p, p dr dr P dr. The projecton s a D array of sze (n + ) (n + ). The coordnates of each element n the array correspond to a pont p tr P tr. The value of the array element s LI p tr Defnton 4.4. Parallel projectons data structure. All the lne ntegrals, requred to reconstruct the mage by the dscrete nverse X-Ray transform ([]), are stored n the array P P ds., p dr f. The frst coordnate n the array s, =,, 3. It represents the man axs X, Y or Z. The followng two coordnates p, q {,..., n + } represent the drecton of the lne ntegral p dr (k, l) where k = (p n/ ) /n and l = (q n/ ) /n. The last two coordnates, u, v {,..., n + }, represent the translaton of the lne ntegral, p tr(s, t) where s = (u n ) /n and t = (v n ) /n. Formally, P P ds (, p, q, u, v) = LI p tr (s,t), p dr (k,l)f. (4.) For specfc, p, q and all u, v {,..., n + }, the collecton of P P ds (, p, q, u, v) values s the parallel projectons XR P tr, p dr (k,l)f where k = (p n/ ) /n and l = (q n/ ) /n. In order to compute a parallel projecton n a gven drecton, the emtter and the detector have to be postoned at (n + ) locatons. It means that each parallel projecton requres (n + ) operatons. For each man axs A, =,, 3, there are (n+) parallel projectons that correspond to dfferent drectons. Thus, computng the data structure P P ds requres 3 (n + ) (n + ) operatons. Therefore, the total number of operatons s O(n 4 ) where n s the resoluton of each dmenson. The mages were computed by two methods:. Analytcally.. The projectons were computed analytcally and the reconstructon was done va the applcaton of the nverse dscrete X-Ray transform ([]). The dfferences between the projectons computed by. and. were computed. The numercal results from ths reconstructon are presented n secton 5.4. 5 Pyramd-Beam (PB) reconstructon Emtters are more expensve than detectors. Therefore, the PB data acquston geometry suggests to add detectors n order to collect smultaneously the lne ntegrals n multple drectons. Usually, the use of only one emtter s common n X-Ray tomography to reduce the data acquston costs. PB geometry s used n X-Ray transform when one emtter s present. It becomes more effcent for data acquston than parallel geometry. In PB geometry based data acquston, the lne ntegrals 3
n all drectons can be calculated smultaneously. Therefore, the number of operatons requred to collect the projected data has to be dvded by O(n ). In ths secton, a famly of methods, whch are based on PB geometry, s descrbed. For two constants c, c R, c c, PB projectons are computed by locatng the emtter on the plane P (c ) (see defnton 4.) and the detectors on the plane P (c ). For dfferent PB methods, c and c have dfferent values. Ths geometry allows a smultaneous computaton of multple dfferent lne ntegrals that pass through the same pont and have dfferent drectons. The planes P (c ) and P (c ) are orthogonal. Therefore, the lnes partcpatng n each PB projecton form a shape of a square pyramd (see Fg. 5.). PB projecton s defned n a smlar way to defnton 3.5. Defnton 5.. PB projecton. A PB projecton of the X-Ray transform s a collecton of all the computed lne ntegrals that pass through a specfc pont p R 3 and have arbtrary drectons d θ,φ or dpdr where θ, φ R, p dr R 3. Ths projecton s denoted by XR p θ,φ f or by XRp p dr f. A PB projecton XR p tr, P f, =,, 3, s a collecton of lne ntegrals defned by a specfc pont dr from the set p tr and by all the ponts from the set p dr P dr (see defnton 3.6). The man goal of ths paper s to fnd an effcent method to collect smultaneously multple lne ntegrals. In order to reconstruct the mage by the nverse X-Ray transform ([]), the PB projectons have to be transformed nto the P P ds data structure defned by Eq. 4.. Ths transformaton s called reorderng. Each data acquston method has ts own verson of reorderng algorthm. The dea s that the algorthm n [] s effcent and accurate and each acquston method wth dfferent PB geometres has to be transformed nto a parallel projecton methodology descrbed n defnton 3.5. The reorderng algorthm algorthm does not necesstate any operatons. Several PB methods called CP B, BAP B, SBAP B and MP B are presented here (see also secton ). For each method, ts data acquston geometry and ts reorderng algorthm are descrbed and ts complexty s analyzed. 5. Centered Pyramd-beam (CP B) acquston geometry The CP B geometry s based on the sets P tr and P dr (see defnton 3.6). Assume s, t {j/n j = n,..., n} and =,, 3. For each pont p tr(s, t) P tr, the emtter s located at the pont p G (, s, t) (see defnton 4.). The detectors are located at the ponts p G (, s k, t l) where k, l {j/n j = n/,..., n/}. The emtter s postons are located at ponts from the set P tr. Accordng to Lemma 3. and the fact that f(x, y, z) s bounded by [, ] at every axs, these postons are all located n a square wth coordnate A =, and the coordnates of S and S are taken from the nterval [, ]. Therefore, 4
the detectors postons have the coordnate A = and the coordnates S and S n the nterval [ 3, 3]. It means that on the plane P () there are (3n + ) detectors. For each emtter s poston p G (, s, t), only (n + ) detectors, whose secondary coordnates S and S vary between [s, s + ] and [t, t + ], respectvely, are of nterest. These detectors values are lne ntegrals of lnes from the set P P (see defnton 4.3). The other detectors values are not needed by the nverse dscrete X-Ray transform. Ther values are lne ntegrals of lnes wth drectons that do not belong to P dr. Accordng to defnton 5., n order to compute a CP B projecton XR p tr (s,t), P dr p tr(s, t) P tr, s, t {j/n j = n,..., n}, the emtter s placed at the pont p G (, s, t). The projecton s an array that contans the values of the detectors wth coordnate A =. S are n the nterval [s, s + ] and S coordnates are n the nterval [t, t + ]. f for the pont coordnates For each pont p tr P tr, a CP B projecton s computed. The projecton s result s a D array of sze (n + ) (n + ). The coordnates of each element n the array correspond to a pont p dr P dr and ts value s LI p tr f., p dr Defnton 5.. CP B data structure. All the lne ntegrals, whch are requred to reconstruct the mage by the dscrete nverse X-Ray transform ([]), are stored n the array CP B ds. The frst coordnate s, =,, 3. It represents the man axs X, Y or Z. The followng two coordnates, u, v {,..., n + }, represent the translaton of the lne ntegral p tr(s, t) where s = (u n ) /n and t = (v n ) /n. The last two coordnates, p, q {,..., n + }, represent the drecton of the lne ntegral p dr (k, l) where k = (p n/ ) /n and l = (q n/ ) /n. Formally, CP B ds (, u, v, p, q) = LI p tr (s,t), p dr (k,l)f. (5.) For specfc, u, v and all p, q {,..., n + }, the collecton of the values CP B ds (, u, v, p, q) s the CP B projecton XR p tr (s,t), p dr f where s = (u n ) /n and t = (v n ) /n. Defntons 4.4 and 5. descrbe the coordnates of the data structures of P P ds and CP B ds, respectvely. Lemma 5.3 shows how to swtch between these two data structures. The reorderng algorthm swtches between pars of coordnates n the CP B ds data structure. Lemma 5.3. Reorderng the CP B data structure. The data n the CP B ds data structure s reordered nto parallel projectons by P P ds (, p, q, u, v) = CP B ds (, u, v, p, q), where =,, 3, p, q {,..., n + } and u, v {,..., n + }. The pars (p, q) represent the lne ntegrals drectons p dr (k, l), k = (p n/ ) /n and l = (q n/ ) /n. The pars (u, v) represent the lne ntegrals translatons p tr(s, t), s = (u n ) /n and t = (v n ) /n. Proof. The range of the forth and ffth coordnates n P P ds s the same as the range of the second and thrd coordnates n CP B ds. Each element n the data structure P P ds (, p, q, u, v) contans the lne ntegral LI p tr (s,t), p dr (k,l)f (see defnton 3.4), where k = (p n/) /n, l = (q n/) /n, s = (u n) 5
/n and t = (v n) /n. From defnton 5., each element n the data structure CP B ds (, u, v, p, q) contans the same lne ntegral. Therefore, the equalty P P ds (, p, q, u, v) = CP B ds (, u, v, p, q) holds for every selecton of u, v. In order to compute a CP B projecton, the emtter has to be located at one pont. For each man axs A, =,, 3, there are (n + ) CP B projectons that correspond to dfferent lne translatons. The reorderng does not requre any operatons as was mentoned at the begnnng of ths secton. Therefore, the computaton of the data structure CP B ds requres 3 (n + ) operatons, whch sums to O(n ) operatons. Thus, t accelerates the parallel data acquston geometry by O(n ). Fgure 5. vsualzes the lne ntegrals that partcpate n the CP B projectons from dfferent emtter s postons..5.5.5 Fgure 5.: Locatons of he emtter n the CP B geometry. (a) Emtter located at p 3 tr(.,.). (b)emtter located at p 3 tr(.5,.). (c) Emtter located at p 3 tr(.,.) Usually, the scanned objects are sold. They are placed around the orgn. Ths prevents from placng the emtter on the planes P () n contradcton to the CP B gudelnes. Even f the object s nether sold nor placed n the orgn, we stll cannot utlze the CP B methodology to reconstruct the mage. The reason for that s the fact that the outputs from lne ntegrals represent only porton from the lnes between the planes P () and P (). Thus, they do not represent the whole lne ntegral through the scanned object. In order to overcome these problems, secton 5. proposes a method where the emtter s placed on the planes P (). Secton 5.3 on the other hand, proposes a method that calculates lne ntegrals on both sdes of the emtter s locaton by addng detectors on the plane P (). Ths overcomes only the second problem. 5. Boundary algned Pyramd-beam (BAPB) acquston geometry As mentoned n secton 5., the CP B method assumes that the emtter s placed on the planes P () (see defnton 4.). Ths lmts the scenaros for whch ths method s useful. In order to overcome ths lmtaton, the BAP B proposes to move the emtter s locaton to the boundary of the object beng 6
scanned. Specfcally, t recommends to place the emtter on the planes P (). Ths dsplacement affects the geometry and the projectons results. Ths also dctates a change to be made n the reorderng algorthm. To detect lne ntegrals wth dfferent drectons that pass trough the same pont n the BAP B geometry, the emtter s located at dfferent postons n the plane P (). Multple detectors are located on equally spaced grd n a square n the plane P (). Then, only a subset of the detectors values, whch correspond to lne ntegrals whose tangents are bounded by [, ], are stored n the BAP B data structure (see defnton 5.8). Fgure 5. dsplays the transformaton that s appled to the CP B emtter s locatons and ts effect on the pyramd geometry where (a) s the orgnal geometry of the CPB and (b) and (c) are the geometres of PAPB. It s possble to see that the tp of the pyramd n (a) s on the plane P () and n (b) and (c) P. Moreover, the pyramd base n (b) and (c) s twce the sze n (a)..5.5 4 4 4 4 4 4 4 4 (a) CP B projecton (b) BAP B projecton (c) Emtter located at p 3 G(., 3., 3.) Fgure 5.: Dfferent emtter s postons n the BAP B geometry. The two ponts p tr(s, t) P tr and p dr (k, l) P dr defne a lne. Ths lne ntersects the plane P () at the pont p G (, s + k, t + l). A second lne wth a dfferent drecton p dr (q, r), whch passes through the pont p G (, s+k, t+l), ntersects the planes P () and P () at the ponts p G (, s+k q, t+l r) and p G (, s + k q, t + l r), respectvely. Therefore, the second lne s defned by the ponts p tr(s + k q, t + l r) and p dr (q, r). Snce q, r, the detectors secondary coordnates satsfy s + k S s + k + and t + l S t + l +, s, t and k, l. Therefore, the locatons of the detectors, whch are requred to collect the lne ntegrals n the set P P, have the coordnates A =, 5 S, S 5. The emtter s postons have the coordnates A =, 3 S, S 3. Ths geometry requres to poston the emtter n (3n + ) locatons whle spreadng (5n + ) detectors on the plane P (). From these (5n+) detectors, only (n+) detectors values represent lne ntegrals from the set P P. How to select these detectors? Two ponts from the set P dr defne two dfferent lne drectons. Two lne ntegrals wth dfferent lne drectons, whch pass through the same pont, wll be detected by dfferent detectors. The dstance between these two detectors n the 7
BAP B geometry s twce the dstance between the correspondng detectors n the CP B geometry. Lemma 5.4. Dstance between detectors. Assume we have two rays wth dfferent drectons p dr (k, l ) and p dr (k, l ) P dr that are emtted from the same emtter. The dstance between the detectors, whch detect these rays n the BAP B geometry, s twce the dstance n the CP B geometry. Proof. The proof s based on trangles smlarty. In order to smplfy trangles heght calculaton, assume that the S coordnates of the two detectors and the emtter are the same. Accordng to the CP B geometry, the emtter s located at a pont on the plane P (). When the BAP B method s appled, the emtter s located on the plane P (). The lnes drectons are the same n both geometres. Therefore, the locatons of the emtter and the two detectors, n both methods, construct two smlar trangles. From these assumptons, t s easy to calculate the trangles heghts. The CP B trangle heght equals whle the BAP B trangle heght s. Snce these trangles are smlar, the proporton between the heghts s also the proporton between the trangles edges. For both trangles, the edges on the plane P () are the dstances between the detectors. Therefore, the dstance n the BAP B geometry s doubled. In a smlar way, these trangles smlarty can be shown for every two detectors and emtter whch compute the lne ntegrals wth the same two drectons n both geometres. Lemma 5.5. Postonng the emtter and the step sze. In order to collect the lne ntegrals gven n P P (see defnton 4.3), the dstance between two neghborng locatons of the emtter should be the same as n the CP B geometry. Proof. For two lnes n a parallel projecton, whch are defned by the ponts p tr(s, t), p tr(s + /n, t) and p dr (k, l), the emtter n the CP B geometry must be located at p G (, s, t) and p G (, s + /n, t). In the BAP B geometry, the emtter must be located at p G (, s + k, t + l) and p G (, s + k + /n, t + ). Therefore, the dstance between the two ponts n the CP B geometry and the two ponts n the BAP B geometry s the same. Fgure 5.3 vsualzes Lemma 5.5. The emtter s located at two neghborng locatons. The dstance between the closest detectors, whch contan lne ntegrals from P P n the two BAP B pyramd projectons, s the same as n the CP B and as n the parallel geometres. By comparng between the BAP B and the parallel projectons geometres, we get that the emtter n both methods s located on the planes P () wth S and S coordnates satsfyng 3 S, S 3. The BAP B geometry together wth Lemmas 5.4 and 5.5, lead to the followng concluson: Corollary 5.6. Ineffcent detectors utlzaton. The detectors are poorly utlzed n the BAP B method. At each locaton of the emtter, only (n + ) of the (5n + ) detectors, are lne ntegral values from the set P P. Moreover, there are no two neghborng detectors whch contan values from P P. Ether odd or even postoned detectors are used for the reconstructon. 8
Fgure 5.3 vsualzes Corollary 5.6. In Fgs. 5.3(a) Fg. 5.3(b) the detectors are placed n the even and odd postons, respectvely. Fgure 5.3(c) shows all the detectors that collect lne ntegrals whose tangents are bounded by [, ]. 4 4 4 4 4 4 4 4 4 4 4 4 (a) (b) (c) Fgure 5.3: Detectors locatons for two adjacent emtter s locatons n the BAP B geometry. Emtter located at p 3 G (.,.,.). (b) Emtter located at p3 G (.,.5,.5). (c) All lne ntegrals whose tangents are bounded by [, ] (a) As was mentoned before, the lne ntegral defned by the ponts p tr(s, t) P tr and p dr (k, l) P dr, ntersects the plane P () at the pont p G (, s + k, t + l). Ths leads to the followng concluson: Corollary 5.7. Lnes from P P, whch pass through the same ponts, appear n dfferent BAP B pyramd-beam projectons. Two lne ntegrals, whch are defned by two dfferent drectons p dr (k, l ) and p dr (k, l ) and by one translaton p dr (s, t), appear n dfferent BAP B projectons. These lne ntegrals wll appear n the projectons where the emtter s located at the ponts p G (, s + k, t + l ) and p G (, s + k, t + l ). Ths dffers from the CP B method, n whch both lne ntegrals appear n the same projecton where the emtter s located at p G (, s, t). Fgure 5.4 vsualzes Corollary 5.7. It shows two lnes wth a translaton that s defned by p 3 tr(, ). The lnes drectons are defned by p dr 3 (.5,.5) and p 3 tr(.5,.5). In the CP B geometry (left), both lne ntegral are acqured by the same pyramd (centered at (, )). In the BAP B geometry (rght), each lne s acqured by a dfferent BAP B pyramd. 9
.5.5.5 4 4 4 4.5 4 4 4 4 (a) (b) (c) (d) Fgure 5.4: The translaton of the emtter s locaton n BAP B geometry. (a) A lne defned by p tr(., 3.) and p dr 3 (.5,.5) n the CP B geometry. (b) A lne defned by p 3 tr(.,.) and p dr 3 (.5,.5) n the BAP B geometry. (c) A lne defned by p 3 tr(.,.) and p dr 3 (.5,.5) n the CP B geometry. (d) A lne defned by p tr(., 3.) and p dr 3 (.5,.5) n the BAP B geometry The translaton of the emtter n the BAP B geometry does not enable to compute the projectons n a smlar way as was defned n 5.. Instead, the emtter s located n postons where A =, and S, S vary n the nterval [ 3, 3] wth the step /n. For each S = s and S coordnates are s S s + and t S t +, generate the projecton. = t, the detectors, whose The projecton s result s a D array of sze (n + ) (n + ). The coordnates of each element n the array correspond to a par (k, l) where k, l {j/n j = n,..., n}. Ths par represents the drecton of the lne ntegral n the same way as the ponts n the set P dr. Ths par together wth the emtter s poston p G (, s, t), s, t {j/n j = 3n/,..., 3n/}, defne the par (s k, t l). The par (s k, t l), represents the emtter s translaton pont p G (, s k, t l). The ponts p dr (k, l) and p tr(s k, t l), whch are defned by k, l, s and t, can nether be n P dr nor n P tr (see defnton 3.6), respectvely. Ths s due to the fact that half of the detectors values do not represent lne ntegrals from the set P P (see defnton 4.3). The value of the array element s LI p 3.4). tr (s k,t l), p dr (k,l) f (see defnton Defnton 5.8. BAP B data structure. All the lne ntegrals, computed by the BAP B projectons, are stored n BAP B ds. Its frst coordnate s, =,, 3. It represents the man axs X, Y or Z. The followng two coordnates, u, v {,..., 3n + } represent the translaton of the lne ntegral p tr(s, t) where s = (u 3n/) /n and t = (v 3n/) /n. The last two coordnates p, q {,..., n+} represent the drecton of the lne ntegral p dr (k, l) where k = (p n )/n and l = (q n )/n. Formally, BAP B ds (, u, v, p, q) = LI p tr (s,t), p dr (k,l)f. (5.) For specfc, u, v and all p, q {,..., n + }, the collecton of values BAP B ds (, u, v, p, q) s the BAP B projecton XR p tr (s,t), p dr f where s = (u 3n/ ) /n and t = (v 3n/ ) /n. From Corollary 5.7 and the (n+) sze of each projcton n BAP B, we get that a new reorderng algorthm for processng effcently the BAP B projectons s needed.
Lemma 5.9. Reorderng of the BAPB dataset. The data n the BAP B ds s reordered to ft the parallel projectons geometry by P P ds (, p, q, u, v) = BAP B ds (, u p+n+, v q+n+, p, q), where =,, 3, p, q {,..., n + } and u, v {,..., n + }. The par (p, q) represents the parallel projectons drectons p dr (k, l) where k = (p n/) /n and l = (q n/) /n. The par (u, v) represents the parallel projectons translaton p tr(s, t), where s = (u n) /n and t = (v n) /n. Proof. A lne wth the drecton p dr (k, l), whch passes through the pont p tr(s, t), ntersects the plane P () at the pont p G (, s k, t l). Therefore, the coordnates u, v {,..., n+} are transformed to the coordnates n BAP B ds representng the poston p G (, s k, t l). The range of the translaton ndces n P P ds s [, n + ]. The range of the emtter s coordnates S and S s [, ]. The range of the drectons ndces n P P ds s [, n + ]. The tangents range are bounded by [, ]. A lnear mappng of these ranges shows that s = (u n)/n and k = (p n/)/n. Therefore, s k = (u n)/n (p n/)/n = /n(u p n/). Translaton of the range [ 3, 3] of the emtter s postons back to the range [, 3n + ] of the ndces, shows that the new ndex, whch represents the coordnate S of the emtter, s /n(u p n/) n/ + 3n/ + = u p + n +. The second coordnate s transformed smlarly. The ndces, whch represent the BAP B ds drecton, are doubled and then s subtracted snce only the odd ndces of the projecton, belong to the set of lne ntegrals n P P partcpate as was shown n Corollary 5.6. For each man axs A there are (3n+) BAP B projectons. At each emtter s locaton, all (5n+) lne ntegrals are measured smultaneously. Only (n + ) lne ntegrals from each projecton are used. The reorderng s assumed to take no tme. Therefore, computng the data structure BAP B ds requres 3 (3n + ) operatons,.e. O(n ) operatons. Even f the computatons of CP B ds and BAP B ds take O(n ) operatons, the BAP B ds computaton requres about 9/4 tmes more operatons whle usng about 5/9 tmes more detectors. In order to overcome the low detectors utlzaton n the BAP B method, a varaton of the method s suggested. Ths varaton uses only (n + ) detectors. These detectors are located on a movng board. The dstances between the detectors are doubled n order to collect only lne ntegrals from the set P P. In order to collect the correct data when the emtter moves to ts next poston, the board wth the detectors moves together wth the emtter. Therefore, the detectors coordnates S and S change at the same amount as the emtter s coordnates S and S. Ths settng reduces the number of requred detectors by a factor of 5 and thus t provdes a full utlzaton of the detectors. The CP B geometry also utlzes the detectors neffcently. At each projecton, only (n + ) from (3n + ) detectors on the planes P () are used. A smlar varaton can be appled to the CP B method to reduce the number of requred detectors by a factor of 9. In ths settng, the dstance
between detectors stays the same as the orgnal dstance. Ths settng s called Sldng Boundary Algned Pyramd Beam (SBAP B). Fgure 5.5 shows detectors (marked n red) whch are placed on a movng board (marked as gray rectangle). The detectors dstances are doubled. The rght fgure shows how the board moves together wth the emtter (marked n blue). 4 4 4 4 4 4 4 4 (a) (b) Fgure 5.5: Sldng detectors postoned on a moble board that moves together wth the emtter. (a) Emtter located at p 3 G (.,.,.) n the SBAP B geometry. (b) Emtter located at p 3 G (.,.5,.5) n the SBAP B geometry When SBAP B s used, the projectons become (n+) (n+) arrays. All the data elements n these arrays contan valuable data. The reorderng transform becomes: P P ds (, p, q, u, v) = BAP B ds (, u p + n +, v q + n +, p, q) where, p, q, u and v are the same as n Lemma 5.9. The tme complexty of the SBAP B data acquston method s the same as the BAP B complexty snce there s no dfference between these methods except for the number of lne ntegral beng calculated smultaneously. The memory complexty s also O(n ) but t s reduced by a factor 5. 5.3 Mrrored Pyramd Beam (M P B) acquston geometry from multple objects The CP B geometry does not allow to move the emtter on the planes P () when the scanned object s sold. Ths problem does not exst for multple objects that are placed nether n chambers around the man axes nor for scanned non-sold objects. On the other hand, as mentoned n secton 5., only porton of the lne ntegrals between the planes P () and P () s computed (see Fg. 5.6(a)). Here we extend the CP B geometry that overcomes ths problem. Another set of detectors s placed on the planes P (). Ths set represents the mrror mage of the orgnal set n respect to planes P (). The rays emtted from the emtter, whch are detected by ths new set of detectors, form a mrror mage of the orgnal pyramd (the gray pyramd n Fg. 5.7(a)). Due to the symmetry of the orgnal pyramd, each lne ntegral n the orgnal pyramd has ts lne extenson n the mrrored pyramd. The
sum of the lne ntegrals s the complete lne ntegral through the scanned object (see Fgs. 5.6(b) and 5.6(c)). Two CP B projectons are computed for each pont p tr P tr. One projecton uses the orgnal set of detectors and the other uses the new set of detectors. Each projecton s result s a D array of sze (n + ) (n + ). The coordnates of each element n the arrays correspond to a pont p dr (k, l) P dr. The value of an array element n the projecton, computed by the orgnal CP B pyramd (black pyramd n Fg. 7(a)), s LI p tr, p dr (k,l)f, where f s a porton of the functon f(x, y, z) between the planes P () and P (). Smlarly, the value of the array element n the projecton, whch was computed by the mrrored pyramd (gray pyramd n fg. 7(a)), s LI p tr, p dr (k,l)f +, where f + of the functon f(x, y, z) between the planes P () and P (). s a porton Defnton 5.. M P B data structure. All the lne ntegrals, whch are requred to reconstruct the mage by the dscrete nverse X-Ray transform [], are stored n the arrays MP B ds and MP B+ ds. The frst coordnate n each array, =,, 3, represents the man axs X, Y or Z. The followng two coordnates u, v {,..., n + } represent the translaton of the lne ntegral, p tr(s, t) where s = (u n ) /n and t = (v n ) /n. The last two coordnates, p, q {,..., n + } represent the drecton of the lne ntegral p dr (k, l) where k = (p n/ ) /n and l = (q n/ ) /n. Formally, MP B tr ds (, u, v, p, q) = LI p (s,t), p dr (k,l)f. MP B + tr ds (, u, v, p, q) = LI p (s,t), p dr (k,l)f +. (5.3) For specfc, u, v and all p, q {,..., n + }, the collecton of MP B ds (, u, v, p, q) values s the CP B projecton XR p tr (s,t) f., p dr Lemma 5.. Reorderng the MP B data structure. The data n the MP B ds and MP B+ ds data structures s reordered nto parallel projectons by P P ds (, p, q, u, v) = MP B ds (, u, v, p, q) + MP B + ds (, u, v, p, q) where =,, 3, p, q {,..., n+} and u, v {,..., n+}. The par (p, q) represents the parallel projectons drectons p dr (k, l). k = (p n/) /n and l = (q n/) /n. The par (u, v) represents the lne ntegrals translatons p tr(s, t). s = (p n) /n and t = (q n) /n. Collectng the lne ntegrals wth the MP B method requres O(n ) operatons. O(n ) addtons are requred to compute the full lne ntegrals through f. In contrast to other methods descrbed n ths paper, the reorderng algorthm does take tme snce after the data acquston, two portons of each lne are added. The memory complexty stays O(n ) whle the number of detectors s doubled. The MP B geometry s based on the sets of ponts P tr and P dr (see defnton 3.6). Ths method can be used to scan smultaneously eght objects wth lower resolutons. Puttng an object n one of 3