Poly-energetic Reconstructions in X-ray CT Scanners

Transcription

1 Indan Socety for Non-Destructve Testng Hyderabad Chapter Proc. Natonal Senar on Non-Destructve Evaluaton Dec. 7-9, 26, Hyderabad Poly-energetc Reconstructons n X-ray CT Scanners V.S. Venuadhav Vedula, Ntn Jan 2, K. Muraldhar 2, Prabhat Munsh 2, S. Lukose 3, M.P. Subrananan 3 and C. Muraldhar 3 Presently wth GE Bangalore 2 Nuclear Engneerng & Technology Prograe, Indan Insttute of Technology, Kanpur Non-Destructve Evaluaton Dvson, Defence Research and Developent Laboratory, Hyderabad e-al: punsh@tk.ac.n Abstract Bea-hardenng s an artfact, whch produces false ntegrals f polychroatc x-ray sources are used. It s due to the photon energy dependence of the attenuaton coeffcent. The present work proposes an algorth for bea-hardenng correcton ncorporatng the nherent error forula developed at IIT Kanpur. The effect of bea hardenng and ts reoval along wth nherent error s shown on both sulated and experental data set. It s copared fro the pont-of-vew of nearness of the corrected polychroatc projecton data to the desred onochroatc projecton data. The results ndcate that the algorth, proposed orgnally for edcal applcatons, s gvng encouragng results for non-edcal objects though the physcal stuatons are vastly dfferent. Keywords: Toography, Bea-hardenng, Inherent error. Introducton Toography has becoe a routne part n edcne and ts use n nondestructve evaluaton s ncreasng day by day. Measureent n x-ray toography can only be used to estate the lne ntegrals of the absorpton coeffcent of photons. Inaccuraces n these estates are due to wdth of the x-ray bea, hardenng of the bea and photon statstcs. When x-rays are passed through an object, ther attenuaton depends on the densty dstrbuton and energy spectru of the bea. As a consequence of polychroatc x-ray source, the attenuaton s no longer a lnear functon of absorber thckness. The attenuaton at a fxed pont s generally greater for photons of lower energy and thus energy spectru of x-rays hardens as t passes through the ateral. X-ray beas reachng at partcular pont nsde the ateral fro dfferent drectons are lkely to have dfferent spectra and therefore these rays attenuate dfferently at that pont and t becoes dffcult to nterpret age quanttatvely. Bea hardenng effect has to be copensated to prevent reconstructed age fro corrupton by cuppng artfacts [-3]. In the present study Convoluton Back Projecton (CBP) s used for the reconstructon of the projecton data and wth any flter functon n CBP wll lead to nherent error n the reconstructon process [4,5]. In the present work, correctons for the cuppng artfact and the reducton of the nherent error n the ages are dscussed. NDE-26

2 V.S.V. Vedula et al. 2. Theory 2. Bea Hardenng (BH) Correcton The lnear x-ray coeffcent at a pont nsde a cross secton of the object depends on the poston of the pont ( x, y ) and on energy e. It can be denoted as µ ( x, y, e) In case of onochroatc bea t can be wrtten as L = µ ( x, y, e) dl () L In case of polychroatc bea result wll not be L but rather an estate for the ore coplcated ntegral PL = ln τ ( e) exp µ ( x, y, e) dl de L (2) Where τ ( e) s the probablty that the detected photon s at energy e [2]. It s assued that the spectru of the x-ray bea can be approxated by a dscrete spectru consstng of J energes e(), e(2).., e(j ) and that e( j ) t s the probablty that a detected photon s at energy e(j). Let us dvde the cross secton nto I pxels. We try to estate the lnear attenuaton coeffcent n each of the I pxels. Thus we can get the dscretzed verson of ( and 2) I = µ e Z (3) = J I p= ln τ e( j) exp µ e( j) Z j= = (4) The least expensve type of the bea hardenng correcton can be done by usng a functon f, whch s such that, for source/detector par f ( p ) s a reasonable estate of. Let us refer to the reconstructon fro the so corrected polychroatc data { f ( p )} as the frst reconstructon. It s a set of I nubersµ e, representng the estated lnear attenuaton coeffcent at energy e of the ateral n the th of a total of I pxels. We see that approxate to, and p approxate to p and hence f ( p ) approxate to f ( p ). Furtherore, snce the lne ntegrals n equatons ( and 2) are approxated n the sae way n Eqs. (3 and 4), t appears lkely that the errors, and f ( p) f ( p) wll be slar,.e. the dfference between these errors wll be consderably saller than ether of these errors. The ter, f ( p) + f ( p), s an approxaton to and s superor to the use of just f ( p ). Ths s true n the sense that ({ f ( p) + f ( p)},{ }) < ({ f ( p)},{ }) Where represents the root ean square error. The second reconstructon s one obtaned fro the data f ( p) + f ( p). Snce the second reconstructon s presuably ore accurate than the frst one, ths process can be repeated [6,7]. 2.2 Inherent Error Correcton Projecton data obtaned fro the fnal teraton of BH correcton s free fro bea hardenng artfacts can be further processed to reduce nherent error. Frst Kanpur Theore (KT-) s appled to reove nherent error caused by flter functon [4-5]. Intally factor η s calculated usng followng equaton. NMAX η = NMAX NDE-26

3 Poly-energetc Reconstructons Where NMAX and NMAX 2 are axu gray level values of onoenergetc and BH corrected data respectvely. KT- s used to odfy the convolvng functon by the factor η after that fnal reconstructon s done usng odfed convolvng functon. Bea hardenng and nherent error correcton s suarzed n a cobned nuercal algorth as stated below:. Reconstruct the polyenergetc projecton data of test phanto usng CBP. The functon f s estated wth respect to ths specen, whch fors our ntal guess O. 2. Collectng a new set of relevant nforaton ncludng geoetry, sze of specen fro the reconstructed age and coeffcents of lnear attenuaton for the partcular aterals used, generate specens X at dfferent energes fro the x-ray source spectru. 3. Fro the generated specens X, evaluate pseudo onochroatc ray sus fro the equaton gven below: I = µ z e = 4. Generate pseudo polychroatc ray su, p usng equaton gven below wth τ e( j ) as the probablty that a detected photon of the x-ray bea s at energy e ( j ). τ e( j ) can be calculated fro the x-ray source spectru. E D p= - ln τ e exp - µ e(z) dz de 5. Get the correlaton functons f s, utlzng curve fttng strategy between and p. The ost nexpensve curvefttng route s to adopt a polynoal functon for f, and deterne ts coeffcents, by least squares technque. f s can be obtaned by, f (p) 2 3 = a + a p+ a 2 p + a 3 p Apply correlaton functon f to the actual easured data p recorded n the experent. f (p) 7. For the second step of BH correcton, a ore superor functon s gven below where the R.M.S. error s nzed. = - f ( p ) + ( p ) f 8. Reconstruct obtaned fro above step and copare wth the ntal guesso. Iprove the ntal guess fro and repeat above steps tll cuppng artfact and dark bands are reduced consderably. Ths copletes the BH correcton. 9. Calculate factorη, gven by the equaton below. η = NMAX NMAX 2. Usng KT-, odfy the convolvng functon (here H54) used n CBP algorth by the factorη. Now reconstruct all the s usng ths odfed flter functon. Ths copletes the nherent error correcton. NDE

4 V.S.V. Vedula et al Mn = -.27 Max =.643 LAvg =.3289 AAvg = (a) Pxels Mn =. Max =.46 LAvg =.33 AAvg = (b) Pxels Mn = -. Max =.458 LAvg =.277 AAvg = Pxels (c) Fg. : (a) Polyenergetc reconstructon of sulated specen (S) (b) Monoenergetc reconstructon of sulated specen at 6Kev (c) BH corrected data after applyng KT- for sulated specen 296 NDE-26

5 Poly-energetc Reconstructons Mn =.823 Max = 3.32 LAvg =.2887 AAvg = Pxel (a) Mn =. Max =.242 LAvg =.742 AAvg = Pxels (b) Mn = -.6 Max =.243 LAvg =.679 AAvg = Pxels (c) Fg. 2: (a) Polyenergetc reconstructon of specen-s2 (b) Monoenergetc reconstructon of specen-s2 at 2Kev (c) Reconstructon of BH corrected data after applyng KT- for specen-s2 NDE

6 V.S.V. Vedula et al Mn = Max =.4723 LAvg =.235 AAvg = Pxels (a) Mn =. Max =.242 LAvg =.654 AAvg = Pxels (b) Mn = Max =.242 LAvg =.572 AAvg = Pxels (c) Fg. 3: (a) Polyenergetc reconstructon of specen-s3 (b) Monoenergetc reconstructon of specen-s3 at 2Kev (c) Reconstructon of BH corrected data after applyng KT- for specen-s3 298 NDE-26

7 Poly-energetc Reconstructons 3. Specens Detals a) Specen- (S): Ths s coputer generated specen whch contans aterals of three dfferent denstes. The object consdered s a crcle ade up of ateral a wth three crcular holes, one flled wth ateral b and two flled wth ateral c. A crack (of densty zero) s ntroduced n the rght nner crcular hole wth ateral c. b) Specen-2 (S2): The test phanto consdered here s a Perspex cylnder of 6 radus wth fve holes ebedded n t. There s a central hole of 2.5 radus and the reanng four holes each of 7.5 radus are placed on ether sde of the central hole perpendcularly. Here the central hole s flled wth a unfor ld steel cylnder and the reanng four holes are unflled. c) Specen-3 (S3): The test phanto consdered here s sae as the specen-2 but wth all the holes flled wth ld steel. Thus here t s a Perspex cylnder wth fve ld steel pns ebedded n t. Snce there s lot of attenuaton for ths specen, hgh energy X-rays should be used for scannng. Ths specen s chosen to check for cuppng artfact along wth dark bands n between the steel pns. 4. Results Bea Hardenng and Inherent error correcton has been appled to three specens. Projecton data s acqured n fan bea ode at DRDL Hyderabad, wth source to center dstance of 32.7 for 52 vews and 256 rays for the specens 2-3. Fan bea projecton data s converted to parallel bea ode. X-ray source spectru s dscretsed nto fve energy levels and the probabltes for each of the energy levels are calculated. Monoenergetc data sets for the above specens are sulated at the dscrete energy levels. The flter functon used n all the reconstructons of CBP s Hang 54, that resolves well the sooth varatons n the attenuaton coeffcent and hence the densty. Fgures -3 show the onoenergetc; polyenergetc and BH corrected ages after applyng KT- theore wth correspondng densty profles for the specens S-S3 respectvely. Results are gven n the above secton for all the specens. Snce sulated specen s generated for 28 rays, t s reconstructed for a grd sze of 28. Slarly, specens S2 and S3 are reconstructed for the grd sze of 256. Densty profles are drawn for the specens for versus the pxel nubers. Bea hardenng correcton s done by fttng second order polynoal n the least squares sense. 5. Dscusson Investgatng above results t s depcted that all the polyenergetc reconstructons have hgh NMAX values copared to ther correspondng onoenergetc ones. Monoenergetc projectons havng hgh probablty are consdered to gve better solutons for bea-hardenng correcton. Hence, all the onoenergetc reconstructons consdered for least squares curve fttng (BH correcton) are at the ean energy level. Sulaton of the polyenergetc reconstructons should be done wth good accuracy to ensure better BH correcton, devaton of whch ay lead to dstorted ages. It can be notced fro fgures -3 that ages alost atch wth the onoenergetc ones and cuppng artfact reduces consderably at the fnal teraton. Fg. 2 shows that BH corrected data of specen S2 s well approxated to ts onoenergetc data. Ths ndcates that algorth works equally well for object wth ore than two aterals. Dark bands forng brdges between steel pns are NDE

8 V.S.V. Vedula et al. clearly vsble fro Fg. 3(b), polyenergetc age of specen S3. Reoval of dark bands at the fnal teraton for specen S3 can be notced. Thus algorth s checked for all the specens. Table- gves the error estates for the sulated and experental specens at each teraton of the bea hardenng correcton algorth, before and after applyng nherent error correcton. The error presented here s the relatve error and should approach zero for the deal case. It can be observed that error n the ages s ltng towards zero after processng the for nherent error correcton. Table : Relatve errors n the ages Specen Error n Polyenergetc data Before KT- After KT- Error n 2 nd BH teraton Before KT- After KT- S S S Conclusons Algorth works well for both hoogenous and heterogeneous crosssectons. For objects wth hgh densty aterals, cuppng artfact and dark bands appeared n the polyenergetc reconstructon can also be reduced to a great extent. Frst Kanpur error theore effcently reduced nherent errors and technque used for these error reoval s qute encouragng, applyng whch the NMAX values for experental and onoenergetc data are n well agreeent. Inherent error for real data s donated by other experental errors and there s only 4%-6% of change n relatve error after applyng KT-. Nuercal algorth has been checked for all the coplextes of bea-hardenng, nherent error and dfferent geoetres. The proposed algorth found to be qute robust and s workng effcently for the sulated and experental data. 7. References. Heran G. T., Correcton for Bea Hardenng n Coputed Toography, Phys. Med. Bol. 24, 8-6, (979). 2. Heran G. T., Iage Reconstructon fro Projectons: The Fundaentals of Coputerzed Toography, Acadec Publshers New York (98). 3. Heran G. T., and Trved S. S., A Coparatve Study of Two Post reconstructon Bea Hardenng Correcton Methods. IEEE Trans. Med. Iagng, MI-2(3), (983). 4. Munsh P., Error Analyss of Toographc Flters I : Theory, NDT & E Internatonal 25(4/5), 9-94, (992). 5. Munsh P., Rathore R. K. S., Ra K. S. and Kalra M. S., Error Analyss of Toographc Flters II : Results, NDT & E Internatonal, 26(5), , (993). 6. Raakrshna K., Muraldhar K., Munsh P., Bea Hardenng n Sulated X-ray Toography, NDT&E nternatonal, 39(6), (26). 7. Manzoor M. F., Yadav P., Muraldhar K. and Munsh P., Iage reconstructon of sulated specens usng convoluton back projecton, Defense Scence Journal, 5(2), 75-87, (2). 3 NDE-26