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PET Image Reconstruction Using Physical and Mathematical Modelling For Time of Flight PET-MR Scanners in the STIR Library
Journal Pre-proofs PET Image Reconstruction Using Physical and Mathematical Modelling For Time of Flight PET-MR Scanners in the STIR Library Palak Wadhwa, Kris Thielemans, Nikos Efthimiou, Kristen Wangerin, Nicholas Keat, Elise Emond, Timothy Deller, Ottavia Bertolli, Daniel Deidda, Gaspar Delso, Michel Tohme, Floris Jansen, Roger N. Gunn, William Hallett, Charalampos Tsoumpas PII: DOI: Reference:
S1046-2023(19)30193-8 https://doi.org/10.1016/j.ymeth.2020.01.005 YMETH 4847
To appear in:
Methods
Received Date: Revised Date: Accepted Date:
31 July 2019 15 November 2019 14 January 2020
Please cite this article as: P. Wadhwa, K. Thielemans, N. Efthimiou, K. Wangerin, N. Keat, E. Emond, T. Deller, O. Bertolli, D. Deidda, G. Delso, M. Tohme, F. Jansen, R.N. Gunn, W. Hallett, C. Tsoumpas, PET Image Reconstruction Using Physical and Mathematical Modelling For Time of Flight PET-MR Scanners in the STIR Library, Methods (2020), doi: https://doi.org/10.1016/j.ymeth.2020.01.005
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© 2020 Published by Elsevier Inc.
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AM i?Bb T T2`- i?2 r2HH FMQrM Q`/2`2/ bm#b2i 2tT2+@ i iBQM K tBKBb iBQM UPa1JV (R3) Bi2` iBp2 BK ;2 `2+QM@ bi`m+iBQM H;Q`Bi?K Bb 2ti2M/2/ iQ BM+Hm/2 iBKBM; BM7Q`K @ iBQM UBX2X hP6@Pa1JV M/ J_ BM7Q`K iBQM UBX2X hP6@ E1JVX h?2b2 H;Q`Bi?Kb `2 mb2/ iQ `2+QMbi`m+i i?2 K2 @ bm`2/ bBMQ;` K yt b K2 bm`2/ #v i?2 b+ MM2`X *m``2MiHv QMHv bBMQ;` K@# b2/ `2+QMbi`m+iBQMb `2 /2KQMbi` i2/X
kXjX :1 aA:L S1hfJ_ a+ MM2` +[mBbBiBQM . i BM ahA_ AK ;2 `2+QMbi`m+iBQM Q7 / i +[mB`2/ rBi? i?2 :1 b+ M@ M2` mbBM; ahA_ HB#` `v `2[mB`2b i?2 BM+Q`TQ` iBQM Q7 BMTmi M/ QmiTmi BM7Q`K iBQM i? i +QMp2`i i?2 / i BMiQ M +@ +2bbB#H2 / i 7Q`K iX aT2+B}+ HHv- i?2 QmiTmi }H2b b p2/ #v i?2 b+ MM2` +QMbQH2 i i?2 2M/ Q7 S1h b+ M BM+Hm/2,
kXkXRX hP6@Pa1J H;Q`Bi?K h?2 hP6@Pa1J H;Q`Bi?K r b /2b+`B#2/ BM (Rj)X H@ i?Qm;? i?2 hP6 2tT MbBQM Q7 i?2 ahA_ HB#` `v r b `2@ +2MiHv BKTH2K2Mi2/ M/ p HB/ i2/ QM bBKmH i2/ / i b2ib (Rj)- i?Bb Bb i?2 }`bi iBK2 i? i i?2 BKTH2K2Mi iBQM Bb i2bi2/ QM +[mB`2/ / i b2ibX h?2 hP6@Pa1J H;Q`Bi?K Bb /2b+`B#2/ b 7QHHQrb, xmj+1 =
xmj s∈S j
a sm
s∈S j
a st;m
y st j k a st;k xk + b st
,
RX GBbiKQ/2 UGJV M/ bBMQ;` K ` r / i }H2b U_.6V +QKT`2bb2/ rBi? i?2 :1 T`QT`B2i `v +QKT`2bbBQM H@ ;Q`Bi?K kX _ r / i + HB#` iBQM }H2b BM+Hm/BM; MQ`K HBb iBQM- ;2@ QK2i`B+ M/ r2HH +QmMi2` + HB#` iBQM 7 +iQ`b jX S1h AK ;2b 6Q` ii2Mm iBQM USA6V }H2b UBX2X T`Q@ +2bb2/ S1h ii2Mm iBQM +Q``2+iBQM U*V BK ;2b Q#@ i BM2/ #v +QMp2`iBM; J_ BK ;2b +[mB`2/ mbBM; J_ b2[m2M+2b BM+Hm/BM; x2`Q 2+?Q iBK2 M/ .BtQMV 9X _2+QMbi`m+i2/ S1h BK ;2b rBi? `2+QMbi`m+iBQM b2i@ iBM;b b2H2+i2/ i i?2 iBK2 Q7 b+ M
UjV
r?2`2 xmj `2T`2b2Mib i?2 BMi2MbBiv Q7 mth pQt2H BM BK ;2 mT/ i2/ Qp2` i?2 bm#@Bi2` iBQM j- y st `2T`2b2Mib i?2 bm#b2i s Q7 hP6 T`QD2+iBQM / i 7Q` hP6 #BM t- a st;m `2T`2b2Mib i?2 T`Q# #BHBiv Q7 M 2p2Mi Q++m``BM; BM pQt2H K #2BM; /2i2+i2/ #v /2i2+iQ` T B` s rBi?BM i?2 iBKBM; #BM t- b st `2T`2b2Mib i?2 +Q``2+iBQM i2`K iQ ++QmMi 7Q` i?2 bmK Q7 ` M/QKb M/ b+ ii2` 2p2Mib UQ` b rBHH #2 `272``2/ iQ # +F;`QmM/ i2`KV 7Q` bm#b2i s M/ iBKBM; #BM t M/ S j `2T`2b2Mib i?2 bm#b2i Q7 /2i2+iQ` T B`bX LQi2 i? i QM2 7mHH Bi2` iBQM- Q` bBKTHv K2MiBQM2/ b Bi2` iBQM- +Q``2bTQM/b iQ 7mHH +v+H2 Q7 bm#@Bi2` iBQMb +`Qbb HH bm#b2ib UBX2X Bi2` iBQMb 4 bm#@ Bi2` iBQMbfbm#b2ibVX
HH i?2 QmiTmi ` r / i }H2b `2 BM :1 T`QT`B2i `v / i 7Q`K i M/ + M #2 T`Q+2bb2/ Q|BM2 bQH2Hv rBi? i?2 :1Ƕb T`QT`B2i `v `2+QMbi`m+iBQM bQ7ir `2X :1 biQ`2b / i M/ ?2 /2` BM7Q`K iBQM BM >B2` `+?B+ H . i 6Q`K i 8 U>.68V }H2 7Q`K iX >.68 }H2b BM+Hm/2, RX aBMQ;` K _.6 ;2M2` HHv + HH2/ Ƕ`/7XRXRǶ kX GJ }H2 ;2M2` HHv + HH2/ ǶGAahyyyyX"G6Ƕ jX LQ`K HBb iBQM / i }H2- ǶMQ`Kj/Ƕ- M/ ;2QK2i`B+ +Q`@ `2+iBQM / i }H2- Ƕ;2Qj/Ƕ
kXkXkX hP6@E1J H;Q`Bi?K h?2 E1J H;Q`Bi?K r b }`bi T`QTQb2/ #v q M; 2i HX (RN) M/ r b /2p2HQT2/ BM ahA_ HB#` `v #v .2B// 2i HX(Rk- ky)X h?Bb ? b #22M MQr 2ti2M/2/ 7m`i?2` iQ ++QK@ KQ/ i2 hP6 / i b r2HHX h?2 H;Q`Bi?K bbmK2b i? i M BK ;2 Bb /2b+`B#2/ #v i?2 bmK Q7 MmK#2` Q7 r2B;?i2/ F2`M2Hb- M/ Bi i`B2b iQ `2+QMbi`m+i i?2 r2B;?i Q7 2 +? F2`@ M2H #v bbmKBM; Bi FMQrb i?2 F2`M2H 7`QK MQi?2` bQm`+2 Q7 BM7Q`K iBQM U2X;X J_AV, xm =
Nm
h?2b2 }H2b `2 mb2/ b BMTmi iQ p HB/ i2 i?2 BKTH2K2Mi @ iBQM Q7 i?2 +[mBbBiBQM KQ/2H 7Q` i?Bb b+ MM2` BM ahA_X >Qr2p2`- ahA_ ? b Bib QrM M iBp2 bBMQ;` K / i 7Q`@ K i- mb2/ b BMTmi 7Q` i?2 Bi2` iBp2 M/ M HviB+ H `2+QM@ bi`m+iBQM H;Q`Bi?KbX h?mb- HH i?2 >.68 }H2b M22/ iQ #2 +QMp2`i2/ BMiQ ahA_ M iBp2 BMi2`}H2 7Q`K i (Rd)X r` T@ T2` +H bb r b +`2 i2/ iQ BKTQ`i i?2 >.68 }H2b /m`BM; i?Bb rQ`FX >.68r` TT2` +H bb ? b 7mM+iBQMb BKTH2K2Mi2/ iQ ++2bb M/ `2 / i?2 `2[mB`2/ BM7Q`K iBQM r?B+? Bb 7m`i?2` KQ/2HH2/ rBi?BM i?2 bvbi2K K i`Bt b /Bb+mbb2/ BM b2+iBQM kX9X
U9V
λ f k f m,
f =1
r?2`2 p Hm2 i pQt2H m Q7 BK ;2 xm `2T`2b2Mib i?2 HBM2 ` +QK#BM iBQM Q7 i?2 F2`M2H K i`Bt Uk f m `2T`2b2Mib Bib f mth 2H2K2MibV- Nm `2T`2b2Mib i?2 MmK#2` Q7 72 im`2 p2+iQ`b M/ F2`M2H r2B;?ib λX h?2 F2`M2H r2B;?ib `2 `2+QMbi`m+i2/ b 7QHHQrb, λmj+1 =
λmj
l klm
s∈S j
a st;m
l
klm
s∈S j
a st;m
l
a st;l
kX9X AKTH2K2Mi iBQM Q7 i?2 +[mBbBiBQM KQ/2H BM ahA_ kX9XRX hP6 +[mBbBiBQM . i BM ahA_ hP6 BM7Q`K iBQM Bb BKTH2K2Mi2/ BM ahA_ rBi?BM i?2 bvbi2K K i`Bt b,
y st f
, k f l λ jf + b st U8V
a pt;m = a pm T ptm . j
UeV
h?Bb 2[m iBQM KQ/2Hb i?2 hP6 BM7Q`K iBQM b iBK2 bT`2 / 7mM+iBQM- T ptm 7Q` 2 +? iBK2 #BM t M/ b2T ` i2b Bi 7`QK i?2 b2MbBiBpBiv BM7Q`K iBQM- a pm X h?2 BKTH2K2M@ i iBQM ? b #22M `2+2MiHv T`2b2Mi2/ M/ p HB/ i2/ BM bBKm@ H i2/ / i (Rj)- #mi MQi v2i 7Q` +[mB`2/ / i X h?2 b+ MM2` b p2b i?2 /2i2+iQ` T B` M/ iBKBM; BM7Q`@ K iBQM BM GJ M/ bBMQ;` K _.6bX h?2 / i b p2/ BM i?2 GJ >.68 }H2 Bb ++2bbB#H2 #v M pB; iBM; iQ i?2 }2H/ + HH2/ ǵfGBbi. i fGBbi. i Ƕ (kR- Rd)X GJ }H2b biQ`2 2 +? 2p2Mi Q7 M +[mBbBiBQM b 6 #vi2 bi`m+im`2X h?Bb BM@ +Hm/2b BM7Q`K iBQM bm+? b i?2 +`vbi H B/2MiB}2`b UA.bV r?B+? `2;Bbi2`2/ i?2 +QBM+B/2Mi T?QiQMb M/ i?2 iBKBM; #BMX h?2 /2i2+iQ` T B` TQbBiBQMb `2 ? M/H2/ b tB H M/ i` Mb tB H +QQ`/BM i2b rBi?BM i?2 dz*GBbiJQ/2. i :1a@ A:LǴ +H bb Q7 ahA_X h?2 bB;M2/ hP6 #BM Bb ? M/H2/ rBi?BM i?2 dz*GBbi_2+Q`/:1aA:LǴ +H bbX h?Bb Bb /QM2 #v `2 /BM; i?2 hP6 BM7Q`K iBQM HQM; rBi? i?2 hP6 ǴK b?@ BM;Ǵ BM7Q`K iBQM M/ mb2 hP6 T`QD2+iBQM / i - yt rBi?BM i?2 `2+QMbi`m+iBQM 7` K2rQ`FX AM i?2 +m``2Mi BKTH2K2M@ i iBQM- hP6 #BMb + M #2 K b?2/ iQ;2i?2` iQ ;2i MQM@hP6 / i X h?2 / i Bb `2 `` M;2/ iQ #2 ++m` i2Hv `2 / BM ahA_ #v TTHvBM; i` Mb7Q`K iBQMb 7`QK b+ MM2` bT +2 iQ ahA_ bT +2X h?2 M2ti bi2T iQ `2+QMbi`m+i S1h BK ;2b rBi? ahA_ `2[mB`2b i?2 + H+mH iBQM Q7 i?2 b2MbBiBpBiv BK ;2X h?Bb BM@ +Hm/2b KQ/2HHBM; Q7 /2i2+iQ` 2{+B2M+v- ;2QK2i`B+ M/ i@ i2Mm iBQM 7 +iQ`bX h?2 b2MbBiBpBiv BK ;2b /2KQMbi` i2 i?2 bT iB H p `B iBQM Q7 2tT2+i2/ +QBM+B/2M+2b T2` mMBi +iBpBiv +QM+2Mi` iBQMX
kX9XjX SA6@# b2/ ii2Mm iBQM *Q``2+iBQM ii2Mm iBQM +Q``2+iBQM r b + ``B2/ Qmi mbBM; b+ MM2` 2ti` +i2/ SA6 BK ;2bX h?Bb r b +?B2p2/ #v +`2 iBM; +mbiQK b+`BTib i? i BKTQ`i2/ i?2 SA6 BK ;2b rBi?BM i?2 ahA_ `2+QMbi`m+iBQM 7` K2rQ`F (Rd)X 6m`i?2`KQ`2- # +F;`QmM/ i2`K M22/b iQ #2 + H+mH i2/ iQ +Q``2+i 7Q` b+ ii2` M/ ` M/QK 2p2MibX kX9X9X AKTH2K2Mi iBQM Q7 _ M/QKb *Q``2+iBQM lbBM; aBM@ ;H2b _ M/QKb `2 2biBK i2/ 7`QK bBM;H2b T2` +`vbi H 7Q` i?2 :1 aA:L (k9- k8), RXY = 2τS X S Y /T,
r?2`2 RXY Bb i?2 MmK#2` Q7 ` M/QK 2p2Mib UQ` +QmMibV /2@ i2+i2/ #v i?2 /2i2+iQ` T B` X M/ Y- τ `2T`2b2Mib i?2 +QBM+B@ /2M+2 iBK2 rBM/Qr U4.57 Mb 7Q` i?Bb b+ MM2`V- T `2T`2b2Mib i?2 iQi H +[mBbBiBQM iBK2- S X M/ S Y `2T`2b2Mi i?2 bBM;H2 2p2Mib /2i2+i2/ #v +`vbi H X M/ Y- `2bT2+iBp2HvX aBM;H2 ` i2b 7Q` 2 +? +`vbi H `2 2tTQ`i2/ 7`QK i?2 GJ >.68 }H2 M/ +QMp2`i2/ BMiQ ` M/QK ` i2b mbBM; i?2 ahA_ miBHBiv ǵ+QMbi`m+in` M/QKbn7`QKn:1bBM;H2bǶ BKTH2K2Mi2/ /m`@ BM; i?Bb rQ`FX aBM;H2b T2` b2+QM/ `2 /2i2+i2/ 7Q` 2p2`v +`vbi H M/ i?2M biQ`2/ b / i b2i rBi? /BK2MbBQMb 45 × 448 BM GJ >.68 }H2X 1 +? +2HH Q7 i?2 / i b2i +Q``2bTQM/b iQ i?2 MmK#2` Q7 bBM;H2b /2i2+i2/ #v i?2 +`vbi H MmK#2` (X, Y) r?2`2 X +Q``2bTQM/b iQ i?2 `BM; MmK#2` M/ Y +Q``2bTQM/b iQ i?2 i` Mb tB H +`vbi H MmK#2`X 6Q` M +[mBbBiBQM Q7 iBK2 Q7 N b2+QM/b- i?2`2 Bb HBbi Q7 N + 1 b KTH2bX h?2`2 `2 N + 1 b KTH2b b i?2 T2`BQ/B+ / i `2 bvM+?`QMQmb rBi? i?2 bi `i Q7 S1h b+ M r?B+? M22/b iQ #2 i F2M BMiQ ++QmMi /m`BM; / i + H+mH iBQM #v r2B;?iBM; i?2 }`bi M/ H bi b KTH2b Q7 i?2 T2`BQ/B+ / i TT`QT`B i2HvX 1[m iBQM U3V + H+mH i2b i?2 ` M/QKb +Q``2+iBQM bBMQ;` K- r?B+? Bb H i2` mb2/ 7Q` 2p2`v hP6 #BM 7i2` /BpB/BM; Bi rBi? i?2 iQi H MmK#2` Q7 iBKBM; #BMbX AM i?2 +m``2Mi ahA_ BKTH2@ K2Mi iBQM- /2 /@iBK2 M/ BbQiQT2 /2+ v KQ/2HHBM; `2 MQi +QMbB/2`2/ rBi?BM ` M/QKb +Q``2+iBQMX
kX9XkX AKTH2K2MiBM; LQ`K HBb iBQM *Q``2+iBQM 7Q` :1 aA:L BM ahA_ h?2 ;2QK2i`B+ M/ /2i2+iQ` 2{+B2M+v 7 +iQ`b `2 KmH@ iBTHB2/ iQ ;2i i?2 MQ`K HBb iBQM +Q``2+iBQM 7 +iQ`b T2` /2@ i2+iQ` #BM b #2HQr(kk- kj), nXY =
X Y , gXY
U3V
UdV
kX9X8X a+ ii2` *Q``2+iBQM LQM@hP6 b+ ii2` +Q``2+iBQM bBMQ;` K r b 2ti` +i2/ 7`QK i?2 p2M/Q`Ƕb `2+QMbi`m+iBQM iQQH#Qt 7Q` b+ ii2` +Q`@ `2+iBQMX hP6@b+ ii2` Bb +m``2MiHv MQi BKTH2K2Mi2/- r?2`2 b MQM@ hP6 b+ ii2` +Q``2+iBQM bBMQ;` Kb r2`2 TT`QtBK i2/ iQ hP6 T`QD2+iBQM / i rBi? TT`QT`B i2 b+ HBM; 7 +iQ`b 7Q` 2 +? hP6 #BMbX h?2 b+ HBM; 7 +iQ`b r2`2 2ti` +i2/ mbBM; i?2 2KBbbBQM T`QD2+iBQM / i #v + H+mH iBM; i?2 7` +iBQM Q7 +QmMib BM 2 +? hP6 #BM rBi? `2bT2+i iQ iQi H MmK#2` Q7 +QmMibX h?2 ` M/QKb M/ b+ ii2` +Q``2+iBQM hP6 T`QD2+iBQM / i r2`2 //2/ iQ + H+mH i2 i?2 # +F;`QmM/ i2`K- bt X
r?2`2 nXY `2T`2b2Mib i?2 MQ`K HBb iBQM +Q``2+iBQM 7 +iQ` 7Q` i?2 /2i2+iQ` T B`- j r?B+? Bb +QKT`Bb2/ Q7 /2i2+iQ` A.b X M/ Y- X M/ Y `2T`2b2Mib i?2 +`vbi H 2{+B2M+v 7 +iQ`b M/ gXY `2T`2b2Mib i?2 ;2QK2i`B+ +Q``2+iBQM 7 +iQ`bX h?2 b+ MM2` 2tTQ`ib >.68 }H2b- r?B+? `2 i?2M mb2/ b BMTmib 7Q` + H+mH iBM; i?2 MQ`K HBb iBQM 7 +iQ`b (Rd)X *`vb@ i H 2{+B2M+v 7 +iQ`b `2 biQ`2/ BM ǵMQ`Kj/Ƕ M/ ;2QK2i@ `B+ 7 +iQ`b `2 biQ`2/ BM ǵ;2Qj/ǶX h?2 MQ`Kj/ >.68 }H2 biQ`2b +`vbi H 2{+B2M+v 7 +iQ`b r?B+? `2 `2 / b M `` v Q7 448 × 45 BM ahA_X h?2 j. ;2QK2i`B+ +Q``2+iBQM 7 +iQ`b `2 biQ`2/ b T `i Q7 i?2 T`QD2+iBQM / i M/ QMHv 16 pB2rb `2 biQ`2/ BX2X 7Q` QM2 KQ/mH2X h?2b2 pB2r / i b2ib `2 Q7 /BK2MbBQMb 1981 × 357X h?2 2tTQ`i2/ ;2QK2i`B+ 7 +iQ`b `2 `2T2 i2/ 14 iBK2b rBi?BM ahA_ iQ ;2i 7mHH ;2@ QK2i`B+ +Q``2+iBQM T`QD2+iBQM / i X .2 /@iBK2 +Q``2+iBQM ? b MQi #22M BKTH2K2Mi2/ BM i?Bb bim/vX
kX8X . i +[mBbBiBQM- AK ;2 _2+QMbi`m+iBQM M/ M HvbBb kX8XRX S? MiQK +[mBbBiBQM h?2 7QHHQrBM; irQ T? MiQKb r2`2 b+ MM2/, 9
TQ`i2/ SA6 BK ;2b b /2KQMbi` i2/ BM };m`2 RX _ M@ /QKb +Q``2+iBQM bBMQ;` K r b + H+mH i2/ mbBM; ǵ+QM@ bi`m+in` M/QKbn7`QKn:1bBM;H2bǶ miBHBivX a+ ii2` +Q``2+@ iBQM r b /QM2 mbBM; i?2 bBMQ;` K 2ti` +i2/ 7`QK i?2 p2M@ /Q`Ƕb `2+QMbi`m+iBQM iQQH#QtX 6Q` hP6 BK ;2 `2+QMbi`m+@ iBQMb- i?2 hP6 T`QD2+iBQM / i r2`2 ?BbiQ;` K@ K2/ mbBM; i?2 GJ }H2X _ M/QKb M/ b+ ii2` +Q``2+iBQM bBMQ;` Kb r2`2 i` MbH i2/ BMiQ hP6 T`QD2+iBQM / i #v /2p2HQTBM; +mbiQK miBHBiB2b M/ TT`QT`B i2 b+ HBM; Q7 / i b /2@ b+`B#2/ #Qp2 BM b2+iBQMb kX9X9 M/ kX9X8X ii2Mm iBQM M/ MQ`K HBb iBQM bBMQ;` Kb r2`2 mMB7Q`K BM iBKBM; #BMb M/ r2`2 KmHiBTHB2/ rBi? i?2 # +F;`QmM/ i2`K iQ +`2 i2 //B@ iBp2 bBMQ;` Kb 7Q` ahA_ BKTH2K2Mi iBQMbX AK ;2 `2+QM@ bi`m+iBQMb r2`2 + ``B2/ Qmi mbBM; Pa1J M/ hP6@Pa1J H;Q`Bi?Kb rBi? M/ rBi?Qmi TQBMi bT`2 / 7mM+iBQM USa6V KQ/2HHBM;X hP6@E1J `2+QMbi`m+iBQMb r2`2 HbQ + ``B2/ Qmi 7Q` i?2 T iB2Mi / i b2iX h?2 J_ BK ;2 b?QrM BM };@ m`2 R r b mb2/ b iQ /2`Bp2 i?2 F2`M2HbX HH `2+QMbi`m+iBQMb r2`2 + ``B2/ Qmi rBi? k3 bm#b2ibX
6B;m`2 R, S iB2Mi . i b2i, h` Mbp2`b2 M/ +Q`QM H bHB+2b Q7 i?2 2t@ TQ`i2/ `2b KTH2/ SA6 BK ;2 U#QiiQK `QrV 7`QK J_* UiQT `QrV mbBM; +mbiQK b+`BTib b ahA_ BMi2`}H2X J_* BK ;2 mb2/ b M M iQKB+ H T`BQ` 7Q` F2`M2H K i`Bt 2biBK iBQM 7Q` hP6@E1J BK ;2 `2+QMbi`m+iBQMbX SA6 BK ;2 mb2/ 7Q` *X
kX8X9X AK ;2 _2+QMbi`m+iBQM rBi? o2M/Q`Ƕb _2+QMbi`m+@ iBQM hQQH#Qt :1 T`QT`B2i `v `2+QMbi`m+iBQM H;Q`Bi?Kb BM+Hm/BM; ol1 SQBMi >. UQ` 7mHHv j. Bi2` iBp2 `2+QMbi`m+iBQM r?B+? Bb 7m`i?2` `272``2/ iQ Pa1J@:1V- ol1 SQBMi 6s UQ` 7mHHv j. hP6 Bi2` iBp2 `2+QMbi`m+iBQM r?B+? Bb 7m`i?2` `272``2/ iQ hP6@Pa1J@:1V- ol1 TQBMi >. a? `TA_ UQ` Sa6@ Pa1J@:1V M/ ol1 TQBMi 6s a? `TA_ UQ` Sa6@hP6@ Pa1J@:1V r2`2 mb2/ iQ `2+QMbi`m+i oZ*- >QzK M M/ T iB2Mi / i b2ib rBi? 28 bm#b2ib (kN- jy)X
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h?2 T iB2Mi BK ;2b `2+QMbi`m+i2/ rBi? i?2 irQ bQ7ir `2 T +F ;2b rBi? hP6@Pa1J H;Q`Bi?Kb TT2 ` iQ ? p2 /B7@ 72`2M+2b /m2 iQ i?2 `2 bQMb 7Q`2K2MiBQM2/X hP6 BK ;2b ? / /Bz2`2M+2b BM MQBb2 r?B+? Bb /m2 iQ i?2 H +F Q7 BKTH2@ K2Mi iBQM Q7 hP6 b+ ii2` bBKmH iBQM BM i?2 +m``2Mi p2`@ bBQM Q7 i?2 ahA_ HB#` `vX alo_ +QKT `BbQMb 7Q` hP6 `2+QMbi`m+iBQMb 7m`i?2` b?Qrb i?2 2z2+i Q7 H +F Q7 hP6 b+ ii2` BKTH2K2Mi iBQM b i?2`2 `2 ;`2 i2` /Bz2`2M+2b #2@ ir22M ahA_ M/ p2M/Q`Ƕb `2+QMbi`m+iBQM bQ7ir `2 [m M@ iB}+ iBQMbX hP6@E1J alo_ +QKT `BbQMb /BbTH v p2`v bHB;?i BKT`Qp2K2Mi BM i2`Kb Q7 i?2 [m MiB}+ iBQMX Ai + M #2 b22M BM };m`2 Ry i? i Sa6@hP6@E1J ? b i?2 #2bi mMB7Q`KBiv 7Q` bTH22M M/ hP6@E1J ? b i?2 #2bi mMB@ 7Q`KBiv 7Q` HBp2`X h?Bb b?Qrb i?2 2z2+i Q7 i?2 J_ M iQK@ B+ H BM7Q`K iBQM BM S1h `2+QMbi`m+iBQMbX hP6@E1J *Qo p Hm2b BKT`Qp2b b i?2 MmK#2` Q7 Bi2` iBQMb BM+`2K2MibX h?2 ;HQ# H aaAJ p Hm2 Bb #2ir22M yX39@yX38 7Q` HH +QK@ T `BbQMb T2`7Q`K2/ QM `2+QMbi`m+i2/ BK ;2b Qp2` e Bi2`@ iBQMbX h?Bb b?Qrb ?B;? bi`m+im` H bBKBH `Biv #2ir22M ahA_ M/ :1 `2+QMbi`m+i2/ BK ;2bX h?2 /Bz2`2M+2 #2@ ir22M irQ BK ;2b Bb K DQ`Hv /m2 iQ BMi2MbBiv /Bz2`2M+2 #2ir22M i?2 irQ BK ;2bX h?Bb bim/v /Q2b MQi i F2 BMiQ ++QmMi hP6 b+ ii2` +Q``2+iBQM r?B+? ;Bp2b `Bb2 iQ /B7@ 72`2M+2b BM MQBb2 T`QT2`iB2b #2ir22M ahA_ M/ :1 `2@ +QMbi`m+iBQMbX h?2 ` M/QKb +Q``2+iBQM KQ/2HHBM; BKTH2@ K2Mi2/ BM ahA_ Bb MQi B/2MiB+ H iQ i? i BM :1 iQQH#QtX :1 `2+QMbi`m+iBQMb `2 b+ H2/ #v ;HQ# H M/ + HB#` @ iBQM 7 +iQ` i? i Bb MQi +m``2MiHv i F2M BMiQ ++QmMi rBi?BM i?2 ahA_ `2+QMbi`m+iBQMbX 3
H iBQM (j9) M/ iQ BMi2;` i2 Bi rBi? i?2 M2ti `2H2 b2 Q7 aA_6 UavM2`;BbiB+ AK ;2 _2+QMbi`m+iBQM 6` K2rQ`FV (j8)X eX +FMQrH2/;2K2Mib SX q /?r Bb 7mM/2/ #v J2/B+ H _2b2 `+? *QmM@ +BH AM/mbi`B H *a1 S?. a+?QH `b?BT UJ_fJyRd9esfRVX .` hbQmKT b Bb bTQMbQ`2/ #v _Qv H aQ+B2iv AM/mbi`v 62HHQrb?BT UA6RdyyRRVX h?2 T`QD2+i ? b #22M T `iB HHv bmTTQ`i2/ #v i?2 1Sa_* *QHH #Q` iBp2 *QKTmi iBQM H S`QD2+i U1SfJykk83dfRV M/ Bib ~ ;b?BT `2b2 `+? ;` Mi U1SfSykkkyyfRVX q2 rQmH/ HBF2 iQ i? MF S`Q7X ai27 M o M/2M#2`;?2- S`Q7X JB+?2H EQQH2 M/ Jb 1bi2` .Ƕ>Q2 7Q` i?2B` BMBiB H bmTTQ`i M/ / i 7Q` i?Bb rQ`FX q2 rQmH/ HbQ HBF2 iQ i? MF S`Q7X . pB/ "m+FH2v 7Q` ?Bb bm#bi MiB H bmTTQ`i 7Q` i?2 2MiB`2 +Qm`b2 Q7 i?Bb rQ`FX 1i?B+b MmK@ #2` RdfqJfyy39 rBi? T2`KBbbBQM 7`QK +HBMB+ H bim/v T2`7Q`K2/ i AMpB+`QX
(a)
(b) 6B;m`2 e, S iB2Mi . i b2i, h` Mbp2`b2 M/ +Q`QM H bHB+2 +QKT `BbQMb 7Q` U V Pa1J@ahA_ M/ U#V Pa1J@:1X *QKT `BbQMb `2 K /2 7Q` BK ;2b `2+QMbi`m+i2/ rBi? k3 bm#b2ib M/ j Bi2` iBQMbX
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6B;m`2 d, S iB2Mi . i b2i, h` Mbp2`b2 M/ +Q`QM H bHB+2 +QKT `BbQMb 7Q` U V Sa6@Pa1J@ahA_ M/ U#V Sa6@Pa1J@:1X *QKT `BbQMb `2 K /2 7Q` BK ;2b `2+QMbi`m+i2/ rBi? k3 bm#b2ib M/ j Bi2` iBQMbX
8X *QM+HmbBQM AM i?Bb bim/v- i?2 +[mBbBiBQM T`Q+2bb 7Q` :1 aA:L S1hfJ_ b+ MM2` Bb KQ/2HH2/ rBi?BM i?2 QT2M bQm`+2 HB@ #` `v- ahA_X HH p HB/ iBQMb r2`2 K /2 #v +QKT `BM; BK ;2b `2+QMbi`m+i2/ rBi? Bi2` iBp2 `2+QMbi`m+iBQM H;Q@ `Bi?Kb p BH #H2 BM i?2 HB#` `v 7Q` +[mB`2/ T? MiQK M/ +HBMB+ H / i b2ibX _2+QMbi`m+i2/ BK ;2b TT2 ` iQ #2 i +QKT ` #H2 H2p2H rBi? i?2 QM2b T`QpB/2/ #v i?2 K Mm7 +@ im`2`X .2 /@iBK2 +Q``2+iBQM- #b2M+2 Q7 /2i2+iQ` ; T KQ/@ 2HHBM; M/ hP6 b+ ii2` +Q``2+iBQM ? p2 MQi #22M KQ/2HH2/ BM i?Bb rQ`F- M/ K v ++QmMi 7Q` i?2 Q#b2`p2/ /Bb+`2T@ M+B2bX h?2 K BM /p Mi ;2 Q7 KQ/2HHBM; i?2 +[mBbBiBQM T`Q+2bb2b Q7 i?Bb b+ MM2` BM i?2 ahA_ HB#` `v Bb i? i i?2 b+B2MiB}+ +QKKmMBiv + M T2`7Q`K i?2B` `2b2 `+? BM BK ;2 `2+QMbi`m+iBQM rBi? +QKTH2i2 +QMi`QH Q7 ?Qr i?2 / i Bb K MBTmH i2/X h?Bb rBHH ?2HT iQ T p2 i?2 r v 7Q` ? `KQMB@ b iBQM Q7 K2i?Q/QHQ;v +`Qbb p `BQmb S1h b+ MM2`b mb2/ BM i?2 +HBMB+X 6mim`2 rQ`F rBHH HbQ BM+Hm/2 + H+mH iBM; b+ ii2` +Q``2+iBQM rBi? ahA_ mbBM; bBM;H2 *QKTiQM b+ ii2` bBKm@ N
(RR)
(Rk)
(Rj)
(R9)
(R8)
(Re)
(Rd)
(R3)
(RN) (ky)
(kR)
(kk) (kj)
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S1046-2023(19)30193-8 https://doi.org/10.1016/j.ymeth.2020.01.005 YMETH 4847
To appear in:
Methods
Received Date: Revised Date: Accepted Date:
31 July 2019 15 November 2019 14 January 2020
Please cite this article as: P. Wadhwa, K. Thielemans, N. Efthimiou, K. Wangerin, N. Keat, E. Emond, T. Deller, O. Bertolli, D. Deidda, G. Delso, M. Tohme, F. Jansen, R.N. Gunn, W. Hallett, C. Tsoumpas, PET Image Reconstruction Using Physical and Mathematical Modelling For Time of Flight PET-MR Scanners in the STIR Library, Methods (2020), doi: https://doi.org/10.1016/j.ymeth.2020.01.005
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© 2020 Published by Elsevier Inc.
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BM KQMBiQ`BM; i?2 T`Q;`2bbBQM Q7 /Bb2 b2b M/ i?2 T iB2MiǶb `2bTQMb2 iQ i?2` Tv (N)X "2BM; Q7 BMi2`2bi BM #Qi? i?2 +HBMB+ H M/ `2b2 `+? +QKKmMBiB2b- TH i7Q`K i? i + M `2 /BHv T`QpB/2 M QT2M 7` K2rQ`F iQ `2+QMbi`m+i / i Bb Q7 ?B;? bB;MB}@ + M+2X h?2 `2+QMbi`m+iBQM bQ7ir `2 mb2/ BM i?Bb rQ`F Bb M QT2M bQm`+2 HB#` `v # b2/ QM Q#D2+i@Q`B2Mi2/ T`Q;` K@ KBM; + HH2/ ǵaQ7ir `2 7Q` hQKQ;` T?B+ AK ;2 _2+QMbi`m+@ iBQMǶ UahA_, ?iiT,ffbiB`Xb7XM2iV (Ry)X ahA_ Bb rB/2Hv mb2/ *YY HB#` `v T`QpB/BM; iQQHb iQ `2+QMbi`m+i 2KBbbBQM iQ@ KQ;` T?v BK ;2b rBi? HH / i +Q``2+iBQMb bm+? b MQ`@ K HBb iBQM- ii2Mm iBQM- ` M/QKb M/ b+ ii2` 2biBK iBQMX ahA_ HbQ HHQrb K MBTmH iBQM Q7 T`QD2+iBQM / i M/ BK@ ;2b- BK ;2 M HvbBb M/ M HviB+ H bBKmH iBQMbX Ai Bb C Mm `v kN- kyky
/2bB;M2/ iQ #2 KQ/mH ` M/ ? b M BM?2`Bi M+2 #2ir22M +H bb2bX h?Bb HHQrb +H bb 2ti2MbBQM M/ i?mb + M BM+Q`TQ@ ` i2 MQp2H H;Q`Bi?Kb M/ b+ MM2` ;2QK2i`B2b +QMp2MB2MiHv rBi?Qmi i?2 M22/ iQ `2@BKTH2K2Mi 2tBbiBM; ii`B#mi2b M/ 7mM+iBQMbX h?Bb bi`m+im`2 T`QpB/2b TH i7Q`K iQ i?2 S1h `2b2 `+? +QKKmMBiv r?2`2 i?2 HB#` `v Bb 2tT M/ #H2 iQ dz}i i?2 M22/bǴ Q7 M2r mb2`b b i?2 }2H/ T`Q;`2bb2b #v /2`Bp@ BM; M2r +H bb2b 7`QK # bB+ #mBH/BM; #HQ+FbX 6Q` BMbi M+2iQ BKTH2K2Mi MQp2H Bi2` iBp2 `2+QMbi`m+iBQM H;Q`Bi?Kb- M 2tBbiBM; `2+QMbi`m+iBQM # b2 +H bb bm+? b P`/2`2/ am#b2i J tBKmK SQbi2`BQ`B PM2 ai2T G i2 + M #2 2ti2M/2/ M/ Qi?2` ahA_ 7mM+iBQM HBiB2b + M #2 `2mb2/ (RR)X _2+2Mi +QH@ H #Q` iBp2 2zQ`ib rBi?BM i?2 ahA_ /2p2HQT2`Ƕb +QKKmMBiv ? p2 H2/ iQ i?2 2ti2MbBQM Q7 i?2 HB#` `v iQ MQp2H Bi2` iBp2 `2+QMbi`m+iBQM H;Q`Bi?Kb bm+? b i?2 F2`M2H 2tT2+i iBQM K tBKBx iBQM UE1JV (Rk) Q` KQ/2HHBM; iBKBM; BM7Q`K iBQM 7Q` iBK2@Q7@~B;?i UhP6V S1h (Rj)X S1h b+ MM2`b K2 bm`2 / i r?B+? `2 /B{+mHi iQ `2+QM@ bi`m+i- 2bT2+B HHv rBi? bm{+B2Mi ++m` +v rBi?Qmi FMQrBM; HH M2+2bb `v BM7Q`K iBQM T`QpB/2/ #v i?2 K Mm7 +im`2`X .2bTBi2 i?2 rB/2 ` M;2 Q7 MQp2H Bi2` iBp2 `2+QMbi`m+iBQM H@ ;Q`Bi?Kb p BH #H2 rBi?BM ahA_- Bi + M #2 +? HH2M;BM; 7Q` mb2`b iQ ++m` i2Hv `2+QMbi`m+i / i K2 bm`2/ 7`QK +HBMB+ H Q` T`2+HBMB+ H S1h b+ MM2`bX PM2 M22/b iQ + `27mHHv BKTH2@ K2Mi T`QT`B2i `v b+ MM2` / i 7Q`K ib ++m` i2Hv rBi?BM ahA_ b Bi +m``2MiHv bmTTQ`ib QMHv *hA 1*he- 1*hd M/ aB2K2Mb "BQ;` T? KJ_ }H2 7Q`K ibX
2.8 KKVX 6m`i?2`- i?2`2 `2 4 #HQ+Fb T2` mMBi M/ 2 +? #HQ+F ? b 4 × 9 +`vbi HbX 1 +? #HQ+F 7m`i?2` ? b 3 × 6 `` v Q7 aBSJbX h?2 b+ MM2` Bb +QM};m`2/ #v /2i2+iQ` KQ/mH2b TH +2/ M2ti iQ 2 +? Qi?2` 7Q`KBM; +vHBM/`B+ H #Q`2 rBi? /2i2+iQ` mMBib HQM; i?2 tB H /B`2+iBQMX h?mbi?2 b+ MM2` ? b 45 /2i2+iQ` +`vbi Hb HQM; i?2 tB H /B`2+@ iBQM M/ 448 /2i2+iQ`b `QmM/ i?2 b+ MM2` K FBM; iQi H Q7 ky-Rey /2i2+iQ`bX h?2 KQ/mH2b `2 TH +2/ rBi? KQ/mH2 0 +2Mi2`2/ #Qp2 i?2 BbQ+2Mi2` Q7 i?2 b+ MM2` r?B+? BK@ THB2b i? i +`vbi H 0 Bb MQi p2`iB+ HHv #Qp2 i?2 +2Mi2` Q7 i?2 b+ MM2`X *`vbi H y Bb HQ+ i2/ i M Qzb2i Q7 −5.23◦ 7`QK i?2 p2`iB+ H +2Mi2` r?2M pB2rBM; 7`QK i?2 # +F Q7 i?2 b+ MM2`X h?2 GuaP +`vbi Hb `2 5.3 KK HQM; i?2 tB H /B`2+iBQM3.95 KK HQM; i?2 i` Mb tB H /B`2+iBQM M/ 25 KK /22TX qBi? i?2 7 bi /2i2+iBQM 2H2+i`QMB+b Q7 i?2 b+ MM2`- Bib iBK@ BM; `2bQHmiBQM Bb 390 TbX h?2 b+ MM2` ? b +QBM+B/2M+2 iBKBM; rBM/Qr Q7 4.57 Mb M/ iBKBM; H2 bi bB;MB}+ Mi #Bi Q7 13.02 Tb /m` iBQM r?B+? H2 /b iQ 351 iBKBM; #BMbX h?2 b+ MM2` b p2b i?2b2 b bB;M2/ iBKBM; #BMb UBX2X MmK#2`2/ 7`QK @Rd8 iQ YRd8 #BMbV r?B+? i?2 bQ7ir `2 +QK#BM2b BM ;`QmTb Q7 13 iBKBM; #BMb- T`Q/m+BM; 27 bBMQ;` K iBKBM; #BMbX h?2 +QBM+B/2M+2 2p2Mib `2 biQ`2/ BM bBMQ;` K Q7 /BK2MbBQMb 224 × 1981 × 27 × 357- r?2`2 224 +Q``2bTQM/b iQ i?2 iQi H MmK#2` Q7 pB2rb- 1981 +Q``2bTQM/b iQ i?2 iQi H MmK#2` Q7 tB H TQbBiBQMb- 27 +Q``2bTQM/b iQ i?2 MmK#2` Q7 dzK b?2/Ǵ iBKBM; #BMb M/ 357 +Q``2bTQM/b iQ i?2 MmK@ #2` Q7 i M;2MiB H TQbBiBQMbX h?2 MmK#2` Q7 pB2rb `272`b iQ QM2@? H7 Q7 i?2 iQi H MmK#2` Q7 /2i2+iQ`b HQM; i?2 `BM; #v +QMp2MiBQMX h?2 b+ MM2` T ` K2i2`b `2 BM+Hm/2/ BM i?2 b+ MM2` i2K@ TH i2 +H bb Ǵa+ MM2`X+ttǴ (Rd)X h?Bb +H bb biQ`2b i?2 `2H2@ p Mi ;2QK2i`B+ BM7Q`K iBQM 7Q` S1h b+ MM2`bX
RXRX P#D2+iBp2b h?2 BKb Q7 i?Bb rQ`F `2, RX JQ/2HHBM; Q7 i?2 :1 aA:L S1hfJ_ b+ MM2`Ƕb +@ [mBbBiBQM T`Q+2bbX kX 1tT M/ ahA_ HB#` `v iQ :1 7Q`K ibX jX o HB/ iBQM Q7 i?2 BK ;2b `2+QMbi`m+i2/ #v ahA_ M/ i?2 p2M/Q`Ƕb bQ7ir `2X 9X .2KQMbi` iBQM Q7 `2+QMbi`m+iBQM mbBM; E1JX 8X .2KQMbi` iBQM Q7 hP6 `2+QMbi`m+iBQM mbBM; ahA_ HB@ #` `v 7Q` K2 bm`2/ / i X
kXkX a+ MM2` JQ/2H h?2 ` /BQ +iBp2 /Bbi`B#miBQM Q7 i?2 Q#D2+i TH +2/ rBi?BM i?2 b+ MM2` Bb K i?2K iB+ HHv KQ/2HH2/ b 7QHHQrb, y = Ax + b,
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r?2`2 y `2T`2b2Mib i?2 +[mB`2/ bBMQ;` K K2 bm`2K2MiA `2T`2b2Mib i?2 bvbi2K K i`Bt r?2`2 A = [a pm ] - x `2T@ `2b2Mib i?2 BK ;2 Q7 i?2 ` /BQ +iBpBiv /Bbi`B#miBQM Q7 i?2 Q#D2+i M/ b `2T`2b2Mib i?2 # +F;`QmM/ +QBM+B/2M+2b U2X;X ` M/QKb M/ b+ ii2`VX h?2 bvbi2K K i`Bt KQ/2Hb i?2 b2M@ bBiBpBiv i 2 +? pQt2H m Q7 i?2 b+ MM2` #v KQ/2HHBM; i?2 T`Q# #BHBiv Q7 TQbBi`QM MMB?BH i2/ BM pQt2H m M/ #2BM; /2i2+i2/ b T?QiQM@T B` +QBM+B/2M+2 #v i?2 /2i2+iQ` T B` pX h?2 b2MbBiBpBiv Q7 2 +? pQt2H m +QK#BM2b i?2 /2i2+iQ` 2{+B2M+v- ;2QK2i`B+ M/ ii2Mm iBQM 2z2+ibX aT2+B}+ HHv- i?2 hP6 BM7Q`K iBQM 7Q` i?2 K2 bm`2/ bBMQ@ ;` K / i Bb KQ/2HH2/ rBi?BM i?2 7Q`r `/ KQ/2H b,
kX J2i?Q/b M/ J i2`B Hb kXRX :1 aA:L S1hfJ_ a+ MM2` AM ahA_ :1 aA:L hP6 S1hfJ_ b+ MM2` BMbi HH2/ i AMpB+`QAKT2`B H *QHH2;2 GQM/QM- > KK2`bKBi? >QbTBi H- GQM@ /QM- lE- r b mb2/ BM i?Bb bim/vX Ai Bb jh J_ b+ MM2` i? i ? b M BMi2;` i2/ J_@+QKT iB#H2 S1h b+ MM2` (R9)X h?2 S1h b+ MM2` ? b }2H/ Q7 pB2r Q7 60 +K M/ 25 +K BM i` Mb tB H M/ tB H /B`2+iBQMb- `2bT2+iBp2HvX Ai Bb +QK@ TQb2/ Q7 Hmi2iBmK@vii`BmK QtvQ`i?QbBHB+ i2 UGuaPV +`vb@ i Hb- bBHB+QM T?QiQKmHiBTHB2`b UaBSJV M/ Qi?2` T`Q+2bbBM; 2H2+i`QMB+b (R8)X h?2 aBSJ iQ;2i?2` rBi? i?2 GuaP +`vb@ i Hb HHQr 7Q` hP6 BK ;BM; (Re)X h?2 S1h +QKTQM2Mi Q7 :1 aA:L +QKT`Bb2b Q7 28 /2i2+iQ` KQ/mH2b M/ 2 +? /2@ i2+iQ` KQ/mH2 ? b 5 /2i2+iQ` mMBib U2 +? /2i2+iQ` mMBib `2 b2T ` i2/ 7`QK i?2 M2ti mMBi #v +`vbi H@7`22 ; T Q7 bBx2
yt = Σm a ptm x + bt ,
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/2i2+i2/ #v /2i2+iQ` T B` p rBi?BM i?2 bB;M2/ iBKBM; #BM MmK#2`- tX
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AM i?Bb T T2`- i?2 r2HH FMQrM Q`/2`2/ bm#b2i 2tT2+@ i iBQM K tBKBb iBQM UPa1JV (R3) Bi2` iBp2 BK ;2 `2+QM@ bi`m+iBQM H;Q`Bi?K Bb 2ti2M/2/ iQ BM+Hm/2 iBKBM; BM7Q`K @ iBQM UBX2X hP6@Pa1JV M/ J_ BM7Q`K iBQM UBX2X hP6@ E1JVX h?2b2 H;Q`Bi?Kb `2 mb2/ iQ `2+QMbi`m+i i?2 K2 @ bm`2/ bBMQ;` K yt b K2 bm`2/ #v i?2 b+ MM2`X *m``2MiHv QMHv bBMQ;` K@# b2/ `2+QMbi`m+iBQMb `2 /2KQMbi` i2/X
kXjX :1 aA:L S1hfJ_ a+ MM2` +[mBbBiBQM . i BM ahA_ AK ;2 `2+QMbi`m+iBQM Q7 / i +[mB`2/ rBi? i?2 :1 b+ M@ M2` mbBM; ahA_ HB#` `v `2[mB`2b i?2 BM+Q`TQ` iBQM Q7 BMTmi M/ QmiTmi BM7Q`K iBQM i? i +QMp2`i i?2 / i BMiQ M +@ +2bbB#H2 / i 7Q`K iX aT2+B}+ HHv- i?2 QmiTmi }H2b b p2/ #v i?2 b+ MM2` +QMbQH2 i i?2 2M/ Q7 S1h b+ M BM+Hm/2,
kXkXRX hP6@Pa1J H;Q`Bi?K h?2 hP6@Pa1J H;Q`Bi?K r b /2b+`B#2/ BM (Rj)X H@ i?Qm;? i?2 hP6 2tT MbBQM Q7 i?2 ahA_ HB#` `v r b `2@ +2MiHv BKTH2K2Mi2/ M/ p HB/ i2/ QM bBKmH i2/ / i b2ib (Rj)- i?Bb Bb i?2 }`bi iBK2 i? i i?2 BKTH2K2Mi iBQM Bb i2bi2/ QM +[mB`2/ / i b2ibX h?2 hP6@Pa1J H;Q`Bi?K Bb /2b+`B#2/ b 7QHHQrb, xmj+1 =
xmj s∈S j
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RX GBbiKQ/2 UGJV M/ bBMQ;` K ` r / i }H2b U_.6V +QKT`2bb2/ rBi? i?2 :1 T`QT`B2i `v +QKT`2bbBQM H@ ;Q`Bi?K kX _ r / i + HB#` iBQM }H2b BM+Hm/BM; MQ`K HBb iBQM- ;2@ QK2i`B+ M/ r2HH +QmMi2` + HB#` iBQM 7 +iQ`b jX S1h AK ;2b 6Q` ii2Mm iBQM USA6V }H2b UBX2X T`Q@ +2bb2/ S1h ii2Mm iBQM +Q``2+iBQM U*V BK ;2b Q#@ i BM2/ #v +QMp2`iBM; J_ BK ;2b +[mB`2/ mbBM; J_ b2[m2M+2b BM+Hm/BM; x2`Q 2+?Q iBK2 M/ .BtQMV 9X _2+QMbi`m+i2/ S1h BK ;2b rBi? `2+QMbi`m+iBQM b2i@ iBM;b b2H2+i2/ i i?2 iBK2 Q7 b+ M
UjV
r?2`2 xmj `2T`2b2Mib i?2 BMi2MbBiv Q7 mth pQt2H BM BK ;2 mT/ i2/ Qp2` i?2 bm#@Bi2` iBQM j- y st `2T`2b2Mib i?2 bm#b2i s Q7 hP6 T`QD2+iBQM / i 7Q` hP6 #BM t- a st;m `2T`2b2Mib i?2 T`Q# #BHBiv Q7 M 2p2Mi Q++m``BM; BM pQt2H K #2BM; /2i2+i2/ #v /2i2+iQ` T B` s rBi?BM i?2 iBKBM; #BM t- b st `2T`2b2Mib i?2 +Q``2+iBQM i2`K iQ ++QmMi 7Q` i?2 bmK Q7 ` M/QKb M/ b+ ii2` 2p2Mib UQ` b rBHH #2 `272``2/ iQ # +F;`QmM/ i2`KV 7Q` bm#b2i s M/ iBKBM; #BM t M/ S j `2T`2b2Mib i?2 bm#b2i Q7 /2i2+iQ` T B`bX LQi2 i? i QM2 7mHH Bi2` iBQM- Q` bBKTHv K2MiBQM2/ b Bi2` iBQM- +Q``2bTQM/b iQ 7mHH +v+H2 Q7 bm#@Bi2` iBQMb +`Qbb HH bm#b2ib UBX2X Bi2` iBQMb 4 bm#@ Bi2` iBQMbfbm#b2ibVX
HH i?2 QmiTmi ` r / i }H2b `2 BM :1 T`QT`B2i `v / i 7Q`K i M/ + M #2 T`Q+2bb2/ Q|BM2 bQH2Hv rBi? i?2 :1Ƕb T`QT`B2i `v `2+QMbi`m+iBQM bQ7ir `2X :1 biQ`2b / i M/ ?2 /2` BM7Q`K iBQM BM >B2` `+?B+ H . i 6Q`K i 8 U>.68V }H2 7Q`K iX >.68 }H2b BM+Hm/2, RX aBMQ;` K _.6 ;2M2` HHv + HH2/ Ƕ`/7XRXRǶ kX GJ }H2 ;2M2` HHv + HH2/ ǶGAahyyyyX"G6Ƕ jX LQ`K HBb iBQM / i }H2- ǶMQ`Kj/Ƕ- M/ ;2QK2i`B+ +Q`@ `2+iBQM / i }H2- Ƕ;2Qj/Ƕ
kXkXkX hP6@E1J H;Q`Bi?K h?2 E1J H;Q`Bi?K r b }`bi T`QTQb2/ #v q M; 2i HX (RN) M/ r b /2p2HQT2/ BM ahA_ HB#` `v #v .2B// 2i HX(Rk- ky)X h?Bb ? b #22M MQr 2ti2M/2/ 7m`i?2` iQ ++QK@ KQ/ i2 hP6 / i b r2HHX h?2 H;Q`Bi?K bbmK2b i? i M BK ;2 Bb /2b+`B#2/ #v i?2 bmK Q7 MmK#2` Q7 r2B;?i2/ F2`M2Hb- M/ Bi i`B2b iQ `2+QMbi`m+i i?2 r2B;?i Q7 2 +? F2`@ M2H #v bbmKBM; Bi FMQrb i?2 F2`M2H 7`QK MQi?2` bQm`+2 Q7 BM7Q`K iBQM U2X;X J_AV, xm =
Nm
h?2b2 }H2b `2 mb2/ b BMTmi iQ p HB/ i2 i?2 BKTH2K2Mi @ iBQM Q7 i?2 +[mBbBiBQM KQ/2H 7Q` i?Bb b+ MM2` BM ahA_X >Qr2p2`- ahA_ ? b Bib QrM M iBp2 bBMQ;` K / i 7Q`@ K i- mb2/ b BMTmi 7Q` i?2 Bi2` iBp2 M/ M HviB+ H `2+QM@ bi`m+iBQM H;Q`Bi?KbX h?mb- HH i?2 >.68 }H2b M22/ iQ #2 +QMp2`i2/ BMiQ ahA_ M iBp2 BMi2`}H2 7Q`K i (Rd)X r` T@ T2` +H bb r b +`2 i2/ iQ BKTQ`i i?2 >.68 }H2b /m`BM; i?Bb rQ`FX >.68r` TT2` +H bb ? b 7mM+iBQMb BKTH2K2Mi2/ iQ ++2bb M/ `2 / i?2 `2[mB`2/ BM7Q`K iBQM r?B+? Bb 7m`i?2` KQ/2HH2/ rBi?BM i?2 bvbi2K K i`Bt b /Bb+mbb2/ BM b2+iBQM kX9X
U9V
λ f k f m,
f =1
r?2`2 p Hm2 i pQt2H m Q7 BK ;2 xm `2T`2b2Mib i?2 HBM2 ` +QK#BM iBQM Q7 i?2 F2`M2H K i`Bt Uk f m `2T`2b2Mib Bib f mth 2H2K2MibV- Nm `2T`2b2Mib i?2 MmK#2` Q7 72 im`2 p2+iQ`b M/ F2`M2H r2B;?ib λX h?2 F2`M2H r2B;?ib `2 `2+QMbi`m+i2/ b 7QHHQrb, λmj+1 =
λmj
l klm
s∈S j
a st;m
l
klm
s∈S j
a st;m
l
a st;l
kX9X AKTH2K2Mi iBQM Q7 i?2 +[mBbBiBQM KQ/2H BM ahA_ kX9XRX hP6 +[mBbBiBQM . i BM ahA_ hP6 BM7Q`K iBQM Bb BKTH2K2Mi2/ BM ahA_ rBi?BM i?2 bvbi2K K i`Bt b,
y st f
, k f l λ jf + b st U8V
a pt;m = a pm T ptm . j
UeV
h?Bb 2[m iBQM KQ/2Hb i?2 hP6 BM7Q`K iBQM b iBK2 bT`2 / 7mM+iBQM- T ptm 7Q` 2 +? iBK2 #BM t M/ b2T ` i2b Bi 7`QK i?2 b2MbBiBpBiv BM7Q`K iBQM- a pm X h?2 BKTH2K2M@ i iBQM ? b #22M `2+2MiHv T`2b2Mi2/ M/ p HB/ i2/ BM bBKm@ H i2/ / i (Rj)- #mi MQi v2i 7Q` +[mB`2/ / i X h?2 b+ MM2` b p2b i?2 /2i2+iQ` T B` M/ iBKBM; BM7Q`@ K iBQM BM GJ M/ bBMQ;` K _.6bX h?2 / i b p2/ BM i?2 GJ >.68 }H2 Bb ++2bbB#H2 #v M pB; iBM; iQ i?2 }2H/ + HH2/ ǵfGBbi. i fGBbi. i Ƕ (kR- Rd)X GJ }H2b biQ`2 2 +? 2p2Mi Q7 M +[mBbBiBQM b 6 #vi2 bi`m+im`2X h?Bb BM@ +Hm/2b BM7Q`K iBQM bm+? b i?2 +`vbi H B/2MiB}2`b UA.bV r?B+? `2;Bbi2`2/ i?2 +QBM+B/2Mi T?QiQMb M/ i?2 iBKBM; #BMX h?2 /2i2+iQ` T B` TQbBiBQMb `2 ? M/H2/ b tB H M/ i` Mb tB H +QQ`/BM i2b rBi?BM i?2 dz*GBbiJQ/2. i :1a@ A:LǴ +H bb Q7 ahA_X h?2 bB;M2/ hP6 #BM Bb ? M/H2/ rBi?BM i?2 dz*GBbi_2+Q`/:1aA:LǴ +H bbX h?Bb Bb /QM2 #v `2 /BM; i?2 hP6 BM7Q`K iBQM HQM; rBi? i?2 hP6 ǴK b?@ BM;Ǵ BM7Q`K iBQM M/ mb2 hP6 T`QD2+iBQM / i - yt rBi?BM i?2 `2+QMbi`m+iBQM 7` K2rQ`FX AM i?2 +m``2Mi BKTH2K2M@ i iBQM- hP6 #BMb + M #2 K b?2/ iQ;2i?2` iQ ;2i MQM@hP6 / i X h?2 / i Bb `2 `` M;2/ iQ #2 ++m` i2Hv `2 / BM ahA_ #v TTHvBM; i` Mb7Q`K iBQMb 7`QK b+ MM2` bT +2 iQ ahA_ bT +2X h?2 M2ti bi2T iQ `2+QMbi`m+i S1h BK ;2b rBi? ahA_ `2[mB`2b i?2 + H+mH iBQM Q7 i?2 b2MbBiBpBiv BK ;2X h?Bb BM@ +Hm/2b KQ/2HHBM; Q7 /2i2+iQ` 2{+B2M+v- ;2QK2i`B+ M/ i@ i2Mm iBQM 7 +iQ`bX h?2 b2MbBiBpBiv BK ;2b /2KQMbi` i2 i?2 bT iB H p `B iBQM Q7 2tT2+i2/ +QBM+B/2M+2b T2` mMBi +iBpBiv +QM+2Mi` iBQMX
kX9XjX SA6@# b2/ ii2Mm iBQM *Q``2+iBQM ii2Mm iBQM +Q``2+iBQM r b + ``B2/ Qmi mbBM; b+ MM2` 2ti` +i2/ SA6 BK ;2bX h?Bb r b +?B2p2/ #v +`2 iBM; +mbiQK b+`BTib i? i BKTQ`i2/ i?2 SA6 BK ;2b rBi?BM i?2 ahA_ `2+QMbi`m+iBQM 7` K2rQ`F (Rd)X 6m`i?2`KQ`2- # +F;`QmM/ i2`K M22/b iQ #2 + H+mH i2/ iQ +Q``2+i 7Q` b+ ii2` M/ ` M/QK 2p2MibX kX9X9X AKTH2K2Mi iBQM Q7 _ M/QKb *Q``2+iBQM lbBM; aBM@ ;H2b _ M/QKb `2 2biBK i2/ 7`QK bBM;H2b T2` +`vbi H 7Q` i?2 :1 aA:L (k9- k8), RXY = 2τS X S Y /T,
r?2`2 RXY Bb i?2 MmK#2` Q7 ` M/QK 2p2Mib UQ` +QmMibV /2@ i2+i2/ #v i?2 /2i2+iQ` T B` X M/ Y- τ `2T`2b2Mib i?2 +QBM+B@ /2M+2 iBK2 rBM/Qr U4.57 Mb 7Q` i?Bb b+ MM2`V- T `2T`2b2Mib i?2 iQi H +[mBbBiBQM iBK2- S X M/ S Y `2T`2b2Mi i?2 bBM;H2 2p2Mib /2i2+i2/ #v +`vbi H X M/ Y- `2bT2+iBp2HvX aBM;H2 ` i2b 7Q` 2 +? +`vbi H `2 2tTQ`i2/ 7`QK i?2 GJ >.68 }H2 M/ +QMp2`i2/ BMiQ ` M/QK ` i2b mbBM; i?2 ahA_ miBHBiv ǵ+QMbi`m+in` M/QKbn7`QKn:1bBM;H2bǶ BKTH2K2Mi2/ /m`@ BM; i?Bb rQ`FX aBM;H2b T2` b2+QM/ `2 /2i2+i2/ 7Q` 2p2`v +`vbi H M/ i?2M biQ`2/ b / i b2i rBi? /BK2MbBQMb 45 × 448 BM GJ >.68 }H2X 1 +? +2HH Q7 i?2 / i b2i +Q``2bTQM/b iQ i?2 MmK#2` Q7 bBM;H2b /2i2+i2/ #v i?2 +`vbi H MmK#2` (X, Y) r?2`2 X +Q``2bTQM/b iQ i?2 `BM; MmK#2` M/ Y +Q``2bTQM/b iQ i?2 i` Mb tB H +`vbi H MmK#2`X 6Q` M +[mBbBiBQM Q7 iBK2 Q7 N b2+QM/b- i?2`2 Bb HBbi Q7 N + 1 b KTH2bX h?2`2 `2 N + 1 b KTH2b b i?2 T2`BQ/B+ / i `2 bvM+?`QMQmb rBi? i?2 bi `i Q7 S1h b+ M r?B+? M22/b iQ #2 i F2M BMiQ ++QmMi /m`BM; / i + H+mH iBQM #v r2B;?iBM; i?2 }`bi M/ H bi b KTH2b Q7 i?2 T2`BQ/B+ / i TT`QT`B i2HvX 1[m iBQM U3V + H+mH i2b i?2 ` M/QKb +Q``2+iBQM bBMQ;` K- r?B+? Bb H i2` mb2/ 7Q` 2p2`v hP6 #BM 7i2` /BpB/BM; Bi rBi? i?2 iQi H MmK#2` Q7 iBKBM; #BMbX AM i?2 +m``2Mi ahA_ BKTH2@ K2Mi iBQM- /2 /@iBK2 M/ BbQiQT2 /2+ v KQ/2HHBM; `2 MQi +QMbB/2`2/ rBi?BM ` M/QKb +Q``2+iBQMX
kX9XkX AKTH2K2MiBM; LQ`K HBb iBQM *Q``2+iBQM 7Q` :1 aA:L BM ahA_ h?2 ;2QK2i`B+ M/ /2i2+iQ` 2{+B2M+v 7 +iQ`b `2 KmH@ iBTHB2/ iQ ;2i i?2 MQ`K HBb iBQM +Q``2+iBQM 7 +iQ`b T2` /2@ i2+iQ` #BM b #2HQr(kk- kj), nXY =
X Y , gXY
U3V
UdV
kX9X8X a+ ii2` *Q``2+iBQM LQM@hP6 b+ ii2` +Q``2+iBQM bBMQ;` K r b 2ti` +i2/ 7`QK i?2 p2M/Q`Ƕb `2+QMbi`m+iBQM iQQH#Qt 7Q` b+ ii2` +Q`@ `2+iBQMX hP6@b+ ii2` Bb +m``2MiHv MQi BKTH2K2Mi2/- r?2`2 b MQM@ hP6 b+ ii2` +Q``2+iBQM bBMQ;` Kb r2`2 TT`QtBK i2/ iQ hP6 T`QD2+iBQM / i rBi? TT`QT`B i2 b+ HBM; 7 +iQ`b 7Q` 2 +? hP6 #BMbX h?2 b+ HBM; 7 +iQ`b r2`2 2ti` +i2/ mbBM; i?2 2KBbbBQM T`QD2+iBQM / i #v + H+mH iBM; i?2 7` +iBQM Q7 +QmMib BM 2 +? hP6 #BM rBi? `2bT2+i iQ iQi H MmK#2` Q7 +QmMibX h?2 ` M/QKb M/ b+ ii2` +Q``2+iBQM hP6 T`QD2+iBQM / i r2`2 //2/ iQ + H+mH i2 i?2 # +F;`QmM/ i2`K- bt X
r?2`2 nXY `2T`2b2Mib i?2 MQ`K HBb iBQM +Q``2+iBQM 7 +iQ` 7Q` i?2 /2i2+iQ` T B`- j r?B+? Bb +QKT`Bb2/ Q7 /2i2+iQ` A.b X M/ Y- X M/ Y `2T`2b2Mib i?2 +`vbi H 2{+B2M+v 7 +iQ`b M/ gXY `2T`2b2Mib i?2 ;2QK2i`B+ +Q``2+iBQM 7 +iQ`bX h?2 b+ MM2` 2tTQ`ib >.68 }H2b- r?B+? `2 i?2M mb2/ b BMTmib 7Q` + H+mH iBM; i?2 MQ`K HBb iBQM 7 +iQ`b (Rd)X *`vb@ i H 2{+B2M+v 7 +iQ`b `2 biQ`2/ BM ǵMQ`Kj/Ƕ M/ ;2QK2i@ `B+ 7 +iQ`b `2 biQ`2/ BM ǵ;2Qj/ǶX h?2 MQ`Kj/ >.68 }H2 biQ`2b +`vbi H 2{+B2M+v 7 +iQ`b r?B+? `2 `2 / b M `` v Q7 448 × 45 BM ahA_X h?2 j. ;2QK2i`B+ +Q``2+iBQM 7 +iQ`b `2 biQ`2/ b T `i Q7 i?2 T`QD2+iBQM / i M/ QMHv 16 pB2rb `2 biQ`2/ BX2X 7Q` QM2 KQ/mH2X h?2b2 pB2r / i b2ib `2 Q7 /BK2MbBQMb 1981 × 357X h?2 2tTQ`i2/ ;2QK2i`B+ 7 +iQ`b `2 `2T2 i2/ 14 iBK2b rBi?BM ahA_ iQ ;2i 7mHH ;2@ QK2i`B+ +Q``2+iBQM T`QD2+iBQM / i X .2 /@iBK2 +Q``2+iBQM ? b MQi #22M BKTH2K2Mi2/ BM i?Bb bim/vX
kX8X . i +[mBbBiBQM- AK ;2 _2+QMbi`m+iBQM M/ M HvbBb kX8XRX S? MiQK +[mBbBiBQM h?2 7QHHQrBM; irQ T? MiQKb r2`2 b+ MM2/, 9
TQ`i2/ SA6 BK ;2b b /2KQMbi` i2/ BM };m`2 RX _ M@ /QKb +Q``2+iBQM bBMQ;` K r b + H+mH i2/ mbBM; ǵ+QM@ bi`m+in` M/QKbn7`QKn:1bBM;H2bǶ miBHBivX a+ ii2` +Q``2+@ iBQM r b /QM2 mbBM; i?2 bBMQ;` K 2ti` +i2/ 7`QK i?2 p2M@ /Q`Ƕb `2+QMbi`m+iBQM iQQH#QtX 6Q` hP6 BK ;2 `2+QMbi`m+@ iBQMb- i?2 hP6 T`QD2+iBQM / i r2`2 ?BbiQ;` K@ K2/ mbBM; i?2 GJ }H2X _ M/QKb M/ b+ ii2` +Q``2+iBQM bBMQ;` Kb r2`2 i` MbH i2/ BMiQ hP6 T`QD2+iBQM / i #v /2p2HQTBM; +mbiQK miBHBiB2b M/ TT`QT`B i2 b+ HBM; Q7 / i b /2@ b+`B#2/ #Qp2 BM b2+iBQMb kX9X9 M/ kX9X8X ii2Mm iBQM M/ MQ`K HBb iBQM bBMQ;` Kb r2`2 mMB7Q`K BM iBKBM; #BMb M/ r2`2 KmHiBTHB2/ rBi? i?2 # +F;`QmM/ i2`K iQ +`2 i2 //B@ iBp2 bBMQ;` Kb 7Q` ahA_ BKTH2K2Mi iBQMbX AK ;2 `2+QM@ bi`m+iBQMb r2`2 + ``B2/ Qmi mbBM; Pa1J M/ hP6@Pa1J H;Q`Bi?Kb rBi? M/ rBi?Qmi TQBMi bT`2 / 7mM+iBQM USa6V KQ/2HHBM;X hP6@E1J `2+QMbi`m+iBQMb r2`2 HbQ + ``B2/ Qmi 7Q` i?2 T iB2Mi / i b2iX h?2 J_ BK ;2 b?QrM BM };@ m`2 R r b mb2/ b iQ /2`Bp2 i?2 F2`M2HbX HH `2+QMbi`m+iBQMb r2`2 + ``B2/ Qmi rBi? k3 bm#b2ibX
6B;m`2 R, S iB2Mi . i b2i, h` Mbp2`b2 M/ +Q`QM H bHB+2b Q7 i?2 2t@ TQ`i2/ `2b KTH2/ SA6 BK ;2 U#QiiQK `QrV 7`QK J_* UiQT `QrV mbBM; +mbiQK b+`BTib b ahA_ BMi2`}H2X J_* BK ;2 mb2/ b M M iQKB+ H T`BQ` 7Q` F2`M2H K i`Bt 2biBK iBQM 7Q` hP6@E1J BK ;2 `2+QMbi`m+iBQMbX SA6 BK ;2 mb2/ 7Q` *X
kX8X9X AK ;2 _2+QMbi`m+iBQM rBi? o2M/Q`Ƕb _2+QMbi`m+@ iBQM hQQH#Qt :1 T`QT`B2i `v `2+QMbi`m+iBQM H;Q`Bi?Kb BM+Hm/BM; ol1 SQBMi >. UQ` 7mHHv j. Bi2` iBp2 `2+QMbi`m+iBQM r?B+? Bb 7m`i?2` `272``2/ iQ Pa1J@:1V- ol1 SQBMi 6s UQ` 7mHHv j. hP6 Bi2` iBp2 `2+QMbi`m+iBQM r?B+? Bb 7m`i?2` `272``2/ iQ hP6@Pa1J@:1V- ol1 TQBMi >. a? `TA_ UQ` Sa6@ Pa1J@:1V M/ ol1 TQBMi 6s a? `TA_ UQ` Sa6@hP6@ Pa1J@:1V r2`2 mb2/ iQ `2+QMbi`m+i oZ*- >QzK M M/ T iB2Mi / i b2ib rBi? 28 bm#b2ib (kN- jy)X
RX oQHmK2i`B+ Zm HBiv *QMi`QH UoZ*V S? MiQK, *QK@ T`Bb2/ Q7 }p2 0.7 J"[ 68 :2 bT?2`2b Q7 RN KK BM /B@ K2i2` rBi? R9 KK +iBp2 /B K2i2` UrBi? ? H7 KK iQH2` M+2V 2M+ b2/ rBi?BM J_@pBbB#H2 im#2b K /2 Q7 LB*H2 X h?2 2MiB`2 +QM};m` iBQM Bb 2M+ b2/ rBi?BM 7Q K +m#2X h?2 T? MiQK r b b+ MM2/ 7Q` 10 KBM K/ 5 × 106 T`QKTib r2`2 `2;Bbi2`2/X kX >QzK M S? MiQK, a+ MM2/ 7Q` 20 KBM M/ 1.5 × 108 T`QKTib r2`2 `2;Bbi2`2/ (ke)X
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h?2 T iB2Mi BK ;2b `2+QMbi`m+i2/ rBi? i?2 irQ bQ7ir `2 T +F ;2b rBi? hP6@Pa1J H;Q`Bi?Kb TT2 ` iQ ? p2 /B7@ 72`2M+2b /m2 iQ i?2 `2 bQMb 7Q`2K2MiBQM2/X hP6 BK ;2b ? / /Bz2`2M+2b BM MQBb2 r?B+? Bb /m2 iQ i?2 H +F Q7 BKTH2@ K2Mi iBQM Q7 hP6 b+ ii2` bBKmH iBQM BM i?2 +m``2Mi p2`@ bBQM Q7 i?2 ahA_ HB#` `vX alo_ +QKT `BbQMb 7Q` hP6 `2+QMbi`m+iBQMb 7m`i?2` b?Qrb i?2 2z2+i Q7 H +F Q7 hP6 b+ ii2` BKTH2K2Mi iBQM b i?2`2 `2 ;`2 i2` /Bz2`2M+2b #2@ ir22M ahA_ M/ p2M/Q`Ƕb `2+QMbi`m+iBQM bQ7ir `2 [m M@ iB}+ iBQMbX hP6@E1J alo_ +QKT `BbQMb /BbTH v p2`v bHB;?i BKT`Qp2K2Mi BM i2`Kb Q7 i?2 [m MiB}+ iBQMX Ai + M #2 b22M BM };m`2 Ry i? i Sa6@hP6@E1J ? b i?2 #2bi mMB7Q`KBiv 7Q` bTH22M M/ hP6@E1J ? b i?2 #2bi mMB@ 7Q`KBiv 7Q` HBp2`X h?Bb b?Qrb i?2 2z2+i Q7 i?2 J_ M iQK@ B+ H BM7Q`K iBQM BM S1h `2+QMbi`m+iBQMbX hP6@E1J *Qo p Hm2b BKT`Qp2b b i?2 MmK#2` Q7 Bi2` iBQMb BM+`2K2MibX h?2 ;HQ# H aaAJ p Hm2 Bb #2ir22M yX39@yX38 7Q` HH +QK@ T `BbQMb T2`7Q`K2/ QM `2+QMbi`m+i2/ BK ;2b Qp2` e Bi2`@ iBQMbX h?Bb b?Qrb ?B;? bi`m+im` H bBKBH `Biv #2ir22M ahA_ M/ :1 `2+QMbi`m+i2/ BK ;2bX h?2 /Bz2`2M+2 #2@ ir22M irQ BK ;2b Bb K DQ`Hv /m2 iQ BMi2MbBiv /Bz2`2M+2 #2ir22M i?2 irQ BK ;2bX h?Bb bim/v /Q2b MQi i F2 BMiQ ++QmMi hP6 b+ ii2` +Q``2+iBQM r?B+? ;Bp2b `Bb2 iQ /B7@ 72`2M+2b BM MQBb2 T`QT2`iB2b #2ir22M ahA_ M/ :1 `2@ +QMbi`m+iBQMbX h?2 ` M/QKb +Q``2+iBQM KQ/2HHBM; BKTH2@ K2Mi2/ BM ahA_ Bb MQi B/2MiB+ H iQ i? i BM :1 iQQH#QtX :1 `2+QMbi`m+iBQMb `2 b+ H2/ #v ;HQ# H M/ + HB#` @ iBQM 7 +iQ` i? i Bb MQi +m``2MiHv i F2M BMiQ ++QmMi rBi?BM i?2 ahA_ `2+QMbi`m+iBQMbX 3
H iBQM (j9) M/ iQ BMi2;` i2 Bi rBi? i?2 M2ti `2H2 b2 Q7 aA_6 UavM2`;BbiB+ AK ;2 _2+QMbi`m+iBQM 6` K2rQ`FV (j8)X eX +FMQrH2/;2K2Mib SX q /?r Bb 7mM/2/ #v J2/B+ H _2b2 `+? *QmM@ +BH AM/mbi`B H *a1 S?. a+?QH `b?BT UJ_fJyRd9esfRVX .` hbQmKT b Bb bTQMbQ`2/ #v _Qv H aQ+B2iv AM/mbi`v 62HHQrb?BT UA6RdyyRRVX h?2 T`QD2+i ? b #22M T `iB HHv bmTTQ`i2/ #v i?2 1Sa_* *QHH #Q` iBp2 *QKTmi iBQM H S`QD2+i U1SfJykk83dfRV M/ Bib ~ ;b?BT `2b2 `+? ;` Mi U1SfSykkkyyfRVX q2 rQmH/ HBF2 iQ i? MF S`Q7X ai27 M o M/2M#2`;?2- S`Q7X JB+?2H EQQH2 M/ Jb 1bi2` .Ƕ>Q2 7Q` i?2B` BMBiB H bmTTQ`i M/ / i 7Q` i?Bb rQ`FX q2 rQmH/ HbQ HBF2 iQ i? MF S`Q7X . pB/ "m+FH2v 7Q` ?Bb bm#bi MiB H bmTTQ`i 7Q` i?2 2MiB`2 +Qm`b2 Q7 i?Bb rQ`FX 1i?B+b MmK@ #2` RdfqJfyy39 rBi? T2`KBbbBQM 7`QK +HBMB+ H bim/v T2`7Q`K2/ i AMpB+`QX
(a)
(b) 6B;m`2 e, S iB2Mi . i b2i, h` Mbp2`b2 M/ +Q`QM H bHB+2 +QKT `BbQMb 7Q` U V Pa1J@ahA_ M/ U#V Pa1J@:1X *QKT `BbQMb `2 K /2 7Q` BK ;2b `2+QMbi`m+i2/ rBi? k3 bm#b2ib M/ j Bi2` iBQMbX
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6B;m`2 d, S iB2Mi . i b2i, h` Mbp2`b2 M/ +Q`QM H bHB+2 +QKT `BbQMb 7Q` U V Sa6@Pa1J@ahA_ M/ U#V Sa6@Pa1J@:1X *QKT `BbQMb `2 K /2 7Q` BK ;2b `2+QMbi`m+i2/ rBi? k3 bm#b2ib M/ j Bi2` iBQMbX
8X *QM+HmbBQM AM i?Bb bim/v- i?2 +[mBbBiBQM T`Q+2bb 7Q` :1 aA:L S1hfJ_ b+ MM2` Bb KQ/2HH2/ rBi?BM i?2 QT2M bQm`+2 HB@ #` `v- ahA_X HH p HB/ iBQMb r2`2 K /2 #v +QKT `BM; BK ;2b `2+QMbi`m+i2/ rBi? Bi2` iBp2 `2+QMbi`m+iBQM H;Q@ `Bi?Kb p BH #H2 BM i?2 HB#` `v 7Q` +[mB`2/ T? MiQK M/ +HBMB+ H / i b2ibX _2+QMbi`m+i2/ BK ;2b TT2 ` iQ #2 i +QKT ` #H2 H2p2H rBi? i?2 QM2b T`QpB/2/ #v i?2 K Mm7 +@ im`2`X .2 /@iBK2 +Q``2+iBQM- #b2M+2 Q7 /2i2+iQ` ; T KQ/@ 2HHBM; M/ hP6 b+ ii2` +Q``2+iBQM ? p2 MQi #22M KQ/2HH2/ BM i?Bb rQ`F- M/ K v ++QmMi 7Q` i?2 Q#b2`p2/ /Bb+`2T@ M+B2bX h?2 K BM /p Mi ;2 Q7 KQ/2HHBM; i?2 +[mBbBiBQM T`Q+2bb2b Q7 i?Bb b+ MM2` BM i?2 ahA_ HB#` `v Bb i? i i?2 b+B2MiB}+ +QKKmMBiv + M T2`7Q`K i?2B` `2b2 `+? BM BK ;2 `2+QMbi`m+iBQM rBi? +QKTH2i2 +QMi`QH Q7 ?Qr i?2 / i Bb K MBTmH i2/X h?Bb rBHH ?2HT iQ T p2 i?2 r v 7Q` ? `KQMB@ b iBQM Q7 K2i?Q/QHQ;v +`Qbb p `BQmb S1h b+ MM2`b mb2/ BM i?2 +HBMB+X 6mim`2 rQ`F rBHH HbQ BM+Hm/2 + H+mH iBM; b+ ii2` +Q``2+iBQM rBi? ahA_ mbBM; bBM;H2 *QKTiQM b+ ii2` bBKm@ N
(RR)
(Rk)
(Rj)
(R9)
(R8)
(Re)
(Rd)
(R3)
(RN) (ky)
(kR)
(kk) (kj)
(k9)
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(k8) (ke)
(kd) (k3)
(kN) (jy)
(jR)
(jk)
(jj) (j9)
(j8)
Ry
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RR
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Rk
(b)
(c)
(d)
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