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KLT picker: Particle picking using data-driven optimal templates
Journal Pre-proofs KLT picker: Particle picking using data-driven optimal templates Amitay Eldar, Boris Landa, Yoel Shkolnisky PII: DOI: Reference:
S1047-8477(20)30035-6 https://doi.org/10.1016/j.jsb.2020.107473 YJSBI 107473
To appear in:
Journal of Structural Biology
Received Date: Revised Date: Accepted Date:
11 December 2019 21 January 2020 24 January 2020
Please cite this article as: Eldar, A., Landa, B., Shkolnisky, Y., KLT picker: Particle picking using data-driven optimal templates, Journal of Structural Biology (2020), doi: https://doi.org/10.1016/j.jsb.2020.107473
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© 2020 Published by Elsevier Inc.
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T `iB+H2 TB+FBM; bi2T- b r2 /2KQMbi` i2 QM i?`22 /Bz2`@ 2Mi / i b2ib BM a2+iBQM jX AM Q`/2` iQ 2p Hm i2 i?2 TB+F@ k
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6B;m`2 R, h?2 }`bi Re 7mM+iBQMb ψn U7Q` i?2 H `;2bi λn V Q#i BM2/ 7`QK bQHpBM; U9V mbBM; i?2 _Sa. 2biBK iBQM T`Q+2/m`2 /2b+`B#2/ BM a2+iBQM kXkX h?2 2B;2M7mM+iBQMb M/ 2B;2Mp Hm2b r2`2 +QKTmi2/ 7`QK bBM;H2 ivTB+ H KB+`Q;` T? Q7 i?2 1JSA_@Ryyk3 / i b2i (kd)- M/ r2`2 MQ`K HBx2/ bQ i? i i?2 H `;2bi 2B;2Mp Hm2 Bb QM2X
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iBQMb iQ URV- BM TQH ` +QQ`/BM i2b- `2 Q7 i?2 7Q`K 1 ψm,k (r, θ) = √ Rm,k (r)eimθ , 2π
m ∈ Z, k ∈ N,
UjV
r?2`2 Rm,k (r) Bb i?2 kǶi? 2B;2M7mM+iBQM Q7 i?2 BMi2;` H 2[m @ iBQM λm,k Rm,k (r) =
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a
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0
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M/ hm (r, r ) Bb ;Bp2M #v 0
hm (r, r0 ) =
Z
c
G(ρ)Jm (ρr0 )Jm (ρr)r0 ρdρ,
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0
r?2`2 Jm Bb i?2 mǶi? Q`/2` "2bb2H 7mM+iBQM Q7 i?2 }`bi FBM/-
#v P1 = A(1 : l, 1 : l)- i?2 b2+QM/ #v P2 = A(1 : l, 1 + d : l + d)
c Bb i?2 # M/@HBKBi Q7 f - M/ G Bb i?2 _Sa. Q7 f - ;Bp2M 2tTHB+BiHv b G(ρ) =
1 2π
Z
2π
E| fˆ(ρ, θ)|2 dθ,
M/ bQ QM U7Q` /2i BH2/ 2tTH M iBQM QM ?Qr l M/ d `2 +?Qb2M b22 TT2M/Bt "VX h?2M- 7Q` 2 +? T i+? r2 2biBK i2 BiǶb _Sa. i l 2[m HHv@bT +2/ TQBMib b 2tTH BM2/ BM (Rj) M/ 7Q`K i?2 K i`Bt S ∈ Rl×M r?Qb2 i0 i? +QHmKM Si Bb +
U8V
0
i?2 _Sa. +QKTmi2/ 7`QK i?2 iǶi? T i+?X q2 +QMbB/2` i?2
r?2`2 fˆ = F ( f ) Bb i?2 irQ@/BK2MbBQM H 6Qm`B2` i` Mb7Q`K Q7 f X Ai Bb rQ`i?r?BH2 iQ TQBMi Qmi i? i ψm,k Q7 UjV `2
7QHHQrBM; KQ/2H 7Q` i?2 _Sa. Q7 i?2 T i+?2b BM ;Bp2M KB+`Q;` T?,
bi22` #H2 7mM+iBQMb (kk)- bBM+2 +Q``2H iBM; i2KTH i2 Q7 i?2 7Q`K Q7 UjV rBi? M `#Bi` `BHv `Qi i2/ BK ;2 ? b #bQHmi2 p Hm2 r?B+? Bb BM/2T2M/2Mi Q7 i?2 `Qi iBQMX
Si = ν + αi γ,
AM Q`/2` iQ bQHp2 U9V M/ 2p Hm i2 Rm,k - QM2 M22/b iQ
UeV
r?2`2 ν ∈ Rl+ `2 i?2 b KTH2b Q7 i?2 MQBb2Ƕb _Sa.- M/ γ ∈ Rl+ `2 i?2 b KTH2b Q7 G Q7 U8V Ui?2 T `iB+H2Ƕb _Sa.VX
FMQr i?2 T `iB+H2Ƕb _Sa. GX AM i?2 7QHHQrBM; b2+iBQM r2 /2b+`B#2 K2i?Q/ iQ 2biBK i2 Bi 7`QK ;Bp2M KB+`Q;` T?X
q2 bbmK2 i? i ν `2K BMb i?2 b K2 +`Qbb HH T i+?2b M/ i? i γ Bb KmHiBTHB2/ #v 7 +iQ` 0 ≤ αi ≤ 1 BM 2 +? T i+?- r?B+? ++QmMib 7Q` i?2 T`QTQ`iBQM Q7 i?2 T `iB+H2
h?2M- 2[mBTT2/ rBi? M 2biBK i2 7Q` G- r2 mb2 Lvbi`ƺKǶb K2i?Q/ (R) BM Q`/2` iQ bQHp2 U9V MmK2`B+ HHvX hQ bBKTHB7v bm#b2[m2Mi MQi iBQM- r2 bQ`i HH 2B;2Mp Hm2b λm,k U7Q` HH
+QMi BM2/ BM i?2 iǶi? T i+?X h? i Bb- T i+? r?B+? +QMi BMb
m M/ kV BM MQM@BM+`2 bBM; Q`/2`- M/ /2MQi2 i?2 bQ`i2/
MQ T `iB+H2 /KBib α = 0- T i+? r?B+? 7mHHv +QMi BMb T `iB+H2 /KBib α = 1- M/ T i+? r?B+? +QMi BMb T `i Q7
2B;2Mp Hm2b #v λn X q2 bQ`i i?2 2B;2M7mM+iBQMb ++Q`/BM;HvX 6B;m`2 R BHHmbi` i2b i?2 }`bi Re i2KTH i2b ψn UbQ`i2/ +@ +Q`/BM; iQ i?2B` +Q``2bTQM/BM; 2B;2Mp Hm2b λn V Q#i BM2/
T `iB+H2 /KBib α 2[m H iQ i?2 T`QTQ`iBQM Q7 i?2 T `iB+H2 BM i?2 T i+?X ++Q`/BM; iQ UeV- r2 + M 2biBK i2 ν- γ M/ M {αi }i=1 #v bQHpBM;
7`QK i?2 1JSA_@Ryyk3 / i b2i (kd)X 6B;m`2 k b?Qrb # ` THQi Q7 i?2 500 H `;2bi 2B;2Mp Hm2b UkV 7Q` i?2 b K2 / i b2i U+QKTmi2/ pB U9VVX
{ˆγ, νˆ , α} ˆ = arg min γ˜ ,˜ν,α˜
bXiX γ˜ , ν˜ ∈ j
M X i=1
Rl+ ,
||Si − ν˜ − α˜ i γ˜ ||2 , α˜ i ∈ [0, 1],
UdV
r?2`2 I ∈ Rb2a+1c Bb ~ ii2M2/ KB+`Q;` T? T i+?- N ∈ 2 2 Rb2a+1c Bb ~ ii2M2/ BK ;2 Q7 i?2 MQBb2 QMHv- P ∈ Rb2a+1c Bb 2
6B;m`2 j, S `iB+H2 M/ MQBb2 _Sa. 2biBK iBQM T`Q+2/m`2X q2 /BpB/2 i?2 KB+`Q;` T? BMiQ UTQbbB#Hv Qp2`H TTBM;V l × l T i+?2b r?2`2 l Bb H2bb i? M Q` 2[m H iQ i?2 K tBK H /B K2i2` Q7 i?2 T `iB+H2X 6Q` 2 +? T i+? r2 +QKTmi2 Bib _Sa. M/ 7 +iQ`Bx2 Bi b bmK Q7 i?2 MQBb2 _Sa. M/ i?2 T `iB+H2 _Sa. KmHiBTHB2/ #v T`QTQ`iBQM 7 +iQ` 0 ≤ α ≤ 1X
~ ii2M2/ T `iB+H2 BK ;2 rBi?Qmi MQBb2 M/ rBi? x2`Q p H@
m2b QmibB/2 Q7 aD- M/ b·c Bb i?2 ~QQ` 7mM+iBQMX q2 bbmK2 i? i i?2 MQBb2 Bb r?Bi2- r?B+? Bb 2M7Q`+2/ #v T`2r?Bi2MBM; i?2 KB+`Q;` T? mbBM; i?2 b KTH2b Q7 i?2 2biBK i2/ MQBb2
6B;m`2 9, >2 i K T Q7 {αˆ i } Qp2` i?2 KB+`Q;` T? BiǶb 2biBK i2/ 7`QKX h?2 p Hm2b Q7 {αˆ i } r2`2 Q#i BM2/ #v bQHpBM; UdV 7Q` ivTB+ H KB+`Q@ ;` T? Q7 i?2 1JSA_@Ryyk3 / i b2i (kd)- M/ r2`2 MQ`K HBx2/ bQ i? i i?2B` H `;2bi p Hm2 Bb RX AM Q`/2` iQ +`2 i2 i?2 ?2 i K T- αˆ i r b mb2/ iQ +QHQ` i?2 iǶi? T i+?X b 2tTH BM2/ BM a2+iBQM kXk- H `;2 p Hm2b 7Q` αˆ i U#`B;?i +QHQ`V BM/B+ i2 i?2 T`2b2M+2 Q7 T `iB+H2- M/ bK HH p Hm2b U/ `F +QHQ`V BM/B+ i2 Bib #b2M+2X
_Sa. νˆ UQ#i BM2/ BM a2+iBQM kXkVX :Bp2M i?2 KQ/2H U3V- r2 i2bi 7Q` i?2 ?vTQi?2b2b H0 M/ H1 #v i?2 r2HH@FMQrM GBF2@ HB?QQ/ _ iBQ h2bi UG_hV (ke)X q2 /2`Bp2 i?2 G_h 7Q` i?2 + b2 r?2`2 P M/ N `2 MQ`K HHv /Bbi`B#mi2/- M/ /2KQM@ bi` i2 i?2 T` +iB+ H 2z2+iBp2M2bb Q7 i?2 `2bmHiBM; i2bi BM
r?2`2 ||·|| Bb i?2 bi M/ `/ 1m+HB/2 M MQ`KX a22 6B;m`2 j 7Q`
a2+iBQM jX hQ /2`Bp2 i?2 G_h BM i?Bb + b2- r2 M22/ i?2 +Q@ h i p `B M+2 K i`B+2b 7Q` #Qi? ?vTQi?2b2b- M K2Hv E IIT |H0 i h M/ E IIT |H1 X bbmKBM; i? i P M/ N `2 BM/2T2M/2Mi-
M BHHmbi` iBQM Q7 i?2 _Sa. 2biBK iBQM T`Q+2/m`2X LQi2
i? i UdV + M #2 pB2r2/ b M BMbi M+2 Q7 LQM@L2; iBp2 J i`Bt 6 +iQ`Bx iBQM ULLJ6V (e) Q7 i?2 K i`Bt SX AM T@
Bi 7QHHQrb i? i
T2M/Bt "- r2 /2b+`B#2 K2i?Q/ 7Q` 2biBK iBM; i?2 bQHm@
h i h i Σ ≡ E IIT |H1 = E PPT + σ2 I,
iBQM Q7 UdV #v mbBM; Hi2`M iBM; H2 bi@b[m `2b (Ry)- r?B+? Bb TQTmH ` TT`Q +? 7Q` LLJ6X q2 BHHmbi` i2 ivTB+ H
r?2`2 σ Bb i?2 bi M/ `/ /2pB iBQM Q7 i?2 MQBb2- M/ I Bb i?2
p Hm2b Q7 αˆ i 7`QK i?2 7 +iQ`Bx iBQM UdV BM 6B;m`2 9X Ai Bb 2p@ B/2Mi i? i αˆ i Bb H `;2` BM `2 b Q7 i?2 KB+`Q;` T? r?B+? `2
b2a + 1c2 × b2a + 1c2 B/2MiBiv K i`BtX aBM+2 σ Bb ivTB+ HHv
mMFMQrM- r2 T`QTQb2 K2i?Q/ 7Q` 2biBK iBM; Bi 7`QK ;Bp2M KB+`Q;` T? BM TT2M/Bt *X hQ 2p Hm i2 UNV Bi i?2`2@ i h 7Q`2 `2K BMb iQ 2biBK i2 E PPT X G2i U #2 M 2[m HHv@
/2Mb2Hv T +F2/ rBi? T `iB+H2b- BM/B+ iBM; i? i i?2 7 +iQ`@ Bx iBQM 2ti` +ib i?2 _Sa. Q7 i?2 T `iB+H2b 7`QK i?2 +Q``2+i `2;BQMb Q7 i?2 KB+`Q;` T?X
bT +2/ U* `i2bB MV ;`B/ rBi? b2a + 1c × b2a + 1c TQBMib Qp2` 2 i?2 b[m `2 b−a, ac × b−a, ac- M/ /2MQi2 #v U ∈ Rb2a+1c i?2
kXjX S `iB+H2 /2i2+iBQM
~ ii2M2/ p2`bBQM Q7 UX "v b2iiBM; i?2 p Hm2b Q7 f QmibB/2 Q7 aD iQ #2 x2`Q- r2 ? p2 i? i Pi = f (i) 7Q` 2p2`v i ∈ UX a2iiBM; HbQ i?2 p Hm2b Q7 ψn Q7 URV QmibB/2 Q7 aD iQ x2`Q-
AM i?2 T`2pBQmb b2+iBQMb- r2 /2b+`B#2/ ?Qr iQ +QKTmi2 b2i Q7 i2KTH i2b {ψn } UiQ;2i?2` rBi? i?2B` 2B;2Mp Hm2b {λn } /2b+`B#BM; i?2B` BKTQ`i M+2V / Ti2/ iQ i?2 T `iB+H2
M/ /2MQiBM; #v Ψ ∈ Rb2a+1c
i?2 K i`Bt r?Qb2 nǶi? +QH@ mKM +QMbBbib Q7 i?2 ~ ii2M2/ b KTH2b Q7 ψn QM U- H2 /b iQ i?2 7QHHQrBM; TT`QtBK iBQM Q7 i?2 T `iB+H2Ƕb +Qp `B M+2
BK ;2b BM i?2 KB+`Q;` T?X >Qr2p2`- i i?Bb TQBMi Bi Bb
mM+H2 ` ?Qr iQ K F2 mb2 Q7 i?2 /Bz2`2Mi i2KTH i2bX AM T `iB+mH `- bBM+2 bBM;H2 i2KTH i2 /Q2b MQi + Tim`2 i?2 #2 mb2/X q2 b?QmH/ i?2`27Q`2 /2i2`KBM2 ?Qr K Mv i2K@ TH i2b iQ mb2 M/ ?Qr iQ +QK#BM2 i?2K 7Q` T `iB+H2 TB+F@
URyV
N r?2`2 Λ Bb /B ;QM H K i`Bt rBi? i?2 2B;2Mp Hm2b {λn }n=1 QM Bib K BM /B ;QM HX h?2 T ` K2i2` N Bb +?Qb2M bm+? i? i i?2 2``Q` BM i?2 TT`QtBK iBQM URyV Bb bK HH- r?B+?
M/ /2`Bp2 b+?2K2 r?2`2 /Bz2`2Mi i2KTH i2b ψn `2 +QK@ #BM2/ ++Q`/BM; iQ i?2B` `2H iBp2 BKTQ`i M+2 Ui?`Qm;? i?2 2B;2Mp Hm2b λn VX aT2+B}+ HHv- r2 M2ti /2`Bp2 T`Q+2/m`2
2bb2MiB HHv /2T2M/b QM i?2 /2+ v ` i2 Q7 i?2 2B;2Mp Hm2b
7Q` /Bb+`BKBM iBM; #2ir22M M BK ;2 Q7 Tm`2 MQBb2- r?B+? Bb /2}M2/ b i?2 MmHH ?vTQi?2bBb M/ /2MQi2/ #v H0 - M/
λn Ub22 TT2M/Bt . 7Q` /2i BH2/ 2tTH M iBQM QM ?Qr i?2 T ` K2i2` N Bb /2i2`KBM2/VX :Bp2M i?2 +Qp `B M+2 K i`Bt Σ Q7 UNV U+QKTmi2/ pB i?2
M BK ;2 Q7 T `iB+H2 THmb MQBb2- r?B+? Bb /2}M2/ b i?2 Hi2`M iBp2 ?vTQi?2bBb M/ /2MQi2/ #v H1 X h? i Bb- r2 +QMbB/2` i?2 KQ/2H
H1 : T `iB+H2 THmb MQBb2-
×N
i h E PPT ≈ ΨΛΨT ,
BM;X hQ i? i 2M/- r2 im`M iQ +H bbB+ H ?vTQi?2bBb i2biBM;
H0 : Tm`2 MQBb2-
2
K i`Bt Ub22 TT2M/Bt .V
p `B #BHBiv Q7 HH T `iB+H2 BK ;2b- KmHiBTH2 i2KTH i2b Kmbi
N, I= P + N,
UNV
TT`QtBK iBQM URyVV- r2 mb2 b+Q`2 # b2/ QM i?2 G_h 7Q` +QKT `BM; #2ir22M irQ KmHiBp `B i2 MQ`K H /Bbi`B#miBQMb b 7QHHQrbX 1 +? KB+`Q;` T? T i+? Ii, j Q7 bBx2 b2a + 1c ×
U3V
b2a+1c U~ ii2M2/ iQ p2+iQ`V Bb +QMbB/2`2/ b `2 HBx iBQM
9
H;Q`Bi?K R h?2 EGh TB+F2`
6B;m`2 8, 1t KTH2 Q7 ivTB+ H b+Q`BM; K i`Bt b ?2 i K T U`B;?i BK ;2V M/ T `iB+H2b TB+F2/ 7`QK Bi UH27i BK ;2V rBi? i?`2b?QH/ Q7 x2`QX h?2 / i b2i Bb 1JSA_@Ryyk3 (kd)X
R,
Q7 I M/ Bb bbB;M2/ b+Q`2
k,
! p(Ii, j |H1 ) b+Q`2(i, j) := log URRV p(Ii, j |H0 ) ! ! T |Σ| 1 = log 2b2a+1c2 + Ii, j σ−2 I − Σ−1 Ii, j , 2 σ
j, 9, 8,
r?2`2 | · | Bb K i`Bt /2i2`KBM MiX a22 TT2M/Bt . 7Q`
M 2H #Q` i2 /2`Bp iBQM Q7 i?Bb b+Q`2 M/ M 2{+B2Mi H;Q@
`Bi?K iQ +QKTmi2 BiX >B;? p Hm2 Q7 b+Q`2(i, j) BM/B+ i2b i?2 T`2b2M+2 Q7 T `iB+H2 BM i?2 (i, j) T i+?X q2 +QMbB/2` i?2
e,
b+Q`2b b K i`Bt r?Qb2 2H2K2Mib `2 b+Q`2(i, j)X 1ti` +i@
d,
BM; i?2 T `iB+H2Ƕb +QQ`/BM i2b Qmi Q7 i?2 b+Q`BM; K i`Bt Bb /QM2 #v mbBM; p2`bBQM Q7 i?2 MQM@K tBKmK bmTT`2b@
3,
bBQM H;Q`Bi?K (Rk)X h?Bb ;`22/v H;Q`Bi?K }`bi +?QQb2b i?2 ?B;?2bi b+Q`2 BM i?2 b+Q`BM; K i`Bt b i?2 }`bi TB+F2/ T `iB+H2Ƕb +2Mi2`X h?2M- BM Q`/2` iQ T`2p2Mi M2 `#v TBt@
N,
_2[mB`2/, JB+`Q;` T? M/ i?2 T `iB+H2bǶ ` /Bmb aX S `iBiBQM i?2 KB+`Q;` T? BMiQ l × l T i+?2b rBi? bi`B/2 d BM 2 +? /BK2MbBQM U#v /27 mHi- l Bb +?Qb2M iQ #2 3yW Q7 i?2 K tBK H /B K2i2` Q7 i?2 T `iB+H2 M/ d = lb22 TT2M/Bt " 7Q` KQ`2 /2i BHbVX *QKTmi2 i?2 _Sa. Q7 2 +? T i+? b /2b+`B#2/ BM b2+@ iBQM kXkX 1biBK i2 i?2 T `iB+H2 M/ MQBb2 _Sa. γˆ , νˆ - `2bT2+iBp2Hv #v bQHpBM; UdVX S`2r?Bi2M i?2 KB+`Q;` T? mbBM; νˆ X *QKTmi2 i?2 b2i Q7 2B;2M7mM+iBQMb {ψˆn } M/ 2B;2Mp Hm2b {λn } Q7 URV #v bQHpBM; U9V mbBM; γˆ M/ i?2 Lvbi`ƺK K2i?Q/X 1biBK i2 i?2 T `iB+H2Ƕb +Qp `B M+2 K i`Bt E[PPT ] +@ +Q`/BM; iQ URyV M/ i?2 MQBb2 bi M/ `/ /2pB iBQM σ- b /2i BH2/ BM TT2M/Bt *Xi h *QKTmi2 Σ = E IIT |H1 Ub22 UNVVX S `iBiBQM i?2 KB+`Q;` T? BMiQ [2a + 1] × [2a + 1] T i+?2b rBi? bi`B/2 1 BM 2 +? /BK2MbBQM M/ +QKTmi2 i?2 G_h b+Q`2 7Q` 2 +? T i+? ++Q`/BM; iQ URRVX h?2 Qmi+QK2 Bb b+Q`BM; K i`BtX 1ti` +i i?2 T `iB+H2bǶ +QQ`/BM i2b 7`QK i?2 b+Q`BM; K @ i`Bt b /2b+`B#2/ i i?2 2M/ Q7 a2+iBQM kXjX
2Hb 7`QK #2BM; +QMbB/2`2/ b T `iB+H2b- HH TBt2Hb rBi?BM mb2`@/2}M2/ ` /Bmb `2 2t+Hm/2/ 7`QK i?2 TB+FBM;- M/ i?2 T`Q+2/m`2 `2bmK2b rBi? i?2 M2ti H `;2bi b+Q`2- M/ bQ QMX
i?2 ?B;?2bi 7`2[m2M+B2b r2`2 2HBKBM i2/ bBM+2 i?2v +QMi BM KQbiHv MQBb2X h?`Qm;?Qmi HH Q7 i?2 2tT2`BK2Mib- r2 mb2/ i?2 7QHHQrBM; T ` K2i2`b 7Q` i?2 EGh TB+F2`, a M/ l `2
q2 b2i i?2 2t+HmbBQM ` /Bmb iQ #2 i?2 T `iB+H2 bBx2- ?Qr2p2`bK HH2` p Hm2b K v ;Bp2 #2ii2` `2bmHib 7Q` +HQb2Hv T +F2/B``2;mH `Hv b? T2/ T `iB+H2bX h?2 TB+FBM; +QMiBMm2b mMiBH i?2 `2K BMBM; b+Q`2b `2 #2HQr x2`Q Q` mMiBH i?2 H;Q`Bi?K
kyW M/ 3yW- `2bT2+iBp2Hv- Q7 i?2 K tBK H /B K2i2` Q7
TB+Fb T`2/2}M2/ MmK#2` Q7 T `iB+H2bX aiQTTBM; r?2M i?2 `2K BMBM; b+Q`2b `2 M2; iBp2 Bb # b2/ QM i?2 Q#b2`p iBQM
bK HH2` i? M ? H7 Q7 i?2 2biBK i2/ T `iB+H2Ƕb bBx2- b i?Bb
i?2 T `iB+H2- d = l- M/ b 2tTH BM2/ #Qp2- i?2 /2i2+@ iBQM i?`2b?QH/ Bb b2i iQ x2`QX LQi2 i? i a Bb i F2M iQ #2
i? i b+Q`2(i, j) < 0 ?QH/b r?2M2p2` p(Ii, j |H1 ) < p(Ii, j |H0 )-
H2 /b- BM T` +iB+2- iQ #2ii2` +2Mi2`2/ T `iB+H2 b2H2+iBQMbX h?2 Q#i BM2/ T `iB+H2 +QQ`/BM i2b r2`2 BKTQ`i2/ BMiQ _1@
KTH2 Q7 b+Q`BM; K i`Bt M/ i?2 T `iB+H2b TB+F2/ 7`QK
BM Q`/2` iQ Q#i BM j. `2+QMbi`m+iBQMX h?2 1J." (R8) KQ/2H +Q``2bTQM/BM; iQ 2 +? / i b2i r b mb2/ b i?2 #
r?B+? K2 Mb i? i HH Q7 i?2 `2K BMBM; T i+?2b ? p2 ?B;?2` T`Q# #BHBiv iQ +QMi BM MQBb2 QMHvX a22 6B;m`2 8 7Q` M 2t@
GAPL (R3) M/ i?2 `QmiBM2 /2b+`B#2/ BM (Rd) r b 2t2+mi2/
BiX a22 6B;m`2 e M/ H;Q`Bi?K R 7Q` /B ;` K M/ M H;Q`Bi?K i? i bmKK `Bx2 i?2 T `iB+H2 TB+FBM; T`Q+2bbX
BMBiBQ KQ/2H 7Q` i?2 j. `2}M2K2Mi T`Q+2bbX h?2 T`Q+2/m`2 /2b+`B#2/ #Qp2 Bb /2KQMbi` i2/ QM i?2 7QHHQrBM; / i b2ib Q7 i?2 1JSA_ `2TQbBiQ`v (N), 1JSA_@ Ryyk3 USH bKQ/BmK 6 H+BT `mK 3ya `B#QbQK2V (kd)- 1JSA_@
jX 1tT2`BK2Mi H `2bmHib
RyyeR Uβ@: H +iQbB/ b2V (k)- M/ 1JSA_@Ryy9N UavM T@ iB+ _:R@_:kV (Re)X _2bmHib 7Q` / i b2i Q7 .mM HB2HH
AM i?Bb b2+iBQM- r2 /2KQMbi` i2 i?2 T2`7Q`K M+2 Q7 Qm` H;Q`Bi?K QM i?`22 /Bz2`2Mi / i b2ib- b 7QHHQrbX 6B`bi- i?2
a HBM S?QiQbvbi2K A amT2`+QKTH2t `2 b?QrM BM i?2 amT@ TH2K2Mi `v J i2`B HX h?Bb H ii2` / i b2i +Q``2bTQM/b iQ
KB+`Q;` T?b r2`2 /QrMb KTH2/ iQ kyW@8yW Q7 i?2B` Q`B;B@ M H bBx2 M/ r2`2 MQ`K HBx2/ iQ x2`Q K2 M M/ p `B M+2 Q7
M Q#D2+i r?Qb2 bT2+i ` iBQ Bb Km+? H `;2` i? M Q7 i?2
QM2X AM //BiBQM- 2 +? KB+`Q;` T? r b # M/T bb }Hi2`2/-
i?`22 #Qp2@K2MiBQM2/ 1JSA_ / i b2ibX Pm` H;Q`Bi?K r b BKTH2K2Mi2/ BM J iH #- M/ HH i?2
2HBKBM iBM; 8W Q7 Bib HQr2bi M/ ?B;?2bi ` /B H 7`2[m2M@ +B2bX h?2 HQr2bi 7`2[m2M+B2b r2`2 2HBKBM i2/ bBM+2 i?2v /Q
2tT2`BK2Mib r2`2 2t2+mi2/ QM GBMmt K +?BM2 rBi? Re
MQi +QMi BM 2MQm;? b KTH2b 7Q` 2biBK iBM; i?2 _Sa.- M/ 8
6B;m`2 e, 6Q` 2 +? KB+`Q;` T?- r2 2biBK i2 i?2 T `iB+H2 M/ MQBb2 _Sa. b 2tTH BM2/ BM a2+iBQM kXkX h?2M- r2 T`2r?Bi2M i?2 KB+`Q;` T? M/ 2biBK i2 i?2 i2KTH i2b mbBM; U9VX G bi- r2 +QKTmi2 i?2 G_h b+Q`BM; K i`Bt M/ 2ti` +i i?2 T `iB+H2bǶ +QQ`/BM i2b 7`QK Bi- b /2b+`B#2/ i i?2 2M/ Q7 a2+iBQM kXjX
+Q`2b `mMMBM; i kXR:>x- de3:" Q7 K2KQ`v- M/ M LpB/B
#v +QKT `BM; i?2 Qmi+QK2 Q7 _1GAPLǶb TQbi@T`Q+2bbBM;
hBi M :SlX h?2 +QKTH2i2 bQm`+2 +Q/2 7Q` i?2 EGh TB+F2`
i bF iQ i?2 ;`QmM/@i`mi?X Ai Bb BKTQ`i Mi iQ MQi2 i? i
Bb p BH #H2 i ?iiTb,ff;Bi?m#X+QKf KBi v2H/ `fEGhTB+F2`r?B+? HbQ BM+Hm/2b HBMF iQ bi M/@ HQM2 2t2+mi #H2 BK@
i?2 KB+`Q;` T?bǶ ` r KQpB2b r2`2 MQi p BH #H2 iQ mb- M/ bQ #2ii2` `2bmHib K v #2 +?B2p2/ #v 7m`i?2` TQHBb?BM;X AM
TH2K2Mi iBQM Q7 i?2 H;Q`Bi?KX
+QKT `BbQM- (k) TB+F2/ 93, 686 T `iB+H2b mbBM; M miQK iB+
jXRX SH bKQ/BmK 6 H+BT `mK 3ya
K2i?Q/ # b2/ QM +Q``2H iBQM rBi? : mbbB M `BM;X h? M j. +H bbB}+ iBQM BM 6_1GA:L (d) r b mb2/ iQ /Bb+ `/
h?2 SH bKQ/BmK 6 H+BT `mK 3ya / i b2i +QMbBbib Q7
# / b2H2+iBQMb- M/ 41, 123 T `iB+H2b r2`2 `2i BM2/ 7Q` i?2
1081 KB+`Q;` T?b Q7 bBx2 4096 × 4096 TBt2Hb- rBi? TBt2H bBx2
j. `2}M2K2Mi bi2T Ui? i r b 2t2+mi2/ BM i?2 b K2 T`Q@ ;` KVX h?2 `2+QMbi`m+iBQM BM (k) `2 +?2/ ;QH/ bi M/ `/
Q7 1.34 X h?2 TT`QtBK i2 bBx2 Q7 i?2 T `iB+H2 Bb 360 × 360 TBt2Hb U ++Q`/BM; iQ (kd)VX 1t KTH2b Q7 i?2 EGh TB+FBM;
6a* `2bQHmiBQM Q7 2.2X 6B;m`2 N T`2b2Mib bm`7 +2 pB2rb Q7 i?2 j. KQ/2HbX h?2 `mMMBM; iBK2 Q7 i?2 EGh TB+F2` 7Q`
`2bmHib `2 b?QrM BM 6B;m`2 RyX h?2 EGh TB+F2` TB+F2/ 141, 679 T `iB+H2b 7`QK i?2 KB+`Q;` T?bX AM Q`/2` iQ Tm`;2 i?2 b2i Q7 TB+F2/ T `iB+H2b- k. +H bb p2` ;BM; bi2T r b
bBM;H2 KB+`Q;` T?- mbBM; :Sl- r b TT`QtBK i2Hv 33
2t2+mi2/ UmbBM; 30 +H bb2bV BM _1GAPL M/ 87, 439 T `@
b2+QM/b U7QHHQrBM; QM2@iBK2 T`2T`Q+2bbBM; bi2T Q7 TT`Qt@ BK i2Hv 40 b2+QM/bVX
iB+H2b r2`2 `2i BM2/X 1t KTH2b Q7 i?2 b2H2+i2/ k. +H bb2b `2 b?QrM BM 6B;m`2 dX h?2 j. `2+QMbi`m+iBQM `2 +?2/
jXjX avM TiB+ _:R@_:k h?2 avM TiB+ _:R@_:k +QMbBbib Q7 e3y KB+`Q;` T?b-
;QH/ bi M/ `/ 6Qm`B2` a?2HH *Q``2H iBQM U6a*V `2bQHmiBQM Q7 3.05X 6B;m`2 3 T`2b2Mib i?2 6a* +m`p2 T`Q/m+2/ #v +QK@
2 +? Q7 bBx2 3838 × 3710 TBt2Hb- rBi? TBt2H bBx2 Q7 RXkjX
h?2 TT`QtBK i2 bBx2 Q7 i?2 T `iB+H2 ++Q`/BM; iQ (Re) Bb 192 × 192 TBt2HbX h?2 EGh TB+F2` TB+F2/ 163, 470 T `iB@
T `BM; i?2 Qmi+QK2 Q7 _1GAPLǶb TQbi@T`Q+2bbBM; i bF iQ i?2 ;`QmM/@i`mi?X AM +QKT `BbQM- 158, 212 T `iB+H2b r2`2 TB+F2/ #v (kd) mbBM; _1GAPLǶb miQK i2/ TB+FBM; iQQH-
+H2bX 1t KTH2b Q7 i?2 `2bmHib Q7 i?2 TB+FBM; `2 b?QrM BM
6B;m`2 RkX 7i2` 2t2+miBM; k. +H bb p2` ;BM; bi2T Umb@ BM; 30 +H bb2bV BM _1GAPL- 34, 100 T `iB+H2b r2`2 `2i BM2/X
M/ 7i2` k. M/ j. +H bb p2` ;BM; bi2Tb- 105, 247 T `iB@ +H2b `2K BM2/X h?2 `2+QMbi`m+iBQM BM (kd) `2 +?2/ ;QH/ bi M/ `/ 6a* `2bQHmiBQM Q7 3.20X 6B;m`2 N T`2b2Mib bm`7 +2
1t KTH2b Q7 i?2 b2H2+i2/ k. +H bb2b `2 b?QrM BM 6B;m`2 dX h?2 j. `2+QMbi`m+iBQM `2 +?2/ ;QH/ bi M/ `/ 6a* `2bQ@ HmiBQM Q7 3.87X 6B;m`2 3 T`2b2Mib i?2 6a* +m`p2 T`Q/m+2/
pB2rb Q7 Qm` j. `2+QMbi`m+i2/ KQ/2H M/ i?2 `2+QMbi`m+i2/ KQ/2H Q7 (kd)X h?2 `mMMBM; iBK2 Q7 i?2 EGh TB+F2` 7Q` bBM;H2 KB+`Q;` T?- mbBM; :Sl- r b TT`QtBK i2Hv 35 b2+QM/b U7QHHQrBM; QM2@iBK2 T`2T`Q+2bbBM; bi2T Q7 TT`Qt@
#v +QKT `BM; i?2 Qmi+QK2 Q7 _1GAPLǶb TQbi@T`Q+2bbBM; i bF iQ i?2 ;`QmM/@i`mi?X q2 MQi2 i? i bBKBH `Hv iQ i?2 β@
BK i2Hv 40 b2+QM/bVX
: H +iQbB/ b2 / i b2i- i?2 ` r KQpB2b r2`2 mM p BH #H2 iQ
jXkX β@: H +iQbB/ b2
mb- ?2M+2 i?2 `2bmHib `2 bm#@QTiBK HX AM +QKT `BbQM- (Re) mb2/ b2KB@ miQK iB+ K2i?Q/ 2t2+mi2/ BM aJl1G (RN)
h?2 β@: H +iQbB/ b2 / i b2i +QMbBbib Q7 R8jN KB+`Q@
7Q` i?2 T `iB+H2 TB+FBM; bi2TX AMBiB HHv- 2, 000 T `iB+H2b r2`2
;` T?b- 2 +? Q7 bBx2 7676 × 7420 TBt2Hb- rBi? TBt2H bBx2 Q7 yXe9X h?2 TT`QtBK i2 bBx2 Q7 i?2 T `iB+H2 ++Q`/BM; iQ (k)
K Mm HHv b2H2+i2/ M/ +H bbB}2/ #v k. +H bb p2` ;BM; bi2TX :QQ/ +H bb2b r2`2 K Mm HHv +?Qb2M M/ i?2 T `iB+H2b
Bb 768 × 768 TBt2HbX EGh TB+F2` TB+F2/ 100, 241 T `iB+H2bX
BM i?Qb2 +H bb2b r2`2 mb2/ b i2KTH i2b 7Q` M miQK iB+ TB+FBM; T`Q+2bbX b /2b+`B#2/ BM (Re)- //BiBQM H K Mm H 2zQ`i r b `2[mB`2/ iQ /Bb+ `/ # / b2H2+iBQMb- r?B+? `2@
1t KTH2b Q7 i?2 `2bmHib Q7 i?2 TB+FBM; `2 b?QrM BM 6B;@ m`2 RRX 7i2` 2t2+miBM; k. +H bb p2` ;BM; bi2T UmbBM; 30 +H bb2bV BM _1GAPL- 13, 000 T `iB+H2b r2`2 `2i BM2/X 1t@
bmHi2/ BM }M H / i b2i Q7 89, 158 T `iB+H2bX h?2 j. `2@
KTH2b Q7 i?2 b2H2+i2/ k. +H bb2b `2 b?QrM BM 6B;m`2 dX h?2 j. `2+QMbi`m+iBQM `2 +?2/ ;QH/ bi M/ `/ 6a* `2bQ@
}M2K2Mi bi2T r b 2t2+mi2/ BM _1GAPL- M/ i?2 `2+QM@ bi`m+iBQM `2 +?2/ ;QH/ bi M/ `/ 6a* `2bQHmiBQM Q7 3.4X
HmiBQM Q7 3.01X 6B;m`2 3 T`2b2Mib i?2 6a* +m`p2 T`Q/m+2/
6B;m`2 N T`2b2Mib bm`7 +2 pB2rb Q7 i?2 j. KQ/2HbX h?2 e
`mMMBM; iBK2 Q7 i?2 EGh TB+F2` 7Q` bBM;H2 KB+`Q;` T?mbBM; :Sl- r b TT`QtBK i2Hv 50 b2+QM/b U7QHHQrBM;
EGh TB+F2` Bb BM #2ir22M #Qi? + i2;Q`B2bX PM QM2 ? M/Bi `2HB2b QM i?2 bbmKTiBQM i? i i?2 T`QD2+iBQM BK ;2b + M
QM2@iBK2 T`2T`Q+2bbBM; bi2T Q7 TT`QtBK i2Hv 40 b2+QM/bVX
#2 TT`QtBK i2/ #v # M/@HBKBi2/- `Qi iBQM HHv BMp `B Mi-
9X .Bb+mbbBQM
rB/2 b2Mb2 bi iBQM `v T`Q+2bb `2bi`B+i2/ iQ /BbFX h?Bb bbmKTiBQM Bb MQM@`2bi`B+iBp2 BM T` +iB+2- b 2t2KTHB}2/ #v Qm` `2bmHibX PM i?2 Qi?2` ? M/- i?2 i2KTH i2b Q7 i?2
9XRX _2pB2r Q7 i?2 `2bmHib
EGh TB+F2` `2 / Ti2/ iQ 2p2`v KB+`Q;` T? b2T ` i2Hvi?`Qm;? i?2 _Sa. Q7 i?2 T `iB+H2b M/ Q7 i?2 MQBb2X
h?2 / i b2ib mb2/ iQ /2KQMbi` i2 Qm` K2i?Q/ `2T@ `2b2Mi /Bz2`2Mi ivT2b Q7 KB+`Q;` T?bX h?2 SH bKQ/BmK 6 H+BT `mK 3ya Bb i?2 H2 bi +? HH2M;BM;- b i?2 K +`Q@
8X *QM+HmbBQM
KQH2+mH2 Bb H `;2 U4.2 J. V M/ i?2 T `iB+H2b `2 +H2 `Hv
AM i?Bb T T2`- r2 T`2b2Mi2/ i?2 EGh TB+F2`- M m@ iQK i2/- / i /`Bp2M T `iB+H2 TB+F2`- i? i `2[mB`2b QMHv
pBbB#H2X h?2 β@: H +iQbB/ b2 Bb KQ`2 +? HH2M;BM;- b i?2 K +`Q@KQH2+mH2 Bb bK HH2` U0.465 J. V- M/ Bi Bb ? `/2` iQ
QM2 BMTmi r?B+? Bb i?2 2biBK i2/ T `iB+H2Ƕb bBx2X h?2 EGh
b22 i?2 T `iB+H2b QM i?2 KB+`Q;` T?bX 6BM HHv- i?2 avM T@ iB+ _:R@_:k Bb i?2 KQbi +? HH2M;BM; / i b2i- b Bi +Q``2bTQM/b iQ bK HH2` K +`Q@KQH2+mH2 U0.38 J. V M/
TB+F2` Bb /2bB;M2/ 2bT2+B HHv iQ ? M/H2 HQr aL_ KB+`Q@ ;` T?b- Bi Bb 2 bv iQ mb2- 7 bi- M/ K i?2K iB+ HHv `B;Q`QmbX
? b HQr aL_X qBi? HH i?2b2 / i b2ib- r2 +?B2p2/ ?B;? `2bQHmiBQMb rBi? KBMBKmK K Mm H 2zQ`iX AM //BiBQM- i?2 bm`7 +2 pB2rb BM 6B;m`2 N b?Qr i? i i?2 j. `2+QMbi`m+iBQMb
/Bz2`2Mi / i b2ibX AM HH Q7 i?2K r2 +?B2p2/ ?B;?@[m HBiv `2bmHib rBi? KBMBK H K Mm H 2zQ`iX
q2 2p Hm i2/ i?2 T2`7Q`K M+2 Q7 i?2 EGh TB+F2` QM i?`22
h?2 H;Q`Bi?K + M #2 7m`i?2` BKT`Qp2/ b 7QHHQrbX JB@
r2 ;Qi `2 +QKT iB#H2 rBi? i?2 Tm#HBb?2/ QM2bX
+`Q;` T?b i2M/ iQ +QMi BM +QMi KBM iBQMb Q7 p `BQmb 7Q`Kb7`QK bK HH +QMi KBM iBQMb iQ r?QH2 `2 b +QMi BMBM; 2B@
h?`Qm;?Qmi Qm` 2MiB`2 `2+QMbi`m+iBQM T`Q+2bb- i?2 QMHv K Mm H bi2T r b i?2 b2H2+iBQM Q7 +H bb2b Qmi Q7 _1GAPLǶb
i?2` B+2 Q` i?2 2/;2b Q7 i?2 bmTTQ`iBM; ;`B/X h?2b2 `@
k. +H bbB}+ iBQM QmiTmiX q2 /B/ MQi mb2 j. +H bbB}+ iBQM-
2 b + M /2;` /2 i?2 2biBK iBQM Q7 i?2 _Sa.- r?B+? Bb i?2 2bb2M+2 Q7 Qm` K2i?Q/X .2pBbBM; M H;Q`Bi?K iQ miQK i@
MQ` Mv Qi?2` K2i?Q/ iQ }Hi2` i?2 EGh TB+F2`Ƕb TB+F2/ T `iB+H2bX Ai Bb HbQ BKTQ`i Mi iQ MQi2 i? i i?2 T ` K2i2`b
B+ HHv B/2MiB7v i?2b2 `2 b K v BKT`Qp2 i?2 TB+FBM;X .2Mb2 T `iB+H2 +Hmbi2`b + M TQb2 T`Q#H2K iQ i?2 EGh TB+F2`- b
BM HH Q7 i?2 2tT2`BK2Mib r?2`2 +?Qb2M miQK iB+ HHv # b2/ QM i?2 T `iB+H2Ƕb bBx2- bQ i?2`2 Bb MQ M22/ iQ /Dmbi i?2K K Mm HHvX
i?2 b+Q`BM; K i`Bt- BMi`Q/m+2/ BM a2+iBQM kXj- i2M/b iQ b@ bB;M ?B;? b+Q`2b iQ i?2 BMi2`7 +2 #2ir22M T `iB+H2b BMbi2 / Q7 iQ i?2 +2Mi2` Q7 2 +? T `iB+H2 b2T ` i2Hv- b22 M 2t K@
AM irQ Qmi Q7 i?2 i?`22 2tT2`BK2Mib #Qp2- i?2 `2bQ@ HmiBQM +?B2p2/ mbBM; T `iB+H2b TB+F2/ #v Qm` H;Q`Bi?K Bb BM72`BQ` iQ i?2 Tm#HBb?2/ QM2X h?Bb BKTHB2b i? i 7mHHv m@
TH2 BM 6B;m`2 RjX PM2 TQbbB#H2 bQHmiBQM iQ i?Bb T`Q#H2K Bb iQ `2TH +2 i?2 +m``2Mi H;Q`Bi?K 7Q` 2ti` +iBM; T `iB+H2br?B+? mb2b i?2 BM/BpB/m H p Hm2b Q7 i?2 b+Q`BM; K i`Bt U/2@
iQK iB+ TB+FBM; K v +QK2 i T`B+2X u2i- i?2 `2bmHib Q7 Qm` TB+FBM; + M #2 mb2/ iQ ;2M2` i2 M BMBiB H j. bi`m+@ im`2- r?B+? +QmH/ i?2M #2 mb2/ iQ T`Q/m+2 `272`2M+2b 7Q`
b+`B#2/ i i?2 2M/ Q7 a2+iBQM kXjV- rBi? M H;Q`Bi?K i? i +QMbB/2`b i?2 2MiB`2 M2B;?#Q`?QQ/ Q7 T `iB+H2 + M/B/ i2X
b2+QM/ `QmM/ Q7 `272`2M+2@# b2/ TB+FBM;X aBM+2 `272`2M+2@ # b2/ TB+FBM; /Q2b MQi `2[mB`2 K Mm H BMi2`p2MiBQM- i?2 2MiB`2 `2bmHiBM; TB+FBM; T`Q+2bb Bb biBHH 7mHHv miQK iB+X
+FMQrH2/;K2Mib
9XkX *QKT `BbQM rBi? Qi?2` T `iB+H2 TB+FBM; K2i?Q/b
h?Bb `2b2 `+? r b bmTTQ`i2/ #v i?2 1m`QT2 M _2b2 `+? *QmM+BH U1_*V mM/2` i?2 1m`QT2 M lMBQMǶb >Q`BxQM kyky
S `iB+H2 TB+FBM; K2i?Q/b + M #2 /BpB/2/ BMiQ irQ + i2@ ;Q`B2bX h?2 }`bi `2HB2b QM T`BQ` bbmKTiBQMb QM i?2 T `iB@ +H2Ƕb bi`m+im`2- bm+? b : mbbB M b? T2 (k9) Q` `2b2K#H M+2
`2b2 `+? M/ BMMQp iBQM T`Q;` KK2 U;` Mi ;`22K2Mi dkjNNR @ *_uPJh>VX
iQ Qi?2` FMQrM K +`Q@KQH2+mH2b (k8- k3)X h?2 b2+QM/ `2HB2b bQH2Hv QM BM7Q`K iBQM 7`QK i?2 / i Bib2H7- r?B+? + M #2 b2i Q7 K Mm HHv TB+F2/ T `iB+H2b (R3- j)- Q` b2i Q7 KB+`Q@ ;` T? T i+?2b i? i r2`2 miQK iB+ HHv b2H2+i2/ (3)X h?2 d
6B;m`2 d, 1t KTH2b Q7 9 k. +H bb p2` ;2b Q7 SH bKQ/BmK 6 H+BT `mK 3ya UH27iV- β@: H +iQbB/ b2UKB//H2V- M/ avM TiB+ _:R@_:k U`B;?iV T`Q/m+2/ #v _1GAPL 7`QK T `iB+H2b TB+F2/ #v i?2 EGh TB+F2`X
6B;m`2 3, 6a* +m`p2b Q7 SH bKQ/BmK 6 H+BT `mK 3ya UH27iV- β@: H +iQbB/ b2 UKB//H2V- M/ avM TiB+ _:R@_:k U`B;?iVX
TT2M/Bt X .2`BpBM; UjV M/ U9V
i?mb
h?2 E `?mM2M GQ2p2 h?2Q`2K (RR) bi i2b i? i i?2 bQHm@ iBQMb ψn iQ URV `2 i?2 2B;2M7mM+iBQMb Q7 i?2 miQ+Q``2H iBQM QT2` iQ` K(x, y) = E f (x) f (y) - i? i Bbλn ψn (x) =
Z
K(x, y)ψn (y)dy.
λm,k Rm,k (r)eimθ Z Z 1 0 0 = G(ρ)ei(ρr cos(φ−θ)−ρr cos(φ−θ )) (2π)2 aD cD 0
UXRV
=
aD
lM/2` i?2 bbmKTiBQMb i? i f Bb # M/@HBKBi2/- rB/2 b2Mb2
1 (2π)2
Z
0
aZ
Z
× Rm,k (r0 )eimθ r0 ρdr0 dθ0 dρdφ ! π i(mθ0 −ρr0 cos(φ−θ0 )) 0 e dθ G(ρ)eiρr cos(φ−θ)
−π
cD
× Rm,k (r0 )r0 ρdr0 dφdρ.
bi iBQM `v- M/ i? i K Bb ` /B HHv@bvKK2i`B+ Ui? i Bb- K(x, y) = UX8V K(kx − yk)V- i?2 qB2M2`ĜE?BM+?BM i?2Q`2K (8) bi i2b i? i *? M;BM; p `B #H2b BM i?2 BMM2`KQbi BMi2;` H BM UX8V iQ i?2 ` /B H TQr2` bT2+i` H /2MbBiv U_Sa.V Q7 f - ;Bp2M #v U8Vθ0 = φ − α − π2 ;Bp2b Bb i?2 6Qm`B2` i` Mb7Q`K Q7 i?2 miQ+Q``2H iBQM KX >2M+2Z π Z π π 0 0 0 0 #v i FBM; i?2 BMp2`b2 6Qm`B2` i` Mb7Q`K Q7 Gei(mθ −ρr cos(φ−θ )) dθ0 = eim(φ− 2 ) ei(ρr sin α−mα) dα −π −π UXeV Z im(φ− π2 ) 0 1 iω·(x−y) 2πJ (ρr ), = e m G(kωk)e dω, UXkV K(kx − yk) = (2π)2 cD r?2`2 Jm Bb i?2 mǶi? Q`/2` "2bb2H 7mM+iBQM Q7 i?2 }`bi FBM/X am#biBimiBM; # +F UXeV BM UX8V ;Bp2b
r?2`2 c Bb i?2 # M/@HBKBi Q7 f X .2MQi2 x = (r cos θ, r sin θ)y = (r0 cos θ0 , r0 sin θ0 )- ω = (ρ cos φ, ρ sin φ) M/ r`Bi2 i?2 bQHmiBQMb Q7 UXRV b 1 ψm,k (r, θ) = √ Rm,k (r)eimθ , 2π
m ∈ Z, k ∈ N.
UXjV
λm,k Rm,k (r)eimθ =
Z
a 0
Z
c
Z
0 0
π
π
ei(m(φ− 2 )+ρr cos(φ−θ)) dφ
−π
× Jm (ρr )G(ρ)Rm,k (r0 )r0 ρdr0 dρ.
am#biBimiBM; UXjV BMiQ UXRV vB2H/b Z
1 2π
λm,k Rm,k (r)eimθ =
!
UXdV
*? M;BM; p `B #H2b BM i?2 BMM2`KQbi BMi2;` H BM UXdV iQ φ = θ − α + π2 ;Bp2b
K(r, θ, r0 , θ0 )Rm,k (r0 )eimθ r0 dr0 dθ0 . UX9V 0
aD
Z
LQi2 i? i UXkV BM TQH ` +QQ`/BM i2b Bb ;Bp2M #v
π
π
ei(m(φ− 2 )+ρr cos(φ−θ)) dφ = eimθ −π
= K(r, θ, r0 , θ0 ) = Z 1 0 0 0 0 G(ρ)ei(ρ cos φ,ρ sin φ)·(r cos θ−r cos θ ,r sin θ−r sin θ ) ρdρdφ = (2π)2 cD Z 1 0 0 G(ρ)ei(ρr cos(φ−θ)−ρr cos(φ−θ )) ρdρdφ, (2π)2 cD
π
Z
ei(ρr sin α−mα) dφ
−π imθ 2πe Jm (ρr).
UX3V
lbBM; UX3V BM UXdV r2 ;2i λm,k Rm,k (r) =
Z
=
Z
a 0
c
! Jm (ρr)Jm (ρr )G(ρ)r ρdρ Rm,k (r0 )dr0 0
0
0
a
hm (r, r0 )Rm,k (r0 )dr0 , 0
r?2`2 hm (r, r0 ) =
3
Z
Rc 0
Jm (ρr)Jm (ρr0 )G(ρ)r0 ρdρX
6B;m`2 N, am`7 +2 pB2rb i F2M 7`QK i?2 j. `2+QMbi`m+iBQMb Q7 SH bKQ/BmK 6 H+BT `mK 3ya UH27iV- β@: H +iQbB/ b2 UKB//H2V- M/ avM TiB+ _:R@_:k U`B;?iVX h?2 iQT bm`7 +2 pB2rb +Q``2bTQM/ iQ `2+QMbi`m+iBQMb 7`QK T `iB+H2b TB+F2/ #v i?2 EGh TB+F2`- M/ i?2 #QiiQK QM2b iQ i?2 `2+QMbi`m+iBQM Q7 (kd)- (k)- M/ (Re)- `2bT2+iBp2HvX h?2 pB2rb r2`2 ;2M2` i2/ mbBM; i?2 l*a6 *?BK2` bQ7ir `2 (R9)X
6B;m`2 Ry, 1t KTH2b Q7 TB+FBM; `2bmHib 7Q` i?2 SH bKQ/BmK 6 H+BT `mK 3ya / i b2iX h?2 KB+`Q;` T?b `2 b?QrM BM i?2 H27i +QHmKM- TB+F2/ T `iB+H2b rBi? i?`2b?QH/ 2[m H iQ x2`Q `2 BM i?2 KB//H2 +QHmKMb- M/ i?2 iQT 8y TB+F2/ T `iB+H2b `2 BM i?2 `B;?i +QHmKMX
TT2M/Bt "X 1biBK iBM; i?2 bQHmiBQM Q7 UdV
h?mb-
AM i?Bb b2+iBQM- r2 QmiHBM2 M H;Q`Bi?K 7Q` 2biBK iBM;
{i0 , j0 } = argmax
Si − S j
1 = argmax
αi − α j γ
i?2 bQHmiBQM Q7 UdV- i? i Bb Q7 {ˆγ, νˆ , α} ˆ = arg min γ˜ ,˜ν,α˜
bXiX γ˜ , ν˜ ∈
i, j
M X i=1
Rl+ ,
i, j
= argmax |αi − α j | kγk1 , i, j
2
||Si − ν˜ − α˜ i γ˜ || ,
U"XRV
r?B+? BKTHB2b
α˜ i ∈ [0, 1].
γˆ (0) = (αK t − αKBM ) · γ.
1[m iBQM U"XRV + M #2 pB2r2/ b M BMbi M+2 Q7 MQM@ M2; iBp2 K i`Bt 7 +iQ`Bx iBQM T`Q#H2K ULLJ6V (e)- r?B+? Bb LS@? `/X PM2 K2i?Q/ 7Q` 2biBK iBM; bQHmiBQMb Q7 LLJ6
aBM+2 r2 2tT2+i i H2 bi QM2 T i+? iQ BM+Hm/2 +QKTH2i2 T `iB+H2- M/ i H2 bi QM2 T i+? iQ +QMi BM QMHv MQBb2- r2 ;2i αK t ≈ 1, αKBM ≈ 0- r?B+? vB2H/b
T`Q#H2Kb Bb #v Hi2`M iBM; H2 bi@b[m `2b UGaV (Ry)X h?2 B/2 #2?BM/ i?Bb Bi2` iBp2 K2i?Q/ Bb bQHpBM;- BM 2 +? bi2Ti?2 LLJ6 KBMBKBx iBQM T`Q#H2K 7Q` QM2 Q7 i?2 p `B #H2b-
γˆ (0) = (αK t − αKBM ) · γ ≈ (1 − 0) · γ = γ.
r?BH2 i`2 iBM; i?2 Qi?2`b b +QMbi MibX A7 i?2 K2i?Q/ +QM@
h? i Bb γˆ (0) Bb BMBiB HBx2/ +HQb2 iQ Bib i`m2 p Hm2X b 7Q` νˆ (0) - U"X9V BKTHB2b
p2`;2b- i?2M Bi +QMp2`;2b iQ HQ+ H KBMBKmK Q7 i?2 LLJ6 T`Q#H2KX AM Qm` + b2- i?2 p `B #H2b `2 i?2 p2+iQ`b γˆ , νˆ , αˆ M/ i?2 Ga Bi2` iBQMb `2 ;Bp2M #v U7Q` Bi2` iBQM iV
{k0 } = argmin kSk k1 = argmin kν + αk γk1 k
{αˆ } = arg min
M X
||Sk − νˆ
{ˆγ(i) } = arg min
M X
||Sk − νˆ (i−1) − αˆ (i) ˜ ||2 , k γ
{ˆν(i) } = arg min
M X
||Sk − ν˜ − αˆ (i) ˆ (i) ||2 , k γ
(i)
α˜
γ˜
ν˜
s.t γ˜ , ν˜ ∈
k=1
k=1
k=1
Rl+ ,
(i−1)
− α˜ k γˆ
νˆ (0) = Sk0 ,
|| ,
k
k
r?2`2 i?2 i?B`/ 2[m HBiv ?QH/b /m2 iQ i?2 TQbBiBpBiv Q7 HH `;mK2MibX b r`Bii2M #Qp2- αKBM ≈ 0 M/ ?2M+2
U"XkV
νˆ (0) ≈ ν + 0 · γ = ν, i? i Bb- νˆ (0) Bb HbQ BMBiB HBx2/ +HQb2 iQ Bib i`m2 p Hm2X L2ti- r2 /2`Bp2 M 2tTHB+Bi 7Q`KmH 7Q` i?2 bQHmiBQM
α˜ k ∈ [0, 1].
Q7 U"XkVX lbBM; # bB+ H;2#` - r2 + M `2r`Bi2 i?2 bmK BM
{i0 , j0 } = argmax
Si − S j
1 ,
U"XjV
i, j
{k0 } = argmin kSi k1 , i
P r?2`2 kuk1 = li=1 |ui | 7Q` u ∈ Rl X hQ DmbiB7v i?Bb +?QB+2`2+ HH i? i S Bb KQ/2H2/ b Si = ν + αi γ,
k
= argmin kνk1 + αk kγk1 = argmin αk ,
(i−1) 2
b i?2 BMBiB H p2+iQ`b γˆ (0) , νˆ (0) r2 b2i γˆ (0) = Si0 − S j0 ,
1
1 ≤ i ≤ M.
U"X9V
N
6B;m`2 RR, 1t KTH2b Q7 TB+FBM; `2bmHib 7Q` i?2 β@: H +iQbB/ b2 / i b2iX a22 6B;m`2 Ry 7Q` /2i BHbX
6B;m`2 Rk, 1t KTH2b Q7 TB+FBM; `2bmHib 7Q` i?2 avM TiB+ _:R@_:k / i b2iX a22 6B;m`2 Ry 7Q` /2i BHbX
r?2`2 7Q` r ∈ R
6B;m`2 Rj, M 2t KTH2 Q7 /2Mb2 T `iB+H2b +Hmbi2` 7Q` r?B+? i?2 EGh TB+F2` TB+F2/ i?2 BMi2`7 +2 #2ir22M T `iB+H2b BMbi2 / Q7 i?2 +2Mi2` Q7 2 +? T `iB+H2X JB+`Q;` T? i F2M 7`QK i?2 1JSA_@Ryyk3 / i b2i (kd)X
[r]+,≤1
U"XRV BM i?`22 2[mBp H2Mi 7Q`Kb M X i=1
M/
U"X8V
kSi − ν˜ − α˜ i γ˜ k2
M M l X M X X 2 X 2 2 (S ) ˜ k = α˜ i γ˜ k − 2 α˜ i ν˜ k + α˜ i Ski γ˜ k + ki − ν k=1
i=1
i=1
i=1
k=1
0 ≤ r ≤ 1, r > 1,
0 [r]+ = r
r < 0, r ≥ 0.
BM U"XeV- Bi ?QH/b i? i
i=1
l l M X l X X X 2 2 2 (Ski − ν˜ k ) . = γ˜ k α˜ i − 2 ν˜ k γ˜ k + γ˜ k Ski α˜ i + i=1
r < 0,
PM2 + M T`Qp2 i? i B7 S + M #2 7 +iQ`Bx2/ b BM U"X9V- M/ i?2 Bi2` iBQMb Q7 U"XeV +QMp2`;2- i?2M BM i?2 HBKBi Q7 i → ∞
i=1
M l M X X X 2 (Ski − α˜ i γ˜ k ) = M ν˜ k − 2 α˜ i γ˜ k + Ski ν˜ k + k=1
0 = r 1
k=1
γˆ = aγ,
k=1
αˆ k =
1 (αk + b) , a
νˆ = ν − bγ,
U"XdV
r?2`2 a, b ∈ R + M #2 2biBK i2/ 7`QK i?2 MQBb2 p `B M+2X
h`2 iBM; γ˜ b p `B #H2 M/ ν˜ , α˜ b +QMbi Mib BM i?2 b2+@ {˜γk }lk=1 X h?2 KBMBKmK Q7 i?Bb bmK- mM/2` i?2 +QMbi` BMi γ˜ k ≥ 0 Bb ii BM2/ i i?2 KBMBKmK Q7 i?2b2 T ` #QH b r?2M@
h?Bb BKTHB2b i? i r2 + M `2+Qp2` γ, α, νX q2 QKBi i?2 /2@ i BHb Q7 i?2 T`QQ7X AM TT2M/Bt *- r2 /2b+`B#2 K2i?Q/ iQ 2biBK i2 i?2 MQBb2 p `B M+2 7`QK ;Bp2M KB+`Q;` T?X h?2 T`Q+2/m`2 7Q` bQHpBM; U"XRV Bb bmKK `Bx2/ BM H;Q`Bi?K kX
r?2M i`2 iBM; ν˜ b i?2 p `B #H2 BM i?2 i?B`/ HBM2 U M/ i?2 Qi?2` b +QMbi MibVX h`2 iBM; α˜ b i?2 p `B #H2 BM i?2 7Q`i?
H;Q`Bi?K k Ga H;Q`Bi?K
QM/ HBM2- ;Bp2b bmK Q7 +QMp2t T ` #QH b 7Q` i?2 p `B #H2b
2p2` i?2v `2 TQbBiBp2 M/ i γ˜ k = 0 2Hb2X h?2 b K2 TTHB2b
HBM2- BKTHB2b i? i i?2 KBMBKmK Bb ;Bp2M b 7QHHQrX 6Q` 2 +? T ` #QH BM i?2 bmK U"X8V- B7 Bib KBMBKmK Bb Q#i BM2/ i
R,
M2; iBp2 α˜ k - i?2M i?2 KBMBKmK Q7 i?2 bmK Bb ii BM2/ i α˜ k = 0X A7 i?2 KBMBKmK Bb Q#i BM2/ i α˜ k > 1- r2 b2i α˜ k = 1X Pi?2`rBb2- i?2 KBMBKmK Q7 i?2 bmK Bb ii BM2/
k, j, 9, 8,
i i?2 KBMBKmK Q7 i?2 T ` #QH X hQ +QM+Hm/2- r2 ;2i i?2 7QHHQrBM; 7Q`KmH b 7Q` i?2 bQHmiBQMb Q7 i?2 iǶi? Bi2` iBQM
e,
BM U"XkV Pl k=1 γˆ k(i−1) · Sk, j − νˆ k(i−1) αˆ (i) , 1 ≤ j ≤ M, Pl j = (i−1) γ ˆ k=1 k +,≤1 P M (i) k=1 αˆ k · S j,k − νˆ (i−1) j (i) , γˆ j = 1 ≤ j ≤ l, P M (i) ˆk k=1 α + PM k=1 S j,k − αˆ (i) ˆ (i) j k γ (i) , νˆ j = 1 ≤ j ≤ l, M
_2[mB`2/, J i`Bt SX *QKTmi2 i?2 BMBiB H p2+iQ`b νˆ (0) , γˆ (0) mbBM; U"XjV M/ αˆ (0) , αˆ (1) , νˆ (1) , γˆ (1) mbBM; U"XeVX i=1
r?BH2
νˆ (i) − νˆ (i−1)
,
γˆ (i) − γˆ (i−1)
,
αˆ (i) − αˆ (i−1)
≥ /Q *QKTmi2 νˆ (i+1) , γˆ (i+1) , αˆ (i+1) mbBM; U"XeV i=i+1 2M/ r?BH2 PmiTmi, νˆ (i) , γˆ (i) , αˆ (i)
6BM HHv- r2 `2im`M iQ i?2 +QKTmi iBQM Q7 i?2 +QHmKMb Q7 SX 1 +? +QHmKM Bb i?2 _Sa. Q7 KB+`Q;` T? T i+?
Q7 bBx2 l × l rBi? bi`B/2 d- b /2b+`B#2/ i i?2 #2;BMMBM; Q7 a2+iBQM kXkX h?2b2 T ` K2i2`b `2 /2i2`KBM2/ b 7QH@
U"XeV
HQrbX AM U"X9V r2 bbmK2 i? i S + M #2 7 +iQ`Bx2/ b
+QK#BM iBQM Q7 QMHv ν, γ, αX >Qr2p2`- i?Bb KQ/2H Bb QMHv M TT`QtBK iBQM- M/ BM T` +iB+2 Bi ?QH/b i? i
+
S = ν + αT γ + E, Ry
U"X3V
r?2`2 E Bb M 2``Q` i2`K K i`BtX lM/2` i?2 bbmKTiBQMb i? i i?2 2tT2+i M+v Q7 i?2 `Qrb Q7 E Bb x2`Q- M/ i? i i?2
Ub22 HbQ a2+iBQM kXjVX h?2 E `?mM2M GQ2p2 h?2Q`2K (RR) bi i2b i? i
+QHmKMb Q7 E `2 mM+Q``2H i2/ M/ rBi? i?2 b K2 p `B@
M+2- QM2 + M T`Qp2 i? i i?2 HBKBi Q7 U"XeV r?2M mbBM; S Q7 U"X3V bi vb i?2 b K2 b BM U"XdV- #mi rBi? /Bz2`2Mi p H@
f (x) =
∞ X
zi ψi (x),
i=1
m2b 7Q` a M/ b Ur2 QKBi i?2 T`QQ7VX h?2b2 p Hm2b + M #2
h i r?2`2 ψi `2 ;Bp2M #v URV- M/ KQ`2Qp2`- E zi z j = δi, j λi X h?2`27Q`2- i?2 miQ+Q``2H iBQM K(x, y) = E f (x) f (y) + M #2 r`Bii2M b ∞ ∞ X X E f (x) f (y) = E zi ψi (x) z j ψ j (y)
2biBK i2/ 7`QK i?2 MQBb2 p `B M+2- M/ i?2`27Q`2- r2 + M `2+Qp2` γ, α, ν 7Q` i?2 KQ`2 ;2M2` H KQ/2H U"X3VX LQr- i F@ BM; d < l +H2 `Hv #`2 Fb i?2 bbmKTiBQMb 7Q` E M/ i FBM;
d > l /Q2b MQi mb2 HH Q7 i?2 / i - ?2M+2 r2 mb2/ d = lX b 7Q` i?2 T ` K2i2` l- B/2 HHv r2 rQmH/ HBF2 T i+? iQ +QM@
i=1
i BM MQ KQ`2 i?2M QM2 +QKTH2i2 T `iB+H2X h FBM; l Km+? #B;;2` i?2M i?2 T `iB+H2 /B K2i2` + M + mb2 T i+?2b iQ +QM@ i BM irQ Q` KQ`2 T `iB+H2b- M/ i FBM; Bi Km+? bK HH2` rBHH
=
∞ X ∞ X
=
∞ X ∞ X
i=1 j=1
MQi HHQr T i+?2b iQ +QMi BM 2MQm;? Q7 i?2 T `iB+H2X AM T` +iB+2- Bi b22Kb i? i i FBM; l 2[m H 3yW Q7 i?2 T `iB+H2 /B K2i2` vB2H/b ;QQ/ `2bmHibX
j=1
h i ψi (x)ψ j (y)E zi z j ψi (x)ψ j (y)δi, j λi
i=1 j=1
=
∞ X
ψi (x)ψi (y)λi .
i=1
TT2M/Bt *X 1biBK iBM; i?2 MQBb2 p `B M+2
lbBM; U.XRV r2 ;2i
h?2 MQBb2 p `B M+2 Bb M22/2/ iQ `2+Qp2` γ, α, ν 7`QK U"XdV M/ iQ 2p Hm i2 UNVX bbmKBM; i? i #Qi? i?2 MQBb2 M/
h i E PPT
i?2 T `iB+H2 7mM+iBQM `2 ;2M2` i2/ 7`QK rB/2 b2Mb2 bi @ iBQM `v ` M/QK T`Q+2bb M/ i? i i?2v `2 mM+Q``2H i2/- Bi 7QHHQrb i? i i?2 p `B M+2 Q7 ;Bp2M KB+`Q;` T? T i+? Bb
(i, j)
∞ h i X = E Pi P j = E f (i) f ( j) = ψk (i)ψk ( j)λk . k=1
U.XkV
AM Q`/2` iQ /2}M2 i`mM+ iBQM `mH2 7Q` i?2 BM}MBi2 bmK
i?2 bmK Q7 i?2 MQBb2 p `B M+2 M/ 7` +iBQM Q7 i?2 T `iB@
BM U.XkV- r2 +QMbB/2` i?2 `2H iBp2 2``Q` i2`K
+H2Ƕb p `B M+2- r?B+? Bb T`QTQ`iBQM H iQ i?2 7` +iBQM Q7 i?2 T `iB+H2 +QMi BM2/ BM i?2 T i+?X b i?2 p `B M+2 Bb TQb@
P
P∞ PN
k=1 ψk ψk λk − k=1 ψk ψk λk
∞ k=N+1 ψk ψk λk
P∞
=
P∞
k=1 ψk ψk λk
k=1 ψk ψk λk
P∞ λk , = Pk=N+1 ∞ λ k=1 k
BiBp2 MmK#2`- T i+?2b rBi? HQr2` p `B M+2 `2 KQ`2 HBF2Hv iQ +QMi BM QMHv MQBb2X qBi? i? i BM KBM/- r2 T `iBiBQM i?2 KB+`Q;` T? BMiQ Qp2`H TTBM; T `iB+H2@bBx2 T i+?2b rBi? bi`B/2 2[m H iQ 1X h?2M- r2 TT`QtBK i2 i?2 p `B M+2 Q7 2 +? T i+? b 2tTH BM2/ BM (ke) M/ }M HHv 2biBK i2 i?2
U.XjV U.X9V
r?2`2 || · || Bb i?2 L2 (aD × aD) bi M/ `/ MQ`K M/ U.X9V 7QHHQrb 7`QK U.XjV /m2 iQ Q`i?Q;QM HBiv Q7 ψk X h?2M- r2 /2}M2 i?2 7QHHQrBM; i`mM+ iBQM `mH2X 6B`bi r2 i`mM+ i2 i?2
MQBb2 p `B M+2 #v p2` ;BM; x% Q7 i?2 HQr2bi p `B M+2 `2@ bmHibX *?QQbBM; x /2T2M/b QM i?2 MmK#2` Q7 T `iB+H2b BM i?2 KB+`Q;` T?- r?B+? Bb mMFMQrM iQ mbX q2 ? p2 Q#@
BM}MBi2 bmK i M i2`Kb r?2`2 M Bb +?Qb2M bm+? i? i
b2`p2/ 2tT2`BK2Mi HHv i? i i FBM; x = 25% T`QpB/2b b iBb@
λ M − λ M+1 ≤ 10−6 , λM
7 +iQ`v `2bmHib BM HQr@aL_ +QM/BiBQMbX
M/ i?2M- r2 i`mM+ i2 Bi ; BM iQ N i2`Kb bm+? i? i TT2M/Bt .X .2`BpBM; i?2 TT`QtBK iBQM URyV
PM
k=N+1
AM i?Bb b2+iBQM- r2 /2`Bp2 M TT`QtBK iBQM 7Q` i?2 T `@ h i iB+H2Ƕb +Qp `B M+2 K i`Bt E PPT - r?2`2 Pi = f (i),
i∈U
PM
k=1
U.XRV RR
λk
λk
≤ 0.01.
U.X8V
"v bi M/ `/ HBM2 ` H;2#` - Bi + M #2 b?QrM i? i i?2 i`mM@ + iBQM `mH2 Q7 U.X8V vB2H/b i?2 /2bB`2/ TT`QtBK iBQM Q7 URyV
K i`Bt M/ 7 +iQ`Bx2 Bi mbBM; i?2 Z_ /2+QKTQbBiBQM (kR) b
∞ N X h i X E PPT = ψk (i)ψk ( j)λk ≈ ψk (i)ψk ( j)λk = ΨΛΨT k=1
k=1
Ψ = QR,
U.XeV
r?2`2 Ψ M/ Λ `2 /2}M2/ BM a2+iBQM kXjX
r?2`2 Q ∈ Rb2a+1c
2
×b2a+1c2
TT2M/Bt 1X .2`Bp iBQM M/ +QKTmi iBQM Q7 URRV Pm` ;Q H Bb iQ +QKTmi2 i?2 b+Q`2 ! p(Ii, j |H1 ) . p(Ii, j |H0 )
U1XRV
LQi2 i? i RΛRT + σ2 I Bb #HQ+F /B ;QM H K i`Bt Q7 i?2 7Q`K H 0 T 2 , RΛR + σ I = 0 σ 2 I2
iBp `B i2 MQ`K H- M/ i? i i?2 MQBb2 Bb r?Bi2- i?2 T`Q# @ #BHBiv /2MbBiv 7mM+iBQM US.6V Q7 KB+`Q;` T? T i+? +QM@ i BMBM; T `iB+H2 THmb MQBb2 Bb 1 T −1 1 e− 2 I Σ I , |2πΣ|
˜ R˜ T + σ2 I1 Bb Q7 bBx2 N × N- I1 Bb i?2 A/2MiBiv r?2`2 H = RΛ
U1XkV
K i`Bt Q7 i?2 b K2 bBx2 b H- M/ I2 Bb i?2 B/2MiBiv K i`Bt
Q7 bBx2 (b2a + 1c2 − N) × (b2a + 1c2 − N)X h?2 /2i2`KBM Mi Q7 Σˆ Bb ˆ = |H|σb2a+1c2 −N , U1X3V |Σ|
r?2`2 | · | Bb K i`Bt /2i2`KBM MiX h?2 S.6 Q7 T i+? rBi? MQBb2 QMHv Bb
p (I|H0 ) = p
1
1
(2πσ2 )b2a+1c2
T
e− 2σ2 I I .
U1XjV
M/ Bib BMp2`b2 Bb
am#biBimiBM; i?2 #Qp2 S.6b BM U1XRV M/ `2TH +BM; i?2 ` M/QK p `B #H2 I rBi? Bib (i, j) T i+? `2 HBx iBQM Ii, j ;Bp2b
ˆ −1
Σ
Ub22 HbQ URRVV b+Q`2(i, j) =
!
AM //BiBQM-
!
T 1 |Σ| log 2b2a+1c2 + Ii, j σ−2 I − Σ−1 Ii, j . 2 σ U1X9V
Σ = E PP
i
am#biBimiBM; i?2 H ii2` # +F BM U1X8V vB2H/b
ˆ + σ I ≈ ΨΛΨ + σ I := Σ. 2
2
T
σ−2 I1 − H −1 0 T Q I IT σ−2 I − Σˆ −1 I = IT Q 0 0 U1XNV T σ−2 I1 − H −1 0 T T Q I. = Q I 0 0
ˆ pB i?2 bBM;mH ` p Hm2b /2+QKTQbB@ *QKTmiBM; Σˆ −1 M/ |Σ|
iBQM Uao.V (kR) i F2b O(b2a + 1c2 N 2 ) QT2` iBQMb T2` KB+`Q@ ;` T?- r?2`2 N Bb ;Bp2M #v i?2 i`mM+ iBQM `mH2 Q7 U.X8VX h?2 +QKTmi iBQM H +Qbi Q7 /B`2+iHv +QKTmiBM; Pi, j =
ITi, j
−2
ˆ −1
σ I−Σ
0 T Q . σ−2 I2
I1 0 H −1 0 T −1 −2 ˆ Q σ I − Σ = σ − Q 0 I2 0 σ−2 I2 σ−2 I1 − H −1 0 T Q . = Q 0 0
i?2 T `iB+H2Ƕb +Qp `B M+2 K i`Bt- i? i BbT
H −1 = Q 0
−2
q2 2biBK i2 Σ 7Q` U1X9V mbBM; i?2 TT`QtBK iBQM URyV 7Q`
h
Bb M Q`i?Q;QM H K i`Bt M/ R ∈
Σˆ = Q RΛRT + σ2 I QT .
lM/2` i?2 bbmKTiBQMb i? i i?2 T `iB+H2 Bb /Bbi`B#mi2/ KmH@
p (I|H1 ) = √
U1XdV
Rb2a+1c ×N Bb M mTT2` i`B M;mH ` K i`Bt r?Qb2 H bi b2a + R˜ 1c2 − N `Qrb `2 x2`QX q2 /2MQi2 R = r?2`2 R˜ ∈ RN×N 0 M/ bm#biBimi2 U1XdV BM U1XeV- `2bmHiBM; BM 2
b+Q`2(i, j) := log
U1XeV
Σˆ = ΨΛΨT + σ2 I,
Ii, j ,
.2MQi2 i?2 }`bi N +QQ`/BM i2b Q7 i?2 p2+iQ` QT I #v vX
U1X8V
h?2M- i?2 H bi 2tT`2bbBQM Bb bBKTHB}2/ iQ
Bb O(b2a + 1c6 ) QT2` iBQMb T2` T i+?X AM Q`/2` iQ `2/m+2
IT σ−2 I − Σˆ −1 I = σ−2 vT v − vT H −1 v.
i?2 H ii2` +QKTH2tBiv- +QMbB/2` i?2 TT`QtBK i2 +Qp `B M+2 Rk
U1XRyV
am#biBimiBM; U1X3V M/ U1XRyV BM U1X9V M/ +? M;BM; I iQ Ii, j ;Bp2b b+Q`2(i, j) =
! ! 1 σ2N log + σ−2 vT v − vT H −1 v . 2 |H|
(Rk)
U1XRRV
(Rj)
h?2 +QKTH2tBiv H pB i?2 Z_ /2+QKTQbB@ Q7 +QKTmiBM; iBQM Q7 Σˆ Bb O b2a + 1c2 N 2 QT2` iBQMbX h?2M- +QKTmiBM; H −1 M/ |H| pB S V D `2[mB`2b O(N 3 ) QT2` iBQMb- #Qi? +QK@ Tmi2/ T2` KB+`Q;` T?X *QKTmiBM; vT H −1 v `2[mB`2b O(N 3 ) QT2` iBQMb- M/ +QKTmiBM; vT v `2[mB`2b O(N 2 ) QT2` iBQMb#Qi? +QKTmi2/ T2` T i+?X 6BM HHv- BM Q`/2` iQ +QKTmi2 i?2 p2+iQ` v- r2 M22/ iQ +QKTmi2 QMHv i?2 }`bi N +QQ`/BM i2b Q7 QT IX h?Bb + M #2 BKTH2K2Mi2/ mbBM; O b2a + 1c2 N QT2` iBQMb T2` T i+?X hQ +QM+Hm/2- i?2 T2` T i+? +QKTm@ 6 i iBQM H +QKTH2tBiv ? b #22M `2/m+2/ 7`QK O(b2a + 1c ) iQ 2 3 O b2a + 1c N + N X AM T` +iB+2- i?Bb bT22/b mT i?2 H;Q@ `Bi?K #v i?`22 Q`/2`b Q7 K ;MBim/2X
(R9)
(R8) (Re)
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(Rd)
(R3)
(RN)
(ky)
(kR) (kk) (kj) (k9)
(k8)
(ke) (kd)
Rj
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7 H+BT `mK 3ya `B#QbQK2 #QmM/ iQ i?2 MiB@T`QiQxQ M /`m; 2K2@ iBM2X 2GB72- j- CmM2 kyR9X AaaL ky8y@y39sX /QB, RyXd889f2GB72X yjy3yX l_G ?iiT,ff2m`QT2TK+XQ`;f `iB+H2bfSJ*9y3ekd8X (k3) uB7 M sB Q M/ :m M;r2M u M;X 7 bi K2i?Q/ 7Q` T `iB+H2 TB+FBM; BM +`vQ@2H2+i`QM KB+`Q;` T?b # b2/ QM 7 bi _@*LLX AM AS +QM72`2M+2 T`Q+22/BM;b- pQHmK2 R3je- T ;2 ykyy3yX AS Sm#HBb?BM;- kyRdX
R9
S1047-8477(20)30035-6 https://doi.org/10.1016/j.jsb.2020.107473 YJSBI 107473
To appear in:
Journal of Structural Biology
Received Date: Revised Date: Accepted Date:
11 December 2019 21 January 2020 24 January 2020
Please cite this article as: Eldar, A., Landa, B., Shkolnisky, Y., KLT picker: Particle picking using data-driven optimal templates, Journal of Structural Biology (2020), doi: https://doi.org/10.1016/j.jsb.2020.107473
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© 2020 Published by Elsevier Inc.
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mi?Q` 1K BH //`2bb2b, KBi v2H/ `!K BHXi mX +XBH UKBi v 1H/ `V#Q`BbXH M/ !v H2X2/m U"Q`Bb G M/ V- vQ2Hb?!i m2tXi mX +XBH UuQ2H a?FQHMBbFvV S`2T`BMi bm#KBii2/ iQ 1Hb2pB2`
a2KB@ miQK iB+ H;Q`Bi?Kb `2[mB`2 i?2 mb2` iQ b2H2+i K Mm HHv ?mM/`2/b iQ i?Qmb M/b Q7 T `iB+H2b r?B+? `2 mb2/ BM p `BQmb r vb BM Q`/2` iQ miQK iB+ HHv /2i2+i KQ`2 T `iB+H2bX _1GAPL (R3) M/ aA:Lhl_1 (9) mb2 K Mm@ HHv TB+F2/ T `iB+H2b b i2KTH i2b M/ +Q``2H i2 i?2K ; BMbi i?2 KB+`Q;` T?b BM Q`/2` iQ +QKTmi2 bi iBbiB+ H bBKBH `Biv K2 bm`2bX h?2 K BM B/2 Bb i? i +`Qbb@+Q``2H iBQM #2ir22M i2KTH i2 M/ T `iB+H2 BK ;2 rBi? //2/ MQBb2 rBHH #2 H `;2` i? M +`Qbb@+Q``2H iBQM rBi? MQBb2 QMHvX MQi?2` T@ T`Q +?- ; BMBM; TQTmH `Biv `2+2MiHv- Bb i?2 mb2 Q7 K2i?Q/b 7`QK i?2 }2H/ Q7 /22T H2 `MBM;X hPSw (j) mb2b K Mm HHv b2H2+i2/ T `iB+H2b iQ i` BM +QMpQHmiBQM H M2m` H M2irQ`Fr?2`2 sKBTT (ky) mb2b i?2K iQ i` BM M 2Mb2K#H2 Q7 +H b@ bB}2`bX AM #Qi? + b2b- i?2 M2irQ`F M/ i?2 2Mb2K#H2 `2 H i2` mb2/ iQ /2i2+i KQ`2 T `iB+H2bX *QKKQM iQ HH b2KB@ miQK iB+ H;Q`Bi?Kb Bb i?2 M22/ 7Q` ?B;? [m HBiv b2i Q7 K Mm HHv TB+F2/ T `iB+H2bX JQbi Q7 i?2K +QmH/ HbQ #2M2}i 7`QK +QHH2+iBQM Q7 BK ;2b i? i +QMi BM QMHv MQBb2X miQK iB+ H;Q`Bi?Kb /Q MQi `2[mB`2 i?2 mb2` iQ K M@ m HHv b2H2+i T `iB+H2bX L2p2`i?2H2bb- i?2v /Q `2[mB`2 bQK2 T`BQ` FMQrH2/;2 #Qmi i?2 K +`Q@KQH2+mH2X JQbi Q7 i?2 H;Q`Bi?Kb `2[mB`2 i H2 bi M 2biBK i2 Q7 i?2 T `iB+H2 bBx2 UBX2X i?2 K tBK H /B K2i2` Q7 i?2 T `iB+H2V M/ bQK2 Q7 i?2K `2[mB`2 i?2 mb2` iQ /Dmbi HH bQ`ib Q7 T ` K2i2`bX h?2 .Q: TB+F2` (k9) Bb # b2/ QM +QMpQHpBM; /Bz2`2M+2 Q7 : mbbB M 7mM+iBQMb rBi? KB+`Q;` T? BK ;2X h?Bb K2i?Q/ Bb bmBi #H2 7Q` B/2MiB7vBM; #HQ#b BM i?2 BK ;2 M/ iQ bQ`i 62#`m `v 8- kyky
i?2K #v i?2B` bBx2bX h?2 SSG1 TB+F2` (3) mb2b +`Qbb@ +Q``2H iBQM #2ir22M KB+`Q;` T? T i+?2b BM Q`/2` iQ /2i2+i
BM; `2bmHib- r2 T2`7Q`K j. `2+QMbi`m+iBQM 7Q` 2 +? / i b2i- M/ +QKT `2 Qm` `2+QMbi`m+iBQM iQ i?2 Tm#HBb?2/ QM2
`2 b i? i `2 KQ`2 HBF2Hv iQ +QMi BM T `iB+H2X h?2b2 `@
mbBM; i?2 ;QH/@bi M/ `/ 6Qm`B2` a?2HH *Q``2H iBQM U6a*V
2 b `2 i?2M #2BM; mb2/ iQ i` BM bmTTQ`i p2+iQ` K +?BM2r?B+? Bb mb2/ iQ /2i2+i T `iB+H2bX .22T H2 `MBM; K2i?@
+`Bi2`BQM (kj)X h?2 / i b2ib mb2/ `2 SH bKQ/BmK 6 H+B@ T `mK 3ya `B#QbQK2 (kd)- β@: H +iQbB/ b2 (k) M/ avM TiB+
Q/b `2 HbQ ; BMBM; TQTmH `Biv BM i?2 miQK iB+ + i2;Q`vX
_:R@_:k (Re)X h?2 +Q/2 7Q` i?2 EGh TB+F2` Bb p BH@
.22TSB+F2` (k8) M/ 7 bi _@*LL (k3) mb2 j. `2+QMbi`m+@ iBQMb Q7 FMQrM K +`Q@KQH2+mH2b b H #2H2/ / i b2ib iQ
#H2 i ?iiTb,ff;Bi?m#X+QKf KBi v2H/ `fEGhTB+F2`X
i` BM M2m` H M2irQ`Fb i? i `2 H i2` mb2/ iQ TB+F T `iB+H2bX
kX J i2`B Hb M/ K2i?Q/b
AM i?Bb T T2`- r2 T`2b2Mi i?2 EGh UE `?mM2M GQ2p2 h` Mb7Q`KV TB+F2`- M miQK iB+ T `iB+H2 TB+FBM; K2i?Q/
q2 M2ti /2i BH i?2 /Bz2`2Mi bi2Tb BM Qm` T `iB+H2 TB+FBM;
BM r?B+? i?2 QMHv T ` K2i2` `2[mB`2/ 7`QK i?2 mb2` Bb
H;Q`Bi?KX
M 2biBK i2 Q7 i?2 T `iB+H2Ƕb bBx2X Pm` K2i?Q/ Bb # b2/ QM +QKTmiBM; 7Q` 2 +? KB+`Q;` T? b2i Q7 / i @ / TiBp2
kXRX *QKTmiBM; / i / TiBp2 T `iB+H2 i2KTH i2b G2i a ∈ R+ #2 i?2 T `iB+H2Ƕb ` /Bmb M/ H2i f ∈ L2 (R2 ) #2 7mM+iBQM `2T`2b2MiBM; +2Mi2`2/ T `iB+H2 r?Qb2 2M2`;v Bb
i2KTH i2b- r?B+? `2 QTiBK H BM i?2 b2Mb2 i? i i?2v /KBi K tBK H +Q``2H iBQM rBi? i?2 T `iB+H2 BK ;2bX //BiBQM@ HHv- i?2v `2 M im` HHv / Ti2/ iQ +QTBM; rBi? i?2 mM@
+QM+2Mi` i2/ BM /BbF Q7 ` /Bmb aX q2 +QMbB/2` f iQ #2 ` M/QK 7mM+iBQM- rBi? bQK2 T`Q# #BHBiv /Bbi`B#miBQM + T@ im`BM; i?2 p `B #BHBiv Q7 T `iB+H2 BK ;2bX 6Q` i?2 Tm`TQb2
FMQrM BM@TH M2 `Qi iBQMb Q7 i?2 T `iB+H2b- b i?2 #bQHmi2 p Hm2 Q7 i?2B` +Q``2H iBQM rBi? T `iB+H2 BK ;2 Bb `Qi iBQM@ HHv BMp `B MiX h?2 #Qp2@K2MiBQM2/ T`QT2`iB2b Q7 Qm` i2K@
Q7 T `iB+H2 TB+FBM;- r2 b22F i?2 b2i Q7 Q`i?Q;QM H 7mM+iBQMb i? i K tBKBx2 i?2B` +Q``2H iBQM rBi? i?2 T `iB+H2 BK ;2bX aT2+B}+ HHv-
TH i2b 2p2Mim HHv HHQr iQ /2i2+i T `iB+H2b mM/2` HQr aL_ +QM/BiBQMbX h?2 2biBK iBQM Q7 i?2b2 i2KTH i2b Bb +?B2p2/ i?`Qm;? i?2 E `?mM2M GQ2p2 h` Mb7Q`K (RR) #v miBHBxBM;
D E ψn = argmax E f, ψˆ
i?2 ` /B H TQr2` bT2+i` H /2MbBiv U_Sa.V (Rj) Q7 i?2 T `@
ˆ 2 (R2 ) ψ∈L
iB+H2bX lbBM; i?2b2 i2KTH i2b ;Bp2b `Bb2 iQ b+Q`2 7Q` 2 +? T i+? BM i?2 KB+`Q;` T?- # b2/ QM i?2 r2HH FMQrM GBF2@ HB?QQ/ _ iBQ h2bi UG_hV (ke)X aT2+B}+ HHv- i?2 H;Q`Bi?K T `iBiBQMb 2 +? KB+`Q;` T? BMiQ Qp2`H TTBM; T i+?2b- M/
D E
ˆ ψj bXiX
ψˆ
L2 (aD) = 1, ψ,
L2 (aD)
L2 (aD)
2 .
URV
= 0, j < n ∈ N,
r?2`2 aD Bb i?2 /BbF Q7 ` /Bmb a- h·, ·iL2 (aD) /2MQi2b i?2 bi M@ p / `/ BMM2` T`Q/m+i QM L2 (aD)- M/ k·kL2 (aD) = h·, ·iL2 (aD) Bb i?2 TT`QT`B i2 MQ`KX AM rQ`/b- ψ1 Bb i?2 7mM+iBQM r?B+?
bbB;Mb iQ 2 +? Q7 i?2K b+Q`2 i? i `2~2+ib i?2 T`Q# #BHBiv i? i Bi +QMi BMb T `iB+H2X h?2 i?`2b?QH/ T`Q# #BHBiv 7Q` /2i2+iBM; T `iB+H2b Bb b2i 2Bi?2` iQ i?2 i?2Q`2iB+ H i?`2b?QH/
K tBKBx2b i?2 +Q``2H iBQM rBi? i?2 T `iB+H2 7mM+iBQM f - ψ2 Bb i?2 7mM+iBQM r?B+? K tBKBx2b i?2 +Q``2H iBQM rBi? f
U2tTH BM2/ #2HQrV- Q`- i?2 mb2` + M /Dmbi i?2 i?`2b?QH/ iQ
r?BH2 #2BM; Q`i?Q;QM H iQ ψ1 - M/ bQ QMX h?2b2 7mM+@
TB+F bK HH2` MmK#2` Q7 T i+?2b rBi? i?2 ?B;?2bi b+Q`2b U?B;? T`Q# #BHBiv 7Q` i?2 T`2b2M+2 Q7 T `iB+H2VX AM //B@
iBQMb `2 i?2 `2bmHib Q7 i?2 E `?mM2M GQ2p2 h` Mb7Q`K UEGhV (RR) Q7 i?2 T `iB+H2 7mM+iBQM f X q2 /2MQi2 i?2 K t@ BK H 2tT2+i2/ p Hm2 ii BM2/ BM i?2 `B;?i@? M/ bB/2 Q7 URV
iBQM- i?2 i?`2b?QH/ T`Q# #BHBiv + M #2 b2i iQ TB+F T i+?2b i? i `2 KQ`2 HBF2Hv iQ +QMi BM QMHv MQBb2 UT i+?2b rBi? i?2 HQr2bi b+Q`2bVX PM2 Q7 i?2 /p Mi ;2b Q7 Qm` K2i?Q/ Bb
#v λn - BX2X
i? i Bi + M +QKTH2K2Mi Qi?2` b2KB@ miQK iB+ H;Q`Bi?Kb-
2 λn := E h f, ψn iL2 (aD) .
i? i `2[mB`2 M BMBiB H b2i Q7 i` BMBM; BK ;2bX 6Q` 2t KTH2i?2 mb2` + M /2+B/2 iQ TB+F QMHv i?2 Ry ?B;?2bi M/ HQr@
UkV
λn /2b+`B#2b i?2 BKTQ`i M+2 Q7 2p2`v 7mM+iBQM ψn BM b2`p@ BM; b i2KTH i2 7Q` T `iB+H2 TB+FBM;- b +Q``2H iBM; ψn +Q``2bTQM/BM; iQ H `;2 λn ; BMbi MQBbv T `iB+H2 BK ;2
2bi b+Q`2/ T i+?2b Qmi Q7 2 +? KB+`Q;` T? M/ mb2 i?2K BMbi2 / Q7 M BMBiB H K Mm H TB+FBM; bi2TX L2p2`i?2H2bb-
Bb 2tT2+i2/ iQ T`Q/m+2 H `;2` p Hm2 UQM p2` ;2V i? M
i?2 EGh TB+F2` Bb 7mHHv + T #H2 Q7 T`27Q`KBM; i?2 2MiB`2
+Q``2H iBM; Bi ; BMbi Tm`2 MQBb2 Ur?B+? rQmH/ T`Q/m+2 i?2 b K2 +Q``2H iBQM `2; `/H2bb Q7 i?2 i2KTH i2 mb2/VX
T `iB+H2 TB+FBM; bi2T- b r2 /2KQMbi` i2 QM i?`22 /Bz2`@ 2Mi / i b2ib BM a2+iBQM jX AM Q`/2` iQ 2p Hm i2 i?2 TB+F@ k
aQHpBM; URV UmM/2` bbmKTiBQMb /2i BH2/ #2HQrV `2[mB`2b i?2 ` /B H TQr2` bT2+i` H /2MbBiv U_Sa.V (Rj) Q7 f X AM i?2
6B;m`2 R, h?2 }`bi Re 7mM+iBQMb ψn U7Q` i?2 H `;2bi λn V Q#i BM2/ 7`QK bQHpBM; U9V mbBM; i?2 _Sa. 2biBK iBQM T`Q+2/m`2 /2b+`B#2/ BM a2+iBQM kXkX h?2 2B;2M7mM+iBQMb M/ 2B;2Mp Hm2b r2`2 +QKTmi2/ 7`QK bBM;H2 ivTB+ H KB+`Q;` T? Q7 i?2 1JSA_@Ryyk3 / i b2i (kd)- M/ r2`2 MQ`K HBx2/ bQ i? i i?2 H `;2bi 2B;2Mp Hm2 Bb QM2X
M2ti b2+iBQM- r2 T`2b2Mi K2i?Q/ iQ 2biBK i2 i?2 _Sa. mM/2` i?2 bbmKTiBQMb i? i f Bb # M/@HBKBi2/ M/ Bb ;2M2`@ i2/ 7`QK rB/2 b2Mb2 bi iBQM `v ` M/QK T`Q+2bbX lM/2` i?2b2 bbmKTiBQMb- r2 b?Qr BM TT2M/Bt i? i i?2 bQHm@
6B;m`2 k, h?2 }`bi 8yy 2B;2Mp Hm2b λn UbQ`i2/ BM MQM@BM+`2 bBM; Q`@ /2`V Q#i BM2/ 7`QK bQHpBM; U9V mbBM; i?2 _Sa. 2biBK iBQM T`Q+2/m`2 /2b+`B#2/ BM a2+iBQM kXkX h?2 `2/ / b?2/ HBM2 Bb MQ`K HBx2/ +m@ KmH iBp2 bmK Q7 i?2 2B;2Mp Hm2bX h?2 2B;2Mp Hm2b r2`2 +QKTmi2/ 7`QK bBM;H2 ivTB+ H KB+`Q;` T? Q7 i?2 1JSA_@Ryyk3 / i b2i (kd) M/ r2`2 MQ`K HBx2/ bQ i? i i?2 H `;2bi 2B;2Mp Hm2 Bb QM2X Ai Bb 2pB@ /2Mi i? i i?2 2B;2Mp Hm2b /2+ v ` TB/Hv- M/ 300 2B;2M7mM+iBQMb `2 H`2 /v 2MQm;? iQ + Tim`2 KQ`2 i? M 90% Q7 i?2 p `B #BHBiv BM i?2 T `iB+H2 BK ;2b UbBM+2 i?2 MQ`K HBx2/ +mKmH iBp2 bmK Q7 i?2 }`bi jyy 2B;2Mp Hm2b 2t+22/b 0.9VX
iBQMb iQ URV- BM TQH ` +QQ`/BM i2b- `2 Q7 i?2 7Q`K 1 ψm,k (r, θ) = √ Rm,k (r)eimθ , 2π
m ∈ Z, k ∈ N,
UjV
r?2`2 Rm,k (r) Bb i?2 kǶi? 2B;2M7mM+iBQM Q7 i?2 BMi2;` H 2[m @ iBQM λm,k Rm,k (r) =
Z
a
hm (r, r0 )Rm,k (r0 )dr0 ,
U9V kXkX _ /B H TQr2` bT2+i` H /2MbBiv 2biBK iBQM
0
rBi? {λm,k }k #2BM; i?2 2B;2Mp Hm2b +Q``2bTQM/BM; iQ {Rm,k }k -
AM i?Bb b2+iBQM- r2 /2b+`B#2 K2i?Q/ 7Q` 2biBK iBM; i?2 T `iB+H2Ƕb _Sa. G Ub22 U8VV 7`QK ;Bp2M KB+`Q;` T?X hQ@ r `/b i?Bb 2M/- r2 T `iBiBQM i?2 KB+`Q;` T? BMiQ M UTQb@
M/ hm (r, r ) Bb ;Bp2M #v 0
hm (r, r0 ) =
Z
c
G(ρ)Jm (ρr0 )Jm (ρr)r0 ρdρ,
bB#Hv Qp2`H TTBM;V T i+?2b Q7 bBx2 l×l rBi? bi`B/2 dX aT2+B7@ B+ HHv- B7 A Bb KB+`Q;` T?- i?2M i?2 }`bi T i+? Bb /2}M2/
0
r?2`2 Jm Bb i?2 mǶi? Q`/2` "2bb2H 7mM+iBQM Q7 i?2 }`bi FBM/-
#v P1 = A(1 : l, 1 : l)- i?2 b2+QM/ #v P2 = A(1 : l, 1 + d : l + d)
c Bb i?2 # M/@HBKBi Q7 f - M/ G Bb i?2 _Sa. Q7 f - ;Bp2M 2tTHB+BiHv b G(ρ) =
1 2π
Z
2π
E| fˆ(ρ, θ)|2 dθ,
M/ bQ QM U7Q` /2i BH2/ 2tTH M iBQM QM ?Qr l M/ d `2 +?Qb2M b22 TT2M/Bt "VX h?2M- 7Q` 2 +? T i+? r2 2biBK i2 BiǶb _Sa. i l 2[m HHv@bT +2/ TQBMib b 2tTH BM2/ BM (Rj) M/ 7Q`K i?2 K i`Bt S ∈ Rl×M r?Qb2 i0 i? +QHmKM Si Bb +
U8V
0
i?2 _Sa. +QKTmi2/ 7`QK i?2 iǶi? T i+?X q2 +QMbB/2` i?2
r?2`2 fˆ = F ( f ) Bb i?2 irQ@/BK2MbBQM H 6Qm`B2` i` Mb7Q`K Q7 f X Ai Bb rQ`i?r?BH2 iQ TQBMi Qmi i? i ψm,k Q7 UjV `2
7QHHQrBM; KQ/2H 7Q` i?2 _Sa. Q7 i?2 T i+?2b BM ;Bp2M KB+`Q;` T?,
bi22` #H2 7mM+iBQMb (kk)- bBM+2 +Q``2H iBM; i2KTH i2 Q7 i?2 7Q`K Q7 UjV rBi? M `#Bi` `BHv `Qi i2/ BK ;2 ? b #bQHmi2 p Hm2 r?B+? Bb BM/2T2M/2Mi Q7 i?2 `Qi iBQMX
Si = ν + αi γ,
AM Q`/2` iQ bQHp2 U9V M/ 2p Hm i2 Rm,k - QM2 M22/b iQ
UeV
r?2`2 ν ∈ Rl+ `2 i?2 b KTH2b Q7 i?2 MQBb2Ƕb _Sa.- M/ γ ∈ Rl+ `2 i?2 b KTH2b Q7 G Q7 U8V Ui?2 T `iB+H2Ƕb _Sa.VX
FMQr i?2 T `iB+H2Ƕb _Sa. GX AM i?2 7QHHQrBM; b2+iBQM r2 /2b+`B#2 K2i?Q/ iQ 2biBK i2 Bi 7`QK ;Bp2M KB+`Q;` T?X
q2 bbmK2 i? i ν `2K BMb i?2 b K2 +`Qbb HH T i+?2b M/ i? i γ Bb KmHiBTHB2/ #v 7 +iQ` 0 ≤ αi ≤ 1 BM 2 +? T i+?- r?B+? ++QmMib 7Q` i?2 T`QTQ`iBQM Q7 i?2 T `iB+H2
h?2M- 2[mBTT2/ rBi? M 2biBK i2 7Q` G- r2 mb2 Lvbi`ƺKǶb K2i?Q/ (R) BM Q`/2` iQ bQHp2 U9V MmK2`B+ HHvX hQ bBKTHB7v bm#b2[m2Mi MQi iBQM- r2 bQ`i HH 2B;2Mp Hm2b λm,k U7Q` HH
+QMi BM2/ BM i?2 iǶi? T i+?X h? i Bb- T i+? r?B+? +QMi BMb
m M/ kV BM MQM@BM+`2 bBM; Q`/2`- M/ /2MQi2 i?2 bQ`i2/
MQ T `iB+H2 /KBib α = 0- T i+? r?B+? 7mHHv +QMi BMb T `iB+H2 /KBib α = 1- M/ T i+? r?B+? +QMi BMb T `i Q7
2B;2Mp Hm2b #v λn X q2 bQ`i i?2 2B;2M7mM+iBQMb ++Q`/BM;HvX 6B;m`2 R BHHmbi` i2b i?2 }`bi Re i2KTH i2b ψn UbQ`i2/ +@ +Q`/BM; iQ i?2B` +Q``2bTQM/BM; 2B;2Mp Hm2b λn V Q#i BM2/
T `iB+H2 /KBib α 2[m H iQ i?2 T`QTQ`iBQM Q7 i?2 T `iB+H2 BM i?2 T i+?X ++Q`/BM; iQ UeV- r2 + M 2biBK i2 ν- γ M/ M {αi }i=1 #v bQHpBM;
7`QK i?2 1JSA_@Ryyk3 / i b2i (kd)X 6B;m`2 k b?Qrb # ` THQi Q7 i?2 500 H `;2bi 2B;2Mp Hm2b UkV 7Q` i?2 b K2 / i b2i U+QKTmi2/ pB U9VVX
{ˆγ, νˆ , α} ˆ = arg min γ˜ ,˜ν,α˜
bXiX γ˜ , ν˜ ∈ j
M X i=1
Rl+ ,
||Si − ν˜ − α˜ i γ˜ ||2 , α˜ i ∈ [0, 1],
UdV
r?2`2 I ∈ Rb2a+1c Bb ~ ii2M2/ KB+`Q;` T? T i+?- N ∈ 2 2 Rb2a+1c Bb ~ ii2M2/ BK ;2 Q7 i?2 MQBb2 QMHv- P ∈ Rb2a+1c Bb 2
6B;m`2 j, S `iB+H2 M/ MQBb2 _Sa. 2biBK iBQM T`Q+2/m`2X q2 /BpB/2 i?2 KB+`Q;` T? BMiQ UTQbbB#Hv Qp2`H TTBM;V l × l T i+?2b r?2`2 l Bb H2bb i? M Q` 2[m H iQ i?2 K tBK H /B K2i2` Q7 i?2 T `iB+H2X 6Q` 2 +? T i+? r2 +QKTmi2 Bib _Sa. M/ 7 +iQ`Bx2 Bi b bmK Q7 i?2 MQBb2 _Sa. M/ i?2 T `iB+H2 _Sa. KmHiBTHB2/ #v T`QTQ`iBQM 7 +iQ` 0 ≤ α ≤ 1X
~ ii2M2/ T `iB+H2 BK ;2 rBi?Qmi MQBb2 M/ rBi? x2`Q p H@
m2b QmibB/2 Q7 aD- M/ b·c Bb i?2 ~QQ` 7mM+iBQMX q2 bbmK2 i? i i?2 MQBb2 Bb r?Bi2- r?B+? Bb 2M7Q`+2/ #v T`2r?Bi2MBM; i?2 KB+`Q;` T? mbBM; i?2 b KTH2b Q7 i?2 2biBK i2/ MQBb2
6B;m`2 9, >2 i K T Q7 {αˆ i } Qp2` i?2 KB+`Q;` T? BiǶb 2biBK i2/ 7`QKX h?2 p Hm2b Q7 {αˆ i } r2`2 Q#i BM2/ #v bQHpBM; UdV 7Q` ivTB+ H KB+`Q@ ;` T? Q7 i?2 1JSA_@Ryyk3 / i b2i (kd)- M/ r2`2 MQ`K HBx2/ bQ i? i i?2B` H `;2bi p Hm2 Bb RX AM Q`/2` iQ +`2 i2 i?2 ?2 i K T- αˆ i r b mb2/ iQ +QHQ` i?2 iǶi? T i+?X b 2tTH BM2/ BM a2+iBQM kXk- H `;2 p Hm2b 7Q` αˆ i U#`B;?i +QHQ`V BM/B+ i2 i?2 T`2b2M+2 Q7 T `iB+H2- M/ bK HH p Hm2b U/ `F +QHQ`V BM/B+ i2 Bib #b2M+2X
_Sa. νˆ UQ#i BM2/ BM a2+iBQM kXkVX :Bp2M i?2 KQ/2H U3V- r2 i2bi 7Q` i?2 ?vTQi?2b2b H0 M/ H1 #v i?2 r2HH@FMQrM GBF2@ HB?QQ/ _ iBQ h2bi UG_hV (ke)X q2 /2`Bp2 i?2 G_h 7Q` i?2 + b2 r?2`2 P M/ N `2 MQ`K HHv /Bbi`B#mi2/- M/ /2KQM@ bi` i2 i?2 T` +iB+ H 2z2+iBp2M2bb Q7 i?2 `2bmHiBM; i2bi BM
r?2`2 ||·|| Bb i?2 bi M/ `/ 1m+HB/2 M MQ`KX a22 6B;m`2 j 7Q`
a2+iBQM jX hQ /2`Bp2 i?2 G_h BM i?Bb + b2- r2 M22/ i?2 +Q@ h i p `B M+2 K i`B+2b 7Q` #Qi? ?vTQi?2b2b- M K2Hv E IIT |H0 i h M/ E IIT |H1 X bbmKBM; i? i P M/ N `2 BM/2T2M/2Mi-
M BHHmbi` iBQM Q7 i?2 _Sa. 2biBK iBQM T`Q+2/m`2X LQi2
i? i UdV + M #2 pB2r2/ b M BMbi M+2 Q7 LQM@L2; iBp2 J i`Bt 6 +iQ`Bx iBQM ULLJ6V (e) Q7 i?2 K i`Bt SX AM T@
Bi 7QHHQrb i? i
T2M/Bt "- r2 /2b+`B#2 K2i?Q/ 7Q` 2biBK iBM; i?2 bQHm@
h i h i Σ ≡ E IIT |H1 = E PPT + σ2 I,
iBQM Q7 UdV #v mbBM; Hi2`M iBM; H2 bi@b[m `2b (Ry)- r?B+? Bb TQTmH ` TT`Q +? 7Q` LLJ6X q2 BHHmbi` i2 ivTB+ H
r?2`2 σ Bb i?2 bi M/ `/ /2pB iBQM Q7 i?2 MQBb2- M/ I Bb i?2
p Hm2b Q7 αˆ i 7`QK i?2 7 +iQ`Bx iBQM UdV BM 6B;m`2 9X Ai Bb 2p@ B/2Mi i? i αˆ i Bb H `;2` BM `2 b Q7 i?2 KB+`Q;` T? r?B+? `2
b2a + 1c2 × b2a + 1c2 B/2MiBiv K i`BtX aBM+2 σ Bb ivTB+ HHv
mMFMQrM- r2 T`QTQb2 K2i?Q/ 7Q` 2biBK iBM; Bi 7`QK ;Bp2M KB+`Q;` T? BM TT2M/Bt *X hQ 2p Hm i2 UNV Bi i?2`2@ i h 7Q`2 `2K BMb iQ 2biBK i2 E PPT X G2i U #2 M 2[m HHv@
/2Mb2Hv T +F2/ rBi? T `iB+H2b- BM/B+ iBM; i? i i?2 7 +iQ`@ Bx iBQM 2ti` +ib i?2 _Sa. Q7 i?2 T `iB+H2b 7`QK i?2 +Q``2+i `2;BQMb Q7 i?2 KB+`Q;` T?X
bT +2/ U* `i2bB MV ;`B/ rBi? b2a + 1c × b2a + 1c TQBMib Qp2` 2 i?2 b[m `2 b−a, ac × b−a, ac- M/ /2MQi2 #v U ∈ Rb2a+1c i?2
kXjX S `iB+H2 /2i2+iBQM
~ ii2M2/ p2`bBQM Q7 UX "v b2iiBM; i?2 p Hm2b Q7 f QmibB/2 Q7 aD iQ #2 x2`Q- r2 ? p2 i? i Pi = f (i) 7Q` 2p2`v i ∈ UX a2iiBM; HbQ i?2 p Hm2b Q7 ψn Q7 URV QmibB/2 Q7 aD iQ x2`Q-
AM i?2 T`2pBQmb b2+iBQMb- r2 /2b+`B#2/ ?Qr iQ +QKTmi2 b2i Q7 i2KTH i2b {ψn } UiQ;2i?2` rBi? i?2B` 2B;2Mp Hm2b {λn } /2b+`B#BM; i?2B` BKTQ`i M+2V / Ti2/ iQ i?2 T `iB+H2
M/ /2MQiBM; #v Ψ ∈ Rb2a+1c
i?2 K i`Bt r?Qb2 nǶi? +QH@ mKM +QMbBbib Q7 i?2 ~ ii2M2/ b KTH2b Q7 ψn QM U- H2 /b iQ i?2 7QHHQrBM; TT`QtBK iBQM Q7 i?2 T `iB+H2Ƕb +Qp `B M+2
BK ;2b BM i?2 KB+`Q;` T?X >Qr2p2`- i i?Bb TQBMi Bi Bb
mM+H2 ` ?Qr iQ K F2 mb2 Q7 i?2 /Bz2`2Mi i2KTH i2bX AM T `iB+mH `- bBM+2 bBM;H2 i2KTH i2 /Q2b MQi + Tim`2 i?2 #2 mb2/X q2 b?QmH/ i?2`27Q`2 /2i2`KBM2 ?Qr K Mv i2K@ TH i2b iQ mb2 M/ ?Qr iQ +QK#BM2 i?2K 7Q` T `iB+H2 TB+F@
URyV
N r?2`2 Λ Bb /B ;QM H K i`Bt rBi? i?2 2B;2Mp Hm2b {λn }n=1 QM Bib K BM /B ;QM HX h?2 T ` K2i2` N Bb +?Qb2M bm+? i? i i?2 2``Q` BM i?2 TT`QtBK iBQM URyV Bb bK HH- r?B+?
M/ /2`Bp2 b+?2K2 r?2`2 /Bz2`2Mi i2KTH i2b ψn `2 +QK@ #BM2/ ++Q`/BM; iQ i?2B` `2H iBp2 BKTQ`i M+2 Ui?`Qm;? i?2 2B;2Mp Hm2b λn VX aT2+B}+ HHv- r2 M2ti /2`Bp2 T`Q+2/m`2
2bb2MiB HHv /2T2M/b QM i?2 /2+ v ` i2 Q7 i?2 2B;2Mp Hm2b
7Q` /Bb+`BKBM iBM; #2ir22M M BK ;2 Q7 Tm`2 MQBb2- r?B+? Bb /2}M2/ b i?2 MmHH ?vTQi?2bBb M/ /2MQi2/ #v H0 - M/
λn Ub22 TT2M/Bt . 7Q` /2i BH2/ 2tTH M iBQM QM ?Qr i?2 T ` K2i2` N Bb /2i2`KBM2/VX :Bp2M i?2 +Qp `B M+2 K i`Bt Σ Q7 UNV U+QKTmi2/ pB i?2
M BK ;2 Q7 T `iB+H2 THmb MQBb2- r?B+? Bb /2}M2/ b i?2 Hi2`M iBp2 ?vTQi?2bBb M/ /2MQi2/ #v H1 X h? i Bb- r2 +QMbB/2` i?2 KQ/2H
H1 : T `iB+H2 THmb MQBb2-
×N
i h E PPT ≈ ΨΛΨT ,
BM;X hQ i? i 2M/- r2 im`M iQ +H bbB+ H ?vTQi?2bBb i2biBM;
H0 : Tm`2 MQBb2-
2
K i`Bt Ub22 TT2M/Bt .V
p `B #BHBiv Q7 HH T `iB+H2 BK ;2b- KmHiBTH2 i2KTH i2b Kmbi
N, I= P + N,
UNV
TT`QtBK iBQM URyVV- r2 mb2 b+Q`2 # b2/ QM i?2 G_h 7Q` +QKT `BM; #2ir22M irQ KmHiBp `B i2 MQ`K H /Bbi`B#miBQMb b 7QHHQrbX 1 +? KB+`Q;` T? T i+? Ii, j Q7 bBx2 b2a + 1c ×
U3V
b2a+1c U~ ii2M2/ iQ p2+iQ`V Bb +QMbB/2`2/ b `2 HBx iBQM
9
H;Q`Bi?K R h?2 EGh TB+F2`
6B;m`2 8, 1t KTH2 Q7 ivTB+ H b+Q`BM; K i`Bt b ?2 i K T U`B;?i BK ;2V M/ T `iB+H2b TB+F2/ 7`QK Bi UH27i BK ;2V rBi? i?`2b?QH/ Q7 x2`QX h?2 / i b2i Bb 1JSA_@Ryyk3 (kd)X
R,
Q7 I M/ Bb bbB;M2/ b+Q`2
k,
! p(Ii, j |H1 ) b+Q`2(i, j) := log URRV p(Ii, j |H0 ) ! ! T |Σ| 1 = log 2b2a+1c2 + Ii, j σ−2 I − Σ−1 Ii, j , 2 σ
j, 9, 8,
r?2`2 | · | Bb K i`Bt /2i2`KBM MiX a22 TT2M/Bt . 7Q`
M 2H #Q` i2 /2`Bp iBQM Q7 i?Bb b+Q`2 M/ M 2{+B2Mi H;Q@
`Bi?K iQ +QKTmi2 BiX >B;? p Hm2 Q7 b+Q`2(i, j) BM/B+ i2b i?2 T`2b2M+2 Q7 T `iB+H2 BM i?2 (i, j) T i+?X q2 +QMbB/2` i?2
e,
b+Q`2b b K i`Bt r?Qb2 2H2K2Mib `2 b+Q`2(i, j)X 1ti` +i@
d,
BM; i?2 T `iB+H2Ƕb +QQ`/BM i2b Qmi Q7 i?2 b+Q`BM; K i`Bt Bb /QM2 #v mbBM; p2`bBQM Q7 i?2 MQM@K tBKmK bmTT`2b@
3,
bBQM H;Q`Bi?K (Rk)X h?Bb ;`22/v H;Q`Bi?K }`bi +?QQb2b i?2 ?B;?2bi b+Q`2 BM i?2 b+Q`BM; K i`Bt b i?2 }`bi TB+F2/ T `iB+H2Ƕb +2Mi2`X h?2M- BM Q`/2` iQ T`2p2Mi M2 `#v TBt@
N,
_2[mB`2/, JB+`Q;` T? M/ i?2 T `iB+H2bǶ ` /Bmb aX S `iBiBQM i?2 KB+`Q;` T? BMiQ l × l T i+?2b rBi? bi`B/2 d BM 2 +? /BK2MbBQM U#v /27 mHi- l Bb +?Qb2M iQ #2 3yW Q7 i?2 K tBK H /B K2i2` Q7 i?2 T `iB+H2 M/ d = lb22 TT2M/Bt " 7Q` KQ`2 /2i BHbVX *QKTmi2 i?2 _Sa. Q7 2 +? T i+? b /2b+`B#2/ BM b2+@ iBQM kXkX 1biBK i2 i?2 T `iB+H2 M/ MQBb2 _Sa. γˆ , νˆ - `2bT2+iBp2Hv #v bQHpBM; UdVX S`2r?Bi2M i?2 KB+`Q;` T? mbBM; νˆ X *QKTmi2 i?2 b2i Q7 2B;2M7mM+iBQMb {ψˆn } M/ 2B;2Mp Hm2b {λn } Q7 URV #v bQHpBM; U9V mbBM; γˆ M/ i?2 Lvbi`ƺK K2i?Q/X 1biBK i2 i?2 T `iB+H2Ƕb +Qp `B M+2 K i`Bt E[PPT ] +@ +Q`/BM; iQ URyV M/ i?2 MQBb2 bi M/ `/ /2pB iBQM σ- b /2i BH2/ BM TT2M/Bt *Xi h *QKTmi2 Σ = E IIT |H1 Ub22 UNVVX S `iBiBQM i?2 KB+`Q;` T? BMiQ [2a + 1] × [2a + 1] T i+?2b rBi? bi`B/2 1 BM 2 +? /BK2MbBQM M/ +QKTmi2 i?2 G_h b+Q`2 7Q` 2 +? T i+? ++Q`/BM; iQ URRVX h?2 Qmi+QK2 Bb b+Q`BM; K i`BtX 1ti` +i i?2 T `iB+H2bǶ +QQ`/BM i2b 7`QK i?2 b+Q`BM; K @ i`Bt b /2b+`B#2/ i i?2 2M/ Q7 a2+iBQM kXjX
2Hb 7`QK #2BM; +QMbB/2`2/ b T `iB+H2b- HH TBt2Hb rBi?BM mb2`@/2}M2/ ` /Bmb `2 2t+Hm/2/ 7`QK i?2 TB+FBM;- M/ i?2 T`Q+2/m`2 `2bmK2b rBi? i?2 M2ti H `;2bi b+Q`2- M/ bQ QMX
i?2 ?B;?2bi 7`2[m2M+B2b r2`2 2HBKBM i2/ bBM+2 i?2v +QMi BM KQbiHv MQBb2X h?`Qm;?Qmi HH Q7 i?2 2tT2`BK2Mib- r2 mb2/ i?2 7QHHQrBM; T ` K2i2`b 7Q` i?2 EGh TB+F2`, a M/ l `2
q2 b2i i?2 2t+HmbBQM ` /Bmb iQ #2 i?2 T `iB+H2 bBx2- ?Qr2p2`bK HH2` p Hm2b K v ;Bp2 #2ii2` `2bmHib 7Q` +HQb2Hv T +F2/B``2;mH `Hv b? T2/ T `iB+H2bX h?2 TB+FBM; +QMiBMm2b mMiBH i?2 `2K BMBM; b+Q`2b `2 #2HQr x2`Q Q` mMiBH i?2 H;Q`Bi?K
kyW M/ 3yW- `2bT2+iBp2Hv- Q7 i?2 K tBK H /B K2i2` Q7
TB+Fb T`2/2}M2/ MmK#2` Q7 T `iB+H2bX aiQTTBM; r?2M i?2 `2K BMBM; b+Q`2b `2 M2; iBp2 Bb # b2/ QM i?2 Q#b2`p iBQM
bK HH2` i? M ? H7 Q7 i?2 2biBK i2/ T `iB+H2Ƕb bBx2- b i?Bb
i?2 T `iB+H2- d = l- M/ b 2tTH BM2/ #Qp2- i?2 /2i2+@ iBQM i?`2b?QH/ Bb b2i iQ x2`QX LQi2 i? i a Bb i F2M iQ #2
i? i b+Q`2(i, j) < 0 ?QH/b r?2M2p2` p(Ii, j |H1 ) < p(Ii, j |H0 )-
H2 /b- BM T` +iB+2- iQ #2ii2` +2Mi2`2/ T `iB+H2 b2H2+iBQMbX h?2 Q#i BM2/ T `iB+H2 +QQ`/BM i2b r2`2 BKTQ`i2/ BMiQ _1@
KTH2 Q7 b+Q`BM; K i`Bt M/ i?2 T `iB+H2b TB+F2/ 7`QK
BM Q`/2` iQ Q#i BM j. `2+QMbi`m+iBQMX h?2 1J." (R8) KQ/2H +Q``2bTQM/BM; iQ 2 +? / i b2i r b mb2/ b i?2 #
r?B+? K2 Mb i? i HH Q7 i?2 `2K BMBM; T i+?2b ? p2 ?B;?2` T`Q# #BHBiv iQ +QMi BM MQBb2 QMHvX a22 6B;m`2 8 7Q` M 2t@
GAPL (R3) M/ i?2 `QmiBM2 /2b+`B#2/ BM (Rd) r b 2t2+mi2/
BiX a22 6B;m`2 e M/ H;Q`Bi?K R 7Q` /B ;` K M/ M H;Q`Bi?K i? i bmKK `Bx2 i?2 T `iB+H2 TB+FBM; T`Q+2bbX
BMBiBQ KQ/2H 7Q` i?2 j. `2}M2K2Mi T`Q+2bbX h?2 T`Q+2/m`2 /2b+`B#2/ #Qp2 Bb /2KQMbi` i2/ QM i?2 7QHHQrBM; / i b2ib Q7 i?2 1JSA_ `2TQbBiQ`v (N), 1JSA_@ Ryyk3 USH bKQ/BmK 6 H+BT `mK 3ya `B#QbQK2V (kd)- 1JSA_@
jX 1tT2`BK2Mi H `2bmHib
RyyeR Uβ@: H +iQbB/ b2V (k)- M/ 1JSA_@Ryy9N UavM T@ iB+ _:R@_:kV (Re)X _2bmHib 7Q` / i b2i Q7 .mM HB2HH
AM i?Bb b2+iBQM- r2 /2KQMbi` i2 i?2 T2`7Q`K M+2 Q7 Qm` H;Q`Bi?K QM i?`22 /Bz2`2Mi / i b2ib- b 7QHHQrbX 6B`bi- i?2
a HBM S?QiQbvbi2K A amT2`+QKTH2t `2 b?QrM BM i?2 amT@ TH2K2Mi `v J i2`B HX h?Bb H ii2` / i b2i +Q``2bTQM/b iQ
KB+`Q;` T?b r2`2 /QrMb KTH2/ iQ kyW@8yW Q7 i?2B` Q`B;B@ M H bBx2 M/ r2`2 MQ`K HBx2/ iQ x2`Q K2 M M/ p `B M+2 Q7
M Q#D2+i r?Qb2 bT2+i ` iBQ Bb Km+? H `;2` i? M Q7 i?2
QM2X AM //BiBQM- 2 +? KB+`Q;` T? r b # M/T bb }Hi2`2/-
i?`22 #Qp2@K2MiBQM2/ 1JSA_ / i b2ibX Pm` H;Q`Bi?K r b BKTH2K2Mi2/ BM J iH #- M/ HH i?2
2HBKBM iBM; 8W Q7 Bib HQr2bi M/ ?B;?2bi ` /B H 7`2[m2M@ +B2bX h?2 HQr2bi 7`2[m2M+B2b r2`2 2HBKBM i2/ bBM+2 i?2v /Q
2tT2`BK2Mib r2`2 2t2+mi2/ QM GBMmt K +?BM2 rBi? Re
MQi +QMi BM 2MQm;? b KTH2b 7Q` 2biBK iBM; i?2 _Sa.- M/ 8
6B;m`2 e, 6Q` 2 +? KB+`Q;` T?- r2 2biBK i2 i?2 T `iB+H2 M/ MQBb2 _Sa. b 2tTH BM2/ BM a2+iBQM kXkX h?2M- r2 T`2r?Bi2M i?2 KB+`Q;` T? M/ 2biBK i2 i?2 i2KTH i2b mbBM; U9VX G bi- r2 +QKTmi2 i?2 G_h b+Q`BM; K i`Bt M/ 2ti` +i i?2 T `iB+H2bǶ +QQ`/BM i2b 7`QK Bi- b /2b+`B#2/ i i?2 2M/ Q7 a2+iBQM kXjX
+Q`2b `mMMBM; i kXR:>x- de3:" Q7 K2KQ`v- M/ M LpB/B
#v +QKT `BM; i?2 Qmi+QK2 Q7 _1GAPLǶb TQbi@T`Q+2bbBM;
hBi M :SlX h?2 +QKTH2i2 bQm`+2 +Q/2 7Q` i?2 EGh TB+F2`
i bF iQ i?2 ;`QmM/@i`mi?X Ai Bb BKTQ`i Mi iQ MQi2 i? i
Bb p BH #H2 i ?iiTb,ff;Bi?m#X+QKf KBi v2H/ `fEGhTB+F2`r?B+? HbQ BM+Hm/2b HBMF iQ bi M/@ HQM2 2t2+mi #H2 BK@
i?2 KB+`Q;` T?bǶ ` r KQpB2b r2`2 MQi p BH #H2 iQ mb- M/ bQ #2ii2` `2bmHib K v #2 +?B2p2/ #v 7m`i?2` TQHBb?BM;X AM
TH2K2Mi iBQM Q7 i?2 H;Q`Bi?KX
+QKT `BbQM- (k) TB+F2/ 93, 686 T `iB+H2b mbBM; M miQK iB+
jXRX SH bKQ/BmK 6 H+BT `mK 3ya
K2i?Q/ # b2/ QM +Q``2H iBQM rBi? : mbbB M `BM;X h? M j. +H bbB}+ iBQM BM 6_1GA:L (d) r b mb2/ iQ /Bb+ `/
h?2 SH bKQ/BmK 6 H+BT `mK 3ya / i b2i +QMbBbib Q7
# / b2H2+iBQMb- M/ 41, 123 T `iB+H2b r2`2 `2i BM2/ 7Q` i?2
1081 KB+`Q;` T?b Q7 bBx2 4096 × 4096 TBt2Hb- rBi? TBt2H bBx2
j. `2}M2K2Mi bi2T Ui? i r b 2t2+mi2/ BM i?2 b K2 T`Q@ ;` KVX h?2 `2+QMbi`m+iBQM BM (k) `2 +?2/ ;QH/ bi M/ `/
Q7 1.34 X h?2 TT`QtBK i2 bBx2 Q7 i?2 T `iB+H2 Bb 360 × 360 TBt2Hb U ++Q`/BM; iQ (kd)VX 1t KTH2b Q7 i?2 EGh TB+FBM;
6a* `2bQHmiBQM Q7 2.2X 6B;m`2 N T`2b2Mib bm`7 +2 pB2rb Q7 i?2 j. KQ/2HbX h?2 `mMMBM; iBK2 Q7 i?2 EGh TB+F2` 7Q`
`2bmHib `2 b?QrM BM 6B;m`2 RyX h?2 EGh TB+F2` TB+F2/ 141, 679 T `iB+H2b 7`QK i?2 KB+`Q;` T?bX AM Q`/2` iQ Tm`;2 i?2 b2i Q7 TB+F2/ T `iB+H2b- k. +H bb p2` ;BM; bi2T r b
bBM;H2 KB+`Q;` T?- mbBM; :Sl- r b TT`QtBK i2Hv 33
2t2+mi2/ UmbBM; 30 +H bb2bV BM _1GAPL M/ 87, 439 T `@
b2+QM/b U7QHHQrBM; QM2@iBK2 T`2T`Q+2bbBM; bi2T Q7 TT`Qt@ BK i2Hv 40 b2+QM/bVX
iB+H2b r2`2 `2i BM2/X 1t KTH2b Q7 i?2 b2H2+i2/ k. +H bb2b `2 b?QrM BM 6B;m`2 dX h?2 j. `2+QMbi`m+iBQM `2 +?2/
jXjX avM TiB+ _:R@_:k h?2 avM TiB+ _:R@_:k +QMbBbib Q7 e3y KB+`Q;` T?b-
;QH/ bi M/ `/ 6Qm`B2` a?2HH *Q``2H iBQM U6a*V `2bQHmiBQM Q7 3.05X 6B;m`2 3 T`2b2Mib i?2 6a* +m`p2 T`Q/m+2/ #v +QK@
2 +? Q7 bBx2 3838 × 3710 TBt2Hb- rBi? TBt2H bBx2 Q7 RXkjX
h?2 TT`QtBK i2 bBx2 Q7 i?2 T `iB+H2 ++Q`/BM; iQ (Re) Bb 192 × 192 TBt2HbX h?2 EGh TB+F2` TB+F2/ 163, 470 T `iB@
T `BM; i?2 Qmi+QK2 Q7 _1GAPLǶb TQbi@T`Q+2bbBM; i bF iQ i?2 ;`QmM/@i`mi?X AM +QKT `BbQM- 158, 212 T `iB+H2b r2`2 TB+F2/ #v (kd) mbBM; _1GAPLǶb miQK i2/ TB+FBM; iQQH-
+H2bX 1t KTH2b Q7 i?2 `2bmHib Q7 i?2 TB+FBM; `2 b?QrM BM
6B;m`2 RkX 7i2` 2t2+miBM; k. +H bb p2` ;BM; bi2T Umb@ BM; 30 +H bb2bV BM _1GAPL- 34, 100 T `iB+H2b r2`2 `2i BM2/X
M/ 7i2` k. M/ j. +H bb p2` ;BM; bi2Tb- 105, 247 T `iB@ +H2b `2K BM2/X h?2 `2+QMbi`m+iBQM BM (kd) `2 +?2/ ;QH/ bi M/ `/ 6a* `2bQHmiBQM Q7 3.20X 6B;m`2 N T`2b2Mib bm`7 +2
1t KTH2b Q7 i?2 b2H2+i2/ k. +H bb2b `2 b?QrM BM 6B;m`2 dX h?2 j. `2+QMbi`m+iBQM `2 +?2/ ;QH/ bi M/ `/ 6a* `2bQ@ HmiBQM Q7 3.87X 6B;m`2 3 T`2b2Mib i?2 6a* +m`p2 T`Q/m+2/
pB2rb Q7 Qm` j. `2+QMbi`m+i2/ KQ/2H M/ i?2 `2+QMbi`m+i2/ KQ/2H Q7 (kd)X h?2 `mMMBM; iBK2 Q7 i?2 EGh TB+F2` 7Q` bBM;H2 KB+`Q;` T?- mbBM; :Sl- r b TT`QtBK i2Hv 35 b2+QM/b U7QHHQrBM; QM2@iBK2 T`2T`Q+2bbBM; bi2T Q7 TT`Qt@
#v +QKT `BM; i?2 Qmi+QK2 Q7 _1GAPLǶb TQbi@T`Q+2bbBM; i bF iQ i?2 ;`QmM/@i`mi?X q2 MQi2 i? i bBKBH `Hv iQ i?2 β@
BK i2Hv 40 b2+QM/bVX
: H +iQbB/ b2 / i b2i- i?2 ` r KQpB2b r2`2 mM p BH #H2 iQ
jXkX β@: H +iQbB/ b2
mb- ?2M+2 i?2 `2bmHib `2 bm#@QTiBK HX AM +QKT `BbQM- (Re) mb2/ b2KB@ miQK iB+ K2i?Q/ 2t2+mi2/ BM aJl1G (RN)
h?2 β@: H +iQbB/ b2 / i b2i +QMbBbib Q7 R8jN KB+`Q@
7Q` i?2 T `iB+H2 TB+FBM; bi2TX AMBiB HHv- 2, 000 T `iB+H2b r2`2
;` T?b- 2 +? Q7 bBx2 7676 × 7420 TBt2Hb- rBi? TBt2H bBx2 Q7 yXe9X h?2 TT`QtBK i2 bBx2 Q7 i?2 T `iB+H2 ++Q`/BM; iQ (k)
K Mm HHv b2H2+i2/ M/ +H bbB}2/ #v k. +H bb p2` ;BM; bi2TX :QQ/ +H bb2b r2`2 K Mm HHv +?Qb2M M/ i?2 T `iB+H2b
Bb 768 × 768 TBt2HbX EGh TB+F2` TB+F2/ 100, 241 T `iB+H2bX
BM i?Qb2 +H bb2b r2`2 mb2/ b i2KTH i2b 7Q` M miQK iB+ TB+FBM; T`Q+2bbX b /2b+`B#2/ BM (Re)- //BiBQM H K Mm H 2zQ`i r b `2[mB`2/ iQ /Bb+ `/ # / b2H2+iBQMb- r?B+? `2@
1t KTH2b Q7 i?2 `2bmHib Q7 i?2 TB+FBM; `2 b?QrM BM 6B;@ m`2 RRX 7i2` 2t2+miBM; k. +H bb p2` ;BM; bi2T UmbBM; 30 +H bb2bV BM _1GAPL- 13, 000 T `iB+H2b r2`2 `2i BM2/X 1t@
bmHi2/ BM }M H / i b2i Q7 89, 158 T `iB+H2bX h?2 j. `2@
KTH2b Q7 i?2 b2H2+i2/ k. +H bb2b `2 b?QrM BM 6B;m`2 dX h?2 j. `2+QMbi`m+iBQM `2 +?2/ ;QH/ bi M/ `/ 6a* `2bQ@
}M2K2Mi bi2T r b 2t2+mi2/ BM _1GAPL- M/ i?2 `2+QM@ bi`m+iBQM `2 +?2/ ;QH/ bi M/ `/ 6a* `2bQHmiBQM Q7 3.4X
HmiBQM Q7 3.01X 6B;m`2 3 T`2b2Mib i?2 6a* +m`p2 T`Q/m+2/
6B;m`2 N T`2b2Mib bm`7 +2 pB2rb Q7 i?2 j. KQ/2HbX h?2 e
`mMMBM; iBK2 Q7 i?2 EGh TB+F2` 7Q` bBM;H2 KB+`Q;` T?mbBM; :Sl- r b TT`QtBK i2Hv 50 b2+QM/b U7QHHQrBM;
EGh TB+F2` Bb BM #2ir22M #Qi? + i2;Q`B2bX PM QM2 ? M/Bi `2HB2b QM i?2 bbmKTiBQM i? i i?2 T`QD2+iBQM BK ;2b + M
QM2@iBK2 T`2T`Q+2bbBM; bi2T Q7 TT`QtBK i2Hv 40 b2+QM/bVX
#2 TT`QtBK i2/ #v # M/@HBKBi2/- `Qi iBQM HHv BMp `B Mi-
9X .Bb+mbbBQM
rB/2 b2Mb2 bi iBQM `v T`Q+2bb `2bi`B+i2/ iQ /BbFX h?Bb bbmKTiBQM Bb MQM@`2bi`B+iBp2 BM T` +iB+2- b 2t2KTHB}2/ #v Qm` `2bmHibX PM i?2 Qi?2` ? M/- i?2 i2KTH i2b Q7 i?2
9XRX _2pB2r Q7 i?2 `2bmHib
EGh TB+F2` `2 / Ti2/ iQ 2p2`v KB+`Q;` T? b2T ` i2Hvi?`Qm;? i?2 _Sa. Q7 i?2 T `iB+H2b M/ Q7 i?2 MQBb2X
h?2 / i b2ib mb2/ iQ /2KQMbi` i2 Qm` K2i?Q/ `2T@ `2b2Mi /Bz2`2Mi ivT2b Q7 KB+`Q;` T?bX h?2 SH bKQ/BmK 6 H+BT `mK 3ya Bb i?2 H2 bi +? HH2M;BM;- b i?2 K +`Q@
8X *QM+HmbBQM
KQH2+mH2 Bb H `;2 U4.2 J. V M/ i?2 T `iB+H2b `2 +H2 `Hv
AM i?Bb T T2`- r2 T`2b2Mi2/ i?2 EGh TB+F2`- M m@ iQK i2/- / i /`Bp2M T `iB+H2 TB+F2`- i? i `2[mB`2b QMHv
pBbB#H2X h?2 β@: H +iQbB/ b2 Bb KQ`2 +? HH2M;BM;- b i?2 K +`Q@KQH2+mH2 Bb bK HH2` U0.465 J. V- M/ Bi Bb ? `/2` iQ
QM2 BMTmi r?B+? Bb i?2 2biBK i2/ T `iB+H2Ƕb bBx2X h?2 EGh
b22 i?2 T `iB+H2b QM i?2 KB+`Q;` T?bX 6BM HHv- i?2 avM T@ iB+ _:R@_:k Bb i?2 KQbi +? HH2M;BM; / i b2i- b Bi +Q``2bTQM/b iQ bK HH2` K +`Q@KQH2+mH2 U0.38 J. V M/
TB+F2` Bb /2bB;M2/ 2bT2+B HHv iQ ? M/H2 HQr aL_ KB+`Q@ ;` T?b- Bi Bb 2 bv iQ mb2- 7 bi- M/ K i?2K iB+ HHv `B;Q`QmbX
? b HQr aL_X qBi? HH i?2b2 / i b2ib- r2 +?B2p2/ ?B;? `2bQHmiBQMb rBi? KBMBKmK K Mm H 2zQ`iX AM //BiBQM- i?2 bm`7 +2 pB2rb BM 6B;m`2 N b?Qr i? i i?2 j. `2+QMbi`m+iBQMb
/Bz2`2Mi / i b2ibX AM HH Q7 i?2K r2 +?B2p2/ ?B;?@[m HBiv `2bmHib rBi? KBMBK H K Mm H 2zQ`iX
q2 2p Hm i2/ i?2 T2`7Q`K M+2 Q7 i?2 EGh TB+F2` QM i?`22
h?2 H;Q`Bi?K + M #2 7m`i?2` BKT`Qp2/ b 7QHHQrbX JB@
r2 ;Qi `2 +QKT iB#H2 rBi? i?2 Tm#HBb?2/ QM2bX
+`Q;` T?b i2M/ iQ +QMi BM +QMi KBM iBQMb Q7 p `BQmb 7Q`Kb7`QK bK HH +QMi KBM iBQMb iQ r?QH2 `2 b +QMi BMBM; 2B@
h?`Qm;?Qmi Qm` 2MiB`2 `2+QMbi`m+iBQM T`Q+2bb- i?2 QMHv K Mm H bi2T r b i?2 b2H2+iBQM Q7 +H bb2b Qmi Q7 _1GAPLǶb
i?2` B+2 Q` i?2 2/;2b Q7 i?2 bmTTQ`iBM; ;`B/X h?2b2 `@
k. +H bbB}+ iBQM QmiTmiX q2 /B/ MQi mb2 j. +H bbB}+ iBQM-
2 b + M /2;` /2 i?2 2biBK iBQM Q7 i?2 _Sa.- r?B+? Bb i?2 2bb2M+2 Q7 Qm` K2i?Q/X .2pBbBM; M H;Q`Bi?K iQ miQK i@
MQ` Mv Qi?2` K2i?Q/ iQ }Hi2` i?2 EGh TB+F2`Ƕb TB+F2/ T `iB+H2bX Ai Bb HbQ BKTQ`i Mi iQ MQi2 i? i i?2 T ` K2i2`b
B+ HHv B/2MiB7v i?2b2 `2 b K v BKT`Qp2 i?2 TB+FBM;X .2Mb2 T `iB+H2 +Hmbi2`b + M TQb2 T`Q#H2K iQ i?2 EGh TB+F2`- b
BM HH Q7 i?2 2tT2`BK2Mib r?2`2 +?Qb2M miQK iB+ HHv # b2/ QM i?2 T `iB+H2Ƕb bBx2- bQ i?2`2 Bb MQ M22/ iQ /Dmbi i?2K K Mm HHvX
i?2 b+Q`BM; K i`Bt- BMi`Q/m+2/ BM a2+iBQM kXj- i2M/b iQ b@ bB;M ?B;? b+Q`2b iQ i?2 BMi2`7 +2 #2ir22M T `iB+H2b BMbi2 / Q7 iQ i?2 +2Mi2` Q7 2 +? T `iB+H2 b2T ` i2Hv- b22 M 2t K@
AM irQ Qmi Q7 i?2 i?`22 2tT2`BK2Mib #Qp2- i?2 `2bQ@ HmiBQM +?B2p2/ mbBM; T `iB+H2b TB+F2/ #v Qm` H;Q`Bi?K Bb BM72`BQ` iQ i?2 Tm#HBb?2/ QM2X h?Bb BKTHB2b i? i 7mHHv m@
TH2 BM 6B;m`2 RjX PM2 TQbbB#H2 bQHmiBQM iQ i?Bb T`Q#H2K Bb iQ `2TH +2 i?2 +m``2Mi H;Q`Bi?K 7Q` 2ti` +iBM; T `iB+H2br?B+? mb2b i?2 BM/BpB/m H p Hm2b Q7 i?2 b+Q`BM; K i`Bt U/2@
iQK iB+ TB+FBM; K v +QK2 i T`B+2X u2i- i?2 `2bmHib Q7 Qm` TB+FBM; + M #2 mb2/ iQ ;2M2` i2 M BMBiB H j. bi`m+@ im`2- r?B+? +QmH/ i?2M #2 mb2/ iQ T`Q/m+2 `272`2M+2b 7Q`
b+`B#2/ i i?2 2M/ Q7 a2+iBQM kXjV- rBi? M H;Q`Bi?K i? i +QMbB/2`b i?2 2MiB`2 M2B;?#Q`?QQ/ Q7 T `iB+H2 + M/B/ i2X
b2+QM/ `QmM/ Q7 `272`2M+2@# b2/ TB+FBM;X aBM+2 `272`2M+2@ # b2/ TB+FBM; /Q2b MQi `2[mB`2 K Mm H BMi2`p2MiBQM- i?2 2MiB`2 `2bmHiBM; TB+FBM; T`Q+2bb Bb biBHH 7mHHv miQK iB+X
+FMQrH2/;K2Mib
9XkX *QKT `BbQM rBi? Qi?2` T `iB+H2 TB+FBM; K2i?Q/b
h?Bb `2b2 `+? r b bmTTQ`i2/ #v i?2 1m`QT2 M _2b2 `+? *QmM+BH U1_*V mM/2` i?2 1m`QT2 M lMBQMǶb >Q`BxQM kyky
S `iB+H2 TB+FBM; K2i?Q/b + M #2 /BpB/2/ BMiQ irQ + i2@ ;Q`B2bX h?2 }`bi `2HB2b QM T`BQ` bbmKTiBQMb QM i?2 T `iB@ +H2Ƕb bi`m+im`2- bm+? b : mbbB M b? T2 (k9) Q` `2b2K#H M+2
`2b2 `+? M/ BMMQp iBQM T`Q;` KK2 U;` Mi ;`22K2Mi dkjNNR @ *_uPJh>VX
iQ Qi?2` FMQrM K +`Q@KQH2+mH2b (k8- k3)X h?2 b2+QM/ `2HB2b bQH2Hv QM BM7Q`K iBQM 7`QK i?2 / i Bib2H7- r?B+? + M #2 b2i Q7 K Mm HHv TB+F2/ T `iB+H2b (R3- j)- Q` b2i Q7 KB+`Q@ ;` T? T i+?2b i? i r2`2 miQK iB+ HHv b2H2+i2/ (3)X h?2 d
6B;m`2 d, 1t KTH2b Q7 9 k. +H bb p2` ;2b Q7 SH bKQ/BmK 6 H+BT `mK 3ya UH27iV- β@: H +iQbB/ b2UKB//H2V- M/ avM TiB+ _:R@_:k U`B;?iV T`Q/m+2/ #v _1GAPL 7`QK T `iB+H2b TB+F2/ #v i?2 EGh TB+F2`X
6B;m`2 3, 6a* +m`p2b Q7 SH bKQ/BmK 6 H+BT `mK 3ya UH27iV- β@: H +iQbB/ b2 UKB//H2V- M/ avM TiB+ _:R@_:k U`B;?iVX
TT2M/Bt X .2`BpBM; UjV M/ U9V
i?mb
h?2 E `?mM2M GQ2p2 h?2Q`2K (RR) bi i2b i? i i?2 bQHm@ iBQMb ψn iQ URV `2 i?2 2B;2M7mM+iBQMb Q7 i?2 miQ+Q``2H iBQM QT2` iQ` K(x, y) = E f (x) f (y) - i? i Bbλn ψn (x) =
Z
K(x, y)ψn (y)dy.
λm,k Rm,k (r)eimθ Z Z 1 0 0 = G(ρ)ei(ρr cos(φ−θ)−ρr cos(φ−θ )) (2π)2 aD cD 0
UXRV
=
aD
lM/2` i?2 bbmKTiBQMb i? i f Bb # M/@HBKBi2/- rB/2 b2Mb2
1 (2π)2
Z
0
aZ
Z
× Rm,k (r0 )eimθ r0 ρdr0 dθ0 dρdφ ! π i(mθ0 −ρr0 cos(φ−θ0 )) 0 e dθ G(ρ)eiρr cos(φ−θ)
−π
cD
× Rm,k (r0 )r0 ρdr0 dφdρ.
bi iBQM `v- M/ i? i K Bb ` /B HHv@bvKK2i`B+ Ui? i Bb- K(x, y) = UX8V K(kx − yk)V- i?2 qB2M2`ĜE?BM+?BM i?2Q`2K (8) bi i2b i? i *? M;BM; p `B #H2b BM i?2 BMM2`KQbi BMi2;` H BM UX8V iQ i?2 ` /B H TQr2` bT2+i` H /2MbBiv U_Sa.V Q7 f - ;Bp2M #v U8Vθ0 = φ − α − π2 ;Bp2b Bb i?2 6Qm`B2` i` Mb7Q`K Q7 i?2 miQ+Q``2H iBQM KX >2M+2Z π Z π π 0 0 0 0 #v i FBM; i?2 BMp2`b2 6Qm`B2` i` Mb7Q`K Q7 Gei(mθ −ρr cos(φ−θ )) dθ0 = eim(φ− 2 ) ei(ρr sin α−mα) dα −π −π UXeV Z im(φ− π2 ) 0 1 iω·(x−y) 2πJ (ρr ), = e m G(kωk)e dω, UXkV K(kx − yk) = (2π)2 cD r?2`2 Jm Bb i?2 mǶi? Q`/2` "2bb2H 7mM+iBQM Q7 i?2 }`bi FBM/X am#biBimiBM; # +F UXeV BM UX8V ;Bp2b
r?2`2 c Bb i?2 # M/@HBKBi Q7 f X .2MQi2 x = (r cos θ, r sin θ)y = (r0 cos θ0 , r0 sin θ0 )- ω = (ρ cos φ, ρ sin φ) M/ r`Bi2 i?2 bQHmiBQMb Q7 UXRV b 1 ψm,k (r, θ) = √ Rm,k (r)eimθ , 2π
m ∈ Z, k ∈ N.
UXjV
λm,k Rm,k (r)eimθ =
Z
a 0
Z
c
Z
0 0
π
π
ei(m(φ− 2 )+ρr cos(φ−θ)) dφ
−π
× Jm (ρr )G(ρ)Rm,k (r0 )r0 ρdr0 dρ.
am#biBimiBM; UXjV BMiQ UXRV vB2H/b Z
1 2π
λm,k Rm,k (r)eimθ =
!
UXdV
*? M;BM; p `B #H2b BM i?2 BMM2`KQbi BMi2;` H BM UXdV iQ φ = θ − α + π2 ;Bp2b
K(r, θ, r0 , θ0 )Rm,k (r0 )eimθ r0 dr0 dθ0 . UX9V 0
aD
Z
LQi2 i? i UXkV BM TQH ` +QQ`/BM i2b Bb ;Bp2M #v
π
π
ei(m(φ− 2 )+ρr cos(φ−θ)) dφ = eimθ −π
= K(r, θ, r0 , θ0 ) = Z 1 0 0 0 0 G(ρ)ei(ρ cos φ,ρ sin φ)·(r cos θ−r cos θ ,r sin θ−r sin θ ) ρdρdφ = (2π)2 cD Z 1 0 0 G(ρ)ei(ρr cos(φ−θ)−ρr cos(φ−θ )) ρdρdφ, (2π)2 cD
π
Z
ei(ρr sin α−mα) dφ
−π imθ 2πe Jm (ρr).
UX3V
lbBM; UX3V BM UXdV r2 ;2i λm,k Rm,k (r) =
Z
=
Z
a 0
c
! Jm (ρr)Jm (ρr )G(ρ)r ρdρ Rm,k (r0 )dr0 0
0
0
a
hm (r, r0 )Rm,k (r0 )dr0 , 0
r?2`2 hm (r, r0 ) =
3
Z
Rc 0
Jm (ρr)Jm (ρr0 )G(ρ)r0 ρdρX
6B;m`2 N, am`7 +2 pB2rb i F2M 7`QK i?2 j. `2+QMbi`m+iBQMb Q7 SH bKQ/BmK 6 H+BT `mK 3ya UH27iV- β@: H +iQbB/ b2 UKB//H2V- M/ avM TiB+ _:R@_:k U`B;?iVX h?2 iQT bm`7 +2 pB2rb +Q``2bTQM/ iQ `2+QMbi`m+iBQMb 7`QK T `iB+H2b TB+F2/ #v i?2 EGh TB+F2`- M/ i?2 #QiiQK QM2b iQ i?2 `2+QMbi`m+iBQM Q7 (kd)- (k)- M/ (Re)- `2bT2+iBp2HvX h?2 pB2rb r2`2 ;2M2` i2/ mbBM; i?2 l*a6 *?BK2` bQ7ir `2 (R9)X
6B;m`2 Ry, 1t KTH2b Q7 TB+FBM; `2bmHib 7Q` i?2 SH bKQ/BmK 6 H+BT `mK 3ya / i b2iX h?2 KB+`Q;` T?b `2 b?QrM BM i?2 H27i +QHmKM- TB+F2/ T `iB+H2b rBi? i?`2b?QH/ 2[m H iQ x2`Q `2 BM i?2 KB//H2 +QHmKMb- M/ i?2 iQT 8y TB+F2/ T `iB+H2b `2 BM i?2 `B;?i +QHmKMX
TT2M/Bt "X 1biBK iBM; i?2 bQHmiBQM Q7 UdV
h?mb-
AM i?Bb b2+iBQM- r2 QmiHBM2 M H;Q`Bi?K 7Q` 2biBK iBM;
{i0 , j0 } = argmax
Si − S j
1 = argmax
αi − α j γ
i?2 bQHmiBQM Q7 UdV- i? i Bb Q7 {ˆγ, νˆ , α} ˆ = arg min γ˜ ,˜ν,α˜
bXiX γ˜ , ν˜ ∈
i, j
M X i=1
Rl+ ,
i, j
= argmax |αi − α j | kγk1 , i, j
2
||Si − ν˜ − α˜ i γ˜ || ,
U"XRV
r?B+? BKTHB2b
α˜ i ∈ [0, 1].
γˆ (0) = (αK t − αKBM ) · γ.
1[m iBQM U"XRV + M #2 pB2r2/ b M BMbi M+2 Q7 MQM@ M2; iBp2 K i`Bt 7 +iQ`Bx iBQM T`Q#H2K ULLJ6V (e)- r?B+? Bb LS@? `/X PM2 K2i?Q/ 7Q` 2biBK iBM; bQHmiBQMb Q7 LLJ6
aBM+2 r2 2tT2+i i H2 bi QM2 T i+? iQ BM+Hm/2 +QKTH2i2 T `iB+H2- M/ i H2 bi QM2 T i+? iQ +QMi BM QMHv MQBb2- r2 ;2i αK t ≈ 1, αKBM ≈ 0- r?B+? vB2H/b
T`Q#H2Kb Bb #v Hi2`M iBM; H2 bi@b[m `2b UGaV (Ry)X h?2 B/2 #2?BM/ i?Bb Bi2` iBp2 K2i?Q/ Bb bQHpBM;- BM 2 +? bi2Ti?2 LLJ6 KBMBKBx iBQM T`Q#H2K 7Q` QM2 Q7 i?2 p `B #H2b-
γˆ (0) = (αK t − αKBM ) · γ ≈ (1 − 0) · γ = γ.
r?BH2 i`2 iBM; i?2 Qi?2`b b +QMbi MibX A7 i?2 K2i?Q/ +QM@
h? i Bb γˆ (0) Bb BMBiB HBx2/ +HQb2 iQ Bib i`m2 p Hm2X b 7Q` νˆ (0) - U"X9V BKTHB2b
p2`;2b- i?2M Bi +QMp2`;2b iQ HQ+ H KBMBKmK Q7 i?2 LLJ6 T`Q#H2KX AM Qm` + b2- i?2 p `B #H2b `2 i?2 p2+iQ`b γˆ , νˆ , αˆ M/ i?2 Ga Bi2` iBQMb `2 ;Bp2M #v U7Q` Bi2` iBQM iV
{k0 } = argmin kSk k1 = argmin kν + αk γk1 k
{αˆ } = arg min
M X
||Sk − νˆ
{ˆγ(i) } = arg min
M X
||Sk − νˆ (i−1) − αˆ (i) ˜ ||2 , k γ
{ˆν(i) } = arg min
M X
||Sk − ν˜ − αˆ (i) ˆ (i) ||2 , k γ
(i)
α˜
γ˜
ν˜
s.t γ˜ , ν˜ ∈
k=1
k=1
k=1
Rl+ ,
(i−1)
− α˜ k γˆ
νˆ (0) = Sk0 ,
|| ,
k
k
r?2`2 i?2 i?B`/ 2[m HBiv ?QH/b /m2 iQ i?2 TQbBiBpBiv Q7 HH `;mK2MibX b r`Bii2M #Qp2- αKBM ≈ 0 M/ ?2M+2
U"XkV
νˆ (0) ≈ ν + 0 · γ = ν, i? i Bb- νˆ (0) Bb HbQ BMBiB HBx2/ +HQb2 iQ Bib i`m2 p Hm2X L2ti- r2 /2`Bp2 M 2tTHB+Bi 7Q`KmH 7Q` i?2 bQHmiBQM
α˜ k ∈ [0, 1].
Q7 U"XkVX lbBM; # bB+ H;2#` - r2 + M `2r`Bi2 i?2 bmK BM
{i0 , j0 } = argmax
Si − S j
1 ,
U"XjV
i, j
{k0 } = argmin kSi k1 , i
P r?2`2 kuk1 = li=1 |ui | 7Q` u ∈ Rl X hQ DmbiB7v i?Bb +?QB+2`2+ HH i? i S Bb KQ/2H2/ b Si = ν + αi γ,
k
= argmin kνk1 + αk kγk1 = argmin αk ,
(i−1) 2
b i?2 BMBiB H p2+iQ`b γˆ (0) , νˆ (0) r2 b2i γˆ (0) = Si0 − S j0 ,
1
1 ≤ i ≤ M.
U"X9V
N
6B;m`2 RR, 1t KTH2b Q7 TB+FBM; `2bmHib 7Q` i?2 β@: H +iQbB/ b2 / i b2iX a22 6B;m`2 Ry 7Q` /2i BHbX
6B;m`2 Rk, 1t KTH2b Q7 TB+FBM; `2bmHib 7Q` i?2 avM TiB+ _:R@_:k / i b2iX a22 6B;m`2 Ry 7Q` /2i BHbX
r?2`2 7Q` r ∈ R
6B;m`2 Rj, M 2t KTH2 Q7 /2Mb2 T `iB+H2b +Hmbi2` 7Q` r?B+? i?2 EGh TB+F2` TB+F2/ i?2 BMi2`7 +2 #2ir22M T `iB+H2b BMbi2 / Q7 i?2 +2Mi2` Q7 2 +? T `iB+H2X JB+`Q;` T? i F2M 7`QK i?2 1JSA_@Ryyk3 / i b2i (kd)X
[r]+,≤1
U"XRV BM i?`22 2[mBp H2Mi 7Q`Kb M X i=1
M/
U"X8V
kSi − ν˜ − α˜ i γ˜ k2
M M l X M X X 2 X 2 2 (S ) ˜ k = α˜ i γ˜ k − 2 α˜ i ν˜ k + α˜ i Ski γ˜ k + ki − ν k=1
i=1
i=1
i=1
k=1
0 ≤ r ≤ 1, r > 1,
0 [r]+ = r
r < 0, r ≥ 0.
BM U"XeV- Bi ?QH/b i? i
i=1
l l M X l X X X 2 2 2 (Ski − ν˜ k ) . = γ˜ k α˜ i − 2 ν˜ k γ˜ k + γ˜ k Ski α˜ i + i=1
r < 0,
PM2 + M T`Qp2 i? i B7 S + M #2 7 +iQ`Bx2/ b BM U"X9V- M/ i?2 Bi2` iBQMb Q7 U"XeV +QMp2`;2- i?2M BM i?2 HBKBi Q7 i → ∞
i=1
M l M X X X 2 (Ski − α˜ i γ˜ k ) = M ν˜ k − 2 α˜ i γ˜ k + Ski ν˜ k + k=1
0 = r 1
k=1
γˆ = aγ,
k=1
αˆ k =
1 (αk + b) , a
νˆ = ν − bγ,
U"XdV
r?2`2 a, b ∈ R + M #2 2biBK i2/ 7`QK i?2 MQBb2 p `B M+2X
h`2 iBM; γ˜ b p `B #H2 M/ ν˜ , α˜ b +QMbi Mib BM i?2 b2+@ {˜γk }lk=1 X h?2 KBMBKmK Q7 i?Bb bmK- mM/2` i?2 +QMbi` BMi γ˜ k ≥ 0 Bb ii BM2/ i i?2 KBMBKmK Q7 i?2b2 T ` #QH b r?2M@
h?Bb BKTHB2b i? i r2 + M `2+Qp2` γ, α, νX q2 QKBi i?2 /2@ i BHb Q7 i?2 T`QQ7X AM TT2M/Bt *- r2 /2b+`B#2 K2i?Q/ iQ 2biBK i2 i?2 MQBb2 p `B M+2 7`QK ;Bp2M KB+`Q;` T?X h?2 T`Q+2/m`2 7Q` bQHpBM; U"XRV Bb bmKK `Bx2/ BM H;Q`Bi?K kX
r?2M i`2 iBM; ν˜ b i?2 p `B #H2 BM i?2 i?B`/ HBM2 U M/ i?2 Qi?2` b +QMbi MibVX h`2 iBM; α˜ b i?2 p `B #H2 BM i?2 7Q`i?
H;Q`Bi?K k Ga H;Q`Bi?K
QM/ HBM2- ;Bp2b bmK Q7 +QMp2t T ` #QH b 7Q` i?2 p `B #H2b
2p2` i?2v `2 TQbBiBp2 M/ i γ˜ k = 0 2Hb2X h?2 b K2 TTHB2b
HBM2- BKTHB2b i? i i?2 KBMBKmK Bb ;Bp2M b 7QHHQrX 6Q` 2 +? T ` #QH BM i?2 bmK U"X8V- B7 Bib KBMBKmK Bb Q#i BM2/ i
R,
M2; iBp2 α˜ k - i?2M i?2 KBMBKmK Q7 i?2 bmK Bb ii BM2/ i α˜ k = 0X A7 i?2 KBMBKmK Bb Q#i BM2/ i α˜ k > 1- r2 b2i α˜ k = 1X Pi?2`rBb2- i?2 KBMBKmK Q7 i?2 bmK Bb ii BM2/
k, j, 9, 8,
i i?2 KBMBKmK Q7 i?2 T ` #QH X hQ +QM+Hm/2- r2 ;2i i?2 7QHHQrBM; 7Q`KmH b 7Q` i?2 bQHmiBQMb Q7 i?2 iǶi? Bi2` iBQM
e,
BM U"XkV Pl k=1 γˆ k(i−1) · Sk, j − νˆ k(i−1) αˆ (i) , 1 ≤ j ≤ M, Pl j = (i−1) γ ˆ k=1 k +,≤1 P M (i) k=1 αˆ k · S j,k − νˆ (i−1) j (i) , γˆ j = 1 ≤ j ≤ l, P M (i) ˆk k=1 α + PM k=1 S j,k − αˆ (i) ˆ (i) j k γ (i) , νˆ j = 1 ≤ j ≤ l, M
_2[mB`2/, J i`Bt SX *QKTmi2 i?2 BMBiB H p2+iQ`b νˆ (0) , γˆ (0) mbBM; U"XjV M/ αˆ (0) , αˆ (1) , νˆ (1) , γˆ (1) mbBM; U"XeVX i=1
r?BH2
νˆ (i) − νˆ (i−1)
,
γˆ (i) − γˆ (i−1)
,
αˆ (i) − αˆ (i−1)
≥ /Q *QKTmi2 νˆ (i+1) , γˆ (i+1) , αˆ (i+1) mbBM; U"XeV i=i+1 2M/ r?BH2 PmiTmi, νˆ (i) , γˆ (i) , αˆ (i)
6BM HHv- r2 `2im`M iQ i?2 +QKTmi iBQM Q7 i?2 +QHmKMb Q7 SX 1 +? +QHmKM Bb i?2 _Sa. Q7 KB+`Q;` T? T i+?
Q7 bBx2 l × l rBi? bi`B/2 d- b /2b+`B#2/ i i?2 #2;BMMBM; Q7 a2+iBQM kXkX h?2b2 T ` K2i2`b `2 /2i2`KBM2/ b 7QH@
U"XeV
HQrbX AM U"X9V r2 bbmK2 i? i S + M #2 7 +iQ`Bx2/ b
+QK#BM iBQM Q7 QMHv ν, γ, αX >Qr2p2`- i?Bb KQ/2H Bb QMHv M TT`QtBK iBQM- M/ BM T` +iB+2 Bi ?QH/b i? i
+
S = ν + αT γ + E, Ry
U"X3V
r?2`2 E Bb M 2``Q` i2`K K i`BtX lM/2` i?2 bbmKTiBQMb i? i i?2 2tT2+i M+v Q7 i?2 `Qrb Q7 E Bb x2`Q- M/ i? i i?2
Ub22 HbQ a2+iBQM kXjVX h?2 E `?mM2M GQ2p2 h?2Q`2K (RR) bi i2b i? i
+QHmKMb Q7 E `2 mM+Q``2H i2/ M/ rBi? i?2 b K2 p `B@
M+2- QM2 + M T`Qp2 i? i i?2 HBKBi Q7 U"XeV r?2M mbBM; S Q7 U"X3V bi vb i?2 b K2 b BM U"XdV- #mi rBi? /Bz2`2Mi p H@
f (x) =
∞ X
zi ψi (x),
i=1
m2b 7Q` a M/ b Ur2 QKBi i?2 T`QQ7VX h?2b2 p Hm2b + M #2
h i r?2`2 ψi `2 ;Bp2M #v URV- M/ KQ`2Qp2`- E zi z j = δi, j λi X h?2`27Q`2- i?2 miQ+Q``2H iBQM K(x, y) = E f (x) f (y) + M #2 r`Bii2M b ∞ ∞ X X E f (x) f (y) = E zi ψi (x) z j ψ j (y)
2biBK i2/ 7`QK i?2 MQBb2 p `B M+2- M/ i?2`27Q`2- r2 + M `2+Qp2` γ, α, ν 7Q` i?2 KQ`2 ;2M2` H KQ/2H U"X3VX LQr- i F@ BM; d < l +H2 `Hv #`2 Fb i?2 bbmKTiBQMb 7Q` E M/ i FBM;
d > l /Q2b MQi mb2 HH Q7 i?2 / i - ?2M+2 r2 mb2/ d = lX b 7Q` i?2 T ` K2i2` l- B/2 HHv r2 rQmH/ HBF2 T i+? iQ +QM@
i=1
i BM MQ KQ`2 i?2M QM2 +QKTH2i2 T `iB+H2X h FBM; l Km+? #B;;2` i?2M i?2 T `iB+H2 /B K2i2` + M + mb2 T i+?2b iQ +QM@ i BM irQ Q` KQ`2 T `iB+H2b- M/ i FBM; Bi Km+? bK HH2` rBHH
=
∞ X ∞ X
=
∞ X ∞ X
i=1 j=1
MQi HHQr T i+?2b iQ +QMi BM 2MQm;? Q7 i?2 T `iB+H2X AM T` +iB+2- Bi b22Kb i? i i FBM; l 2[m H 3yW Q7 i?2 T `iB+H2 /B K2i2` vB2H/b ;QQ/ `2bmHibX
j=1
h i ψi (x)ψ j (y)E zi z j ψi (x)ψ j (y)δi, j λi
i=1 j=1
=
∞ X
ψi (x)ψi (y)λi .
i=1
TT2M/Bt *X 1biBK iBM; i?2 MQBb2 p `B M+2
lbBM; U.XRV r2 ;2i
h?2 MQBb2 p `B M+2 Bb M22/2/ iQ `2+Qp2` γ, α, ν 7`QK U"XdV M/ iQ 2p Hm i2 UNVX bbmKBM; i? i #Qi? i?2 MQBb2 M/
h i E PPT
i?2 T `iB+H2 7mM+iBQM `2 ;2M2` i2/ 7`QK rB/2 b2Mb2 bi @ iBQM `v ` M/QK T`Q+2bb M/ i? i i?2v `2 mM+Q``2H i2/- Bi 7QHHQrb i? i i?2 p `B M+2 Q7 ;Bp2M KB+`Q;` T? T i+? Bb
(i, j)
∞ h i X = E Pi P j = E f (i) f ( j) = ψk (i)ψk ( j)λk . k=1
U.XkV
AM Q`/2` iQ /2}M2 i`mM+ iBQM `mH2 7Q` i?2 BM}MBi2 bmK
i?2 bmK Q7 i?2 MQBb2 p `B M+2 M/ 7` +iBQM Q7 i?2 T `iB@
BM U.XkV- r2 +QMbB/2` i?2 `2H iBp2 2``Q` i2`K
+H2Ƕb p `B M+2- r?B+? Bb T`QTQ`iBQM H iQ i?2 7` +iBQM Q7 i?2 T `iB+H2 +QMi BM2/ BM i?2 T i+?X b i?2 p `B M+2 Bb TQb@
P
P∞ PN
k=1 ψk ψk λk − k=1 ψk ψk λk
∞ k=N+1 ψk ψk λk
P∞
=
P∞
k=1 ψk ψk λk
k=1 ψk ψk λk
P∞ λk , = Pk=N+1 ∞ λ k=1 k
BiBp2 MmK#2`- T i+?2b rBi? HQr2` p `B M+2 `2 KQ`2 HBF2Hv iQ +QMi BM QMHv MQBb2X qBi? i? i BM KBM/- r2 T `iBiBQM i?2 KB+`Q;` T? BMiQ Qp2`H TTBM; T `iB+H2@bBx2 T i+?2b rBi? bi`B/2 2[m H iQ 1X h?2M- r2 TT`QtBK i2 i?2 p `B M+2 Q7 2 +? T i+? b 2tTH BM2/ BM (ke) M/ }M HHv 2biBK i2 i?2
U.XjV U.X9V
r?2`2 || · || Bb i?2 L2 (aD × aD) bi M/ `/ MQ`K M/ U.X9V 7QHHQrb 7`QK U.XjV /m2 iQ Q`i?Q;QM HBiv Q7 ψk X h?2M- r2 /2}M2 i?2 7QHHQrBM; i`mM+ iBQM `mH2X 6B`bi r2 i`mM+ i2 i?2
MQBb2 p `B M+2 #v p2` ;BM; x% Q7 i?2 HQr2bi p `B M+2 `2@ bmHibX *?QQbBM; x /2T2M/b QM i?2 MmK#2` Q7 T `iB+H2b BM i?2 KB+`Q;` T?- r?B+? Bb mMFMQrM iQ mbX q2 ? p2 Q#@
BM}MBi2 bmK i M i2`Kb r?2`2 M Bb +?Qb2M bm+? i? i
b2`p2/ 2tT2`BK2Mi HHv i? i i FBM; x = 25% T`QpB/2b b iBb@
λ M − λ M+1 ≤ 10−6 , λM
7 +iQ`v `2bmHib BM HQr@aL_ +QM/BiBQMbX
M/ i?2M- r2 i`mM+ i2 Bi ; BM iQ N i2`Kb bm+? i? i TT2M/Bt .X .2`BpBM; i?2 TT`QtBK iBQM URyV
PM
k=N+1
AM i?Bb b2+iBQM- r2 /2`Bp2 M TT`QtBK iBQM 7Q` i?2 T `@ h i iB+H2Ƕb +Qp `B M+2 K i`Bt E PPT - r?2`2 Pi = f (i),
i∈U
PM
k=1
U.XRV RR
λk
λk
≤ 0.01.
U.X8V
"v bi M/ `/ HBM2 ` H;2#` - Bi + M #2 b?QrM i? i i?2 i`mM@ + iBQM `mH2 Q7 U.X8V vB2H/b i?2 /2bB`2/ TT`QtBK iBQM Q7 URyV
K i`Bt M/ 7 +iQ`Bx2 Bi mbBM; i?2 Z_ /2+QKTQbBiBQM (kR) b
∞ N X h i X E PPT = ψk (i)ψk ( j)λk ≈ ψk (i)ψk ( j)λk = ΨΛΨT k=1
k=1
Ψ = QR,
U.XeV
r?2`2 Ψ M/ Λ `2 /2}M2/ BM a2+iBQM kXjX
r?2`2 Q ∈ Rb2a+1c
2
×b2a+1c2
TT2M/Bt 1X .2`Bp iBQM M/ +QKTmi iBQM Q7 URRV Pm` ;Q H Bb iQ +QKTmi2 i?2 b+Q`2 ! p(Ii, j |H1 ) . p(Ii, j |H0 )
U1XRV
LQi2 i? i RΛRT + σ2 I Bb #HQ+F /B ;QM H K i`Bt Q7 i?2 7Q`K H 0 T 2 , RΛR + σ I = 0 σ 2 I2
iBp `B i2 MQ`K H- M/ i? i i?2 MQBb2 Bb r?Bi2- i?2 T`Q# @ #BHBiv /2MbBiv 7mM+iBQM US.6V Q7 KB+`Q;` T? T i+? +QM@ i BMBM; T `iB+H2 THmb MQBb2 Bb 1 T −1 1 e− 2 I Σ I , |2πΣ|
˜ R˜ T + σ2 I1 Bb Q7 bBx2 N × N- I1 Bb i?2 A/2MiBiv r?2`2 H = RΛ
U1XkV
K i`Bt Q7 i?2 b K2 bBx2 b H- M/ I2 Bb i?2 B/2MiBiv K i`Bt
Q7 bBx2 (b2a + 1c2 − N) × (b2a + 1c2 − N)X h?2 /2i2`KBM Mi Q7 Σˆ Bb ˆ = |H|σb2a+1c2 −N , U1X3V |Σ|
r?2`2 | · | Bb K i`Bt /2i2`KBM MiX h?2 S.6 Q7 T i+? rBi? MQBb2 QMHv Bb
p (I|H0 ) = p
1
1
(2πσ2 )b2a+1c2
T
e− 2σ2 I I .
U1XjV
M/ Bib BMp2`b2 Bb
am#biBimiBM; i?2 #Qp2 S.6b BM U1XRV M/ `2TH +BM; i?2 ` M/QK p `B #H2 I rBi? Bib (i, j) T i+? `2 HBx iBQM Ii, j ;Bp2b
ˆ −1
Σ
Ub22 HbQ URRVV b+Q`2(i, j) =
!
AM //BiBQM-
!
T 1 |Σ| log 2b2a+1c2 + Ii, j σ−2 I − Σ−1 Ii, j . 2 σ U1X9V
Σ = E PP
i
am#biBimiBM; i?2 H ii2` # +F BM U1X8V vB2H/b
ˆ + σ I ≈ ΨΛΨ + σ I := Σ. 2
2
T
σ−2 I1 − H −1 0 T Q I IT σ−2 I − Σˆ −1 I = IT Q 0 0 U1XNV T σ−2 I1 − H −1 0 T T Q I. = Q I 0 0
ˆ pB i?2 bBM;mH ` p Hm2b /2+QKTQbB@ *QKTmiBM; Σˆ −1 M/ |Σ|
iBQM Uao.V (kR) i F2b O(b2a + 1c2 N 2 ) QT2` iBQMb T2` KB+`Q@ ;` T?- r?2`2 N Bb ;Bp2M #v i?2 i`mM+ iBQM `mH2 Q7 U.X8VX h?2 +QKTmi iBQM H +Qbi Q7 /B`2+iHv +QKTmiBM; Pi, j =
ITi, j
−2
ˆ −1
σ I−Σ
0 T Q . σ−2 I2
I1 0 H −1 0 T −1 −2 ˆ Q σ I − Σ = σ − Q 0 I2 0 σ−2 I2 σ−2 I1 − H −1 0 T Q . = Q 0 0
i?2 T `iB+H2Ƕb +Qp `B M+2 K i`Bt- i? i BbT
H −1 = Q 0
−2
q2 2biBK i2 Σ 7Q` U1X9V mbBM; i?2 TT`QtBK iBQM URyV 7Q`
h
Bb M Q`i?Q;QM H K i`Bt M/ R ∈
Σˆ = Q RΛRT + σ2 I QT .
lM/2` i?2 bbmKTiBQMb i? i i?2 T `iB+H2 Bb /Bbi`B#mi2/ KmH@
p (I|H1 ) = √
U1XdV
Rb2a+1c ×N Bb M mTT2` i`B M;mH ` K i`Bt r?Qb2 H bi b2a + R˜ 1c2 − N `Qrb `2 x2`QX q2 /2MQi2 R = r?2`2 R˜ ∈ RN×N 0 M/ bm#biBimi2 U1XdV BM U1XeV- `2bmHiBM; BM 2
b+Q`2(i, j) := log
U1XeV
Σˆ = ΨΛΨT + σ2 I,
Ii, j ,
.2MQi2 i?2 }`bi N +QQ`/BM i2b Q7 i?2 p2+iQ` QT I #v vX
U1X8V
h?2M- i?2 H bi 2tT`2bbBQM Bb bBKTHB}2/ iQ
Bb O(b2a + 1c6 ) QT2` iBQMb T2` T i+?X AM Q`/2` iQ `2/m+2
IT σ−2 I − Σˆ −1 I = σ−2 vT v − vT H −1 v.
i?2 H ii2` +QKTH2tBiv- +QMbB/2` i?2 TT`QtBK i2 +Qp `B M+2 Rk
U1XRyV
am#biBimiBM; U1X3V M/ U1XRyV BM U1X9V M/ +? M;BM; I iQ Ii, j ;Bp2b b+Q`2(i, j) =
! ! 1 σ2N log + σ−2 vT v − vT H −1 v . 2 |H|
(Rk)
U1XRRV
(Rj)
h?2 +QKTH2tBiv H pB i?2 Z_ /2+QKTQbB@ Q7 +QKTmiBM; iBQM Q7 Σˆ Bb O b2a + 1c2 N 2 QT2` iBQMbX h?2M- +QKTmiBM; H −1 M/ |H| pB S V D `2[mB`2b O(N 3 ) QT2` iBQMb- #Qi? +QK@ Tmi2/ T2` KB+`Q;` T?X *QKTmiBM; vT H −1 v `2[mB`2b O(N 3 ) QT2` iBQMb- M/ +QKTmiBM; vT v `2[mB`2b O(N 2 ) QT2` iBQMb#Qi? +QKTmi2/ T2` T i+?X 6BM HHv- BM Q`/2` iQ +QKTmi2 i?2 p2+iQ` v- r2 M22/ iQ +QKTmi2 QMHv i?2 }`bi N +QQ`/BM i2b Q7 QT IX h?Bb + M #2 BKTH2K2Mi2/ mbBM; O b2a + 1c2 N QT2` iBQMb T2` T i+?X hQ +QM+Hm/2- i?2 T2` T i+? +QKTm@ 6 i iBQM H +QKTH2tBiv ? b #22M `2/m+2/ 7`QK O(b2a + 1c ) iQ 2 3 O b2a + 1c N + N X AM T` +iB+2- i?Bb bT22/b mT i?2 H;Q@ `Bi?K #v i?`22 Q`/2`b Q7 K ;MBim/2X
(R9)
(R8) (Re)
(R) E2M/ HH 1 iFBMbQMX h?2 MmK2`B+ H bQHmiBQM Q7 6`2/?QHK BMi2@ ;` H 2[m iBQMb Q7 i?2 b2+QM/ FBM/X aAJ CQm`M H QM LmK2`B+ H M HvbBb- 9UjV,jjdĜj93- RNedX (k) H#2`iQ " `i2b ;?B- H M J2`F- aQQD v " M2`D22- .Q`22M J ii?B2b- sBQM;rm qm- C +[m2HBM2 G a JBHM2- M/ a`B` K am#@ ` K MB KX kXk `2bQHmiBQM +`vQ@1J bi`m+im`2 Q7 β@; H +iQbB/ b2 BM +QKTH2t rBi? +2HH@T2`K2 Mi BM?B#BiQ`X a+B2M+2 UL2r uQ`FLXuXV- j93UekjNV,RR9děRR8R- CmM2 kyR8X AaaL yyje@3yd8X /QB, RyXRRkefb+B2M+2X #R8deX l_G ?iiT,ff2m`QT2TK+XQ`;f `iB+H2bf SJ*e8Rkjj3X (j) h`Bbi M "2TH2`- M/`2r JQ`BM- H2t C LQ#H2- CmHB "` b+?G r`2M+2 a? TB`Q- M/ "QMMB2 "2`;2`X SQbBiBp2@mMH #2H2/ +QM@ pQHmiBQM H M2m` H M2irQ`Fb 7Q` T `iB+H2 TB+FBM; BM +`vQ@2H2+i`QM KB+`Q;` T?bX AM _2b2 `+? BM +QKTmi iBQM H KQH2+mH ` #BQHQ;vpQHmK2 Ry3Rk- T ;2b k98Ĝk9d- kyR3X (9) C K2b w *?2M M/ LBFQH mb :`B;Q`B2zX aA:Lhl_1, bBM;H2@ T `iB+H2 b2H2+iBQM bvbi2K 7Q` KQH2+mH ` 2H2+i`QM KB+`Qb+QTvX CQm`M H Q7 bi`m+im` H #BQHQ;v- R8dURV,Re3ĜRdj- kyydX (8) G2QM *Q?2MX h?2 ;2M2` HBx iBQM Q7 i?2 qB2M2`@E?BM+?BM i?2Q@ `2KX AM S`Q+22/BM;b Q7 i?2 RNN3 A111 AMi2`M iBQM H *QM72`2M+2 QM +QmbiB+b- aT22+? M/ aB;M H S`Q+2bbBM;- A*aaSǶN3 U* iX LQX N3*>jeR3RV- pQHmK2 j- T ;2b R8ddĜR83yX A111- RNN3X (e) LB+QH b :BHHBbX AMi`Q/m+iBQM iQ MQMM2; iBp2 K i`Bt 7 +iQ`Bx iBQMX `sBp T`2T`BMi `sBp,RdyjXyyeej- kyRdX (d) LBFQH mb :`B;Q`B2zX 6_1GA:L, ?B;?@`2bQHmiBQM `2}M2K2Mi Q7 bBM;H2 T `iB+H2 bi`m+im`2bX CQm`M H Q7 bi`m+im` H #BQHQ;v- R8dURV, RRdĜRk8- kyydX (3) v2H2i >2BKQrBix- CQ FBK M/ûM- M/ KBi aBM;2`X TTH2 TB+F2`, miQK iB+ T `iB+H2 TB+FBM;- HQr@2zQ`i +`vQ@1J 7` K2@ rQ`FX CQm`M H Q7 bi`m+im` H #BQHQ;v- ky9UkV,kR8Ĝkkd- kyR3X (N) X Am/BM- SXEX EQ`B`- CX a H p2`i@hQ``2b- :XCX EH2vr2;i- M/ X S ir `/? MX 1JSA_, Tm#HB+ `+?Bp2 7Q` ` r 2H2+i`QM KB+`Qb+QTv BK ;2 / i X L im`2 J2i?Q/b- Rj- kyReX (Ry) >vmMbQQ EBK M/ > 2bmM S `FX LQMM2; iBp2 K i`Bt 7 +iQ`@ Bx iBQM # b2/ QM Hi2`M iBM; MQMM2; iBpBiv +QMbi` BM2/ H2 bi b[m `2b M/ +iBp2 b2i K2i?Q/X aAJ DQm`M H QM K i`Bt M HvbBb M/ TTHB+ iBQMb- jyUkV,dRjĜdjy- kyy3X (RR) *H m/BQ J ++QM2X bBKTH2 BMi`Q/m+iBQM iQ i?2 EGh
(Rd)
(R3)
(RN)
(ky)
(kR) (kk) (kj) (k9)
(k8)
(ke) (kd)
Rj
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7 H+BT `mK 3ya `B#QbQK2 #QmM/ iQ i?2 MiB@T`QiQxQ M /`m; 2K2@ iBM2X 2GB72- j- CmM2 kyR9X AaaL ky8y@y39sX /QB, RyXd889f2GB72X yjy3yX l_G ?iiT,ff2m`QT2TK+XQ`;f `iB+H2bfSJ*9y3ekd8X (k3) uB7 M sB Q M/ :m M;r2M u M;X 7 bi K2i?Q/ 7Q` T `iB+H2 TB+FBM; BM +`vQ@2H2+i`QM KB+`Q;` T?b # b2/ QM 7 bi _@*LLX AM AS +QM72`2M+2 T`Q+22/BM;b- pQHmK2 R3je- T ;2 ykyy3yX AS Sm#HBb?BM;- kyRdX
R9