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tÎ;24:<245OÛI58¼6Ct¹DÎh@+Óæ€Ö

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¼4½"@8Ä692^Îh@8¹D24:\¾^¹t9{J8J

[r, c] ˆ ext

¼4½"@8Ä69+:¼6Î;2€@87¼^B¼4½"@8ÄD2€@ 9àR@8589;¿e9!Ê424:¹69^Ì ß Ä67<JO@89;¿eÀ6:<2€@89 7<JOEp¼^:<Ct¹DÎ|24BDC\9;ÊR7B69J|Ó×XÖj@8Ä6589;9|ÀD245K24¿e9@89;5KJI589@87BD24:j¿«24À6À67B6¾>¿e¼tCt9;:

M (r f , ω, α)

Ç

ÉÄ69;589Wn

R

r f

7<JO@8Ä69!½Â¼XÊ^9{24:5K2^Ct7¹DJ;Ç

R

ω

7<JO@8Ä69+589{Î9;Àt@87Ê^9|ÑD9;:<C¼€Ê^9;58:<24À§½Œ2^Îh@8¼^5{Ç624BDC

R

α

7<J\@8Ä69b589{Î9;Àt@87Ê^9%ÑD9;:<CJA7ê;9Ë]@8¼4Ëx9{Î;Î9;Bg@8587<Î7@UÝ|5K2€@87¼DÇgÒÅ@8Ä69OJA7ê;9I¼4½t@8Ä69p589{Î9Àt@87<Ê49IÑD9;:<C

7<J"@8Ä69&Ct7<24¿e9@89;5 ¼4½D@8Ä69589{Î9;Àt@87Ê49pÑD9;:<C>Î75KÎ:9^Ç424BDC@8Ä69O9{Î;Î9;Bg@8587<Î7@Uݼ4½j2|589{Î9;Àt@87Ê^9

ÑD9;:<C7<J;ÇR7B¼^¹65OÎ;2^JA9^Çg@8Ä69!5K2^Ct7<24:jCt7<JG@K24BDÎ9 ½Â58¼^¿ @8Ä69!Î9;Bg@89;5O¼4½@8Ä69|589{Î9;Àt@87Ê^9 ÑD9;:<C

@8¼e@8Ä692^Îh@8¹D24:¾^¹69;J8J

[r, c] ˆ ext

¼4½"@8Ä69+:¼tÎ;2€@87¼^B@8Ä69+9àR@8589;¿e9!ʀ24:¹69XÙhÌ

ß

Ä69b7B67@87<24:t2^ÎK@K¹624:g¾^¹69{J8J\¼4½t@8Ä69p:¼tÎ;2€@87¼^B+¼4½t@8Ä69p9àR@8589;¿e9IÊ424:¹69bÎ;24B3n9O24BgÝ!9;:9;¿e9;B^@ ¼4½

L F

ÒÅ@8Ä69|¹DJA9|¼4½@8Ä69!Î9;Bg@89;5p¼4½\¿«2^J8Jb¼4½@8Ä69!Ct¼^¿«247B§7<JOJA¹6¾^¾^9;JG@K9;CDÙhÌRÍ6¼^::¼€É7B6¾Ep¼^:ÅCR¹DÎ 24BDC\9;ÊR7B69JI¿e¼tCt9;:]Çg589{Î9;Àt@87Ê^9ÑD9;:<C§Î9;Bg@89;5KJb24589 24585K24B6¾^9{Ce24:¼^B6¾+5K2XÝtJI¼^587¾^7BD2€@89{Ce2€@

[r, c] ˆ ext

Ç624BDC§24:¼^B6¾>Î75KÎ:9{Jp24:<JA¼À6:<2^Î9{C§2€@p@8Ä69!J824¿e9|:¼tÎ;2€@87¼^B£Ò R9;9|HbènÌjÔËUÕ¼4½bÓ×XÖj½Â¼^5 Ct9@K247:<J!¼4½b@8Ä69«Ct9;587Ê42€@87¼^B£¼4½b@8Ä69eÀn¼gJA7@87¼^B67B6¾ÀD245K24¿e9@89;5KJ

θ

24BDC

k

ÇÉÄ69;589

θ

7<J @KÄt9 24B6¾^:9+3n9@GÉb9;9;B?2^C€FG2^Î9;Bg@&5K2XÝtJ24BDC

k

7<JO@KÄt95K2€@87¼«¼4½ 5K2^Ct77\¼4½I2^C€FG2^Î9;B^@|Î75KÎ:9{JKÙhÌ ß Ä69 Î75KÎ:9¼4½%5K2^Ct7¹DJ

r nr

7<J JA¹DÎKÄ@8ÄD2€@|7@|7<J&@8Ä69Î75KÎ:9¼4½I¿«2€àt7¿¹6¿5K2^Ct7¹DJ½Â¼^5 ÉÄ67<Î8Ä£2€@

:9{2^JG@¼^B69eÀn¼^7Bg@3n9;:¼^B6¾gJ|@8¼

L F

Ì ß

Ä69e½Â¹6BDÎh@87¼^B

F

7<Jμ^¿eÀ6¹t@89{C£½Â¼^59{2^Î8ÄåÎ9;B^@89;5¼4½

589{Î9;Àt@87Ê^9 ÑD9;:<C6JOÀ6:<2^Î9{C§¼^B§@8Ä69+587B6¾gJp24BDC§½Â¼^5O9{2^ÎKÄ9;:9;¿e9;Bg@3n9;:¼^B6¾^7B6¾@8¼@8Ä69|½Â¼€Ê^9{26Ì

ß

Ä69 :¼tÎ;2€@87¼^B«¼4½

f ext

¾^¹69{J8JA9{C>½Â¼^5I@8Ä69 B69àR@p7@89;5K2€@87¼^Be7ÅJI¾^7Ê^9;B«3gÝ>@8Ä69 :¼tÎ;2€@87¼^B«¼4½@8Ä69 9àR@8589;¿e9Oʀ24:¹69

f ˆ ext

¼4½

F

μ^¿eÀ6¹t@89{C>¹6B^@87:6B6¼€É+̀Ðx½j¿e¼^589b@8ÄD24Be¼^B69O9;:9;¿e9;B^@I¼4½

L F

Î;24B

3n9&@8Ä69!2^Îh@8¹D24:D¾^¹69{J8J;Ç4@8Ä69;B¼^B69&¼4½\@8Ä69¿á7<Jb5K24BDCt¼^¿e:ÝeJA9;:9{Îh@89{CÌ

ß

Ä69|24:¾^¼^587@8Ä6¿ JG@8¼^ÀDJ

ÉÄ69;B¢@8Ä69ºÊ€24:¹69{Je¼4½

[r, c] ˆ ext

3j9@UÉp9;9;BÁ@UÉp¼åμ^BDJA9{ιt@87Ê^97@89;5K2€@87¼^BDJeCt¼.B6¼4@§ÎKÄD24B6¾^9

(7)

24BRÝR¿e¼^589^Ì ß Ä69Îh¼^BRÊ^9;58¾^9;BDÎ9|7<JO¾^¹D245K24Bg@89;9{CÌ

H&ÌÌÈÌÌnÄD2^J2e¿«2€àt7¿«24:\À658¼tÎ9{J8JA7B6¾>@87¿e9¼4½

O(n r f + nr ∗ (round( 2∗π θ )))

Àn9;5

7@89;5K2€@87¼^B\ÇRÉÄ69;589

n r f

7<Jb@8Ä69!BR¹6¿3j95¼4½"9;:9;¿e9;B^@KJO¼4½\@8Ä69|½Â¼€Ê^9{26Ç

nr

7<Jp@8Ä69!Bg¹6¿3n9;5¼4½ 587B6¾gJ24BDC

round(( 2∗π θ ))

7<JO@8Ä69BR¹6¿3j9;5&¼4½ 589{Î9;Àt@87Ê^9!ÑD9:ÅCtJ&À6:<2^Î9{C¼^B9{2^Î8Ä587B6¾DÌ6Ðx@

ÄD2^J!2§½Œ2^JG@+μ^BRÊ^9;58¾^9;B6Î93n9{Î;24¹DJA9^Çn½Â¼^5!9{2^ÎKÄ£7@89;5K2€@87¼^B\Çj2:¼6Î;24:"B69;7¾^ÄR3n¼^58Ä6¼R¼6C¼4½b@8Ä69

9{JG@87¿«2€@89{C¿e7B67¿¹6¿Ê424:¹69μ^¿eÀn¼gJA9{C¼4½ @8Ä699;:9;¿e9;B^@KJ&¼4½ @8Ä69+½Â¼€Ê49{2e7<J 24BD24:ÝRê;9{C7B

Ct9@K247:]Çt2€@p@8Ä69!J824¿e9 @87¿e9|¼4@8Ä69;5pÀD245A@KJp¼4½\@8Ä69|ÉÄ6¼^:9|Ct¼^¿«247B24589&@K24Ã^9;B§7Bg@8¼e2^Î;μ^¹6B^@

3RÝJ824¿eÀ6:7B6¾@8Ä69+Î9;Bg@89;5KJp¼4½"589{Î9;Àt@87Ê49|ÑD9;:<C6JpÀ6:<2^Î9{C¼€Ê49;5O587B6¾gJp¼4½"7BDÎ589{2^JA7B6¾5K2^Ct7¹DJ

2€@&2e:¼^¾g24587@8Ä6¿e7<Î|J8Î;24:9^Ì

R¼^¿e9!9à624¿eÀ6:9{JO¼4½"¿«24À6À67B6¾e@89;¿eÀt:Å2€@89{J24589+JAÄ6¼XÉB7BºÍ"7¾^¹6589+Õ6Ì

ß Ä69pH&ÌÌÈÌÌ4ÄD2^J"3n9;9;B@89{JG@89{C¹DJA7B6¾ ¸ 7:<JA¼^B J =º¼tCt9;:jÓ×^×XÖtÀD245K24¿e9@89;5KJbÒ

ω = 0.5

24BDC

α = 0.25

Ù¼^B:<2€@A@87<Î9{J ¼4½Ô{ç^ç 3gݧÔ{ç^ç&9;:9;¿e9;Bg@KJ;Ì ß Ä69b½Â¼€Ê^9{24:R5K2^Ct7¹DJÉO2^J"7BDÎ589{2^JA9{C

½Â58¼^¿ Ô%@8¼ 4ç6Ì;Í6¼^5\9{2^ÎKÄ!½Â¼XÊ^9{24:^5K2^Ct7¹DJ;Çh@8Ä69b24:¾^¼^587@8Ä6¿ ÉO2^Jj@K9;JG@K9;C×4çp@87¿e9{J;Ì ß Ä69I7B67@87<24:

¾^¹69{J8J\¼4½t@8Ä69b:¼tÎ;2€@87¼^B+¼4½R@8Ä69b9àR@8589;¿e9bʀ24:¹69IÉO2^J\5K24BDCt¼^¿e:Ý!JA9h@XÌ ß Ä69b¿e7B67¿¹6¿ Ê424:¹69{J

¼4½"@8Ä69|½Â¼^::<¼XÉ7B6¾>½Â¹6BDÎh@87¼^BDJÉb9;589|½Â¼^¹6BDC_n

›R—  mnNKaxTGNKc>Z<¦gWg`8TG_Y{Wga

bSh}

ì

¦gWg`8TG_Y{W MOY{VQ;_W

ï

Z = (X 2 + Y 2 ) 1 2 L Z = {[r, c]| − 99 ≤ r ≤ 0, −99 ≤ c ≤ 0}

w

Z = (X 2 + Y 2 ) L Z = {[r, c]| − 49 ≤ r ≤ 50, −49 ≤ c ≤ 50}

Z = 1 − 1 (X 2 +Y 2 +1 ) L Z = {[r, c]| − 49 ≤ r ≤ 50, −49 ≤ c ≤ 50}

í

Z = 1 − 1 (X 2 +Y 2 +1 ) L Z = {[r, c]| − 89 ≤ r ≤ 10, −69 ≤ c ≤ 30}

Í6¼^5I9{2^ÎKÄe½Â¹6BDÎh@87¼^B24BDC>½Â¼XÊ^9{24:D5K2^Ct7¹DJ;ǀ@8Ä69 BR¹6¿3n9;5p¼4½9{JG@87¿«2€@89{Ce½Â¹6B6Îh@87<¼^B«Ê424:¹69{J

24BDC£@8Ä69§2XÊ^9;5K24¾49>BR¹6¿3n9;5¼4½O7@89;5K2€@87¼^BDJ!7<JJAÄ6¼€ÉB»7B¤Í"7¾DÌ×tÌ &¼4@89e@8ÄD2€@+½Â¼^524::%@8Ä69

@89{JG@87B6¾»½Â¹6BDÎh@87¼^BDJ;Ç%3j9;7B6¾»@8Ä69ºÎ¼^BRÊ^9;58¾^9BDÎ9¾^¹D245K24B^@89;9{CÇ"@8Ä69ºÀn9;5KÎ9;Bg@K24¾^9¼4½|9;Ê424:¹tË

2€@89{CºÀj¼^7B^@KJ|7BDÎ589{2^JA9{J&¹6B6B69{Î9;J8J824587<:ÝÉÄ69;Bº@8Ä69½Â¼€Ê49{24:"5K2^Ct7¹DJ 7BDÎ589{2^JA9{J;Ì ß Ä69>Î;2^JA9{J

½Â¼^5ÉÄ67<ÎKÄ@8Ä69|½Â¹6BDÎh@87¼^Bº7<JOB6¼4@ JA¿e¼R¼4@8Ä\Ç6μ^Bg@89àR@8¹D24:7Bt½Â¼^58¿«2€@87¼^Bμ^¹6:<C3n9+¹DJA9½Â¹6:]Ì

¯g° ·U±6·KLKA·b°k °b±

R¼^¿e9Ct9ÑDB67@87¼^BDJ@8¼?3j9§¹DJA9{Cå7Bå@8Ä69À6589{JA9;B^@eJA9{Îh@87¼^B¤24589eÑD5KJG@>7B^@858¼6Ct¹DÎ9{CÌ ½Å@89;5AË

ÉO245KC6J;Çt24Bº24:¾^¼^587@8Ä6¿ @8¼«Î;245858Ý«¼^¹t@@8Ä69+7B67@87<24:24:7¾^B6¿e9;B^@&7<JOÀ658¼^Àn¼gJA9{CÌ

1|7Ê49;B?2§JA9@ ¼4½IJA9;¾^¿e9;Bg@89{CÀD2€@A@89;58BDJ&¼4½I2e¾^7Ê^9;Bº:9{JA7¼^B\Ç67Bº@8Ä697B67@87<24: 24:7¾^B6¿e9;B^@

JG@K24¾^9^Çb¾^:¼^3D24:&@85K24BDJA:<2€@K7¼^BDJ§24BDCÏ58¼4@K2€@87¼^BDJ§9âj9{Îh@KJ3n9@GÉb9;9;B ÀD2475KJ§¼4½ÀD2€@A@89;58BDJ24589

589;¿e¼€Ê49{CÌjÍ6¼^5!μ^¿eÀD24587<JA¼^BºÀ6¹t58Àj¼gJA9{J;ÇD7@!7<J!2^J8JA¹6¿e9{C@8ÄD2€@|@8Ä69eCt7<JAÀn9;5KJA7¼^B?¼4½%@8Ä69>9;:Ë

(8)

rf=5 theta=60 rf=5 theta=40 rf=5 theta=30

rf=10 theta=60 rf=10 theta=40 rf=10 theta=30

rf=15 theta=60 rf=15 theta=40 rf=15 theta=30

rf=5 theta=60 rf=5 theta=40 rf=5 theta=30

rf=10 theta=60 rf=10 theta=40 rf=10 theta=30

rf=15 theta=60 rf=15 theta=40 rf=15 theta=30

 R deQ;PgPg_Wgk|TGNKV+PgiQTGNKa

m fgN8r>QSGNaUfgYhŸIWeY{W«Q!c^Y{VQ;_W>Y;Z ïhz{z!€r§ïhz{z|Pg_ª^NKiaIY;Z\aU_KNOŸI_TGf«c^_¥6N8SGNKW€T%Z<YqXNhQ;iDSAQ{c^_¦ga

r f = 5, 10, 15

Pg_ª^NKia%Q;WRc>c^_¥6N8SGNKW€TbQ;Wgk{iNKabtN8T]Ÿ NKNKWeQ{c xQ;`KNKW€TbSAQ8r^aIY;Z

θ = 60, 40, 30

c^NKk;SGNKNKa"Z<Y;SbTGfgNOPtY{aU_TG_Y{Wg_Wgk+PRQSAQ;V+N8TGN8S

k = 0.4

¡gSGaxTITGf^SGNKNpSGYhŸIaIQ;WRc

k = 1

iQ;axTITGf^SGNKN

(9)

0 5 10 15 20 25 30 35 40 45 50 0

20 40 60 80 100

Foveal Radius

Percentage of Evaluated Points

0 5 10 15 20 25 30 35 40 45 50

2 3 4 5 6

Foveal Radius

Iterations

  mnNKaxTGNhc?Z<¦gWg`8TG_Y{WgaQ;WRc?SGNKaU¦giTGa+Y;ZpTGfgN ª4TUSGNKV+N"Q;i¦gNeMON8TGNK`8TG_Y{W ž ik{Y;SG_TGfgVZ<Y;S

Z<YqXNhQ;iDSAQ{c^_¦gaIq;QSUr^_Wgk|ZSGY{V ïOTGYô;z4}

Ct9ÑDB67@87¼^BDJO½Â¼^::¼XÉan

!$+./4/2143U1K'*-:W3,18

A

9 -:

B

3Bk8E'+

n a = n b

1K'*/J :W3,T 9 '*-M8

L A

9 -:

L B

8 @ CFJ6/J

9 /

∃t ∧ θ ˆ

BE3 = 1JO' CFJMP

L B = Rot[T rans[L A , t], θ] ˆ

ÒGÔX×^Ù

1JM+

=+

Rot

9 -:

T rans

9,= +U/JM+ = 3,/9 /2' 3,- 9 -:&/=A9 -M8 >9 /2' 3,-lB @ -C./2' 3,- = +E8Y5 +FC./2'*),+

>S :W+

?

ZK-+F: '*-

9W?

:,'D8QC

=

+./+a8Y5

9

CF+

e

Rot

9 -:

T rans

9,= + CA3,TUT @ / 9 /2'*),+ e

Í6¼^5b9{2^ÎKħÉ7BDCt¼€É+ÇgÉb9!μ^¿eÀ6¹t@89|2+@89àR@8¹6589|Ct9{J8Î587Àt@8¼^5{Çg24BDC«@8Ä69;B§@K24Ã49|2¿e9{2^JA¹6589

¼4½ Ct7<JG@K24BDÎ9|3j9@UÉp9;9;BÀD2475KJp¼4½"É7BDCt¼€É&J;Ì ß Ä69!¿e9{2^JA¹6589|¼4½ CR7ÅJG@K24BDÎh9!¹DJA9{C7<J=?24B6ÄD2€@AË

@K24B»Ct7<JG@K24BDÎ9ÒR9;9>Hbèg¹D2€@87¼^B»ÕgÙhÌ ß Ä69@89àR@8¹6589eCt9{J8Î587Àt@8¼^5 7<J @8Ä69>Ê^¼^:¹6¿e9¹6BDCt9;5|@8Ä69

JA¹65A½Œ2^Î9p½Â¼^58¿e9{C>3RÝ@8Ä69B6¼^58¿«24:7ê;9{CÊ424:¹69{J ¼4½n@8Ä69À67àt9;:<J%¼4½n@8Ä69&É7BDCt¼€É+̀ÐUBe@8Ä67<J%ÉO2XÝ^Ç

@8Ä69èg¹D24Bg@87ê{2€@87¼^B:9;Ê49;:\7Bt½Â¼^58¿«2€@87¼^B7<J589;¿e¼€Ê49{CÇtÉÄ67:9!@8Ä69!@89àR@8¹6589!7<JOÀ6589{JA9;58Ê^9{CÌ

!_$_JM+6:,' + =

+.-CF+63BV),3

>@ TV+

dV (A),(B)

mF+./214+F+.- /JM+&1K'*-:W3,18

A

9 -:

B

'D8 ; '*-6N+.-+ =A9,>; :W+2ZK-+F: 9 8QP

dV (A),(B) = dV (B),(A) = 1 2

n A∩B

X

i=1

|w (A) [i] − w (B) [i]|

ÒGÔ{ÕgÙ

1JM+ + '*-H/JM+ 5 +E8Q+.-7/G143

(10)

w (J) [i] = x (J) [i]

P n J

i=1 x (J) [i]

ÒGÔ;ØRÙ

x (J) [i]

'D8c/JM+c) 9,>@ +L3B /JM+45!'O+ >>3QC 9 /+F: 9 /K/JM+

i − th

5M3 8E'*/2' 3,-H3B /JM+

J − th

1K'*-:W3,1

3B

n J

+ >

+QTV+.-7/ 8

e

<3,/+L/J

9 /

0 ≤ dV ≤ 1

e

È ¹6587B6¾?@8Ä697B67@87<24:24:7¾^B6¿e9;Bg@{Ç 7@e7<Je2^J8JA¹6¿e9{Cǽ¼^5«2?¾^7Ê^9;B¢:9{JA7¼^B\Ç@8ÄD2€@eHbèg¹D2€Ë

@87¼^BåÔX×>7<J@858¹69+½Â¼^5@8Ä697Bg@89;5KJA9{Îh@87¼^Bº¼4½"@8Ä69@85K24BDJA:<2€@89{C24BDC58¼4@K2€@89{CÊ49;5KJA7¼^BDJ¼4½ @8Ä69

ÀD2475|¼4½IÀD2€@A@89;58B»Ct¼^¿«247BDJ;ÇD½Â¼^5|ÉÄ67<Î8Ä?@8Ä69eCt7<JG@K24BDÎ9ҌJA9;9>Hbèg¹D2€@87¼^B¤Ô{ÕgÙ3n9@GÉp9;9;B?@8Ä69

μ^58589{JAÀn¼^BDCt7B6¾£À67àt9;:&ʀ24:¹69{Je7<Je¿e7B67¿«24:]ÌIÐx@§7<J«24:<JA¼.2^J8JA¹6¿e9{C¤@8ÄD2€@e@KÄt9º¼^36FU9{Îh@KJ>@8¼

3n924:7¾^B69{CÄD2XÊ^9|@8Ä69J824¿e9J8Î;24:9^Ì6Æ&¼€Ép9;Ê49;5{ÇD2^JO¿e9;Bg@87¼^B69{C3n9½Â¼^589^Çt@8Ä69+7B6À6¹t@ ¼4½ @8Ä69

7B67@87<24: 24:7¾^B6¿e9;Bg@!JG@K24¾^9+7<J|2JA9@|¼4½IJA9;¾^¿e9;Bg@89{CºÀD2€@A@89;58BDJÒÅ@8Ä67<J|¿e9{24BDJ;Ç6@8Ä69¼€Ê^9;58:<24ÀDJ

3n9@GÉp9;9;B?JA9;¾^¿e9;Bg@89{C¼^3tFU9{Îh@KJ 7BDCR7ÅÎ2€@K7B6¾§:9{JA7¼^Bº24BDC@8Ä69¼^587¾^7BD24:C62€@K2gÙhÇDÉÄ67<Î8Ä?¿«2{Ý

B6¼4@eÄD2{Ê^9§@8Ä69ºJ824¿e924589{26ÌpÒ R9;9^Ç"½Â¼^5>7BDJG@K24BDÎ9^Ç%Í"7¾^¹6589{Je×@8¼»Ø£7B ÓÔ;Ø4ÖÂÙhÌ

ß

Ä69¿e¼gJG@

7¿eÀn¼^5A@K24Bg@e589{JA¹6:@>À658¼tCt¹DÎ9{C¤3gÝ.@8Ä67<J«JG@K24¾^97<J@8Ä69ºJAÀn9{Î7ÑnÎ;2€@87¼^B¤¼4½&@8Ä69BR¹6¿3n9;5e¼4½

:<24BDCt¿«2458ÃtJb@8¼«3n9+589;¾^7<JG@89;589{C§7Bº2>B69àR@ JG@K24¾^9^ÇR@8Ä67<J¿e9{24BDJ;ÇR@8ÄD2€@@8Ä6924589{2¼4½"@8Ä69+¼^3tË

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|B − G|

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(13)

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µ p,l,b = 1 n s n c

n s

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i=1 n c

X

j=1

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E[corr(X p,l,s,c,b , X p,l,:,:,b )] = 1 n s n c − 1 (

n s

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X

j=1

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n s

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(14)

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(15)

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kxk Y kx2

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