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LF¼4½b2�:<2�@A@87<Î�9

L ={[r, c]|1 ≤ r ≤ M, 1 ≤ c ≤ N}

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7<J[r, c]ext

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7<JOEp¼^:<Ct¹DÎ|24BDC� \9;ÊR7 B69�� J|Ó ×XÖj@8Ä6589;9|ÀD245K24¿e9�@89;5KJI589�@87 BD24:j¿«24À6À67 B6¾>¿e¼tCt9;:M(rf , ω, α)

ÇÉ�Ä69;589WnR rf

7<JO@8Ä69!½Â¼XÊ^9{24:�5K2^Ct7 ¹DJ;ÇR ω

7<JO@8Ä69+589{Î�9;Àt@87 Ê^9|ÑD9;:<C�¼�Ê^9;58:<24À§½�2^Îh@8¼^5{Ç624BDCR α

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θ24BDC

kÇ�É�Ä69;589

θ7<J @KÄt9

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Î�7 5KÎ�: 9�¼4½%5K2^Ct7 ¹DJrnr

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LFÌ ß Ä69e½Â¹6BDÎh@87 ¼^B

F7<J�Î�¼^¿eÀ6¹t@89{C£½Â¼^5�9{2^Î8ÄåÎ�9;B^@89;5�¼4½

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9�àR@8589;¿e9OÊ�24: ¹69f̂ext

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LFÎ;24B

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[A} d�} d�}tmDNK`AfgWg_ `hQ;inbNKPtY;SUT��p¦gV|�tN8S�� ñ

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H&Ì ��Ì È�Ì ��ÌnÄD2^J�2e¿«2�àt7 ¿«24:\À658¼tÎ�9{J8JA7 B6¾>@87 ¿e9�¼4½O(nrf

+ nr ∗ (round( 2∗πθ

)))Àn9;5

7 @89;5K2�@87 ¼^B\ÇRÉ�Ä69;589nrf

7<Jb@8Ä69!BR¹6¿�3j9�5�¼4½"9;: 9;¿e9;B^@KJO¼4½\@8Ä69|½Â¼�Ê^9{26Çnr

7<Jp@8Ä69!Bg¹6¿�3n9;5�¼4½587 B6¾gJ�24BDC

round(( 2∗πθ

))7<JO@8Ä69�BR¹6¿�3j9;5&¼4½ 589{Î�9;Àt@87 Ê^9!ÑD9�:ÅCtJ&À6:<2^Î�9{C�¼^B9{2^Î8Ä587 B6¾DÌ6Ðx@

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�R¼^¿e9!9�à624¿eÀ6: 9{JO¼4½"¿«24À6À67 B6¾e@89;¿eÀt:Å2�@89{J�24589+JAÄ6¼XÉ�B�7 BºÍ"7 ¾^¹6589+Õ6Ì

ß Ä69pH&Ì ��Ì È�Ì ��Ì4ÄD2^J"3n9;9;B�@89{JG@89{C�¹DJA7 B6¾ ¸ 7 :<JA¼^B � J =º¼tCt9;:jÓ ×^×XÖtÀD245K24¿e9�@89;5KJbÒω = 0.524BDC

α = 0.25Ù�¼^B�:<2�@A@87<Î�9{J ¼4½�Ô{ç^ç 3gÝ§Ô{ç^ç&9;: 9;¿e9;Bg@KJ;Ì ß Ä69b½Â¼�Ê^9{24:R5K2^Ct7 ¹DJ�ÉO2^J"7 BDÎ�589{2^JA9{C

½Â58¼^¿ Ô%@8¼ �4ç6Ì;Í6¼^5\9{2^ÎKÄ!½Â¼XÊ^9{24:^5K2^Ct7 ¹DJ;Çh@8Ä69b24: ¾^¼^587 @8Ä6¿ ÉO2^Jj@K9;JG@K9;C�×4çp@87 ¿e9{J;Ì ß Ä69I7 B67 @87<24:¾^¹69{J8J\¼4½t@8Ä69b: ¼tÎ;2�@87 ¼^B+¼4½R@8Ä69b9�àR@8589;¿e9bÊ�24: ¹69IÉO2^J\5K24BDCt¼^¿e: Ý!JA9h@XÌ ß Ä69b¿e7 B67 ¿�¹6¿ Ê424: ¹69{J¼4½"@8Ä69|½Â¼^: :<¼XÉ�7 B6¾>½Â¹6BDÎh@87 ¼^BDJ�Éb9;589|½Â¼^¹6BDC_n

� �R�� mnNKaxTGNKc>Z<¦gWg`8TG_ Y{Wga�bSh} ì ¦gWg`8TG_ Y{W MOY{V�Q;_ Wï

Z = (X2 + Y 2)−1

2 LZ = {[r, c]| − 99 ≤ r ≤ 0,−99 ≤ c ≤ 0}wZ = (X2 + Y 2) LZ = {[r, c]| − 49 ≤ r ≤ 50,−49 ≤ c ≤ 50}�

Z = 1− 1(X2+Y 2+1

) LZ = {[r, c]| − 49 ≤ r ≤ 50,−49 ≤ c ≤ 50}íZ = 1− 1

(X2+Y 2+1) LZ = {[r, c]| − 89 ≤ r ≤ 10,−69 ≤ c ≤ 30}

Í6¼^5I9{2^ÎKÄe½Â¹6BDÎh@87 ¼^B�24BDC>½Â¼XÊ^9{24:D5K2^Ct7 ¹DJ;Ç�@8Ä69 BR¹6¿�3n9;5p¼4½�9{JG@87 ¿«2�@89{Ce½Â¹6B6Îh@87<¼^B«Ê424: ¹69{J24BDC£@8Ä69§2XÊ^9;5K24¾49>BR¹6¿�3n9;5�¼4½O7 @89;5K2�@87 ¼^BDJ!7<J�JAÄ6¼�É�B»7 B¤Í"7 ¾DÌ�×tÌ �&¼4@89e@8ÄD2�@+½Â¼^5�24: :%@8Ä69@89{JG@87 B6¾»½Â¹6BDÎh@87 ¼^BDJ;Ç%3j9;7 B6¾»@8Ä69ºÎ�¼^BRÊ^9;58¾^9�BDÎ�9�¾^¹D245K24B^@89;9{C�Ç"@8Ä69ºÀn9;5KÎ�9;Bg@K24¾^9�¼4½|9;Ê424: ¹tË2�@89{CºÀj¼^7 B^@KJ|7 BDÎ�589{2^JA9{J&¹6B6B69{Î�9;J8J824587<: Ý�É�Ä69;Bº@8Ä69�½Â¼�Ê49{24:"5K2^Ct7 ¹DJ 7 BDÎ�589{2^JA9{J;Ì ß Ä69>Î;2^JA9{J½Â¼^5�É�Ä67<ÎKÄ�@8Ä69|½Â¹6BDÎh@87 ¼^Bº7<JOB6¼4@ JA¿e¼R¼4@8Ä\Ç6Î�¼^Bg@89�àR@8¹D24:�7 Bt½Â¼^58¿«2�@87 ¼^BÎ�¼^¹6:<C�3n9+¹DJA9�½Â¹6:]Ì

� ¯g° ·U±6· �K�L�K�A·��b°k� �°b±

�R¼^¿e9�Ct9�ÑDB67 @87 ¼^BDJ�@8¼?3j9§¹DJA9{Cå7 Bå@8Ä69�À6589{JA9;B^@eJA9{Îh@87 ¼^B¤24589eÑD5KJG@>7 B^@858¼6Ct¹DÎ�9{C�Ì ��½Å@89;5AËÉO245KC6J;Çt24Bº24: ¾^¼^587 @8Ä6¿ @8¼«Î;245858Ý«¼^¹t@�@8Ä69+7 B67 @87<24:�24: 7 ¾^B6¿e9;B^@&7<JOÀ658¼^Àn¼gJA9{C�Ì

1|7 Ê49;B?2§JA9�@ ¼4½IJA9;¾^¿e9;Bg@89{CÀD2�@A@89;58BDJ&¼4½I2e¾^7 Ê^9;Bº: 9{JA7 ¼^B\Ç67 Bº@8Ä69�7 B67 @87<24: 24: 7 ¾^B6¿e9;B^@JG@K24¾^9^Çb¾^: ¼^3D24:&@85K24BDJA:<2�@K7 ¼^BDJ§24BDCÏ58¼4@K2�@87 ¼^BDJ§9�âj9{Îh@KJ�3n9�@GÉb9;9;B ÀD247 5KJ§¼4½�ÀD2�@A@89;58BDJ�24589589;¿e¼�Ê49{C�ÌjÍ6¼^5!Î�¼^¿eÀD24587<JA¼^BºÀ6¹t58Àj¼gJA9{J;ÇD7 @!7<J!2^J8JA¹6¿e9{C�@8ÄD2�@|@8Ä69eCt7<JAÀn9;5KJA7 ¼^B?¼4½%@8Ä69>9;: Ë9;¿e9;Bg@KJ ¼4½ @8Ä69>C62�@K2�ËUJA9�@KJ�7<J ¼4½I7 B^@89;589{JG@{ÇnÉ�Ä67 : 9+@8Ä69�¿e9{24B£Ct7 âj9;589;BDÎ�9{J 24589�B6¼4@{Ì �R¼^¿e9

y deQ;i N8TUTG_nN8TpQ;iÅ}

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






�� R� deQ;PgPg_ Wgk|TGNKV+Pgi Q�TGNKam fgN8r>Q�SGN�aUfgYh�IWeY{W«Q!c^Y{V�Q;_ W>Y;Z ïhz{z!��r§ïhz{z|Pg_ ª^NKi aIY;Z\aU_ �KNO�I_ TGf«c^_ ¥6N8SGNKW�T%Z<Y�qXNhQ;iDSAQ{c^_ ¦ga�rf = 5, 10, 15

Pg_ ª^NKi a��%Q;WRc>c^_ ¥6N8SGNKW�TbQ;Wgk{i NKab�tN8T]� NKNKWeQ{c�xQ;`KNKW�TbSAQ8r^aIY;Z �θ = 60, 40, 30c^NKk;SGNKNKa��"Z<Y;SbTGfgNOPtY{aU_ TG_ Y{Wg_ Wgk+PRQ�SAQ;V+N8TGN8S

k = 0.4� ¡gSGaxTITGf^SGNKNpSGYh�Ia��IQ;WRc

k = 1� i Q;axTITGf^SGNKN

SGYh�Ia��A}


0 5 10 15 20 25 30 35 40 45 500

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�� mnNKaxTGNhc?Z<¦gWg`8TG_ Y{Wga�Q;WRc?SGNKaU¦gi TGa+Y;ZpTGfgN �ª4TUSGNKV+N��"Q;i ¦gNeMON8TGNK`8TG_ Y{W � i k{Y;SG_ TGfgV�Z<Y;SZ<Y�qXNhQ;iDSAQ{c^_ ¦gaIq;Q�SUr^_ Wgk|Z SGY{V ïOTGY�ô;z4}

Ct9�ÑDB67 @87 ¼^BDJO½Â¼^: : ¼XÉan

�� !��$��+./4/2143U1K'*-�:W3,1�8A 9 -�:

B3�Bk8E' ��+

na = nb1K'*/�J :W3,T 9 '*-M8

LA 9 -�:LB

8 @ CFJ6/�J 9 / ∃t ∧ θ̂ BE3 = 1�JO' CFJMP

LB = Rot[Trans[LA, t], θ̂]ÒGÔX×^Ù

1�JM+ = + Rot 9 -�:Trans 9,= +U/�JM+ = 3,/ 9 /2' 3,- 9 -�:&/ =A9 -M8 > 9 /2' 3,-lB @ -�C./2' 3,- = +E8Y5+FC./2'*),+ > S :W+ ?ZK-�+F: '*- 9��W? :,'D8QC = +./�+a8Y5 9 CF+ e Rot 9 -�:

Trans 9,= + CA3,TUT @ / 9 /2'*),+ eÍ6¼^5b9{2^ÎKÄ§É�7 BDCt¼�É+ÇgÉb9!Î�¼^¿eÀ6¹t@89|2+@89�àR@8¹6589|Ct9{J8Î�587 Àt@8¼^5{Çg24BDC«@8Ä69;B§@K24Ã49|2�¿e9{2^JA¹6589

¼4½ Ct7<JG@K24BDÎ�9|3j9�@UÉp9;9;B�ÀD247 5KJp¼4½"É�7 BDCt¼�É&J;Ì ß Ä69!¿e9{2^JA¹6589|¼4½ CR7ÅJG@K24BDÎh9!¹DJA9{C�7<J�=?24B6ÄD2�@AË@K24B»Ct7<JG@K24BDÎ�9�Ò �R9;9>Hbèg¹D2�@87 ¼^B»ÕgÙhÌ ß Ä69�@89�àR@8¹6589eCt9{J8Î�587 Àt@8¼^5 7<J @8Ä69>Ê^¼^: ¹6¿e9�¹6BDCt9;5|@8Ä69JA¹65A½�2^Î�9p½Â¼^58¿e9{C>3RÝ�@8Ä69�B6¼^58¿«24: 7 ê;9{C�Ê424: ¹69{J ¼4½n@8Ä69�À67 àt9;:<J%¼4½n@8Ä69&É�7 BDCt¼�É+Ì�ÐUBe@8Ä67<J%ÉO2XÝ^Ç@8Ä69�èg¹D24Bg@87 ê{2�@87 ¼^B�: 9;Ê49;:\7 Bt½Â¼^58¿«2�@87 ¼^B�7<J�589;¿e¼�Ê49{C�ÇtÉ�Ä67 : 9!@8Ä69!@89�àR@8¹6589!7<JOÀ6589{JA9;58Ê^9{C�Ì

�� !��_$� _JM+6:,' � + = +.-�CF+63�BV),3 > @ TV+dV(A),(B)

mF+./214+F+.- /�JM+&1K'*-�:W3,1�8A 9 -�:

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

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

2

nA∩B∑

i=1

|w(A)[i]− w(B)[i]|ÒGÔ{ÕgÙ

1�JM+ = + ; '*-H/�JM+ 5 = +E8Q+.-7/G143 =A\ ;

ïhz deQ;i N8TUTG_nN8TpQ;iÅ}

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

∑nJ

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