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Page 1: NUMERICAL OPTIMIZATION METHODS FOR IMAGE

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46+ , + , ���#� 1 +0+�+%+%+�+%+%+0+�+%+%+�+%+%+�+0+%+%+�+%+%+�+0+%+%+�+%+%+�+0+%+%+�+%+ 0�q46+ , + o , � +%+%+0+�+%+%+�+%+%+0+�+%+%+�+%+%+�+0+%+%+�+%+%+�+0+%+%+�+%+%+�+0+%+%+�+%+ 0bo46+ , +10 AC�3iTlP\����aiF���b_����$. ���Hn c �aeSg�� e cr_�+0+%+%+�+%+%+�+0+%+%+�+%+%+�+0+%+%+�+%+ 0bo

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9 ��� ��� )��� �+� ����� B��:D+ , 1�15h ,�iF���ac�� +0+�+%+%+�+%+%+0+�+%+%+�+%+%+�+0+%+%+�+%+%+�+0+%+%+�+%+%+�+0+%+%+�+%+ 0 ;

� ��� ��� )��� �+��� ����=�� �!B< + , ' h%1�A & +�+%+%+0+�+%+%+�+%+%+0+�+%+%+�+%+%+�+0+%+%+�+%+%+�+0+%+%+�+%+%+�+0+%+%+�+%+ 4G0

< + , + , 1Qc*���bi%�Acbn���� +�+%+%+0+�+%+%+�+%+%+�+0+%+%+�+%+%+�+0+%+%+�+%+%+�+0+%+%+�+%+ 4G:< + , + o 1Qc*���bi%�Acbn���� +�+%+%+0+�+%+%+�+%+%+�+0+%+%+�+%+%+�+0+%+%+�+%+%+�+0+%+%+�+%+ 4G:< + , +10 .5� `bcbn e����Hg +%+%+�+%+%+0+�+%+%+�+%+%+�+0+%+%+�+%+%+�+0+%+%+�+%+%+�+0+%+%+�+%+ 4 <

< + o 1D����n�� eV_a`'ZcreS_ ��+0+�+%+%+�+%+%+0+�+%+%+�+%+%+�+0+%+%+�+%+%+�+0+%+%+�+%+%+�+0+%+%+�+%+ 4�;< +10 � _ai+���bj&� � i � � cr_#,�iF���acD��� +0+�+%+%+�+%+%+�+0+%+%+�+%+%+�+0+%+%+�+%+%+�+0+%+%+�+%+ 4 @< +54 15jF���SeV_a` +%+�+%+%+0+�+%+%+�+%+%+0+�+%+%+�+%+%+�+0+%+%+�+%+%+�+0+%+%+�+%+%+�+0+%+%+�+%+ :�q< +1: hpe���`bcr_����/' n i<jkcr_��He�� e cr_aiTn-� +0+�+%+%+�+%+%+�+0+%+%+�+%+%+�+0+%+%+�+%+%+�+0+%+%+�+%+ :bo< + < - iT`r^�����n e�� �� e cr_ +�+%+%+�+%+%+0+�+%+%+�+%+%+�+0+%+%+�+%+%+�+0+%+%+�+%+%+�+0+%+%+�+%+ :�0< + ; ' n i �3eSj&� cbn"! A]cbn n i<j&� cbn;,�iF���ac�� +�+%+%+�+%+%+�+0+%+%+�+%+%+�+0+%+%+�+%+%+�+0+%+%+�+%+ : << + @ " �]c! � iF�bi � ����� 1 +%+%+�+%+%+0+�+%+%+�+%+%+�+0+%+%+�+%+%+�+0+%+%+�+%+%+�+0+%+%+�+%+ : <

� ��� ��� )��� �+� C & �!�� ���

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� C ��� �� ��� = E��������E+��� �������'���+� "D�����!� �$%@ + , , � ^�� eS_a` 1�15h +�+%+%+�+%+%+0+�+%+%+�+%+%+�+0+%+%+�+%+%+�+0+%+%+�+%+%+�+0+%+%+�+%+ ; ,@ + o , � ^�� eS_a`�' h%1�A & +%+�+%+%+0+�+%+%+�+%+%+�+0+%+%+�+%+%+�+0+%+%+�+%+%+�+0+%+%+�+%+ ;�o@ +10 , � ^�� eS_a`�,#� � & 1 +%+%+�+%+%+0+�+%+%+�+%+%+�+0+%+%+�+%+%+�+0+%+%+�+%+%+�+0+%+%+�+%+ ;�4@ +54 AC�aiTlQ\����aiF���b_�����.;���3n c �aeSg�� e cr_�+�+%+%+�+0+%+%+�+%+%+�+0+%+%+�+%+%+�+0+%+%+�+%+ ;�4

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< + , "Mn ^ai%�b_�� '&h%1�A & n i ��� cbn-�� e cr_ �/cbn i �bj-��� +0+%+%+�+%+%+�+0+%+%+�+%+%+�+0+%+%+�+%+ :�:< + o ?5� ifc��(� �]c! � iF�bi �Zg iF���ac�� +%+0+�+%+%+�+%+%+�+0+%+%+�+%+%+�+0+%+%+�+%+%+�+0+%+%+�+%+ <#,< +10 15eS_3`*� i�� c*���biTn �b_�� � �]c! � iF�bi �/� c*���biTn 3 i+�6�bg���� i , 7 +�+%+%+�+0+%+%+�+%+%+�+0+%+%+�+%+ < 4< +54 15eS_3`*� i�� c*���biTn �b_�� � �]c! � iF�bi �/� c*���biTn 3 i+�6�bg���� i?0*7 +�+%+%+�+0+%+%+�+%+%+�+0+%+%+�+%+ < :< +1: 15eS_3`*� i�� c*���biTn �b_�� � �]c! � iF�bi �/� c*���biTn 3 i+�6�bg���� i 4 7 +�+%+%+�+0+%+%+�+%+%+�+0+%+%+�+%+ <�<

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' 3 �82 ��7 ' 3 � � 2 �67 � �ai<_aiF�biTn � � � g c���� 2' 3 �82 ��7 ' 3 �32 � � 7 � �ai<_aiF�biTn � � � g cD��� 2� �aiTn i g cD�+e������ai����)*�� � �"c��7iTn �� cbn + � _ g�� n e'� _ac����� e cr_ � : � 0 ���;�aiTn i 0 �b_�� : ��n i�bi<j&� cbn e��Ti �KeSg��`bi � �b_��

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)+*******,� 3 oD+1:*7

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4�#

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$ �� �&% � $ ��� ���'% � $ � �3�1�'% � ����� $ � � �)( ���&% �$ �����&% � $ �� �&% � $ � � ���'% � ����� $ � � �)($�1�&% �$ ���1�&% � $ � ���&% � $ �� �&% � ����� $ � � �)(+* �&% �!!! ! ! ! $ � � ���&% �

$ ��� �$���'% � $ ���#�����&% � ����� $ �,%���� � $ ����'% �

)+*******,�

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Page 21: NUMERICAL OPTIMIZATION METHODS FOR IMAGE

� � � � � � � ���&� �������� �,����� ��������� ��� A

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������ � �� � �� � �� � �

)+***, �

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)+***,

������ � �� � �� � �� � �

)+***, �

/@iTn i ' ��� � e��5���ai jki<_ � iTn�c��@���ai `re��bi<_ '�1 �@+4� � ���ai `re��bi<_ 'C1 �+e��p_ac���eS_ ���3e �Lcbn-�3iTn �Ee��pgm^����IlBin iTcbn-�3iTn i ��� c������� ' ��� � e��������bjki � ��C���ai 3 , 2 , 7C�7c*� e�� e cr_ + � e `r^an i�oD+ , ���ac � �����ai�`bi<_aiTn-���C_3cr_ �TiTn c�"� n ^Hj&� ^an i c�� � � �ai<_����ai�� e��Ti c��2���ai eSg��`bi e�� @ � @ + �@�bj-� l2� cQj-�me�� l2�b_��ai � ����� e�� ���ai��9^����Q[ " " [g�� n e���e���� i ��� +�d%�bg g �b_�� � ��`b\ # o , $Zi ����lBcbn-�� i cr_ ���ai��"� n ^3j&� ^an i c�� l��b_��ai ��[ " "L[ g�� n eSjki ��/n crg ���ai%'C1 ��+�9'i0������� �C^6� e��Se��Ti ���ai�l2�b_��ai ��[ " " [ _acr_ �TiTn c�����"� iTn _�eS_����3e����"� ^��a\�+

0 10 20 30 40 50 60

0

10

20

30

40

50

60

nz = 2304

Nonzero structure of H

� e `r^an i oD+ , ECBIi<_aiTn ���E_acr_ �TiTn c��"� n ^3j&� ^an i�c�� � 7 ���.7 g�� n e'� �

.5�����acr^a`*�:���ai��� `biTl3n-�beVj i+�D�3n i ��� e cr_Fc��C���ai �3iT`bn-������ e cr_g cD�ai � �ScPc ���%� eSg��� i�� � �3eSn i<j&��� c!�S^6� eScr_ � c�`biF� 0��]cr^����FeS_3jT^3n5��� n i<g i<_��acr^���jkcrg��?^6���� e cr_���� jkc*�"���/cbnfeSg��`bi �pc����3n-�bj&� eVjF��� � e��Ti�+�acbn i+���bg��� i�� e�� � �o�: < � � e�� c��$� e��Ti�o�: < � � o�: < � � � !� < :�:�0 < � < :�:�0 < + 1�� �5eS_a` � i+�����SeSj8!e���� \ �&cr^�� �"n i �P^3eSn i ��lBcr^6� 0�:K`reS`*��lP\D� i � e��#�&i �"� cbn i ������:�����acr^al��Si��3n i<jTe�� e cr_ +Hc*�� eS_3`! �7creV_ �_Q^3g lBiTn-� + " �P^�� �Qe�� e�� _ac��C�3n �bj&� eSjF���4� c���� �bi � i+�����SeVjTe���� \�+(� _��"� i ��� � �&i����3cr^�����n i � \ cr_�e�� iTn �� e��bi

Page 22: NUMERICAL OPTIMIZATION METHODS FOR IMAGE

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g iF���ac���� eS_(� �HeSj-��� e �Mn iT`*��n-�ai ����� �b_�� !���� � ���/cbn cbl����beS_3eV_a` �3n c��3^3j&��� � � �b_�� � ) �mj �ai ���� \�/cbn���n l?e�� n-��n \��bi<j&� cbn-� � �b_���� +�/Lc �&iF�biTn �6���ai@jkcrg��2� i+�5e�� \ c�� ���3e��)�3n cbl��Si<g jF�b_�l7iLn i �3^3jki ��jkcr_�!� e��3iTn-��l�� \ lQ\����*�QeV_a` ���6��b_ ����`bimc�� �b_ i+��� i<_��ai � �/cbn g�c�� � ������ �����@jTe n jT^����b_ ���3n c��BiTn�� eSi �F+ 9'ijF�b_ �ac ���ai g�� n e'�D! �bi<j&� cbnfc��BiTn-�� e cr_��#�@e����#���3i i+��� i<_��ai � � �Q^3eSj-�D�S\�lP\%^�� eS_a`����3e�� jkn iF� i o=! h�acr^an e iTn � n-�b_��"�/cbn g �b_�� e���� eS_ �biTn � i�+/9'iLi+�D� i<_�����ai$�3e�� jkn iF� i��9^3_3j&� eScr_�� 0 3 �82 ��7 �b_�� ' 3 �32 �67/� cflBi�BiTn e c��3eSj �Le���� � crg i �BiTn eScD� � + � �&e�� �5_ac��@_������ � � � � � � , o � � , � c�� �bcre��(eS_ � iTn��/iTn i<_3jki�/n crg ��� �"�bjki<_ �C�BiTn e c�����# , ; $ + 9'i � iF� � o � �/cbn jkcr_ �bi<_3e i<_Hjki�+ �acbn�� �]c�3e `re�� e��Ti � � � � eVg��`bi �0 3 �82 ��7��b_�� ' 3 �82 ��7 � �]i%�/cbn g � � � �BiTn eScD�3eVjLi+��� i<_�� e cr_���0�� 3 �82 ��7C�b_�� ' � 3 �32 �67 E

0�� 3 �32 �67 ���� 0 3 �82 ��7 q � � � , �b_�� q � � � ,q �� � � � , cbn �� � � � ,

�b_�� � eSg e�����n � \��Acbn ' � 3 �32 �67 +@� _�g �� n e'��_ac����� e cr_ �

0�� 3 �32 �67��� 0 3 �32 �67 q . .q . . q . .�� 2 ' � 3 �82 ��7 �� ' 3 �32 �67$q . .

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:�� 3 �32 �67 �� � ������

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=

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Page 23: NUMERICAL OPTIMIZATION METHODS FOR IMAGE

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$ ��� �����'% � $ ��� �����'% � ����� $ � %������ $ ����'% �

)+*******,� 3 oD+ < 7

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=

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Page 24: NUMERICAL OPTIMIZATION METHODS FOR IMAGE

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Page 25: NUMERICAL OPTIMIZATION METHODS FOR IMAGE

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0 50 100 150 200 250

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150

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Nonzero structure of BCCB

� e `r^an i oD+ oDECBIi<_aiTn ���E_acr_ �TiTn c��"� n ^3j&� ^an i�c��(� 7�=>=37 g �� n e'� =

& ��� � ���)* ��� 2�� # ) � (�*�� *+(�� � ) �

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Page 26: NUMERICAL OPTIMIZATION METHODS FOR IMAGE

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Page 27: NUMERICAL OPTIMIZATION METHODS FOR IMAGE

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Page 28: NUMERICAL OPTIMIZATION METHODS FOR IMAGE

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# 3 0 7 $ A ,� � � �h ��� � � �j ��� 0Mh j e � # �gf � A<hil � � j l � � 2 C 2 �� q 2 , 2 o 2 � � � 2 3 � � , 7��

"$�ai o=! h h � " e�� i �Q^3e����� i<_ ��� c������� \5eS_a`����ai , ! h h � " � c����ai jkc*�S^Hg _����b_�� ���3i<_ ���aifn c � � c�����ai g �� n e'�4+,"$�ai �acr^an e iTn � n-�b_��"�/cbn g�c�� � o=! h eSg��`bi 0 3 �32 �67 jF�b_ ���ai<_�lBifi+���3n i �-� i � ^�� eV_a`����aid�n cr_3i<j��biTn �Hn cD�H^3j&� c���b � E

3 b � 68b � 7 �!��� 3 0 7 �$��� 3 b � 0/b � ) 7 �$��� 3 b � 0/b � 7��

15eS_Hjki@b � 0 b � e������ai o=! h h � " c�� 0 3 �82 ��7 �3e��C�/c*��� c��;�@������ FD� 3"3 b � 6 b � 7 �$��� 3 0 7"7]e � i �P^3e�� !��� i<_ �$� c���ai o=! h h � " c�� 0 3 �32 �67 +

��������� � R ���+�� ���� R�� �)��R �� a��� ���������1R�S� ������ ��'TU�)�*SR V.I_6\'jTe n jT^����b_ �fg �� n e'� � �(�ai<_ac�� i �lQ\ �G����� 3�� 7 �G����� 3�� � 2 � 2 � � � 2 � � � &7 �ZjF�b_XlBi�3e���`bcr_����Se��Ti �lQ\����ai �acr^an e iTn@g�� n e'� # , q�$ E

� b � � �3e���` 3�� 7 b � 2 3 oD+ , q 7

� �aiTn i5b � e��C���ai �3cr^an e iTnLg �� n e'��c�� cbn-�3iTn � �b_��

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E

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Page 29: NUMERICAL OPTIMIZATION METHODS FOR IMAGE

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�/c*��� c��;��# , q�$ E

= 3 b � 68b � 7 � � 3 b � 68b � 7�2 3 oD+ ,�, 7

� �aiTn i � � � � ��� 3�� � 6 � 7�2 � � �3e ��` 3 , 2;X 2;X � 2 � � � 2;X � � 7 +�"$�ai �3e ��`bcr_���� i � i<g i<_ ���fc���3e���`bcr_����Cg�� n e'� � jF�b_%lBiIi+���3n i �-� i � eS_�� iTn g � c��/���3ifi � i<g i<_ ��� c�� � o=! h �acr^an e iTn�� n-�b_��"�/cbn g�c��' � 3 �82 ��7 � 4 3Dc 2;d67 # , ;%$ E

�He���` 3 � 7� � 3 �$��� 3 '� 7"7 � 3 b � 68b � 7 �$��� 3 '� 7�� 3 oD+ , o*7

"$�ai �?n-�"� � i � i<g i<_ ������n i 4 3 q 2 q 7 � 4 3 , 2 q 7 � � � �=��4 3 � � , 2 q 7 7;���ai�_ai+�D�F�"4 3 q 2 , 7 ��4 3 , 2 , 7 �� � � � 4 3 � � , 2 , 7 7D�b_�� � cLcr_ �*�Le����0���aiC�����"� � �He���`bcr_����ri �Si<g i<_ ���ZlBi<eS_a` 4 3 q 2 � � , 7 � 4 3 , 2 � � , 7 �� � � ��4 3 � � , 2 � � , 7]g ^���� e����Se i � lP\ � + "$�3i �He���`bcr_����Cg�� n e��(jkcr_ ���beS_3eS_a`����ai�i<eS`bi<_ ����S^ai �F� � �e��C���ai o=! h h � " c�� ' � 3 �32 �67 �@e���� � jkcr_��"���b_ �F+

��������� �1���i��� T �k�*SR V�� �����*�+ ����� &��"�*�k��R��� � c�� �]i0���3c�� �ac � �]ifjF�b_%jkcrg��?^6� i � 0�i ��jTeSi<_ ��� \(lQ\�g �*�QeS_3` ^�� i�c��

= 0�� 3 b � 68b � 7 � � 3 b � 68b � 7 0�� 2 0�� �$��� 3 0�� 3 �82 ��7"7

eS_����ai0�Ac*���Sc��LeS_3` �/cr^an �"� iF�2�F+

3 , 7 0�� 3 b � 68b � 7 0��Le������bi<j&� cbn e��Ti �#�biTn � e cr_�c��(���ai o=! h h � " c�� 0�� 3 �32 �67 +

3 o*7 � iF� � � 0�� � "$�ai �bi<j&� cbn � e�� jkcrg��H^�� i � ��� � � 0�� � � � 0�� 2 � �aiTn i5���3i�c��7iTn �� cbn�� � e�����3ifi � i<g i<_ � �Le � i��3n c��3^3j&� c�� � �]c �bi<j&� cbn-� +

3 0*7��0 � 3 b � 68b � 7 � � e��C���3i eS_ �biTn-� i o=! h h � " c�� � +

3 4 7�.;�����S\ "D� 3 �0 � 7C�b_�� i+�D� n-�bj&�;���ai%� i ���3eV_a` � � � � ^al6! l2� cQj-� �/n crg ���ai�i+��� i<_��ai �%cr^6�"�H^6�F+

15eSg e�����n-� \��)�&imjF�b_%jkcrg��H^�� i

= ) 0 = � 0 3 b � 68b � 7 � � � 3 b � 68b � 7 0i ��jTe i<_ ��� \���� �]i ��� + �an crg ���ai%� �bj&�$������ = e���� n i ���Eg �� n e'�4� = ) = � �ac*�����F+

Page 30: NUMERICAL OPTIMIZATION METHODS FOR IMAGE

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"Mc `biF� ���3i jkcrg��H^6���� eScr_����-jkc*�"�Lc��(�He n i<j&�Ljkcrg��?^6���� e cr_ � 0 �Acbn5��n lHe�� n-��n \ 0 � �]i0�"����n�� �Le�����b_�i+�6�bg���� i�c��*4 � 4�eSg��`bi �F+ 9'i��ai<_ac�� i%���3i -! ���%jkc*�V^3g _�c��&0KlP\<0 # + "$�Q^��F� � 0�l7i<jkcrg i �

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n i �Q^3e n i � , � o � 0 � 4� 3 4 ��:*7 � o� , q"gm^���� e����SeSjF�� eScr_��F+ � _�`bi<_aiTn-��� � � 0 _3iTi ��� , � o �0 ��� � � � � � � 3 � � , 7 � oXl�� c5j���gm^���� e����SeSjF�� eScr_��F� �b_��#cr_3i l�� c5j��#g ^���� e����VeSjF�� e cr_"n i �P^He n i �, �+o � 0 � � � �(� � � � 3 � � , 7 � o(gm^���� e����SeSjF�� eScr_��F+ "$�aiTn iF�/cbn i������3i � c������ �]cbn���c�� ���ai �He n i<j&�jkcrg��H^����� e cr_'e�� � 3 � � � 4 7pc �biTn-����� + & _ ���ai c����3iTn%���b_�� � ���ai�� c������@�bg cr^3_ �Ic�� �]cbn���^�� eS_a` = 0 �e�� � 3 � � � cb` � 3 � 7"7 l7i<jF�b^�� i � 3 � � � cb` � 3 � 7"7 e�� _3iTi �ai � �/cbnLi �bj-�Ko=! h h � " �b_��!�]i ^�� i�cr_aimo=! hh � " �b_���cr_ai�eS_ �biTn-� ifo=! h h � "�+2A crg��H^6� eS_3` ���aipi<e `bi<_ ����S^ai�g�� n e'� � n i �Q^3e n i �����aifo=! h h � "lBi<jF�b^�� ifi<eS`bi<_ ����S^ai �;��n ifcbl����beS_3i � �/n crg ���ai o=! h h � " c�� ' + /Lc �&iF�biTn � � eV_3jki �&i jkcrg��H^6� i0���3e��cr_�� \�cr_3jki lBiF�/cbn i%�"����n�� eV_a`����aifg�beS_#��� `bcbn e����3g �D���3i jkcrg��H^6���� eScr_ e�� _aiT`*�SeS`re l�� i�+

��������� ���� �8� ��+&(��*��R � � � � � � �� _ �����3e�� eScr_�� c ���aifl�� c5j���jTe n jT^����b_ �C�3n c��7iTn-� \�� = �����$� crg ifeSg�7cbn-���b_ ���3n c��7iTn-� e i �F+

3 , 7 �@�bj-��n c�� 3 jkc*�S^3g _27 � ^3g�e�� � .A � � .# � '�A # , + "$�Q^��F� = ) e e � = e e + �3^an����aiTn g cbn i��i �bj �%i �Si<g i<_ �@e�� _acr_3_aiT`*�� e��bi�7��ai<_3jki = e��/* ��� � ����� �" ���%� �� "F+

3 o*7 = ) = � lBi<jF�b^�� i = e�� � n i ���Eg�� n e'� +

3 0*7 = e � �#_3cbn g���pg�� n e'�4� � ! �� = ) = = = ) + /@i<_3jki�� = e���^3_3e�����n e�� \��3e���`bcr_����Ve�� ��l�� i�E= b � � b"2 b � b ��+

3 4 7 = ) = = = ) = � = 3 b � � � b 7 3 b � � b 7 b � 3 � � � 7 b�+ /@i<_3jki�� = ) = e�������� c��7�=>=37 g �� n e'�4+

3 :*7 = b � � b e�� _ac�� � \Qg g iF� n eSj+ = b � � b�� = ) = � b � � � b�+

Page 31: NUMERICAL OPTIMIZATION METHODS FOR IMAGE

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50 100 150 200 250

50

100

150

200

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Single Restoration

50 100 150 200 250

50

100

150

200

250

Two−level Restoration

50 100 150 200 250

50

100

150

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0

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15(b) Noisy Gong: SNR = 1

0 0.2 0.4 0.6 0.8 1−10

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0 0.2 0.4 0.6 0.8 1−10

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0 0.2 0.4 0.6 0.8 1−10

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0 0.2 0.4 0.6 0.8 1−10

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