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一种改进的Canny算法

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一种改进的Canny算法 456 Proceedings of the 27th Chinese Control Conference July 16-18, 2008, Kunming,Yunnan, China �����Canny��* 潘大夫, 汪渤 ������� ���� �������� 100081 E-mail: pandaful@bit.edu.cn � ���Canny����������,� !"#$%&'()+,-./%&012345678%&����'( 9:;�1?0@%&�ABCD.%&1<>E...

一种改进的Canny算法
456 Proceedings of the 27th Chinese Control Conference July 16-18, 2008, Kunming,Yunnan, China �����Canny��* 潘大夫, 汪渤 ������� ���� �������� 100081 E-mail: pandaful@bit.edu.cn � ���Canny����������,� !"#$%&'()+,-./%&012345678%&����'( 9:;�1<,=>?0@%&�ABCD.%&1<>E& !�FGH0I,JKLMNOGPQR��,S>TUV W�GX�YZ !��,[\]^_`,��ab�MN%&���Canny��,���X�H��.�5 �� � �C�, ��������,%&01, Canny�� An Improved Canny Algorithm Pan Dafu, Wang Bo Department of Automatic Control, School of Information Science and Technology, Beijing Institute of Technology, Beijng 100081,P. R. China E-mail: pandaful@bit.edu.cn Abstract: Image is analysed under multi-scale filtering in Canny algorithm frame. Firstly, scale multiplication is defined by filter responses multiplication of two different scales. Then the modulus and orientation is determined by the similarity of ad- jacent scales. Thirdly, a threshold value is chosen to eliminate false edge. Finally, the edge is obtained after non-maximum suppression. Experimental results show the superior performance of the proposed algorithm in noise restrain and edge local- ization than the traditional Canny algorithm. Key Words: edge detection, scale multiplication, Canny algorithm 1 ��(Introduction) ��� !���&��S����),�� !)������� !)+H��� !"#,� ����$%,&'(:>)�*�+,-.,/0, ����+41�234H !5�67S89� :;<=>N,?@:;AB�CDE$����� �[1-3],FG,H��������I�J� !�N KLMKN3,���W�GLOPQRYZ !� �,ST,Canny[2]U-�CD�V�����W�X Y:$�����C��$���.�C��Z[� R��,?\CK9���S]��9�^_'(9 �NKN3, �`1Canny����9a,QD��'(9� ���,�N� �� ����'(%&,�'Q !���8������.��&H��C,=�, ?�Canny��b�MN%&"#��X�H��� �, �� 56��:%&U�����V���, %&U������]�,�N��,����ZN6 � �!"%&,#$$7��56��%&��] ^%&'(, )�Canny��MN'(%&�7*,+,�- $%&1<,��5678%&'9:;�1<./ Canny���M%&'(:;,=>?@A%&0B * 01��YZ2��34516�789 , 16:;< : 51405030104BQ0171, C=�D.%&1<>��>IHFG,J?MNO GPQR��,S>TUVW�GX�YZ !� �,����? !����8@,�AB�X���, ��%&�3KL� ,�� M,��� ���, 2 Canny ��(Canny Theory) �����:;'(9FIR4f(x),���[-T,T], �������4: W(x)=G(x)+n(x),��n(x)~N(0, σ2) ������,G(x)�FG4A�� ��,G(x)�f(x) � : ; ( ) ( )T T dGH G x t f t t − = −∫ ,n(x) � f(x) � : ; ( ) ( )dTTH n x t f t tn −= −∫ ,W(x) � f(x) � : ; ( ) ( ) ( ) W G H x H x H xn= + ,��4HW(x)�!"W�G,#$ Canny%- C&����9�'()*, 2.1 �����(Good Detection) ����x=0+,��aSNR,-.�, 2( ) ( )d ( )d (2.1)T TT TSNR E x f x x f x xσ− − ⎛ ⎞⎜ ⎟⎝ ⎠= −∫ ∫ ��,E(x)�/��0D��,σ����E1�,2� ��G(x)3D45Y: 0 2( ) d ( ) d (2.2)TT TSNR A f x x f x xσ− − ⎛ ⎞⎜ ⎟⎝ ⎠= ∫ ∫ Authorized licensed use limited to: Xi'an University of Technology. Downloaded on March 15, 2009 at 01:58 from IEEE Xplore. Restrictions apply. 457 2.2 �����(Good Localization) ��Z���;,-.6A[F�����,� y�HW(x)�!"W�G�,Canny%- ' "( ) / (0)Gy H y Hn≈ , .7C.)*4y��3,=�,TagareHFigueiredo[4] 8-$59�8N7�b: ' ( )H xn ; ' (0)Hnζ = − , " "(0) (0)GH Hnξ = + ,*ζHξ4H0I G���J?@�,� ζ μ , ξ μ , 2 ζ σ H 2 ξ σ )�4��� �GH��, y� �)�23 : ( )21 e /Pr( ) 1 / e / (2.3)2 c b ay b a erf b aaζ ξσ σ ⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠ − = ⋅ + π ⋅ ⋅ π ��,erf(x) ��23, 2 2 2 1 2 2 ya ζ ξσ σ = + , 2 22 2 y b ζ ξ ζ ξ μ μ σ σ = − − , 2 2 2 22 2 c ζ ξ ζ ξ μ μ σ σ = + ,��4 [-T,T],��y��� 2 2E[ ] Pr( )dTTy y y y−= ∫ ,��,�� �.�� : 2 1 Pr( )dTT L y y y − = ∫ (2,4) 2.3������(Low Spurious Response) �� ������,����97;� $6 W�G, �S��& X�R��,����f(x)� ���:;�,50@W�G���xmax(f)�f’(x)�� �������xzc(f)��: max ( ) ( )zcx f x f= (2,5) ��� '2 ''2( ) ( )d ( )dTTT Tx f f x x f x xzc − −= π ∫ ∫ Canny 4S!����9;"�#R��:; �$��,%YSNRHL�1�,��,'(�)$%&'(9�*+�, ,�%&01���, 3.1 ��!�(Scale Multiplication) ?%&s2f(x)-.4 ( ) ( / )/f x f x s ss = , fs(x)��@� 4[-Ts,Ts],Ts =sT, W(x)�:; ( ) ( ) ( )W G s s sH x H x H xn= + , )AK-FDOG23 2 / 2( ) e xf x x −= − E4��'(9, ( 2 2 2/(2 )( ) e / x sf x x ss − = − ⋅ ,%&s1Hs2������W(x)� :;)�4 1 ( )sWH x H 2 ( )sWH x ,%&1<./4: 1 2( ) ( ) ( ) (3.1)s sW W WP x H x H x= ⋅ ? 1�D,fs1(x)�'D.�����,&;� N/RWG; fs2(x)B0� RWG,&1#C���� �.�K&,PW(x)D2Cfs1(x)Hfs2(x)�*�,7/� 'D.�����,3�"� X�C��,%&1< '(E^�*4M%&'(, (a)N������� (b) fs1(x)Hfs2(x) (c):;� 1 ( )W s H x H 2 ( )W s H x (d) %&1Y#: 1 24 (3.6) cos sin s sASNRp σ α α α π = ⋅ + 2r3.4,ρ4I3�α3.,�^A/ρ7�,SNRp �5a� s,�^6 ( )f xs 7Ff(x),�: 4( / ) 4/SNR s A σ= ⋅ ⋅ π ,SNR/5a� s, 3.3��!��������(Localization Criterion) 6z4PW(x)�x=08A�WG�,� 1 2 1 2' '' ( ) ( ) ( ) ( ) ( ) 0.W W W W W s s s s P z H z H z H z H z= ⋅ + ⋅ = TU Taylor97,:6 ' (0) 0 G sH = J; 2( )o z Y: 1 2 2 1 1 2 2 1 1 2 ' '(0) (0) (0) (0) (3.7)'' '' ' '(0) (0) (0) (0) 2 (0) (0) W W W W W W s s s sH H H Hn nz s s s s s sH H H H H Hn n − ⋅ − ⋅ ≈ ⋅ + ⋅ + ⋅ 4< �8=,� (0)W siHiς = , ' (0)W siHiζ = , '' (0)W siHiη = , 1, 2i = , � 1 1 2 2 1 1 2 2 1 22 z ζ ς ζ ς η ς η ς ζ ζ − − ≈ + + , 6X=[ς1 ς2 ζ1 ζ2 η1 η2]T,�X��GH��: M=E[X], Ω=E[(X-M)(X-M)T], X�K>)�23�: ( ) ( )T 131 1Pr( ) exp (3.8)2(2π) ⎛ ⎞⎜ ⎟⎝ ⎠ − = − − −Ω Ω X X M X M ?�ςi, ζiHηi�0LM?�,FPr(X)-@A4: 1 2 1 2 1 2Pr( ) Pr( , ) Pr( , ) Pr( , )ς ς ζ ζ η η= ⋅ ⋅X ( )W siP x ��@�[-Ts,Ts],� T 2 T 2E[ ] Pr( )dz z z z − = ∫ ,�^9Bz�K>)�23,� N:�#E[z2],=Cz�ςi,ζiHηi�D223,�EFG YPr(z),�4-TUz�L;�YE[z2]: 1 2 2 1 2 1 2 1 2E[ ] Pr( )d d d d d dsz z Xx T ς ς ζ ζ η η= ∫ ∫∫ ∫ ∫ ∫ M (3.9) (2.4)0B,%&01�.H���: 21 E[ ]L zP = (3.10) 3.4 SNRp,Lp1SNR, L�2� (Numerical Com- parison of SNRp, Lp with SNR, L) SNRP! (3.6)+O��� ( )f xs 7�+ (2.2)> , 4( / ) 4/SNR s A σ= ⋅ ⋅ π,LPHL)�!+(3.10)H(2.4) O�,�@I4[-4,4], fs(x)��@�[-4s,4s],+(2.3) ,�<3μζ)μξ)σζHσξO���: 'E[ (0)] 0Hnζσ = = 2 2 3'E[ ' (0)] ( )d 3 /(4 )TsTsH f x x ssnξσ σ σ−= = = π∫ 2" 'E[ (0)] (0) /GH Af A ssζμ = = = − 2 2 5" ''E[ (0)] ( )d 15 /(8 )Ts Ts H f x x sn sξμ σ σ−= = = π∫ 6Σ1HΣ2_"fs1(x)Hfs2(x)�SNR,Λ1HΛ24fs1(x) Hfs2(x)�L, ΣPHΛP�%&1<�SNRPHLP,�.' )+,,Is2= 2·s1,=>)�O�Σ1)Σ2)Λ1)Λ2)ΣP HΛP,J��.�)+]^� 2, (a) s1=20,s2=21�Σ1�Σ2HΣP (b) s1=21,s2=22�Σ1�Σ2HΣP (c) s1=20,s2=21�Λ1�Λ2HΛP (d) s1=21,s2=22�Λ1�Λ2HΛP (e)s1=20,s2=21�1<�� (f)s1=21,s2=22�1<�� "2 �2� 34 3.5 5$"6����� (Scale Multiplication of Image) � !,,� ( , )xf x ys H ( , ) yf x ys _�xHy ��'(9 ,��� ! I�:;�4 ( , )sH x yx H ( , )sH x yy ,�%&1<234: 1 2( , ) ( , ) ( , ) s sxP x y H x y H x yx xI = ⋅ H 1 2( , ) ( , ) ( , ) s syP x y H x y H x yy yI = ⋅ . ?�0@%&:;��0BC ,����� 0 0( , )x y , ( , ) sH x yx � s1H s2@A%&����0 8, ( , )sH x yy �=,�� 0 0( , ) xP x yI H 0 0( , ) yP x yI �4V�,% &01]^��� �� ,& � 1 0 0( , ) s H x yx H Authorized licensed use limited to: Xi'an University of Technology. Downloaded on March 15, 2009 at 01:58 from IEEE Xplore. Restrictions apply. 459 1 0 0( , ) s H x yy ���,�V���,�^ ( , ) 0 xP x yI < (!� ( , ) 0yP x yI < ),� 4 �� 0,�(x,y)�FGH �I4: ( , ) ( , ) ( , )yxM x y P x y P x yI I I= + 1 1 sign ( , ) ( , ) ( , ) arctan (3.11) sign ( , ) ( , ) s yH x y P x yy IA x y s xH x y P x yx I ⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎜ ⎟⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠ ⋅ = ⋅ �8Canny��, !�����%&1<]^ ���W�G�,?�%&01��X���) �, �7��Canny��b��OG,��MOG��� MI(x,y)PQR��,S>TUVW�GX��I� !��, 4 7834(Experiment Results) "s2= r·s1,TIIr>1,r !�7"#$%G&s1 Hs25%&�� ,r ��'(#56%&�0) C,�*[\_`,s2= 2s1�����+^S$,$� -.[%, 3b�/0��&P1Canny��"#� a,[\]^_`,'�%&01��I]^2�($ �����C"H.34&,aMN%&Canny�� �I���$, (a)&) ! (b)!%&s1����I]^ (c)�%&s2����I]^ (d)/0�����I]^ "3 9:"6��;<34 5 3=(Conclusion) /0�$%&)+�'56,��%&01�� �7"P1Canny��,TU5678%&'(9� :;1<8)N6*�'(:;,+X�9 �,- :9��]K,=>?'(:;�@A%&6�0B C,;��%&1<>�FGH �I,S>b�VW �GX��I� !��,�.)+H[\]^_`, 7"���Canny�������9<�=� /` 01�M%&�C"��,'�%&01�Canny7 "��,2�2$�X� �H=D.3���1�, ����� !������^ , >?@A(References) [1] MARR D� HILDRETH E.Theory of Edge Detection[J]. Proceeding of Royal Soc. London, 1980, B(27): 187-217. [2] CANNY J. A computational app roach to edge detection[J]. IEEE Trans. on Pattern Analysis and Machine Intelligence, 1986, 8(6): 679-698. [3] DEMIGNY D. On Optimal Linear Filtering for Edge De- tection[J]. IEEE Trans. on Image Processing, 2002, 11(7): 728-737. [4] TAGARE H D, FIGUEIREDO R J P,etal.On the Localiza- tion Performance Measure and Optimal Edge etection[J]. IEEE Trans. on Pattern Analysis and Machine Intelligence, 1990, 12(12): 1186-1190. [5] KOPLOWITZ J, GRECO V. On the Edge Location Error for Local Maximum and Zero-Crossing Edge Detectors[J]. IEEE Trans. on Pattern Analysis and Machine Intelligence, 1994, 16(12): 1207-1212. Authorized licensed use limited to: Xi'an University of Technology. Downloaded on March 15, 2009 at 01:58 from IEEE Xplore. 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