Conic Fitting Using the Geometric Distance

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HAL Id: inria-00384284 https://hal.inria.fr/inria-00384284 Submitted on 14 May 2009 HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés. Conic Fitting Using the Geometric Distance Peter Sturm, Pau Gargallo To cite this version: Peter Sturm, Pau Gargallo. Conic Fitting Using the Geometric Distance. ACCV 2007 - 8th Asian Conference on Computer Vision, Nov 2007, Tokyo, Japan. pp.784-795, 10.1007/978-3-540-76390- 1_77. inria-00384284

Transcript of Conic Fitting Using the Geometric Distance

Page 1: Conic Fitting Using the Geometric Distance

HAL Id: inria-00384284https://hal.inria.fr/inria-00384284

Submitted on 14 May 2009

HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, estdestinée au dépôt et à la diffusion de documentsscientifiques de niveau recherche, publiés ou non,émanant des établissements d’enseignement et derecherche français ou étrangers, des laboratoirespublics ou privés.

Conic Fitting Using the Geometric DistancePeter Sturm, Pau Gargallo

To cite this version:Peter Sturm, Pau Gargallo. Conic Fitting Using the Geometric Distance. ACCV 2007 - 8th AsianConference on Computer Vision, Nov 2007, Tokyo, Japan. pp.784-795, �10.1007/978-3-540-76390-1_77�. �inria-00384284�

Page 2: Conic Fitting Using the Geometric Distance

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×�7�+,¹:°D´�´,*R¶�+,°DE87�µ0Ã!*�9Y*~´,¶�-,°D»_*iEm12-K3^*�+,¹2EY9;²£E8-¡3^°F72°D3^°FÖR°F7:@�+4¹2*?@8*�Em3^*�+,-4° ¶�5CB_9Y° ´e+ 5C7:¶�*»:58´,*R9�¶�Em´c+O²£127r¶j+,°DE87#¿0Ý¡¹:*/¼8*RÎ^EC²�+4¹2*/3^*�+4¹2EY9�° ´K+,¹2*i³:5C- 5C3^*�+4*�-4°FÖ~5n+4°FEm7�EC²'+,¹:*i³2-4E8»2ÕBF*R3�¿'×�7Í+,¹:*�72*�Â0+�³:5C- 5C@m-458³2¹�µrÃK*^Ã�°DBFB$Ù:- ´c+�92*R´4¶�-4°F»_*�+4¹2*�³:58-4583�*�+,*R-,°DÖR5C+,°DE87�µr»r*�²£E8-4*´c¹2EnÃ�°D72@�+,¹:5C+�°X+�°D7:9Y*R*R9¸5CBDBFEnÃ�´i+,E�3^°D72°F3^°DÖ�*�@m*�E83^*�+,-4°D¶R5CB$9Y°D´c+4587:¶�*m¿ � ²�+4*�-�+4¹2°D´Rµ'Ã!**�ÂY³2B 5C°D7�¹2En÷+,E?°D72°F+,° 5CBD°FÖR*¡+,¹2*�³:58-4583^*�+,*R-4´$587:9^9Y*R´4¶�-4°F»_*¡¹2EnÃØ+4E�´cEmBF½m*O+4¹2*�72Em7YÕIBD°D72*R58-E8³Y+4°F3^°DÖR5n+4°FEm7�³:-,Em»2BF*R3Ä°D7�5^¶�E83^³21Y+ 5n+,°DE87r5CBDBFÎ�*�æ�¶�°D*�7A+�Ã!5�Îm¿

9�òUó :1�%�������A÷;�<�nü>=��2÷Rü�õ_úÝ¡¹2*�³:5C- 5C3^*�+,*�-4°DÖR5n+4°FEm7�*�ÂY³2B 5C°D72*~9;°D7�+,¹2*?²£EmBFBDEnÃ�°F7:@:µY° ´¡°FBDBF1r´e+4-45C+,*R9�°D7�Ùr@812-4*^É8¿¯2E8-K*R5m¶ ¹�EC²'+4¹2*

n3^*~58´,12-,*~9^³rEm°F7A+ ´

qp

µAÃK*i³:5C- 5C3^*�+4*�-4°FÖR*�5�³_E8°D7A+qp

µY´c1:¶ ¹�+,¹:5C+5CBDB

qp

BF°D*^E87¾5�¶�E872° ¶C¿�Ý¡¹:*^´,°D3�³:BF*~´e+�á5�Î�+,E>9YE>´cE�° ´�+,E>¶ ¹2E0Em´,*�+4¹2*^1272°F+�¶�°D-4¶�BF*�5m´´c12³:³rEm-c+~µY°F7�Ã�¹2° ¶ ¹>¶�58´,*�ÃK*�3�5�Î�³:58-4583�*�+,*R-,°DÖ�*i+,¹2*

qp

»0Î;5C7�5C72@mBF*αp

*R5m¶ ¹�ß

qp =

cosαp

sin αp

1

¯212-,+,¹2*R-,3^Em-,*mµ�ÃK*�°D7:¶�BF1:92*;°D7±E812-�³:5C- 5C3^*�+4*�-4°FÖ~5n+4°FEm7¸5ºÔ8Úij2-4ECçe*~¶j+4°F½m*^+,- 5C7:´c²£E8-,Õ3^5C+,°DE87�µYEm-RµY¹:E83^E8@m-458³2¹0Î8µ

P¿

�Í*?+4¹2*�7�á5C7A+K+4E�´cEmBF½m*/+4¹2*?²£E8BDBDEnÃ�°F72@^Em³Y+,°D3^°FÖ~5n+,°DE87�³2-4E8»2BD*�3�ß

minP,α1···αn

n∑

p=1

dist (qp, Pqp)2 â�ÇAã

×�7;´c*~¶j+4°FEm7�Ç2¿gÔYµ8Ã!*/´,¹2EnÃ�+,¹:5C+K+4¹2° ´O³:58-4583^*�+,*R-,°DÖR5C+,°DE87�°F7r9Y*�*~9�5CBDBFEnÃ�´�+,E�3�°D72°D3^°FÖR*+,¹2*�9Y*~´c°D-4*R9�¶�Em´c+¡²£127:¶�+,°DE87�»:58´,*R9�E87�³_E8°D7m+,ÕU+4ECÕ�¶�Em72°D¶?9Y° ´c+45C7r¶�*R´R¿

� +OÙ:- ´e+¡´c°D@8¹A+~µC+,¹:°D´K³:5C- 5C3^*�+,*�-4°DÖR5n+4°FEm7^¹:58´O+,¹2*�9Y- 5�Ã�»:58¶ ¼�EC²#5�3�1:¶ ¹�B 5C-4@8*�-O7A1:3�Õ»r*R-/EC²�³r5C- 5C3^*�+4*�- ´!+,¹:587Á72*R¶�*R´4´,58-,Îxß

n + 8â�+4¹2*

n5872@8BD*R´

αp

5C7r98³:5C- 5C3^*�+,*�- ´¡²£Em-

Pã�°D7:´c+,*R5m9ºE8²

5+,¹r5n+�Ã!E812B 9Í´,1Yæ�¶�*�+4E>³:5C- 5C3^*�+4*�-4°FÖR*�5>¶�E872° ¶C¿��¸*�Ã�°FBDB�´,¹2EnÃ�¹2EnáÕ

*�½8*R-Rµx°D7¸´,*R¶�+,°DE87¸Ç2¿ È:µ_+,¹:5C+?+,¹2*�Em³Y+,°D3^°FÖ~5n+,°DE87¸¶�587º72*R½8*�-,+,¹:*�BD*R´4´i»_*�¶�58-,-4°F*~9ÁEm1Y+�°D7¾5¶�E83^³212+45n+4°FEm7:5CBDBDκ*�æ�¶�°F*R7m+�3^587272*R-Rµ#9212*�+,EÁ+,¹2*;´,³:58-4´,°X+eκEC²¡+,¹2*;72E8-43�5CB�*~Åm1r5n+,°DE87r´58´4´cEY¶�° 5n+4*R9�+,E�E81:-!BD*R5m´e+/´4ÅA1:5C-4*R´¡¶�Em´c+!²£1:7:¶j+4°FEm7�¿

?�³�+,EM72EnÃ�µ�Ã!*Á¹:5�½8*Á¶�E87:´,° 9Y*�-4*R9P5m´�5±@m*�72*R-458B/ÔCÚ�¹:E83^E8@m-458³2¹0Î8µ<Ã�¹2° ¶ ¹Ð°D´

¶�BD*R5C-4BDÎ�5C7�En½8*R-,³:58-4583^*�+,*R-,°DÖR5C+,°DE87�¿��Í*�9YE�5m¶j+,1r5CBDBFÎ�³r5C- 5C3^*�+4*�-4°FÖR*P3^°D72°D3^58BFBDÎxß

P ∼ R Σ = R9Y°D58@

(a, b, c)Ã�¹2*�-4*

R° ´�587¾E8-,+,¹2Em72E8-43�5CB<3�5n+4-,°F¸587:9

a, b5C7:9

c58-,*�´4¶�58BD58-4´R¿��Í*�´c¹2Enà °D7±+,¹2*

²£E8BDBFEnÃ�°D72@�´,*R¶�+,°DE87�+,¹:5C+K+4¹2° ´¡³:5C- 5C3^*�+4*�-4°FÖ~5n+4°FEm7�°D´¡´,1Yæ�¶�°D*�7A+RµY°I¿ *8¿Y°F+�58BFBDEnÃ�´O+,E�*�ÂY³2-4*R´4´5CBDB'³2-,Em³r*R-�-,*~5CB�¶�E872° ¶�´R¿

Page 5: Conic Fitting Using the Geometric Distance

@BADC }E0} GIV�V�lAZ�d�bc[�def�aRQ^aR��aRlmb�W0[�bc[Rp�S4deS,befXo [�def�a~Q:�

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3²£Em-

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.0°F7r¶�*¡+,¹:*�´,¶R5CB 5C- ´�58-,*¡E87:BFÎ�-4*�BD*�½n587m+<12³�+,E�5�@mBFEm»:5CBY´4¶�5CBD*!²�58¶�+,E8-~µnÃ!*!3�5�Î�Ù:Â�E872*�587:9+,¹01:´�-4*R9Y1r¶�*?+,¹:*�7A1:3�»_*�-�E8²�³:58-4583^*�+,*R-4´K²£Em-

P+,E�+4¹2*�3^°F7:°F3�1236E8²

5¿2À>E8-4*�9Y*�+ 5C°DBD´

E87�+,¹2*�³:58-4583^*�+,*R-,°DÖR5C+,°DE87�EC²#+4¹2*�E8-,+,¹:E872Em-,3�5CBx3�5n+4-,°FÂR58-,*�@8°D½8*R7�°D7>´c*~¶j+4°FEm7�Ç:¿ Èr¿

9�ò�ñ ô�õ�����FG�m÷�0úH�0ö~ö�õ ¡÷;I���:1�3� �%�J�m÷�<��üG=��Y÷~ü�õ_ú

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R, a, b587:9

câa, b, c 6= 0

ãjµ_+,¹2*^58´4´,E0¶�°D5C+,*~9>¹2E83^Em@8- 5C³2¹0Î�Ã�°FBDB3^58³�+,¹2*�³_E8°D7A+4´

qp

+4E�5�´c*�+�EC²$³rEm°F7A+4´�+,¹:5C+�BF°D*�E87�5�¶�Em72°D¶C¿:Ý¡¹2° ´�° ´�E8»0½0°FEm1:´¡´c°D7:¶�*

³rEm°F7A+cÕ�¶�Em72° ¶�°D7:¶�° 9Y*�7r¶�*�°D´K°F70½n5C-4° 5C7A+�+,E�³2-4ECçe*~¶j+,°D½8*�+,- 5C7:´c²£E8-43�5n+,°DE87r´O5C7r9�´c°D7:¶�*�+,¹:*qpBF°D*�E87�5^¶�E87:°D¶?5n+�+4¹2*�E812+4´,*�+�â�+4¹2*�1272°F+i¶�°D- ¶�BD*~ãj¿2Ý¡¹:*�-4*R´,12BF+,°D72@�¶�Em72°D¶

C° ´�@m°F½m*�7;»0Îxß

C ∼ P−T

11−1

︸ ︷︷ ︸M&N&O P QRO S>QRT U

P−1 ∼ R

1/a2

1/b2

−1/c2

RT

�Í*�72Enà ´c¹:Enà +,¹r5n+/5C70Î�³:-,Em³r*R-�-,*~5CB�¶�Em72° ¶C′¶R5C7�»_*1'e-4*R5m¶ ¹2*R9<)�»AÎ;E812-�³:58-4583�Õ

*�+,*R-,°DÖR5C+,°DE87�µ:°U¿ *8¿Y+4¹:5n+/+,¹2*R-,*�*�Â0° ´c+i5C7�E8-,+,¹:E872Em-,3�5CBx3�5n+4-,°FÂR5C7r9�´4¶�58BD58-4´

a, b587:9

c´c1:¶ ¹�+,¹r5n+C ∼ C′

¿:Ý�E�9YE�´cErµYÃK*�¶�E87:´,°D92*�-!+,¹2*�*R°F@m*�7:92*R¶�Em3^³rEA´c°F+,°DE87�EC²C′ß

C′ = R

a′

b′

c′

R′T

Ã�¹2*�-4*R′° ´�5C7;Em-c+4¹2E87:E8-43^58B:3�5n+4-,°FÂ'µ0Ã�¹2Em´,*/-4EnÃ�´K58-,*�*�°D@8*R70½8*R¶�+,Em-4´OEC²

C′µ2587:9

a′, b′5C7:9c′5C-4*;°F+4´�*�°D@8*R7A½n58BF12*~´;âU5C70ÎØ´cÎ03^3^*�+4-,° ¶�3�5n+4-,°F±3�5�θ»_*>9Y*R¶�E83^³_Em´,*R9±°D7Ø+4¹2°D´

Ã!5�Î2ãj¿�Ý¡¹:*�¶�E87:92°X+4°FEm7M²£Em-C′»_*�°D72@Ø5¸³2-4E8³_*�-�¶�Em72° ¶�° ´^+4¹:5n+�°F+4´^+4¹2-,*R*�*�°D@8*�70½n5CBD12*~´

Page 6: Conic Fitting Using the Geometric Distance

5C-4*�72E87YÕ�Ö�*R-,Erµ�5C7:9Í+4¹2*;¶�Em7:9Y°F+,°DE87¾+4¹:5n+�°F+�° ´�5�³2-4E8³_*�- �2úH� -,*~5CBO¶�E872° ¶^° ´?+,¹:5C+�E87:**�°D@8*�70½n5CBD12*má ´¡´,°F@m7�° ´�E8³:³rEA´c*~9�+4E�+4¹:5n+�E8²�+4¹2*?+eÃ!E�E8+,¹2*R-4´R¿×�²�²£Em-�*�Â2583�³:BF*mµ

c′°D´�+,¹2° ´�*R°F@m*�70½n5CBD12*8µ<+4¹2*�7ÛÃ�°X+4¹

R = R′, a = 1/

|a′|, b =

1/√

|b′|5C7r9

c = 1/√

|c′|µ~Ã!*!¹:5�½8*KE8»0½0°FEm1:´cBDÎ

C ∼ C′¿�×�²2+,¹2*B'c°F7r9Y°F½0° 9Y1:58BV)O*R°F@m*�70½n5CBD12*

°D´a′°D7:´c+,*R5m9'µY+4¹2*?²£E8BDBFEnÃ�°D72@^´cEmBF12+,°DE87�¹2E8B 92´Rß

a =1

|c′|b =

1√

|b′|c =

1√

|a′|R =

1−1

1

R′

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+,°DE87�EC²_587�¹:E83^E8@m-458³2¹0Î�½0° 5?5C7�Em-c+4¹2E87:E8-43^58Bm3�5C+,-4°XÂ�587:9�+,¹2-4*�*�´4¶�5CB 5C- ´RµR° ´<¶�Em3�³:BF*�+,*8¿�Í*�7:Enø´,¹2EnÃÍ+,¹:5C+#+,¹:*O58´4´cEY¶�°D5C+,*R9?¶�Em´c+'²£127:¶j+4°FEm7;â�ÇAãjµR°D´#*RÅA12°D½�58BF*R7A+'+,E�+4¹2*K9Y*~´c°D-4*R9

¶�Em´c+�²£127:¶�+,°DE87ØâcÉ~ãjµr°U¿ *8¿:+,¹:5C+�+,¹:*�@8BDE8»:58B�3^°F7:°F3�5�E8²�»_EC+4¹º¶�EA´e+�²£127r¶j+,°DE87r´/¶�Em-,-4*R´,³_E87:9+,E�+4¹2*^´,583�*�¶�E87:°D¶�⣰F²K5�127:°DÅA12*�@8BDE8»:58B#3^°D72°F3�123�*�ÂY° ´e+ ´�E8²K¶�Em12-4´,*~ã�¿rý#*�+

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µvp

»_*+,¹2*�¶�BDEm´,*R´c+�³rEm°F7A+�E87

C′¿'×�²¡3^E8-4*�+4¹:5C7¸E872*^³_E8°D7A+?Em7

C′58-,*�*RÅA12° 9Y° ´e+ 5C7A+i²£-4E83

qp

µ³2°D¶ ¼�5C70Î�E87:*�E8²�+4¹2*�3�¿

�Í*!¹:5�½8*¡´,¹2EnÃ�7�58»rEn½m*�+,¹r5n+�+,¹2*R-,*¡*�ÂY° ´e+R, a, b

5C7:9cµC´c1r¶ ¹�+,¹r5n+

P3�58³:´�+,¹2*¡1272°F+

¶�°D-4¶�BF*?+4EC′¿_ý�*�+

wp = P−1vp

¿_.0°D7:¶�*vp

BD°D*R´�Em7C′µ:°F+/²£EmBFBDEnÃ�´¡+4¹:5n+

wp

BF°D*R´/E87�+,¹:*1272°F+�¶�°F- ¶�BD*8¿xå�*R7:¶�*mµ_+,¹2*R-,*�*�ÂY° ´e+ ´�587Í5872@8BD*

αp

´c1:¶ ¹º+,¹:5C+wp ∼ (cosαp, sinαp, 1)

T ¿ÌKE87:´,*RÅA12*R7m+4BFÎmµA+,¹2*R-,*�*�ÂY° ´e+ ´!5�´,*�+�EC²#³r5C- 5C3^*�+4*�- ´

R, a, b, c, α1, . . . , αn

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½�58BF1:*R´�²£E8-R, a, b

5C7r9câ�+4¹01:´�µ�²£E8-

Pã�58-,*�E8»2+45C°D72*~9M°D7M+4¹2*>á5�αE81Y+4BF°D72*R9�°D7�+,¹2*

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� ´!²£E8-¡+4¹2*�5872@8BD*R´αp

µYÃ!*�9Y*�+,*�-43^°F7:*i+4¹2*�¶�BDEm´,*R´c+�³rEm°F7A+ ´�E87C′+,E^+,¹:*�3�*~58´,12-4*R9

qp

µ$»0ÎØ´,E8BD½A°D72@º+,¹2*�ÈC+4¹MEm-49Y*R-�³_E8BDÎ072E83^° 5CB!3^*�7A+,°DE872*~9Ø°D7M+4¹2*�°F7A+,-4EY9Y1:¶�+,°DE87MEm-�587*RÅA12°D½�58BF*R7A+?³:-,Em»2BF*R3 â�´,*�*�ÆDÉ��Ë$²£Em-�9Y*�+ 5C°DBD´ ãj¿��¸*�+,¹2*R7¸3�58³º+4¹2*R´,*�+,E�+,¹2*^127:°X+�¶�°F- ¶�BD*1:´c°D72@

P587:9;*�Â0+,- 58¶j+¡+4¹2*�5C72@mBF*~´

αp

µ:5m´�9Y*R´4¶�-4°F»_*R9�°D7�+4¹2*�³2-4*�½0°DE81:´¡´,*R¶j+4°FEm7�¿

9�òDY Z���÷RüG�Øü>=[�Y÷RüUõrú�Í*�7:EnÃþ92*R´4¶�-4°F»_*�¹2EnÃþÃK*�Em³Y+,°D3^°FÖR*�+,¹:*�¶�EA´e+�²£1:7:¶j+4°FEm7Ûâ�ÇAãj¿ � 70ÎÍ72Em7YÕ�BF°D72*R58-�E8³Y+4°XÕ3�°DÖR5C+,°DE87�3^*�+,¹2EY9�3�5�Î�»r*�1:´,*R9'µA»21Y+�´,°F7:¶�*/Ã!*i9Y*~5CBrÃ�°X+4¹�5�7:E87YÕ�BF°D72*~5C-OBD*R5m´e+¡´,ÅA1:58-,*~´³2-,Em»2BD*�3�µ:Ã!*�1:´,*�+,¹2*�ý�*�½m*�70»_*�-4@CÕ�ÀÁ5C- ÅA1:5C- 90+¡3�*�+,¹2EY9¾ÆV\~Ë�¿:×�7�+4¹2*�²£EmBFBDEnÃ�°F7:@:µ2Ã!*�9Y*�Õ´,¶�-,°D»r*/¹2EnÃÐÃK*i9Y*R58BrÃ�°F+,¹�+4¹2*/-4EC+ 5n+4°FEm7:5CB_¶�E83^³_E872*R7A+4´�E8²�E812-K³:5C- 5C3^*�+,*�-4°DÖR5n+4°FEm7�â£+,¹2*E8-,+,¹2Em72E8-43�5CB'3�5n+4-,°FÂ

R5C7r9�+,¹2*�5C72@mBF*~´

αp

ã�µ:Ã!*�+4¹2*�7Á*�ÂY³2BF° ¶�°F+,BDÎ�@8°D½8*�+4¹2*K]A58¶�E8»2° 5C7EC²�+,¹2*�¶�Em´c+!²£1:7:¶j+4°FEm7�µ:5C7r9�´,¹2Enà ¹:Enà +,¹:*�72E8-43�5CB'*RÅA1:5C+,°DE87:´Rá2´,³:5C- ´c°F+eÎ�3�5�Î�»_*�1:´c*~9+,E�´cEmBF½m*�+,¹2*R3 *�æ�¶�°F*R7m+4BFÎm¿

Page 7: Conic Fitting Using the Geometric Distance

^ �����Y÷�Ðõ�`_�õ:÷�Y÷RüUõrú��%F7:��3� �%���A÷;�<�nömò Ý�EÍ5�½mE8° 9±´c°D72@m12BD58-,°F+,°D*R´�°F7M+,¹2*�³:58-4583�*�Õ+,*�-4°DÖR5n+4°FEm7ºE8²�+,¹2*^Em-c+4¹2E872Em-,3�58B#3�5n+4-,°FÂ

Rµ'ÃK*�*R´c+,°D3�5n+,*mµ'58´�°D´�+eÎ0³2° ¶�58B$³2-45m¶j+4°D¶�*�*8¿ @:¿

°F7�³2¹2E8+,Em@8- 5C3^3^*�+,-4Î>ÆDÉ�Ë�µ25�Ù:-4´c+�E8- 9Y*�-�58³2³2-4E�ÂY°F3�5n+4°FEm7�EC²<5C7�Em-c+4¹2E872Em-,3�58BL'c12³x925n+4*a)3^5C+,-4°XÂ�5n+�*R5m¶ ¹�°F+,*R-45C+,°DE87�µ25m´!²£E8BDBFEnÃ�´RßÉ8¿�ý�*�+

R0

»_*?+,¹2*�*~´e+4°F3�5C+,°DE87�EC²R5n²�+4*�-�+4¹2*�³2-4*�½0°FEm1:´!°X+4*�- 5n+4°FEm7�¿

ÔY¿�ý�*�+R1 = R0∆

»_*�+4¹2*�*~´e+4°F3�5n+4°FEm7�+,E�»_*�Em»Y+458°F7:*R9>5n²�+4*�-/+,¹2*�¶�12-4-,*R7m+i°X+4*�- 5n+,°DE87#¿å�*R-,*mµmÃ!*/Em72BFÎ^5CBDBDEn÷+4¹2*�12³_9:5n+,*i3�5n+,-4°FÂ

∆+4E�½n5C-4Î8µm°U¿ *8¿

R0

° ´O¼m*�³Y+KÙ2ÂY*R9'¿L?/´,°F7:@+,¹2*��O12BF*R-/5C7:@8BD*R´

α, β, γµ2Ã!*�3�5�Î�³r5C- 5C3^*�+4*�-4°FÖR*

∆58´!²£E8BDBDEnÃ�´�ß

∆ =

cosβ cos γ sin α sin β cos γ − cosα sin γ cosα sin β cos γ + sin α sin γcosβ sin γ sin α sin β sin γ + cosα cos γ cosα sin β sin γ − sin α cos γ− sinβ sin α cosβ cosα cosβ

â£È0ãÇ2¿�Ý¡¹2*�12³x925C+,*K5872@8BD*R´

α, β5C7:9

γÃ�°DBDBm1:´,1:58BFBDÎ/»_*O½m*�-4Îi´,3^58BFBIµ�°I¿ *m¿RÃ!*<¹:5�½m*

cosα ≈ 15C7:9sin α ≈ α

¿�×�7:´c+,*R5m9¸EC²¡E8³Y+4°F3^°DÖ�°D72@Á9Y°D-,*~¶j+,BDκEn½m*�-�+,¹:*;5C72@mBF*~´�µ'Ã!*^+,¹01:´�1:´c*+,¹2*?Ùr-4´c+�E8- 9Y*�-�58³2³2-4E�Â0°D3�5n+4°FEm7�EC²

Ƨ

∆′ =

1 −γ βγ 1 −α−β α 1

È:¿¡×�7>+,¹:*�¶�EA´e+/²£127:¶j+4°FEm7Mâ�ÇAãjµ_ÃK*�+,¹01:´�-4*�³:BD5m¶�*R»0Î

R0∆′µr587:9>*~´e+4°F3�5n+4*

α, β587:9

γ¿ � +�+,¹2*¸*�7r9ÐEC²�+4¹2*¸°X+4*�- 5n+,°DE87#µKÃ!*º12³x925C+,*Í+,¹2*¸*R´c+,°D3�5n+,°DE87ÏEC²

R¿¡×�7 E8- 9Y*�-

+,E�¼8*R*�³REm-c+4¹2E87:E8-43^58BUµ_Ã!*�9YE�EC²!¶�E81:-4´,*�72E8+�12³_9:5n+,*^°F+�1r´c°D72@�+,¹2*�Ù:- ´e+?E8- 9Y*R-

5C³2³:-,E�ÂY°D3^5C+,°DE87�µR°U¿ *8¿�58´R1 = R0∆

′¿~×�7:´e+4*R5m9'µ�ÃK*!¶�E83^³212+,*K587�*�Â25m¶j+�Em-c+4¹2E87:E8-43^58B

12³x925n+4*�3�5C+,-4°XÂ∆1:´,°F7:@�*~Åm1r5n+,°DE87·â£ÈAãi5C7r9>+,¹:*�*~´e+4°F3�5n+4*R9Í5C72@mBF*~´�µx5C7:9º12³x925n+4*

+,¹2*�-4EC+ 5n+4°FEm7�½0° 5R1 = R0∆

¿�Y¿¡×�+�°D´�°D3�³_E8-,+4587A+i+4E�7:EC+,*�+,¹r5n+�5n+?+,¹2*^72*�Â0+�°F+,*R-45C+,°DE87�µ

R1

Ã�°DBDB<»r*^¼8*R³Y+�Ù2ÂY*R9ÁEm7°X+ ´�+412-,7#µr5C7r9�72*�à â�´,3�5CBDB ã�12³_9:5n+,*�5C72@mBF*~´�Ã�°FBDB#»_*�*~´e+4°F3�5n+4*R9'¿_Ý¡¹01:´�µ:+,¹2*�°F72°F+,° 5CB½n5CBD12*R´!EC²�+,¹2*?12³x925C+,*�5C72@mBF*~´K5C+�*R58¶ ¹;°X+4*�- 5n+,°DE87#µ058-,*?5CBDá5�Î0´OÖ�*R-,ErµAÃ�¹:°D¶ ¹�@8-4*R5n+4BFδc°D3^³2BD°XÙ:*~´¡+,¹:*�5C7:58BFÎA+,° ¶�58B'¶�E83^³212+45n+4°FEm7�E8²$¶�Em´c+¡²£127:¶j+4°FEm7�á ´b]m5m¶�Em»2°D587�¿Ý¡¹2*^³rEm°F7A+4´

qp

E87Á+4¹2*�1272°F+�¶�°D- ¶�BD*8µ'5C-4*�1:³_925C+,*~9Í°D7¾5�´,°D3�°DB 5C-?3�5C7272*R-Rµ'1r´c°D72@�5ÉRÚ¤-4EC+ 5n+,°DE87�3�5n+,-4°FÂ;*R5m¶ ¹;²£Em-¡+,¹2*�12³x925C+,*8ß

Ψp =

cos ρp − sin ρp 0sin ρp cos ρp 0

0 0 1

5C7:9�°X+ ´¡Ù:- ´e+�Em-49Y*R-�5C³2³:-,E�ÂY°D3^5C+,°DE87�ß

Ψ′

p =

1 −ρp 0ρp 1 00 0 1

)�Em°F7A+ ´�58-,*;+,¹01:´�12³_9:5n+,*~9·5m´�²£EmBFBDEnÃ�´�ßqp → Ψpqp

â�Ã�¹2*�-4*8µ$58´�²£E8-Rµ�+,¹2*�12³x925n+4*

5C72@mBF*~´�5C-4*i*~´e+4°F3�5n+4*R9�1:´,°F7:@�+4¹2*?Ù:- ´e+�Em-49Y*R-�5C³2³2-4E�ÂY°F3�5C+,°DE87:´Ψ′

p

ãj¿

Page 8: Conic Fitting Using the Geometric Distance

ô�õ_öR÷¸ø�ù�ú�ûC÷~ü�õ_úc�:úH�cde�2ûmõ�f$ü>�2úKò ý�*�+�+4¹2*�3^*R58´,12-4*R9±³_E8°D7A+4´�»r*�@m°F½m*�7Ø»0Îqp =

(xp, yp, 1)Tµ�5C7:9Ø+,¹2*�¶�1:-,-4*�7A+^*R´c+,°D3^5C+,*�EC²�+4¹2*

qp

»0Îqp = (xp, yp, 1)T

¿ � +�*~58¶ ¹°X+4*�- 5n+,°DE87#µYÃK*?¹:5�½m*i+4E�´cEmBF½m*/+4¹2*�³2-4E8»2BD*�3�ß

mina,b,c,α,β,γ,ρ1···ρn

n∑

p=1

d2(qp, R∆

′ΣΨ

pqp

) âG�mã

Ý¡¹2°D´¡¹r58´�5�BD*R58´c+/´4Åm1r5C-4*R´K²£E8-43�µ2°U¿ *8¿:ÃK*�3�5�Î�²£Em-,3�12BD5C+,*�+,¹:*�¶�Em´c+�²£127:¶�+,°DE87�1:´,°F7:@2n-4*R´,°D921:5CBx²£127r¶j+,°DE87r´�ß

2n∑

j=1

r2

j

Ã�°X+4¹r2i−1 = xi −

(R∆′ΣΨ

iqi)1(R∆′ΣΨ′

iqi)3

5C7r9r2i = yi −

(R∆′ΣΨ

iqi)2(R∆′ΣΨ′

iqi)3� ´!²£E8-¡+4¹2*(]A58¶�Em»2° 5C7;E8²�+4¹2*�¶�EA´e+¡²£127r¶j+,°DE87#µ2°X+�° ´�9Y*�Ù:72*R9�5m´�ß

J =

∂r1

∂a∂r1

∂b∂r1

∂c∂r1

∂α∂r1

∂β∂r1

∂γ∂r1

∂ρ1

∂r1

∂ρ2

· · · ∂r1

∂ρn¿¿¿ ¿¿¿ ¿¿¿ ¿¿¿ ¿¿¿ ¿¿¿ ¿¿¿ ¿¿¿ ¿ ¿ ¿ ¿¿¿∂r2n

∂a∂r2n

∂b∂r2n

∂c∂r2n

∂α∂r2n

∂β∂r2n

∂γ∂r2n

∂ρ1

∂r2n

∂ρ2

· · · ∂r2n

∂ρn

×�+�¶R5C7�»r*�¶�Em3^³21Y+,*~9�587:5CBDÎA+,° ¶�58BFBDÎ8µ058´K²£E8BDBDEnÃ�´�¿YÚi12*�+,E�+,¹2*�²�5m¶j+¡+,¹:5C+�»r*�²£E8-4*i*~58¶ ¹°X+4*�- 5n+,°DE87#µ�+,¹:*Í12³x925C+,*Í5872@8BD*R´

α, β, γ, ρ1, . . . , ρn

5C-4*º58BFBiÖ�*�-4E:µO+,¹2*¸*�7A+4-,°D*R´;EC²�+,¹:*]m5m¶�E8»:°D587�µ0*�½n5CBD1:5n+4*R9�5n+�*~58¶ ¹;°F+,*R-45C+,°DE87�µY¹r5�½8*�+,¹2*?²£EmBFBDEnÃ�°D72@�½m*�-4Î;´,°F3^³2BD*�²£E8-43�ß

0

B

B

B

B

B

@

x1u11 y1u12 u13 (by1u13 − cu12) (cu11 − ax1u13) (ax1u12 − by1u11) u14 · · · 0x1v11 y1v12 v13 (by1v13 − cv12) (cv11 − ax1v13) (ax1v12 − by1v11) v14 · · · 0��� ��� ��� ��� ��� ��� ��� � � � ���xnun1 ynun2 un3 (bynun3 − cun2) (cun1 − axnun3) (axnun2 − bynun1) 0 · · · un4

xnvn1 ynvn2 vn3 (bynvn3 − cvn2) (cvn1 − axnvn3) (axnvn2 − bynvn1) 0 · · · vn4

1

C

C

C

C

C

A

Ã�°X+4¹

ui = s2

i

0

B

B

@

bR23yi − cR22

cR21 − aR23xi

aR22xi − bR21yi

c(bR21xi + aR22yi) − abR23

1

C

C

A

vi = s2

i

0

B

B

@

cR12 − bR13yi

aR13xi − cR11

bR11yi − aR12xi

abR13 − c(bR11xi + aR12yi)

1

C

C

A

5C7:9si = (aR31xi + bR32yi + cR33)

−1 ¿� ´?²£Em-�+,¹2*;-,*~´c° 9Y1:58B<²£127:¶�+,°DE87:´�+,¹2*R3^´,*�BD½8*~´�µ#Ã�°X+4¹

α, β, γ, ρ1, . . . , ρn

»_*�°D72@ÁÖR*�-4E»r*�²£E8-4*�*~58¶ ¹�°X+4*�- 5n+4°FEm7�µ0+,¹2*RÎ;*R½�58BF1r5n+,*�+4E:ß

r2i−1 = xi − si (aR11xi + bR12yi + cR13)

r2i = yi − si (aR21xi + bR22yi + cR23)�аX+4¹�+,¹2*i*�ÂY³2BD°D¶�°X+!*�ÂY³2-4*R´4´c°DE87r´$²£E8-O+,¹2*g]m5m¶�E8»:°D587^587:9^+4¹2*/-4*R´,°D921:5CBY²£1:7:¶j+4°FEm7:´�µAÃ!*

¹:5�½8*�@8°D½8*�7¸5CBDB<°F7:@8-4*R9Y°D*�7A+4´i-4*RÅA12°D-,*~9>+,E�E8³2+,°D3�°DÖ�*�+4¹2*�¶�Em´c+�²£127:¶j+4°FEm7Í1r´c°D72@�*8¿ @:¿x+,¹:*ý�*�½m*�70»_*�-4@CÕ�ÀÁ5C- ÅA1:5C- 90+�E8-�=?581:´,´cÕRh�*�á+4E87Ø3^*�+,¹2EY92´R¿<×�7Ø+4¹2*�²£E8BDBFEnÃ�°D72@¸³:5C- 5C@m-458³2¹�µÃK*?´c¹:Enà ¹2EnÃÐ+4E�»_*�72*�Ù2+�²£-4E83þ+,¹:*?´,³:58-4´,°X+eÎ^E8²�+,¹2*�]A58¶�E8»2° 5C7ºâ�72*R58-,BDÎ�5CBDB'9Y*�-4°D½�5C+,°D½8*~´Ã�°X+4¹�-4*R´,³_*R¶j+¡+4E^+,¹2*

ρp

5C-4*��*�-4EA�

Page 9: Conic Fitting Using the Geometric Distance

i �Aö~ö~ü>�2úKò Ý¡¹2*?»:5m´c° ¶?5C³2³:-,E�ÂY°D3^5C+,°DE87�+4E�+,¹2*�å/*R´4´c° 5C7�3�5n+,-4°FÂ;1:´,*R9�°F7�BD*R58´c+�´,ÅA1:58-,*~´E8³Y+4°F3^°DÖ�*�- ´$´c1r¶ ¹�5m´$=?581:´4´eÕRh�*Rá+,E87�°D´

H = JT J¿j�O5m¶ ¹�°F+,*�- 5n+4°FEm7�E8²r´,1:¶ ¹�5i72Em7YÕIBD°D72*R58-

3�*�+,¹2EY9�¶�Em3^*R´�9YEnÃ�7�+,E�´,E8BD½A°D72@�5�BF°D72*~5C-�*RÅA1:5C+,°DE87�´cÎY´c+,*R3ÄEC²�+,¹2*�²£EmBFBDEnÃ�°F7:@�²£Em-,3�ß(

A6×6 B6×n

BT

Dn×n

) (x6

yn

)

=

(a6

bn

)

Ã�¹2*�-4*D° ´Rµ�9Y12*�+,EÁ´,³:5C- ´c°F+eÎÁE8²!+4¹2*k]A58¶�E8»2° 5C7�â�´,*�*�³2-4*�½0°DE81:´�³:5C- 5C@m-458³2¹rãi5>9Y°XÕ

5C@8Em7:5CB�3�5n+,-4°FÂx¿xÝ¡¹2*�-,°D@8¹A+,ÕI¹:587:9>´,°D9Y*�° ´/1:´,1:58BFBDÎ�+,¹2*�72*�@A5n+4°F½m*�@8- 5892°F*R7m+iEC²�+4¹2*�¶�Em´c+²£127:¶j+4°FEm7�µYÃ�¹2° ¶ ¹;²£Em-¡BF*~58´c+¡´4ÅA1:5C-4*R´!³2-4E8»2BD*�3�´!¶�587�58BD´,E�»_*�¶�Em3�³:1Y+,*~9;58´

−JTrµr»_*�Õ

°F72@�+4¹2*^½8*R¶�+,Em-iE8²O+,¹:*2n-4*R´,°D9Y1r5CB ´�92*�Ù:72*~9Í58»rEn½m*8¿ � ´�´,12@8@m*R´c+,*~9Á°D7·ÆDÉ�È8ËIµxÃ!*�3�5�Î

-,*~9Y1:¶�*�+4¹2°D´(6 + n) × (6 + n)

³2-4E8»2BD*�3 +4E�56× 6³2-,Em»2BD*�3�µ258´!²£EmBFBDEnÃ�´�ß

É8¿�Ý¡¹2*�BDEnÃK*R-¡´,*�+�E8²�*~Åm1r5n+,°DE87r´¡@8°D½8*8ß

y = D−1

(b− B

Tx) â�ÊAã

ÔY¿ml�*�³:BD5m¶�°D72@�+,¹2° ´�°D7�+4¹2*�12³2³_*�-�´,*�+�E8²�*~ÅA1:5n+4°FEm7:´�µ0Ã!*?@8*�+~ß

(A − BD

−1B

T)x = a − BD

−1bân\8ã

Ç2¿�.0°D7:¶�*D°D´�92°D58@8E87r5CBIµr°F+4´�°D70½8*R-4´,°FEm7º° ´�+,-4°F½0° 5CBIµ�5C7:9º+,¹01:´�+,¹2*�¶�E0*�æ�¶�°F*R7A+4´?EC²!+,¹:*

*RÅA1:5C+,°DE87�´cÎY´c+,*�3ân\Cã!3�5�Î�»_*�¶�Em3^³21Y+,*~9�*�æ�¶�°D*�7A+,BDκâ�°F7�+,°D3^*�5C7:9�3^*�3^Em-,Î2ãj¿È:¿1oi7:¶�*

x°D´!¶�Em3^³21Y+,*~9�»0Î�´cEmBF½0°D72@?+,¹2*

6× 6´,ÎY´e+4*�3�ân\8ãjµ

y° ´OEm»Y+458°F72*~9^1:´,°F72@�â�ÊAãj¿

å�*�7r¶�*8µr+,¹:*�3^Em´c+�¶�E83^³2BD*�Âqpsr tjpsu;p>tjv�w xxE8³_*�- 5n+4°FEm7Á5n+�*R58¶ ¹º°X+4*�- 5n+,°DE87º°D´/+,¹2*^´45C3^*58´�+4¹:5n+�°F7·°X+4*�- 5n+4°F½m*�3�*�+,¹2EY92´�3^°F7:°F3^Ö�°D72@¾5CBD@8*R»2-458°D¶�92°D´c+4587:¶�*^Ò¾°F70½m*�-,+,°D72@Á5

6 × 6´cÎ03^3^*�+,-4° ¶i3�5C+,-4°XÂ�Em-Rµ0*RÅA12°D½n5CBD*�7A+,BDÎ8µ2´,E8BD½0°F7:@�5�BD°F72*~5C-¡*RÅA1:5C+,°DE87�´cÎY´c+,*�3 EC²#+4¹2*�´45C3^*´c°DÖ�*m¿8×�7^³2- 58¶j+4°D¶�*8µmÃK*�-4*R9Y1r¶�*¡+,¹:*�E8-4°F@m°F7r5CB2³2-4E8»:BF*R3ï+,E

(5+n)× (5+n)µn-4*R´,³_*R¶j+4°F½m*�BDÎ

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