determining the epipolar geometry and its uncertainty: a review · 2018-01-04 · issn 0249-6399...

30
Determining the Epipolar Geometry and its Uncertainty: A Review Zhengyou Zhang To cite this version: Zhengyou Zhang. Determining the Epipolar Geometry and its Uncertainty: A Review. RR- 2927, INRIA. 1996. <inria-00073771> HAL Id: inria-00073771 https://hal.inria.fr/inria-00073771 Submitted on 24 May 2006 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 destin´ ee au d´ epˆ ot et ` a la diffusion de documents scientifiques de niveau recherche, publi´ es ou non, ´ emanant des ´ etablissements d’enseignement et de recherche fran¸cais ou ´ etrangers, des laboratoires publics ou priv´ es.

Upload: others

Post on 17-Jul-2020

6 views

Category:

Documents


0 download

TRANSCRIPT

Page 1: Determining the Epipolar Geometry and its Uncertainty: A Review · 2018-01-04 · ISSN 0249-6399 apport de recherche INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE

Determining the Epipolar Geometry and its

Uncertainty: A Review

Zhengyou Zhang

To cite this version:

Zhengyou Zhang. Determining the Epipolar Geometry and its Uncertainty: A Review. RR-2927, INRIA. 1996. <inria-00073771>

HAL Id: inria-00073771

https://hal.inria.fr/inria-00073771

Submitted on 24 May 2006

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, estdestinee au depot et a la diffusion de documentsscientifiques de niveau recherche, publies ou non,emanant des etablissements d’enseignement et derecherche francais ou etrangers, des laboratoirespublics ou prives.

Page 2: Determining the Epipolar Geometry and its Uncertainty: A Review · 2018-01-04 · ISSN 0249-6399 apport de recherche INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE

ISS

N 0

249-

6399

ap por t de r ech er ch e

INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE

Determining the Epipolar Geometry and itsUncertainty: A Review

Zhengyou Zhang

N˚ 2927

July 1996

THEME 3

Page 3: Determining the Epipolar Geometry and its Uncertainty: A Review · 2018-01-04 · ISSN 0249-6399 apport de recherche INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE
Page 4: Determining the Epipolar Geometry and its Uncertainty: A Review · 2018-01-04 · ISSN 0249-6399 apport de recherche INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE

����������� �� ������������������������! "���#�������%$&����'()��*,+-��.����/����0�#�1$324 5 �768���9

:�;<>=?A@CBADE:�;GFA=�?

H�IKJ LNM�O#P�QSRUTVMXWZY>[�TZ\^]UR�IK]ULNLNM0_`LaYb[cId\eRKM>f\gLaYih>M j flkd]>RdRKm M j fd[�]URdRdYi\gjnjZY>Rd[�MXj

opWZ]iqSM0T�rs]>t1]bTVul\gj

rsYbvdv1]>WZTwkKMxWZM [cIKMXWZ[cIKMyR�z>{b|>{>}~P��>�K�e�3� |>|b��P��i�NvdYbhbM j

����� � ���d�b�b� H7��]s\gLaYihbMXj%]b�1Y�jV\gRKhb�eM7jZ[0M RKM���]>tKqSM [�T�YbWZM�WZM0�gYiTZM k�tU�xTZIdM�MXvK\gv�]>�eY>W%h>M0]>LNM TZWZ�bf��wIK\g[cI[ YbR�t�M�kKM jn[0WZ\gt�MXk�tU��Y�Oi�dO�jZ\eRKhU�K�gYbWNLNYbTZWZ\���[ Yi�e�^MXk�TnIKM�M jnjVMXR>TZ\eYb��LaYbTZWZ\���\^�s\gLNYbhbMXj0�\gR>TZM WnRdYi�v�YbWnYbLNM0TZM Wnj8YbWZM��lRK]��wR�fw]UW�TnIKM3� ��RdkdYbLNMXR>TnYi�xLaYiTnWV\^�¡]>TZIKMXWV��\gjZMb¢£Q¤T�[0Y>vKTZ��WVMXj�Yi�e��hbM0]ULNM0TZWZ\g[\gRl�¥]UWZLaYiTZ\^]UR�[�]URUTZYi\gRKMXk�\gR�T¤��]�\gLaYihbMXj0f¦YbRdk§\^Tnj¨kdM0TVMXWZLN\gRdYiTZ\^]UR�\gj¨ubM WZ�©\eLav1]>WZTZY>R>T¨\gRªLaYbRU�Y>vdvK�e\e[ YiTZ\^]URdj%jn�d[cI#Y>j�jn[�MXRKM�LN]KkKM �^\gRKh¦Y>Rdk�u>M IK\g[��eM�R�Y�uU\eh>YbTV\e]>R�¢�H�IK\ej�vdY>v�MXW�h>\^u>M j�YbRx\gRUTZWZ]lk��d[�TZ\^]URTZ]�TZIdM�MXvK\ev1]b�gYbW�hbM ]>LNM0TnWV�>f7Y>Rdk�vdWZ]�ul\ekdM j8Y�[�]>LavK�eM0TZM�WZM0ul\^M �«]i��TZIdM�[0��WZWZM RUT8TZM [cIdRK\g¬l�KM j8�¥]>WMXjVTZ\eLaYbTV\gRKhxTZIKMw� �dRdkdY>L�MXRUTZYi�1LNYbTZWZ\��aYbRdka\eTZj��dRd[0M WZTZYi\gRUT¤�b¢�­®��M �^�^_¯�¥]>�dRdkdM kaLNM Ybjn�dWZM�\ejpvdWV]Uv�]UjVMXkTZ]�[�]ULNv�YbWZM~TZIKMXjVM3TZM [cIdRd\e¬l�KMXj0¢°opWZ]iqSM [0TV\eubM~WZM [�]URdjZTZWn�d[�TZ\^]UR�\gj8Yb�ejZ]�WVM ul\^M ��MXk�¢ªH�IKM�jV]b�¥T¤��Y>WVMXj�wId\e[cI���MxIdY�u>MxkKM0u>M0�e]>v1M k��¥]>W7TZId\ejwWZM0ul\eM0�°YbWZM�Y�u�Yi\e�gYbtK�eM¦]UR¨TZIKMxQSRUTVMXWZRdM0T0¢±�²U³C´nµ#¶ ��·��b�(¸ vK\gv1]b�gYbW�¹¦M0]UL�M TZWZ�bf�ºA�dRdkdY>L�MXRUTZYi�G»3YbTZWZ\���f�¼�Yb�^\gtdWnYiTV\e]>R/f�rwM [0]>RdjZTZWn�d[�TZ\^]UR�fCoGY�½WnYbLNM TVM W ¸ jZTV\gLaYiTZ\^]UR�fKrw]>t��djVTwH�M [cI�RK\e¬l�KMXj0fl¾�Rd[�MXWVTnYi\gR>T¤��¼�IdYbWnYb[0TVM WZ\e¿ YiTZ\^]UR�fUo�MXWS�¥]UWZLaY>Rd[�M ¸ u�Yi�g�l½YbTV\e]>R�f1Àl]i�¥T¤�7YbWZM

ÁÃÂ/ÄVÅ`Æ Ç�Ä7È�ÉeÅ`Ê`Ë�Ì

Unite de recherche INRIA Sophia-Antipolis2004 route des Lucioles, BP 93, 06902 SOPHIA-ANTIPOLIS Cedex (France)

Telephone : (33) 93 65 77 77 – Telecopie : (33) 93 65 77 65

Page 5: Determining the Epipolar Geometry and its Uncertainty: A Review · 2018-01-04 · ISSN 0249-6399 apport de recherche INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE

� ��� � .���������6�����*,6����¨������*%�w6����� *���!����.�����.��� '¨� ��� � �x����0.��� �#��'���� �w�#���#�0�

�� ���� ��� YNhbm ]>LNm0TnWV\eM¦mXvK\gv�]>�eYb\eWZMxkKm [ WV\eT��eYaWZM0�gYiTZ\^]UR�MXR>TnWVM�kdM �l��\gLaYih>M j�k�� �dRKM�L�� LNM�jn[�JXRKMbf¬l�K\GMXjVTx[ YbWnYb[�TZm WZ\ejZm0M�vdYbW �dRKM LaYiTnWV\g[�MaO��dO¨jV\gRKhU�K�^\eJ WZMb¢ ¼�M0TZTVM LaYiTZWZ\g[�MNM jZT [0]>RdRl�KMajZ]>�dj��eMaRd]>LkdM LaYbTZWZ\e[0M¦MXjZjZM RUTV\eM0�e�eM�jZ\��^MXj7vdYbWnYbLNJ0TnWVMXjp\gRUTZWZ\eR�jVJX¬U�dM j7kKM j7\gLaYih>M j7jV]URUT�[0]>RdRl�dj fKkKMxLNYbTZWZ\e[0Ms�¥]URl½k�YbLNM RUTZYb�^M�j0� \^�gj�RKM��eM�jZ]>RUT¦vdY>j0¢ ¸ �^�eM�[0]>RUTV\eM RUT¦TV]>�dTVM js�^MXjs\gRl�¥]UWZLaYiTZ\^]URdjwh>m0]UL�m TZWZ\e¬l�KMXjwkKMXj¦kKMX�l�\gLaYih>M j ¢�ÀU]>R3M jZTV\gLNYbTV\e]>R~MXjVT¦TZWZJ js\eLav1]>WZTZYbRUTZM�v1]>�dW�t�MXYb�d[0]>�dv~k/� Y>vdvK�e\e[ YiTV\e]>R�j¦[0]>LaLNM��eY�LN]lkdm�½�e\gjZYbTV\e]>R kKM�jn[�J RdM j M0T#�^M8kKmXvK�gYb[�MXLNM RUT�k�� �dR WV]Ut�]>T�L�]UtK\e�^M8Y>�KTV]URK]>LNMb¢�¼�M0T�YbWZTV\g[��eMa�¥]>��WZRK\eT��dRKM\gRUTZWZ]Kkd�d[�TZ\^]UR��a�gYahbm ]>LNm0TnWV\eM�m vK\gv�]>�eYb\eWZMbf�M0T7� Yi\eTw�dRKM�WZM0uK�KM�[0]>LavK�eJ0TZM kdM j�TZM [cIdRK\g¬l�KM jwM��K\ejZTZY>RUTVM jk/� MXjVTZ\eLaYiTZ\^]UR�kKMN�gY�LaYiTnWV\g[�M��¥]>R�kdYbLNM RUTnYi�eM#M0TxkKMajZ]>R~\gRd[�MXWVTZ\^Tn�dkKM>¢��s]U�dj¦\eRUTZWZ]Kkd�K\gjV]URdj �dRKMaLNm�½TnIK]KkKM�kdM�LNM jn�dWZM�v1M WnL�M TVTnYbRUT¦kKMN[�]ULNv�YbWZM WwY�u>M [�vdWZm [0\ejZ\e]>R~[�MXjsTZM [cIdRd\e¬l�KMXj0¢ � Y8WZM [0]>RdjZTZWn�d[�TZ\^]URv�WV]bqSM [�TZ\^u>M M jZTwYb�djnjV\/v�YbjnjVm MxM R�WVM uK�KMb¢ � M j��e]bh>\e[0\^M �ej�¬l�KM�Rd]>�djwY�u>]>Rdj�kdm0ubM �^]Uvdv1m j�v1]>�dW�M��AMX[�Tn�KM W[0M0TZTVM�WZM0uK�KMxjV]URUT7kd\ejnv�]URK\gtK�^Mxjn�dW7�¯� QSRUTVMXWZRdM0T0¢� ¶ �X� ´ ��� � ¹¦m0]>LNm TZWZ\^M mXvK\ev1]b�gYi\gWZMbfLNYbTZWZ\e[0M �¥]>R�kdYbLNM RUTnYi�eMbf[0Yi�e\gtdWZYbTV\e]>R�f�WVMX[�]URdjVTnWZ��[�TV\e]>R/f%M jV½TZ\eLaYbTV\e]>R©kKM�vdY>WZY>LNJ0TZWZM j f�TVM [cI�RK\e¬l�KMXj8WZ]>td�djZTVMXj0f7[0Y>WZY>[�TZm WZ\ejnYiTZ\^]UR�k�� \eR�[�M WZTV\eTZ��kKMbf�m u�Yb�e��YiTV\e]>R©kKMv1M WV�¥]>WnLaYbRd[0Mbfl�e]bhb\g[�\eM0�

Page 6: Determining the Epipolar Geometry and its Uncertainty: A Review · 2018-01-04 · ISSN 0249-6399 apport de recherche INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE

������������� ��������������� ��������� �!�"�#�$���%��& ���(')��+* ,���-.�/���0���� �+&�13254��/67���8 O

9 �������w����*

:<;�= �X� ¶ ·���i�7> ¶ = ?

@BA�C > C ¶ � �d�ED ²l¶ � ² � � ³ � = ·GF � ¶ �� ² �IH/� �K� ² � ² = � J{d¢^� �¦]bTnYiTV\e]>R�¢x¢�¢x¢ ¢x¢x¢ ¢x¢ ¢x¢ ¢x¢x¢�¢x¢ ¢x¢ ¢x¢x¢ ¢x¢ ¢x¢�¢x¢ ¢x¢x¢ ¢x¢ ¢x¢ ¢x¢x¢�¢x¢ �{d¢ { ¸ vK\ev1]b�gYbWs¹¦M ]>LNM0TnWV��YbRdk¨º1��RdkdYbLNMXR>TnYi�/»3YbTZWZ\��(¢x¢ ¢x¢�¢x¢ ¢x¢x¢ ¢x¢ ¢x¢ ¢x¢x¢�¢x¢ }{d¢ O ­£¹¦MXRKM WnYi��º1]>WnL,]i� ¸ vK\gv�]>�eY>W ¸ ¬l�dYbTV\e]>R��¥]>W�­sRU��opWV]bqSM [0TV\e]>R¨»~]KkKM0�!¢x¢ ¢x¢x¢�¢x¢ |{d¢ K o�WV]UtK�^MXL!ÀUTnYiTZM LNM RUT8¢ ¢x¢ ¢x¢ ¢x¢x¢�¢x¢ ¢x¢ ¢x¢x¢ ¢x¢ ¢x¢�¢x¢ ¢x¢x¢ ¢x¢ ¢x¢ ¢x¢x¢�¢x¢��7L

MBN ² �PO = >RQ � ² ��S ¶ � A �0�7> ���K�7> =UT �PO ²WV � = ·�� � ² = � � � � �K�X�P>�X : :O�¢^� ¸ �KY>[�T¦ÀU]>�e�dTV\e]>R���\^TnI~}#o]b\gRUTw»3YiTn[cIKM j¨¢x¢ ¢x¢x¢ ¢x¢ ¢x¢�¢x¢ ¢x¢x¢ ¢x¢ ¢x¢ ¢x¢x¢�¢x¢��>�O�¢ { ­¦RdYi�e�lTV\g[x»~M0TnIK]Kk¨��\^TnIEYN]UW�»~]>WZM¦o]>\eRUTw»3YbTZ[cIKMXjx¢ ¢x¢�¢x¢ ¢x¢x¢ ¢x¢ ¢x¢ ¢x¢x¢�¢x¢���{

O�¢ {d¢^� � \gRKM Y>W � M Y>jVTV_SÀl¬l�dY>WVMXj7H%MX[cIdRK\g¬U�dM�¢ ¢x¢x¢ ¢x¢ ¢x¢�¢x¢ ¢x¢x¢ ¢x¢ ¢x¢ ¢x¢x¢�¢x¢���{O�¢ {d¢ { ¸ \^h>M R¨­sRdYb�^�KjZ\ej ¢ ¢x¢ ¢x¢x¢�¢x¢ ¢x¢ ¢x¢x¢ ¢x¢ ¢x¢�¢x¢ ¢x¢x¢ ¢x¢ ¢x¢ ¢x¢x¢�¢x¢��XOO�¢ {d¢ O QSLNv1]>jZ\gRKhaTZIKMxrsY>Rd�>_¤{a¼�]>R�jVTnWZYb\eRUT ¢x¢x¢ ¢x¢ ¢x¢�¢x¢ ¢x¢x¢ ¢x¢ ¢x¢ ¢x¢x¢�¢x¢��ZKO�¢ {d¢ K ¹¦M0]>LNM TZWZ\e[�QSRUTVMXWZvdWZM0TnYiTZ\^]UR8]i�TZIdM � \gRKM Y>W�¼�WZ\eTVM WZ\e]>R ¢x¢x¢ ¢x¢ ¢x¢ ¢x¢x¢�¢x¢��ZKO�¢ {d¢ � �s]>WnLaYi�e\^¿ \eRdh�QSRdv��KT3[�YbTZY ¢�¢x¢ ¢x¢ ¢x¢x¢ ¢x¢ ¢x¢�¢x¢ ¢x¢x¢ ¢x¢ ¢x¢ ¢x¢x¢�¢x¢����

O�¢ O ­¦RdYi�e�lTV\g[x»~M0TnIK]Kk¨��\^TnI�r¦YbRd�>_S{N¼�]URdjVTnWZYb\eRUT�¢x¢ ¢x¢ ¢x¢�¢x¢ ¢x¢x¢ ¢x¢ ¢x¢ ¢x¢x¢�¢x¢���}O�¢ K �¦]>RK�e\gRKM Y>W�»~M0TnIK]Kk¨»3\eRK\gLN\^¿ \eRdh)[¦\gjVTnYbRd[0M j7]b��o]b\gRUTZj�TZ] ¸ vK\ev1]b�gYbW � \eRdM j ¢x¢x¢�¢x¢���}

O�¢ K1¢^� Q¤TVM WnYiTZ\^u>M � \eRdM YbW�»3M0TZId]lk�¢�¢x¢ ¢x¢ ¢x¢x¢ ¢x¢ ¢x¢�¢x¢ ¢x¢x¢ ¢x¢ ¢x¢ ¢x¢x¢�¢x¢��7YO�¢ K1¢ { �s]>Rd�^\gRKM Y>W7»~\eRd\eLN\e¿ YiTZ\^]UR¨\eR¨oGYbWnYbLNM0TZM WwÀlvdY>[�M�¢�¢x¢ ¢x¢x¢ ¢x¢ ¢x¢ ¢x¢x¢�¢x¢��7Y

O�¢ � ¹�WnYbkd\^MXR>TV_`t�YbjZM k8TZM [cIdRd\e¬l�KM ¢ ¢x¢x¢�¢x¢ ¢x¢ ¢x¢x¢ ¢x¢ ¢x¢�¢x¢ ¢x¢x¢ ¢x¢ ¢x¢ ¢x¢x¢�¢x¢�{�LO�¢ � �¦]>RK�e\gRKM Y>W%»~M TZIK]Kk�»~\gRK\gLN\^¿ \eRKh\[¦\gjVTnYbRd[0M j^]�M T¤��M M R`_�tdjZM WZu�YiTZ\^]UR�Y>Rdk rwMXvdWV]bqSM [0TV\e]>R£{d�O�¢ } rs]>td�djZTw»~M TZIK]Kkdj ¢x¢ ¢x¢ ¢x¢ ¢x¢x¢�¢x¢ ¢x¢ ¢x¢x¢ ¢x¢ ¢x¢�¢x¢ ¢x¢x¢ ¢x¢ ¢x¢ ¢x¢x¢�¢x¢�{>{

O�¢ }K¢^� »�_ ¸ jVTZ\eLaYiTZ]>Wnj~¢x¢ ¢x¢ ¢x¢x¢�¢x¢ ¢x¢ ¢x¢x¢ ¢x¢ ¢x¢�¢x¢ ¢x¢x¢ ¢x¢ ¢x¢ ¢x¢x¢�¢x¢�{bOO�¢ }K¢ { � M Y>jVT�»~MXkK\gYbR�]i�pÀl¬l�dYbWZM jba � »~M kAÀdc�¢x¢ ¢x¢ ¢x¢�¢x¢ ¢x¢x¢ ¢x¢ ¢x¢ ¢x¢x¢�¢x¢�{b�

O�¢ Y ¼�I�YbWnYb[�TZM WZ\^¿ \eRdh�TnIKMx¾¦Rd[0M WZTZYb\eRUT¤� ]i�º1��RdkdYbLNMXR>TnYi�/»3YbTZWZ\��3¢ ¢x¢x¢ ¢x¢ ¢x¢ ¢x¢x¢�¢x¢�{b|O�¢ Y�¢^� H�IKM#ÀUTZYbTV\gjVTZ\e[ Yi�%»~M TZIK]Kk ¢�¢x¢ ¢x¢ ¢x¢x¢ ¢x¢ ¢x¢�¢x¢ ¢x¢x¢ ¢x¢ ¢x¢ ¢x¢x¢�¢x¢"OeLO�¢ Y�¢ { H�IKMx­sRdYb�^�lTV\g[0Yb�%»~M TZIK]Kk ¢�¢x¢ ¢x¢ ¢x¢x¢ ¢x¢ ¢x¢�¢x¢ ¢x¢x¢ ¢x¢ ¢x¢ ¢x¢x¢�¢x¢"OeL

O�¢ | _¦TnIKM W7TZM [cIdRd\e¬l�KMXj£¢x¢ ¢x¢ ¢x¢ ¢x¢x¢�¢x¢ ¢x¢ ¢x¢x¢ ¢x¢ ¢x¢�¢x¢ ¢x¢x¢ ¢x¢ ¢x¢ ¢x¢x¢�¢x¢"O>O

f � =gA X/� � C � ²¨¶ S V � = ·�� � ² = � � � � �K�X�7>RX A �0�7> ���d�Z> ¶ =ih >¥�7Okj ¶ � C ���7> � ¶ = MlfK1¢^� ­E»3M Ybjn�dWZM¦]b� ¼�]ULavdYbWZ\ejZ]>Rm]7M0T¤��M0M R�º1��RdkdYbLNMXR>TnYi�/»3YbTZWZ\e[0M j�¢x¢ ¢x¢ ¢x¢ ¢x¢x¢�¢x¢"Ol}

?Bn >¥�X� ��� �7> ¶ = � f�o�d¢^� ÀK�dLaLNY>WV� ¢�¢x¢ ¢x¢x¢ ¢x¢ ¢x¢ ¢x¢x¢�¢x¢ ¢x¢ ¢x¢x¢ ¢x¢ ¢x¢�¢x¢ ¢x¢x¢ ¢x¢ ¢x¢ ¢x¢x¢�¢x¢pK�L�d¢ {q[�M0h>M RKMXWZYbTVMx[�]URsr�hU�dWZYbTV\e]>Rdj£¢ ¢x¢x¢�¢x¢ ¢x¢ ¢x¢x¢ ¢x¢ ¢x¢�¢x¢ ¢x¢x¢ ¢x¢ ¢x¢ ¢x¢x¢�¢x¢pKl{�d¢ O ­!t8RKM�[0Y>L�MXWZY>j&¢x¢x¢ ¢x¢ ¢x¢ ¢x¢x¢�¢x¢ ¢x¢ ¢x¢x¢ ¢x¢ ¢x¢�¢x¢ ¢x¢x¢ ¢x¢ ¢x¢ ¢x¢x¢�¢x¢pKl{�d¢ K ¼�Y>LNM WnYbj���\^TnI � M R�ju[¦\gjVTZ]>WZTV\e]>R�¢x¢�¢x¢ ¢x¢ ¢x¢x¢ ¢x¢ ¢x¢�¢x¢ ¢x¢x¢ ¢x¢ ¢x¢ ¢x¢x¢�¢x¢pKUO�d¢ � »��K�^TZ\evd�^M�[ YbLNM WnYbj©¢x¢ ¢x¢ ¢x¢ ¢x¢x¢�¢x¢ ¢x¢ ¢x¢x¢ ¢x¢ ¢x¢�¢x¢ ¢x¢x¢ ¢x¢ ¢x¢ ¢x¢x¢�¢x¢pKUO

v(vxw�y"z"{.z.|

Page 7: Determining the Epipolar Geometry and its Uncertainty: A Review · 2018-01-04 · ISSN 0249-6399 apport de recherche INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE

K � ���/�s�#&������ �s�#���

� F � ¶���² �b�Z>�� ² ��² � ¶ = � � � �%�b�Z> ¶ = f f­#¢^��o�WV]bqSM [�TZ\^u>M#ÀUTZWn�d[�Tn�dWZM¦� WZ]>L H7��]a¾¦R�[0Yi�e\gtdWZYbTVMXk�QSLaYih>M j ¢x¢ ¢x¢x¢ ¢x¢ ¢x¢ ¢x¢x¢�¢x¢pKeK­#¢ { ¼�]ULavd�KTV\gRKh�¼�Y>L�MXWZY�opWZ]iqSMX[�TV\e]>R¨»3YbTZWZ\e[0M j ¢x¢x¢ ¢x¢ ¢x¢�¢x¢ ¢x¢x¢ ¢x¢ ¢x¢ ¢x¢x¢�¢x¢pKl�

­#¢ {d¢^� º1Yb[0TV]UWV\e¿ YbTV\e]>R�»3M0TZId]lk ¢x¢�¢x¢ ¢x¢ ¢x¢x¢ ¢x¢ ¢x¢�¢x¢ ¢x¢x¢ ¢x¢ ¢x¢ ¢x¢x¢�¢x¢pKU�­#¢ {d¢ { ¼�IK]l]>jZ\eRKhaYNopWZ]iqSM [0TV\eubM ]�YbjZ\ej�¢ ¢x¢ ¢x¢x¢ ¢x¢ ¢x¢�¢x¢ ¢x¢x¢ ¢x¢ ¢x¢ ¢x¢x¢�¢x¢pKU�

­#¢ O rsM [�]URdjZTZWn�d[�TZ\^]UR�H�M [cIdRd\e¬l�KMXj�¢ ¢x¢x¢�¢x¢ ¢x¢ ¢x¢x¢ ¢x¢ ¢x¢�¢x¢ ¢x¢x¢ ¢x¢ ¢x¢ ¢x¢x¢�¢x¢pKK}­#¢ O�¢^� � \gRKM Y>W�»3M0TZId]lk�j�¢ ¢x¢ ¢x¢x¢�¢x¢ ¢x¢ ¢x¢x¢ ¢x¢ ¢x¢�¢x¢ ¢x¢x¢ ¢x¢ ¢x¢ ¢x¢x¢�¢x¢pKK}­#¢ O�¢ { Q¤TVM WnYiTZ\^u>M � \eRdM YbW�»3M0TZId]lk�j ¢x¢ ¢x¢ ¢x¢x¢ ¢x¢ ¢x¢�¢x¢ ¢x¢x¢ ¢x¢ ¢x¢ ¢x¢x¢�¢x¢pKK}­#¢ O�¢ O �s]>Rd�^\gRKM Y>W7»~M0TnIK]Kkdj�¢ ¢x¢x¢�¢x¢ ¢x¢ ¢x¢x¢ ¢x¢ ¢x¢�¢x¢ ¢x¢x¢ ¢x¢ ¢x¢ ¢x¢x¢�¢x¢pK�Y

� � CUC � ¶ X > ���K� ² A � �Z> ���K�7> ¶ = ¶ S V � = ·�� � ² = �X� � � �K� �P>�XgS`� ¶ � D ² = ² �X� � � �K� �P>�X f

� 0*�� � � � 0��������*

� H�IdMxM vK\gv�]>�eY>W7hbM ]>LNM0TnWV� ¢ ¢x¢ ¢x¢x¢�¢x¢ ¢x¢ ¢x¢x¢ ¢x¢ ¢x¢�¢x¢ ¢x¢x¢ ¢x¢ ¢x¢ ¢x¢x¢�¢x¢ }{ Q¤�e�g�djVTnWZYbTV\e]>R�]i�GYatd�d[c�iM TV\gRKhaTVMX[cIdRK\g¬U�dM ¢x¢ ¢x¢x¢ ¢x¢ ¢x¢�¢x¢ ¢x¢x¢ ¢x¢ ¢x¢ ¢x¢x¢�¢x¢�{�YO QSRUTZM WZu�Yb�/YbRdk¨td�d[c�bM0TwLaYbv�vK\eRdh ¢x¢�¢x¢ ¢x¢ ¢x¢x¢ ¢x¢ ¢x¢�¢x¢ ¢x¢x¢ ¢x¢ ¢x¢ ¢x¢x¢�¢x¢�{b|K QSLaYbhbMavdYi\gW �djVMXk3�¥]UWx[�]ULNv�YbWZ\eRKh¨kK\ �AMXWVMXR>T MXjVTZ\eLaYiTZ\^]UR3TZM [cIdRK\g¬l�KM j ]b��TnIKMN� �dRdkdYi½

LNMXR>TnYi�%LaYbTZWZ\���¢ ¢x¢x¢ ¢x¢ ¢x¢ ¢x¢x¢�¢x¢ ¢x¢ ¢x¢x¢ ¢x¢ ¢x¢�¢x¢ ¢x¢x¢ ¢x¢ ¢x¢ ¢x¢x¢�¢x¢"O�K� ¸ vK\ev1]b�gYbW%hbM ]>LNM0TnWV�¦M jZTV\gLNYbTVMXkxTZIdWZ]>�dh>Ix[��gYbjnjZ\e[ Yi�ijZTVM WZM0]¦[0Yi�e\gtdWZYbTV\e]>R�fX�wIK\g[cIxjVMXWVu>M j

Y>j7TZIKMxhUWV]U�dRdk�TZWn�KTZI¨¢ ¢x¢ ¢x¢ ¢x¢x¢�¢x¢ ¢x¢ ¢x¢x¢ ¢x¢ ¢x¢�¢x¢ ¢x¢x¢ ¢x¢ ¢x¢ ¢x¢x¢�¢x¢"OU�� ¸ vK\ev1]b�gYbW7h>M0]UL�M TZWZ�8M jZTV\gLaYiTZM k���\^TnI�TnIKMx�e\eRKMXYbW�LNM0TnIK]Kk ¢x¢ ¢x¢x¢ ¢x¢ ¢x¢ ¢x¢x¢�¢x¢"O>�} ¸ vK\ev1]b�gYbWwhbM0]ULNM0TZWZ�8MXjVTZ\eLaYiTZM k���\eTZI�TnIKMx�^\gRKMXYbWwLNM0TnIK]Kk���\^TnI~vdWZ\^]UW�kdYiTnY RK]>WnLaY�½

�e\e¿ YiTZ\^]UR ¢x¢�¢x¢ ¢x¢x¢ ¢x¢ ¢x¢ ¢x¢x¢�¢x¢ ¢x¢ ¢x¢x¢ ¢x¢ ¢x¢�¢x¢ ¢x¢x¢ ¢x¢ ¢x¢ ¢x¢x¢�¢x¢"Ol}Y ¸ vK\ev1]b�gYbW7h>M0]UL�M TZWZ�8M jZTV\gLaYiTZM k���\^TnI�TnIKM�RK]URK�^\gRKMXYbW�LNM0TnIK]Kk£¢x¢x¢ ¢x¢ ¢x¢ ¢x¢x¢�¢x¢"OeY| ¼�]ULavdYbWZ\ejZ]>R�t1M0T¤��M0M R�TnIKM ¸ vK\gv1]b�gYbW/h>M0]UL�M TZWZ�wM jZTV\gLaYiTZM k TZI�WV]U�Kh>I�[��gYbjnjV\g[0Yb�ijVTZM WZM0]

[ Yi�e\et�WZYbTV\e]>R a jnIK]��wR \gR8rsM k1��[�Y>WZ���^\gRKMXj.cpYbRdk8TZIdYbT�MXjVTZ\eLaYiTZM k���\^TnI8TZIdM¦RK]URK�^\gRKMXYbWLNM TZIK]Kk a jnIK]��wR�\eR3¹�WZM0M RA�b¹�WZM0� �e\eRKMXj.c�¢x¢ ¢x¢x¢ ¢x¢ ¢x¢�¢x¢ ¢x¢x¢ ¢x¢ ¢x¢ ¢x¢x¢�¢x¢"OeY

�7L [�M/rARK\^TZ\^]UR¡]i� TZIKM3kK\ �CM WZM Rd[0M�t1M0T¤��M0MXR¡T¤��] � �dRdk�YbLNM RUTZYb�wLaYiTZWZ\g[�M ja\gR¡TVMXWZLaja]i�\gLaYih>M kd\ejZTZY>Rd[�MXj�¢x¢x¢ ¢x¢ ¢x¢ ¢x¢x¢�¢x¢ ¢x¢ ¢x¢x¢ ¢x¢ ¢x¢�¢x¢ ¢x¢x¢ ¢x¢ ¢x¢ ¢x¢x¢�¢x¢"O>|

� 0*�� � � � ��������*

� ¼�]ULavdYbWZ\ejZ]>R�]i�GkK\ �AMXWVMXR>T�LNM0TnIK]Kkdj7�¥]UW7M jZTV\gLNYbTV\gRKhaTZIKM � ��RdkdYbLNMXR>TnYi�%LaYbTZWZ\�� ¢�¢x¢"OU�{ [�\gjVTnYbRd[0M j�t1M0T¤��M0M R¨TnIKM � �dRdkdY>LNM RUTZYb�/LNYbTZWZ\e[0M j7MXjVTZ\eLaYiTZM k�tU��kK\ �AMXWVMXRUT�TVM [cI�RK\e¬l�KMXj K�L

���������

Page 8: Determining the Epipolar Geometry and its Uncertainty: A Review · 2018-01-04 · ISSN 0249-6399 apport de recherche INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE

������������� ��������������� ��������� �!�"�#�$���%��& ���(')��+* ,���-.�/���0���� �+&�13254��/67���8 �

� � �#����� ' ��.p������

H7��]�v1M Wnjnv�MX[�TV\eubMN\gLNYbhbMXjs]b��Y�jZ\gRKhb�eM�WZ\^h>\ek�]>tlqSMX[�Tc��jn[�M RdM�YbWZM�WZM0�gYiTZM k3tU�~TnIKM jV]b_`[ Yi�e�^MXk �0�� ������������.���)�/�%��&�f��wIK\e[cI¡[0Y>R�t1M~kKM jn[0WZ\gt�MXk�tU��Y������ jV\gRKhU�K�eY>WaLNYbTZWZ\���¢¡Q`��TnIKM�\eRUTZM WnRdYi� a \eRUTZWZ\gRdjV\g[7cv�YbWnYbLNM0TZM Wnja]i�xTZIKM�\eLaYbhbM jWa Mb¢ h�¢ TnIKM3�¥]K[0Yb�w�eM RKh>TZI�f�TnIKM�[0]U]UWZkd\eRdYbTVMXj ]i�xTZIdM�vdWZ\eRd[0\ev�Yi�¦v1]b\gRUT0fM TZ[PcwY>WVMN�URd]��wR�fC��Ma[0Y>R~��]UWZ�¨��\eTZI�TZIKM ���#���$��� ��7�.' ��$�Z�e� -.�Z�#� '���(���0��* a ºAYb�Kh>M WnYbj��X|b|>O�c)fCYbR�kTnIKM�LNYbTZWZ\���\gja�lRK]��wR�Y>j#TnIKM ��*.*��/� �+���� �$�#�+�� a � ]>Rdh>�KM TS_��s\ehbhb\gRdj�� |eYd�Pc� p]>TZIKMXWV��\gjVM>fG��M�IdY�u>MTZ]���]>Wn����\eTZI�TnIKMW��s���\��$�Z�e� -.�7��� '#������0�/*)f�YbRdk©TnIKM�LaYbTZWZ\��©\ej¨�lRK]��wR©Ybj�TZIKM�� ����'��#�$���l�0�#��)���%�� a � �K]URKh��X|b|U{KfAº1Yb�dhbM WnYbj�� |>|>�df � �K]>RKh�YbR�k�ºAYb�Kh>M WnYbj�� |b|>��cc¢�Q¤T�[�]UR>TnYi\gRdj Yi�e�hbM0]ULNM0TZWZ\g[\gRl�¥]UWZLaYiTZ\^]UR�TnIdYiTa\gjNRdM [�MXjZjnYbWZ� �¥]>WNM jZTZY>tK�e\ejnIK\gRKh [0]>WnWVMXjZv1]>R�kKM Rd[0M j�t�M T¤��M M R�T¤��]�\eLaYih>M j fG� WZ]>L�wId\e[cI�TnIdWVM M�_¤kK\gL�MXRdjZ\^]URdYi�%jZTZWn�d[0TZ�dWZM�]i�GTnIKM�v1M Wn[�M \^u>M k�jZ[0M RKM�[0YbR~t1M#\gRl�¥MXWZWZM k�¢�QSR~Y�jVTZM WZM0]�ul\ejZ\e]>RjZ�KjVTZM L-�wIKM WZM TZIKM8[ YbLNM WnY¨hbM ]>LNM0TnWV�3\gj�[ Yi�e\etdWnYiTZM k�f�\^T#\ej�v1]>jnjZ\etK�eM TV]3[ Yi�g[0�K�gYiTZM8jZ��[cI Y~LaYbTZWZ\��� WZ]>L°TnIKM�[ YbLNM WnYsv1M Wnjnv�MX[�TV\eubM�vdWZ]iqSM [0TV\e]>R�LaYbTZWZ\e[0M j�TZIdWZ]>�KhUI�[0Yb�^\gtdWnYiTZ\^]UR$a ­w�UYb[cIKM¦� |>|d�bf º1Y>�KhbMXWZY>j�X|b|>O�c)¢���IKM R TnIKMa\eRUTZWZ\gRdjV\g[avdYbWnYbLNM TVM Wnj�YbWZMN�lRK]��wR td�KTxTnIKMaM��KTZWZ\eR�jV\g[N]>RKMXja TZIKM8WZ]bTnYiTZ\^]UR�YbR�kTnWZY>RdjZ�eYbTV\e]>R t1M0T¤��M0M R8TZIKM¦T¤��]�\eLaYih>M j�cYbWZMsRK]>T0flTZIdM�vdWZ]>td�^MXL£\gj7�URd]��wR Ybj�L�]>TV\e]>R8Y>Rdk8jZTZWn�d[�Tn�dWZM� WZ]>L LN]bTZ\^]UR�fiYbR�k#IdY>j�t1M0MXR�M��KTZM RdjZ\^u>M0�e�xjVTn�dkK\eM k�\gRN¼�]ULavd�KTVMXW��¦\ejZ\e]>R� iT¤��]sM0�d[�M0�e�eM RUT�WZM0ul\eM0�wj�YbWZMYb�eWZM Y>kK��Y�u�Yb\^�gYbtd�^M�\eR�TZIK\gjxkK]>LaYb\eR a¯­wh>h>YbWZ�7Yi��Y>Rdk ��YbRdk�IdYb�l�dLaYbW�� |eY�Y�f��¦��YbRKh�YbRdk �sM TZWnY�u�Yb�^\�X|b|�Ksc)¢���M YbWZMs\gRUTVMXWVMXjVTZM k8IKMXWVM¦\eR�kK\ �AMXWVMXR>T�TVMX[cIdRK\g¬U�dM j��¥]>W�M jZTV\gLaYiTV\gRKh�TZIKM¦� �dRdkdY>LNM RUTZYb�CLaYbTZWZ\��� WZ]>L°T¤��]¦�dR�[0Yi�e\gtdWZYbTVMXkx\eLaYbhbM j f�\¯¢ M>¢%TZIdM7[ YbjZMp�wIKMXWVM�t�]>TZI#TZIKM�\eRUTnWV\gRdjZ\e[�YbRdk�M��KTZWZ\eR�jV\g[�v�YbWnYbLNM0TZM Wnj]b��TnIKM�\gLaYihbMXj�YbWZMx�dRd�lRK]��wR�¢�º1WZ]>L TnIK\gjwLaYiTZWZ\^�Cf���M�[ YbR�WVMX[�]URdjVTnWZ��[�T�Y vdWZ]iqSM [0TV\eubMxjZTZWn�d[�Tn�dWVMx]b�TnIKM�jn[�M RdMbfdkKM�r1RKM k��dv¨TZ]aY������8LaYiTZWZ\^��TZWnYbRdjV�¥]>WnLaYiTZ\^]UR�¢

H�IKM�jZTZ�dkK� ]b�s�dRd[0Yb�^\gtdWnYiTZM k \gLaYihbMXj#IdY>j#LaYbRU� \gLNv1]>WZTZY>RUT�Ybv�vK�^\g[0YbTV\e]>Rdj ¢�H�IdM�WZM YbkdM W#LaY����]>R�kKM W�TZIKM �djZM�� �K�gRKM jnj�]i��jn�d[cI�Y�vdWV]bqSM [0TV\eubM�jZTZWn�d[�Tn�dWZMb¢���M [0Y>RdRK]bT�]>tKTnYi\gR�YbRU�®LNM0TnWV\g[3\gRl½�¥]UWZLaYbTV\e]>R � WZ]>L�Y�v�WV]bqSM [�TZ\^u>M�jZTZWn�d[�Tn�dWZM��8LNM Y>jZ�dWZM LNMXR>Tnj#]b�w�^MXRKhbTnIdjaYbRdk�YbRKh>�^MXj�kK] RK]bTaLaYb�bMjZM R�jVM>¢��s]���M0u>M W f�Y¨v�WV]bqSM [�TZ\^u>M�jZTZWn�d[�Tn�dWZMNjZTV\e�^��[�]>RUTnYi\gRdj¦WZ\g[cI�\gRl�¥]>WnLaYiTZ\^]UR�f/jZ��[cI�Ybjx[0]>vK�gYbR�YbWZ\^T¤�>f[0]b�e�^\gRKMXYbWZ\^T¤�>f1YbR�k3[0WZ]>jnjwWnYiTV\e]>j`a WnYiTZ\^]8]b�pWZYbTV\e]>j�]b�pkK\gjVTnYbRd[0M j�c)fA�wIK\e[cI3\ej�jV]ULNM0TV\gLNM j�jZ��t [0\^MXRUT��¥]>WY>WVTZ\�rA[�\gYi� jZ�ljZTVMXLaj0f�jZ�d[cI Ybj#WV]Ut�]>TZj f/TV]3v1M WV�¥]>WnL�TnYbjn�Ujxjn�d[cI Ybj�R�Y�uU\eh>YbTV\e]>R�YbRdk ]>tKqSM [�T�WZM [0]bhURK\�½TZ\^]UR a¤ÀlIdY>jZIl�dY¨� |>|#K1f �CM0�e�^MXW�YbRdk�º1Y>�KhbMXWZY>j¦�X|b|�K�fs]7M YbWnkdjZ�^M �bf��A\gjZjZM WnLaYbR�YbRdk¨»3�dWnWZY��~�X|b|�Ksc)¢QSR©LaYbRU��Y>vdvK�e\e[ YiTV\e]>R�jNjn�d[cI¡YbjaTnIKM~WVMX[�]>R�jVTnWZ�d[0TV\e]>R�]i� TZIKM�M RUul\gWV]URdLNM RUTN� WV]UL Y jVMX¬U�dM Rd[0M

]b��uU\gkKM ]3\eLaYbhbM j#�wIKM WZM TZIdM�vdY>WZY>L�M TVMXWZjx]b��TZIKM�uU\gkKM ]3�^MXRdj�\ej�jZ�dt�L�\eTVTZM k TZ]�[0]>RUTV\gRl�K]>�dj#L�]KkK\^½rA[0YiTZ\^]UR�f�[0Y>L�MXWZY [ Yi�e\et�WZYbTV\e]>R¡\eR©TnIKM�[0�eY>jZjZ\g[0Yi�¦jVMXRdjVM~\gj�Rd]bT�v1]>jnjZ\etK�eMb¢!� M�[0Y>RdRK]>T8M��dYb[0T�YbRU�LNM TZWZ\e[ \gRl�¥]UWZLaYiTZ\^]UR�ftd�KT#Y3v�WV]bqSM [�TZ\^u>M jVTnWZ�d[0TZ�dWZM8\gj#jZTV\e�^��v�]UjZjZ\gtK�^M \^�wTZIdM�[0Y>L�MXWZY~[ YbR t1M�[�]URdjZ\�½kdM WZM k¡YbjaY vK\eR�IK]b�eMb¢©º1��WVTnIKM WnLN]>WZMbfp\^� ��M~[0Y>R�\gRUTZWZ]Kkd�d[�M~jZ]>LNM��URd]����^MXkKhbM¨]i� TZIKM~jn[�MXRKM�\eRUTV]TnIKM vdWZ]iqSM [0TV\eubMNjVTnWZ�d[0TZ�dWZMbfC��M [0YbR�]>tKTnYi\gR�LN]>WZM�jnv1M [�\ r1[ajZTZWn�d[�Tn�dWVM�]i��TZIKM jn[�M RdMb¢�º1]>W¦M��dYbLavK�eMbftU� jZv1M [0\��¥�l\gRKh�Y�vK�eY>RKM�YbT#\gRsr1RK\eT¤� a \gR�vdWnYb[0TV\g[�Mbf��M�RKM M k ]URK�^� TV] jZv1M [0\��¥� Y�vK�eY>RKM�jn�st8[�\eM RUTV�e�� Y>W�Y��7Y��dccf%YbR Y#t RdM8jVTnWZ�d[0TZ�dWZM8[0Y>R t�M�[0]>Lavd�KTZM k�f�wIK\g[cI vdWZM jZM WZubM j vdY>WZYb�^�eM0�e\gjZL YbRdk WnYiTV\e]>j ]b�kd\ejZTZY>Rd[�MXj!a�"���YbR�� |>|bO�f�º1Yb�dhbM WnYbjw� |b|U�ecc¢��¦Y>WVTZ�^M �bfd¹��dvKTnY#YbR�k�¼�IdYbRKhmaV� |>|>{�c^r1WZjZTpWZM [0]>RdjZTZWn�d[�T Yv�WV]bqSM [�TZ\^u>MxjVTnWZ�d[0TZ�dWZMbfAYbRdk�TnIKM R��djVMYah>WZ]>�dRdk�WZM��¥MXWVMXRd[�M�v1]b\gRUTZjwTZ]8]>tdTZYi\gR�TZIdM ¸ �d[0�^\gkKM Y>R~jVTnWZ��[c½Tn�dWZMsYbRdkaTnIKM¦[ YbLNM WnY vdYbWnYbLNM TVM Wnj ¢�»~]UIdW0f�]7]>�l� Y>LaYxYbRdk$]�WnYbRdkEaV�X|b|>O�c�M L�t�MXk8[�]URdjZTZWnYi\gR>TnjGjZ��[cIY>j�^]K[ YiTV\e]>R�]i��v1]b\gR>Tnj0fUvdYbWnYi�e�eM0�e\ejnL YbR�k�u>M WZTV\g[0Yb�dvK�eY>RKM jua Mb¢ h�¢���Yb�^�gj�c�kK\eWZM [0TV�e�#\gRUTV]xYxLN\eRK\gLN\^¿XYiTZ\^]URv�WV]K[�MXkd�dWZM�TV]�kKM TVM WnLN\eRdM�Y ¸ �d[��e\gkKM Y>R�jZTZWn�d[�Tn�dWVM>¢¦rs]>t1M WZT¦YbR�k3º1Y>�KhbMXWZY>j aZ� |b|>O�c7jnIK]��ETnIdYiT�TZIKMO�[§[�]URUubM��#Il�K�e�d]b�CY>R�]UtlqSM [0T [0Y>R�t1Ms[�]ULavd�KTVMXk�� WV]UL YxvdYi\gW]i�C\gLNYbhbMXj�wIK]>jZM7M vK\gv�]>�eY>WhbM ]>LNM0TnWV�\gj��lRK]��wR�¢

v(vxw�y"z"{.z.|

Page 9: Determining the Epipolar Geometry and its Uncertainty: A Review · 2018-01-04 · ISSN 0249-6399 apport de recherche INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE

� � ���/�s�#&������ �s�#���

Q`����MaYbjnjZ��L�M�TZIdYbT TZIKMa[0Y>L�MXWZY�vdYbWnYbLNM TVM Wnjskd]¨RK]>T�[cI�YbRKh>M�t1M0T¤��M0M R�jZ��[0[�MXjZjZ\eubM#ul\eM0�wj f�TZIKMv�WV]bqSM [�TZ\^u>M7\eRUu�YbWZ\eY>RUTZj[0Y>R�M ubMXRNt1Ms�djZM kNTZ]�[0Yi�e\gtdWZYbTVM7TZIdMs[0Y>L�MXWZY>j�\eRaTnIKMw[��gYbjnjV\g[0Yb�djVMXRdjZM���\^TnIK]>�KT��jV\gRKh~YbRU�3[0Yb�^\gtdWnYiTZ\^]UR�Ybv�vdYbWnYiTn�dj�a �lRK]��wR Ybj*���� ��� -.��� ����"���%���� c$a »�Y��lt�YbRd�~Y>Rdk�ºAYb�Kh>M WnYbjN� |>|>{dfºAYb�Kh>M WnYbj f � �K]>RKh�YbRdk8»�Y��lt�YbRd��� |>|>{Kf � �K]URKh8�X|b|>{df �CI�YbRKh1f � �K]>RKh�YbRdk ºAYb�Kh>M WnYbjs� |b|>�df ¸ Rd[�\gjV]�X|b|U�ec)¢rwMX[�MXR>TZ�^�>f���MpI�Y�ubM jnIK]��wR$a��CI�YbRKh��X|b|>�bY�c�TnIdYiT�M0ubMXR�\eRxTnIKM�[ YbjZMG�wIKM WZM \gLaYihbMXj%Y>WVM [ Yi�e\etdWnYiTZM k�f

LN]UWVM¨WVM �^\gYbtK�eM�WVMXjZ�K�eTZja[ YbR�t1M¨]>tKTnYi\gRKM k�\�����M��djZM�TnIKM~[�]>R�jVTnWZYb\eRUTZjNYbWZ\gjV\gRKh3� WZ]>L ��Rd[0Yb�^\gtdWnYiTVMXk\gLaYih>M j7Ybj�Y>R�\gRUTVM WnLNM kK\gYiTZM�jVTZM v�¢

H�IK\gj#v�Ybv1M W�h>\^u>M jxY>R \eRUTZWZ]Kkd�d[0TV\e]>R TV]3TnIKM8MXvK\ev1]b�gYbW�hbM0]ULNM0TZWZ�bf�vdWZ]�uU\gkKMXj�Y~RKM0� �¥]UWZL��K�eY�]b�TnIKM¦� �dRdkdY>L�MXRUTZYi�/LaYbTZWZ\��8�wIK\g[cI8\gj�u�Yi�e\ek8�¥]>W7t�]>TZI¨v�MXWZjnv1M [�TZ\^u>M�Y>Rdk�Y#t8RKM�[ YbLNM WnYbj flYbRdk�WVM uU\eM0�wjkd\ �CM WZM RUTpLNM0TnIK]Kkdj WZM v1]>WZTVMXk�\gRaTZIdMw�^\eTVMXWZYbTZ�dWZM7�¥]>W M jZTV\gLNYbTV\gRKh�TnIKM�� �dRdk�YbLNM RUTZYb��LaYiTnWV\^��¢�ºA�dWVTnIKM WV½LN]UWVM>f%Y~RKM0� LNM0TnIK]Kk�\gj�kKMXjZ[ WV\gt1M k TV]3[0]>LavdYbWZMaT¤��]�MXjVTZ\eLaYiTZ\^]URdj ]i�7TZIdMa� �dRdkdY>L�MXRUTZYi��LNYbTZWZ\���¢Q¤T�\gj�tdY>jVMXk�]UR�Y LNM Y>jZ��WVM�]>tKTnYi\gRKM k�TZI�WV]U�Kh>I¨jZY>LavK�^\gRKhaTZIdMx�wIK]b�eM uU\gjZ\etK�eMxO#[£jnvdY>[�Mb¢opWZ]iqSMX[�TZ\^u>MWZM [0]>RdjZTZWn�d[0TV\e]>Rx\gjYi�gjV] WVM ul\^M ��MXk�¢�H�IKM�jZ]i�¥T¤�7YbWZMp[0Yb�^�eM k������ �����¦�wId\e[cI#\eLavK�eM LNM RUTnjTZIKM�WZM0ul\^M ��MXkLNM TZIK]Kkdj YbR�k3TZIdMajZ]i�¥T¤�7YbWZM#[ Yi�e�^MXk�� ���������wIK\g[cI�[0]>Lavd�KTZM j�TZIKMakK\ �AMXWVMXRd[�Mat1M0T¤��M0M R�T¤��]�� �dRdk�Y�½LNMXR>TnYi�/LaYiTnWV\g[�MXj�YbWZM�t1]bTnI�Y�u�Yb\^�gYbtd�^M�� WZ]>L L#��IK]>LNMxvdYbhbM��� �������������������! �"���#�$��� �%��&�'&%�(���)����+*%�+)�&� % +* ,���-�- � �� .��-�- � �� .�/0*� .�� � 012,����% �"����kdM0TVMX[�Tnjs� Yb�ejZM�LaYbTZ[cIKMXj0fC[�]ULNv��KTVMXjsTnIKM#� ��RdkdYbLNMXR>TnYi�LaYiTZWZ\^�~Y>Rdk~\eTZj �dRd[0M WZTZYb\eRUT¤�bfCYbR�kTnIKM�vdWZ]iqSMX[�TV\eubM WZM [�]URdjZTZWn�d[�TZ\^]UR8]i�TZIdM�v�]>\eRUTZj�Y>j7��M �^�¯¢

3 ��������������« "���#��������$&�#��'54E���������w� 6N��� ���w���w�#�

798;: <>=@? AB?C;=#D­ [0YbLNMXWZY¨\gj#kKMXjZ[ WV\gt�MXk t>� TZIdM8��\ekdM0�e� �djZM k vd\eRdId]b�eM8L�]KkKM � ¢~H�IKM�[�]l]>WnkK\gRdYiTZM j ]i�wY~O�[ v1]b\gR>TEGFIH JLKNM�K�O0PRQ \gR8Y���]>WZ�gkN[0]l]>WnkK\eR�YiTVM¦jV�KjZTVM L YbR�k \^Tnj�WZM0TZ\eR�Yi��\gLaYih>Ms[�]l]>WnkK\gRdYiTZM j#S FIH TBK�U%PVQ YbWZMWZM0�gYiTZM k¨tU�

W XY T U Z![\]F_^ X``Y J M O Z [baa\ K�wIdM WZM W \ejaY>R�YbWntK\eTZWnYbWZ� jn[0Yb�^M>f YbRdk ^ \ejNY � � � LNYbTZWZ\���f [ Yi�e�^MXk TZIKM�v1M WnjZv1M [0TV\eubM�v�WV]bqSM [�TZ\^]URLaYbTZWZ\���¢ [¦MXRK]bTZ\eRdh�TnIKM IK]>LN]bh>M RKM ]>�dj¦[�]l]>WnkK\gRdYiTZM j¦]i��Y�ubMX[�TV]UWdc FeH JLK�M2Kgfhfhf;PVQ tU�jic f�\ ¢ Mb¢efkic FH JLKNM2KhfhfgfkK Z PVQ fd��MxIdY�u>M W iS Fl^ i E ¢H�IKM�LaYbTZWZ\�� ^ [0Y>R�t1M�kKMX[�]>Lav1]>jZM k¨Ybj^mFonIH prqsPtK

�wIdM WZM n \gj�Y � ��� LaYbTZWZ\���fdLaYbvdvd\eRKhaTnIKM�RK]>WnLaYi�e\^¿ M k�\eLaYih>Mx[�]l]>WnkK\gRdYiTZM j�TV] TnIKM�WVM TV\gRdYi��\gLaYih>M[0]l]>WnkK\eR�YiTVMXj0f�YbRdkvu pjKNq�w \gj�TZIKM¨O#[ kd\ejnvK�gYb[�MXLNM RUT�a WZ]bTnYiTZ\^]UR YbRdk TnWZY>RdjV�gYiTZ\^]UR cs� WZ]>L TZIKM���]UWV�gk[0]l]>WnkK\eR�YiTVM jZ�ljZTVMXL,TV] TnIKM�[0Y>LNM WnY�[0]U]UWZkd\eRdYbTVM jV�KjZTVMXL�¢

���������

Page 10: Determining the Epipolar Geometry and its Uncertainty: A Review · 2018-01-04 · ISSN 0249-6399 apport de recherche INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE

������������� ��������������� ��������� �!�"�#�$���%��& ���(')��+* ,���-.�/���0���� �+&�13254��/67���8 }

H�IKM ¬l�dY>R>TZ\^TZ\^MXjxWVM �eYbTVMXk3TV]�TnIKM jVMX[�]>R�k�[0Y>LNM WnY�\gj�\eRdkd\e[ YiTVMXk�tU��� ¢Nº�]>W�M0�KY>LavK�^M>f/\^��S��7\gj�Yv1]b\gRUT7\eR¨TZIdM r1WZjZT�\gLNYbhbM>f S��� kdM RK]>TVM j7\eTZjs[�]>WnWZM jnv�]URdkK\gRKhNv1]b\gRUT7\eR¨TZIdM�jVMX[�]>R�k�\gLaYihbM>¢­y�^\gRKM��p\gR3TnIKMN\eLaYih>M�vdY>jZjZ\gRKh8TZI�WV]U�Kh>I3v1]b\gRUT S F H TBK�U%P Q \gj¦kKMXjZ[ WV\gt1M k�tU��M ¬l�dYbTV\e]>R�� T�

U�� �tF�� ¢ � M0T�� F H � K K�� PVQ fATnIKM R�TZIKMNM ¬l�dYbTV\e]>R3[ YbR3t1MaWVM �wWV\eTVTZM R3Y>j�� Q iS F�� ]UW�iS Q � F�� ¢»��K�^TZ\evd�^�l\gRKh ��tU��YbRU�¡Rd]>Rl_`¿0MXWV]�jn[0Yb�eY>W8��\e�^�xkKM�r1RKM TZIKM jnYbLNM {#[��e\gRKMb¢,H�Il�dj f¦Y�{#[��^\gRKM \ejWZM v�WVMXjVMXR>TZM k3tU�3Y�IK]UL�]>hbMXRKM0]U�djsO�[,ubM [0TV]UW0¢�H�IKM kK\gjZTZYbR�[�M�� WV]UL"v�]>\eRUTtS�� FeH T � K�U � PRQ TV]��e\eRKM� F H � K K���PRQ \ejwhb\eubM R¨tU�

� u S�� K � w F � T � � U � ���� ��� � � �

�¦]bTZM�TnIdYiT���MxIKM WZMx�djZMxTZIKM$*�������.'¨kK\ejZTZY>Rd[�M>¢

798 7 ��� C�� = ��A"!$#&%L= '(%k?)!)* A@D�+-,�. D�+dA/'0%LD ?�A1�32 AB?4!"C65H�IdMNMXvK\gv�]>�eY>W hbM0]ULNM0TZWZ�~M��K\gjVTnj�t1M0T¤��M0M R YbRU�3T¤��]~[0Y>L�MXWZY¨jZ�ljZTVMXLaj0¢�¼�]>R�jV\gkKM W TnIKM8[0Y>jVMN]i�7T¤��][ YbLNM WnYbjxY>j�jnIK]��wR \eR º\eh�¢��b¢ � M T37!YbRdk�7 � t�M�TZIKM�]>vKTZ\e[ Yi�p[0M RUTVMXWZj#]i��TZIKM�r1WZjZT#Y>Rdk jVMX[�]URdk

E

7��8 8 �

S S���:9<; �=�9

>?�

@

7

>

u p�K�q�w

º\eh>�dWZMa���GH�IKMxM vd\ev1]b�gYbW�hbM ]>LNM0TnWV�

[ YbLNM WnYbj f%WZM jnv�MX[�TZ\^u>M0�e�b¢3¹¦\^u>M R Y3v1]b\gR>T S&\gR TZIKM�r1WZjZT�\gLaYihbM>f%\eTZj�[�]>WnWZM jnv�]URdkK\gRKh~v�]>\eRUT�\gR TZIKMjZM [0]>Rdk#\eLaYbhbM7\ejG[0]>RdjZTZWnYi\gRKMXk�TV]x�e\eM�]UR�Y��e\eRdMw[0Yi�e�eM k�TZIKM � � �d�#���#�b� ��(��]b�2S¡fbkKMXRK]bTZM katU�A�=�9 ¢�H�IKMv(vxw�y"z"{.z.|

Page 11: Determining the Epipolar Geometry and its Uncertainty: A Review · 2018-01-04 · ISSN 0249-6399 apport de recherche INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE

Y � ���/�s�#&������ �s�#���

�e\gRKM �:�9 \gjsTnIKMN\eRUTVMXWZjZM [0TV\e]>R~]b�pTZIdMNvK�gYbRdMA@#f/kdM/r1RKMXk�tU� S¡f 7,YbR�k 7�� a �lRK]��wR3Y>jsTnIKMW�0�� ����������(�����(��c)f ��\eTZI TnIKM�jZM [0]>Rdk \gLaYihbM�vK�eY>RKMA>?�¯¢�H�Id\ej#\ej�t�MX[0Yb��jVM8\gLaYih>M�v�]>\eRUTGS�LaY�� [�]UWZWZM jnv�]URdkTZ]8YbR~Y>WZtK\eTZWnYbWZ�8v1]b\gR>Ts]>R¨TZIKM�jVMXL�\^_`�^\gRKMA7 E a E LNY���t�M#YbTw\gRsr1RK\eT¤�lc7YbRdk�TnIdYiT�TnIKM#vdWZ]iqSMX[�TZ\^]UR�]b�7 E ]UR >?�\gj�TZIKMa�e\eRdM �=�9 ¢�º1��WVTnIKM WnLN]>WZMbfC]>RKMN]>t�jVMXWVu>M jwTnIdYiT Yi�e�M vK\gv1]b�gYbW��^\gRKMXj¦]b�pTZIdMav1]b\gRUTZj�\gRTnIKMr1WZjZT�\eLaYih>M�vdY>jZj�TnIdWV]U�Kh>I�Y�[0]>LaLN]>R3v1]b\gRUT 8 � f/�wIK\g[cI~\ej [ Yi�e�^MXk3TZIdM � � �d�#� ��¢ ¸ vK\gv�]>�^M 8 ��\ejTnIKM�\gRUTVM WnjZM [�TZ\^]UR¨]i�GTZIKM#�^\gRKM37�7�����\^TnI�TZIKM#\eLaYih>M#vK�gYbRKM<> � ¢�H�IK\gjw[ YbR~t1M�M Y>jV\e�^�¨�dRdkKMXWZjZTV]l]Kk~Ybj�¥]>�^�e]��wj ¢�º1]>W MXYb[cI v1]b\gRUT9S � \gR TZIdM!r1WnjVT�\eLaYbhbM >�fl\eTZj�M vK\gv1]b�gYbW �e\gRKM �=�9�� \gR >?�A\gjpTnIKMs\gRUTVM WnjZM [�TZ\^]UR]b��TZIKM�vd�eY>RKM@ � fkdM/r1RKMXk�tU� S � f?7«Y>Rdk�7�� f��\eTZI \gLaYih>M8vK�eY>RKMA>?�¯¢~­w�e��M vK\gv�]>�eY>W�vK�gYbRKMXj @ �TnIl�dj7�¥]>WnL!Y v�MXRd[�\e�%]b�Gvd�eY>RKM jw[0]>RUTZYb\eRd\eRKhaTnIKM��e\eRKM 7�7�� ¢7H�IKM �¨L��djVT�\gRUTVMXWZjZM [0T?>?��YbTwY8[�]ULaL�]URv1]b\gRUT0fA�wIK\g[cI�\ej 8 � ¢�º\eR�Yi�e�^�>f1]URKM#[0Y>R�M Y>jV\e�^�¨jVM M�TZIdM�jZ�lLaLNM0TnWV�¨]i�GTnIKM#MXvK\ev1]b�gYbWwhbM0]ULNM0TZWZ�b¢�H�IKM[0]>WnWVMXjZv1]>R�kK\eRdh¦v1]b\gR>TG\gRNTZIKMurAWZjZT\eLaYih>M7]i�1MXYb[cINv�]>\eRUT#S�� � �^�l\gRKh ]>R �=�9�� L��djVTG�e\eM�]>R�TZIKM�MXvK\ev1]b�gYbW�e\gRKM � 9 ; � f%�wIK\e[cI�\ej TnIKMa\eRUTZM WnjVMX[�TV\e]>R�]i�pTnIKM8jnYbLNMNvd�eY>RKM3@ � ��\eTZI�TZIdMrAWZjZT \eLaYih>M�vK�gYbRdM�>w¢�­w�e�MXvK\gv�]>�eY>W �^\gRKM j��¥]>WnL-Y~v1M Rd[0\^��[�]UR>TnYi\gRK\gRKh�TZIKM8MXvK\gv�]>�^M 8 f��wIK\e[cI \gjxTZIdM8\eRUTZM WnjVMX[�TV\e]>R ]b��TnIKM8�e\eRKM7�7�����\eTZI TZIdM�\gLNYbhbM8vK�gYbRdMA>�¢~H�IKM�jZ�KLNLNM TZWZ���eM Y>kdjxTV]�TZIKM �¥]>�^�e]���\gRKh~]>tdjZM WZu�YiTZ\^]UR�¢�Q`�dS a Yv1]b\gRUTx\eR > c YbRdk S��\a Y~v1]b\gR>T#\eR�>?�Rc [0]>WnWVMXjZv1]>Rdk�TV]�Y~jZ\gRKhb�eM8vdIU�KjV\g[0Yb� v�]>\eRUT E \eR jnvdYb[0Mbf%TnIKM RS¡f2S��¥f 7 YbRdk 7���L#�djZTw�e\^Mx\gR3Y jZ\eRKh>�^M�vd�eY>RKMb¢7H�IK\gjw\gjwTZIdM���M �^�^_¤�URd]��wR -.�0� �(�����(������%&E-.���l*��+� �#��l�\gR�jZ]b�eul\eRdh�LN]bTZ\^]UR8YbRdk�jVTnWZ��[�TZ��WVMs� WV]UL,L�]>TV\e]>R�vdWV]UtK�eM Lajp�wIdM R8TnIKM�\eRUTZWZ\gRdjV\g[�vdYbWnYbLNM0TZM Wnj ]b�%TZIKM[ YbLNM WnYbj7Y>WVM �lRK]��wR a � ]URKh>�KM TS_��s\ehbh>\eRdj��X|�Yd��c)¢H�IKM�[�]ULNv��KTZYbTV\e]>RdYb��jV\eh>Rd\�r1[ YbRd[0M�\eR�LNYbTZ[cIK\gRKh#kd\ �CM WZM RUT ul\eM0�wj \gj TnIdYiTp�¥]>W�Yxv�]>\eRUTp\gR TnIKM3r1WnjVT

\gLaYih>Mbf�\eTZj�[0]>WnWVMXjZv1]>RdkdM Rd[0M#\gR�TnIKM8jVMX[�]URdk3\gLaYihbMaL#��jVTx�e\^Ma]UR�TnIKMaM vK\gv�]>�eY>W��^\gRKM \gR�TnIKM8jVMX[�]URdk\gLaYih>Mbf%Y>Rdk�TnIKM R TnIKM8jVMXYbWn[cI�jnvdYb[0MN�¥]>WxY�[�]UWZWZM jnv�]URdkKMXRd[�M�\ej�WZM k��d[�MXk3� WZ]>L-{¨kK\gLNM RdjZ\e]>Rdj TZ] �kd\eLNM R�jV\e]>R�¢�H�IK\gj¦\gj�[0Yi�e�eM k~TnIKME� � �d�#���#�m-.���d*��%� ���� � ¢�­w�ehbMXtdWZYb\e[ Yi�e�^�>f�\gR3]UWZkKMXW��¥]UW S \gR3TZIdM r1WnjVT\gLaYih>M�Y>Rdk S��C\gR�TZIKMxjZM [�]URdk8\gLaYih>M�TZ]Nt1M�LaYiTn[cIKM k�fKTnIKM��¥]b�e�^]���\gRKhNM ¬l�dYbTV\e]>R�L��djZT7t1MxjZYbTV\gj r1M k��iS Q�� iS � F$� ��\eTZI � Fon�� Q H qhPkp n � ��� K aV�Pc

�wIdM WZM�u pjKNq�w \ej%TZIKMpWZ\ehb\gk TZWnYbRdjV�¥]>WnLaYiTV\e]>Ra WZ]bTnYiTZ\^]URxYbRdk TnWZY>RdjV�gYiTZ\^]UR c1�wIK\g[cIxtdWZ\eRKhUj/v1]b\gR>Tnj/M0�Kv�WVMXjS½jZM k�\gR�TZIdM)*��"-.�#��'�[0Y>LNM WnY [�]l]>WnkK\gRdYiTZM�jV�KjZTVM L TV]8TnIKM rAWZjZT�]>RKM>f1Y>Rdk H qhP \gjwTnIKM�Y>RUTV\gjV�KLaLNM0TZWZ\g[LaYbTZWZ\���kKM�r1RKM k¡tU� q jn�d[cI�TZIdYbT H qsP c F q � c®�¥]UWaYi�e�7O�[«ubM [0TV]UW cG¢�H�Id\ejNM ¬l�dYbTV\e]>R�[0YbR�t�MkdM WZ\^u>M k¨Ybj��¥]b�e�e]��wj0¢ �®\eTZId]>�KTw�e]>jnj7]i�hbMXRKM WnYi�e\eT¤�bfK��M Y>jZjn�dLNMxTZI�YiT�TZIdMx��]UWV�gk�[0]l]>WnkK\eR�YiTVMxjZ�KjVTZM L[0]b\gRd[�\gkKMXj7��\^TnI�TnIKM$*��.-"�#��'�[0Y>L�MXWZYN[�]l]UWZkK\gRdYbTVMxjV�KjZTVMXL�¢ºAWV]UL,TZIKM�vd\eRdId]b�eM�L�]KkKM � f���MxIdY�u>MW iS FlnIH p qhP i E � YbRdk W � iS � Fvn � H ��P i E � �¸ �^\gLN\eRdYbTV\gRKh E � f W Y>Rdk W �C\gR8TZIdM Y>t�]�u>MsT¤��]�MX¬l�dYiTZ\^]URdj0fl��M¦]UtKTZYb\eR�M ¬l�dYbTV\e]>R aZ�Pc)¢�¹¦M ]>LNM0TnWV\g[0Yb�^�e�bf� iS���kdM/r1RKMXj7TZIKM MXvK\ev1]b�gYbW7�e\gRKM��:9 ; ]i�v�]>\eRUTdS��C\gR�TnIKMbr1WnjVT7\gLNYbhbM>¢ ¸ ¬U��YiTV\e]>R aV��c�jZY��Kj7RK] LN]>WZMTnIdYbR�TZIdYbT�TnIKM�[0]>WnWVMXjZv1]>RdkdM Rd[0M \eR�TZIKM r1WnjVT�\eLaYih>M ]i��v1]b\gRUT S��>�e\^MXj%]URxTZIKM7[�]UWZWZM jnv1]>RdkK\gRKh�MXvK\ev1]b�gYbW�e\gRKM � 9 ; ¢�H�WnYbRdjnv1]>jZ\eRKh aV�Pcw�l\^M �ek�j¦TnIKMajV�KLaLNM0TZWZ\g[�WZM0�gYiTZ\^]UR�� WV]UL!TZIKM�r1WnjVT¦\eLaYbhbM#TZ]�TnIKM jVMX[�]URdk\gLaYih>Mb¢

H�IKM � � �aLaYiTZWZ\^� � \gj�[0Yb�^�eM k�TZIdM � ����'��#�$���l�0�#�^�$�#�+��>¢pÀU\gRd[�M������hu H qhP wBF$� f

�����hu � w F$� �a¯{ec

���������

Page 12: Determining the Epipolar Geometry and its Uncertainty: A Review · 2018-01-04 · ISSN 0249-6399 apport de recherche INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE

������������� ��������������� ��������� �!�"�#�$���%��& ���(')��+* ,���-.�/���0���� �+&�13254��/67���8 |

� \ej�]b��WnYbRd�~{K¢]7M jZ\ekKMXj0f�\eT \ej ]URK�e�3kKM/r1RdM k �dv�TZ]¨Y�jn[0Yb�eY>Ww� Y>[�TV]UW0f/t1M [ Yb�djZMN\�� � \gj�L��K�eTV\gvK�^\eM ktU�3Y>R�YbWntK\eTZWnYbWZ�¨jn[0Yb�eY>W0fCMX¬U��YiTV\e]>R aV��cwjZTV\e�^� Id]b�gkdj0¢�H�IKM WZM��¥]UWVM>fCY�� �dRdk�YbLNM RUTZYb� LaYiTZWZ\^��IdYbj¦]>RK�e�jZM0u>M R~kKM h>WZM0MXj�]i�� WZM0M kd]>L¨¢sH�IKMXWVM#Y>WVMx]URK�e�~} \eRdkdM v1M RdkKMXRUT¦vdY>WZY>L�M TVMXWZjwY>LN]>RKh TnIKM�| M �^MXLNM RUTZj]b��TnIKM � �dRdkdY>LNM RUTZYb�%LaYiTZWZ\^�C¢

� ��� 6P���l�%���� ���#���71 � M��djZM�TnIKM�jZM [�]URdk�[ YbLNM WnYN[�]l]>WnkK\gRdYiTZM�jZ�KjVTZM L Y>j�TnIKMx��]>WZ�ek�[�]l]>WnkK\gRdYiTZMjZ�KjVTZM L¨fl�wIK\g[cI8\gj�\gR�Y>[0[0]>WnkdYbRd[0Mw��\eTZI�TZIKM [�]UR>u>M RUTV\e]>R��djVMXk8\gRWa º1Yb�dhbM WnYbj¦� |>|bOsc)¢ÀlM0ubMXWZYb�AWZM jZM Y>WS½[cIdM Wnj�v�WVM0�¥M W7TV] �djZMxTZIdM�rAWZjZTw[0Y>L�MXWZYN[�]l]UWZkK\gRdYbTVM jV�KjZTVM L¨fKTnIKM R aV��c�t�MX[�]UL�MXj iS�� Q � �giS F$� ��\eTZI� � F H q � P p � f/�wIKM WZM u p � K�q � w TnWZY>RdjV�¥]>WnLNj�v1]b\gR>Tnj�� WV]UL"TZIdMrAWZjZT�[ YbLNM WnY¨[�]l]>WnkK\gRdYiTZM�jZ�ljZTVMXL"TV]TnIKM�jZM [�]URdk�¢ H�IKM�WZM0�gYiTZ\^]UR�t1M0T¤��M0MXR u pjKNq�w Y>Rdk u p � K�q � w \ej�h>\^u>M R¨tU� p � Flp Q f�Y>Rdk q � F��dp QLq ¢H�IdM WZM Y>kKM W7[ YbR¨M Y>jV\e�^� u>M WZ\��¥��TZIdYbT � F � � Q ¢798�� � # %LD<% !+A/��,d=?! ' =�� ��� C�� = �;A/! �� . AB?C;= D��=?!��oD *� !=� �%���?C�=#D-2 =��

+�% �QSRaTnIK\gj�jZM [0TV\e]>Ra��M���\^�e�AkdM WZ\^u>MwYxhbMXRKM WnYi�K�¥]UWZLy]i��MXvK\ev1]b�gYbW M ¬l�dYiTZ\^]URa�wIK\g[cI kK]lM jpRd]bTpY>jZjn�dLNMsYbRU�v�YbWZTV\g[0�K�gYbW7vdWZ]iqSMX[�TV\e]>R�LN]lkdM0� a��¦��YbRdk �CI�YbRKh¨� |>|b��cc¢­ v1]b\gR>T S \gR3TnIKM`rAWZjZT¦\gLaYih>M#\gj�LaYbTZ[cIKMXk3TV]�Y�v1]b\gRUTtS���\gR3TnIKMajVMX[�]>R�k3\gLNYbhbM>¢�ºAWV]UL«TZIKM

[ YbLNM WnY�v�WV]bqSM [�TZ\^]UR�LN]KkKM0� a ]UWVTnIK]bhUWZY>vdIK\g[if��M Yb� v1M Wnjnv�MX[�TV\eubM>fpY�t8RKM>f ]>W#� �K�e�wv�MXWZjnv1M [�TZ\^u>MPccf ��MI�Y�ubM W iS F_^ i E � YbRdk W �giS�� Fv^ � i E � K �wIKM WZM ^ YbRdk ^ �AY>WVM � ���NLNYbTZWZ\e[0M j ¢�­¦R�\gLaYih>M¦v1]b\gR>TS���kKM�r1RKM jaY>[�TZ��Yi�e�^� YbR�]>vKTZ\e[ Yi�7WZY��>f�]UR �wIK\g[cI�M ubM WZ� jZvdY>[�M¨v�]>\eRUT i E �7vdWZ]iqSMX[�TZj#]>R�TZIKM�jVMX[�]URdk\gLaYih>M YbT iS � ¢ H�IK\gj7]>vdTV\g[0Yi�/WnY���[ YbR¨t�Mx�wWZ\^TZTVMXR�\gR�vdYbWnYbLNM0TnWV\g[s�¥]>WnL Ybji E � F W � ^ ��� iS � ��� ��� K a O�c

�wIdM WZM ^ ����\ej7TnIKM�vdjZM �dkd]i_`\eRUubMXWZjZM�]i��LaYbTZWZ\�� ^ � �^ ��� F_^ � Q u ^ � ^ � Q w ��� K a�Ksc

Y>Rdk � ����\ejsYbRU�)Ki_`ubMX[�TV]UW�TZIdYbT�\ej�v1M Wnv�MXRdkK\g[0�K�gYbW7TZ]8Yi�e��TZIKM)�"��8§u>M [0TV]>Wnj�]i� ^ � f�\ ¢ Mb¢^ � � ��� F � �H�Il�dj f � ����\ej YxRl�K�e�du>M [�TZ]>W]b� ^ �¯¢­sj YxLaYiTZTVMXWG]i�A� Yb[0T0f � ����\eR�kK\e[ YiTZM jTZIdMsv�]UjV\eTV\e]>RN]b�ATnIKM�]>vKTZ\e[ Yi�[0M RUTVMXW�a TV]��wId\e[cI�Yb�^��]>vKTZ\e[ Yi�7WZY��Kj�[0]>RUubMXWVh>MPcc¢���M¨jZIK]�� �gYiTZM W#Id]�� TZ]�kKM TVMXWZLN\gRKM � ����¢�º1]>W#Yv�YbWZTV\g[0�K�gYbW�u�Yi�g�KM W � f�M ¬l�dYbTV\e]>R a Osc�[�]>WnWZM jnv�]URdkdj�TV]~Y~v1]b\gR>Tx]UR�TnIKM ]>vdTV\g[0Yi��WZY��3kKM�r1RKMXk tU� S�� ¢¸ ¬U��YiTV\e]>R a O�cw\gj¦MXYbjZ\^�e�NqV�djZTV\ r�M k tU��v�WV]bqSM [�TZ\eRdh E �%]URUTV]�TnIKM jVMX[�]URdk~\gLNYbhbM>f1�wIK\g[cI3\gRdkKM M k�hb\eubM jS�� ¢

ÀU\gLN\^�gYbWZ�e�bfdYbR�\gLNYbhbM v�]>\eRUT S \gR�TZIKMbr1WnjVT7\gLaYihbM kKM�r1RKM j�Yb�ejZ]aYbR�]>vKTZ\e[ Yi��WZY��>¢�rsM ¬l�K\gWV\gRKhaTZIKMT¤��]aWZY��Kj�TV]a\gR>TZM WnjVMX[�T7\gR¨jnvdY>[�M \eLavK�e\eM j7TZI�YiTwYav�]>\eRUT E �/[�]UWZWZM jnv1]>RdkK\gRKh�TV] YavdYbWZTV\g[0�K�gYbW W ��\gR a O�cL��djZT7v�WV]bqSM [�T7]UR>TZ]aTZIKM r1WnjZT�\gLaYih>M YbT S¡fdTZIdYbT�\ejW iS F W � ^ ^ ��� iS � � ^�� ��� �

v(vxw�y"z"{.z.|

Page 13: Determining the Epipolar Geometry and its Uncertainty: A Review · 2018-01-04 · ISSN 0249-6399 apport de recherche INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE

�7L � ���/�s�#&������ �s�#���

oMXWS�¥]UWZLN\gRKhNYa[ WV]UjZj7vdWZ]Kkd�d[�T���\eTZI ^�� �����l\eM0�gkdjW u ^�� � � w �>iS F W � u ^�� � � w � u ^ ^ ��� iS � w �¸ �^\gLN\eRdYbTV\gRKh W YbRdk W � tU�NL#�K�eTV\gvK�e�U\gRKh iS Q � WZ]>L£TZIKMs�^M0�¥T\a MX¬l�K\^u�Yb�^MXR>TGTZ]#YxkK]>T vdWV]Kkd��[�T.ccf>��MwIdY�u>MiS Q � iS � F �GK a¯�ec

�wIdM WZM � \gj�Y�� � �aLaYiTZWZ\^�Cfd[ Yi�e�^MXk � ����'��#�$���l�0�#� �$�#�+����

� F H ^�� � � P ^ ^ ��� � a ��c

Q¤T�[0YbRxYb�ejZ]wt1MpjZId]��wR¦TnIdYiT�TZIK\gj/M0�dvdWVMXjZjZ\e]>R¦\ej%M ¬l�K\eu�Yb�^MXRUT/TV] aZ�Pc1�¥]UW�TZIdM � �K�e�>v1M WnjZv1M [0TV\eubMpv�WV]bqSM [�TZ\^]URa¯jVM M ����Y>Rdk �CI�YbRKh � |>|b��ccfAt��KT \^T�\ejxLN]UWVM�hbM RdM WnYi�¯¢�QSR�kKM0MXk�f aV��csY>jZjn�dLNM j¦TZI�YiT�TZIKM�r1WnjVT � � �jn�dtl_¤LaYiTnWV\^��]b� ^ ��\ejw\eRUubMXWVTZ\etd�^M>f�YbRdk�TnIl�djw\gj�]>Rd�^�¨u�Yi�e\ek��¥]>W�� �K�^�v1M WnjZv1M [0TV\eubM�vdWZ]iqSMX[�TV\e]>R�td�dTsRK]bT�¥]UW7Y�t8RKM [0YbLNMXWZY>j\a¯jVM MxÀUMX[�T ¢n�K¢ O�ccfK�wIK\e�^M�a �scpLaY>�iM j��djZM ]i�%TnIKMxvdjVMX�dkK]>\eRUubMXWZjZMs]b�%TZIdM v�WV]bqSM [�TZ\^]URLaYbTZWZ\���f��wIK\g[cIx\ej�u�Yi�e\ekx�¥]>W�t�]>TZI � �K�e�Kv�MXWZjnv1M [�TZ\^u>M�vdWZ]iqSMX[�TZ\^]URxYbj/��M0�e�lYbj�Y�t8RKM�[ YbLNM WnYbj ¢/H�IKMXWVM0�¥]>WZMTnIKM�M ¬l�dYiTZ\^]UR kK]lM j�RK]bT7kKMXv�MXRdk8]UR8YbRU�ajZv1M [0\�r1[��lRK]����eM kKh>Mw]i�%vdWZ]iqSMX[�TZ\^]UR8LN]lkdM0�¯¢�rsM vK�gYb[0\eRKh�TZIKMv�WV]bqSM [�TZ\^]UR�LNYbTZWZ\��~\gR~TZIdM#M ¬l�dYbTV\e]>R~tU��jZv1M [0\�r1[�vdWZ]iqSM [0TV\e]>R~LaYiTnWV\^�¨�¥]>WwMXYb[cI~jnv�MX[�\ r1[#v�WV]bqSM [�TZ\^]URLN]KkKM �^a M>¢ h1¢]>WZTZIK]>h>WnYbvdIK\g[ifb��MXYb�Nv�MXWZjnv�MX[�TZ\^u>Mbf>Y#t8RKMs]UW� �K�e�Cv1M Wnjnv�MX[�TV\eubM�cvdWZ]Kkd�d[�MXj TnIKMsMXvK\ev1]b�gYbWMX¬l�dYiTZ\^]UR8�¥]UW7TZIdYbTwjZv1M [0\�r1[ vdWZ]iqSMX[�TZ\^]UR�LN]KkKM0�¯¢�ÀUM M�a��¦�~Y>Rdk ��IdYbRKh��X|b|b�sc�¥]UW�LN]>WZM�kKM TZYb\^�gj0¢

H�IKM u>M [�TZ]>W � � ��jVTZ\^�e�/RKM0MXkdj�TV]at�M kKM TVM WnLN\eRdM k�¢ � Mbr1WZjZT7RK]bTZM¦TnIdYiT7jZ��[cI�Y�ubM [0TV]UW�L��djVT�M��K\gjVTt1M [ Yb�djZM8TnIKM�kK\ �CM WZM Rd[0M�t�M T¤��M M R TnIKM�WZ]�� kd\eLNM R�jV\e]>R Y>Rdk TZIdM�[�]>�e�dLaR�kd\eLNM R�jV\e]>R \gj�]URKMbfGYbR�kTnIdYiT�TnIKM�WZ]��§ubMX[�TZ]>Wnj�Y>WVM h>M RKMXWZYb�^�e� \eR�kKM v1M RdkdM RUT7� WV]UL MXYb[cI�]bTZIdM W ¢GQSRdkKM0MXk�fd]URKM ��Y��8TZ]a]>tKTnYi\gR� ����\ej

� � � F u � ^ ��� ^ � w � K a`}�c

�wIdM WZM � \gjwYbR~Y>WZtK\eTZWnYbWZ��Kb_¯u>M [�TZ]>W ¢pH�]�jZId]��ªTnIdYiT � ����\ej¦v�MXWZv1M R�kK\e[ �K�gYbW�TZ]8M Y>[cI¨WZ]��°]i� ^ � f1��ML��K�eTV\gvK�^� � ��� tU� ^ �K� WZ]>L TnIKMw�eM��¥T � ^ � � ��� F u ^ � � ^ � ^ � Q u ^ � ^ � Q#w ��� ^ � w � F � K �wIK\g[cIN\ejp\eR�kKM0MXkYa¿ M WZ]aubM [0TV]UW0¢pH�IdM#Yb[0TV\e]>R¨]i� � ^ � � ^ �/\gj�TV] TnWZY>RdjS�¥]UWZL YbR�YbWntK\eTZWnYbWZ� ubMX[�TV]UW7TV]�Yau>M [0TV]>W7TnIdYiT\gj�v1M Wnv�MXRdkK\g[0�K�gYbW�TV]~M0u>M WZ��WZ]��£ubMX[�TZ]>W ]i� ^ �¯¢�Q`� ^ �G\gj�]b�7WnYbRd��O a �wIK\g[cI \ej�TZIKM8[ YbjZMa�¥]>W�t1]bTnIv1M Wnjnv�MX[�TV\eubMxY>Rdk�Y#t8RKM�[0Y>L�MXWZY>j.ccflTZIKMXR � ��� \ej��dRd\e¬l�KM��dv¨TZ]aY jn[0Yb�^M¦� Yb[0TV]UW0¢

798�� � !=�� ��% ' �9?�AB?)% '(% D#?H�IdM#vdWZ]>tK�eM L [�]>R�jV\gkKM WZM k�\gR�TZIdM#jVMX¬l�KM0��\ejsTZIKM�M jZTV\gLNYbTV\e]>R¨]i� � � WZ]>L Y8jZ��t [0\^MXRUTV�e���gYbWZhbM�jZM0Tw]b�v1]b\gRUT¦[0]>WnWVMXjZv1]>RdkdM Rd[0M j ����u S�� K S��� w��9F Z K � � � K� �� f1�wIKMXWVM ���� ¢¦H�IdM�v1]b\gR>T�[�]UWZWZM jnv�]URdkKMXRd[�MXjt1M0T¤��M0MXR T¤��]�\eLaYih>M j�[ YbR t1M�MXjVTnYbtK�e\ejnIKMXk tU� Y~TVMX[cIdRK\g¬l�KM�jn�d[cI Ybj�TZIdYbT�kKMXjZ[ WV\gt�MXk \eRka��CIdY>RKh�f[�M WZ\g[cIKMbfUº1Y>�KhbMXWZY>jpYbR�k � �K]URKh��X|b|U�ec)¢���M Yi�e�e]��xfUIK]���M0u>M W fiTZI�YiT7Yx� WnYb[�TZ\^]UR ]i�/TnIKM LNYbTZ[cIKMXj�LaY��t1Mx\gRd[�]UWZWZM [0TV�e�8vdYi\gWVMXk�fdYbR�k�TnIU��j�TZIKMxMXjVTZ\eLaYbTV\e]>R�TVM [cI�RK\e¬l�KMXj�jZId]>�K�gk�t�M�WZ]>td��jVT ¢

���������

Page 14: Determining the Epipolar Geometry and its Uncertainty: A Review · 2018-01-04 · ISSN 0249-6399 apport de recherche INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE

������������� ��������������� ��������� �!�"�#�$���%��& ���(')��+* ,���-.�/���0���� �+&�13254��/67���8 �>�

� � ��.G���¨ � ����* � ��� ��*��� � �x��0��������� � ����'¨�#� �w�#������� � �����

� M0T�Y~v1]b\gRUT�S�� F H T � K�U � PVQ \eR TnIKM)r1WnjVTx\gLaYih>M8t�M8LaYiTn[cIKM k TZ]3Y~v�]>\eRUT�S��� F H T �� KNU �� PRQ \gR TZIKMjZM [0]>Rdka\gLaYih>Mb¢H�IKM0� L#��jVT�jZYbTV\gjS�¥�NTZIKM�M vd\ev1]b�gYbWpM ¬l�dYiTZ\^]UR aV��c)fU\ ¢ Mb¢GiS Q� � iS��� F � ¢GH�IK\gj�M ¬l�dYiTZ\^]UR[ YbRNt�M7�wWZ\^TZTVMXRNY>jY¦�e\gRKM Y>WGYbRdkNIK]>LN]>hbM RdM0]>��j%MX¬U��YiTV\e]>R�\gR#TnIKMw| �dRd�lRK]��wRN[�]lM/t8[�\eM RUTnj�]b�CLaYbTZWZ\��� �

� Q��� F$� K a+Y�c

�wIdM WZM

� � F H T � T �� K�T � U �� K�T � K�U � T �� K�U � U �� KNU � KNT �� K�U �� K Z P Q� F H � � � K� � � K� �� K� � � K�� ��� K�� � K�� � K�� � K�� P Q K

�wIdM WZM � �� \gj�TZIdMxM0�eM LNM RUT7]i� � YbT�WV]�� � YbRdk¨[�]>�e��LNR��1¢Q`���M�Y>WVM�hb\eubMXR v1]b\gR>TwLaYbTZ[cIKMXj0fdtU��jZTZY>[c�>\gRKh a+Ysc)fK��M�IdY�ubM�TZIdM �¥]b�e�^]���\gRKha�^\gRKMXYbW�jZ�ljZTVMXL,TV]

jZ]b�eubM �

��� � F �jK�wIdM WZM

��� FIH � � KhfhfgfkK � � P Q �H�Id\ej�jVM T¦]b� �^\gRKMXYbW�IK]>LN]bh>M RKM ]>�djsM ¬l�dYbTV\e]>Rdj f1TZ]bh>M0TZIdM W¦��\^TnI3TZIdM�WnYbRd��[�]URdjZTZWnYi\gR>Ts]i�pTnIKMNLaYbTZWZ\��� f1Yi�e�^]��§��j�TV]aM jZTV\gLaYiTZM�TnIKMxM vd\ev1]b�gYbW7h>M0]>LNM TZWZ�b¢

��8;: � 5�A ��? �9= � .�?C�=#D�� C$?���� � =#C�D ? 2 AB? ����%��­¦j�kdM jn[0WZ\et1M k�\eR�ÀlM [�T ¢n{d¢ {df Y~� �dRdk�YbLNM RUTZYb��LNYbTZWZ\�� � IdY>j#]URK�e��}3kKM0hUWVM M j�]i��� WZM0MXkK]>L¨¢ H�IU��j0f}�\gj TZIKM8LN\gRK\gL#�dL RU��L#t1M W ]i�7v�]>\eRUT�LaYiTn[cIKM j WZM ¬l�K\gWVMXk3�¥]UWxIdY�ul\eRdh¨Y�jZ]b�g�KTV\e]>R�]i��TZIKM MXvK\ev1]b�gYbWh>M0]UL�M TZWZ�b¢QSR�TZIK\gj7[0YbjZMbf �F � YbRdk�WZY>Rd��u���� w F � ¢GH�IdWZ]>�KhUI8jV\gRKhU�K�eY>Wpu�Yb�e�KM�kKM [0]>Lav1]>jZ\^TZ\^]UR�fl��M¦]>tKTnYi\gR

u>M [0TV]>Wnj � � YbR�k � � �wId\e[cI�jZvdY>R TZIdM�Rl�K�e��jZvdY>[�M¦]i�����i¢GH�IKM Rl�K�e�CjnvdY>[�M�\ej�Y#�e\gRKM Y>W�[0]>L#td\eRdYbTV\e]>R ]b�� � YbR�k � � f>�wIK\g[cI#[�]UWZWZM jnv1]>Rdk#TV]xLaYiTZWZ\g[�M j � � YbR�k � � fbWVMXjZv1M [0TV\eubM �^�>¢^]7M [ Yb�djZM�]b�1\eTZjGIK]ULN]bhbMXRKM0\eT¤�bfTnIKM�� ��RdkdYbLNMXR>TnYi��LNYbTZWZ\���\gjsY ]URKM�_¤vdYbWnYbLNM TVM W�� Y>LN\^�e��]i� LaYiTZWZ\g[�M j! � � � u Z � w � � ¢�ÀU\gRd[�M�TZIKMkdM0TVMXWZLN\gRdYbRUT7]i� � L��djZTwt�M�Rl�K�e� f�\ ¢ Mb¢

����� H � � � u Z � w � � P2F ��K��M�]UtKTZYb\eR3Y�[0�dtK\g[#v1]b�e�KRK]>LN\gYi��\eR" �¢ H�IKM�LaY��K\gL#�dL!Rl�dL#t1M W¦]i�pWZM Yb�%jZ]b�g�KTV\e]>Rdj¦\gjsOd¢¦º�]UWwM Y>[cIjZ]b�g�KTZ\^]UR� �fdTZIdM � �dRdkdY>L�MXRUTZYi�/LaYbTZWZ\���\gj�TnIKM R¨hb\eubMXR�tU�

� F � � � u Z � w � � �v(vxw�y"z"{.z.|

Page 15: Determining the Epipolar Geometry and its Uncertainty: A Review · 2018-01-04 · ISSN 0249-6399 apport de recherche INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE

��{ � ���/�s�#&������ �s�#���

­¦[�Tn�dYi�e�^�>fTZIK\gj�TZM [cIdRK\g¬l�KM�IdY>j#Yb�eWZM Y>kK��t1M0MXR �djVMXk \eR�M jZTV\gLNYbTV\gRKh3TnIKM8M jnjZM RUTV\gYi�pLaYbTZWZ\�� �wIKM R�}v1]b\gRUT LaYiTZ[cIdM j\gRNRd]>WnLNYb�^\e¿0MXk#[0]U]UWZkd\eRdYbTVMXjGYbWZM�Y�u�Yi\e�gYbtK�eM a �¦��YbRKhxYbR�k��¦M0TnWZY�u�Yi�e\�� |>|#K�c)¢/Q¤TG\gj Yi�gjZ]LNMXR>TZ\^]URKM k�\gR a¯H%]UWZW �X|b|U�ecG�¥]>W7M jZTV\gLaYiTZ\eRKhaTnIKM � �dRdkdY>LNM RUTZYb�/LNYbTZWZ\���¢­¦j�Y LaYiTVTZM W ]b�¦� Y>[�T f�TnIKM�WZM jn�K�eT TZIdYbT TZIKMXWVM3LaY���IdY�u>M�TZIdWZM0M3jZ]b�g�KTV\e]>R�j hb\eubMXR©} LaYiTn[cIKM j

I�Ybj�t1M0MXR �URd]��wR jZ\gRd[�M~�7Y�L�L�� j$a �sMXjZjZM~�ZY>�bOdf�ÀlTZ�dWnL �7Yb�>|�c)¢�ÀUTZ�dWnL¨� j�Yi�ehb]UWV\eTZIdL a`ÀUTn�dWZL �7Yb�b|sc[0]>Lavd�KTZM jpTZIKM�M vK\gv1]b�eM jpY>Rdk TnIKMsMXvK\ev1]b�gYbWpTZWnYbRdjV�¥]>WnLaYiTZ\^]UR a¯jVM M ÀUMX[�T ¢n{K¢ {ec� WV]UL }�v�]>\eRUT�LNYbTZ[cIKMXj0¢Q¤T�\ejNtdY>jVMXk ]>R TnIKM�]UtdjZM WZu�YbTV\e]>R TnIdYiT�TZIKM¨M vK\gv1]b�gYbW��e\gRKM j�\eR�TZIKM�T¤��]�\eLaYbhbM j�YbWZM�WZM0�gYiTVMXk tU� YId]>LN]bhUWZY>vdIU�bf>Y>Rdk8TnIl�dj�TZIKMx[0WZ]>jnjV_`WnYiTZ\^]UjG]i�/�¥]U�dW�M vK\gv1]b�gYbW��^\gRKM j7\gj�\eRUu�YbWZ\gYbRUT0¢�QSR8MXYb[cI8\gLaYih>MbflTZIKM}¨v1]b\gRUTZj�kKM/r1RdM�}¨�e\eRdM j�h>]b\gRKh�TZIdWZ]>�dh>I TZIdM��dRd�lRK]��wR MXvK\ev1]b�eMbfTnIU��j#vdWZ]�ul\ekK\gRKh K3\gRdkKMXv�MXRdkKM RUT[ WV]UjZjV_¤WZYbTV\e]>j ¢�ÀU\gRd[�M¦TZIKMXjVM [0WZ]>jnjS_¤WnYiTV\e]>jGjnIK]U�K�ek�WVMXLaYi\gR TZIKM jnYbLNMs\gR8TnIKM¦T¤��]�\gLaYihbMXj0fU]>RdM¦]UtKTZYb\eRdjK�[ �dtK\g[wv1]b�e�lRd]>LN\eYb��MX¬U��YiTV\e]>R�jG\eRaTnIKMs[0]U]UWZkd\eRdYbTVMXjG]i��TZIdMwM vK\gv1]b�eM j!a+K�\gRdkKM v1M R�kKM RUTpvdY>WZY>LNM0TVMXWZj�c)¢Q¤T�\gj�jnIK]��wR�TnIdYiT�TnIKM WZMxLaY��8M��K\gjVT��dv¨TV] TnIdWZM0MxjV]>�e�dTV\e]>Rdj��¥]UW7TZIKMxMXvK\ev1]b�eM j ¢

��8 7 �oDdA1�6*�?C � 2 %k?��d= + � C$?���� =?!&2 =?! % � =#CND ? 2 A ? � �<%��QSR�vdWZY>[�TZ\e[0MbfA��MNYbWZM�hb\eubMXR~L�]UWVM#TZIdY>R }aLaYiTn[cIKM j ¢sQ`����M�\eh>Rd]>WZM�TZIKMaWnYbRd�>_S{a[�]>R�jVTnWZYb\eRUT0fA��M#[0Y>R��jVMxYN�^MXYbjZTS_¤jZ¬l�dY>WVMXj�LNM TZIK]Kk¨TV]8jZ]b�eubM

������ �u�iS Q� � iS �� w � K a |�c

�wId\e[cI¨[0Y>R�t1M�WVM �wWV\eTVTZM R�Ybj �

����� � ��� � � � � aZ�ZL�c

H�IdMNu>M [0TV]>W � \ej ]URK�^��kKM/r1RdM k �dv TV]~YbR �dR��URd]��wR�jn[0Yb�^MN� Yb[0TV]UW0¢ H�IKMaTZWZ\euU\gYi� jZ]b�g�KTZ\^]UR � TZ]~TZIKMY>t�]�u>M#vdWZ]>tK�eM L \gj � F � fC�wIK\g[cI~\gj¦Rd]bTs�wI�YiTs��M#�7YbRUT ¢¦H�]�Y�ub]>\ek�\eT0fC��M�RKM M k~TZ]�\gLav�]UjVMajV]ULNM[0]>RdjZTZWnYi\gRUT7]>R¨TZIKM�[�]lM/t8[�\eM RUTnj�]i�GTZIdMx� �dRdkdY>LNM RUTZYb�%LaYiTZWZ\^�C¢wÀUM ubM WnYi�/LNM TZIK]KkdjwY>WVM�v1]>jnjZ\etK�eM�YbR�kY>WVM�vdWVMXjVMXRUTVM k~t1M0�e]��x¢ � MN��\^�e��[ Yi�e�%TnIKM L �+�s����� ���#��l����� �e�#���������f/Yi�eTZIK]U�Kh>I3LN]UWVM�TnIdYbR Y�v1]b\gR>TLaYbTZ[cIKMXj�[0YbR¨t1M��djVMXk�¢

M�� @���: � > = ² �d� � ² �d�0� ´ H^Q �%��� ² � N ² �PO = >RQ � ²H�IdM`r1WnjZT¦LNM0TnIK]Kk�jZM0TZj�]URKM#]b�pTZIKMa[�]lM�t [0\^MXRUTZj¦]i� � TZ]��>f�YbRdk~TnIKM R jZ]b�eubM jsTZIKMaYbt1]�ubMNvdWZ]>tK�eM L��jV\gRKh~�e\eRKMXYbWx�eM Y>jVTV_`jn¬l�dYbWZM j�TZM [cIdRK\g¬l�KM j ¢ �®\eTZId]>�KT��^]UjZj ]b�7h>M RKMXWZYb�^\eT¤�bf%��M�Y>jZjn�dLNM TZI�YiTxTZIdM8�eY>jVTM �^MXLNM RUT�]i�GubMX[�TZ]>W � a \ ¢ Mb¢���� F � c�\gjsRd]bT�M ¬l�dYb�/TV]�¿0M WZ]�f1YbRdk¨TZIl�djs��M�[ YbR�jZM0T���� F �

Z¢�H�Id\ej

h>\^u>M j

� ��� � � � F � � �� � � ��� � � � F � � Q � � Q� � �� � � ����� Q� � �� � � ��� Q� � � K

���������

Page 16: Determining the Epipolar Geometry and its Uncertainty: A Review · 2018-01-04 · ISSN 0249-6399 apport de recherche INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE

������������� ��������������� ��������� �!�"�#�$���%��& ���(')��+* ,���-.�/���0���� �+&�13254��/67���8 �XO

�wIdM WZM � �� \gj¦TnIKM ���¨LNYbTZWZ\���[0]>Lav1]>jZM k~]b�pTZIKM�r1WnjZTbY¨[�]>�e�dLaRdj�]b� � � f�YbRdk � �N\ej�TnIKMNRd\eRUTZI[0]b�g�dLaR�]b� � � ¢ H�IKM�jZ]b�g�KTV\e]>R�\gj7]>tKTnYi\gRKM k�tU��WZM ¬l�K\gWV\gRKhNTZIKM r1WnjZT7kdM WZ\^u�YiTZ\^u>M TV]8t1Mx¿0M WZ]�fl\¯¢ M>¢

� � � � � � �� � � F � �]7��kdM/r1RK\eTV\e]>R¨]b��ubMX[�TZ]>W�kKMXWV\eu�YiTV\eubMXj0f � u�� Q c w�� � c F ��fK�¥]UW�Yi�e��ubMX[�TV]UW���¢ � MxTZIl�dj�IdY�u>M

� � � Q� � �� � � ��� � � Q� � � F � K ]UW � � F� � � Q� � ��� � � � � Q� � � �H�IdM�vdWZ]>td�^MXLy��\eTZI�TZId\ej7LNM0TnIK]Kk8\gj�TnIdYiT7��M¦kK]aRK]>T��lRK]���YNvdWZ\^]UWV\1�wIK\e[cI�[�]lM/t8[0\^MXR>T�\gj�Rd]bT�¿0M WZ]�¢Q`����MajZM0TxYbR�M0�eM LNM RUT TV] �a�wIK\e[cI�\ejxY>[�Tn�dYi�e�^�~¿ M WZ]�]UW�L��d[cI jZLaYi�e�eM W¦TZIdY>R�TnIKMa]bTZIdM W�M0�eM LNM RUTnj0fTnIKM WVMXjZ�K�eTp��\e�e�At1M�[ YiTnYbjZTZWZ]>vdIK\g[i¢­§WZM LNMXkK�a\ej�TV]NTZWZ� Yi�e�ARK\gRKM v�]UjZjZ\gtK\^�e\eTV\eM j�t>� jZM0TZTV\gRKh#]URKMs]b�%TZIKMRd\eRKM�[0]lM/t8[�\eM RUTZj7]b� � TV]���YbR�k�WZM0TnYi\gR�TnIKM�t�MXjVT�MXjVTZ\eLaYbTV\e]>R�¢

M�� @�� @ A > T ² = � = � � ³ �7>¥�H�IdM8jVMX[�]>R�k�LNM0TnIK]Kk [�]>R�jV\gjVTnj \eR \gLav�]UjV\gRKh~Y�[�]>R�jVTnWZYb\eRUT ]>R�TZIKM�RK]>WnL"]i� � fYbR�k�\gR vdYbWZTV\g[0�K�gYbW��M¦[ YbR�jVM T � � � F Z

�¼�]ULNv�YbWZM k TZ]#TZIdM�vdWZM0ul\e]>�dj�LNM0TnIK]Kk�fdRK]�[0]lM/t8[�\eM RUT�]i� � v�WVM u�Yb\^�gj�]�ubM WpTZIKM

]>TZIKMXWZj ¢QSR¨TZIK\gj�[0Y>jVM>fKTZIKM�vdWZ]>td�^MXL aV�7L�c�t�MX[�]UL�MXj�Y [��gYbjnjV\g[0Yb�C]>RKM �

������ � � � � � � jZ��tlqSM [0T�TZ] � � � F Z�

aZ�b�Pc

Q¤T¦[0YbR~t1M#TnWZY>RdjV�¥]>WnL�MXk¨\eRUTV]�YbR~�dRd[0]>RdjZTZWnYi\gRKMXk¨LN\gRK\gL�\e¿ YbTV\e]>R~vdWZ]>td�^MXL TnIdWV]U�Kh>I � Yih>WnYbRdhbMxL#�K�^½TZ\evd�^\eM Wnj �

��� �� � u � K�� w#K aZ�X{ec

�wIdM WZM

� u � K�� w@F � � � � � � ��� u Z � � � � � w aZ� O�c

Y>Rdk � \ej�TZIdM � Yih>WnYbRdhbMwL#�d�^TZ\evK�e\eM W ¢�]7� WZM ¬l�K\gWV\gRKhxTZIdM\r1WnjZT�kKMXWV\eu�YiTV\eubMs]i� � u � K�� w ��\eTZI�WVMXjZv1M [0TpTV]� TV] t�Mx¿ M WZ]�fK��MxIdY�ubM

� Q� � � � F�� � �H�Il�dj fTZIKM¨jV]>�e�dTV\e]>R � L��djZT�t1M�Y3�dRd\^T�M0\ehbM RUu>M [�TZ]>W ]b�wTZIKM�� ���3LaYiTnWV\^� � Q� � � Y>Rdk � \ej�TZIKM[0]>WnWVMXjZv1]>R�kK\eRdh M0\ehbM RUu�Yi�g�KM>¢ ÀU\gRd[�MaLaYiTnWV\^� � Q� � �3\ej�jV�KLaLNM0TnWV\g[#Y>Rdk3v1]>jZ\^TZ\^u>M�jZM LN\�_¤kKM�r1RK\eTVMbf�Yb�^�\eTZj7M \^h>M RUu�Yi�g�KM j7Y>WVM WVMXYi��YbR�k�v1]>jZ\^TZ\^u>M ]>W�¿0MXWV]1¢ �®\eTZId]>�KT��e]>jnj�]i��h>M RKMXWZYb�^\eT¤�bfl��MxYbjnjZ�dLNM�TZIKM�Rd\eRKMM \^h>M RUu�Yi�g�KM j�]b� � Q� ����YbWZM¦\gR�RK]>Rl_`\gRd[0WZM Y>jV\gRKh#]UWZkdM W �

� � �_fgfhf ��� � �>fhfgf ��� � �&� �

v(vxw�y"z"{.z.|

Page 17: Determining the Epipolar Geometry and its Uncertainty: A Review · 2018-01-04 · ISSN 0249-6399 apport de recherche INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE

�ZK � ���/�s�#&������ �s�#���

� M¨TZIKMXWVM0�¥]>WZM�IdY�u>M�|�v1]bTZM RUTV\gYi��jZ]b�g�KTV\e]>R�j � �lF � �¦�¥]>W ��F Z K � � � K ��¢ ]�Yb[c� jZ�dt�jVTZ\^Tn�KTV\gRKh TZIKMjZ]b�g�KTZ\^]UR�TZ]EaZ� Osc h>\^u>M j

� u � K�� � w F � � �Àl\eRd[0M ��M�YbWZM jZM0M �U\gRKhwTZ]sLN\eRK\gLN\^¿ M � u � K�� w fiTnIKM�jZ]b�g�KTZ\^]UR TV] aZ�b�PcC\ej�M0ul\gkKM RUTV�e��TZIKM��dRK\eT%M0\ehbMXRUubM [0TV]UW]b��LaYiTnWV\^��� Q� � � YbjnjV]K[�\gYiTZM k�TV]aTZIdM$*.�)�#��� ��*��7M0\ehbMXR>u�Yb�e�KM>fK\ ¢ Mb¢ � �i¢M�� @�� M ; � C ¶ �7> =^T �7O ² � � = � ´ @ j ¶ = �0�X�X�l> = �H�IdMwYbkKu�YbRUTnYihbM7]b�CTnIKMw�e\eRdM YbWp[0WZ\^TZM WZ\^]UR#\gjGTZI�YiT \eT �l\^M �ek�jpYbRaY>RdYi�e�lTV\g[�jV]>�e�KTZ\^]UR�¢ �s]���M0ubMXW0f���MwIdY�u>M�¥]U�dRdk�TnIdYiT�\eTw\gjw¬l�K\^TZM#jZM RdjZ\^TZ\^u>MxTV]8RK]>\ejZMbf�M0ubMXR¨��\^TnI~Ya�eY>WVh>M jZM0Ts]i�GkdYiTnY v�]>\eRUTZj ¢u_�RdM�WVMXYbjZ]>R�\ejTnIdYiT�TnIKM�WnYbRd�>_S{#[0]>RdjZTZWnYi\gRUTba \ ¢ Mb¢ ����� � F � cp\gjwRK]bTwjnYiTZ\ej r�MXk�¢ ��M�[ YbR¨\gLNv1]>jZMxTZId\ejw[0]>RdjZTZWnYi\gRUT�Yv1]>jZTVMXWV\e]>WZ\¯¢¦H�IKMNL�]UjVT�[�]UR>u>M RK\eM RUT¦��Y��¨\ej¦TV]�WZM vK�gYb[0M�TZIKMaLaYiTnWV\^� � M jZTV\gLNYbTVMXk3��\eTZI�Y>RU��]b�pTZIKMY>t�]�u>MsLNM0TnIK]Kkdj7t>�aTnIKM�LaYbTZWZ\����� �wIK\g[cI�LN\gRK\eLN\e¿0MXj�TnIKM�º1WZ]>t1M RK\g�dj�RK]>WnL a jZM0M ÀlM [�T ¢ ]\c ]i� � � ��jn�dtlqSMX[�T7TV] TnIKM�[�]URdjZTZWnYi\gR>T �������� F � ¢ � M0T

� F ����� Qt1M�TnIKM�jV\gRKhU�K�eY>W7u�Yi�g�KM�kKM [0]>Lav1]>jZ\^TZ\^]UR�]b�LaYbTZWZ\�� � f��wIKMXWVM� F kK\gYih�u� � K � K w \gjsY kK\gYih>]>RdYb�LaYbTZWZ\��~jZYbTV\gjS�¥�l\gRKh� � � � � a �p\gj¦TZIKM � � � jV\gRKh>�d�eY>Wwu�Yi�g�KMPccfAY>Rdk � YbRdk�� Y>WVMx]UWVTnIK]bh>]>RdYb�LaYbTZWZ\e[0M j ¢�Q¤Tw[ YbR¨t�M�jnIK]��wR�TZIdYbT

�� F ������ Q��\eTZI��� F kd\eYbh�u � K � K �w LN\gRK\eLN\e¿0MXj7TZIKM�º1WZ]>t1M RK\g�dj7RK]UWZL ]b� � � �� a¯jVM M¦TnIKMxYbvdv1M Rdkd\��¨ÀUMX[�T ¢ ]�¥]UWGTZIdMsvdWZ]U]b�.c)¢ a H�Id\ej�L�M TZIK]Kka�7Ybj ��jVMXkNtU�NH�jZYb\�YbRdk ���dYbRdh)aV� |eY#K�c/\gRaM jZTV\gLaYiTZ\eRKhxTnIKMwMXjZjZM RUTV\gYi�LaYbTZWZ\���fdYbR�k�\gj�\gR>TnWV]Kkd��[�M k¨tU� �¦Y>WVTZ�^M �WaV�X|b|U�ec TV] M jZTV\gLaYiTZM�TnIKM � �dRdk�YbLNM RUTZYb�%LaYiTnWV\^��¢ c

M�� @�� f D ²U¶ � ² �X�7> � ;�= � ² � C � ² �X�K�Z> ¶ = ¶ S��PO ² � > = ² ��� j��P>Ã� ² �P> ¶ =­¦RK]bTnIKM W v�WV]UtK�^MXL���\^TnI¡TZIKM��e\eRKMXYbWa[0WZ\eTVM WZ\e]>R�\gjaTZI�YiTaTZIKM3¬l�dY>RUTV\eT¤� ��M~YbWZM�L�\gRK\gLN\^¿ \eRKh \gj RK]bTv�I>�KjZ\e[ Yi�e�^�¨LNM YbRd\eRKhb� �K�¯¢7­£vdIU�ljZ\g[0Yi�e�e��LNM Y>RK\eRdhi� �K��¬U��YbRUTV\eT¤��jnIK]>�d�ek~t1M#jZ]>LNM0TnIK\eRdh�LNM Ybjn�dWZM k¨\gRTnIKM#\gLaYih>M�vK�eY>RKMbfCt1M [0Y>�djZM�TZIKMNY�u�Yi\e�eY>tK�^Mx\gRl�¥]UWZLaYiTZ\^]UR a¯{�[ v1]b\gR>Tnj.cwYbWZMxM��KTZWnYb[0TVMXk¨� WV]UL \gLaYihbMXj0¢_�RdMajn�d[cI�¬U��YbRUTV\eT¤�3\gj TZIKM kK\gjVTnYbRd[0MN� WV]UL�Y�v�]>\eRUTtS � TV]�\^Tnj�[�]UWZWZM jnv1]>RdkK\gRKh�M vK\gv�]>�eY>W��^\gRKM � � F� iS ���� H � � K�� � K�� P Q fd�wIK\g[cI�\gj�hb\eubMXR�tU� a¯jVM M#ÀUM [0T0¢n{d¢^��c

� u S � K � � w F iS Q� �=�� � �� ��� ��

F Z� � iS Q� � iS �� K aZ��Ksc

�wIdM WZM � � F � � ����� �� ¢ H�IU��j0fKTnIKM�[0WZ\^TZM WZ\^]UR a¯|�c�[0Y>R�t1M�WZM0�wWZ\^TZTVM R�Y>j

��� ������ �

� �� � � u S � K � � w ����������

Page 18: Determining the Epipolar Geometry and its Uncertainty: A Review · 2018-01-04 · ISSN 0249-6399 apport de recherche INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE

������������� ��������������� ��������� �!�"�#�$���%��& ���(')��+* ,���-.�/���0���� �+&�13254��/67���8 ���

H�Id\ej�LNM YbR�jpTnIdYiT���M�YbWZMsLN\eRd\eLN\e¿0\gRKhNRd]bTp]URK�e�aY�vdIU�KjV\g[0Yb�A¬l�dY>R>TZ\^T¤� � u S�� K �=� w fdtd�dT�Yb�ejZ] � ���wIK\g[cI\gj7RK]bT7vdIU�KjV\g[0Yb�^�e�aL�MXYbRK\gRKhb� �K� ¢ � �K]>RdhmaV�X|b|U{ecjnIK]��wj�TZIdYbT�TnIKM �^\gRKM Y>W�[ WV\eTVMXWV\e]>R8\eRUTZWZ]Kkd�d[0M j�Y#tK\gYbjY>Rdk�TVM R�kdj�TV] tdWZ\gRKhaTZIKMxMXvK\ev1]b�eM jwTV]��7YbWnkdjpTnIKMx\gLNYbhbMx[�MXRUTVM W ¢

M�� @�� ? � ¶ � ��� �R>���> =^T ;�=UC �/� n �K� ���YbWZTV�eM0� aV� |>|>��c%I�Ybj Y>RdYi�e�l¿0M k/fi� WZ]>L£Rl�dLNMXWV\g[0Yb��[�]ULNv��KTZYbTV\e]>Rav�]>\eRUT ]b��uU\eM0�xf>TZIKM�IK\eh>IN\eR�jVTnYbtK\e�^\eT¤�]b�1TnIK\ej�e\gRKM Y>W�LNM TZIK]Kk�\���vK\^�lM �d[�]l]>WnkK\gRdYiTZM j�YbWZM�kd\eWZM [0TV�e�#�djZM k�fbY>Rdk#v�WV]Uv�]UjVMXk�TV]xv1M WV�¥]>WnLEY jV\gLavK�^MRd]>WnLNYb�^\e¿ YbTV\e]>R�]i� \gRdvd�KT�kdYiTnY8vdWZ\^]UW�TV]�WZ�dR�RK\eRdh8TZIKM�Y�_¤v�]>\eRUTwYb�^h>]>WZ\^TnIdL¨¢7H�Id\ejsTVM [cI�RK\e¬l�KM�\eR�kKM0MXkv�WV]Kkd�d[0M j�L��d[cI�t1M0TZTVMXWwWVMXjZ�d�^Tnj0fdY>Rdk�\gjwjn�dLaLaYbWZ\^¿ M k¨t�M �^]��x¢

Àl��vdv�]UjVM#TZIdYbTw[�]l]>WnkK\gRdYiTZM j S��G\gR�]>RKMx\gLaYih>M�YbWZM WZM vd�eY>[�M k�tU���S�� F�� iS��SfAYbRdk�[�]l]UWZkK\gRdYbTVM jS��� \gR8TnIKM ]bTZIdM W�\eLaYbhbM YbWZM�WZM vK�gYb[0M k8tU���S��� F�� �giS��� fK�wIKMXWVM � YbRdk � ��YbWZM YbRU� ��� ��LaYiTnWV\g[�MXj0¢ÀK�dtdjZTV\eTZ�KTZ\eRKh~\gR TZIKM�M ¬l�dYiTZ\^]URIiS Q� � iS��� F0� f/��M8kKMXWV\eubMaTZIdM M ¬l�dYiTZ\^]UR�S Q� � � Q � � � ��� �S��� F � ¢H�Id\ej�WVM �eYbTV\e]>R \gLavK�e\^MXjpTnIdYiT � � Q � � � � � \gj�TnIKM¦� �dRdkdY>L�MXRUTZYi�CLaYiTnWV\^�8[�]>WnWZM jnv�]URdkK\gRKhxTV]NTZIKM v1]b\gR>T[0]>WnWVMXjZv1]>R�kKM Rd[0M j�S���� �S �� ¢GH�Il�dj fKYbR¨Yi�eTVM WnRdYbTV\eubM¦LNM TZIK]Kk�]i� r1R�kK\eRdh�TnIKM�� �dRdkdY>L�MXRUTZYi�/LaYbTZWZ\��\gj�Ybj��¥]>�^�e]��wj ��b¢�H�WnYbR�jS�¥]UWZL TZIKM�\eLaYih>M�[�]l]>WnkK\gRdYiTZM jwY>[0[0]>WnkK\eRdh TV]8TnWZY>RdjS�¥]UWZLaYbTV\e]>Rdj �S�� F�� iS���YbRdk��S��� F� �!iS��� ¢

{K¢7º\eRdk¨TnIKM � �dRdkdY>LNM RUTZYb�%LaYiTZWZ\^� �� [0]>WnWVMXjZv1]>R�kK\eRdh#TV] TnIKM�LaYiTn[cIKM j �S���� �S��� ¢Od¢7rwM0TnWV\eM0u>M TZIKMx]UWV\ehb\gRdYi�A� �dR�kdYbLNM RUTnYi�%LaYiTnWV\^��Y>j � F��tQ �� � � ¢

H�IdM ¬l�KMXjVTZ\^]UR�RK]���\gjwIK]��§TZ] [cIK]l]>jZM�TnIKMxTZWnYbR�jS�¥]UWZLaYiTZ\^]URdj � Y>Rdk � � ¢¼�]URdjV\gkKMXWsTnIKMNjZM [0]>Rdk�L�M TZIK]Kk�kKM jn[0WZ\et1M k�Ybt1]�ubMbfA�wIK\g[cI�[0]>RdjZ\ejZTZj¦\gR r1RdkK\gRKh¨TZIKMNM0\ehbMXRUubM [0TV]UW

]b�%TZIdM ��� ��LaYbTZWZ\���� Q� ����Y>jZjZ]K[�\gYiTZM ka��\eTZI�TZIKM �eM YbjZT�M0\ehbM RUu�Yi�g�KMa¥�¥]>W7jZ\eLavK�e\g[�\eT¤�bfKTZId\ej�ubMX[�TZ]>W�\ej[ Yi�e�^MXk TZIKMb���.�P*��^����e���l6��.-/�0���7\gRxTZIKM�jVMX¬l�KM0�Rc)¢�H�IK\ej�LaYiTZWZ\^� [ YbRxt1MpM��dvdWZM jnjZM k Ybj�� Q� � � F ����� Q f�wIdM WZM � \gj/]UWVTnIK]bh>]>RdYb�iYbRdk��£\gj�kK\gYih>]>RdYb���wIK]>jZMpkK\gYih>]>RdYb�iM RUTnWV\eM j � �^a � F Z K � � � K ��c�YbWZMpYbjnjZ��L�MXkTZ]#t1Ms\gR RK]>Rl_`\gRd[0WZM Y>jV\gRKhx]>WnkKMXW0¢�QSRaTZIK\gj�[ YbjZMbf>TZIKM¦�^MXYbjZT M0\ehbMXR>u>M [0TV]>W]b��� Q� � � \gj TnIKMs�gYbjZTp[0]b�g�dLaR]b� ��¢k[�M RK]>TVM3tU���®TZIKM3WnYiTZ\^] � � � ��� ¢§H�IKM3vdY>WZY>L�M TVMXW��©\ej8TZIKM -.���('���%����x� ��� � ������]i� TZIKMLaYbTZWZ\�� � Q� � � f���M �^���URd]��wR�TZ]3t�M�Y>R \eLav1]>WZTZYbRUT�� Yb[0TV]UW \eR TnIKM8YbRdYb�^�KjZ\ej ]b�7jZTZY>tK\e�^\eT¤��]b���e\eRdM YbWv�WV]UtK�^MXLaj!a`¹¦]>�e��t�YbR�k u�YbR � ]>Y>R�� |�Y>|�cc¢�Q`����\gjp�gYbWZhbM>fbTZIdM R8u>M WZ�NjnLaYi�e��[cIdYbRKh>M jGTZ]�TnIKM¦k�YiTZY�[0Y>R[ Yb�djZMa�eY>WVh>M [cIdYbRKh>M j�TV]~TnIKM�jZ]b�g�KTV\e]>R/¢8H�IKM�jZM RdjZ\^TZ\^ul\eT¤�3]i�7\gR>u�Y>WV\gYbRUTxjZ��tdjZv�Yb[�MXj \ej#kK\ejn[0��jZjZM k \gRkdM0TZYb\^��\gR a`¹¦]>�e�dt�Y>Rdk�u�Y>R � ]>Y>R��X|�Yb|�fUv/¢ K1� O�cc¢H�IKM¨LaY�qS]>W#WVMXYbjZ]>R��¥]>W#TZIKM¨v�]l]>W�[�]URdkK\eTV\e]>R ]i��TnIKM�LaYiTnWV\^� � Q� � � ��� \ej�TZIKM��eY>[c��]b��Id]i½

LN]>hbM RdM0\eT¤�¨\gR3TnIKMN\eLaYih>MN[0]l]>WnkK\eR�YiTVMXj0¢�QSR YbR~\gLaYihbMN]i��kK\eLNMXRdjV\e]>R � � � � � � � f�Y8T¤�KvK\g[0Yb��\gLaYih>Mv1]b\gRUT#��\e�^�7t1M�]i�wTnIKM8�¥]UWZL u Z � �+K Z � �K Z w ¢�Q`�¦t1]bTnI iS � Y>Rdk iS��� YbWZM8]i��TZId\ej#�¥]>WnL¨fGTZIKMXR � � ��\e�^�t1M8]b��TnIKM8�¥]UWZL H Z ����K Z ����K Z � � K Z ���%K Z ����K Z � � K Z � � K Z � � K Z PRQ ¢�H�IKM�[�]URUTZWZ\etd�dTV\e]>R TV]3TnIKM�LaYiTnWV\^� �\gjx]b��TnIKMa�¥]>WnL � � � Q� f��wIK\g[cI���\e�e�p[�]UR>TnYi\gR M RUTZWZ\^MXjxWZY>RKhb\gRKh�t�M T¤��M M R Z � � YbRdk¡�>¢8H�IKM�kK\gYih>]>RdYb�MXRUTZWZ\^MXj¦]b� � ��\^�e� t1MN]b�pTZIdM��¥]UWZL H Z � � K Z � � K Z ����K Z � � K Z � � K Z ����K Z ����K Z ����K Z PRQ ¢aÀK�dLaL�\gRKh¨]�ubMXW¦Yb�^�v1]b\gRUT#LaYiTn[cIKM jx��\e�e��WZM jn�K�^Tx\gR Y3LaYiTnWV\^� � �wIK]>jZM8kK\gYihb]URdYi�GMXRUTZWZ\^MXj�YbWZM8Ybvdv�WV]��K\gLNYbTVM �^�3\gR TZId\ej

"!$#&%('*)+#&, -/.10�243"56'*7989:�;=<?>@;�AB'*.C#&D92E)+F@79G9' #&'*F@7H7$I9JLK�2+%+:9K9IM#�;�<">@;ONB'*.P#1DM2Q093"%13"JR2S#&24%TF"UV'*JR0�F@%(#13"79)+2WD92+%&2@X

� � 7EY�Z�[�Z�\

Page 19: Determining the Epipolar Geometry and its Uncertainty: A Review · 2018-01-04 · ISSN 0249-6399 apport de recherche INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE

�X� � ���/�s�#&������ �s�#���

v�WV]Uv�]UWVTZ\^]UR�¢ ��M�kdM RK]>TVM3tU� ��� TnIKM3TnWZYb\^�e\eRdh������ vdWZ\eR�[�\gvdYi�wjn�dtdLaYiTnWV\^�xa TZIdYbT \ej8TZIKM~�gYbjZT��[0]b�g�dLaRdjsY>Rdk�WV]��wj�c�]i� � fAY>Rdk~tU� � ��u ��� w \eTZj ��� � �gYbWZhbMXjVT�M \^h>M RUu�Yi�g�KMb¢�H�Il�dj � � F � F � Q� � �Y>Rdk � F� � u � � w � � � u � � w ¢�º�\gWnjVT f%��M8[�]>R�jV\gkKM W TnIKM8M \^h>M RUu�Yi�g�KM j ]b� � � ¢�Àl\eRd[0M TZIKM8jn�dL ]i�7TZIKMT¤��]aM0\ehbMXR>u�Yb�e�KMXj7\ejwM ¬l�dYi�%TV] TZIdMxTZWnYb[�M>fK��M�jZM0MxTnIdYiT � � u � � w � � � u � � w@F TZWnYb[0M�u � � w@F Z

��� �Z¢

Àl\eRd[0M�M \^h>M RUu�Yi�g�KM j¦YbWZM�RK]URl_¤RKM0hUYiTV\eubM>f���MN�lRK]�� TnIdYiT � � u � � w�� Z��� �

Z¢�ºAWV]UL!TZIKME��l�0�/�/� ��-�����

��� �.�d�����+&Wa¤¹¦]b�g�dt¨YbRdk¨u�YbR � ]UYbR�� |�Y>|dfKv�¢ K��>�Pc)fl��M�YbWnWV\eubM¦TZIdYbT

� � u � � w�� � �0u � ��w��_fgfhf��� � u � � w�� Z� � �

Z�

_�R¨TnIKMx]bTnIKM WwI�YbRdk�fAYi�gjV]N� WZ]>L TZIKMx\gRUTVM WZ�gYb[�\gRKh vdWZ]>v1M WZT¤�bfd��M��lRK]��§TnIdYiT�TnIKM��gYbWZhbMXjVT7M0\ehbMXRUu�Yb�e�dM]b� � \gj�RK]>T��eM jnj�TZI�YbR TnIKM��gYbWZhbMXjVT�kK\gYih>]>RdYb� M RUTnWV�>f�\¯¢ Mb¢ � � u � � w � Z

� � ¢�H�IKMXWVM0�¥]>WZMbf�TZIKM¨WZYbTV\e]� F � � u � � w � � � u � � w � Z

� � � u Z ����� Z w ¢�QSR � Y>[�T0f � � u � � w ��\^�e�7�djZ��Yi�e�^� t�M�L��d[cI jnLNYb�^�eM W�TZIdY>RZ� � �

ZY>Rdk~TnIKM�[0]>RdkK\eTV\e]>R~Rl�dL�t�MXWw��\e�^�t1M�� YbWsh>WZM YbTVM W ¢�H�IK\gj¦Y>RdYi�e�KjV\gjwjnIK]��wj�TZI�YiT *�-.�#� ���� �+�s�

-.�7��� '#����#���/* *����+�s�#� �+�s��& �#� �$��� �+�s�$�#6P��� �����)�� �l��� �0����RK\^T¤� 8 R���U��\��� ��6����+�s�$-.����'#��+��#� � � �+�s��)���%�� � Q� � � ¢�¦]��°[0]>RdjZ\ekdM W7TZIdMxM �CM [�Ts]i�TZWnYbRdjZ�eYbTV\e]>R�¢ ­£�djZ��Yi�/vdWnYb[0TV\g[�Mx\gj�TV]$rd�¨TZIKMx]UWV\ehb\gR�]i�TZIKM�\gLaYih>M

[0]l]>WnkK\eR�YiTVMXjwYiT¦TZIKM�TV]Uv��^M0�¥T�IdY>Rdk3[0]>WnRKM W�]b�GTZIKMN\eLaYbhbMbfAjV]�TZIdYbT¦Yi�e��TZIKM�\eLaYih>M#[�]l]UWZkK\gRdYbTVM jsYbWZMv1]>jZ\eTV\eubMb¢�QSR TZIK\gj�[0Y>jVM>f �#� ���� � �#6P���$���l� ��i����� -.���('���%����g� �)�+�s�m�)���%�� �)��& �"�E��-/�e���6P�.' �/&�%� �#�d*/�����%����W����� �d���� ��*$*�� �+�s�#�\�+�s�E-.�/� �+� �#�' � �)����� �d���� ��*) *m���b����� �#����#��1¢�QSRl�¥]UWZLaYb�^�e�bf/\^��TZIKMrAWZjZTp\gLaYihbM�[�]l]>WnkK\gRdYiTZM j\a TZIdM T _`[0]l]>WnkK\eR�YiTVMXj.c�]b��Y#jZM0T�]i�%v1]b\gRUTZj7YbWZM � Z � � Z � � K Z � � � � � K � � � � � K � � � � fTnIKM R¨TnIKM�jV\eh>Rd\�r1[ YbRUT�u�Yb�e�dM j�]b��TZIdM�[�]l]>WnkK\gRdYiTZM j7YbWZM ]>tdjn[0�dWZM k�tU��TZIdM [0]l]>WnkK\eR�YiTVM ]��CjZM0T7]i�7�ZLeL�L�¢]7�¨TZWnYbRdjZ�gYiTV\gRKh�tU� �ZLeL�L�fdTZIKMXjVMNRl�dL#t1M Wnj¦Y>WVM�[cIdYbRKh>M k�TZ] �

Z� � K � � � K � Z � � K � � � � ¢�H�IKM�jZ\eh>RK\ r1[0Y>RUTu�Yb�e�KMXj�t�MX[�]UL�MxRK]��°vdWZ]>LN\eRdM RUT0¢

]�YbjZM k ]UR TZIKM�Ybt1]�ubM8YbR�Yi�e�ljZ\gj0f �¦Y>WVTZ�^M � aV�X|b|U�ec�vdWZ]>v1]>jZM8YbR \gjV]>TZWZ]>vK\g[ jn[0Yi�e\gRKh~]i�wTnIKM8\gRdvd�KTk�YiTZY���b¢�­sjwYr1WnjZT7jZTVMXv�fdTnIKM�v�]>\eRUTnjwYbWZM¦TnWZY>RdjZ�eYbTVM k¨jZ]NTnIdYiT�TnIKM0\gW�[�MXRUTZWZ]b\gk�\gj�YiT�TnIKMx]>WZ\^h>\eR/¢{K¢�H�IKM R/fGTZIKM�[�]l]>WnkK\gRdYiTZM j�Y>WVM�jZ[ Yi�eM k�fGjZ]3TZIdYbT#]UR TZIKM¨Y�ubMXWZYbhbM Y�v1]b\gRUT iS��w\gj�]i�wTZIdM8�¥]>WnLiS�� F H Z K Z K Z PVQ ¢aÀl�d[cI Y¨v�]>\eRUT ��\^�e�G�e\^M YiT Y¨kK\ejZTZY>Rd[�M � � � WV]UL"TZIKMa]UWV\ehb\gR�¢�rsYbTZIKMXW�TZIdY>R[cIK]l]>jZ\eRdh8kK\ �CM WZM RUTwjn[0Yb�^M � Y>[�TV]UWZj��¥]>W T Y>Rdk U [0]l]>WnkK\eR�YiTVMXj0f���M�[cIK]l]UjVM TV]�jZ[ Yi�eMxTZIKM#v1]b\gRUTZj\ejZ]bTnWV]UvK\g[0Yi�e�e�3jV]�TZI�YiTxTZIdM8Y�ubM WnYih>M�kK\gjZTZYbR�[�MN� WV]UL"TZIdM ]>WZ\^h>\eR�TV]�TZIdM jZM8v�]>\eRUTZj \gjxMX¬U��Yi�GTV]� � ¢

ÀK�d[cI�YaTnWZY>RdjV�¥]>WnLNYbTV\e]>R \gjwY>vdvK�e\^MXk�TZ] M Yb[cI�]i�TZIKMxT¤��]a\eLaYbhbM j�\gRdkKMXv�MXRdkKM RUTZ�^�>¢­¦R#Yi�eTVMXWZR�YiTV\eubMGTZ]sTnIKMp\gjV]>TZWZ]>vK\g[Gjn[0Yi�e\gRKhw\gj�Y>RxY�t8RKMpTnWZY>RdjV�¥]>WnLNYbTV\e]>R jZ]wTZIdYbT%TnIKMpT¤��]svdWZ\eR�[�\gvdYi�

LN]UL�MXRUTZj�]i��TnIKM7jVM T�]b�1v�]>\eRUTnj�Y>WVM�t�]>TZI�M ¬l�dYb�>TZ]��dRd\^T¤�>¢��¦]���M ubM W f �¦Y>WVTZ�^M ��aZ� |>|>�ecC�¥]>�dR�k�TZI�YiTTZIKMWZM jn�K�eTZjp]>tKTnYi\gRKM ka��M WZM��^\eTVTZ�^M�kK\ �AMXWVMXR>Tp� WV]UL£TnIK]>jZMw]UtKTZYb\eRKMXk �djZ\eRKh#TZIKM¦\ejZ]bTnWV]UvK\e[�jn[0Yb�^\gRKh�LNM0TnIK]Kk�¢]7M Y>WZk�jV�eM0� M T7Yb� ¢ aZ� |>|#Ksc LNM RUTV\e]>R¨YaRK]>WnLaYi�e\^¿XYiTV\e]>R�jZ[cIKMXLNM��wId\e[cI¨YbjnjZ��L�MXj7jV]UL�M �lRK]����eM kKh>M

]b��[0Y>LNM WnY�vdY>WZY>L�M TVMXWZj ¢ ­¦[�Tn�dYi�e�^�>f¦\^�NY>vdvdWZ]��K\eLaYiTZM3\gR>TnWV\gRdjZ\e[ vdY>WZY>L�M TVMXWZj a \¯¢ M>¢ TZIKM \gR>TnWV\gRdjZ\e[LaYbTZWZ\�� n c ]i��Y#[0Y>LNM WnY#YbWZM¦Y�u�Yb\^�gYbtK�eMbf>��M [0Y>R8Ybv�vK�^� TnIKM�TZWnYbRdjV�¥]>WnLaYiTV\e]>R � Fln � � TZ]�]UtKTZYb\eR8Y� ¬l�dYbjZ\^_ ¸ ��[��e\ekKMXYbR��¦� WZY>L�M>¢]7]>�K� YbLaYxYbRdk »3]>IdWuaZ� |b|U�ec��djZM�\gLNvd�^\g[�\eTV�e� kdYiTnYxRK]>WnLaYi�e\^¿XYiTV\e]>RNtU��jZM0�eM [0TV\gRKh K#v1]b\gR>Tnj0f>�wIK\g[cI

Y>WVM �gYbWZhbM �^� jZv�WVMXYbk \eR�TZIKM�\eLaYih>M a \¯¢ M>¢�LN]>jZT#kK\gjVTnYbRUT�� WZ]>L M Y>[cI ]bTZIdM W�c)fTV]~�¥]UWZL�Y3vdWZ]iqSMX[�TZ\^u>Mt�YbjZ\ej ¢

���������

Page 20: Determining the Epipolar Geometry and its Uncertainty: A Review · 2018-01-04 · ISSN 0249-6399 apport de recherche INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE

������������� ��������������� ��������� �!�"�#�$���%��& ���(')��+* ,���-.�/���0���� �+&�13254��/67���8 ��}

��8�� �oDdA1�6*�?C � 2 %k?��d= + � C$?���� A@D����%7�� =#D �%?)!A@C�D ?H�IdMaLNM TZIK]Kk kKM jn[0WZ\et1M k�\eR�TZId\ejxjZM [0TV\e]>R3\gjxkd�KMaTZ]¨ºAYb�Kh>M WnYbj`aV� |>|>��c��wIK\g[cI3\gLav�]UjVMXj TZIKM WnYbRd�>_S{[0]>RdjZTZWnYi\gRUT�kd��WV\gRKh TZIKM~LN\eRd\eLN\e¿ YiTZ\^]UR�td�dTajZTV\e�e���U\eM0�gkdjaYbR�YbRdYb�^�lTZ\e[¨jV]>�e�dTV\e]>R�¢��®\eTZId]>�KTa�e]>jnj�]b�h>M RKMXWZYb�^\eT¤�bf>�^M T � F H �2Q#K � � K � � PRQ fl�wIKMXWVM � \ej7Y�u>M [0TV]>W�[�]URUTZYi\gRK\gRKh�TnIKM\rAWZjZT�jZM0u>M R8[0]>Lav1]>RKMXR>Tnjp]b�� ¢ � M T ��� YbRdk � �wt�M¦TZIKMs�eY>jVT T¤��]�[�]>�e��LNRau>M [0TV]>Wnj]i��� � flYbRdk£t�MwTnIKM � � LaYiTZWZ\^�a[0]>Lav�]UjVMXk]b��TnIKM r1WnjVTwjZM0u>M R�[0]b�g�dLaRdj�]b� � � ¢Gº1WZ]>L � � � F � f���MxIdY�u>M

� F � � � ��� � � � � � �­¦jZjn�dLNM TZIdYbT�TZIKM�WnYbR�� ]i�� \ejs}Kfd��M [ YbR¨jV]>�^u>Ms�¥]UW � tU�8�eM Y>jVTV_`jn¬l�dYbWZM j�Ybj

� F � � � u� Q w ��� Q ��� � ����u Q w � � Q � � �H�IdM�jZ]b�g�KTV\e]>R kKMXv�MXRdkdjN]>R�T¤��]~� WZM0M¨vdYbWnYbLNM TVM Wnj � � YbRdk � � ¢ ­sjN\eR¡ÀUM [0T0¢ZO�¢^�>f��M¨[0YbR��djVM�TZIKM[0]>RdjZTZWnYi\gRUT �����gu � w#F � f1�wIK\g[cI�hb\eubM jsY TZIK\gWnkl_`kdM0h>WZM0MxIK]ULN]bhbMXRKM0]U�dj�MX¬U��YiTV\e]>R¨\gR � � Y>Rdk ����fCYbR�k��M#[ YbR�jV]>�^u>M �¥]>WsTZIKM \eW¦WZYbTV\e]�¢ ]�MX[0Y>�djVM�Y8TZIK\gWZkK_`kKM h>WZM0MxMX¬U��YiTV\e]>R�IdY>jsYbTw�^MXYbjZTw]>RdM�WVMXYi��WZ]l]bT fd��MY>WVMxhU�dYbWnYbRUTZM0M k¨TZ]�]>tdTZYi\gR~YiTs�^MXYbjZTs]>RdM#jV]>�e�dTV\e]>R¨�¥]>W � ¢sH�IK\gj¦jZ]b�g�KTZ\^]UR�\ej¦kKM/r1RdM k3�dv3TV]�Y�jZ[ Yi�eM� Y>[�TZ]>W f/YbRdk���M [ YbR RK]>WnLaYi�e\^¿ M � jn�d[cI�TZIdYbTx\^Tnj ubM [0TV]UW�Rd]>WnL«\ej MX¬U��Yi�GTV] �>¢�Q`�7TZIKMXWVM Y>WVMNTZIdWZM0MWZM Yb�AWZ]l]bTnj0fU��M [cIK]l]>jZMwTnIKM ]>RKM¦TZIdYbT7LN\gRK\gL�\e¿0MXj�TZIKM�ubMX[�TV]UW�Rd]>WnLy]b��� � � jZ�dtKqSM [�T�TV] � � � F Z

¢QSR� Y>[�T f���M�[0YbR�kK] TnIKM#jnYbLNM�[�]ULavd�KTZYbTV\e]>R��¥]>WsYbRU��]b�TnIKM�Ob� [cId]b\g[�M jw]i�GvdYi\gWZjw]i�G[�]l]>WnkK\gRdYiTZM j7]i� �Y>Rdk�[cId]U]UjVM>f�YbLN]>RKh�TZIKM v1]>jnjV\gtK�e� �ZLeY8jV]>�e�dTV\e]>Rdj f�TZIKMa]URKM#TnIdYiTxLN\eRd\eLN\e¿0M j TnIKMavdWVM ul\^]U�djsu>M [0TV]>WRd]>WnL�¢H�IKM¨kK\ �AMXWVMXRd[�M�t1M0T¤��M0MXR TZIK\gj#LNM TZIK]Kk�Y>Rdk TZIK]UjVM�kKM jn[0WZ\et1M k \gR�ÀUMX[�T0¢nOd¢ {¨\ej#TZIdYbT#TnIKM8�gYiTVTZM W

\gLav�]UjVMNTZIdMNWnYbRd�>_S{8[0]>RdjZTZWnYi\gRUT¦Y��¥TZM W�Ybvdvd�^\g[0YbTV\e]>R3]b�pTZIdM��e\eRdM YbW��eM Y>jVTV_`jn¬l�dYbWZM j ¢ � M IdY�ubM#M��dv�MXWV\^½LNMXR>TZM k�TnIK\ej�LNM0TnIK]Kk���\^TnI3Y �e\eLN\eTVM k�RU��L#t1M Ww]b� kdYiTnY jVM TZj fAY>Rdk¨�¥]>�dR�k�TZIKMNWVMXjZ�K�eTZj¦[�]ULNv�YbWnYbtK�eM��\eTZI¨TnIK]>jZM ]>tKTnYi\gRKM k¨tU��TZIdM v�WVM uU\e]>��j7]>RKM>¢

��8�� <>=#D ��C�D�% A/!�2 %k?�� = + 2 C�D C ' C�� C�D����lC��%? A@D � %�� =�� � =#CND#?�� ? =���� C�� = �;A/!� C�D�%��

­¦jwkK\gjZ[ �djnjVMXk�\gR3ÀUM [0T0¢ZO�¢ {d¢ K1fKTZIKMx�e\gRKM Y>W�L�M TZIK]Kk aV�7L�c�kK]lM jsRK]bT�LN\eRd\eLN\e¿0M#YavdIU�KjV\g[0Yb�^�e��LNM Y>RK\gRKhi½� �d�w¬U��YbRUTV\eT¤�b¢�­!RdYbTZ�dWnYi�7\gkKM Y�\ejNTnIKM R�TV] LN\eRd\eLN\e¿0M�TZIKM�kK\gjVTnYbRd[0M jat1M0T¤��M0M R�v�]>\eRUTnj YbRdk�TZIKM \eW[0]>WnWVMXjZv1]>R�kK\eRdh�MXvK\ev1]b�gYbW �e\gRKM j � ��� � ��� � � � u%iS � K � iS��� wtK �wIKMXWVM � u f Khf w \gj hb\eubM R tU� aZ��Kscc¢��s]���M�½u>M W f���RK�^\g�iM�TZIKM¨[0YbjZM8]b�7TnIKM��e\gRKM Y>W#[0WZ\eTVM WZ\e]>R�f�TZIKM�T¤��]3\gLaYih>M j�kK] RK]bT�vK�eY�� Y3jZ�KLNLNM TZWZ\e[8WZ]b�eMb¢H�Id\ej�\gj�t1M [ Yb�djZM�TZIKMxYbt1]�ubM�[0WZ\^TZM WZ\^]UR8kKM TVM WnLN\eRdM j�]>RK�e� TZIdM�MXvK\gv�]>�eY>Wp�e\eRdM j7\gR8TZIdM�rAWZjZT�\gLaYih>Mb¢G­sj��M¦I�Y�ubMsjZM0MXR \eR¨ÀUM [0T0¢n{d¢ {dfKt>�aM0�K[cI�YbRKh>\eRKhxTnIKM WV]>�^Mw]b�%TZIdMsT¤��]#\gLNYbhbMXj0fUTZIKMs� �dRdkdY>LNM RUTZYb�CLaYbTZWZ\��\gjx[cIdY>RKhbMXk�TZ]~\^TnjxTZWnYbRdjnv1]>jZMb¢ H%]~Y�u>]b\gk�TnIKM \eR�[�]>R�jV\gjVTZM Rd[0�3]i�7TZIdM M vK\gv�]>�eY>W hbM ]>LNM0TnWV�3t1M0T¤��M0MXRTnIKMxT¤��]N\gLaYih>M j fU��MxLN\eRK\gLN\^¿ MxTZIKM �¥]>�^�e]���\gRKh [0WZ\^TZM WZ\^]UR

��� �� �� � � u%iS�� K � iS �� w � � � u%iS �� K � Q iS�� w K aZ�X�ec

�wId\e[cI�]>v1M WnYiTZM j�jZ\eL��K�^TnYbRKM ]>�djZ�e� \eR¨TZIdMxT¤��]a\gLNYbhbMXj0¢

v(vxw�y"z"{.z.|

Page 21: Determining the Epipolar Geometry and its Uncertainty: A Review · 2018-01-04 · ISSN 0249-6399 apport de recherche INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE

�7Y � ���/�s�#&������ �s�#���

� M0T��=� F � iS��� � H � � K�� � K�� PVQ YbR�k �=�� F � Q iS�� � H � �� K�� �� K � � PRQ ¢�¾¦jZ\eRdh aZ��K�cwY>Rdk�TnIKMN� Yb[0T�TnIdYiTiS Q� � iS��� F iS�� Q� � Q iS��Sf�TZIKM�[ WV\eTVMXWV\e]>R aV�X��c�[0Y>R�t1M�WVM �wWV\eTVTZM R¨Ybj �������

��� �� u iS Q� � iS �� w � K aZ� ��c

�wIdM WZM

� � F � Z� �� � � ��

�Z

� � �� � � � ���� ���� F � � �� � � �� � � � �� ��� � ��u � �� ��� �� w u � � �� � � � �� w � ��� � �

� M�RK]��°vdWVMXjVMXRUT�T¤��]aL�M TZIK]Kkdj��¥]>W�jZ]b�euU\gRKhaTZId\ej�vdWZ]>td�^MXL�¢

M�� f ��: ; � ² ���K�Z>�� ² � > = ² �d� �E² �PO ¶ ·H�IdM#jV\gLN\^�gYbWZ\eT¤�¨t1M0T¤��M0MXR aV� �sc�Y>Rdk a |sc�[�]URdkd�d[0TZj¦�djwTZ]�jZ]b�eubMxTZIdM�Ybt1]�ubM�v�WV]UtK�^MXL tU��YW8 ����������.'�e\gRKM Y>W �^MXYbjZTS_¤jZ¬l�dY>WVMXjaTVMX[cIdRK\g¬U�dMb¢EQSRdkKM M k�fs\�����M3[0Y>R®[�]>Lavd�dTVM�TnIKM3��M0\eh>IUT � � �¥]>W MXYb[cI®v1]b\gR>TLaYbTZ[cI�f�TZIKM�[0]>WnWVMXjZv1]>R�kK\eRdh��e\eRdM YbW�M ¬l�dYiTZ\^]UR�[ YbR t1M�L#�K�eTV\gvK�e\^MXk tU� � � a �wId\e[cI \gjxM ¬l�K\eu�Yi�eM RUTxTV]WZM vd�eY>[�\gRKh � � \gRWa+YscGtU� � � � � ccfKYbRdkaM0�KY>[�TZ�^�aTZIdM¦jnYbLNM�Yi_`v1]b\gRUTpYi�ehb]UWV\eTZIdL [0Y>R8t1M¦Wn�dR TV]�M jZTV\gLaYiTVMTnIKM � �dRdk�YbLNM RUTZYb�/LaYiTZWZ\^�CfK�wId\e[cI�LN\eRd\eLN\e¿0M j aV� �sc)¢

H�IKM�vdWV]UtK�eM L \gj TnIdYiT8TnIKM3��M0\eh>IUTZj � � kKM v1M R�kdj8TnIKM LajZM0�eubM j8]>R¡TZIKM3� �dRdkdY>L�MXRUTZYi�¦LNYbTZWZ\���¢H�]¨]�ubM Wn[�]ULNM#TZId\ejxkK\ t8[0�K�eT¤�bf���MaYbv�vK�^��YbR�\^TZM WnYiTV\eubMN�e\eRKMXYbW LNM0TnIK]Kk�¢ ��M�r1WnjVTxY>jZjn�dLNM�TnIdYiTxYb�^�

� � F ZYbRdk Wn�dR TZIdMmY�_¤v�]>\eRUT#Yb�^h>]>WZ\^TnIdL�TZ]3]>tdTZYi\gR YbR \gRK\eTV\gYi��M jZTV\gLNYbTV\e]>R ]b�7TnIKM8� ��RdkdYbLNMXR>TnYi�

LaYbTZWZ\���¢ H�IKM#��M0\eh>IUTZj � �7YbWZM�TZIKMXR�[0]>Lavd�KTZM k~� WZ]>L!TnIK\ej�\gRK\eTV\gYi�jV]>�e�KTZ\^]UR�¢ H�IKM���M0\eh>IUTZM k~�e\eRdM YbW�eM Y>jVTV_`jn¬l�dYbWZM j�\gjwTnIKM R~Wn�dR¨�¥]>WsYbR�\gLNv�WV]�u>M k¨jV]>�e�KTZ\^]UR�¢7H�IK\ej¦vdWZ]l[0M kd��WVM�[ YbR�t�M�WVMXv�MXYiTVMXk~jVM ubMXWZYb�TZ\eLNMXj0¢

­s�^TnIK]>�KhUI�TZIK\gj�Yi�ehb]>WZ\eTZIdL \ej¦jV\gLavK�^M#TV]8\gLavK�^MXLNM RUT¦Y>Rdk�L�\gRK\gLN\^¿ M j�Y8vdIU�KjV\g[0Yb�%¬l�dYbRUTV\eT¤�bf1]>�dWM0�dv�MXWV\eM Rd[0MNjnIK]��wj�TZI�YiT TZIKMXWVMa\gjxRK]~jV\eh>Rd\�r1[ YbRUT�\eLavdWZ]�ubMXL�MXRUT¦[�]ULavdYbWZM k�TV]�TZIKMa]UWV\ehb\gRdYi��e\eRdM YbWLNM TZIK]Kk�¢¨H�IKM8LaYi\gR WZM Y>jV]UR3\gjxTZI�YiTxTZIdM�WZY>Rd�>_¤{�[�]URdjZTZWnYi\gR>T ]b��TnIKMa� �dRdkdY>LNM RUTZYb��LaYiTnWV\^��\gj�RK]bTTnYb�bM R�\eRUTV] Y>[0[�]U�dRUT0¢

M�� f � @ � ¶ = �R> = ² �d� � > = > � >��b�d�Z> ¶ = > = Fw���X� � ² � ² �mH C �d� ²ºAWV]UL«TZIKMaYbt1]�ubMNkK\gjZ[ �djZjZ\e]>Rdj f�\^T�\gjx[��eM YbW¦TZI�YiT�TZIKM WZ\^hUI>T�TZId\eRKh¨TV]~kK]¨\gj¦TZ]¨jZM Y>WZ[cI~�¥]>W Y�LaYbTZWZ\��Y>LN]>RKh�TZIdM � � �~LaYiTZWZ\g[�M j ]b�wWZY>Rd��{¨�wIK\g[cI L�\gRK\gLN\^¿ M j)aZ� ��cc¢�H�IKMXWVM8Y>WVM jZM0u>M WnYi� v1]>jnjV\gtK�eMav�Y�½WnYbLNM TVM WZ\e¿ YiTZ\^]URdj��¥]>WwTZIKM#� �dRdkdY>L�MXRUTZYi��LaYiTZWZ\^� a � �K]>Rdh3� |>|>{eccfKMb¢ h�¢7��M#[0Y>R�M0�dvdWVMXjZjw]>RKM#WZ]�� a ]>W[0]b�g�dLaR c�]i��TZIKM�� ��RdkdYbLNMXR>TnYi�wLaYiTnWV\^��Y>jaTZIKM~�e\eRdM YbW [0]>L�tK\eR�YiTV\e]>R�]i��TZIKM~]>TZIKMXWaT¤��] WZ]��wjEa ]>W[0]b�g�dLaRdj�c)¢GH�IKMxvdYbWnYbLNM TVM WZ\e¿ YiTZ\^]UR8kKMXjZ[ WV\gt�MXk�t1M0�e]���\gj�tdYbjZM k�kK\eWZM [0TV�e�8]>R�TZIKMxvdY>WZY>LNM0TVMXWZj�]i��TZIKMMXvK\gv�]>�eY>W�TnWZY>RdjV�¥]>WnLNYbTV\e]>R a jZM0M�ÀUMX[�T0¢){K¢ {ecc¢

���������

Page 22: Determining the Epipolar Geometry and its Uncertainty: A Review · 2018-01-04 · ISSN 0249-6399 apport de recherche INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE

������������� ��������������� ��������� �!�"�#�$���%��& ���(')��+* ,���-.�/���0���� �+&�13254��/67���8 �X|

Fs�d��� � ² � ² �P>��>�K�Z> ¶ = ¶ S3S � = ·�� � ² = � � � ���K� �P>�X � [�M RK]>TV\gRKhaTZIKMx[0]b�g�dLaRdj�]i� � t>��TZIKM ubMX[�TZ]>Wnj� � f � � YbR�k � fK��M�IdY�ubM �

WZY>Rd�+u � w F ����u�� � � K � � K � ��� H Z K � P w u�� � � K � ���� w K � � � � � � � ��� � � �� F � aZ��}�cu��� � �� w K � �� � � � �� F$� �

aZ�ZY�c

¼�]URdkK\eTV\e]>R aV�7Y�c)fiY>jGY¦Rd]>Rl_`M��K\gjVTZM Rd[0M�[0]>RdkK\eTV\e]>R/fb[0Y>RdRK]bTGt1M7M��dvdWZM jnjVMXk#tU�#Y�vdYbWnYbLNM0TZM WZ\^¿XYiTZ\^]UR�����MjnIdYb�^�C]>Rd�^�8�bM0M v�[�]URdkK\eTV\e]>RWaZ��}ec Y>Rdk�jZ]�M0�KTVM R�k�TnIKMxvdYbWnYbLNM0TZM WZ\^¿ M k jZM0T7TV]aYi�e�CTZIKM ��� ��_¤LaYiTnWV\g[�MXj]b�/WZY>Rd��jZTZWZ\e[0TV�e���eM jnjGTZI�YbR Od¢�QSRdkKM M k�fUTZIdM¦WnYbRd�>_S{¦LaYbTZWZ\e[0M jG]b�Sfi�¥]>W M��dYbLavK�eMbfUTZIKMw�¥]b�e�^]���\gRKh �¥]>WnLaj �

� � � � � � � ��� YbRdk � � � � � � Y>Rdk � � � � � � �kd]8RK]bTwIdY�u>M�YbRU��vdY>WZY>LNM0TVMXWV\e¿ YbTV\e]>R�\��G��MxTZY>�iM � � F Z

¢�­ vdYbWnYbLNM TVM WZ\e¿ YiTZ\^]UR�]b� � \gj�TZIdM R�hb\eubMXRtU� u � � K � � K�� � K � � w ¢8H�IK\gj�vdY>WZY>LNM0TVMXWV\e¿ YbTV\e]>R~\gLavK�e\^MXjxTV]~kK\eul\ekKM TnIKM8vdY>WZY>L�M TVMXWV\e¿0MXk�jZM0TxYbLN]>RdhTnIdWZM0MxLaYbvdj fd[�]UWZWZM jnv�]URdkK\gRKhNTV] � � F Z

f � � F � YbR�k � � F �d¢Q`�p��Ma[�]>R�jVTnWZ�d[0T¦Y ��_`ubM [0TV]UWwjZ��[cI~TZIdYbT � � YbRdk � � YbWZM#TnIKM � � � � Y>Rdk � � � � [�]l]>WnkK\gRdYiTZM j¦YbRdk �\gj�TZIKM � � � � [�]l]UWZkK\gRdYbTVMbfUTZIdM R8\eT7\ej�]Ut>ul\e]>�dj�TZI�YiT�TZIK\gj�ubM [0TV]UWp\gj�TZIKM�M0\ehbMXR>u>M [0TV]>Wp]i� � fdY>Rdk8\gj�TZIl�dj

TnIKMaM vK\gv1]b�eM�\gR�TnIKM [0Y>jVMN]i��TZIKMN� �dRdk�YbLNM RUTZYb�GLNYbTZWZ\���¢�¾¦jZ\eRKh�jn�d[cI�Y¨vdYbWnYbLNM TVM WZ\e¿ YiTZ\^]UR~\gLNvd�^\eM jTZ]�[0]>Lavd�KTZM¦kK\gWVMX[�TZ�^�aTZIdMsM vd\ev1]b�eMs�wIK\g[cI8\gj�]i�¥TVMXR8Y#��jVM0� �K�A¬l�dY>RUTV\eT¤�bfU\eRdjZTVMXYbk ]b�/TZIKM LaYiTnWV\^�N\eTZjZM0�^�S¢

H�]�LaYb�bMwTZIdM¦vdWZ]>tK�eM L jZ�KLNLNM TZWZ\e[ Yi�1Y>Rdk8jZ\eR�[�MsTnIKMsMXvK\gv�]>�^M¦\eR TnIKM�]bTZIdM W \gLaYih>Mw\ej�Yi�gjZ]���]UWVTnIt1M0\gRKh#[0]>Lavd�KTZM k�f>TZIKM¦jZY>L�M¦kKM [0]>Lav�]UjV\eTV\e]>RaY>j�¥]>WGTnIKMs[0]b�g�dLaRdjG\gjp�djZM kN�¥]UWGTZIKM¦WV]��wj fi�wIK\g[cI RK]��kd\^ul\gkKM j�TZIdMNvdY>WZY>LNM0TVMXWV\e¿0MXk3jZM0T�\eRUTV]�|¨LNY>vdj f/[�]>WnWZM jnv�]URdkK\gRKh8TV]¨TZIdMa[cId]b\g[�M#]b��Y¨[�]b�g�dLaR�Y>Rdk�YWZ]��®Ybjp�^\gRKM Y>Wp[0]>L#td\eRdYbTV\e]>Rdjp]i�/TnIKMsT¤��]#[�]>�e��LNR�j�Y>Rdk T¤��]#WV]��wjp�^M0�¥T0¢G­°vdYbWnYbLNM0TZM WZ\^¿XYiTZ\^]UR�]b�/TZIKMLaYbTZWZ\��x\gj�TnIKM R#�¥]>WnLNM k�t>�xTnIKM�T¤��] [�]l]>WnkK\gRdYiTZM j J Y>Rdk M ]b��TZIdM rAWZjZT�MXvK\gv�]>�^M>f�TZIKM7T¤��]¦[�]l]UWZkK\gRdYbTVM jJ �YbRdk M ��]i� TZIdM�jZM [�]URdk~MXvK\ev1]b�eM#Y>Rdk~TZIdM��¥]>��WsM �^MXL�MXRUTZj �1f f � YbR�k � �eM��¥T tU� � � �if � � �if � �xYbR�k� ��fK�wId\e[cI�\eR�TZ�dWnR�vdY>WZY>LNM0TVMXWV\e¿0MsTZIKM M vd\ev1]b�gYbW�TZWnYbRdjV�¥]>WnLaYiTZ\^]UR LNY>vdvK\gRKhaYbR MXvK\ev1]b�gYbW��e\eRKM ]b�%TZIKMjZM [0]>Rdk�\gLaYih>MxTV]�\eTZj�[�]UWZWZM jnv1]>RdkK\gRKhaM vd\ev1]b�gYbWs�^\gRKM#\gR~TnIKM r1WnjZTw\eLaYbhbMb¢�QSR3TZIdYbTs�7Y��bf�TnIKM�LaYbTZWZ\��\gj7�wWZ\^TZTVM R/fK�¥]>W7M0�KY>LavK�^M>fl�¥]>W � � F �NY>Rdk � � F � �

� F XY � � � J � � M �� � �<� J � � � M �

� � J ��� M � J � � M u � J � � M � w J � u � J � � � M � w�M [\ �aZ� |�c

­7T��gYbjZT0fCTV]¨TZY>�iM#\gRUTV]�Yb[ [�]>��R>T¦TZIKM�� Yb[0T¦TnIdYiT�TZIdM�� ��RdkdYbLNMXR>TnYi�GLaYiTnWV\^�~\ej kKM�r1RKM k�]>RK�e�~�dv�TV]�Yjn[0Yb�^Ms� Yb[0TV]>W fUTZIKM LaYiTZWZ\^�a\gj7RK]>WnLaYi�e\^¿ M k8tU� kK\eul\ekK\gRKhNTZIKM¦�¥]>��WpM �^MXL�MXRUTZj u � K K��0K � w tU�NTnIKM��eY>WVh>M jZT\gR�YbtdjZ]b�g�KTZM¦u�Yi�g�KM>¢ ��M�IdY�u>M¦TnIl�dj�\gR�TZ]bTnYi�/Ob�aLaYbv�j7TV]8vdY>WZY>L�M TVMXWV\e¿0M¦TZIKM � �dR�kdYbLNM RUTnYi�%LaYiTnWV\^��¢

j�O ¶/¶ �Z> =UT �PO ² � ² � � ��� C � ¹¦\euU\gRKh�Y�LaYiTnWV\^� � YbR�k3TZIdM�MXvK\ev1]b�eM j f�]>W�YbR�YbvdvdWZ]��K\eLaYbTV\e]>R�TV]\eT0fA��M�L#�djZTst1M#YbtK�eM�TZ]�[cIK]l]>jZMbf�Y>LN]>RKh TnIKM#kK\ �AMXWVMXRUTsLaYbvdjs]i�TZIKMNvdYbWnYbLNM0TZM WZ\^¿XYiTZ\^]UR�fKTZIKM�L�]UjVTjn�K\eTZY>tK�^M8�¥]>W � ¢ [¦MXRK]bTZ\eRKh�t>� � �� � TZIKM�ubMX[�TV]UWx]i�wTnIKM�M �^MXLNM RUTZj#]i� � ]>Rd[0M�kKMX[�]>Lav1]>jZM k Y>j�\gRMX¬l�dYiTZ\^]UR aV� |sc)f � � YbR�k"� � YbWZM�[cIK]UjVMXR�\gR ]UWZkKMXWxTV]�LNYi�K\eLN\e¿0M�TnIKM�WnYbRd��]i��TnIKM � � ���>Yb[�]UtK\gYbR

v(vxw�y"z"{.z.|

Page 23: Determining the Epipolar Geometry and its Uncertainty: A Review · 2018-01-04 · ISSN 0249-6399 apport de recherche INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE

{�L � ���/�s�#&������ �s�#���

LaYbTZWZ\����

� F � � � � � �wIKM WZM � F H J K�M2K�J � KNM � K � K K���K � P Q � a`{#L�c

H�Id\ej�\gj�kK]>RKMxtU�8LaY��K\gL�\e¿0\gRKhNTZIKMxRK]UWZL ]b��TnIKM ubM [0TV]UWp�wIK]UjVM [�]l]UWZkK\gRdYbTVM j7Y>WVM¦TZIKM�kKM TVMXWZLN\gRdYbRUTZj]b��TnIKM�RK\gRKM � ���ajn�dtdLaYiTnWV\g[�MXj7]i����¢�­sR¨M Y>jV�8[ Yi�g[0�K�gYiTZ\^]UR�jnIK]��wj�TZIdYbT�TZIK\gjwRK]UWZL \ej�MX¬l�dYi��TZ]u � � � ��w � � J � � M � � Z � J � � � M � � � Z

�­7T�TZIdM�M0�Kv1M R�jVM�]i�1kdM Yi�e\gRKhs��\eTZINkK\ �AMXWVMXR>TLaYbvdj f�TZIdM�Y>t�]�u>M�vdY>WZY>L�M TVMXWV\e¿ YbTV\e]>R ��]>Wn�Uj%M ¬l�dYb�^�e� ��M �^��wIdM0TZIdM WGTnIKMsMXvK\ev1]b�eM jpYbWZMwYiT \gRsr1RK\eT¤�N]>W RK]>T0¢H�IK\gj \gjpRK]>TGTZIKM�[0YbjZM7��\^TnI TZIdMw]>WZ\^h>\eR�Yi�dvdWZ]>v1]>jZ\^TZ\^]UR\gR a � �K]>RKh��X|b|>{�c)¢»~]>WZMxkKM0TnYi\e�ej�[ YbR¨t�M �¥]U�dRdk¨\gR a¤¼�jZ��WZ��Ydf �CM0�e�^MXW0f �CI�YbRKhaYbRdk¨ºAYb�Kh>M WnYbj¦� |b|>��cc¢

� > = > � >��>�K�Z> ¶ = � H�IdMsLN\eRK\gLN\^¿XYiTZ\^]UR�]b� aV� �sc�[0Y>RNRd]��¡t1Msv1M WV�¥]>WnL�MXkat>�NYbRU�#LN\gRK\eLN\e¿ YbTV\e]>R vdWZ]i½[0M kd��WVM>¢�H�IKM � M0u>M Rlt�MXWVhb_`»�YbWn¬U��YbWnkKT�LNM0TnIK]Kk a Y>jG\eLavK�eM LNMXR>TZM ka\gRG�����������¦� WZ]>L ����������a »3]>WZM�X|U}>}�c7Y>Rdk3\gR3TnIKM����dLNM WZ\e[�rwMX[�\gv�MXj¦\gR ¼ a opWZM jnj0f�º�gYbRdRdM WZ�bf�H%MX�d�i]>�ejn�U�~YbRdk��GM TVTZM WZ�^\gRKh � |�YeY�c c\gj �djZM k¡\gR�]U�dWavdWZ]bh>WnYbL¨¢ [���WV\gRKh TZIdM3vdWV]K[0M jnj�]b� LN\gRK\gL�\e¿ YbTV\e]>R�f�TZIdM3vdYbWnYbLNM0TZM WZ\^¿XYiTZ\^]UR ]i� �[ YbR�[cIdY>RKhbM ��H�IKM#v�YbWnYbLNM0TZM WZ\^¿XYiTV\e]>R¨[cIK]UjVMXR¨�¥]>W�TnIKM�LaYiTnWV\^�¨YbTwTZIdM�t1M0hb\gRdRK\gRKh8]b�GTZIKM#v�WV]K[�MXjZjw\ejRd]bTsRKMX[�MXjZjnYbWZ\^�e�8TnIKM�LN]>jZTsjn�K\eTZYbtd�^M �¥]UWwTZIdM r1RdYb��LaYiTnWV\^��¢7H�IdM#RK]>Rd�^\gRKM Y>WwLN\eRd\eLN\e¿ YiTZ\^]UR~LNM0TZId]lkkdM LaYbRdk�jxYbR�\eRK\eTV\gYi�GMXjVTZ\eLaYbTVMa]i��TZIKMa� �dR�kdYbLNM RUTnYi� LaYiTnWV\^��f/�wIK\g[cI�\gj ]>tKTnYi\gRKM k tU��Wn�dRdRK\gRKh�TZIKMYi_`v1]b\gRUT�Yi�ehb]>WZ\eTZIdL¨¢

��8�� # !A1+dC %LD ? � �dA ��% + ?)%�� �dD C � .<%� M0T � � F iS Q� � iS��� ¢¦»~\gRK\gL�\e¿0\gRKh � � � �� kK]lM j�RK]bTw�l\eM0�gk3Y8h>]l]lk�MXjVTZ\eLaYbTV\e]>R�]i� TnIKM�� ��RdkdYbLNMXR>TnYi�LaYbTZWZ\���f�t�MX[0Y>�djVM�TZIKM�u�YbWZ\eY>Rd[�Mp]i�dM Y>[cI � � \gj�RK]bT�TZIKM�jZY>LNMb¢�H�IdM��eM Y>jVTV_`jn¬l�dYbWZM jCTZM [cIdRK\g¬l�KM7vdWV]Kkd��[�M jY>R ]>vKTZ\eLaYb��jZ]b�g�KTV\e]>R \^�wM Y>[cI TVM WnL�IdYbj#TZIKM¨jZY>L�M u�Y>WV\gYbRd[0Mb¢3H�IKMXWVM0�¥]>WZMbf���M�[0YbR LN\gRK\gL�\e¿0M�TZIKM�¥]>�^�e]���\gRKh���M0\eh>IUTZM k¨jZ�dL ]i�jZ¬l�dY>WVMXj �

������ �� �� � ���� K a`{K�Pc

�wIdM WZM� �� � \gj TnIKMwu�YbWZ\gYbRd[0M�]i��� �SflYbR�k�\eTZj�[�]ULNv��KTZYbTV\e]>Ra��\e�^�At�Mwh>\^u>M RajnIK]>WZTV�e�b¢H�IK\gjp[0WZ\eTVM WZ\e]>RaRK]��I�YbjTZIdMskKM jZ\gWZY>tK�^M�vdWZ]>v1M WZT¤� � � � � ��� �¥]b�e�^]��wj fb�dRdkdM WGTnIKMur1WnjVTG]UWZkKMXWGYbvdv�WV]��K\gLNYbTV\e]>R�f�TnIKMwjZTZYbR�kdYbWnk¹�Y>�djnjV\gYbR kK\gjZTZWZ\etd�dTV\e]>R�¢ QSR�vdY>WVTZ\e[ �K�gYbW fYb�^� � � � ��� IdY�u>M8TZIdM¨jnYbLNM�u�Y>WV\gYbRd[0MbfM ¬l�dYb��TZ]��>¢ H�IKMjnYbLNMxvdY>WZY>L�M TVMXWV\e¿ YbTV\e]>R ]b��TnIKM � �dRdkdY>LNM RUTZYb�%LaYiTZWZ\^��Ybj7TnIdYiTwkKMXjZ[ WV\gt1M k¨\eR¨TnIKM�vdWZM0ul\^]U�dj7jVMX[�TZ\^]UR\gj��djZM k�¢]7M [ Yb�djZM�v1]b\gRUTZj�YbWZMpM0�lTnWZY>[�TZM kx\gRdkKMXv�MXRdkKM RUTZ�^��tU�xTZIdM�jnYbLNM�Yb�^h>]>WZ\^TnIdL¨f���M�LNY>�iM�YsWZM YbjZ]>R�YbtK�eM

Y>jZjn�dLavKTZ\^]UR8TZI�YiT7TZIKMx\gLaYih>M�v1]b\gRUTZj�YbWZM [�]>WnWn�dvKTVMXk�tU� \gRdkKM v1M R�kKM RUTwYbR�k�\gkKM RUTZ\e[ Yi�e�^��kd\ejZTZWZ\et��KTVMXk¹�Y>�djnjV\gYbR�RK]b\gjVM>fK\ ¢ Mb¢GTnIKM0\gWw[0]�u�Y>WV\gYbR�[�M¦LaYbTZWZ\e[0M j�YbWZM�hb\eubM R¨tU�

� 9 � F � 9 ;� F � kK\gYih�u Z K Z w K

���������

Page 24: Determining the Epipolar Geometry and its Uncertainty: A Review · 2018-01-04 · ISSN 0249-6399 apport de recherche INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE

������������� ��������������� ��������� �!�"�#�$���%��& ���(')��+* ,���-.�/���0���� �+&�13254��/67���8 {d�

�wIdM WZM \ej¦TZIdM�RK]>\ejZM��eM0ubM � fA�wIK\g[cI3LaY��¨t1M�Rd]bT¦�lRK]��wR/¢�¾¦RdkKMXWsTnIKM rAWZjZTs]UWZkKMXWwYbv�vdWV]��K\gLaYiTV\e]>R/fTnIKMxu�YbWZ\eY>Rd[�M�]i� � �\gj7TZIKMXR¨hb\eubM R¨tU�

�� � F � � �� S�� Q � 9 � � � �� S�� � � � �

� S �� Q � 9 ;� � � �� S ��F � H � �� � � �� � � � �� � � � �� P K�wIdM WZM �=� F � iS��� � H � � K�� � K�� PVQ Y>Rdk �=�� F � Q iS�� � H � �� K�� �� K�� � PRQ ¢sÀU\gRd[0M#L#�K�eTV\gvK�e�l\eRKh�M Yb[cI¨TZM WnL tU�Ya[0]>RdjZTZY>R>T�kK]lMXj�RK]bT�Y �CM [0T�TZIKM�LN\gRK\gL�\e¿ YbTV\e]>R�f�TZIKM�vdWZ]>td�^MXL a¯{d�Pc�t�MX[�]UL�MXj

��� �� �u iS Q� � iS �� w � ��� �� K

�wIdM WZM � � F � � �� ��� �� ��� � �� � � � �� \ej�jZ\eLavK�e��TnIKMxh>WnYbkd\^MXR>T�]i� � �V¢ �s]bTZM TZIdYbT � �GkKM v1M Rdk�j�]>R � ¢Q¤T \gj8jnIK]��wRka � �K]URKh®� |>|>{ecxTZIdYbT � � ��� ��\ej�Y rAWZjZTa]>WnkKM WaYbv�vdWV]��K\gLaYiTV\e]>R�]i��TZIKM~]UWVTnIK]bh>]>RdYb�kd\ejZTZY>Rd[�M�� WZ]>L u S�� K S��� w TV]aTnIKM�¬l�dYbkdWnYiTZ\e[ jn�dWV� Yb[�M kKM�r1RKM k�tU�liS Q � iS�� F$� ¢

��8�� <>=#D ��C�D�% A/! 2 % ? �d= +02 C�D C ' C�� C�D�� �lC��%?�A@D � %�����%k? ��% %LD�� �!��% !�BAB?C;=#DA@D + � % ��!= %���? C�=#DQ`����Mw[0Y>RNY>jZjn�dLNM�TZI�YiT TnIKMw[�]l]>WnkK\gRdYiTZM j]b�CTnIKMw]UtdjVMXWVu>M kNv�]>\eRUTnj YbWZMw[�]UWZWn�dvKTZM kNt>�NYbkdkd\^TZ\^u>MsRK]b\gjZMY>Rdk�TZIdYbT�TnIKM RK]b\gjZM j \gR�kK\ �CM WZM RUTxv�]>\eRUTZjxY>WVMN\gRdkKM v1M R�kKM RUT�td�KT ��\eTZI M ¬l�dYb� jVTnYbRdk�YbWnk�kKM uU\gYiTZ\^]URa TZIKM¦jZY>L�M�Y>jZjn�dLavKTV\e]>RaY>jGTZIdYbT �djZM kN\eRaTnIKMsvdWZM0ul\e]>�djTZM [cIdRd\e¬l�KM�c)fbTnIKM RaTnIKMwLaY��K\gL#�dL£�^\g�iM �^\gIK]l]KkMXjVTZ\eLaYbTV\e]>R�]i�TZIKM�� �dRdk�YbLNM RUTZYb�%LaYiTnWV\^��\gj�]>tKTnYi\gRKM k¨tU��LN\eRd\eLN\e¿0\gRKh TZIKM �¥]>�^�e]���\gRKhN[ WV\eTVMXWV\e]>R �

� u � K�E�wBF � u � S�� �� u � K�E � w � � � � S �� �� � u � KNE � w � � w K a`{b{ec

�wIdM WZM � WZM vdWZM jZM RUTZj%TZIKM�vdY>WZY>LNM0TVMXW%u>M [0TV]>W%]i��TnIKM�� �dRdkdY>LNM RUTZYb�lLaYbTZWZ\��#jn�d[cI�Y>j�TnIKM�]>RKM7kKM jn[0WZ\gt�MXk\gR�ÀUMX[�T0¢nOd¢ K�f E F H E Q � K � � � K�E Q� PRQ Y>WVM#TZIKM#jZTZWn�d[0TZ�dWZM#vdY>WZY>L�M TVMXWZj�]b�GTZIKM v�]>\eRUTZjs\eR�jZvdY>[�M>f��wIK\e�^M u � K�E � w Y>Rdk � u � KNE � w YbWZM�TZIKMavdWZ]iqSMX[�TV\e]>R¨� ��Rd[�TZ\^]URdjs\gR3TnIKM rAWZjZT¦Y>Rdk3jZM [�]URdk~\gLaYihbM��¥]>W�Y8hb\eubMXRjnvdY>[�M7[�]l]>WnkK\gRdYiTZM j E ��Y>Rdk#Y�h>\^u>M Rx� ��RdkdYbLNMXR>TnYi�dLaYiTnWV\^�#t1M0T¤��M0M R�TZIdM�T¤��]�\eLaYbhbM jWZM vdWZM jZM RUTZM k�tU�u>M [0TV]>W � ¢�ÀU\eLavK�e��jnv1M Yb�U\gRKh�f � u � K�E�w \ej7TnIKM�jn�dL,]i�jZ¬l�dY>WVMXk8kK\gjVTnYbRd[0M j�t1M0T¤��M0MXR�]UtdjVMXWVu>M k8v1]b\gRUTZjY>Rdk3TnIKM � � � � �� ��"-��%����d* ]b�pTZIKM [0]>WnWVMXjZv1]>Rdkd\eRKh¨v�]>\eRUTnj \eR�jZvdY>[�M>¢�H�IK\gj�\eLavK�e\^MXj TZIdYbT ��MNM jZTV\gLaYiTVMRd]bTs]URK�e�¨TnIKM�� ��RdkdYbLNMXR>TnYi�LaYiTZWZ\^�~t��KT�Yb�ejZ]8TnIKM�jZTZWn�d[�Tn�dWZM�vdY>WZY>L�M TVMXWZj�]b�pTZIdM�v1]b\gR>Tnjs\gR�jnvdY>[�Mb¢H�IdMxM jZTV\gLNYbTV\e]>R�]b��TnIKM�jZTZWn�d[�Tn�dWZM�vdYbWnYbLNM0TZM Wnj0fK]UW��Z� � �.-.���l*.�%� �l-��+��#��f1\eR�TnIKM#�dRd[ Yi�e\et�WZYbTVM k�[ YbjZM\gjwY>R�\gLav�]UWVTnYbRUTwjn�dtlqSM [0TsYbR�k¨RdM0M k�jwY jVMXvdYbWnYiTZMxjVMX[�TV\e]>R¨TZ]8kKM jn[0WZ\et1Mx\eTw\eR~jn�st8[�\eM RUTwkKM TZYb\^�gj a jZM0MÀlM [�T ¢Z­ c)¢GQSR�TZIKM�WZM LaYi\gRK\gRKh�jn�dtdjZM [0TV\e]>R�fK��MxYbjnjZ��L�M�TZI�YiT7TZIKMXWVM \gj�Y�v�WV]K[�MXkd�dWZM�Y�u�Yb\^�gYbtK�eM¦�¥]>W7O#[WZM [0]>RdjZTZWn�d[0TV\e]>R�¢­§hbM RdM WnYi�e\^¿XYiTV\e]>R�TV])a`{b{�c�\ejpTV]�TnYb�bM�\eRUTV]�Yb[0[0]>�dRUT kK\ �CM WZM RUTp�dRd[0M WZTZYb\eRUTV\eM j fb\^�/Y�u�Yi\e�eY>tK�^M>fi\gR TZIKM

\gLaYih>Msv�]>\eRUTnj0¢Q`��Y�v�]>\eRUT�S��%\gj�Y>jZjn�dLNM k8TV]�t1M [�]>WnWn�dvKTVMXk tU�8YN¹�Yb��jZjZ\eY>R RK]b\gjZMs��\eTZI�LNMXYbR ¿ M WZ]Y>Rdk [�]�u�Y>WV\gYbRd[0M�LNYbTZWZ\�� � 9 � a Y � � � jZ�KLaL�M TZWZ\e[�v1]>jZ\eTV\eubM�_¤kKM�r1RK\eTVMsLaYiTnWV\^� c)fUTZIKMXR TZIdMsLaY��K\eL��dL�e\g�iM0�e\gIK]l]lk�M jZTV\gLaYiTV\e]>R�]i�¦TZIKM¨� �dRdk�YbLNM RUTZYb�wLaYiTZWZ\^��\eja]UtKTZYb\eRKMXk�tU��LN\gRK\eLN\e¿0\gRKh TZIKM¨�¥]>�^�e]���\eRdh

v(vxw�y"z"{.z.|

Page 25: Determining the Epipolar Geometry and its Uncertainty: A Review · 2018-01-04 · ISSN 0249-6399 apport de recherche INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE

{>{ � ���/�s�#&������ �s�#���

[ WV\eTVMXWV\e]>R �

� u � K�E�w F � ��� S Q� � ���9 � � S � � � S � Q� � � �9 ;� � S ������\eTZI � S�� F S�� �� u � K�E � w YbRdk � S �� F S �� �� � u � K�E � w ��¦M WZM3��M�jVTZ\^�e�¦Y>jZjn�dLNM3TnIdYiT8TnIKM�Rd]b\gjVMXj8\gR®kK\ �CM WZM RUT¨v�]>\eRUTZj�YbWZM~\eR�kKM v1M RdkdM RUT0fw�wId\e[cI©\gj�¬l�K\eTVMWZM Y>jV]URdYbtd�^M>¢

��IKMXR3TnIKM�Rl�dL�t�MXWw]i�pv1]b\gRUTZj \ejs�eY>WVh>Mbf1TZIKMNRK]>RK�e\gRKM Y>WsLN\eRd\eLN\e¿ YiTZ\^]UR~]i� � u � K�E�w jZIK]U�K�gk3t�M[ YbWnWV\eM k3]>�KT�\eR Y¨IU�dhbMavdYbWnYbLNM0TZM W�jZv�Yb[�MEa � � � kK\eLNMXRdjV\e]>R�j�t1M [ Yb�djZMNM Yb[cI�jZv�Yb[�Mav1]b\gR>T IdY>j�OkdM0h>WZM0MXj ]i��� WVM M kK]UL�c)f�Y>Rdk TZIKM8[0]>Lavd�KTnYiTZ\^]UR�\gjxu>M WZ�3M��dv1M RdjZ\^u>Mb¢�­¦j�Y~LaYiTZTVM W�]i��� Yb[�T f���M [0Y>RjZM v�YbWnYiTVM¦TZIKM�jZTZWn�d[0TZ�dWZM�v�YbWnYbLNM0TZM Wnjp� WZ]>L TZIKM�� �dRdk�YbLNM RUTZYb�/LaYiTZWZ\^��jZ��[cI8TZI�YiT�TZIdM�]UvKTV\gLN\^¿XYiTZ\^]UR]b� TZIKMNjVTnWZ��[�TZ��WVM#v�YbWnYbLNM0TZM Wnj�\ej�[�]URdkd�d[0TVMXk~\eR3M Yb[cI�]UvKTV\gLN\^¿XYiTZ\^]UR�\^TZM WnYiTV\e]>R¨�¥]UWsTnIKM�v�YbWnYbLNM0TZM Wnj]b��TnIKM � �dRdkdY>LNM RUTZYb�%LaYiTZWZ\^�CfKTnIdYiT�\gj �

��� ���� � ��� �� � � � S � �� u � K�E � w � � � � S �� � � u � KNE � w � � � �a`{iO�c

H�IdM WZM��¥]>WZMbfKY vdWZ]>td�^MXL,]i�GLN\eRK\gLN\^¿XYiTZ\^]UR�]�u>M W u � � ��w _%[yjZvdY>[�M$a¯{>{ec�t1M [�]ULNM j�Y vdWZ]>tK�eM L ]i�GL�\^½Rd\eLN\e¿ YiTZ\^]URa]�ubMXWp}X_�[ªjnvdYb[0MbfU\eRaTnIKMs�gYiTZTVM WpM Y>[cI�\eTVMXWZYbTV\e]>R [0]>RUTZYb\eRdj \eRdkdM v1M RdkKMXRUTp]>vdTV\gL�\e¿ YbTV\e]>Rdj]b�COxjZTZWn�d[0TZ�dWZM�vdYbWnYbLNM0TZM Wnj0¢�H�IKMw[�]ULavd�KTZYbTV\e]>R�\ej TZIl�dj [0]>RdjZ\ekdM WnYbtK�e�#WZM kd�d[0M k�¢­sj ��\^�e��t1MsjVM M RN\gRÀlM [�T ¢Z­#f�TZIKM�]>vKTZ\eLN\e¿ YbTV\e]>Rx]b��jVTnWZ�d[0TZ�dWZM�v�YbWnYbLNM0TZM Wnj�\ej�RK]URK�^\gRKMXYbW ¢%QSRx]UWZkKMXW/TZ]¦jnv�M M k#��v�jVTZ\^�e�KLN]>WZMTnIKM�[0]>Lavd�KTnYiTV\e]>R/f%\eT#[0Y>R t�M8Y>vdvdWZ]��l\gLaYiTZM k t>� YbR Y>RdYi�e�lTV\g[ L�M TZIK]Kk� G�wIdM R TZId\ejx]UvKTV\gLN\^¿XYiTZ\^]URv�WV]K[�MXkd�dWZM�[0]>RUubMXWVh>M j f>��M TZIKMXR�WVMXjVTnYbWZT�\eT���\eTZI¨TZIKM�Rd]>RK�e\eRdM YbW�]UvKTV\gLN\^¿XYiTV\e]>R�LNM0TnIK]Kk�¢H�IKM�\ekdM Y3�dR�kKM WZ�^�l\gRKh~TZId\ejNLNM0TZId]lk�\ej�Yi�gWVMXYbkK����M �^���URd]��wR \eR LN]bTZ\^]UR YbRdk�jVTnWZ�d[0TZ�dWZMa� WZ]>L

LN]>TV\e]>R a º1Y>�KhbMXWZY>j�� |b|>Odf �CIdY>RKh�� |b|U�ecpYbRdk¨[0Y>LNM WnY�[ Yi�e\etdWnYiTZ\^]URWa º1Yb�dhbM WnYbj��X|b|bOsc)¢ ÀU\gL�\e�gYbW7TVMX[cIl½Rd\e¬l�KMXj�IdY�u>M�Yi�gjV]�t�M M R�WZM v1]>WZTVMXk¡�¥]>W���Rd[0Yb�^\gtdWnYiTVMXk®\gLNYbhbMXj a »~]UIdW f �GM0\e�^�e]>R§YbRdk "���YbR � |>|bO�f��YbWZTV�eM0�3�X|b|>O�c)¢

��8 � � =�� . �%? 2 % ? �d= + �¾�v~TV]¨RK]��xfC��M�YbjnjZ�dLNM�TZIdYbT¦v1]b\gR>T LaYiTn[cIKM j¦YbWZM�hb\eubMXR�¢¦H�IKM �~[0YbR�t�MN]>tKTnYi\gRKM k~tU��TVMX[cIdRK\g¬l�KM jjn�d[cI Ybj�[�]UWZWZM0�gYiTZ\^]UR�Y>Rdk WVM �eYi�dYiTV\e]>Ria���IdYbRKh1fU[�M WZ\g[cIKMbf�º1Y>�KhbMXWZY>j�YbRdk � �K]>RKh�� |b|U�ecc¢�H�IKM � Yb�^�M0�dvK�^]>\^T/jZ]>LNMu��� ���� *��%�-�* \gR�]>RKM�¥]UWZL§]>W/Y>RK]bTnIKM W f��¥]UW�M��dYbLavK�eMbf \eRUTZM RdjZ\^T¤� jZ\eLN\e�eY>WV\eT¤�w]>W�WZ\ehb\gk1��Y#t RdMTnWZY>RdjV�¥]>WnLNYbTV\e]>RN\gR8\gLNYbhbM�vK�eY>RKMbfl�wIK\g[cI8Y>WVM�RK]bT�YbvdvK�e\g[0Ybtd�^M¦TV]�LN]UjVT7[0Y>jVMXj0¢­sLN]>Rdh#TZIdM�LaYiTn[cIKM jMXjVTnYbtK�e\gjZIKMXk�fK��M LaY��)rARdk¨T¤��]aT¤�Kv�MXj7]i�!� ����� ����.*xk��KM TV]���d· � ¶ �b�K�7> ¶ = � � QSR¨TnIKM�M jZTV\gLaYiTZ\^]UR�]b��TnIKMx� �dR�kdYbLNM RUTnYi�%LaYiTnWV\^��fdTZIdM��^]K[ YiTV\e]>R�M WnWZ]>W7]i�GY v1]b\gR>T

]i�w\gRUTVM WZM jZT�\gj#Y>jZjn�dLNM k TZ]�M��dIK\gtK\eTa¹�Y>�djZjZ\gYbR t�MXIdY�ul\^]UW0¢3H�Id\ejNYbjnjZ��LNvdTV\e]>R \gj#WZM YbjZ]>R�YbtK�eMjV\gRd[0MNTnIKMaM WnWV]UWs\gR �^]K[ Yi�e\^¿XYiTV\e]>R~�¥]>WxLN]>jZTxv�]>\eRUTZj�]b��\gR>TZM WZM jZT \ejxjnLaYi�e� a ��\^TnIK\gR�]URKMa]>W�T¤��]

���������

Page 26: Determining the Epipolar Geometry and its Uncertainty: A Review · 2018-01-04 · ISSN 0249-6399 apport de recherche INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE

������������� ��������������� ��������� �!�"�#�$���%��& ���(')��+* ,���-.�/���0���� �+&�13254��/67���8 {bO

vK\^�lM �ej�c)ftd�KTNY¨�¥M0�,v1]b\gR>Tnj�YbWZM8v1]>jnjV\gtK�e��\gRd[�]UWZWZM [0TV�e�3�^]K[ Yi�e\^¿ M k a LN]>WZM TZI�YbR TZI�WVM M8vK\��KM �ej�c)¢H�IKMx�gYiTVTZM W�v1]b\gRUTZj���\e�^�/jZM0u>M WZM0�e�8kKM0hUWZY>kKM�TZIKM�Y>[0[0��WZY>[�� ]i�TZIdMxM jZTV\gLNYbTV\e]>R�¢

S`� � � ² ���K� ��O ² � � QSR�TZIKM8MXjVTnYbtK�e\gjZIdLNMXR>T#]i�w[�]UWZWZM jnv�]URdkKMXRd[�MXj0f�]>RK�e� IdM �dWZ\ejZTV\g[0j#IdY�ubM8t1M0MXR��djZM k�¢]7M [0Y>�djZM�TnIKM�]>RK�e��hbM ]>LNM0TnWV\g[¨[�]>R�jVTnWZYb\eRUT0fp\ ¢ Mb¢efpTnIKM�M vK\gv�]>�eY>Wa[�]>R�jVTnWZYb\eRUTN\eR�TVMXWZLaja]i��TZIKM� ����'e���)�/� ����� �$�#�+���fd\gj�RK]>T��bM0T�Y�u�Yi\e�gYbtK�eMbfdLaYbRU�8LaYiTn[cIKM j�Y>WVMxv1]>jnjV\gtK�e� � Yi�gjZMb¢ H�IKMXjVM ��\e�^�[�]ULNvd�^M TVM0�e��jnv1]b\e��TZIKM~MXjVTZ\eLaYiTZ\^]UR©vdWV]K[0M jnj0f�YbRdk�TZIKM rARdYi�wMXjVTZ\eLaYbTVM�]i� TZIKM�� ��RdkdYbLNMXR>TnYi�LaYiTZWZ\^����\^�e�%t1M��djZM0�eM jnj0¢

H�IdMs]>�dTV�e\^MXWZj���\^�e��jVM ubMXWVM �^�aY �CM [0TpTZIdM�vdWZM [0\ejZ\e]>Ra]i��TZIKM¦� �dRdkdY>LNM RUTZYb�ALaYiTnWV\^� \�����M�kK\eWZM [0TV�e� YbvdvK�e�TnIKM�LNM0TnIK]Kkdj�kKM jn[0WZ\gt�MXk¨Y>t�]�u>Mbfl�wIK\g[cI�Y>WVMxYb�^���eM Y>jVTV_`jn¬l�dYbWZM j�TVMX[cIdRK\g¬U�dM j ¢� M YbjZTS_¤jn¬U��YbWZM jCM jZTV\gLaYiTZ]>Wnj%Ybjnjn�dLNM TZIdYbT%TnIKM�RK]>\ejZMp[0]>WnWZ�dvdTV\gRKhwTZIdM�kdYbTZY�\gj%]b�K¿0M WZ]sLNM Y>R�f��wIK\g[cI

�l\eM0�gkdj�YbR ���k�����*/�.'�vdYbWnYbLNM0TZM WpM jZTV\gLNYbTVM>¢�Q`��TZIKM RK]b\gjZMsu�YbWZ\eY>Rd[�Ms\ej7�lRK]��wR�flY�����l�� �����06�������#��-.�v�YbWnYbLNM0TZM W�M jZTV\gLaYiTVM�[ YbR�t1M�]>tKTnYi\gRKM k�tU��[cId]U]UjV\gRKhaYbvdv�WV]UvdWV\gYiTZM ��M \^hUI>Tnj7]>R¨TZIdM�kdYiTnYd¢ ºA�dWVTnIKM WV½LN]UWVM>f �eM Y>jVTV_`jn¬l�dYbWZM jCMXjVTZ\eLaYiTZ]>Wnj�\gLNvd�^\g[�\eTV�e�xYbjnjZ�dLNMGTnIdYiT�TZIKM�M RUTV\gWVM jZM0T�]i�dkdYbTZYw[ YbRxt1Mp\gR>TZM WnvdWZM0TVMXktU�g����� & �#���`���#� ���)��������6��"-��0����]i�wY3h>\^u>M R LN]KkKM0�¯¢ �¦��L�MXWV]U�dj�jZTZ�dkd\^MXj�IdY�u>M8t�M M R�[0]>Rdk��d[�TZM k�f�wId\e[cI¡[��eM Y>WV�e��jZId]�� TZI�YiT �eM YbjZTS_¤jn¬U��YbWZM j�M jZTV\gLNYbTV]UWZjaYbWZM¨ul�d�eRKMXWZY>tK�eM¨TZ] TZIKM~ul\e]b�gYiTV\e]>R�]i��TZIKMXjVMY>jZjn�dLavKTZ\^]URdj0¢~ÀU]>LNM TV\gL�MXj�M ubM R �wIKMXR TZIKM�kdYiTnY3[�]UR>TnYi\gRdj ]>Rd�^� ]>RKM8t�Ybk kdYbTZ�dL¨f��^MXYbjZTS_¤jZ¬l�dY>WVMXjMXjVTZ\eLaYbTVM j�LaY��~t�Ma[�]ULNvd�^M TVM0�e�~v�MXWVTn�dWnt�MXk�¢ [��dWZ\gRKh�TnIKM��gYbjZTsTnIdWZM0MakKM [ YbkKMXj0fCLaYbRU�~WV]Utd�djZTsTVMX[cIl½Rd\e¬l�KMXj�I�Y�ubM¦t1M0MXR�v�WV]Uv�]UjVMXk�f>�wId\e[cI¨YbWZM¦RK]>T�u>M WZ�8jVMXRdjZ\^TZ\^u>MsTV] kKMXvdYbWZTZ��WVMs� WV]UL TZIKMxYbjnjn�dLavKTV\e]>Rdj]UR��wId\e[cI¨TZIdM0��kKMXv�MXRdk�¢rwMX[�MXR>TZ�^�>f1[0]>Lavd�KTZM W�ul\gjV\e]>R�WZM jZM YbWn[cIKMXWZj7IdY�u>M v�Yi\gk¨L��d[cI~YiTZTVMXR>TZ\^]UR¨TV] TZIdM#WV]Utd�djZTZRKMXjZjw]i�uU\^½

jZ\e]>RNYi�ehb]>WZ\eTZIdLajt�MX[0Y>�djVM7TnIKMwkdYiTnY YbWZM��dR�Y�ub]b\gkdY>tK�^�xMXWZWZ]>W�vdWZ]>RKM a �¦YbWnYi�e\g[c�a� |�Y>�df��CIl�dY>RKh�f � YbRKhY>Rdk ��IdYbRKh � |>|>{�c)¢#»3Y>RU�~TZIKM jZ]i_¤[0Yi�e�eM k � ��� �s*��!�"�%��� �/*.*.��#��LNM0TnIK]Kkdj IdY�ubMat�M M R vdWZ]>v1]>jZM k3TnIdYiTY>WVM RK]>TwjV]aM Y>jV\e�e��Y �CM [0TVM k�tU�8]U�KTV�e\^MXWZj a �¦�dt1M W�� |�Y��bflrw]U�djZjZM0MX�U�ªYbR�k � M WZ]��~� |�Yl}�cc¢H�IdM WZM Y>kKM W\gjpWZM��¥MXWZWZM kNTZ]�a rw]U�djnjVM M �U�¡Y>Rdk � MXWV]���� |eYU}lfl¼�IdYbv�¢ �Pc%�¥]UWpY�WZM0ul\eM0�¡]i�/kK\ �AMXWVMXR>T�WV]Utd�djZTpLNM0TZId]lk�j0¢H�IdM�T¤��] L�]UjVTGv1]>vd�d�eY>WGWV]Utd�djZT�LNM0TnIK]KkdjGYbWZM7TZIKM�� � �/*.�%��)���0���.*wYbRdk�TZIdM � �"��*����0�)�.'�����"� � ����* �l��� �/*a � »~MXk1Àdc�LNM TZIK]Kk�fx�wIK\g[cI°��\^�e�#t1M�vdWZM jZM RUTVMXkªt1M0�e]��x¢�rwMX[�MXR>T~��]>Wn�lj¨]>R°TZIKM�YbvdvK�e\g[0YiTZ\^]UR§]b�WZ]>t��djVT�TVM [cI�RK\e¬l�KMXj�TZ]�LN]bTZ\^]UR®jVM h>LNM RUTnYiTV\e]>R©\gRd[0�e�dkdM a H�]>WnW�Y>Rdk�»3��WZWnY��ª�X|b|>Odf _�kK]Ut�M ¿�YbR�k]7]U�KTZIKMXL���� |b|�K�fi­s�bM W flÀK[cIdWV]lM TVM WGY>Rdk$]7\^h��dR¨� |>|#KsccfbYbR�kaTZIK]UjVM�]URaTZIKMsWZM [0]�ubMXWV�x]b�CTnIKMwMXvK\ev1]b�gYbWh>M0]UL�M TZWZ� \eR�[��g�dkKM)a�_¦�gjZM R�� |b|U{Kf�ÀlIdY>vK\gWV]aYbRdkE]�WnYbkK�~�X|b|>�dfKH%]UWZW�� |>|>��c

M�������: �°´ A �0�7> ���d� ¶ �X�� M0T � ��t1M3TZIKM � ��*��'��l�#� ]i� TZIdM ��� � kdYiTn�dL¨f�\ ¢ Mb¢°TZIKM�kK\ �CM WZM R�[�M�t1M0T¤��M0MXR�TnIKM ��� � ]>t�jVMXWVu�YiTZ\^]URY>Rdk~\eTZj\r�TVTZM k~u�Yb�e�KM>¢¦H�IKMNjVTnYbRdk�YbWnk��^MXYbjZTS_¤jZ¬l�dY>WVMXj�L�M TZIK]Kk�TZWZ\^MXjwTV]¨LN\eRK\gLN\^¿ M � � � �� fA�wIK\e[cI3\ej��RdjVTnYbtK�eM�\^� TZIKMXWVM�YbWZM�]>�KTZ�^\eM Wnjsv�WVMXjVMXR>T�\gR~TnIKM�kdYbTZY�¢�_��KTV�e�l\eRKh�kdYiTnY hb\eubM#Y>R�M �CM [0T¦jZ]8jVTnWV]URKh8\gRTnIKMaL�\gRK\gLN\^¿XYiTV\e]>R�TZIdYbT¦TnIKMavdYbWnYbLNM0TZM Wnj�TZIl�dj�MXjVTZ\eLaYiTZM k�YbWZM�kK\gjVTZ]>WZTVMXk�¢�H�IdMN»�_`M jZTV\gLaYiTZ]>WnjwTZWZ�TZ]8WVMXkd�d[0MxTZIKMxM��AMX[�Tw]b��]U�KTV�e\^MXWZjst>�¨WVMXvK�gYb[�\gRKhaTZIdM�jZ¬l�dY>WVMXk�WZM jZ\ek��dYi�gj � �� tU��Y>RK]bTnIKM W�� �dR�[�TV\e]>R�]b�TnIKM�WZM jZ\ekd��Yi�gj0fl�l\^M �ekd\eRKh

��� � ���

u � � w K a`{�Ksc

v(vxw�y"z"{.z.|

Page 27: Determining the Epipolar Geometry and its Uncertainty: A Review · 2018-01-04 · ISSN 0249-6399 apport de recherche INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE

{#K � ���/�s�#&������ �s�#���

�wIdM WZM�\gj�Y�jZ�lLaLNM0TnWV\g[if�v�]UjV\eTV\eubM0_`kdM/r1RK\eTVMN� �dR�[�TV\e]>R ��\eTZI Y��dRK\g¬l�KM8LN\eRK\gL#��L-YiTx¿ M WZ]�f%Y>Rdk�\ej

[cId]>jZM R3TZ]~t�M �eM jnj�\eRd[ WVMXYbjZ\eRdh�TnIdYbR jZ¬l�dY>WVM>¢#QSRdjZTVMXYbk�]i�7jV]>�^ul\gRKh¨kd\eWZM [0TV�e�3TZIK\gj�vdWZ]>td�^MXL�f%��Ma[0Y>R\gLavK�eM LNM RUT�\eTwYbj�YbR�\eTVM WnYiTZM k¨WVM ��M \^hUIUTVM k��^MXYbjZTS_¤jZ¬l�dY>WVMXj�]>RKM>¢�¦]��§�eM0Tw�dj�jZM0M�Id]��x¢� M0T � FeH � � K � � � K����gPRQ t�MaTZIdMav�YbWnYbLNM0TZM W¦ubM [0TV]UWsTZ]¨t1MaM jZTV\gLNYbTVMXk�¢�H�IKMa»�_`M jZTV\gLaYiTV]UWs]b� �t�YbjZM kN]>RNTZIdM�� �dRd[0TV\e]>R

�u � � w \gjGTnIKMsu>M [�TZ]>W � �wIK\e[cIN\gj TnIKMsjZ]b�g�KTV\e]>RN]i��TnIKM��¥]b�e�e]���\eRKh � MX¬l�dYiTZ\^]URdj �

� u � � w � � �� � F$� K �¥]>W � F Z K � � � K�� f a`{b�ec

�wIdM WZM#TZIdMakdM WZ\^u�YiTZ\^u>M � u J w F k�u J w � k J \ej [ Yi�e�^MXk3TZIdME���� �l�/�(-"� � ���(-/�%�����¢�Q`��Rd]��E��MakKM/rARKMNY

8 ����#�e� � ����-��+��#�

� u JLw F � u JLwJ K a`{i��c

TnIKM R ¸ ¬l�dYbTV\e]>R a¯{>�ec�t1M [�]ULNM j

� � u � � w � � � � �� � F$� K �¥]>W � F Z K � � � K�� ¢ a`{>}�c

H�Id\ejN\gj�M0�dYb[�TZ�^� TnIKM¨jZ�KjVTZM L�]b�sM ¬l�dYbTV\e]>Rdj�TZIdYbTa��M�]>tKTnYi\gR�\^�¦��M�jZ]b�eubM�TnIKM��¥]b�e�^]���\gRKh�\eTVMXWZYbTVMXkWZM0��M0\eh>IUTZM k8�eM Y>jVTV_`jn¬l�dYbWZM j�vdWZ]>tK�eM L

����� � � u ��� � � �� w � �� K a`{#Y�c

�wIdM WZM�TnIKM8jZ��v�MXWZjn[0WZ\evdT � � \gRdkK\g[0YbTVM j�TnIKM \^TZM WnYiTZ\^]UR�Rl�dL#t1M W ¢aH�IKMa��M \^hUI>T � u � � � ����� w jnIK]>�d�ek t�MWZM [0]>Lavd�KTZM k¨Y��¥TZM W7M Y>[cI�\eTVMXWZYbTV\e]>R�\eR¨]UWZkKMXW�TV] t�M���jVMXk�\gR¨TZIKMxRKM0�KT�\^TZM WnYiTV\e]>R/¢H�IKM \gR �1�KMXRd[�M�� �dRd[0TV\e]>R � u J w LNM Ybjn�dWZM j�TZIKM \gR �1�KMXRd[�M ]i�NY�kdYbTZ�dL ]UR�TnIKM�u�Yb�e�KM ]i�NTZIKM

v�YbWnYbLNM0TZM WNM jZTV\gLaYiTVM>¢©º�]>WaM0�dYbLavK�eMbf �¥]UWaTZIKM~�eM Y>jVTV_`jn¬l�dYbWZM j���\eTZI�u J w F J � � � f�TZIKM~\gR �1�KMXRd[�M

� ��Rd[�TZ\^]UR \ej � u JLw F J fTZIdYbT�\gj0fTZIdM8\eR��1�KM R�[�M8]b��Y~k�YiTZ��L-]UR TZIKM8MXjVTZ\eLaYbTVM8\gRd[0WZM Y>jVMXj �^\gRKMXYbWZ�^���\eTZI�TZIKM jZ\^¿ M#]b�p\^Tnj�M WnWV]UW0fA�wIK\g[cI�[�]URsr1WnLajsTnIKM RK]>Rl_¤WZ]>td�djZTZRdM jnjw]i��TZIKMN�^MXYbjZTS_¤jZ¬l�dY>WVMXjwMXjVTZ\eLaYiTZMb¢��IdM R�Y>R M jZTV\gLaYiTZ]>W \gj�WV]Utd�djZT0fl\^T�LaY��at1M�\eRl�¥MXWZWZM k8TZIdYbTpTZIdM¦\gR �1�KMXRd[�M�]i�%Y>RU�ajZ\gRKhb�eMw]>t�jVMXWVu�YiTZ\^]URa¯kdYiTn�dL�cC\gj/\gRdjn�st8[�\eM RUT%TZ]w�l\^M �ekxY>R>� jZ\^hURK\�rA[0YbRUT�]��CjZM0T a rwM ���X|�YbOsc)¢%H�IdM WZMpYbWZM jVM ubMXWZYb�i[�]>R�jVTnWZYb\eRUTZjTnIdYiTwYaWZ]>td��jVT��ª_`M jZTV\gLNYbTV]UW�jnIK]U�K�ek¨LNM0M T �

� H�IKM r1WnjVT7\gj�]i�[�]U�dWZjZM�TV]8IdY�u>M�Yat1]>�dRdkdM k¨\eR��1�KM R�[�M � �dRd[0TV\e]>R�¢

� H�IKM8jZM [�]URdk3\gj�RdYbTZ�dWnYi�e�e�~TZIKM8WZM ¬l�K\gWVMXLNM RUT¦]b��TnIKM8WV]Utd�djZT¦MXjVTZ\eLaYbTV]>W TZ]~t�M8�dRd\e¬l�KM>¢aH�Id\ej\eLavK�e\eM j TnIdYiT�TZIKMs]>tlqSMX[�TV\eubM7� ��Rd[�TZ\^]UR ]i�/vdY>WZY>L�M TVMXWubM [0TV]UW � TZ]#t�M¦L�\gRK\gLN\^¿ M k jnIK]>�K�gk IdY�u>MY �dRK\g¬l�KMxL�\gRK\gL#�dL¨¢pH�Id\ejwWZM ¬l�K\gWZM j�TZIdYbT ����� ��('��67�' �l���

�� � ����-��+��#� *�[0]>RUubM0� �� 6P�#����%�����

� ¢aH�IK\gj \ej RKMX[�MXjZjnYbWZ�~t�MX[0Yb��jVMa]>Rd�^�3WZM ¬l�K\gWV\gRKh¨Y�_¯� �dRd[�TZ\^]UR�TZ]¨I�Y�ubMaY��dRd\e¬l�KM LN\eRd\eL��dL

\ej7RK]>T�jn�st8[�\eM RUT ¢H�Id\ej�\gj�TnIKM [0YbjZMs��\eTZI�LaY��K\eLaY��wIKM R�[�]URdjV\gkKMXWV\gRKh#LN\^�lTn�dWZM�kK\gjZTZWZ\etd�dTV\e]>R�

���������

Page 28: Determining the Epipolar Geometry and its Uncertainty: A Review · 2018-01-04 · ISSN 0249-6399 apport de recherche INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE

������������� ��������������� ��������� �!�"�#�$���%��& ���(')��+* ,���-.�/���0���� �+&�13254��/67���8 {>�

TZIKMwjn�dL£]i�A��RK\eLN]KkdYb��vdWZ]>tdY>tK\^�e\eT¤��kK\ejZTZWZ\gtd�KTV\e]>R�j\ejGu>M WZ�x]i�¥TZM RaL#�K�eTV\gLN]KkdYi�¯¢�H�IdMw[�]>RUu>M��K\^T¤�[�]URdjVTnWZYb\eRUT7\gj7M ¬l�K\eu�Yb�^MXRUT�TZ] \eLav1]>jZ\eRdhNTnIdYiT�� � � ��� ��� � \gj�RK]URl_`RdM0h>YbTV\eubM kKM/rARK\^TZMb¢

� H�IKM TZId\eWnk ]>RKM \gj�Y~vdWnYb[0TV\g[0Yi�GWZM ¬l�K\gWVMXLNM RUT0¢ ��IdM RKM ubM W � � � ��� ��� � \gj�jZ\eRdh>�K�gYbW f/TZIdM8]>tlqSMX[�TZ\^u>MjZId]>�K�gk©IdY�ubM~Y h>WnYbkd\^MXR>T f�\¯¢ Mb¢ � � ��� ��� �F � ¢ªH�IK\ej�Y�ub]>\ekdjaIdY�ul\gRKh TV]�jZM Y>WZ[cI�TZIdWZ]>�dh>I¡TZIKM[�]ULNvd�^M TVM�vdY>WZY>L�M TVMXW�jnvdY>[�Mb¢

H�IdM WZM�Y>WVM~Y Rl�dL�t�MXW ]i�xkK\ �AMXWVMXR>T8»~_¯MXjVTZ\eLaYiTZ]>Wnj vdWZ]>v1]>jZM k�\eR¡TZIdM3�^\eTVMXWZYbTZ�dWZMb¢�H�IdM3WVMXYbkKMXWa\ejWZM��¥MXWZWZM k�TV] a �CIdYbRdh��X|b|b�>t c �¥]>W�Ya[�]ULavdWVMXIKM R�jV\eubM WVM ul\^M �x¢Q¤T#jZM0MXLaj�kK\ t [ �K�eT#TV]3jZM0�eM [0T#Y

�_ � �dR�[�TV\e]>R �¥]UWxhbMXRKM WnYi� ��jVM8��\eTZId]>�KT#t1M0\gRKh3WnYiTnIKM W�Y>WZtd\^TnWZY>WV�>¢

H�IdM WZM jn�K�eTwWVMXv�]UWVTZM k�\eR3ÀUMX[�T0¢ K �djVMXjwH��d�iM � � �dRd[0TV\e]>R��

� � F � � �� � Z� � Z � � � �� ��� � \^� � � �&� u � � � � w ]bTnIKM WZ��\gjVM>f

�wIdM WZM E\gj~jZ]>LNM�MXjVTZ\eLaYiTZM k°jVTnYbRdk�YbWnk�kKM0ul\gYiTZ\^]UR�]i��M WnWV]UWZj fsYbR�k � F ��� � � Z \gj�TnIKM TZ�dRd\eRKh[0]>RdjZTZY>RUT0¢GH�IKMx[�]UWZWZM jnv�]URdkK\gRKhN��M \^hUI>T�� �dR�[�TV\e]>R¨\gj

� � F � H Z � u J � ��w � P � \�� � � � � � ]bTnIKM WZ��\ejZMb¢

­¦RK]bTnIKM W�[0]>LaLN]>RK�e���djZM k�� �dRd[0TV\e]>R�\gj7TZIKM �¥]>�^�e]���\gRKh�TnWV\^_`��M \^hUI>T7]URKM��

� � F�� �Z

� � � � � � �� � � � � � � �� � � �

QSR a�_¦�gjZM R8�X|b|U{Kf � �K]URKh��X|b|U{ec)f�TnIK\ej��M0\eh>IUT�� �dRd[0TV\e]>RN��Y>j���jVMXk��¥]UW�TnIKM�M jZTV\gLaYiTZ\^]UR�]i�ATZIKM7MXvK\ev1]b�gYbWh>M0]UL�M TZWZ�b¢QSRdIdM WZM RUT7\eR�TZIKMxkK\ �CM WZM RUT�»�_¯MXjVTZ\eLaYbTV]>Wnj�\ej�TZIdM�jV\gL#�K�eTZY>RKM0]U�dj�M jZTV\gLNYbTV\e]>R�]i� f�TZIKMxjZTZYbR�kdYbWnk

kdM0ul\eYbTV\e]>R�]b�TnIKM�WZM jZ\gkd�dYi��M WnWV]UWZj ¢�Q`� ��M#[ YbR~LaYb�bM#Y h>]U]Kk¨M jZTV\gLaYiTZM#]i�GTnIKM�jZTZY>RdkdYbWnk~kKM uU\gYiTZ\^]UR]b�pTZIdM�MXWZWZ]>Wnjw]b� hb]l]Kk�kdYbTZYWa \eRd�^\eM Wnj.ccf�TZIKMXR�kdYbTZY��wIK]>jZM#MXWZWZ]>Ws\ej��gYbWZhbMXWwTZI�YbR�Y�[�MXWVTnYi\gR3Rl�dL#t1M W]b��jZTZYbR�kdYbWnk kKM0ul\gYiTZ\^]URdj�[ YbR�t�M [�]URdjV\gkKMXWVMXk8Ybj�]>�KTZ�^\eM Wnj ¢H�Il�dj flTZIKM�M jZTV\gLaYiTZ\^]UR8]i���\^TnjVM ����jZIK]U�K�gkt1M#WZ]>td��jVT ¢pH�IKM#WZM jn�K�eTZj7]i�GTnIKM�»�_`M jZTV\gLNYbTV]UWZj7��\e�^��kKM v1M R�k�]>R¨TZIKM�L�M TZIK]Kk��djZM k�TZ]8[�]>Lavd�dTVMx\eT0¢H�IdM � �%� �s*��\*��0����'e��� ' '���6Z��#�+��#�©M jZTV\gLNYbTVM \gj#WZM0�gYiTZM k TV]~TnIKM�LNM kK\gYbR ]b��TnIKM�Ybt�jV]>�e�KTZM u�Yi�g�KM jx]b�TnIKM�WZM jZ\ekd��Yi�gj0fKYbR�k�\gj7hb\eubMXR�tU�

� F Z�� � � � H Z � � � u � �kw P � � � ��� �� � � � a`{i|�c

H�IdM�[�]URdjZTZYbRUT��>¢ K�Y>{b��\gj�Y8[�]lM/t8[0\^MXR>T¦TV]¨Yb[cIK\eM0u>MxTZIKMajZY>LNM�M/t8[�\eM R�[���Ybj�Y �^MXYbjZTS_¤jZ¬l�dY>WVMXj7\eR�TZIKMv�WVMXjVMXRd[�M~]b� ]URK�e�©¹�Yb��jZjZ\eY>R©RK]b\gjVM a Yb[0TZ�dYb�^�e�bf7TnIKM�LNM kd\eY>R©]i�xTZIKM�YbtdjZ]b�g�KTVM~u�Yb�e�KMXj8]b�xWnYbR�kK]>L

v(vxw�y"z"{.z.|

Page 29: Determining the Epipolar Geometry and its Uncertainty: A Review · 2018-01-04 · ISSN 0249-6399 apport de recherche INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE

{b� � ���/�s�#&������ �s�#���

Rl�dL�t�MXWZj¨jnYbLavK�eM k®� WZ]>L TnIKM�¹�Yb�djnjZ\eY>R�RK]>WnLaYi� kK\gjVTnWV\gtd�KTZ\^]UR�� u �K Z w \gj¨M ¬l�dYb��TZ]�� � � u � w��Z�Z�� � � � c � � u � �kw a �wIKM WZM \gj TZIdM3jV\e¿0M~]b��TnIKM3kdYiTnY jVM T8YbRdk � \ej8TZIKM3kK\gLNM RdjZ\e]>R¡]i� TZIKM

v�YbWnYbLNM0TZM W7ubMX[�TZ]>W�c \gjwTZ]8[�]ULNv1M R�jZYbTVMxTZIdM�M �CM [0Tw]i� Y jnLNYb�^��jVM T�]i� kdYiTnYd¢�H�IKM#WZM YbkdM W7\gjwWVM0�¥M WnWVMXkTZ] a rw]U�djZjZM0MX�U�ªYbR�k � M WZ]��~� |�Yl}lfKvdYbhbM�{#LU{ecG�¥]>W7TZIdM�kKM0TnYi\e�ej7]b��TnIKM jZMxLNYbhb\g[ RU��L#t1M Wnj0¢_��dW¦M��dv1M WZ\^MXRd[�M�jZIK]��wjsTZIdYbT�»~_¯MXjVTZ\eLaYiTZ]>WnjsY>WVM#WZ]>t��djVTsTV]�]U�KTV�e\eM Wnj¦kd�dM#TV]¨tdYbk3�^]K[0Yb�^\e¿ YbTV\e]>R�¢

H�IdM0��Y>WVM>f�IK]���M0u>M W fdRK]bTwWZ]>td��jVT�TZ] � Yi�gjVM�LaYbTZ[cIKMXj0f�t1M [ Yb�djZMxTZIdM0��kKM v1M Rdk~IdM Y�ul\^�e��]UR¨TZIKM�\gRK\eTV\gYi�hU�KM jnj f/�wIK\g[cI�\gjx�djn�dYi�e�^��]>tKTnYi\gRKM k tU���eM YbjZTS_¤jn¬U��YbWZM j ¢�H�IK\gj �^MXYbkdjx��j TV]3�djZMa]bTZIdM WxLN]>WZM WV]Utd�djZTTZM [cIdRd\e¬l�KMXj0¢

M������ @ � ² �d� � �E² ·U>¥� = ¶ S\H^Q �%��� ² ��� � �E² · H�H�IdM � »~MXk1À LNM0TnIK]Kk M jZTV\gLaYiTVMXj�TnIKM�vdY>WZY>LNM0TVMXWZjxtU� jV]>�^ul\gRKh~TZIdM�RK]>Rd�^\gRKM Y>W#LN\gRK\eLN\e¿ YbTV\e]>R vdWZ]i½td�^MXL �

����� � ��� ��� �� � �� �H�I�YiTN\ej f TZIdM�MXjVTZ\eLaYiTZ]>WNL#�djZTN�U\eM0�gk�TZIKM�jZLaYb�^�eM jZT�u�Yb�e�KM��¥]>WNTZIdM¨LNM kd\eY>R�]i��jZ¬l�dY>WVMXk�WVMXjV\gkd�dYb�ej[0]>Lavd�KTZM k��¥]>WaTnIKM~M RUTZ\eWZM3kdYiTnY jVM T0¢�Q¤T Tn�dWZR�j ]>�KT TnIdYiT TnIK\gj8LNM0TZId]lk©\gj u>M WZ��WV]Utd�djZT TV] � Yi�gjVMLaYbTZ[cIKMXjsYbjs��M �^�Ybjs]>�KTZ�^\eM Wnjsk��KM�TZ]�tdY>k~�^]K[ Yi�e\^¿XYiTV\e]>R/¢w¾¦RK�e\g�iM#TnIKM�»~_¯MXjVTZ\eLaYiTZ]>Wnj0fAIK]���M0ubMXW0f1TZIKM� »~M kAÀ�v�WV]UtK�^MXL [ YbRdRK]>T8t�M�WVMXkd�d[�MXk¡TV]�Y ��M0\eh>IUTVMXk��eM Y>jVTV_`jn¬l�dYbWZM jNvdWZ]>tK�eM L¨¢ªQ¤T8\ej�vdWV]UtdYbtd�^�\gLav�]UjZjZ\gtK�^MaTZ]~�wWV\eTVM kK]��wR Y�jVTnWZYb\^hUI>TV�¥]>WZ��Y>WZk¨�¥]UWZL��K�gY��¥]UW TZIKM � »~M kAÀ~MXjVTZ\eLaYiTZ]>W ¢aQ¤T�L#�djZT�t�MjZ]b�eubMXk�tU� Y~jVMXYbWn[cI�\gR TZIKM�jnvdY>[�M ]i��v1]>jnjV\gtK�eM M jZTV\gLNYbTVMXjxhbMXRKM WnYiTZM k�� WV]UL�TZIdM�kdYiTnYd¢¨Àl\eRd[0M8TZId\ejjnvdY>[�M�\gjsTZ]l]��gYbWZhbMbf1]>RK�e��Y�WnYbRdkK]ULN�^��[cIK]>jZM R~jn�dtdjZM0T¦]i�pkdYbTZY8[ YbR3t1M�Y>RdYi�e�l¿0MXk�¢sH�IKMNYi�ehb]>WZ\eTZIdL�wId\e[cI���M#IdY�u>M�\eLavK�eM LNMXR>TZM k a TZIKM�]>WZ\^h>\eR�Yi�/u>M WnjV\e]>R���Y>jskKMXjZ[ WV\gt1M k�\gR a��CI�YbRKh1fl[�M WZ\e[cIKM>f � �K]>RdhY>Rdk º1Y>�KhbMXWZY>ja� |>|#K�f [�M WZ\g[cIKMbf �CIdY>RKh�f � �d]>RKh3Y>Rdk º1Y>�KhbMXWZY>ja� |b|�K�f ��IdYbRKh1f^[�M WZ\e[cIdMbfº1Y>�KhbMXWZY>jY>Rdk � �K]URKha� |b|U�ec cC�¥]UWGWV]Utd�djZTV�e��M jZTV\gLaYiTV\gRKh TZIdM7� �dRdkdY>L�MXRUTZYi�dLaYbTZWZ\����¥]>�^�e]��wj�TZIKM�]URKM�jVTnWZ��[�TZ��WVMXk\gR a rw]U�djnjVM M �U�§YbR�k � M WZ]��~� |eYU}lfd¼�IdY>v�¢n��c)fdYbj7]U�KTV�e\gRKM k�t1M0�e]��x¢

¹¦\eubMXR v�]>\eRUT�[0]>WnWVMXjZv1]>R�kKM Rd[0M j � �%u S�� K S��� w � F Z K � � � K� �� f>��M�vdWZ]l[0M0MXkaTZIKMs�¥]b�e�^]���\gRKh�jZTVM v�j ��b¢�­®»~]>RUTZMw¼�Y>WV�e]�T¤�lv1M�TVMX[cIdRK\g¬l�KM�\ejG��jVMXk#TZ]�kdWZY��� WZY>RdkK]ULEjZ�dt�jZY>LNvd�^MXj]i� � F � kK\ �AMXWVMXR>Tv�]>\eRUT [0]>WnWVMXjZv1]>RdkdM Rd[0M j a WZM [ Yi�e�KTZIdYbT�}w\gjGTnIKMwLN\eRK\gL#��LyRl�dL�t�MXWTV]�kKM TVM WnLN\eRdM�TZIKM�MXvK\ev1]b�gYbWhbM ]>LNM0TnWV�lc)¢

{K¢7º�]UW�MXYb[cI®jn�dtdjnYbLavK�eMbfw\gRdkKM0�lMXk�tU� ��fs��M �djVM�TZIKM�TVMX[cIdRK\g¬U�dM kKMXjZ[ WV\gt1M k©\gR°ÀUMX[�T0¢nOd¢e�3TV][�]ULNv��KTVMxTnIKM � �dRdkdY>LNM RUTZYb�/LNYbTZWZ\�� ��� ¢ � M�LNY���IdY�ubM YiT�LN]>jZTwOajV]>�e�KTZ\^]URdj ¢

Od¢7º�]UWwM Y>[cI � � f1��M#[ YbR~kKM TVM WnLN\eRdM�TZIKM�L�MXkK\gYbR�]i�GTnIKM�jn¬l�dYbWZM k�WZM jZ\ekd��Yi�gj0f�kdM RK]>TVM k~tU� � � f��\^TnI�WVMXjZv1M [0T�TV]aTZIKMx�wId]b�eM jZM0T�]b��v1]b\gRUTw[�]>WnWZM jnv�]URdkKM R�[�M j fU\ ¢ Mb¢ef

� � F � ��� � � ���� ��� � � � � � H � � u%iS�� K � � iS �� w � � � u%iS �� K � Q � iS�� w P ��sMXWVM>f1TnIKM�kK\gjZTZYbR�[�M j¦t�M T¤��M M R3v1]b\gRUTZj¦YbRdk~MXvK\gv�]>�eY>W��^\gRKMXj¦Y>WVM��djZM k/fCt��KTs��M#[0Y>R~�djZM#]bTnIKM WM WnWV]UW7L�MXYbjn�dWZM j ¢

K�¢7rwM0TnYi\gR¨TZIKMxMXjVTZ\eLaYiTZM ��� �¥]UW7�wIK\g[cI � � \gjwLN\eRd\eLaYi��YbLN]>RKhaYi�e�� � � � j ¢

���������

Page 30: Determining the Epipolar Geometry and its Uncertainty: A Review · 2018-01-04 · ISSN 0249-6399 apport de recherche INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE

������������� ��������������� ��������� �!�"�#�$���%��& ���(')��+* ,���-.�/���0���� �+&�13254��/67���8 {U}

H�IdMN¬l�KMXjVTZ\^]UR�RK]�� \gj ����������������� ���������� ��®­yjn�dtdjnYbLavK�eM�\gj � h>]l]lk �#\���\^T [�]URdjV\gjZTZj�]i� �h>]l]lkN[�]UWZWZM jnv1]>RdkKMXRd[�MXj0¢�­sjnjZ��L�\gRKhxTnIdYiT TnIKM��wIK]>�^MwjZM0T ]b�C[0]>WnWVMXjZv1]>R�kKM Rd[0M jLaY���[0]>RUTZYb\eRN�dvaTV]�Y� WnYb[0TV\e]>R��x]i��]>�KTZ�^\eM Wnj0fKTnIKM�vdWZ]>tdY>tK\e�^\eT¤� TZIdYbT�YiT7�^MXYbjZT7]>RKM ]i��TZIKM -jn�dtdjnYbLavK�eM j�\gj7hb]l]Kk�\gj7hb\eubMXRtU�

� F Z�oH Z � u Z � � w � P�� �

a¯O�L�c

]7��WZM ¬l�K\gWZ\eRKhNTZI�YiT�� L#�djZT�t�M�RKMXYbW �>fK]>RKMx[0Y>R�kdM0TVMXWZLN\gRKM "�¥]UW7hb\eubM R�u�Yi�g�KM j�]b� � YbR�k � �

F !#"�$ u Z � � w!#"�$ H Z � u Z � � w � P �

QSR�]>�dW�\gLNvd�^MXL�MXRUTZYiTZ\^]UR�fK��M¦Ybjnjn�dLNM�� F � �&% YbRdk�WVMX¬U�d\eWZM�� F$� �� �dfUTZIl�dj F Z � �d¢�s]bTZM¦TnIdYiT

TnIKMxYi�ehb]UWV\eTZIdL,[ YbR¨t�M�jnv1M0M kdM k¨�dv�[�]>R�jV\gkKM WnYbtd�^�8tU��LNM Y>Rdj�]i�GvdYbWnYi�e�eM0��[�]ULNv��KTV\gRKh�fdt1M [ Yb�djZM TZIKMv�WV]K[�MXjZjZ\gRKh#�¥]UW7M Y>[cI�jn�dtdjnYbLavK�eMx[0YbR¨t1M�kK]>RKM \gRdkKMXv�MXRdkKM RUTZ�^�>¢­¦j RK]bTZM k�\eR a rs]>�djnjVM M �U��Y>Rdk � M WZ]��a� |eYU}�ccf�TZIKM � »~M kAÀm�(' -������-�&�\ejGv1]l]>W\gR#TnIKMwvdWZM jZM Rd[0M�]b�

¹�Y>�djnjV\gYbR�RK]b\gjVM>¢GH�IKM M/t8[�\eM Rd[0�8]i�GYNL�M TZIK]Kk¨\gj�kKM/r1RdM k�Ybj�TZIdM�WZYbTV\e]Nt1M0T¤��M0MXR�TnIKMx�^]���M jZT7Yb[cIK\eM�½u�Y>tK�^M#u�Y>WV\gYbR�[�M �¥]>WsTZIKM#M jZTV\gLNYbTVMXk~vdYbWnYbLNM0TZM WnjwYbR�k�TZIKM#Y>[�Tn�dYi��u�Y>WV\gYbR�[�M�vdWZ]�ul\ekKMXk�t>�¨TZIdM#hb\eubMXRLNM TZIK]Kk�¢�H�]~[�]ULNv1M R�jZYbTVMa�¥]>W TnIK\gj�kKM/rA[�\eM Rd[0�bf���MN� �dWZTZIKMXW�[0Y>WZWZ�~]>�dT�Y���M0\eh>IUTZM k��^MXYbjZTS_¤jZ¬l�dY>WVMXjv�WV]K[�MXkd�dWZMb¢GH�IKM$� ��� �s*�� *��0���('����"' 'e�/67����%�����M jZTV\gLNYbTVMx\gj7hb\eubMXR�tU�Wa`{i|�ccfdTZI�YiT7\ej f

� F Z�� � � � H Z � � � u � � w P � � � K

�wIdM WZM � � \gj�TZIdM�LN\eRK\gLaYi��L�MXkK\gYbR MXjVTZ\eLaYiTZM k tU� TZIKM � »~M kAÀA¢^]�YbjZM k ]UR �f��M�[ YbR YbjnjZ\^hUR Y��M0\eh>IUT��¥]>W7MXYb[cI�[�]>WnWZM jnv�]URdkKM R�[�M��

� � F � Z \^� � �� � u � � � � w �� ]bTnIKM WZ��\gjVM K�wIdM WZM

� �� F � � u�iS � K � iS �� w � � � u%iS � K � Q iS �� w �H�IdMp[�]UWZWZM jnv�]URdkKMXRd[�MXj/IdY�ul\eRdh � � F$� Y>WVMG]U�KTV�e\eM Wnj%YbR�kxjZIK]U�K�gkxRK]bT�t�Mp� �dWVTnIKM W%TZY>�iM R \gRUTV]sY>[0[0]>�dRUT0¢� MxTZIl�dj�[0]>Rdk��d[�TwY>R�Y>kdkK\eTV\e]>RdYb�/jVTZM v���K¢7rwM/rARKMxTZIKM � ��RdkdYbLNMXR>TnYi�%LaYbTZWZ\�� � tU��jZ]b�euU\gRKhNTZIKMx��M0\eh>IUTZM k��^MXYbjZTS_¤jZ¬l�dY>WVMXj�vdWZ]>td�^MXL �

���� � � � � �� �

H�IdM�� �dR�kdYbLNM RUTnYi�wLaYiTnWV\^��\ejaRK]�� WV]Utd�djZTV�e��YbRdk¡Yb[ [0�dWnYiTZM0�e� MXjVTZ\eLaYbTVM k�t�MX[0Y>�djVM¨]>�dTV�e\^MXWZjaIdY�u>Mt1M0MXR�kKM0TZM [�TZM k�YbR�k�kK\gjn[0YbWnkKMXk�tU�8TnIKM � »~MXk1À�LNM0TZId]lk/¢

v(vxw�y"z"{.z.|