Euclidian Shape and Motion from Multiple Perspective Views ... · Euclidian Shape and Motion from...

14
HAL Id: inria-00074254 https://hal.inria.fr/inria-00074254 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 destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés. Euclidian Shape and Motion from Multiple Perspective Views by Affne Iterations Stéphane Christy, Radu Horaud To cite this version: Stéphane Christy, Radu Horaud. Euclidian Shape and Motion from Multiple Perspective Views by Affne Iterations. RR-2421, INRIA. 1994. inria-00074254

Transcript of Euclidian Shape and Motion from Multiple Perspective Views ... · Euclidian Shape and Motion from...

Page 1: Euclidian Shape and Motion from Multiple Perspective Views ... · Euclidian Shape and Motion from Multiple Perspective Views by Affine Iterations Stéphane Christy, Radu Horaud To

HAL Id: inria-00074254https://hal.inria.fr/inria-00074254

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

Euclidian Shape and Motion from Multiple PerspectiveViews by Affine Iterations

Stéphane Christy, Radu Horaud

To cite this version:Stéphane Christy, Radu Horaud. Euclidian Shape and Motion from Multiple Perspective Views byAffine Iterations. RR-2421, INRIA. 1994. �inria-00074254�

Page 2: Euclidian Shape and Motion from Multiple Perspective Views ... · Euclidian Shape and Motion from Multiple Perspective Views by Affine Iterations Stéphane Christy, Radu Horaud To

ISS

N 0

249-

6399

ap por t de r ech er ch e

1994

INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE

Euclidian Shape and Motion from MultiplePerspective Views by Affine Iterations

Stephane Christy and Radu Horaud

N˚ 2421Decembre 94

PROGRAMME 4

Robotique,

image

et vision

Page 3: Euclidian Shape and Motion from Multiple Perspective Views ... · Euclidian Shape and Motion from Multiple Perspective Views by Affine Iterations Stéphane Christy, Radu Horaud To
Page 4: Euclidian Shape and Motion from Multiple Perspective Views ... · Euclidian Shape and Motion from Multiple Perspective Views by Affine Iterations Stéphane Christy, Radu Horaud To

������������� � �������� �� � ��������� ������ � ���!���"�#�"�$%�&�(')�*�+�(���",��.-/�"�&01'3254 617 ��� 89�:�;��<���"����' =

>)?A@B�C�DFEHGFBJI D�K�LNM ?!O EHGFPRQ5E9P:SUT�VHKWEXS(PY:Z\[W]^Z`_�ababc�dfe gh[^iX[Wj\kml�noc�pokqa5_r]Wc&c�jhstkmuNkv[^w

Y:Z\[^xNc�j�y#z|{<}h~���}N�:}N���o�F�;g��o�g�_r�o�X[^Z�j<��c<Z\c����oc"Z`�`�ocRw��t��d��t�+e ���c���c"a�ioZ�c��rd e �^� �X_r]Wc�u

�b�)���� �¡o¢^��£¤}�w�j��okmuh�X_r�Xc�Z;¥(c���c�u���Z\kqiHc¦_�abc�j��o[t��§¨[^Z;u©[Wªvstkvwo]bj��oc «:nX�!ªvkm�tc�_rw5Z\c"¬��[^w­u©j\Z�nX��j�kv[^w��®Z\[^i®ªvc�a¯¥hkqj\�°_��Xc�Z`uN�Hc���j\kqs�c��"_rabc�Z±_�ab[t��c�ª�it²bkqwX��Z\c"abc�wtj`_rªqªv² �Xc�Z�¬§¨[^Z�abkvwo]¦_rw°«³n­��ªvk´��c�_rwµZ\c���[WwXu©j�Z\nX�!j\kv[Ww5¥hkvj\�5c�kqj\�oc"Zh_ ¥(c�_�¶�[WZ&_��X_�Z`_ �Xc�Z`uN�Hc���j\kqs�c��_raµc�Z`_hab[t��c�ª¸·9¹ºkvj���Z�c�u©�Xc���j)j\[;[^j\�®c�Z:abc�j��o[t�ou�j\�X_�j���[Wab�onoj�c(uN�X_r�Xc�_�wX��ab[^j�kv[Ww§¨Z\[Wa1_µu©c�l�noc�wX�!c&[r§Fkva»_r]^c�u�¥hkvj���_b�"_rªvkqioZ`_�j\c��b�Xc�Z±u©�Xc���j�kvs�c���_raµc�Z`_tp�j\�®kmuhabc�j\�®[t���[^wts�c�Z�]^c�u&kvw¼_b§¨c�¥%kvj�c�Z`_�j\kv[WwXu�p)kmu&�![^ab�on®j`_rj�kv[WwowX_rªq²�c�½»��kqc�w�j"p�_rw­�#��[�c�u+wo[Wj�uNn�§¨¬§¨c�Z¾§¨Z�[^a¿j\�®c wo[^w¼ªvkqwoc�_rZ�wX_�j\noZ�c�[�§hj\�®c �oZ\[Wioªvc�a°·&¹ºkvj\��Z�c�u©�Xc��!j&j\[b§¸_^�!j\[^Z�kvÀ�_�j\kq[^w_rwX�XÁ�[^Z�_�½bw®c�¬Âkqw�sW_�Z\km_�w�jAabc�j\�®[t�ou�p^j��okmu:abc"j\�o[t� uN[^ªvs�c�uA§¨[WZ�j\�oc;u©kq]^w�~ÂZ\c"s�c�Z`u�_rªÂ�Ã_rab¬iokv]WnokvjN²�kvwµ_�s�c"Z\² uNkvab�®ªvc(¥|_�²Ä_rwX� �oZ\[�stkm��c�u³aÄnX�`�<ab[^Z�c�_^����n®Z`_rj�c¾Z�c���[WwXu©j�Z\nX��j�kv[WwXuZ\c�uNnoªvj±u�·�¹¼c5]^kvs�c»_#��c"j`_rkqªvc��*_^����[Wnowtj�[�§&j��oc5abc�j\�®[t�9p�_rwX_�ªv²®u©c�kvj`u �![^wts�c�Z\]Wc�wX��ciX_^uNc��Å[^wºwtnoabc"Z\km�"_rªh_�wX�Åc�Æt�Xc�Z\kqabc�wtj`_rª|��[^w­u©km�tc�Z`_�j\kv[WwXu�p�_�wX�Åj\c�uNj�kqj`u c�½»��kvc�w­��²iX[^j\��¥hkqj\��u©²twtj\�oc�j�km�&_�wX�bZ\c�_�ª)�o_rj±_�·Ç�È�É�Ê\Ë�ÌX �ÍÎ�^£ �Xc�Z±u©�Xc���j�kvs�c�p�¥(c�_�¶º�Hc"Z`u©�Xc��!j\kvs�cÏp&_rw­�Ð�X_rZ±_Ñ�Xc�Z±u©�Xc���j�kvs�c¼��_�abc�Z`_ab[t��c�ª´u�pt�r¬¸�%«:nX��ªvk´��c�_�w5_rwX�µ_�½µwoc&Z\c���[^wXuNj\Z�nX��j\kq[^w�p®u©�X_��Xc�_�wX�bab[^j�kv[Wwf§¨Z�[^aÒa�noªq¬j\kq�oªvc&stkvc"¥�u�·

ÓmÔ ÕÖØ×NÙ�Ú ÕÖ�ÛÎÜq×NÝÂÞ`ß

àNá�â�ãåä�æ9ç!èÂéhâWê`äÎëWìNì©í<ä¨î�ïrïWç�è¨ð¨ì\ñ&ë�ò ó¸ô�ç�õ�ã\öìNð�öì�÷<öìNèÂç!äÂïWê`ðÂãøê�ùåìNúWû

Unite de recherche INRIA Rhone-Alpes

46 avenue Felix Viallet, 38031 GRENOBLE Cedex 1 (France)

Telephone : (33) 76 57 47 77 – Telecopie : (33) 76 57 47 54

Page 5: Euclidian Shape and Motion from Multiple Perspective Views ... · Euclidian Shape and Motion from Multiple Perspective Views by Affine Iterations Stéphane Christy, Radu Horaud To

� ����:�+�F������� �� ����,Ä�+ �+�Ä�3�¾� �� �� '��A�(���(���������\����� 'A����+��� � ����:����� �������� '�����5���+���h� ��\�� �+'

�*�&�(')���+�F���",�+'���� ������;��<���"����' +7 ���+'

�� Èt������ Ⱥ£ �&_rwXuÄ��c�j¦_rZ�j\km�!ªvcÄwo[^nXuÄ�F�c��!Z\kvs�[WwXu+n®wocba �c�j��o[t��c5�tcbZ®�c�uN[^ªqnoj\kq[^w ��n�oZ\[Wioª��c"abc ��c Z�c���[WwXu©j�Z\nX��j�kv[Ww#c�n­��ªvk´��kvc�w®woc�_�s�c��+now¼ab[t���c�ªqc��Xc�Z±u©�Xc���j�k §|��cÄ��_�aº�c�Z`_c�w¼c�� c���j\n­_rwtj&nowoc Z�c���[^w­u©j\Z�nX��j�kv[^w¼c�nX�!ªvkm�tkvc�wow®c+kqwX��Z®�c�abc�wtj`_�ªvc<_�s�c��<now#aµ[t���c�ªvc¦��c��_ra �c�Z`_(�Xc�Z±u©�Xc���j�kvs�cΧ¸_�kvioªqc�[Wn��X_�Z`_h�Xc�Z`uN�Hc���j\kqs�c�·�Y:_�ZÎZ±_r�o�X[WZ\j��_¾��� _rnoj�Z\c�u�a �c�j��o[t��c�ul n®k���_rª´��noªqc�w�j�ªm_µu©j\Z�nX��j�noZ\c c�j�ªvc aµ[^nos�c�aµc�wtj��_b�X_�Z\j\kqZ<��� nowocÄut�c�l noc"wX��c ��� kva»_r]^c�u�p��c�j�j\c�a �c�j\�o[t�tc ��[Ww�s�c�Z�]^c�c�w¼l n®c�ªml�noc�uhkvj®�c�Z`_�j\kv[WwXu�p c�u©j&c!½5��_W��c���� n®w��H[Wkvwtj���c�stnoc��_rª´��noª´_rj\[WkvZ�c�pÎc�jÄwoc�u©[Wn�� Z\c��X_^ub�tc ª´_�wX_rj�noZ\c�wo[^w ªvkqwh�c�_rkqZ\c5�tnÅ�oZ�[^ioª��c�abcbj�Z`_rkqj®�c�·Y:_rZFZ`_��o�X[^Z\j�_���c�u�abc�j\�®[t��c�u:j\c�ªqªvc�uAl noc;ªm_¾§¸_^��j�[^Z�kmu\_�j\kq[^w5[^nbªqc�u:kvwtsW_rZ\k´_rwtj`u�_�½µwoc�u!pwo[^j�Z\c�a �c"j\�o[t��c�Z®�c�uN[^nojµªvc��®Z\[^i®ª��c�abc���c�ª�� _raÄiokv]Wnokvj®�c5��c¼u©kq]^woc¼��� nowoc5§¸_ ���[Wwºj\Z!�c�uu©kqab�oªvchc"j³§¨[WnoZ\w®kvjF��c�u�Zo�c�uNnoªvj±_rj`u:�oªqnXu��®Zo�c���k´u�·���[^nXuF�(�c���Z�kvs�[^w­u³ª´_&wo[^nos�c"ªvªvcha �c�j\�o[t�tcc�w¼�F�c�j±_rkqª�p _rwX_�ªv²®u©[^w­u�u\_5��[Ww�s�c"Z\]^c"wX��c�u©noZ<ª´_�iX_^uNcb�tcÄ��[^wXuNkm�F�c"Z`_rj�kv[WwXu¾wtnoaº�c�Z�kml�noc�uc�jhc�Æt�h�c�Z�kvabc�wtj`_�ªvc�u)c�jhwo[^n­u�j�c�u©j�[^wXu�u©[Ww�c�½»��_^�!kvj®�c�_�s�c��&��c�u���[^w®wh�c�c�uhuN²�wtj\�|�c"j\kml�noc�uc�jhZ®�c"c�ªvªqc�u�·"%Ì­����Ê`¢$#% ÈÅ£%�Hc"Z`u©�Xc��!j\kvs�cÏp9�Xc�Z`uN�Hc���j\kqs�c+§¸_rkqioªvcÏp9�X_rZ±_ �Xc�Z±u©�Xc���j�kvs�c�pXZ\c��![^wXuNj\Z\n­��j\kq[^w�^� _�½µwoc<c�jhc�nX��ªqkm��kqc�wowocÏp aµ[^nos�c�aµc�wtj:c�j�uNj\Z\n­��j\noZ�c�pXut�c�l noc"wX��c<��� kva5_�]^c�u!·

Page 6: Euclidian Shape and Motion from Multiple Perspective Views ... · Euclidian Shape and Motion from Multiple Perspective Views by Affine Iterations Stéphane Christy, Radu Horaud To

������������� ������������� ����������� ��! "��#$�%�&�'���(�)�*�,+-�� /.0�)�"�����12�435�6��78. 9

: 8���A��� ����F���"��� �� 2�f�<; ��������

=(�oc��oZ\[Wioªvc�a [�§ �![^ab�on®j\kvw®]¼�r¬¸�¿u©�X_��Hc¼_�wX�Åab[Wj\kv[Ww�§¨Z\[^a _Jªq[^wo] u©c�l�noc�wX�!c [�§kva»_r]^c�uf�X_Wu Z�c���c�kqs�c�� _ ªq[^jb[�§Ä_�j\j\c"w�j�kv[^wŧ¨[WZ5j\�oc¼ªm_Wu©j�§¨c�¥ ²�c�_rZ±u�·FY:Z\c�stkv[WnXu5_r�o¬�oZ�[�_^�`�oc�uF_rj�j\c�ab�®j\kvw®]+j�[ uN[^ªvs�c�j\�®kmu��®Z\[^i®ªvc�a §¸_rªqªHkqw�j�[ uNc�s�c�Z`_�ªX��_�j\c�]W[^Z\kqc�u�pt¥h�oc�j��oc�Zj\�®c���_rabc"Z`_hkmu:��_�ªvkqioZ`_�j\c���[WZ�w®[^j�pÏ_rwX�HÁ"[^ZF¥h�oc�j\�®c�Z�_&�oZ\[WxNc���j\kqs�c�[^ZF_rwµ_�½µwochab[t��c"ªkmu�iXc�kvwo]�nXu©c��9·�¹ºkvj\�»_���_�ªvkvi®Z`_rj�c��b��_�abc�Z±_¾[^woc<a5_�²¦��[^ab�®noj\c�«:nX��ªqkm��k´_rw5uN�X_r�Xc&n®�j\[Ä_Äu\�"_rªvc�§¸_^��j�[^Z�nXu©kqwo] c�kvj��oc�Z|_f�Xc�Z±u©�Xc���j�kvs�c&ab[t�tc�ª?> d�@¸pA> ��d�@�pB> �C@�pA> ���C@�p�[WZ&_�ªqkvwoc�_rZab[t��c"ªD>(E2@¸p5> �!E2@�p5> �GFC@�p5> �"��@�p5> ���2@¸p5> �"�2@�p5> �^�G@¸·o¹ºkvj\��_�w�nowX��_�ªvkvi®Z`_rj�c��5��_raµc�Z`_�j��oc�Z�c�¬��[�s�c�Z�c��bu©�­_r�Xc�k´u¾��c�HXwoc��bno�5j\[µ_ �oZ\[^xNc���j\kqs�c¾j�Z`_rw­uا¨[^Z�a5_rj�kv[WwI>KJ!@�pB> �2@¸p5>(LC@�pA> �GL�@¸p�[^Zno��j\[Ä_rw°_�½µwoc�j�Z`_�wXuا¨[WZ\a5_�j\kv[WwM>KJG@¸p5>(LC@�·Hz�woc���_�w�j��oc�Z\c!§¨[^Z\c�_W�o��Z\c�u\uhj\�oc<�oZ�[^ioªqc�a[r§)c"kvj\�®c�Z�«:nX��ªqkm��k´_rw�p�_�½bw®c�p�[^Z(�®Z\[^xNc��!j\kvs�c|u©�X_��Xc+Z�c���[WwXu©j�Z\nX��j�kv[Ww�·�}�wbj\�®kmu:�X_r�Xc�ZF¥(c_rZ�c�kqw�j�c�Z\c�uNj\c��bkqw�«:nX��ªqkm��k´_rw�uN�X_r�Xc Z�c���[^w­u©j\Z�nX��j�kv[^w�¥hkvj��¼_���_�ªvkqioZ`_�j\c�����_rabc"Z`_�·X}�wj\�­_rjF��_^uNc�p�[^woc�a5_�²�n­u©c�c�kvj��oc�Z�_&�Xc�Z`uN�Xc���j\kqs�c���_raµc�Z`_¾aµ[t��c�ªo[WZ(_�w5_�½µwoc�_��o�oZ\[�Ætkv¬a5_�j\kv[Ww��µ[^Z\j��o[^]WZ`_��o�okm�<�oZ\[WxNc���j\kq[^w�pt¥(c�_�¶b�Xc�Z`uN�Xc���j\kqs�c�p�[WZh�X_rZ±_��Xc�Z`uN�Xc���j\kqs�c�·

=(�oc��Hc"Z`u©�Xc��!j\kvs�cFab[t��c�ªt�X_Wu�_Wu\uN[t��km_�j\c���¥hkqj\� kqj�p^kqw ]^c�woc"Z`_rª¸prw®[^wbªqkvwoc�_�ZÎZ�c���[WwXu©¬j\Z�nX��j�kv[^w�j\c��`�owok´l n®c�u�·N=(�ok´u�wX_�j\noZ±_rªqªv²5ªvc�_W�ouhj\[Äwo[^wo¬¨ªvkqwoc�_�Z¾abkqwokvaµk À�_rj\kq[^wÄabc�j��o[t�ou¥h�ok´����Z\c�l�nokvZ�c<u©[Wabc+§¨[WZ\a3[r§³kvwokqj\km_�ªvkqÀ�_rj�kv[^wM> d�@�p5> ��dO@�p?> �2@¸p5> �"�2@�p?> �C@�p5>PL2@�p5> �GL�@¸·X}¸§³j\�ockvw®kvj\k´_rªRQ\]^n®c�u\u"S k´u�j\[�[#§¸_�Z`_�¥h_�²�§¨Z\[Wa j\�®c�j\Z�noc�u©[Wªvnoj�kv[^w j\�oc"wÅj\�®c aµkvwokqabk À�_rj�kv[^w�oZ�[t��c�u\u�kmu¾c�kqj\�oc"Z¾s�c�Z�²5u©ªq[�¥ [WZ<kvj;��[^wts�c�Z\]Wc�uhj\[5_�¥hZ�[^wo]�uN[^ªvn®j\kv[Ww�·���½µwoc�ab[t��c�ª´uªvc�_^�9pXkvw�]Wc�woc�Z±_rª�pXj\[µªvkvw®c�_rZ�Z\c�uN[^ªqnoj\kq[^w�abc�j��o[t�ouR>(EC@�p?> �GFC@�p?> �"�2@�p?> �"��@�p?> �W��@�pXionoj;j\�ocu©[Wªvnoj�kv[Ww�k´u:��c�HXw®c���[^woªq² no� j\[�_<u©kq]^w�~ÂZ�c�s�c�Z`u�_rª¨��_�a�iokq]^nokqjN²�·�y�[^Z�c�[�s�c�Z�p�iX[^j��bj\�®c�u©cjN¥([�u©[Wªvnoj�kv[WwXu�_rZ\cFxNnXu©jF_rwµ_r�®�oZ\[�Ætkva5_�j\kv[WwXu:[r§ j\�oc�j\Z�noc�u©[Wªvnoj�kv[^w ·��o[WZ(c�Æ®_rab�®ªvc�p��Îkq¬]^n®Z\c5� uN�o[�¥�u�§¨[^noZ�«:nX��ªqkm��k´_rw�u©�­_r�Xc�u�·5=(�oc,HXZ`uNj&[^w®c5~Âj\[W�o¬¨ªvc�§¨j!�hkmuhj\�oc j��oc�[WZ\c�j�km��_�ªu©�­_r�Xc�·T=(�oc°u©c��![^wX� [^woc ~¨j\[^�®¬ÂZ�kv]^�tj��¦kmu j\�oc�Z\c���[WwXu©j�Z\nX�!j\c�� u©�­_r�Xc#n­u©kvw®]�_#�Xc�Z±u©¬�Xc���j�kvs�c �"_rabc�Z±_faµ[t��c�ªA_rwX��j\�oc aµc�j\�o[t�5[^noj�ªvkqwoc���kvw�j��okmu��X_r�Xc�Z"·U=(�®c j\�okqZ`��_rw­�§¨[^n®Z\j\�Ä[^w®c�u¾~¨iH[Wj\j�[^ab¬¨ªvc�§¨j³_rwX��iX[Wj\j\[Wab¬¨Z\kv]W��j!�A_rZ\c�j\�®chZ\c���[WwXu©j�Z\nX�!j\c��Äu©�X_��Xc�u�nXuNkvwo]_ ¥(c�_�¶b�Xc�Z`uN�Hc���j\kqs�c<��_raµc�Z`_<ab[t��c�ª¸·

z�woc&¥h_�²bj�[Ä��[^aÄiokqwochj\�oc;�Hc"Z`u©�Xc��!j\kvs�c�_rwX�5_�½µwoc&aµ[t��c�ªmuF��[^n®ªm��iXc<j\[ nXuNc<j\�ocªvkqwoc�_�Z ~�_�½bwocW�&uN[^ªqnoj\kq[^w¼kvw¼[^Z`�tc�Z<j\[5kvwokqj\k´_rªvkqÀ�c�j��ocÄwo[^wo¬¨ªvkqwoc�_�Z+aµkvwokqa�kqÀ�_�j\kv[Ww��oZ�[^¬��c�u�u�_Wu\u©[t�!km_rj�c���¥hkvj����Xc�Z`uN�Hc���j\kqs�c�·5V¾[�¥(c�s�c�Z"p9j\�®c�Z\cµ_rZ�c5u©c�s�c"Z`_rªF��Z±_�¥hiX_^�`¶ou�¥hkqj\�u©n­��� _rw _r�o�®Z\[�_W��� ·(�ÎkqZ`uNj�p:uNnX�`�Å_�Z\c�u©[^ªqnoj\kq[^w j\c����ow®kml�noc5��[�c�u wo[Wj j`_r¶�c5kqw�j�[¼_^��¬��[Wnow�jhj��oc<uNkvab�oªqchªvkqwo¶�j��X_rjhc"Æ�k´u©j`u�iXc�jN¥(c�c�wµj\�oc&�Xc�Z±u©�Xc���j�kvs�c�ab[t��c�ª _rwX�bkqj`u�ªqkvwoc�_rZ_r�®�oZ\[�Ætkva5_�j\kv[WwXu�·9�Ïc���[^w­�9p9j\�®c�Z\c�kmuhwo[ba5_�j\�oc�a»_rj\k´��_rª c�stkm��c�w­��c&j\�­_rj<_�w®[^wo¬¨ªvkqwoc�_rZªvc�_^u©j�¬�u�l n­_rZ\c�uÎabkvw®kvaÄkvÀ"_rj\kq[^w&aµc�j\�o[t�<kmuTQ\¥Fc�ªvªWS�kqwokvj�km_rªqkvÀ�c��¾i�²�_�u©[Wªvnoj�kv[^w j��X_rjAkmu�[^io¬

X5X íZY![]\![C^

Page 7: Euclidian Shape and Motion from Multiple Perspective Views ... · Euclidian Shape and Motion from Multiple Perspective Views by Affine Iterations Stéphane Christy, Radu Horaud To

d �)�0� ������ �� � �� �� .���� �� ��� ������,�� "����

�Îkq]^noZ�c<���=(�ok´uUHX]^noZ�c&uN�o[�¥�u�_�j��oc�[WZ\c�j�km��_�ªo��¬��RuN�X_r�Xc�~Âj�[^�o¬¨ªvc�§¨j!�)_rw­�fj��oZ\c�chZ�c���[WwXu©¬j\Z�nX��j�kv[^w­u([r§Aj\�ok´u(u©�­_r�Xc+§¨Z�[^a ��� s�kqc�¥�u�·N=(�oc HXZ±u©jhZ\c���[^wXuNj\Z�nX��j\kq[^w�~¨j\[W�o¬ÂZ�kv]W��j!��¥h_^u[^i®j`_rkqwoc��5nXu©kqwo]5_ �Hc"Z`u©�Xc��!j\kvs�c���_�abc�Z±_�ab[t��c�ª)_�wX� j��oc�abc"j\�o[t�5��c�u���Z\kqiXc�� kqw�j\�®kmu�X_��Hc"Z�·�=(�oc;u©c���[^wX��Z�c���[WwXu©j�Z\nX��j�kv[Ww5~ÂiX[^j�j\[^aµ¬Âªqc�§¨j��)_�wX� kvj`u)Z\c�s�c�Z±u\_�ª9~¨iH[Wj\j�[^ab¬¨Z\kq]^�tj��¥(c"Z\cÄ[^ioj±_rkvw®c��#nXuNkvwo]¼_5¥(c�_�¶#�Xc�Z±u©�Xc���j�kvs�cÄ��_�abc�Z`_bab[t�tc�ª�_rw­� j��oc�§¸_W��j\[WZ\kvÀ�_rj\kq[^wabc�j��o[t��[�§D=Ã[Wa5_^uNk� �<_rw­_^��c�·

j`_�kvwoc��fªqkvwoc�_�Z\ªq²�·�=(�okqZ`�9p�j\�®c�Z\c;_rZ�c¾jN¥([¦u©[Wªvnoj�kv[^w­u(_^u�u©[t��k´_rj�c���¥hkvj���j��oc<_�½bwoc�ab[t��c"ª_rw­� kvjhk´u(wo[^j��!ªvc�_�Zh¥h�okm�`�b[^w®c�j\[Ä�`�o[�[Ïu©c�·

=(�oc��Hc"Z`u©�Xc��!j\kvs�ch�®Z\[^xNc��!j\kv[WwÄ��_rwµiHc�ab[t��c"ªvªvc��+it² _+�®Z\[^xNc��!j\kvs�cFj\Z±_rwXu�§¨[^Z\a»_rj\kq[^w§¨Z\[Wa1j��ocÄ�r¬¸� �oZ�[^xNc���j�kvs�c�u©�X_W��c j\[5j��ocÄ�r¬¸� �oZ�[^xNc���j�kvs�c<�oªm_�woc�·9¹¼c�_�¶��Xc�Z`uN�Xc���j\kqs�c_rw­� �X_�Z`_��Xc�Z±u©�Xc���j�kvs�c�_rZ�c�j\�®c+aµ[�u©j��![^abab[Ww5_�½µwoc¦_r�o�oZ�[�Æ�kqa5_rj�kv[^w­u([r§��Xc�Z±u©�Xc���¬j\kqs�c�·�¹#c�_�¶Å�Xc�Z±u©�Xc���j�kvs�c�a»_�² ¥Fc�ªvª�iXc#stkvc�¥Fc��Å_Wu5_ À"c�Z\[W¬Â[WZ`��c�Z»_r�o�oZ�[�Æ�kqa5_rj�kv[^w�����Ï~\�����`��� �W·9Y:_�Z`_��Xc�Z±u©�Xc���j�kvs�c > ��@Îk´u¾_ HXZ±u©j&[WZ`��c�Z;_r�®�oZ\[�Ætkva5_�j\kv[Ww5[r§�§¨noªvª��Xc�Z±u©¬�Xc���j�kvs�c��:����~������`���¿�����!·)ghc���c�wtj\ªq²�p9kqw >PF2@h_5abc�j��o[t�J�­_^u�iXc�c�w��®Z\[^�X[Ïu©c��#§¨[^Z��c�j�c�Z\aµkvwokqwo]�j��oc �X[�u©c [�§|_ ��¬�� uN�X_r�XcÄ¥hkvj���Z\c�u©�Xc���j&j�[5_buNkvwo]Wªvc�stkvc"¥Ri�²5kqj\c�Z±_rj\kq¬s�c�ªq²bkvab�®Z\[�s�kqwo] j\�®c �H[Ïu©cÄ��[Wab�onoj�c�� ¥hkqj\�¼_�¥Fc�_r¶5�Xc�Z±u©�Xc���j�kvs�c �"_rabc�Z±_faµ[t��c�ª�j\[��[Ww�s�c�Z�]^c�pÏ_rjhj\�®c¾ªqkvabkqj�p�j\[Ä_+�X[Ïu©c<c�u©j�kva5_�j\kq[^w nXuNkvwo]¦_��Hc"Z`u©�Xc��!j\kvs�c;��_�abc�Z`_�ab[t��c�ª¸·�hj&[^noZ<¶two[�¥hªvc��t]^c�p9j��ocÄabc�j\�®[t����kvj�c���_�iH[�s�c�p�k¸· c�·qp5>(F2@Akmu<_rab[Wwo]5[^wocÄ[r§|j\�oc H­Z`u©j��[Wab�onoj±_rj\kq[^wX_�ªh�X_rZ±_^��kq]^a5u�j��X_rjĪvkqwo¶�ªqkvwoc�_�Z j\c��`�owok´l n®c�u�~¸_^u\uN[t��km_�j\c�� ¥hkvj��º_�½µwoc

�! UX"� ÷

Page 8: Euclidian Shape and Motion from Multiple Perspective Views ... · Euclidian Shape and Motion from Multiple Perspective Views by Affine Iterations Stéphane Christy, Radu Horaud To

������������� ������������� ����������� ��! "��#$�%�&�'���(�)�*�,+-�� /.0�)�"�����12�435�6��78. �

��_�abc�Z`_5ab[t��c�ª´u��<¥hkqj\� _J�Xc�Z±u©�Xc���j�kvs�cbaµ[t��c�ª�·A}�w > �2@�_�wÅc�Ætj\c�w­u©kv[Ww [r§<j��okmu��X_�Z`_�¬��kq]^a j�[<�X_rZ±_+�Xc�Z`uN�Xc���j\kqs�c(k´uÎ�oZ\[W�X[�u©c��9·�=(�oc�_�noj\�®[^Z`uFu©�o[�¥ j\�X_�j�j��oc �*� �� "��� �1C� ���� ��)�� .0���"��� �12�)�X[�uNc&_�ªv]^[WZ\kqj\�oa �X_Wu�iXc�j�j\c�Z��![^wts�c�Z\]Wc�wX��cF�oZ\[W�Hc"Z\j\kqc�u:j\�X_�wbj\�®cR�*� �� "��� �1C�7 �"���,�)�� .0�)�"��� �1C��[Wwoc�·

}�wÐj��okmu5�X_r�Xc�Z�¥(c*��c�u\�!Z\kviXc¼_ woc�¥ «:nX��ªqkm��k´_rwºZ\c���[^wXuNj\Z�nX��j\kq[^w¤abc�j��o[t� j��X_rja5_�¶�c�u�nXuNc<[r§³_�½µwoc<Z\c���[WwXu©j�Z\nX�!j\kv[Ww5kvw�_rw�kvj\c"Z`_rj�kvs�c�a5_rw®woc�ZhuNnX�`� j��X_rj�j\�®kmu(kvj�c�Z`_�¬j\kqs�c(�oZ�[t��c�u�u��![^wts�c�Z\]Wc�u�pW_rj:j��oc�ªvkvaµkvj�p!j\[<_<uNc�j:[r§Ã�r¬¸�Ы:nX�!ªvkm�tkm_rw¦u©�X_��Xc<_rwX� ab[Wj\kv[Ww�X_�Z`_raµc�j\c�Z±u¾j\�­_rj�_rZ�cb��[WwXu©k´u©j�c�w�j<¥hkqj\�¼_ �Xc�Z±u©�Xc���j�kvs�c�ab[t��c�ª¸·A=(�ocÄwo[�s�c�ªvjN²5[r§|j\�ocabc�j��o[t� j��X_rj<¥Fc �oZ\[W�H[Ïu©cÄkmu¾jN¥F[r§¨[^ª´� �:~¨kÂ�hkqj&c�Ætj\c�wX�®u�j��oc kvj�c�Z`_�j\kvs�c��X[�u©cµ��c�j�c�Z\aµkv¬wX_�j\kv[Ww5_rªq]^[WZ\kvj��oa5u���c�u���Z\kqiXc��fkqw >PF2@ _rw­��kvw�> �2@Xj\[¦��c�_�ªH¥hkqj\�Äj\�oc;�oZ\[Wioªvc�a [�§Au©�X_��Xc_rw­�Jab[Wj\kq[^w�§¨Z\[Wa¿a�n®ªvj\kq�oªvc<stkvc�¥�u;_rwX� ~ÂkqkÂ�hkqj+k´u&_b]^c�w®c�Z`_�ªvkvÀ�_rj\kq[^w¼j\[5�Xc�Z`uN�Xc���j\kqs�c[r§)j��och§¸_^��j�[^Z\kqÀ�_rj�kv[Wwbabc�j��o[t�ou-> �!F2@�p > �^��@ _�wX��[r§)j��oc;_�½µwoc�¬¨kvwts^_�Z\k´_rwtj�aµc�j\�o[t�®u > ���C@�·y�[^Z�ch�oZ\c���kmuNc�ªv²�p!j\�ocR��� ����0�*� �] �����12� ]�"� �� &.��� ��N������� �abc"j\�o[t�<j\�X_�j�¥Fc(�oZ�[^�X[�uNc¾�®c�Z\c�X_Wu¾_ w�n®a�iXc�Z�[r§³kvwtj\c�Z�c�u©j�kvwo]<§¨c�_rj�noZ\c�u �

� }�j5uN[^ªvs�c�u�j\�oc¼uNkv]Ww ~¨[^Z5Z\c�s�c"Z`u\_�ªÂ�¦_raÄiokv]WnokvjN²�j��X_rj5k´u�kvwo�®c�Z\c�wtjÄ¥hkvj\�U_�½µwocZ�c���[^w­u©j\Z�nX��j�kv[^w�

� }�j(k´uΧ¸_^u©jhiXc��"_rnXuNc&kvj|��[^wts�c�Z\]Wc�u:kvwµ_+§¨c�¥ºkvj�c�Z`_�j\kq[^wXu�~��+j�[ E�kqj\c�Z±_rj\kq[^wXu`�`p�c�_W���kqj\c�Z±_rj\kq[^w5kvwts�[^ªqs�kqwo]�u©kqab�oªvc�ªvkqwoc�_�Z�_rªv]Wc�ioZ±_���[Wab�onoj±_rj�kv[^w­u�

� ¹¼c�u©�o[�¥Rj��X_rj�j\�oc�l�nX_rªqkvjN²Ä[r§Aj\�oc�«:nX��ªqkm��k´_rw5Z\c��![^wXuNj\Z\n­��j\kq[^w5[^ioj±_rkqwoc�� ¥hkqj\�[WnoZ¾aµc�j\�o[t�bk´u([^woªq² ¥Fc�_r¶tªv²µkvw��Xn®c�wX��c��Äi�²5�"_rabc�Z±_ ��_rªqkvioZ±_rj�kv[^w5c�Z�Z\[WZ`u�· =(�oc[Wwoªv²5kvwtj\Z�kvwXuNkm����_rabc"Z`_��X_�Z`_raµc�j\c�Z�j\�­_rj&�X_Wu&_b��Z�nX��k´_rª)c�� c���j�[^w�j\�®cÄl nX_�ªvkqj©²[�§hj\�®c�Z�c���[^w­u©j\Z�nX��j�kv[^w¼kmu&j��oc�Z±_rj�kv[�iXc�jN¥Fc�c�w�j\�®c��o[WZ\kqÀ�[^wtj`_�ª��®kvÆtc�ª�u©kqÀ�cÄ_rw­�s�c�Z�j\k´��_rªX�okqÆ�c�ª u©kqÀ�c&�ÄZ`_rj�kv[Ä¥h�okm�`�bk´u(¶�wo[�¥hw�j\[ iXc�s�c�Z�²buNj`_�ioªvc > �CJG@ �

� }�j�_rªqªv[�¥�u+j��ocbnXuNcb[r§�c�kqj\�oc"Z+¥Fc�_r¶�[WZ��X_�Z`_��Xc�Z±u©�Xc���j�kvs�cµ��_rabc"Z`_�ab[t�tc�ªF_r�o¬�oZ�[�Æ�kqa5_rj�kv[^w­u�¥h�ok´���5_�Z\c�nXu©c��bkqj\c�Z±_rj\kqs�c�ªv²Ïp�_rwX�

� }�j���_�w iXc���[Wa�iokqwoc�� ¥hkvj���_�ªvab[Ïu©j|_rwt²5_�½µwoc�u©�X_��Xc�_�wX� aµ[^j\kq[^w5_�ªv]W[^Z\kqj\�oa°·}�wº�X_�Z\j\k´��noª´_rZb¥Fc�uN�o[�¥1�o[�¥1[WnoZ5abc�j��o[t�Å��_�wÐiXc*��[^aÄiokqwoc���c�kvj��oc�Zb¥hkqj\�§¸_W��j\[WZ\kvÀ�_rj\kq[^w¤abc�j��o[t�ou > �GF�@�p�> �^��@<[^Z5¥hkvj��R_�½bwoc"¬Âkqw�sW_rZ�km_�w�j5abc�j��o[t�ou > ���C@�p> ���2@¸·

����� �������������������! #"$�&%' #�(�

=(�ocbZ�c�a5_�kvwX��c"Z&[r§;j\�ok´u+�X_��Xc�Z�k´u+[^Z�]�_�wokvÀ�c��Ñ_Wu�§¨[Wªvªv[�¥�u!·���c��!j\kv[Ww ���tc�u\��Z�kviXc�u<j\�ocZ\c"ªm_rj�kv[WwXu©�okq�biXc�jN¥(c�c"wf§¨noªqª�p��X_�Z`_�p®_rwX�b¥Fc�_r¶Ä�Xc�Z`uN�Hc���j\kqs�c�_rwX�bc�uNj`_ri®ªvkmuN�oc�u�j\�oc&Z�c�ªm_�¬j\kq[^wXuN�okv�­u�iXc�jN¥(c"c�w5j\�oc��Hc"Z`u©�Xc��!j\kvs�c;�oZ\[^xNc���j\kq[^w5[r§|_Ä�r¬¸� �X[^kqw�j;_rwX��kqj`u(¥Fc�_r¶�_rw­��X_�Z`_h�Hc"Z`u©�Xc��!j\kvs�cA�oZ\[^xNc���j\kq[^wXu!·r��c��!j\kv[Ww��¾��c�u\��Z�kviXc�uH�®[r¥*j\[��Hc"Z©§¨[^Z�aRZ\c���[^wXuNj\Z�nX��j\kq[^w¥hkvj��Ä_¾�Xc�Z`uN�Xc���j\kqs�ch��_�abc�Z±_(ab[t��c�ªti�²<kqj\c�Z±_rj\kqwo]&c�kqj\�oc"Z:_¾¥Fc�_r¶��Xc�Z`uN�Hc���j\kqs�c�Z�c���[WwXu©¬j\Z�nX��j�kv[^wÄ[^ZF_&�X_�Z`_&�Xc�Z`uN�Hc���j\kqs�chZ\c���[WwXu©j�Z\nX�!j\kv[Wwb_�ªv]W[^Z\kqj\�oa°· ��c���j\kq[^w d<[^noj�ªvkqwoc�u:jN¥([_�½µwoc�Z�c���[^w­u©j\Z�nX��j�kv[^w¤_�ªv]W[^Z\kqj\�oa»ub_rwX�ºj\�®c¼��[^wts�c�Z`uNkv[WwÅ[r§f_�½µwoc�Z\c���[^wXuNj\Z�nX��j\kq[^wkvwtj\[¦«:nX��ªvk´��km_�wbZ\c���[^wXuNj\Z�nX��j\kq[^w5nXuNkvwo] c�kqj\�oc"Z(¥(c�_r¶Ä[^Zh�X_�Z`_ �Hc"Z`u©�Xc��!j\kvs�cÏ·­�Ïc���j\kq[^w E��c�u���Z\kqiXc�u¾�o[�¥%j\[5uN[^ªqs�c�§¨[^Z&j��oc Z\c�s�c"Z`u\_�ª)_raÄiokv]WnokvjN²5_Wu\uN[t��km_�j\c��9p�kqw�]^c"woc�Z`_�ª�p9¥hkqj\�_rw*_�½µwoc¦��_rabc"Z`_�·���c��!j\kv[Ww Fbc�Æt�oªm_�kvwXuh¥h�t²�j\�®c abc�j\�®[t����c�u���Z\kqiXc�� kqw�j\�ok´u¾�X_r�Xc�Z

X5X íZY![]\![C^

Page 9: Euclidian Shape and Motion from Multiple Perspective Views ... · Euclidian Shape and Motion from Multiple Perspective Views by Affine Iterations Stéphane Christy, Radu Horaud To

F �)�0� ������ �� � �� �� .���� �� ��� ������,�� "����

kmu�w®[^jÄu©c"wXu©kqj\kvs�cÄj\[¼�"_rabc�Z±_5��_rªqkvioZ±_rj\kq[^w c�Z�Z\[^Z±u _rwX�Å��c���j\kq[^w J5�oZ�[�s�k´��c�u�_�w _rwX_�¬ªv²®u©k´u&[r§hj��ocµ��[^wts�c�Z\]Wc�wX��c [�§hj��ocb�Xc�Z`uN�Xc���j\kqs�c Z\c��![^wXuNj\Z\n­��j\kq[^w*_rªv]W[^Z�kvj\�®a�·9�)kvwX_�ªvªv²Ïp��c��!j\kv[Ww4Lf]Wkvs�c�uhuN[^abc<Z\c�u©noªqj`u([^i®j`_rkqwoc��5iH[Wj\��¥hkvj\�°u©kqa�noª´_rj�c���_�wX��Z\c�_rª)�o_�j`_�p _rw­���c��!j\kv[Ww5�+�®Z\[�s�k´��c�uF_ u©�o[WZ\j���k´u\��nXu�u©kq[^w5_�wX�5_+�oZ�[^�X[�u�_rª�§¨[^Z(§¨noj�noZ\c&¥F[^Z�¶b_�ªv[^w®] j\�ocªvkqwoc�uF��c�s�c�ªq[^�o�Xc��bkvw5j��okmu(�X_��Xc�Z�·

� � �&�� �5��+��'

�9c�j<nXu&�![^wXuNkm��c�Z�_»�®kvw��o[WªvcÄ��_�abc�Z`_ ab[t��c"ª�·X¹#cÄ��c"wo[^j�c i�²����F_5�r¬¸� �X[Wkvwtj+¥hkqj\�«:nX��ªqkm��k´_rw5��[�[WZ`��kqwX_rj�c�u����Np��¸pX_rwX� ���)kvw�_+§¨Z`_�abc&j\�­_rj�kmuh_rj�j`_W���oc��»j�[ j\�oc<[WioxNc���j�µj\�oc�[^ioxNc���j(§¨Z±_rabcÏ·�=(�oc�[^Z\kq]^kqw�[r§³j\�ok´u|§¨Z±_rabc<a5_�²Ä¥(c�ªqª�iHc�j\�oc<[WioxNc���j��H[Wkvwtj�����·�¾w�[WioxNc���j��H[Wkvwtj����h�oZ\[WxNc���j`u&[Ww�j�[�j\�ocbkqa5_r]Wc�kqw����h¥hkqj\��kqa5_r]Wcb�![�[^Z`�tkvwX_�j\c�u����_rw­� � � _�wX� ¥Fc&�X_�s�cÄ~�� � kmuhj\�®c+s�c���j\[WZ ������ � � � §¨Z\[Wa¯�X[^kvwtj�� � j\[ �X[^kqw�j�� � � �

�� "!

���!���!

#�$%'&

�� )(�*

� �+�

(�,�-+

� � �

#�$%�� � � � �� � � �� � � �

#�$%�����

.0/ 1324 / 1356 / 137�<�<� �

#�$$$$%���� ���������

#�$$$%

=(�oc HXZ`uNj a5_rj�Z\kqÆ���c�u\��Z�kviXc�u+j��oc�_�½µwoc5j\Z±_rwXu�§¨[^Z\a»_rj\kq[^w iXc�jN¥(c�c"w�j\�oc���_�abc�Z±_��[�[WZ`��kqwX_rj�c�uh_rwX�bj��oc¾kqa5_r]Wc�u���[�[^Z±��kvw­_rj\c�u�·)=(�oc<uNc���[WwX��a5_�j\Z\kqÆ ��c�u\��Z�kviXc�u�j��oc&�oZ�[^¬xNc���j�kvs�c:j�Z`_�wXuا¨[WZ\a5_�j\kv[Ww iXc�jN¥(c�c�w<j��oc��r¬¸�Å�oZ\[WxNc���j\kqs�c�u©�­_^��c�_�wX��j\�oc|�r¬¸� �®Z\[^xNc��!j\kvs�ckva»_r]^c��oª´_rwocÏ·�=(�ochj\�®kvZ`�<a5_rj�Z\kqÆ<��c�u���Z\kqiHc�uÎj\�ocFZ\kq]^km��j�Z`_�wXuا¨[WZ\a5_�j\kv[Wwb~¨Z\[^j±_rj�kv[^wµ_rw­�j\Z±_rwXuNªm_�j\kv[Ww9�FiHc"j©¥Fc�c�wbj��oc<[^ioxNc��!j(§¨Z`_raµc<_rwX�5j\�oc���_�abc�Z±_�§¨Z`_�abc�·

�oZ\[^a w®[r¥º[Wwb¥(c�¥hkvªqªoiXc&_Wu\uNnoabkvw®]+j��X_rjFj\�oc�kvwtj\Z�kvwXuNkm�h��_�abc�Z±_¾�X_rZ±_rabc"j\c�Z±u�_�Z\c¶two[�¥hw5_rwX�bj��oc�Z\c!§¨[^Z\c�¥(c���[^wXuNkm��c"Z(j\�®c¾Z�c�ªm_�j\kv[WwXu©�®kv�biXc�jN¥(c"c�wb�"_rabc�Z±_<��[�[^Z±��kvwX_�j\c�u_rw­� kva»_r]^c���[�[^Z±��kvwX_�j\c�u �

��� & (�*98� �:�+ ~\�r��;� & (�,=<� �)�>+ ~�� �

}�w�j\�oc�uNc c�l�nX_rj�kv[^w­u(�*

_rw­�(�,

_rZ\c j��oc s�c�Z�j\km�"_rª)_rw­�J�o[WZ\kqÀ�[^wtj`_�ª:u���_rªqc�§¸_^��j�[^Z`u_rw­� �+�_rwX�?�-+�_�Z\c<j\�oc<kqa5_r]Wc<��[�[^Z±��kvw­_rj\c�u([r§�j��oc&kvwtj\c"Z`u©c���j\kq[^wb[�§:j��oc<[^�oj�km��_�ª)_rÆtkmu¥hkvj��bj\�®c&kva5_�]^c��oªm_�woc�·®}ØjF¥hkvªqªHiXc�u©�®[r¥hw5j��X_rjFj\�oc&Z�c���[WwXu©j�Z\nX��j�kv[Ww aµc�j\�o[t�Ä��c�u���Z\kq¬iXc����oc�Z�ch��c"�Hc"wX�ouÎ[Wwoªv²<[^w j��ochZ`_�j\kq[

(�*�

(�,_�wX� j\�X_�j:j\�ocFZ\c��![^wXuNj\Z\n­��j\kq[^w�[Wioj`_�kvwoc��

¥hkvj���[WnoZ&abc�j��o[t�bkmuhwo[^j�u©c�wXuNkvj�kvs�c&j�[bc�Z\Z�[^Z±u�kqw�� + _rwX�@� + · =(�oc�Z\c�ª´_rj\kq[^wXuN�okv��iXc�j\¬¥(c"c�w5[^ioxNc���jF�X[^kvwtj`u�_rw­����_�abc�Z`_<�X[^kvwtj`uh�"_rw�iXc<¥hZ\kqj\j�c�w�_^u �

8� & .BA ��� � 1326 A �C� � 137 ~�� �

<� & 4DA � � � 1 56 A �C� � 137 ~Âd��

¹#c���kqs�k´��c&iX[^j��5j\�oc<wtnoabc�Z±_rj\[WZh_�wX� j��oc<��c�w®[^abkqwX_rj�[^Z([�§�c�ltu�· ~�� �F_�wX�#~¨d���it²137 ·�¹¼c<kvwtj\Z\[t�tnX��c<j\�oc&§¨[Wªvªv[�¥hkqwo] wo[^j±_rj�kv[^w­u �

�! UX"� ÷

Page 10: Euclidian Shape and Motion from Multiple Perspective Views ... · Euclidian Shape and Motion from Multiple Perspective Views by Affine Iterations Stéphane Christy, Radu Horaud To

������������� ������������� ����������� ��! "��#$�%�&�'���(�)�*�,+-�� /.0�)�"�����12�435�6��78. �

��� & . � 137 kmu:j��oc HXZ`uNj(Z�[r¥ [r§ j\�oc�Z\[Wj`_rj�kv[Wwba5_rj�Z\kqÆ�u���_rªqc���it² j\�oc�À�¬¸��[^ab�X[Wwoc�wtj[�§:j��oc<j\Z`_�wXu©ª´_rj�kv[^w�s�c���j�[^Z �

��� & 4 � 137 k´u j\�®c u©c��![^wX�ÐZ�[�¥ [�§Äj\�oc�Z�[^j±_rj\kq[^w%a5_�j\Z\kqƤu\��_�ªvc��¤i�²¤j\�oc*À�¬��[Wab�X[^woc"w�jF[r§�j��oc<j\Z`_�wXu©ª´_rj�kv[^w�s�c���j�[^Z �

� 8� & 1 2 � 1 7 _�wX�

<� & 1 5 � 1 7 _�Z\c�j��oc �"_rabc�Z±_ ��[�[^Z±��kvw­_rj\c�u¾[r§�� � ¥h�ok´����kmuhj\�oc

�oZ�[^xNc���j�kv[Ww [�§ ���¾�Äj��oc<[^Z\kq]^kqw [�§�j\�®c+[WioxNc���j(§¨Z±_rabc�pt_�wX�� ¹¼c���c�wo[Wj\c<i�² �3��j\�oc&§¨[^ªqªv[�¥hkvw®] Z`_rj�kv[ �

�3� & 6 A � �137 ~ E �

¹#c<a5_�²Äwo[�¥ÐZ�c�¥hZ\kqj\c<j\�®c+�Xc�Z±u©�Xc���j�kvs�c&c�l nX_�j\kq[^wXuh_Wu �

8� & � A ��� � 8

�� � �3� ~ F �

<� & � A ��� � <

�� � �3� ~ J^�

¹º�oc�woc"s�c�Z�j\�oc<[^ioxNc���j(k´u(_rj�uN[^abc���kmuNj`_rw­��c¾§¨Z\[Wa¯j\�®c<��_raµc�Z`_tp j��oc��3�)_�Z\c<uNa5_rªqª��[Wab�X_rZ�c���j\[¼�W·9¹#c a»_�²5j��oc�Z\c!§¨[^Z\c kqw�j�Z\[t��n­��c jN¥([�_r�o�®Z\[�Æ�kqa5_rj�kv[WwXu¾[r§|j\�oc �Xc�Z±u©¬�Xc���j�kvs�c&c�l�nX_�j\kv[WwXu �o¥Fc�_r¶»_rwX�b�X_�Z`_ �Hc"Z`u©�Xc��!j\kvs�cÏ·

�!��� � � ��� ������'� �� %� ���¹¼c�_r¶b�Xc�Z±u©�Xc���j�kvs�c<_Wu\uNnoabc�u(j��X_rjhj��oc<[^ioxNc���jh�X[Wkvwtj`u(ªvkqc&kvw�_ �oªm_�woc<�X_rZ±_rªqªvc�ª j\[Äj\�ockva»_r]^cÄ�oª´_rwocb�­_^u\uNkvwo]�j��oZ\[Wno]^� j��ocb[^Z�kv]Wkvw�[�§�j\�ocb[WioxNc���j<§¨Z`_raµc�p9k�· c�·vp � � ·?=(�okmu<kmuc�l�nokvsW_�ªvc�wtj(j�[Ä_fÀ�c"Z\[^¬¨[^Z±��c�Z�_��o�oZ\[�Ætkva5_�j\kq[^w"�

�� � �3� �%�������������������������

¹ºkvj��5j\�ok´uh_r�o�oZ�[�Æ�kqa5_rj�kv[^w p�c�ltu�· ~ F �|_rw­�#~ JW��iXc���[^aµc��

8�� � 8

� & � A ��� ~ L �

<�� � <

� & � A �C� ~�� �

}�wÅj��oc�u©c�jN¥F[#c�l�nX_�j\kv[WwXu8 ��

_rw­�< ��

_rZ�c�j\�oc°��_rabc"Z`_���[�[WZ`��kqwX_rj�c�ub[r§�j\�oc�¥(c�_�¶�Xc�Z`uN�Xc���j\kqs�c<�oZ\[^xNc���j\kq[^w5[r§Fj\�®c��X[^kqw�j����N·��(²5k´��c�wtj\k HH��_rj�kv[^w5¥hkqj\��c�l�u!·�~ F �|_rw­��~ J^�¥(c�[^ioj±_rkqw�j\�oc Z�c�ªm_�j\kv[WwXu©�®kv��iXc�jN¥Fc�c�w5j\�oc ¥Fc�_r¶5�Xc�Z±u©�Xc���j�kvs�c�_rwX��j��oc �Xc�Z`uN�Xc���j\kqs�c�oZ�[^xNc���j�kv[^w­u([r§ ��� �

8�� & 8

�¸~\� � �3�Â� ~���� �

<�� & <

��~�� � �3�m� ~��^�r�

X5X íZY![]\![C^

Page 11: Euclidian Shape and Motion from Multiple Perspective Views ... · Euclidian Shape and Motion from Multiple Perspective Views by Affine Iterations Stéphane Christy, Radu Horaud To

L �)�0� ������ �� � �� �� .���� �� ��� ������,�� "����

=(�oc�uNc+c�l nX_�j\kq[^wXuh_�ªvªv[�¥¤nXu(j\[µ��c�j�c�Z\aµkvwochj��oc�l nX_�ªvkqj©² [�§:j��oc<¥(c�_�¶b�Xc�Z`uN�Xc���j\kqs�c_r�®�oZ\[�Ætkva5_�j\kv[WwÅ¥hkqj\�ºZ\c�u©�Xc���jbj�[�j\�®c��Hc"Z`u©�Xc��!j\kvs�c5�oZ�[^xNc���j�kv[Ww�·h}�wX��c�c��Ñj\�®c�c�Z\Z�[^ZiXc�jN¥(c�c"wbj\�oc<¥Fc�_r¶b�Xc�Z±u©�Xc���j�kvs�c&�®Z\[^xNc��!j\kv[Ww�_rwX�5j\�oc Q\j�Z\noc Sf�oZ�[^xNc���j�kv[Wwbkmu �

�8� & � 8

�� � 8� � &�� 8

�!�3� � ~���� ��<� & � <

�� � <� � &�� <

� � � � ~���� �

V¾c�w­��c<j\�ocÄl�nX_rªqkvjN²b[�§(j��oc _��o�oZ�[rÆtkva»_rj\kq[^w���c��Xc�w­�ou¾iX[^j��#[Ww�j\�oc�s^_�ªvnoc�[r§ � �������[^w j��och�X[�uNkvj\kq[^w [�§�j\�®c(�X[^kqw�j:kqw j\�ocFkva5_�]^c�·��9c"j��åuA��[^wXuNkm��c"ZΧ¨[^Z:c"Æo_�ab�oªvc³_ E��"��ZEt���kva»_r]^cÏ·9�¾�®�oZ\[�Ætkva5_�j\c<sW_rªvn®c�u�§¨[^Z<j��oc kvwtj\Z�kvwXuNkm�+�­_rZ`_�abc�j�c�Z`u�_rZ�c��

( * & ( , &���^�W�

_rw­� � + & � + & �OE�F�· =(�oc�Z�c�§¨[^Z�c<nXu©kqwo]�c�lH· ~\�r�F_�wX�#~¸� �F¥(c<�X_�s�c����

8� �<� ¤� �å��E

¹#c���[^wX�!ªvnX��c�j\�­_rj�§¨[WZ([WioxNc���j`u:j��X_rj|_rZ\c�l nokqj\c;��ªv[Ïu©c�� j\[�j\�oc;��_�abc�Z`_tp^j\�®c&¥(c�_�¶�Xc�Z`uN�Xc���j\kqs�c�_r�o�oZ�[�Æ�kqa5_rj�kv[^w k´u�uNj\kvªqªhsW_rªvk´�Ñ�oZ�[�s�k´��c���j\�X_�j�j��oc�[WioxNc���jĪvkqc�u�kvw j\�ocwoc�kq]^�tiX[^noZ��o[�[t��[r§�j��oc<[^�oj�km��_�ª)_rÆtkmu!·

�!� � �����'� ������'� �� %� ���Y:_�Z`_��Xc�Z`uN�Xc���j\kqs�c¾a»_�²�iXc+stkvc"¥(c�� _Wuh_ HXZ`uNj\¬¨[^Z`�tc�Z�_r�o�®Z\[�Æ�kqa5_rj�kv[Ww�[�§)�Hc"Z`u©�Xc��!j\kvs�cÏ·}�wX��c�c��9p�¥hkqj\�5j\�oc�_��o�oZ\[�Ætkva5_�j\kq[^w"�

�� � �3� � � � �3� ����������������� � ���

¥(c<[Wioj`_�kvw�j��oc<�X_rZ±_f�Xc�Z±u©�Xc���j�kvs�c&�®Z\[^xNc��!j\kv[Wwb[r§ ��� �8� � & ~ � A � � � 8

�!�`~���� �3�m�� � A � � � 8

� �8� � �

& . A ���1 7 �

8� �

8� 6 A ���137

¥h�oc�Z�c<j\�oc<j\c"Z\aÒkvw¼��� 1� 7 ¥h_^uhwoc�]Wªvc��!j\c��9·N=(�oc"Z\c<kmuh_Äu©kqabkvª´_rZ�c�Æt�oZ\c�u�u©kq[^wb§¨[^Z <�� ·

�Îkvw­_rªvªq²�p�j��oc<�X_rZ±_f�Xc�Z±u©�Xc���j�kvs�c&c�l nX_�j\kq[^wXuh_�Z\c��

8� � � 8

� & . �8� 6137 A �C� ~���d��

<�� � <

� & 4 �<� 61 7 A � � ~��GE �

}�w�[WZ`��c"Z(j\[�[^ioj±_rkqw j��oc�Z\c�ª´_rj\kq[^wXuN�okv�µiHc"j©¥Fc�c�w j��oc&�X_�Z`_��Xc�Z`uN�Xc���j\kqs�c�_rwX�Äj\�oc&�Xc�Z±u©¬�Xc���j�kvs�c&�oZ�[^xNc���j�kv[WwXu([r§ ���)¥(c���_�w5¥hZ\kvj�c<j\�oc�uNc<c�l n­_rj\kq[^wXuh_Wu|§¨[Wªvªq[r¥�u �

8� � & 8

� � � A �C� � 8� �3�

<�� & <

� � � A ��� � <� �3�

�! UX"� ÷

Page 12: Euclidian Shape and Motion from Multiple Perspective Views ... · Euclidian Shape and Motion from Multiple Perspective Views by Affine Iterations Stéphane Christy, Radu Horaud To

������������� ������������� ����������� ��! "��#$�%�&�'���(�)�*�,+-�� /.0�)�"�����12�435�6��78. �

�¾]Ï_rkvw pHit²5km��c�wtj\k�H ��_�j\kv[Ww�¥hkqj\��c�ltu�·�~�F �;_rwX��~0J^�h¥Fc�[Wioj`_�kvw�j\�®c Z\c�ª´_rj\kq[^wXuN�okv��iXc�j\¬¥(c"c�w5j\�oc<�X_�Z`_ �Hc"Z`u©�Xc��!j\kvs�c�_rwX�5j\�oc<�Xc�Z`uN�Xc���j\kqs�c¾�®Z\[^xNc��!j\kv[WwXu([r§ ��� �

8� � & 8

�¸~\� � �3�Â���8� �3� ~��GF �

<�� & <

��~�� � �3�¨���<� �3� ~��2J^�

�&u+kqw�j\�®cb�oZ\c"s�kq[^nXu�u©c���j�kv[Ww�p�¥(c5��_�w�c�_Wu©kvªq²�c�u©j�kva5_�j\cÄj\�ocÄc�Z�Z\[^Z iXc�jN¥Fc�c�w¼j\�oc�X_�Z`_ �Hc"Z`u©�Xc��!j\kvs�c�_rwX�5�Hc"Z`u©�Xc��!j\kvs�c;�oZ\[^xNc���j\kq[^wXu �

�8� & � 8

� � � 8� � & � ~ 8

� �8� � � � � ~��GL �

�<� & � <

�� � <� �-&�� ~ <

� �<��� �3� � ~���� �

¹º�oc�woc"s�c�ZF_rwb[WioxNc���j(�X[Wkvwtj����9kmu:§¸_rZ�§¨Z\[^aÒj\�®c¾[W�oj\k´��_rª _rÆtkmu�j��oc&¥Fc�_r¶ �Xc�Z±u©�Xc���¬j\kqs�c(ab[t�tc�ªokmuA_��X[�[WZh_r�®�oZ\[�Ætkva5_�j\kv[Ww�·�V¾[�¥Fc�s�c�Z�p�_&�oZ�[^�Xc�ZF�`�o[^k´��ch[r§ j\�och[WZ\kv]Wkvw�pWk�· cÏp����p9_�wX��j\�oc�nXu©c�[r§�j��oc��­_rZ`_Ä�Hc"Z`u©�Xc��!j\kvs�c<ab[t�tc�ª)��_rw¼�![^ab�Xc�wXu�_rj�c�_�wX���oZ�[rstkm�tc�_]^[�[t��_��o�oZ�[rÆtkva»_rj\kq[^wbc"s�c�wbk §��3��kmu(w®[^j;u©a5_�ªvª�·� � �+�h����' �:�(���F���"��� 01�!��� ���;�|'Ã���+�F���",Ä� �h �&��

�9c�j&n­u&��[^w­u©km�tc�Z<_r]Ï_rkqw�j\�oc �Xc�Z±u©�Xc���j�kvs�c<c�l�nX_rj�kv[^w­u�~ F��&_�wX��~ JW�`·A=(�®c�u©c�c�l n­_rj\kq[^wXua5_�²ÄiHc�_�ªmu©[Ä¥hZ\kqj\j�c�w�_^u �

8�¸~\� � �3�Â���

8� & � A ��� ~¸�^� �

<��~\� � �3�m���

<� & � A ��� ~¸���r�

�9c�j(n­u�uNnoioj�Z`_^�!j(j\�®c ���� �T�)�� /.0�)�"�����12�T�0�� �# §¨Z�[^aÒiH[Wj\�bj��oc�ªvc�§¨j�_�wX��Z�kv]^�tjFu©k´��c�u[r§³c�l n­_rj\kq[^wXu&~¸�^���h_�wX�#~¸���r�±·o¹#c<[Wioj`_�kvw"�

8�¨~\� � �3�Â���

8� �

8� �137 6 A �C�� ��� �

& �137 .BA ��� � 8� �137 6 A ���

<��~�� � �3�m� �

<���

<� �137 6 A �C�� ��� �

& �137 .BA ��� � <� �137 6 A ���

=(�oc�uNc+c�l nX_�j\kq[^wXu���_rw5iXc�¥hZ�kvj�j\c�wbaµ[^Z\c���[^ab�­_^��j�ªv²µ_^u �

~8� �

8���`~�� � �3�¨� & � � A � � ~¸�^� �

~<� �

<���`~�� � �3�¨� & � � A �C� ~¸�^� �

¥hkvj��"�

� � & . �8� 61 7 ~¸�rd��

� � & 4 �<� 6137 ~¸��E �

X5X íZY![]\![C^

Page 13: Euclidian Shape and Motion from Multiple Perspective Views ... · Euclidian Shape and Motion from Multiple Perspective Views by Affine Iterations Stéphane Christy, Radu Horaud To

��� �)�0� ������ �� � �� �� .���� �� ��� ������,�� "����

=Ã[*u©n®aba5_rZ�kvÀ�cWp9¥Fcb�X_�s�cbjN¥([¼��k � c�Z\c�wtj uNc�j`u�[�§&c�l nX_�j\kq[^wXu�j��X_rjµ��c�u���Z\kqiHcµj\�oc.!��#����Xc�Z±u©�Xc���j�kvs�c���_�abc�Z`_�aµ[t��c�ª�·?=(�oc H­Z`u©jµu©c"j�p�k�· cÏ·vp�c�l�nX_rj�kv[^w­u�~��W� �¦_rwX�R~¸���r�c�uNj`_ri®ªvkmuN��j��oc<ªvkvw®¶biXc�jN¥(c�c�wµ�Hc"Z`u©�Xc��!j\kvs�c�_�wX� ¥Fc�_r¶5�Xc�Z`uN�Xc���j\kqs�c&¥h�okqªvc<j\�oc�uNc���[^w­�u©c"j�p�k�· c�·vp`c�l�nX_�j\kv[WwXu:~��W� � _rwX� ~¸�^� � c�uNj`_ri®ªvkmuN�<j\�oc�ªvkqwo¶�iHc"j©¥Fc�c�w&�Xc�Z±u©�Xc���j�kvs�c�_�wX�+�­_rZ`_�Xc�Z`uN�Xc���j\kqs�c�· }¸§Fj��oc�sW_�ªvnoc�u&[�§ �3�F_rZ�cbuNc�j&j\[5À�c"Z\[5j\�X_�w#¥Fc�[Wioj`_�kvw¼c�l�nX_rj�kv[^w­u�j��X_rj_r�®�oZ\[�Ætkva5_�j\c�j\�oc|��_raµc�Z`_(c�kqj\�oc"Z�¥hkvj\��¥(c�_�¶<�Hc"Z`u©�Xc��!j\kvs�c�[WZÎ¥hkqj\���­_rZ`_��Hc"Z`u©�Xc��!j\kvs�cÏ·=(�oc���Z�nX��km_�ª9�X[^kqw�jhk´u�j��X_rj�k §)j��oc �3� � uh_rZ�c�uNc�jhj\[ÄuN[^abc-HXÆ�c���wo[^w�À�c�Z�[�sW_rªqnoc�u�poj��oc�wj\�®c(c�l�nX_rj�kv[WwXuÎZ\c"a5_rkqw&ªvkqwoc�_rZ"·^}�w��X_rZ�j\km�!noªm_�Z�p�j\�®c�Z\c|_rZ\cFsW_rªvn®c�uç¨[^Z)j\�®c �3� � uç¨[^Z)¥h�ok´���j\�®c�u©c&c�l nX_�j\kq[^wXuF_rZ\c���[^wXuNkmuNj\c�wtj��¦no�5j\[ uN[^abc�abc�_Wu©noZ�c�abc�wtj)wo[^k´u©c¾�Ä¥hkqj\�bj��oc�§¨n®ªvª�Xc�Z`uN�Xc���j\kqs�c¾aµ[t��c�ª�· =(�oc&¶�c�²�km��c�_<[r§)[WnoZ(Z�c���[^w­u©j\Z�nX��j�kv[^wµabc�j\�®[t�fk´u�j\[�kvj�c�Z`_�j\kvs�c"ªv²c�uNj\kva»_rj\c�sW_rªvn®c�u�§¨[^Z<j��oc �3� � u+uNnX�`��j\�X_�j+j��oc �Xc�Z`uN�Xc���j\kqs�c Z\c��![^wXuNj\Z\n­��j\kq[^w��oZ�[^ioªqc�aiXc���[Wabc�u(_�w Q±_�½µwoc<kvj\c"Z`_rj�kvs�cCS<Z\c��![^wXuNj\Z\n­��j\kq[^w5�oZ\[Wioªvc"a�·

�9c�jÄnXu��![^wXuNkm��c�ZÄwo[�¥��*s�kqc�¥�u�[r§<j��oc�u�_rabc�u���c�woc5�X[^kqw�j±u�·:¹¼c�_^u�u©noaµc j��X_rjkva»_r]^c"¬Âj�[^¬¨kva5_�]^c���[WZ\Z�c�u©�X[^w­��c�wX��c�u9�X_�s�c(_�ªvZ�c�_^��²;iHc"c�w<c�u©j±_rioªqkmu©�®c��9· �([^j��<c�l n­_rj\kq[^wXu~��W� �|_rwX�¼~��t�r�F[^Zhc�l�nX_rj�kv[WwXu+~¸�^���h_�wX�#~¸�^� �|��_�w iXc<¥hZ\kqj\j�c�w�_^u �

�=������ ��� ���� &� ���� ��� ��� � ������ ����� ~¸��F �

}�w5j\�ok´u�§¨[^Z\aÄnoª´_+j\�®c�uNnoiXu���Z\kq�oj ��u©j`_�wX�ou�§¨[WZhj\�oc �� ��&�X[^kqw�j;_rwX�5j\�oc�uNnoiXu\�!Z\kv�®j��f§¨[^Zj\�®c��� ��&kva5_�]^c�·�=(�oc��r¬¨s�c���j�[^Z�� ����k´u(c�l�nX_rª j\[�~Â¥(c�_r¶b�Xc�Z±u©�Xc���j�kvs�c^� �

� ��� & �8���"~�� � �3���!���

8���

<���"~�� � �3���!���

<����� ~¸�&J^�

[^Z�j\[5~¨�X_rZ±_f�Xc�Z±u©�Xc���j�kvs�c^� �

�=��� & � ~8��� �

8���!�±~\� � �3�����

~<��� �

<���!�`~�� � �3���!��� ~¸��L �

}�w�j\�oc�u©cbc�l�nX_�j\kv[WwXu��3���"p�k�· c�·vp�c�lX·�~ E �<k´u���c�HXwoc��J§¨[WZ�c�_W���*�H[WkvwtjÄ_rwX��§¨[^Z c�_W���kva»_r]^c �

�3��� & 6 � A � �137�� ~¸�^� �

=(�oc Z\c���[^wXuNj\Z�nX��j\kq[^w��oZ\[Wioªvc�a3k´u�wo[�¥%j\�®c��oZ�[^ioªqc�a3[r§|u©kqa�noªqj`_�woc�[^n­u©ªv²»u©[^ªqs�kqwo]� � � ����c�l nX_�j\kq[^wXuh[r§Aj\�oc<§¨[^Z�a¯[r§�c�lH· ~���F��`·X¹#c<kqw�j�Z\[t��nX�!c+w®[r¥ _�abc�j\�®[t�»j��X_rju©[Wªvs�c�u�j\�oc�u©c&c�l�nX_�j\kv[WwXu�it²Äªvkvw®c�_rZFkvj\c"Z`_rj�kv[WwXu�·®y�[^Z\c;�oZ\c��!kmu©c"ªv²�p^j��okmu�abc�j��o[t�b��_�wbiXcu©n®aba5_rZ�kvÀ�c���i�²bj��oc&§¨[^ªqªv[�¥hkvwo]Ä_�ªv]W[^Z\kqj\�oa ��^· ����������������� � ����_rw­� ��� � � �����������!���<u©c"j � �3��� & �����·#"¾� �o_�j\c<j\�®c+sW_�ªvnoc�uh[r§³� ����n­u©kvw®] �3��� ���·�YÎc"Z©§¨[^Z�a3_rw�«:nX��ªqkm��k´_rw5Z\c���[^wXuNj\Z�nX��j\kq[^w5¥hkvj���_rw�_�½bw®c��"_rabc�Z±_��do· ����������������� � ����_rw­� ��� � � �����������!���&c�uNj\kva»_rj\c;woc�¥¤s^_�ªvnoc�u�§¨[^Z��3��� �E�·&���oc���¶Äj\�oc<sW_rªqnoc�uh[r§��3��� �

�! UX"� ÷

Page 14: Euclidian Shape and Motion from Multiple Perspective Views ... · Euclidian Shape and Motion from Multiple Perspective Views by Affine Iterations Stéphane Christy, Radu Horaud To

������������� ������������� ����������� ��! "��#$�%�&�'���(�)�*�,+-�� /.0�)�"�����12�435�6��78. ���

� � �³~ ��� �­�)j��oc<s^_�ªvnoc�u([r§��3���:xNnXuNjhc�u©j�kva5_�j\c��µ_rjhj\�®kmu(kvj�c�Z`_�j\kq[^w5_�Z\c<km��c"w�j�km��_�ª¥hkqj\�5j\�®c+sW_�ªvnoc�uhc�uNj\kva»_rj\c���_�j¾j��oc<�oZ\c�stkv[WnXu(kvj�c�Z`_�j\kq[^w�p��)�� Äu©j\[W�$�

� �(.C�b]W[�j�[buNj\c����t·=(�oc ab[Ïu©j&kqab�X[^Z\j±_rwtj&uNj\c���[�§hj��okmu&_�ªv]W[^Z\kqj\�oa kmu&uNj\c���d �9c�uNj\kva»_rj\c<woc"¥ sW_rªqnoc�u

§¨[^Z � ��� ·�=(�ok´u(��[Wab�onoj±_rj\kq[^wµ��_rwÄiXc&a5_W��chc�Æt�oªvk´��kvj)k § [^woc;��[^w­u©km�tc�Z`u�kvwtj\[�u©[Wabchab[^Z�c��c�j±_rkqª9uNj\c��µ�+[�§)j\�oc;_rªq]^[WZ\kvj��oaÒ¥h�okm�`�Ä��_rwµiHc�§¸_rZ\j��oc�ZF��c��![^ab�X[�uNc�� kvwtj\[ �X~¨kÂ�A��½µwocZ\c���[^wXuNj\Z�nX��j\kq[^w�_rw­�#~ÂkqkÂ�³«³n­��ªvk´��km_�w Z�c���[WwXu©j�Z\nX��j�kv[Ww�·

=(�ocb�oZ�[^ioªqc�a¿[r§�_�½µwocÄZ\c��![^wXuNj\Z\n­��j\kq[^w�k´u+j\�®c��oZ�[^ioªqc�a [r§;��c�j�c�Z\aµkvwokqwo]biX[^j��� ��_rwX� ����pH§¨[WZ�_rªvª ��_�wX�°§¨[WZ<_rªqª��\pXkvw�c�lX·:~¸��F �±p9¥h�oc�w¼uN[^abc c�u©j\kqa5_rj�c�u([r§�� ����_�Z\c�oZ�[rstkm�tc��9·(��nX�`�º_¼Z\c���[WwXu©j�Z\nX�!j\kv[Wwº��c"j\c�Z�abkvwoc�u�u©�­_r�Xc�_�wX�Åab[Wj\kv[Ww no�ºj\[ _*�r¬¸�_�½µwoc<j\Z±_rwXu�§¨[^Z�a5_rj�kv[^w ·X}Øw­��c�c��9p�§¨[^Z�_�w�²���³��kvwts�c�Z\j�kvioªqc¾a»_rj\Z�kvÆ��Ð¥Fc<�X_�s�c��

� � ��� & � ������� � � �}�w5[^Z`�tc�Zhj\[Ä��[Ww�s�c"Z\jF_�½µwoc�u©�X_��Xc _rw­�»aµ[^j\kq[^wbkqw�j�[Ä«:nX��ªvk´��km_�w5u©�­_r�Xc _�wX�bab[^j�kv[^w p[^w®c w®c�c��ou<j\[¼��[WwXu©k´��c�ZÄu©[Wabcµ«³n­��ªvk´��km_�w ��[^w­u©j\Z±_rkqw�j±u _^u\uN[t��km_�j\c�� c�kvj��oc�Z<¥hkvj�� j\�ocab[Wj\kv[Ww�[�§)j\�oc���_rabc"Z`_&[^ZF¥hkvj\�µj\�oc;u©�X_��Xc+iXc�kqwo]�stkvc"¥(c�� i�² j��oc<��_�abc�Z±_�·���kqwX��c�¥(c��c�_�ªÎ�®c�Z\c�¥hkvj\�°_��"_rªvkqioZ`_�j\c�����_raµc�Z`_tpo¥(c<a5_�²b¥(c"ªvª�nXuNc�Z\kq]^k´� ab[Wj\kv[Ww���[^wXuNj\Z±_rkvwtj`ukvw ��[^w^xNn®wX��j\kq[^w�¥hkqj\�¼¥(c�_�¶#[WZ��X_�Z`_��Xc�Z±u©�Xc���j�kvs�c�> �GFC@�pD> �W�G@�·5=(�oc�Z�c�§¨[^Z�c�p)u©j�c�� � [�§j\�®c<_rªq]^[^Z�kvj��oa¯�oZ�[rstkm�tc�u�iX[^j���«:nX��ªqkm��k´_rw5uN�X_r�Xcµ~�� � ��� � ����F_rw­� «:nX�!ªvkm�tkm_rwµab[^j�kv[^w pk�· c�·vp �Äa5_�j\Z\k´��c�u�[r§Aj\�oc&§¨[^Z�a �

�����

. /� 1 2 �4 /� 1 5 �6 /� 137 ��<�+� �

# $$$$%

�h_^uNc���[^w j��och�X_rZ±_rabc"j\c�Z±uÃ[�§9j��oc�«:nX��ªqkm��k´_rw uN�X_r�Xc;_rwX��aµ[^j\kq[^w�j\�tnXu:��[Wab�onoj�c��[^w®c��"_rw�c�u©j\kqa5_rj�c��3���(§¨[^Z;_rªqª ��_rwX��§¨[^Z;_rªqª���nXuNkvwo]Äc�lX·9~¸�^� ���µu©j�c���d®·

=(�oc HXZ`uNj�kqj\c�Z±_rj�kv[^w [�§&j��oc�_rªv]W[^Z�kvj\�®a �Xc�ZN§¨[^Z�a5u�_¼�r¬¸� Z�c���[^w­u©j\Z�nX��j�kv[^w nXuNkvwo]j\�®c¾kqwokvj�km_rª®kva5_�]^chabc�_Wu©noZ�c�abc�wtj`u�_�wX�b_<¥(c�_�¶ [^Z��X_rZ±_+�Xc�Z±u©�Xc���j�kvs�c���_�abc�Z`_�ab[t��c�ª¸·=(�ok´u H­Z`u©j<Z�c���[^w­u©j\Z�nX��j�kv[^w*_rªqªv[�¥�u<_rw¼c�u©j�kva5_�j\kq[^w�[�§hsW_rªqnoc�u¾§¨[^Z<j��oc �3����� u&¥h�okm�`��kvwj\n®Z\w�_�ªvªq[r¥¤j\�®c+kqa5_r]Wc&s�c���j�[^Z±uh� ���hj�[�iXc # �2�� ��"f~¸u©j\c"��� [r§Aj\�®c _rªq]^[WZ\kvj��oa��±·�=(�oc� ��� � u�_�Z\c�aµ[t��k HXc���_^���![^Z`�tkvwo]¼j\[¼c�lX·h~��&JW�b~m§¨[^ZÄ¥(c�_�¶ �Hc"Z`u©�Xc��!j\kvs�cW�&[^Zµ_^�"��[^Z±��kvwo]j\[5c�lX·)~���L���~ §¨[^Z<�X_�Z`_b�Xc�Z`uN�Xc���j\kqs�c^�|u©nX�`��j\�X_�j<j\�oc�²�iXc�j\j�c�Z HXj<j\�®cÄ_r�o�oZ�[�Æ�kqa5_rj�c����_�abc�Z`_<ab[t��c�ª iXc�kvwo] n­u©c��9·

=(�oc5woc�Ætj<kvj\c"Z`_rj�kv[WwXu�[r§�j��oc5_rªq]^[WZ\kvj��oa �Xc�ZN§¨[^Z\a._��r¬¸�¯Z\c��![^wXuNj\Z\n­��j\kq[^w�nXuNkvwo]~Âk¨�<kva5_�]^cÄs�c���j�[^Z`u�j��X_rjÄ_�Z\c�kvwX��Z�c�abc�wtj`_�ªvª ²�aµ[t��k HXc���_rwX�¤~ÂkqkÂ�<_�¥Fc�_r¶ ~Â[^ZÄ�X_�Z`_ ��Xc�Z`uN�Xc���j\kqs�c<��_�abc�Z`_<ab[t��c�ª¸·

��j���[Ww�s�c"Z\]^c"wX��c�p�j\�oc*c�l n­_rj\kq[^wXu�~¸��F ��_�Z\c¼c�l�nokvsW_rªqc�wtjb¥hkvj��Rj\�oc¼�Xc�Z`uN�Xc���j\kqs�cc�l�nX_rj�kv[WwXu ~¸�^���`p&~¸���r�¦[^Z�~��^���`p&~¸�^���`·F}Øw [^j��oc�ZÄj\c�Z�a5u�p)j\�ok´ub_rªq]^[^Z�kvj��oa u©[^ªqs�c�u+§¨[^Z«:nX��ªqkm��k´_rw Z\c��![^wXuNj\Z\n­��j\kq[^wÅ¥hkqj\� _J�Xc�Z±u©�Xc���j�kvs�c���_raµc�Z`_5i�² kvj�c�Z`_�j\kq[^wXu�[�§ _�wº«:no¬��ªqkm��k´_rwºZ\c���[^wXuNj\Z�nX��j\kq[^w¤abc�j��o[t� ¥hkqj\�U_rw%_�½bw®c���_�abc�Z±_�·D=(�oc�Z\c!§¨[^Z\cÏp�iXc�§¨[WZ\c¼¥(c

X5X íZY![]\![C^