3D Reconstruction from Image Pairs. Reconstruction from Multiple Views. Computing Scene Point from Two Matching Image Points

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1 D Recostructo fro Iage ars Recostructo fro ultple Ves Dael Deetho Fd terest pots atch terest pots Copute fudaetal atr F Copute caera atrces ad fro F For each atchg age pots ad copute pot scee Coputg Scee ot fro o atchg Iage ots e o have coputed ad fro F roble: fd fro ad. Cobe to a for A Solve A usg SVD ad pckg the sgular vector correspodg to the sallest sgular value Note: Nolear ethods geerall gve better results Coputg Scee ot fro o atchg Iage ots Detals) ) ) ) ) ) ) ) ear cobato of other equatos Coputg Scee ot fro o atchg Iage ots Ed) Hoogeeous sste A s ' ' ' - ' s the last colu of V the SVD of A A U D V roectve Recostructo heore Assue e detere atchg pots ad. he e ca copute a uque fudaetal atr F he recovered caera atrces are ot uque: ) ) etc. he recostructo s ot uque: etc. here ests a proectve trasforato H such that H H - H - 5 6

2 roectve Recostructo heore Detals) here ests a proectve trasforato H such that H H - H - Iage Sae Iage C e e C C e e C - - H H H 7 roectve Recostructo heore Cosequeces) e ca copute a proectve recostructo of a scee fro ves based o age correspodeces aloe e do t have to ko athg about the calbrato or poses of the caeras he true recostructo s th a proectve trasforato H of the proectve recostructo: H 8 Recostructo Abgutes If the recostructo s derved fro real ages there s a true recostructo that ca produce the actual pots of the scee Our recostructo a dffer fro the actual oe If the caeras are calbrated but ther relatve pose s uko the agles betee ras are the true agles ad the recostructo s correct th a slart e caot get the scale) Eucldea or etrc recostructo If e do t use calbrato the e get a proectve recostructo Rectfg roectve Recostructo to etrc Copute hoograph H such that E H for fve or ore groud cotrol pots E th ko Eucldea postos H s a hoogeeous atr he the etrc recostructo s - - H ' ' H H 9 Results usg 5 pots Stratfed Recostructo Beg th a proectve recostructo Refe t to a affe recostructo arallel les are parallel; ratos alog parallel les are correct Recostructed scee s the a affe trasforato of the actual scee he refe t to a etrc recostructo Agles ad ratos are correct Recostructed scee s the a scaled verso of actual scee

3 Fro roectve to Affe Recostructo Fd tersectos of sets of les the scee that are supposed to be parallel hese pots defe a plae p Fd a trasforato H that aps the plae p to the plae at ft ) : hs plae cotas all pots at ft: ) z ) H - p ) or H ) p π π π π I Appl H to scee pots H π π p ad to caeras ad π π Eaple of Affe Recostructo Fro Affe to etrc Recostructo Use costrats fro scee orthogoal les Use costrats arsg fro havg the sae caera both ages 5 Drect etrc Recostructo usg Caera Calbrato Fd calbrato atrces K ad K usg vashg pots for orthogoal scee les See hoeork Noralze age pots Copute fudaetal atr usg atched oralzed pots: e get the essetal atr E Select [I ] ad [R ]. he E [] R Fd ad R usg SVD of E Fro ad recostruct scee pots 6 Recostructo fro N Ves roectve or affe recostructo fro a possbl large set of ages roble Set of D pots Set of caeras For each caera set of age pots Fd D pots ad caeras such that 7 Budle Adustet Solve follog zato proble Fd ad that ze d ) eveberg-arquardt algorth robles: a paraeters: per caera per D pot atrces + ) + ) Good talzato requred al used as fal adustet step of the budle of ras 8

4 9 Ital Solutos: Affe Factorzato Algorth oas ad Kaade 99) Affe recostructo Affe caera Ihoogeeous coordates + + Z Y Z Y v u Affe Factorzato ze Choose cetrod of pots as org of scee coordate sste Choose pel ) at age of cetrod he the proble becoes: ze Note: hs requres the sae pots to be vsble all ves )) - )) + - Affe Factorzato Cosder the easureet atr oe ro per age pot) he proecto atr s ze O - Affe Factorzato ze Fd as the SVD of trucated to rak : he a be chose as U D ad as V hs decoposto s ot uque: Recostructo s defed up to a atr A Recostructo s affe o upgrade to a etrc recostructo see above - V D U ) A)A - Ŵ roectve Factorzato has rak. Assue the coeffcets ko O ' v u ) ) he are uko related to the depths of pots caera coordates e dropped the pres o V D U roectve Factorzato. Start th a tal estate of the depths. Fro the easureet atr fd the earest rak approato usg the SVD ad decopose to fd the caera atrces ad D pots. Reproect the pots to each age to obta e estates of the depths ad repeat fro step

5 Recostructo fro Vdeo Sequeces Copute terest pots each age Copute terest pot correspodeces betee age pars Copute fudaetal atr F for each age par Ital recostructo Budle-adust the caeras ad D structure to ze proecto errors 5 Issues for Vdeos Sall basele betee age pars Advatage: havg slar ages facltates fdg pot correspodeces Dsadvatage: D structure s estated poorl for each age par Solutos: Use cosecutve ages for pot correspodeces ad ages far apart for D structure recostructo ake sall batches ad cobe the b least square Use recursve least square ethod 6 Eaples of D Recostructo Eaples of D Recostructo 7 8 Refereces ultple Ve Geoetr Coputer Vso R. Hartle ad A. Zssera Cabrdge Uverst ress. D. Forsth ad J. oce. Coputer Vso: A oder Approach Geoetr of ultple Ves Chapter ) Stereopss Chapter ) Affe Structure fro oto Chapter ) roectve Structure fro oto Chapter 5) 9 5