Jorge Salvador Marques, Stereo Reconstruction

Transcription

1 Jorge Slvdor Mrques, Sereo Reconsrucion

2 roblem Gol: reconsruc he D she of objecs in he scene from or more imges. Jorge Slvdor Mrques,

3 Jorge Slvdor Mrques, secil cse f f f f b model reconsrucion, b - fb, d b f f d

4 eiolr geomery eiolr lne eiolr line Jorge Slvdor Mrques,

5 deh normlized disriy df/b) deh esimes obined from he rue disriy vlue nd from noisy disriy vlues wih. errors. Jorge Slvdor Mrques,

6 ses Sereo reconsrucion involves ses. mching: ssociion of corresonding oins in wo or more) imges. ringulion: Jorge Slvdor Mrques,

7 Jorge Slvdor Mrques, clibred cmers

8 Clibrion mri Is his form of K good enough? non-squre iels digil video) skew rdil disorion Jorge Slvdor Mrques, D Models from Imges nd Video Richrd Szeliski 9

9 Cmer mri Fold inrinsic clibrion mri K nd erinsic ose rmeers R,) ogeher ino cmer mri M = K [R ] u in lower r.h. corner for d.o.f.) Jorge Slvdor Mrques, D Models from Imges nd Video Richrd Szeliski

10 Cmer mri clibrion Direcly esime unknowns in he M mri using known D oins i,y i,z i ) nd mesured feure osiions u i,v i ) Jorge Slvdor Mrques, D Models from Imges nd Video Richrd Szeliski

11 Jorge Slvdor Mrques, reconsrucion roblem: reconsruc, from he rojecions, nd mrices P, P. cmer cmer : : - homogeneous coord. known cmers: P, P ) ) ) )

12 Jorge Slvdor Mrques, lgebric mehod ) ) ) ) 4 equions unknowns M M Les squres: is normlized eigenvecor ssocied o he smlles eigenvlue.

13 Jorge Slvdor Mrques, geomeric mehod C C oicl rys # # P C P C ) ) ] ) [ ) ) ] ) [ C C C C C C find he closes ir of oins from boh lines: # # ) ) P P P P PP P P

14 oicl cener lne = C lne = lne = Cenro óico PC Jorge Slvdor Mrques,

15 Jorge Slvdor Mrques, eiolr geomery

16 eiolr geomery CC eiolr lne l eiolr lines l l,l e,e eiolr lines eioles C e e C Jorge Slvdor Mrques,

17 eiolr geomery ) In generl, wo oicl rys re non-colnr nd do no inersec. o gurnee h hey mee he following condicion mus hold F, F ir of corresonding oins homogeneous coord.) fundmenl mri he geomery of ir of cmers is clled eiolr geomery nd i is chrcerized by F. he eiolr geomery is used in boh ses of sereo reconsrucion. Jorge Slvdor Mrques,

18 Lines in D A line l l l cn be eressed s follows using homogeneous coordines l l R here is comlee symmery beween oins nd lines in D using homogeneous coordines. Inersecion of wo lines: l l line defined by wo oins: l l Jorge Slvdor Mrques,

19 Jorge Slvdor Mrques, Mri reresenion of cross roduc Cross roduc of wo vecors b b b b b b b Cn be obined by b b Is skew symmeric mri

20 Proof hyohesis: clibred cmers K=K =I) nd world coordine sysem cenered in cmr. C l e e l C I P R P s se Define oins belonging o he oicl ry defined by C nd A Jorge Slvdor Mrques, B nd se Projec hem by he nd cmer R R R A B eiolr line l R R

21 Proof ) rd se belongs o he eiolr line R l E essenil mri When he cmers re no clibred we cn relce K, K nd obin RK - F, F K Jorge Slvdor Mrques,

22 eioles e eiolr lines l l e e eiolr lines l F, l F eioles Fe, F e Jorge Slvdor Mrques,

23 Proeries of he Fundmenl Mri F is mri wih rnk de F =) fundmenl roery: F eioles: Fe, F e Eiolr lines: l F, l F linhs eiolres l F k l k line which does no conin he eiole relionshi wih P, P F e P P ) # Skew symmeric mri Jorge Slvdor Mrques,

24 Jorge Slvdor Mrques, unclibred cmers

25 Reconsrucion Mri F cn be obined from irs of corresonding oins in boh imges. F cn be esimed even if he rojecion mrices P, P re unknown Cn we reconsruc he scene knowing mri F? Jorge Slvdor Mrques,

26 Esimion of F F cn be esimed from 8 or more irs of oins 7 re enough). Ech ir of oins defines resricion f f f i i i f f f f f f i Resricions cn be eressed in he form Mf M n n n n n n n n n n n n Jorge Slvdor Mrques, consrins: F = e de F =

27 8 oin lgorihm. normlizion: move he men o he origin nd normlize vrince i i. Deermine fundmenl mri i i les squres: f is he eigenvecor ssocied o he smlles eigenvlue of M M; F is obined from f rnk : Fˆ UDV decomosiion of F. Denormlizion where U, V re obined by singulr vlue nd s D s F F ˆ Jorge Slvdor Mrques,

28 heorem: he scene cn be reconsruced from wo unclibred cmers ece for rojecive rnsformion. he reconsruced oins ˆ i re reled o he rue oins i by H i i Where H is n homogrhy wih 5 degrees of freedom. ˆ he reconsrucion obined in his wy does no reserve ngles, disnces or rllel lines nd i is denoed rojecive reconsrucion Jorge Slvdor Mrques,

29 Reconsrucion: unclibred cmers ) Algorihm esime mri F define ir of cmers comible wih F, e.g., e P [ I ] P F e use sndrd sereo reconsrucion mehods o esime Jorge Slvdor Mrques,

30 Jorge Slvdor Mrques, Reconsrucion: unclibred cmers ) Meric reconsrucion: Knowing irs of corresonding oins ) ˆ, i i Mh n n n n n n n n n M 4 h h h h h h h h h H 4

31 Referêncis R. Hrley, A. Zissermn, Mulile View Geomery, Cmbridge Universiy Press,. D. Forsyh, Ponce, Comuer Vision: Modern roch, Prenice Hll, P. H. S. orr, A Srucure nd Moion oolki in Mlb in Inercive Advenures in S nd Ml, Microsof Reserch, h://reserch.microsof.com/hilorr/, June CV-Online Jorge Slvdor Mrques,