Parameter estimation class 5

Transcription

1 Parameter estmaton class 5 Multple Ve Geometr Comp 9-89 Marc Pollefes

2 Content Background: Projectve geometr (D, 3D), Parameter estmaton, Algortm evaluaton. Sngle Ve: Camera model, Calbraton, Sngle Ve Geometr. o Ves: Eppolar Geometr, 3D reconstructon, Computng F, Computng structure, Plane and omograpes. ree Ves: rfocal ensor, Computng. More Ves: N-Lneartes, Multple ve reconstructon, Bundle adjustment, autocalbraton, Dnamc SfM, Ceralt, Dualt

3 Multple Ve Geometr course scedule (subject to cange) Jan. 7, 9 Intro & motvaton Projectve D Geometr Jan. 4, 6 (no class) Projectve D Geometr Jan., 3 Projectve 3D Geometr (no class) Jan. 8, 3 Parameter Estmaton Parameter Estmaton Feb. 4, 6 Algortm Evaluaton Camera Models Feb., 3 Camera Calbraton Sngle Ve Geometr Feb. 8, Eppolar Geometr 3D reconstructon Feb. 5, 7 Fund. Matr Comp. Structure Comp. Mar. 4, 6 Planes & Homograpes rfocal ensor Mar. 8, ree Ve Reconstructon Multple Ve Geometr Mar. 5, 7 MultpleVe Reconstructon Bundle adjustment Apr., 3 Auto-Calbraton Papers Apr. 8, Dnamc SfM Papers Apr. 5, 7 Ceralt Papers Apr., 4 Dualt Project Demos

4 Projectve 3D Geometr Ponts, lnes, planes and quadrcs ransformatons П, ω and Ω

5 Sngular Value Decomposton A UΣ V UΣΣ V Homogeneous least-squares mn A subject to soluton Vn

6 Parameter estmaton D omograp Gven a set of (, ), compute H ( H ) 3D to D camera projecton Gven a set of (, ), compute P ( P ) Fundamental matr Gven a set of (, ), compute F ( F ) rfocal tensor Gven a set of (,, ), compute

7 Number of measurements requred At least as man ndependent equatons as degrees of freedom requred Eample: λ ' 3 H ndependent equatons / pont 8 degrees of freedom 4 8

8 Appromate solutons Mnmal soluton 4 ponts eld an eact soluton for H More ponts No eact soluton, because measurements are neact ( nose ) Searc for best accordng to some cost functon Algebrac or geometrc/statstcal cost

9 Gold Standard algortm Cost functon tat s optmal for some assumptons Computatonal algortm tat mnmzes t s called Gold Standard algortm Oter algortms can ten be compared to t

10 Drect Lnear ransformaton (DL) H H H 3 H ( ),, A

11 Drect Lnear ransformaton (DL) Equatons are lnear n 3 A A A A Onl out of 3 are lnearl ndependent (ndeed, eq/pt) 3 (onl drop trd ro f ) Holds for an omogeneous representaton, e.g. (,,)

12 Drect Lnear ransformaton Solvng for H (DL) A A A A 3 A4 sze A s 89 or 9, but rank 8 rval soluton s 9 s not nterestng -D null-space elds soluton of nterest pck for eample te one t

13 Drect Lnear ransformaton (DL) Over-determned soluton No eact soluton because of neact measurement.e. nose A A A M An Fnd appromate soluton - Addtonal constrant needed to avod, e.g. A - not possble, so mnmze A

14 DL algortm Objectve Gven n 4 D to D pont correspondences { }, determne te D omograp matr H suc tat H Algortm () () For eac correspondence compute A. Usuall onl to frst ros needed. Assemble n 9 matrces A nto a sngle n9 matr A () Obtan SVD of A. Soluton for s last column of V (v) Determne H from

15 Inomogeneous soluton ' ' ~ ' ' ' ' ' ' ' ' ' ' Snce can onl be computed up to scale, pck j, e.g. 9, and solve for 8-vector ~ Solve usng Gaussan elmnaton (4 ponts) or usng lnear least-squares (more tan 4 ponts) Hoever, f 9 ts approac fals also poor results f 9 close to zero erefore, not recommended Note 9 H 33 f orgn s mapped to nfnt [ ] H H l

16 Degenerate confguratons H? H? 3 (case A) (case B) 4 3 Constrants: H,,3,4 Defne: en, H 4l * l, * ( ),, 3 ( l 4 ) 4 H 4 * H 4 4 k H * s rank- matr and tus not a omograp If H * s unque soluton, ten no omograp mappng (case B) If furter soluton H est, ten also αh * +βh (case A) (-D null-space n stead of -D null-space)

17 Solutons from lnes, etc. D omograpes from D lnes l H A l Mnmum of 4 lnes 3D Homograpes (5 dof) Mnmum of 5 ponts or 5 planes D affntes (6 dof) Mnmum of 3 ponts or lnes Conc provdes 5 constrants Med confguratons?

18 Cost functons Algebrac dstance Geometrc dstance Reprojecton error Comparson Geometrc nterpretaton Sampson error

19 DL mnmzes e A Algebrac dstance A resdual vector e partal vector for eac ( ) algebrac error vector d alg alg (,H ) e algebrac dstance (, ) a a ere a ( a, a, a3 ) (, H ) e A e d + d alg Not geometrcall/statstcall meanngfull, but gven good normalzaton t orks fne and s ver fast (use for ntalzaton)

20 ˆ Geometrc dstance measured coordnates estmated coordnates true coordnates d(.,.) Eucldean dstance (n mage) Error n one mage Ĥ argmn d H H (, H ) Smmetrc transfer error e.g. calbraton pattern ( - ), H + d(,h ) Ĥ argmn d Reprojecton error ( ) Ĥ, ˆ, ˆ argmn d(, ˆ ) + d(, ˆ ) H,ˆ,ˆ subject to ˆ Ĥˆ

21 Reprojecton error d ( - ), H + d(, H) d (, ˆ ) + d(, ˆ )

22 Comparson of geometrc and algebrac dstances e ˆ ˆ ˆ ˆ A Error n one mage ( ),, ( ) H ˆ, ˆ, ˆ ˆ 3 ( ) ( ) ( ) alg ˆ ˆ ˆ ˆ ˆ, d + ( ) ( ) ( ) ( ) ( ) d d + ˆ / ˆ, / ˆ / ˆ ˆ / ˆ / ˆ, alg / tpcal, but not, ecept for affntes ˆ 3 For affntes DL can mnmze geometrc dstance Possblt for teratve algortm

23 Geometrc nterpretaton of reprojecton error Estmatng omograp~ft surface to ponts (,,, ) n 4 H d ˆ ν H represents quadrcs n 4 (quadratc n ) ( ˆ ) + ( ˆ ) + ( ˆ ) + ( ˆ ) d (, ˆ ) + d(, ˆ ) (, ˆ ) + d(, ˆ ) d (, ν ) Analog to conc fttng H (,C) C d alg ( ),C d

24 Sampson error ˆ beteen algebrac and geometrc error ν H ˆ Vector tat mnmzes te geometrc error s te closest pont on te varet to te measurement Sampson error: st order appromaton of ˆ A CH ( ) C H C C H ( + δ ) CH( ) + δ + C H ( ) δ H δ Jδ ˆ C ( ˆ ) H e t J C H δ Fnd te vector tat mnmzes subject to δ Jδ e

25 δ Fnd te vector tat mnmzes subject to δ Jδ e Use Lagrange multplers: mnmze ( Jδ + e) δ δ - λ dervatves δ - λ J ( Jδ + e) δ J λ JJ λ + e λ JJ δ ( ) e ( ) J JJ e ˆ + e ( δ δ δ JJ ) e δ

26 Sampson error ˆ beteen algebrac and geometrc error ν H ˆ Vector tat mnmzes te geometrc error s te closest pont on te varet to te measurement Sampson error: st order appromaton of ˆ A CH ( ) C H C C H ( + δ ) CH ( ) + δ + C H ( ) δ H δ Jδ ˆ C ( ˆ ) H e δ Fnd te vector tat mnmzes subject to δ Jδ e δ ( ) JJ e δ δ e (Sampson error)

27 A fe ponts () () () (v) (v) (v) (v) Sampson appromaton δ e JJ For a D omograp (,,, ) e C H H J C ( ) s te algebrac error vector s a 4 matr, e.g. J ( ) e Smlar to algebrac error n fact, same as Maalanobs dstance Sampson error ndependent of lnear reparametrzaton (cancels out n beteen e and J) Must be summed for all ponts ( ) / + Close to geometrc error, but muc feer unknons e e e e e JJ ( ) JJ e

28 Statstcal cost functon and Mamum Lkelood Estmaton Optmal cost functon related to nose model Assume zero-mean sotropc Gaussan nose (assume outlers removed) Pr Pr Error n one mage (, ) / ( ) ( σ e d ) πσ d ({ } ) (,H ) ( ) / σ H e log Pr Π πσ ({ } H) d(,h ) + constant σ Mamum Lkelood Estmate (,H ) d

29 Statstcal cost functon and Mamum Lkelood Estmaton Optmal cost functon related to nose model Assume zero-mean sotropc Gaussan nose (assume outlers removed) Pr Pr ({ } H) (, ) / ( ) ( σ e d ) πσ Error n bot mages Π πσ e (, ) d (,H ) /( σ ) d + Mamum Lkelood Estmate (, ˆ ) + d(, ˆ ) d

30 Maalanobs dstance General Gaussan case Measurement t covarance matr Σ ( ) ( ) Σ Σ Σ Σ + Error n to mages (ndependent) Σ Σ + Varng covarances

31 Net class: Parameter estmaton (contnued) ransformaton nvarance and normalzaton Iteratve mnmzaton Robust estmaton

32 Upcomng assgnment ake to or more potograps taken from a sngle vepont Compute panorama Use dfferent measures DL, MLE Use Matlab Due Feb. 3