Outlier-tolerant parameter estimation
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- Garry Cameron
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1 Outlr-tolrant paramtr stmaton Baysan thods n physcs statstcs machn larnng and sgnal procssng (SS 003 Frdrch Fraundorfr fraunfr@cg.tu-graz.ac.at Computr Graphcs and Vson Graz Unvrsty of Tchnology
2 Outln Eppolar Gomtry A Baysan Formulaton AP stmaton usng RANSAC Exampl
3 Eppolar Gomtry
4 A Baysan Formulaton Θ ˆ arg max Pr( Θ arg max Θ Θ Pr( Θ Pr( Θ Pr( Θ { } Θ.. st of paramtrs Θˆ.. AP stmat.. data (mag corrspondncs.. hypothszd modl (moton modl.. condtonng nformaton
5 Paramtr st Θ.. 9 paramtrs of fundamntal matrx.. paramtrs of pont corrspondncs.. st of ndcator varabls such that for nlr 0 for outlr } { Θ F 0 x T Fx n y x y x m.. (..n
6 What to stmat (CV? AP stmat of. Rcovrs camra vwpont and 3 structur. Paramtrs of th rlaton Whch matchs ar nlrs 3 coordnats of matchs
7 Lklhood [ 3] Pr( Θ Pr( Pr( pnds only on th paramtrs of th matchs and whch dtrmns whthr or not th matchs ar outlrs or nlrs. W now assum that th lklhood s a mxtur modl of dstrbutons for nlrs and outlrs.
8 Lklhood [ 3] W assum that w only know th frst two momnts of th nlr dstrbuton and only th rang of th outlrs thn th maxmally non nformatv dstrbutons ar Gaussan dstrbuton for nlrs and unform dstrbuton for outlrs. + n j j j j j n y y x x.. ˆ ( ( ˆ xp( ( Pr( π ˆ ( ( ˆ j j j j j y y x x +
9 Lklhood [3 3] Outlr dstrbuton...sarchwndow...bldgross S S L L S S L L v v Pr( + n n v.... ( xp( ( ( ( Pr( π ψ nlr (dstr. outlr (dstr.
10 Pror [ ] Pr( Θ Pr( pror of matchs Pr( Pr( pror of rlaton W assum that and ar ndpndnt (rasonabl for ths CV problm Pr( Pr( Pr( unformly dstrbutd (th paramtrs assocatd wth ach match ar ndpndnt c L L S Pr( L L...Bldgross c S...Sarchrang
11 Pror [ ] W assum that th ar ndpndnt and that th pror probablty of a gvn pont bng an nlr s constant ovr all {Pr( } ; s th pror xpctaton of sng an nlr (xpctd prcntag of nlrs n th data Pr( wll typcally hav lttl ffct as n and won t b ncludd n th postror modl
12 Postror of th modl Pr( Pr( ( xp( ( ( Pr( Pr( Pr( Pr( Pr( Pr( Pr(.. v n π + Θ
13 argnalzaton ovr [ ]... ( n ntgrat th unform dstrbuton Pr( Pr( Pr( ( xp( ( ( ( Pr(.. v n π + Θ v c v ( ( c Pr(
14 argnalzaton ovr [ ]... ( n ntgrat th Gaussan part Th AP stmat xp xp π π c d ( v c d ( xp... π +
15 trmnng ndcator varabls Pont corrspondncs wth an r-projcton rror < T ar consdrd to b nlrs ˆ 0 > T T T c log π v d Th AP stmat margnalzng ovr and d π... ˆ xp c ( + ( ˆ v
16 APSAC: An algorthm for th AP Estmator W want an algorthm for ffcntly fndng mods n th postror. da s to xtnd th RANSAC- Algorthm APSAC axmum a postror sampl consnsus
17 RANSAC [ ] Random Sampl Consnsus vd data ponts nto nlrs and outlrs Us only nlrs to calculat th modl Randomly sampl substs of th data n th hop to gt on subst wth no outlrs
18 RANSAC [ ]. From th data st slct randomly a sampl of d data ponts. Estmat a modl from ths subst.. trmn th st of data ponts d whch ar wthn a dstanc thrshold T of th modl. Ths s th consnsus st of th sampl and contans th nlrs for th currnt modl. 3. f th numbr of nlrs (from d s gratr than som thrshold T trmnat. R-stmat th modl usng only th nlrs. 4. f th numbr of nlrs s lss than T go back to stp. 5. Aftr N trals th largst consnsus st s slctd and th modl s r-stmatd usng only th nlrs f th run hasn t bn trmnatd bfor.
19 Usng RANSAC for AP Estmaton RANSAC fnds th mnumum of a cost functon dfnd as Approxmat th postror by mnmzng C ρ < T pos const T.. 0 ρ C ρ < T T T ρ
20 APSAC-Algorthm [Torr00]
21 Rsults on Eppolar Gomtry stmaton St of 0 pont corrspondncs mnmum numbr of 7 rqurd for Fundamntal atrx stmaton 4 tru nlrs 6 tru outlrs
22 Usng nlrs only
23 Usng nlrs&outlrs
24 Robust stmaton
25 Rfrncs Torr P.H.S.: Baysan odl Estmaton and Slcton for Eppolar Gomtry and Gnrc anfold Fttng Fschlr.A. and Bolls R.C.: Random sampl consnsus: A paradgm for modl fttng wth applcaton to mag analyss and automatd cartography
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