Model Fitting, RANSAC. Jana Kosecka
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1 Model Fttg, RANSAC Jaa Kosecka
2 Fttg: Issues Prevous strateges Le detecto Hough trasform Smple parametrc model, two parameters m, b m + b Votg strateg Hard to geeralze to hgher dmesos a o + a + a a 3 3 Now put s a set of pots Nose the measured feature locatos Etraeous data: clutter (outlers), multple les Mssg data: occlusos
3 Fttg: Overvew If we kow whch pots belog to the le, how do we fd the optmal le parameters? Least squares What f there are outlers? Robust fttg, RANSAC What f there are ma les? Votg methods: RANSAC, Hough trasform What f we re ot eve sure t s a le? Model selecto
4 Least squares le fttg Data: (, ),, (, ) Le equato: m + b Fd (m, b) to mmze Y X XB X db de T T ) ( ) ( ) 2( ) ( ) ( 2 XB XB Y XB Y Y XB Y XB Y XB Y E b m B X Y T T T T +!!! Normal equatos: least squares soluto to XBY b m E 2 ) ( (, ) m+b Y X XB X T T
5 Problem wth vertcal least squares Not rotato-varat Fals completel for vertcal les
6 Total least squares Dstace betwee pot (, ) ad le a+bd (a 2 +b 2 ): a + b d E 2 ( a( + b, d ) a+bd Ut ormal: N(a, b)
7 Total least squares Dstace betwee pot (, ) ad le a+bd (a 2 +b 2 ): a + b d Fd (a, b, d) to mmze the sum of squared perpedcular dstaces E a + b E ( a + b d) 2 2 ( ( d, ) a+bd Ut ormal: N(a, b)
8 Total least squares Dstace betwee pot (, ) ad le a+bd (a 2 +b 2 ): a + b d Fd (a, b, d) to mmze the sum of squared perpedcular dstaces E a + b E ( a + b d) 2 2 ( ( d, ) a+bd Ut ormal: N(a, b) E a b 2( + ) 0 a b d d a b + + d 2 2 a T E ( a( ) b( )) ( UN) ( UN) +!! b de T 2 ( U U ) N 0 dn Soluto to (U T U)N 0, subject to N 2 : egevector of U T U assocated wth the smallest egevalue (least squares soluto to homogeeous lear sstem UN 0)
9 Total least squares U!! T U U 2 2 ) ( ) )( ( ) )( ( ) ( secod momet matr
10 Total least squares U!! T U U 2 2 ) ( ) )( ( ) )( ( ) ( ), ( N (a, b) secod momet matr ), (
11 Least squares: Robustess to ose Least squares ft to the red pots:
12 Least squares: Robustess to ose Least squares ft wth a outler: Problem: squared error heavl pealzes outlers
13 Robust estmators Geeral approach: fd model parameters θ that mmze ( r (, θ ) σ ) ρ ; r (, θ) resdual of -th pot w.r.t. model parameters θ ρ robust fucto wth scale parameter σ The robust fucto ρ behaves lke squared dstace for small values of the resdual u but saturates for larger values of u
14 Choosg the scale: Just rght The effect of the outler s mmzed
15 Choosg the scale: Too small The error value s almost the same for ever pot ad the ft s ver poor
16 Choosg the scale: Too large Behaves much the same as least squares
17 Robust estmato: Detals Robust fttg s a olear optmzato problem that must be solved teratvel Least squares soluto ca be used for talzato Adaptve choce of scale: appro..5 tmes meda resdual (F&P, Sec. 5.5.)
18 RANSAC Robust fttg ca deal wth a few outlers what f we have ver ma? Radom sample cosesus (RANSAC): Ver geeral framework for model fttg the presece of outlers Outle Choose a small subset of pots uforml at radom Ft a model to that subset Fd all remag pots that are close to the model ad reject the rest as outlers Do ths ma tmes ad choose the best model M. A. Fschler, R. C. Bolles. Radom Sample Cosesus: A Paradgm for Model Fttg wth Applcatos to Image Aalss ad Automated Cartograph. Comm. of the ACM, Vol 24, pp , 98.
19 RANSAC for le fttg eample Source: R. Raguram
20 RANSAC for le fttg eample Least- squares ft Source: R. Raguram
21 RANSAC for le fttg eample. Radoml select mmal subset of pots Source: R. Raguram
22 RANSAC for le fttg eample. Radoml select mmal subset of pots 2. Hpothesze a model Source: R. Raguram
23 RANSAC for le fttg eample. Radoml select mmal subset of pots 2. Hpothesze a model 3. Compute error fuc<o Source: R. Raguram
24 RANSAC for le fttg eample. Radoml select mmal subset of pots 2. Hpothesze a model 3. Compute error fuc<o 4. Select pots cosstet wth model Source: R. Raguram
25 RANSAC for le fttg eample. Radoml select mmal subset of pots 2. Hpothesze a model 3. Compute error fuc<o 4. Select pots cosstet wth model 5. Repeat hpothesze- ad- verf loop Source: R. Raguram
26 RANSAC for le fttg eample Source: R. Raguram. Radoml select mmal subset of pots 2. Hpothesze a model 3. Compute error fuc<o 4. Select pots cosstet wth model 5. Repeat hpothesze- ad- verf loop 39
27 RANSAC for le fttg eample Ucotamated sample Source: R. Raguram. Radoml select mmal subset of pots 2. Hpothesze a model 3. Compute error fuc<o 4. Select pots cosstet wth model 5. Repeat hpothesze- ad- verf loop 40
28 RANSAC for le fttg eample. Radoml select mmal subset of pots 2. Hpothesze a model 3. Compute error fuc<o 4. Select pots cosstet wth model 5. Repeat hpothesze- ad- verf loop Source: R. Raguram
29 RANSAC for le fttg Repeat N tmes: Draw s pots uforml at radom Ft le to these s pots Fd lers to ths le amog the remag pots (.e., pots whose dstace from the le s less tha t) If there are d or more lers, accept the le ad reft usg all lers
30 Choosg the parameters Ital umber of pots s Tpcall mmum umber eeded to ft the model Dstace threshold t Choose t so probablt for ler s p (e.g. 0.95) Zero-mea Gaussa ose wth std. dev. σ: t σ 2 Number of samples N Choose N so that, wth probablt p, at least oe radom sample s free from outlers (e.g. p0.99) (outler rato: e) Source: M. Pollefes
31 Choosg the parameters Ital umber of pots s Tpcall mmum umber eeded to ft the model Dstace threshold t Choose t so probablt for ler s p (e.g. 0.95) Zero-mea Gaussa ose wth std. dev. σ: t σ 2 Number of samples N Choose N so that, wth probablt p, at least oe radom sample s free from outlers (e.g. p0.99) (outler rato: e) ( ( ) ) s N e p N ( ) s ( p) / log ( e) log proporto of outlers e s 5% 0% 20% 25% 30% 40% 50% Source: M. Pollefes
32 Choosg the parameters Ital umber of pots s Tpcall mmum umber eeded to ft the model Dstace threshold t Choose t so probablt for ler s p (e.g. 0.95) Zero-mea Gaussa ose wth std. dev. σ: t σ 2 Number of samples N Choose N so that, wth probablt p, at least oe radom sample s free from outlers (e.g. p0.99) (outler rato: e) ( ( ) ) s N e p N ( ) s ( p) / log ( e) log Source: M. Pollefes
33 Choosg the parameters Ital umber of pots s Tpcall mmum umber eeded to ft the model Dstace threshold t Choose t so probablt for ler s p (e.g. 0.95) Zero-mea Gaussa ose wth std. dev. σ: t σ 2 Number of samples N Choose N so that, wth probablt p, at least oe radom sample s free from outlers (e.g. p0.99) (outler rato: e) Cosesus set sze d Should match epected ler rato Source: M. Pollefes
34 Adaptvel determg the umber of samples Iler rato e s ofte ukow a pror, so pck worst case, e.g. 50%, ad adapt f more lers are foud, e.g. 80% would eld e0.2 Adaptve procedure: N, sample_cout 0 Whle N >sample_cout Choose a sample ad cout the umber of lers Set e (umber of lers)/(total umber of pots) Recompute N from e: N ( ) s ( p) / log ( e) log Icremet the sample_cout b Source: M. Pollefes
35 RANSAC pros ad cos Pros Smple ad geeral Applcable to ma dfferet problems Ofte works well practce Cos Lots of parameters to tue Does t work well for low ler ratos (too ma teratos, or ca fal completel) Ca t alwas get a good talzato of the model based o the mmum umber of samples
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