Fitting. We ve learned how to detect edges, corners, blobs. Now what? We would like to form a. compact representation of
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1 Fttg
2 Fttg We ve leared how to detect edges, corers, blobs. Now what? We would lke to form a hgher-level, h l more compact represetato of the features the mage b groupg multple features accordg to a smple model 9300 Harrs Corers Pkw, Charlotte, NC
3 Fttg Choose a parametrc model to represet a set of features smple model: les smple model: crcles complcated model: car Source: K. Grauma
4 Fttg: Issues Case stud: Le detecto Nose the measured feature locatos Etraeous data: clutter outlers, multple les Mssg data: occlusos
5 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
6 Least squares le fttg Data:,,,, Le equato: m + b Fd m, b to mmze m+b, b m E, b m B X Y M M M XB XB Y XB Y Y XB Y XB Y XB Y E b T T T T + 0 Y X XB X db de T T Normal equatos: least squares soluto to XBY Y X XB X T T
7 Problem wth vertcal least squares Not rotato-varat Fals completel for vertcal les
8 Total least squares Dstace betwee pot, ad le a+bd a +b : a + b d E a+bd Ut ormal: a + b, d Na, b
9 Total least squares Dstace betwee pot, ad le a+bd a +b : a + b d Fd a, b, d to mmze the sum of squared perpedcular dstaces E a + b E a + b d d, a+bd Ut ormal: Na, b
10 Total least squares Dstace betwee pot, ad le a+bd a +b : a + b d Fd a, b, d to mmze the sum of squared perpedcular dstaces E a + b E a + b E + 0 a b d d d E a b + M M de T U U N 0 dn d d, a+bd Ut ormal: Na, b a b a b + + a b UN T UN Soluto to U T UN 0 subject to N : egevector of U T U Soluto to U T UN 0, subject to N : egevector of U T U assocated wth the smallest egevalue least squares soluto to homogeeous lear sstem UN 0
11 Total least squares U M M T U U secod momet matr secod momet matr
12 Total least squares U M M T U U secod momet matr N a b secod momet matr N a, b,,
13 Least squares as lkelhood mamzato Geeratve model: le pots a+bd are sampled depedetl ad corrupted b Gaussa u, v ose the drecto ε perpedcular to the le, u a + ε v b pot o the le ose: sampled from zero-mea Gaussa wth std. dev. σ ormal drecto
14 Least squares as lkelhood mamzato a+bd u v Geeratve model: le pots are sampled depedetl ad corrupted b Gaussa, u, v ε ad corrupted b Gaussa ose the drecto perpedcular to the le + b a v u ε Lkelhood of pots gve le parameters a, b, d: b v + d b a d b a P d b a P ep,,,,,,,,, σ K + d b a d b a L,,,,,, σ K Log-lkelhood:
15 Least squares: Robustess to ose Least squares ft to the red pots:
16 Least squares: Robustess to ose Least squares ft wth a outler: Problem: squared error heavl pealzes outlers
17 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
18 Choosg the scale: Just rght The effect of the outler s mmzed
19 Choosg the scale: Too small The error value s almost the same for ever pot ad the ft s ver poor
20 Choosg the scale: Too large Behaves much the same as least squares
21 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.
22 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 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 4, pp , 98.
23 RANSAC for le fttg eample Source: R. Raguram
24 RANSAC for le fttg eample Least squares ft Source: R. Raguram
25 RANSAC for le fttg eample. Radoml select mmal subset of pots Source: R. Raguram
26 RANSAC for le fttg eample. Radoml select mmal subset of pots. Hpothesze a model Source: R. Raguram
27 RANSAC for le fttg eample. Radoml select mmal subset of pots. Hpothesze a model 3. Compute error fucto Source: R. Raguram
28 RANSAC for le fttg eample. Radoml select mmal subset of pots. Hpothesze a model 3. Compute error fucto 4. Select pots cosstet wth model Source: R. Raguram
29 RANSAC for le fttg eample. Radoml select mmal subset of pots. Hpothesze a model 3. Compute error fucto 4. Select pots cosstet wth model 5. Repeat hpothesze ad verf loop Source: R. Raguram
30 RANSAC for le fttg eample Source: R. Raguram. Radoml select mmal subset of pots. Hpothesze a model 3. Compute error fucto 4. Select pots cosstet wth model 5. Repeat hpothesze ad verf loop 30
31 RANSAC for le fttg eample Ucotamated sample Source: R. Raguram. Radoml select mmal subset of pots. Hpothesze a model 3. Compute error fucto 4. Select pots cosstet wth model 5. Repeat hpothesze ad verf loop 3
32 RANSAC for le fttg eample. Radoml select mmal subset of pots. Hpothesze a model 3. Compute error fucto 4. Select pots cosstet wth model 5. Repeat hpothesze ad verf loop Source: R. Raguram
33 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
34 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 Zero-mea Gaussa ose wth std. dev. σ: t 3.84σ 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
35 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 Zero-mea Gaussa ose wth std. dev. σ: t 3.84σ 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 proporto of outlers e s 5% 0% 0% 5% 30% 40% 50% s N e p s p / log e N log Source: M. Pollefes
36 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 Zero-mea Gaussa ose wth std. dev. σ: t 3.84σ 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 s p / log e N log Source: M. Pollefes
37 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 Zero-mea Gaussa ose wth std. dev. σ: t 3.84σ 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
38 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. 0 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
39 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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