Least squares and motion. Nuno Vasconcelos ECE Department, UCSD

Transcription

1 Las squars ad moo uo Vascoclos ECE Dparm UCSD

2 Pla for oda oda w wll dscuss moo smao hs s rsg wo was moo s vr usful as a cu for rcogo sgmao comprsso c. s a gra ampl of las squars problm w wll also wrap up dscusso o las squars roduc wo ps of moo smao bloc machg dffral mhods wll al abou moo ambgus ad local vs global moo

3 Las squars a las squars problm s o whr w hav wo varabls XY rlad b a uow fuco Y gx a rag s D { } a modl Y f;φ whr Φ s a vcor of paramrs h goal s: o fd h modl paramrs ha lad o h bs appromao o h obsrvd daa.. o drm h caocal ampl s h problm of fg a l o a s of pos hr * ε m Φ [ f Φ ] Φ a b ad f ; a b a + b 3

4 Two ma cass o-lar las squars fφ o lar o Φ.g. lar las squars fφ lar o Φ.g. f ; Φ s f ; Φ s o: all ha mars s lar o Φ boh olar o ohr lar modls: polomals spls ural wors Fourr dcomposos c. 4

5 5 o-lar las squars mos dffcul cas opmal soluo f ad ol f: grad of ε s zro Hssa of ε gav df gral hs has o closd form umrcal soluo.g. grad dsc pc al sma Φ ra < Φ z z z T ε Φ ε ε ε Φ ε ε ε ε ε L L Φ Φ Φ Φ + ε α εφ Φ ε

6 6 Lar las squars closd form soluo wr soluo s gv b ormal quaos.g. for a l f; + Φ Γ Φ Φ f f L T T Γ Γ Γ Φ Γ L

7 7 Vr powrful Q: wha s h bs lar appromao of a po squc b DFT sl poals? o g las squars soluo w d Γ 4 4 L L L

8 8 Bs Fourr appromao hs mas ha hs s orhoormal.. Γ T Γ ad.. h bs appromao ar h DFT coffcs assocad wh h poals Γ 4 4 L L L T T T Γ Γ Γ Γ Φ...

9 Sgal appromao Q: wha s h bad-pass flr h whos oupu bs appromas a sgal h frquc rag Ω? w hav s ha mus hav DFT Y X hc opmal flr has DFT Ω ohrws Ω H ohrws.. s h dal bad-pass flr of bad Ω uv: dal bs appromao LS ss! 9

10 oo smao s a mpora praccal ampl of LS problms ma applcaos: rcogo: ma vs ar characrzd b h p of moo.g. walg vs rug srog clus abou sc srucur.g. wh w roa a 3D obc moo of a pl drmd b how far h 3D po s from camra sgmao hgs ha mov oghr blog o h sam obc algm oc w ow h moo w ca alg mags a squc.g. h ASA paoramas comprsso sma moo alg mags rasm ol rror c

11 oo smao cosdr h followg wo mags m

12 oo smao cosdr h followg wo mags m +

13 oo smao goal: gv mags ad + for ach pl fd uv whch mmzs dffrc [ u v ] D + problm: mpossbl o solv from o pl alo wo uows uv o quao uv -u-v m 3

14 Fudamal law moo ca ol b solvd ovr a ghborhood d a las wo pls mas ss o cosdr mor ad mmz h avrag rror hs s las squars ε [ u v + ] uv -u-v m 4

15 Bloc machg fac s a o-lar las squars problm sc -u-v s a o-lar fuco of uv soluo : bloc machg for ach bloc + do a hausv sarch for h closs mach vr commo comprsso.g. PEG 5

16 Bloc machg s compuaoall sv d o compu h squard rror bw h bloc ad a collco of blocs h prvous mag dos o alwas produc good moo smas.g. ma machs ca b quall good hs s a problm for all moo smao mhods: moo ca b ambguous wh masurd locall.g. b machg wdows? 6

17 oo ambgus clarl w cao drm h moo of a fla ghborhood for a dg ghborhood w ca ol drm o of h wo compos h wo compos ar uqul dfd ol wh h ghborhood coas D mag srucur hs s calld h aprur problm?????? 7

18 8 Dffral mhods w ca a las lma h compl problm b loog for a closd-form soluo o problm: hs s a o-lar fuco of uv soluo: clarl h problm s du o hs quao ca b mad lar o uv b a Talor srs appromao v u + [ ] + d d v u * m ε v u v u

19 9 Dffral mhods whch lads o o: w ow how o compu hs rms A s h dffrc bw coscuv frams B s.. a fuco of h mag grad B A v u T A + T v u B T

20 Dffral mhods w hus hav ad h las squars problm s o: sc s cosa w om hs s ow lar las squars w ca us us our formula rcall ha + u v * ε + [ + u + v ]

21 Lar las squars f h h LS soluo s: wr soluo s gv b ormal quaos [ ] * m Φ Φ f ε Φ Γ Φ Φ f f L T T Γ Γ Γ Φ

22 Las squars soluo for moo sad of w hav ad wr [ ] * Φ f ε [ ] * + + v u ε Φ Φ v u f f

23 3 Las squars soluo h ormal quaos ar ladg o h soluo v u L L L L v u

24 4 Las squars soluo wh s hs wll dfd? o ha s h mar ha w usd o dc corrs s vrbl ol wh s wo gvalus ar o-zro H T wdow

25 prsg Wdows rcall: cosa wdow small gvalus dg wdow o mdum o small flow wdow o larg o small corr wdow wo larg gvalus H H H H hs cofrms wha w had alrad s: moo ca ol b compud uambguousl wh h ghborhood coas D formao.g. corrs 5

26 6 summar [UV] lsm w compu grads - for ach pl l wdow compu ma U u V v rur UV v u { } w w w w + +

27 7 Problms rcall w usd h Talor srs appromao hs s a good appromao ol for small uv o avod hs problm w d o us pramds v u v u +

28 Hrarchcal smao algorhm: do moo smao usg ad o oba u v warp wh u v : wpd -u -v up-sampl b o g wpd warp moo smao u v do moo smao usg ad o oba u v warp wh u v warp moo smao u v c. wpd... 8

29 Hrarchcal smao ach sag mprovs h mach u v soluo: upsampl all u v o full rsoluo add o oba uv u v + o ha small dsplacms a low rsoluo ar larg dsplacms a full rsoluo combs lar wh abl o sma larg dsplacms... 9

30 oo modls so far w hav dal local moo ach pl movs b slf raslao + u v local moo s h mos grc.g. r lavs blowg h wd o mpora alrav cas s ha of global moo moo of all pls sasfs o commo quao usuall du o camra moo: pag roao zoomg zoom roao 3

31 3 mpora cass po a m warpd o po a m + mpora global moos ar raslao b uv roao b θ scalg b s s θ uv s + v u ' ' θ θ θ θ cos s s cos ' ' s s ' '

32 3 Aff rasformaos hs ar all spcal cass of h aff rasformao moo of r mag dscrbd b Φ abcdf T ca accou for raslao roao scalg ad shar + f d c b a ' ' raslao roao uform scal ouform scal sharg

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