Motion Perception Under Uncertainty. Hongjing Lu Department of Psychology University of Hong Kong

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1 Moton Percepton Under Uncertanty Hongjng Lu Department of Psychology Unversty of Hong Kong

2 Outlne Uncertanty n moton stmulus Correspondence problem Qualtatve fttng usng deal observer models Based on sgnal detecton theory Based on Bayesan framework Quanttatve fttng usng Bayesan models wth prors Bayesan model wth generc prors Expectaton-Maxmzaton EM) mplementaton

3 Perceve Moton from Two Frames Frame 1

4 Perceve Moton from Two Frames Frame 2

5 Correspondence Problem Dot postons n the frst frame Dot postons n the second frame Stmulus generaton 4 dots move coherently Correspondence uncertanty

6 Random Dot Knematograms RDK) Common stmul used n psychophyscs and physology, Barlow et al. 1979, 1997); Brtten et al. 1992, 1995); ewsome et al. 1989, 1990), etc.

7 Moton Detecton Task Tral 1, Frame 1

8 Moton Detecton Task Tral 1, Frame 2 Do you perceve coherent moton?

9 Moton Detecton Task Tral 2, Frame 1

10 Moton Detecton Task Tral 2, Frame 2 Do you perceve coherent moton?

11 Moton Detecton Task Tral 1 Tral 2 Do you perceve coherent moton?

12 Random Dot Knematograms RDK) Common stmul used n psychophyscs and physology, Barlow et al. 1979, 1997); Brtten et al. 1992, 1995); ewsome et al. 1989, 1990), etc. =20, C=1.0 =20, C=0.5 =20, C=0.1 : number of dots C: coherence rato, the proporton of dots that move wth the same velocty

13 Psychophyscal Experments Manpulate vsual stmul by changng Dot number, Dot dsplacement between two frames umber of frames Human performance measurement Accuracy hgh accuracy ndcates good performance) Coherence threshold s the coherence rato needed to acheve 75% accuracy n a specfc task low coherence threshold ndcates good performance

14 Correspondence Pars Two dots Three dots =2, Ψ = 4 =3, Ψ = 9 # of possble correspondence pars, Ψ = 2

15 Coherence Threshold # of possble correspondence pars, = 2 Dffculty of coherent moton percepton would be expected to ncrease wth the number of dots n RDK.e. coherence threshold would be expected to ncrease wth the number of dots Ψ Dot umber

16 Human Coherence Threshold Intrgung psychophyscal fndng Barlow & Trpathy, 1997) Human coherence threshold s almost constant wth the number of dots n RDK

17 Outlne Uncertanty n moton stmulus Correspondence problem Qualtatve fttng usng deal observer models Based on sgnal detecton theory Based on Bayesan framework

18 Ideal Observer for Detectng Coherent Moton Ideal observer model provdes optmal performance as a benchmark to evaluate how well humans process moton nformaton on a specfc task Ideal observer usng sgnal detecton theory SDT) Barlow & Trpathy, 1997) Ideal observer model uses expermenter pror,.e. the proporton of dots that move coherently, C velocty of coherent moton, T

19 Ideal Observer Usng SDT Detecton decson s based on umber of matches, ψ,.e. the number of dots n the frst frame that have a correspondng dot wth dsplacement T n the second frame

20 Ideal Observer Usng SDT Observaton Two possbltes Random moton, the two matches are due to chance algnment of randomly movng dots Coherent moton, the two matches are from coherent moton of sgnal dots

21 Ideal Observer Usng SDT Random moton stmulus ψ ~ dots n the dsplay. For each dot n the frst frame, the probablty of a dot at the correspondng poston n the second frame s /Q. Q s the total number of possble postons n the dsplay Moton stmulus wth C dots movng coherently ψ C ~ Bnomal, /Q) Bnomal 1-C), /Q) < ψ 0 >= Q VAR ψ 0) = 1 Q Q <ψ C Assume 2 >= C + 1 C) Q VAR ψ C ) = VAR ψ 0 )

22 Ideal Observer Usng SDT Barlow & Trpathy,, 1997) Compute senstvty and coherence threshold usng SDT Senstvty < ψ C > < ψ 0 > d' = = C Q VAR ψ ) 0 Coherence threshold for d'= 1 C = 1 th Q 1 Q Q >> Ideal observer predcts that coherence threshold s constant wth the number of dots n RDK

23 Bayesan Ideal Observer Lu & Yulle, 2005) Defne correspondence varables Va V a = a a 1, f y = x + T; V = Random moton stmulus otherwse Generatve model for the RDK stmulus Moton stmulus wth C dots movng coherently 0, P { ya},{ x} rand) = P{ ya}) P{ x}) P{ y a },{ x } coh, T ) = V a P{ y a } { x },{ V a }, T ) P{ V wth the constrant a }) P{ x }) V a = C a

24 Bayesan Ideal Observer Generatve model for the RDK stmulus P{ y a },{ x } coh, T ) Pror dstrbuton for the dot postons allows all confguratons of the dots to be equally lkely P{ ya }) = P{ x}) = Q )! Q! 1 Q f Q >> Pror dstrbuton for correspondence varable s unform P{ V a }) = = V a C)!! P{ y a } { x },{ V C)! C)!! Condtonal Dstrbuton gven dot postons n the frst frame Q )! Va P{ ya} { x},{ Va}, T ) = δ ya, x + T )), Q C )! a a }, T ) P{ V a }) P{ x })

25 Bayesan Ideal Observer Bayesan Ideal Observer Generatve model for the RDK stmulus = V a a a a a x P V P T V x y P T coh x y P }) { }) { ) }, },{ } { { ), } },{ { )! )!! )! )!! )! ), )! )! Q Q C C C T x y C Q Q a a V a V a + = δ! )!! )! )!! )! )! )!! )! )! Q Q C C C C C C Q Q = ψ ψ Lkelhood rato )!! )!!! )! ) ), 2 s s Q Q C rand Data P T coh Data P = ψ ψ

26 Ideal Observer Ideal observer usng sgnal detecton theory s an approxmaton to Bayesan deal observer

27 Human Observer vs. Bayesan Ideal Observer Model Ideal observer model predcts the qualtatve trend of human performance Remarkably low human effcency, 0.3~1% Human Coherence Threshold: 30%~50% Ideal Coherence Threshold: 0.5%~1%

28 Why Low Effcency? Ideal observer model uses pror specfc to the experment, because deal observer model knows the proporton of dots that move coherently, C velocty of coherent moton, T 1 P v { x },{ ya}) = P{ ya} { x}, v) P v) Z Ideal observer model uses specfc pror knowledge about the RDK stmulus used n the experment

29 Why Low Effcency? Ideal observer model uses pror specfc to the experment Human observers mght use some generc pror consstent wth moton statstcs n natural scenes 1 P v { x },{ ya}) = P{ ya} { x}, v) P v) Z Ideal observer model uses specfc pror Human observers mght use generc pror

30 Outlne Uncertanty n moton stmulus Correspondence problem Qualtatve fttng usng deal observer models Based on sgnal detecton theory Based on Bayesan framework Quanttatve fttng usng Bayesan models wth prors Bayesan model wth generc prors Expectaton-Maxmzaton EM) mplementaton

31 Generc Pror: Slow & Smooth Prefer slow and spatally smooth moton Slowness: most objects are statc Smoothness: object features tend to move together atural statstcs of moton flow Roth & Black, 2005) log-hstogram of Horzontal velocty, u Vertcal velocty, v

32 Generc Pror: Slow & Smooth Prefer slow and spatally smooth moton P v Lv m= v) e e 0 = 2m mr 2 σ v r 2 λ m!2 m λ m x Yulle & Grzywacz 1988) = Slowness v = 0 Frst-order Smoothness v = 0 x Second-order Smoothness 2 v = 0 2 x Preferred Less Preferred

33 Bayesan Model wth Prors Ideal observer model uses expermenter pror,.e. the proporton of dots that move coherently, C velocty of coherent moton, T Slow & smooth model estmates velocty feld wthout requrng any knowledge about stmulus parameters

34 otaton n Slow & Smooth Model Bnary correspondence varable M M a a = 1, = 0, M 0 =1, M 0 = 0, f x corresponds otherwse Bnary outler varable to y a, x f the -th dot n the frst frame s an outler wthout a correspondng dot n the second frame otherwse wth a constrant that ensures that dot the frst frame s ether unmatched as an outler, or s matched to a dot a n the second frame, Probablstc correspondence m P M = 1) m P M 1) a = a y 0 = 0 = a a= 0 M a wth the constrant = 1 a= 0 m a = 1

35 Bayesan Model wth Slow & Smooth Pror Goal: maxmze Apply Bayes rule wth the constrant = v v P { x},{ ya}) = P,{ M a} { x},{ ya}) 2 M a ya x v x )) / T 2 1 λ Lv / T ξ M Z e = 1a= 1 of a= 0 M M a = 1, v 1 v v P,{ M a} { x},{ ya}) = P{ ya} { x},,{ M a}) P ) P{ M a}) Z Lkelhood e e SS generc pror on velocty feld = 1 0 / T Bas aganst msmatch

36 Expectaton-Maxmzaton EM) Dempster, Lard, & Rubn 1977) The EM algorthm s a general method of estmatng parameters of an underlyng dstrbuton from a gven data set when the data s ncomplete or has mssng values Two man applcatons of EM When the data ndeed has mssng values, due to problems wth or lmtatons of the observaton process When maxmzng the lkelhood or posteror probablty s analytcally ntractable, but t s doable by assumng hdden varables to smplfy the dstrbuton functon

37 Expectaton-Maxmzaton EM) E-step Replace hdden varables by ther expectatons condtonal on the estmated parameter In the current example, E-step s to evaluate M-step m t) t) t1) a = P M a = 1 { x},{ ya}, v t of the hdden varable M a, condtonal on the observed data and the velocty feld from the last M-step estmate parameters by maxmzng expected log posteror dstrbuton as f hdden varables were observed ) t) v = arg max v E m [ log, { },{ })] t ) P v m x y a a

38 Goal: maxmze EM Implementaton 2 2 M a ya x v x )) / T λ Lv = 1a= 1 = 1 P v x y = e { The classcal EM algorthm s equvalent to mnmzng the free energy functon Frey & Hnton, 1996) },{ 2 F[ m, v] = m y x v x )) + λ Lv + ξ = 1 a= 1 a a }) a M 1 Z wth the constrant of 2 = 1 m 0 + T a= 0 = 1 a= 1 wth the constrant m / T ξ M a log m a= 0 m a a 0 M a = 1, = 1

39 Mnmzng free energy E step: EM Implementaton mnmzng ths energy functon wth respect to m a whle holdng v constant M step: 2 F[ m, v] = m y x v x )) + λ Lv + ξ = 1 a= 1 a a mnmzng ths energy functon wth respect to v whle holdng m a constant 2 = 1 m 0 + T = 1 a= 1 wth the constrant m a log m a= 0 m a a = 1

40 EM Implementaton EM Implementaton E step: update the correspondences = = + = + = a T x v x y T T a T x v x y T T x v x y a a a a e e e m e e e m 1 ) / )) / / 0 1 ) / )) / ) / )) ξ ξ ξ = 0 m a F 0 0 = m F

41 EM Algorthm EM Algorthm m M step: update the velocty feld = = = = + a a a j j j a j x x m m x x G ) ), β λδ = = j j j x x G x v 1 ), ) β = ) exp 2 1 ), σ πσ j j x x x x G where = 0 v E

42 EM Iteraton Steps =50, C=0.5

43 Exp1. Coherence Threshold n Detectng Coherent Moton

44 Model Predcton n a Tral Stmulus =50, C=0.4 Model Predcton

45 Interestng Model Predcton Stmulus =50, C=0.1 Model Predcton What would humans perceve?

46 Exp 2: Moton Percepton n a Sngle Tral =50, C=0.4 Stmulus Human perceved global moton drecton: 2 =50, C=0.4 Model predcted global moton drecton: 1

47 Exp 2: Moton Percepton n a Sngle Tral =50, C=0.1 Stmulus Human perceved global moton drecton: 298 =50, C=0.1 Model predcted global moton drecton: 294

48 Expermental Test of Generc Prors Conjecture Percepton wll be largely domnated by prors when data s extremely nosy, e.g. random moton

49 Exp 2: Moton Percepton n a Sngle Tral =50, C=0 Stmulus Human perceved global moton drecton: 57 =50, C=0% Model predcted global moton drecton: 64

50 Tral-by by-tral Correlaton Between Human Percepton and Model Predcton Subject BR r = 0.83 r = 0.83

51 Tral-by by-tral Correlaton Between Human Percepton and Model Predcton Subject BR r = 0.77 r = 0.74

52 Can humans perceve any structure n random moton? Can Bayesan model wth Slow & Smooth pror predct human percepton?

53 Tral-by by-tral Correlaton Between Human Percepton and Model Predcton Subject BR r = 0.73 r = 0.76

54 Tral-by by-tral Correlaton, =50

55 Tral-by by-tral Correlaton, =500

56 Conclusons Synergy between perceptual experments and Bayesan framework Ideal observer wth specfc expermental pror predcts human performance qualtatvely Bayesan model wth Slow & Smooth generc pror predcts human performance both qualtatvely and quanttatvely Model predcts new phenomena, ncludng percevng structure n random moton