CS 4495 Computer Vision Tracking 1- Kalman,Gaussian

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1 CS 4495 Compuer Vision A. Bobick CS 4495 Compuer Vision - KalmanGaussian Aaron Bobick School of Ineracive Compuing

2 CS 4495 Compuer Vision A. Bobick Adminisrivia S5 will be ou his Thurs Due Sun Nov h :55pm Calendar (enaive) done for he ear S6: /4 due /4 S7: /6 due /5 EAM: Tues before Thanksgiving. Covers conceps and basics So no final on Dec.

3 CS 4495 Compuer Vision A. Bobick Slides adaped from Krisen Grauman Deva Ramanan bu mosl from Svelana Lazebnik

4 CS 4495 Compuer Vision A. Bobick Some examples Older examples: Sae of he ar: hp://

5 CS 4495 Compuer Vision A. Bobick Feaure racking So far we have onl considered opical flow esimaion in a pair of images If we have more han wo images we can compue he opical flow from each frame o he nex Given a poin in he firs image we can in principle reconsruc is pah b simpl following he arrows

6 CS 4495 Compuer Vision A. Bobick challenges Ambigui of opical flow Find good feaures o rack Large moions Discree search insead of Lucas-Kanade Changes in shape orienaion color Allow some maching flexibili Occlusions disocclusions Need mechanism for deleing adding new feaures Drif errors ma accumulae over ime Need o know when o erminae a rack

7 CS 4495 Compuer Vision A. Bobick Handling large displacemens Define a small area around a pixel as he emplae Mach he emplae agains each pixel wihin a search area in nex image jus like sereo maching! Use a mach measure such as SSD or correlaion Afer finding he bes discree locaion can use Lucas- Kanade o ge sub-pixel esimae (hink of he emplae as he coarse level of he pramid).

8 CS 4495 Compuer Vision A. Bobick over man frames Selec feaures in firs frame For each frame: Updae posiions of racked feaures Discree search or Lucas-Kanade Sar new racks if needed Terminae inconsisen racks Compue similari wih corresponding feaure in he previous frame or in he firs frame where i s visible This is done b man companies and ssems ofen ad hoc rules ailored o he conex.

9 CS 4495 Compuer Vision A. Bobick Shi-Tomasi feaure racker Find good feaures using eigenvalues of second-momen marix ou ve seen his now wice! Ke idea: good feaures o rack are he ones ha can be racked reliabl From frame o frame rack wih Lucas-Kanade and a pure ranslaion model More robus for small displacemens can be esimaed from smaller neighborhoods Check consisenc of racks b affine regisraion o he firs observed insance of he feaure Affine model is more accurae for larger displacemens Comparing o he firs frame helps o minimize drif J. Shi and C. Tomasi. Good Feaures o Track. CVR 994.

10 CS 4495 Compuer Vision A. Bobick example J. Shi and C. Tomasi. Good Feaures o Track. CVR 994.

11 CS 4495 Compuer Vision A. Bobick wih dnamics Ke idea: Given a model of expeced moion predic where objecs will occur in nex frame even before seeing he image Resric search for he objec Improved esimaes since measuremen noise is reduced b rajecor smoohness

12 CS 4495 Compuer Vision A. Bobick as inference The hidden sae consiss of he rue parameers we care abou denoed. The measuremen is our nois observaion ha resuls from he underling sae denoed Y. A each ime sep sae changes (from - o ) and we ge a new observaion Y. Our goal: recover mos likel sae given All observaions seen so far. Knowledge abou dnamics of sae ransiions.

13 CS 4495 Compuer Vision A. Bobick as inference: inuiion measuremen Belief: predicion Belief: predicion Correced predicion old belief Time Time

14 CS 4495 Compuer Vision A. Bobick Seps of racking redicion: Wha is he nex sae of he objec given pas measuremens? ( Y Y )

15 CS 4495 Compuer Vision A. Bobick Seps of racking redicion: Wha is he nex sae of he objec given pas measuremens? ( Y Y ) Correcion: Compue an updaed esimae of he sae from predicion and measuremens ( Y Y Y )

16 CS 4495 Compuer Vision A. Bobick Seps of racking redicion: Wha is he nex sae of he objec given pas measuremens? ( Y Y ) Correcion: Compue an updaed esimae of he sae from predicion and measuremens (poserior) ( Y Y Y ) can be seen as he process of propagaing he poserior disribuion of sae given measuremens across ime

17 CS 4495 Compuer Vision A. Bobick Simplifing assumpions Onl he immediae pas maers ( ) ( ) dnamics model

18 CS 4495 Compuer Vision A. Bobick Simplifing assumpions Onl he immediae pas maers dnamics model Measuremens depend onl on he curren sae ( Y Y Y ) ( Y ) ( ) ( ) observaion model

19 CS 4495 Compuer Vision A. Bobick Simplifing assumpions Onl he immediae pas maers dnamics model Measuremens depend onl on he curren sae ( Y Y Y ) ( Y ) ( ) ( ) Hmmm. observaion model Y Y Y

20 CS 4495 Compuer Vision A. Bobick as inducion Base case: Assume we have iniial prior ha predics sae in absence of an evidence: ( ) A he firs frame correc his given he value of Y Given correced esimae for frame : redic for frame Correc for frame predic correc

21 CS 4495 Compuer Vision A. Bobick as inducion Base case: Assume we have iniial prior ha predics sae in absence of an evidence: ( ) A he firs frame correc his given he value of Y ) ( ) ( ) ( ) ( ) ( ) ( Y

22 CS 4495 Compuer Vision A. Bobick redicion redicion involves guessing given ( ) ( )

23 CS 4495 Compuer Vision A. Bobick redicion redicion involves guessing given d d d Law of oal probabili - Marginalizaion

24 CS 4495 Compuer Vision A. Bobick redicion redicion involves guessing given d d d Condiioning on

25 CS 4495 Compuer Vision A. Bobick redicion redicion involves guessing given d d d Independence assumpion

26 CS 4495 Compuer Vision A. Bobick Correcion ( ) Correcion involves compuing given prediced value ( ) and

27 CS 4495 Compuer Vision A. Bobick Correcion Correcion involves compuing given prediced value and d Baes rule

28 CS 4495 Compuer Vision A. Bobick Correcion Correcion involves compuing given prediced value and d Independence assumpion (observaion depends onl on sae )

29 CS 4495 Compuer Vision A. Bobick Correcion Correcion involves compuing given prediced value and d Condiioning on Reall a normalizaion

30 CS 4495 Compuer Vision A. Bobick Summar: redicion and correcion redicion: ( ) d dnamics model correced esimae from previous sep

31 CS 4495 Compuer Vision A. Bobick Summar: redicion and correcion redicion: ( ) d Correcion: dnamics model observaion model correced esimae from previous sep prediced esimae ( ) ( ) ( ) ( ) ( ) d

32 CS 4495 Compuer Vision A. Bobick Linear Dnamic Models Dnamics model: sae undergoes linear ransformaion plus Gaussian noise x ( x ) ~ N D Σ d Observaion model: measuremen is linearl ransformed sae plus Gaussian noise ( x Σ ) ~ N M m

33 CS 4495 Compuer Vision A. Bobick Example: Consan veloci (D) Sae vecor is posiion and veloci x p v Measuremen is posiion onl p v v p ( ) v ξ ε p x D x noise v noise (greek leers denoe noise erms) p Mx noise v [ ] noise

34 CS 4495 Compuer Vision A. Bobick Example: Consan acceleraion (D) Sae vecor is posiion veloci and acceleraion Measuremen is posiion onl [ ] noise a v p noise Mx ζ ξ ε ) ( ) ( a a a v v v p p a v p x noise a v p noise D x x (greek leers denoe noise erms)

35 CS 4495 Compuer Vision A. Bobick The Kalman filer Mehod for racking linear dnamical models in Gaussian noise The prediced/correced sae disribuions are Gaussian You onl need o mainain he mean and covariance The calculaions are eas (all he inegrals can be done in closed form)

36 CS 4495 Compuer Vision A. Bobick ropagaion of Gaussian densiies

37 CS 4495 Compuer Vision A. Bobick The Kalman Filer: D sae Make measuremen Given correced sae from previous ime sep and all he measuremens up o he curren one predic he disribuion over he curren sep ( ) Mean and sd. dev. of prediced sae: µ redic Time advances (from o ) Correc Given predicion of sae and curren measuremen updae predicion of sae ( ) Mean and sd. dev. of correced sae: µ

38 CS 4495 Compuer Vision A. Bobick D Kalman filer: redicion Linear dnamic model defines prediced sae evoluion wih noise ( ) ~ N dx d Wan o esimae disribuion for nex prediced sae ( ) ( ) ( ) d

39 CS 4495 Compuer Vision A. Bobick D Kalman filer: redicion Linear dnamic model defines prediced sae evoluion wih noise Wan o esimae disribuion for nex prediced sae ( ) N µ ( ( ) ~ N dx d ) Updae he mean: Updae he variance: ( µ dµ ) ( d d )

40 CS 4495 Compuer Vision A. Bobick D Kalman filer: Correcion Mapping of sae o measuremens: rediced sae: Wan o esimae correced disribuion ~ m mx N Y ) ( N µ d

41 CS 4495 Compuer Vision A. Bobick D Kalman filer: Correcion Mapping of sae o measuremens: Y ( ) ~ N mx m rediced sae: ( N µ ( ) ) We define he correced disribuion o be: ( ) ( ( ) ) N µ

42 CS 4495 Compuer Vision A. Bobick D Kalman filer: Correcion Mapping of sae o measuremens: rediced sae: Wan o esimae correced disribuion Updae he mean: Updae he variance: ~ m mx N Y ) ( N µ ) ( N µ ) ( ) ( m m m m µ µ ) ( ) ( ) ( m m m

43 CS 4495 Compuer Vision A. Bobick D Kalman filer: Correcion From: µ µ m m m m ( ( ) ) redicion of x µ Wha is his? m µ m m m m The weighed average of predicion and measuremen based on variances! Measuremen guess of x Variance of predicion Variance of x compued from he measuremen

44 CS 4495 Compuer Vision A. Bobick redicion vs. correcion µ m ( ) m m µ ( ) m m ( ) m m ( ) ( ) Wha if here is no predicion uncerain µ µ ( ) The measuremen is ignored! Wha if here is no measuremen uncerain µ ( ) m The predicion is ignored! ( )? ( m )?

45 CS 4495 Compuer Vision A. Bobick Recall: consan veloci example posiion measuremen s sae ime Sae is d: posiion veloci Measuremen is d: posiion

46 CS 4495 Compuer Vision A. Bobick Consan veloci model posiion Kalman filer processing ime o sae x measuremen * prediced mean esimae correced mean esimae bars: variance esimaes before and afer measuremens

47 CS 4495 Compuer Vision A. Bobick Consan veloci model posiion Kalman filer processing ime o sae x measuremen * prediced mean esimae correced mean esimae bars: variance esimaes before and afer measuremens

48 CS 4495 Compuer Vision A. Bobick Consan veloci model posiion Kalman filer processing ime o sae x measuremen * prediced mean esimae correced mean esimae bars: variance esimaes before and afer measuremens

49 CS 4495 Compuer Vision A. Bobick Consan veloci model posiion Kalman filer processing ime o sae x measuremen * prediced mean esimae correced mean esimae bars: variance esimaes before and afer measuremens

50 CS 4495 Compuer Vision A. Bobick Kalman filer: General case (> dim) Wha if sae vecors have more han one dimension? REDICT CORRECT Σ x D x D Σ ( ) D Σ x x K M x T d K Σ M ( I K M ) Σ ( T M Σ M Σ ) T Σ m More weigh on residual when measuremen error covariance approaches.

51 CS 4495 Compuer Vision A. Bobick Kalman filer pros and cons ros Simple updaes compac and efficien Cons Unimodal disribuion onl single hpohesis Resriced class of moions defined b linear model Exensions call Exended Kalman Filering So wha migh we do if no Gaussian? Or even unimodal?

52 CS 4495 Compuer Vision A. Bobick Find ou nex ime! (Acuall nex Thurs)