Short course A vademecum of statistical pattern recognition techniques with applications to image and video analysis. Agenda

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1 Short course A vademecum of statistical attern recognition techniques with alications to image and video analysis Lecture 6 The Kalman filter. Particle filters Massimo Piccardi University of Technology, Sydney, Australia Massimo Piccardi, UTS Agenda Recursive Bayesian estimation The Kalman filter Particle filters A mention to robabilistic data association Eamle aers References Massimo Piccardi, UTS 2

2 State sace models In many cases, we are interested in systems whose state is a continuous multivariate r. v. (as oosed to HMM where it is discrete). Such systems are often referred to as state(-)sace models Their evolution in time is described by two equations, the state equation, or system, or rocess, model, and the outut equation, or measurement model Our further assumtions are: i) time-discrete evolution; ii) time-invariant arameters; iii) no control inuts; iv) noise over the state transitions and on the measurements; v) first-order Marov rocess; vi) indeendence of measurements given the state (aa Bayesian tracing hyotheses) Massimo Piccardi, UTS 3 State sace models Under these assumtions, we obtain the following equations: is the state r.v. R n h (, v ) (, n ) f v is the rocess noise r.v. R l is the measurement r.v. R n is the measurement noise r.v. R r Massimo Piccardi, UTS 4

3 Massimo Piccardi, UTS 5 Recursive Bayesian estimation Problem: estimate the state of a system,, from the sequence of received measurements, { i, i K} Recursive estimation: every new new Estimating the state of a system means to estimate its df given the received measurements, ( ) ( ) is nown as the filtering density and is a marginal of the osterior density (X ) Tyical alication: target tracing Massimo Piccardi, UTS 6 Recursive Bayesian estimation How do we recursively estimate ( )? At time, ( - - ) is available Reeatedly aly the Bayes theorem and use the indeendence of observations given the state: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ),,,,,

4 osterior filtering density rior: because Recursive Bayesian estimation (Marov) lielihood: given by the measurement model evidence: ( ) lielihood ( ) ( ) ( ) ( ) ( ) ( ) d ( ) (, ) ( ) (, ) (, ) ( ) ( ) ( ) d d d rior filtering density evidence Massimo Piccardi, UTS 7 Other related roblems filtering rediction smoothing (fied-lag) t (-h) t offline/batch (fied-interval smoothing) t t ( ) t t t (+h) t ( +h ) t t ( -h ) t t t N t ( N ), ( N N ) Massimo Piccardi, UTS 8

5 The Kalman filter Linear/Gaussian assumtions: linear system and measurement models, Gaussian distributions for all random variables If ( ) and noise distributions are assumed Gaussian, subsequent distributions remain Gaussian under linear transformations X ~ N(µ, Σ) Y AX + K Y ~ N(Aµ + K, AΣA T ) An otimal solution eists (R.E. Kalman, 96) Massimo Piccardi, UTS 9 The Kalman filter The assumtions lead to the following equations: + A H + n v is the state r.v. R n v is the rocess noise r.v. R n, indeendent, ~ N(,Q) A is the state transition matri (n n) is the measurement r.v. R n is the measurement noise r.v. R, indeendent, ~ N(,R) H is the state transition matri ( n) Massimo Piccardi, UTS

6 Solution The target is the filtering distribution s df, ( ); however, since it is Gaussian, mean and covariance suffice to identify it fully Usually, the mean at time interval is noted as ˆ, the covariance as P and called the error covariance The solution is obtained in two stes: the time udate, i.e. the alication of the dynamics (system model); a rediction; quantities are noted as, P and called a-riori estimates ˆ the measurement udate, i.e. the use of the measurement (measurement model); a correction to the revious estimates yielding the a-osteriori estimates Massimo Piccardi, UTS A-riori estimates: Time udate equations ˆ P Aˆ AP A T + Q Directly from transformation of Gaussian r.v. Otimal estimate in the absence of measurement Process noise causes P to increase Massimo Piccardi, UTS 2

7 Weighting A weighting matri, K (n ), called the Kalman gain: K T ( HP H + ) T P H R K decides how much the a-riori estimates should be corrected by the -th observation, : the larger the measurement noise, R (uncertainty on the observation), the smaller the correction Massimo Piccardi, UTS 3 Measurement udate equations A-osteriori estimates: ˆ P Hˆ ˆ + K ( I K H ) P ( Hˆ ) is the difference between the actual and redicted observations (aa innovation or measurement residual) Massimo Piccardi, UTS 4

8 Drift model State vector contains only the osition of the target Constant osition assumtion changes only due to the effect of noise -D eamle (u: osition) [ u] A [ ] Massimo Piccardi, UTS 5 Constant velocity model State vector contains osition and velocity of the target Constant velocity assumtion -D eamle (u: osition; v: velocity) u v A t Massimo Piccardi, UTS 6

9 Massimo Piccardi, UTS 7 Constant acceleration model State vector contains osition, velocity and acceleration of the target Constant acceleration assumtion -D eamle (u: osition; v: velocity; a: acceleration) a v u t t A Massimo Piccardi, UTS 8 Periodic motion model State vector contains osition and velocity of the target Periodic motion assumtion: d 2 u/dt 2 -cu -D eamle (u: osition; v: velocity) v u t t c A

10 Massimo Piccardi, UTS 9 Eamle An eamle courtesy of Greg Welch and Gary Bisho, An Introduction to the Kalman Filter, ACM SIGGRAPH 2 tutorial 2D data from a PC tablet A rich resource age at htt:// A Java-based -D Kalman Filter Learning Tool Massimo Piccardi, UTS 2 Process model: Measurement model (: measured osition): Initialiation: Drift model y yy Q Q Q y H H H y u u A yy R R R ˆ H ε ε P

11 Results y co-ordinate: first moving, then still Massimo Piccardi, UTS 2 Results: highlights Massimo Piccardi, UTS 22

12 Massimo Piccardi, UTS 23 Process model: Measurement model (same as before): Constant velocity model y y H H H y y v v u u t t A Massimo Piccardi, UTS 24 Results: highlights

13 Beyond linear/gaussian The Gaussian modelling of distributions roer of the Kalman filter is restrictive is many cases If transformations are not linear, subsequent distributions would become non-gaussian in any case Many other models have been roosed, the main of which are named here: Etended Kalman filter Grid filter Unscented Kalman filter Particle filters A nice unch line: the Kalman filter rovides an eact solution for an aroimate model; we may refer an aroimate solution for an eact model Massimo Piccardi, UTS 25 Particle filters In the following, we will focus on article filters In article filters, distributions are reresented by weighted sets of samles (called articles) Particle filters are not a single algorithm, rather a family of methods, also nown as sequential Monte Carlo methods The main target is again the filtering density, ( ) Distributions are not restricted to be Gaussian The model may be linear or not linear Massimo Piccardi, UTS 26

14 Monte Carlo methods In many ractical cases, it is hard to evaluate certain target distributions, and eectations based on them In such cases, aroimation schemes are needed (either deterministic or robabilistic) Monte Carlo methods are robabilistic methods where a distribution is simly reresented by a set of its samles These techniques were first adoted by hysicists; the name was coined by N. C. Metroolis based on the gambling habits of a colleague s relative Massimo Piccardi, UTS 27 Monte Carlo: eectations An eectation based on the eact integral: E is estimated by: [ f ( ) ] f ( ) ( ) X L L l f l ( ) d where the l are L samles from () (requiring that () can be samled, and ossibly easily) Massimo Piccardi, UTS 28

15 Sequential Monte Carlo methods In article filters, distributions are reresented a la Monte Carlo by a set of their samles Given that observations become available one by one, the filtering density, ( ), is, as usual, estimated recursively This has aved the way for a number of sequential (i.e. recursive) Monte Carlo methods; here, we will see: Sequential Imortance Samling Sequential Imortance Samling with Resamling Samling Imortance Resamling Massimo Piccardi, UTS 29 Imortance samling Before we can jum the gun on article filters, we need to cover two introductory toics: Imortance samling Resamling Imortance samling is a samling technique based on the following assumtions: we want to samle (); yet, it is hard we can easily samle another distribution, q() (nown as the roosal distribution or imortance density) we can easily evaluate both () and q() (which means: given, we can easily comute () and q(), at least u to roortionality) Massimo Piccardi, UTS 3

16 Imortance samling E [ f ( ) ] f ( ) ( ) X f ( ) q X ( ) ( ) q d L l l ( ) ( ) ( ) d f l L l q( ) where the l are samled from q() Quantities ω l ( l )/q( l ) are called the imortance weights The weighted set of samles { l, ω l } is a reresentation for () For efficiency, q() should be large where f()() is Massimo Piccardi, UTS 3 Eamle q() ω l ( l )/q( l ) f() () l Massimo Piccardi, UTS 32

17 Resamling Given a distribution reresented by a weighted set of its samles, { l, ω l }, l: L, we samle it to obtain a relacement set of L samles, l*, all of equal weight, ω l* /L The l* samles tae value in the { l } set Auiliary assumtion: samles can be generated out of a uniform distribution between and, U[,] Massimo Piccardi, UTS 33 Resamling Construct the cdf of the weighted set: c i c i- + ω i c, i L Samle U[,] L times to obtain the u l samles u l falls in the j-th interval l* j u l c c c 2 c j- c j c L- c L ω ω L l* j Massimo Piccardi, UTS 34

18 SIS article filter Sequential imortance samling (SIS) is the sequential etension of imortance samling It can be used to recursively estimate the filtering density, ( ) L samles, l, are drawn out of a roosal distribution, q( l l -, ), at every time The advantage of the sequential aroach is that new weights ω l need only be adjusted from revious weights, ω l -, through the dynamics and measurement models, and then normalised to add u to Massimo Piccardi, UTS 35 Samling: SIS article filter l q( l l -, ) l - Weight udate: Filtering density: l ω ω l l l l ( ) ( ) l l q(, ) L l l ( ) ω δ ( ) l Massimo Piccardi, UTS 36

19 SIS article filter SIS article filter: At every, draw the l from the current roosal distribution udate weights ω l based on formula normalise weights the new set of samles and weights is an aroimation for the desired filtering density, ( ) Massimo Piccardi, UTS 37 SIS article filter: weight degeneracy Degeneracy of the weights: it was shown that the variance of the ω l weights can only increase over time: after a few iterations, all but one article will have weight ω l Useful measure of degeneracy: ˆ N eff l L l ( ω ) There are two racticable countermeasures:. choice of a good roosal function 2. resamling 2 Massimo Piccardi, UTS 38

20 Choice of roosal function It was shown that the otimal roosal function to samle each l is: l (, ) since it minimises the degeneracy; yet, its use is not ossible in most cases Often, the roosal function is taen as: l ( ) this simlifies the weight udate formula greatly; yet, it ignores in the samling of l : it may samle away from useful directions Massimo Piccardi, UTS 39 Resamling Degeneracy can be monitored at every iteration; if > threshold, resamling is alied SIS article filter with resamling: At every, Draw the l from the current roosal distribution udate weights ω l based on formula normalise weights measure degeneracy: if > threshold, resamle Massimo Piccardi, UTS 4

21 SIR article filter The samling imortance resamling (SIR) article filter is a article filter that uses ( l -) as roosal and resamles at every time SIR article filter: At every, draw the l from ( l -) comute weights ω l ; formula simlifies because of roosal and revious weights being all equal to /L normalise weights resamle Massimo Piccardi, UTS 4 SIR article filter: issues The SIR article filter uses a simle roosal distribution and avoids weight degeneracy by frequent resamling Yet, the roosal distribution ignores in the samling of l : it may samle away from useful directions Moreover, a very frequent resamling tends to create samle imoverishment: the samles with highest weights tend to be samled more often and create l that are all equal; the degeneracy of the ω l weights is mollified at the cost of a degeneracy in the l samles Massimo Piccardi, UTS 42

22 Other article filters From Wiiedia (Jan 9): Auiliary article filter Gaussian article filter Unscented article filter Monte Carlo article filter Gauss-Hermite article filter Cost Reference article filter Rao-Blacwellied article filter Massimo Piccardi, UTS 43 Demo Particle filter (PF) classes in the Mobile Robot Programming Toolit (MRPT) htt://babel.isa.uma.es/mrt/inde.h/particle_filters Massimo Piccardi, UTS 44

23 Probabilistic data association At least a mention must go to robabilistic data association (PDA) Corresondence between observations and traced targets may be challenging and outstri the redictive caability of the filter Single target, multile observations: PDA filter (PDAF) Multile targets: joint PDAF (JPDAF) Other algorithms: multile hyothesis tracer (MHT), interacting multile model (IMM, IMMJPDAF), integrated PDA (IPDA), Massimo Piccardi, UTS 45 Eamle aers Alication: tracing of visual objects Tracing of multile targets in videos. Tracing with article filters: M. Isard, A. Blae, CONDENSATION -- conditional density roagation for visual tracing, Int. J. Comuter Vision, vol. 29, no.,. 5-28, cites on Google Scholar, 8 Nov 28 Tracing and data association with EM: H. Tao, H. S. Sawhney, R. Kumar, Object Tracing with Bayesian Estimation of Dynamic Layer Reresentations, IEEE Trans. on Pattern Anal. and Machine Intell., vol. 24, no., , 22. Massimo Piccardi, UTS 46

24 References Recursive Bayesian estimation: M. S. Arulamalam, S. Masell, N. Gordon, and T. Cla, A Tutorial on Particle Filters for On-line Nonlinear/Non-Gaussian Bayesian Tracing, IEEE Trans. on Signal Processing, 5(2):74-88, February 22 S. Dablemont, Université catholique de Louvain, Introduction to article filters, 26 (slide set) K. Murhy, A tutorial on dynamic Bayesian networs, 22 (slide set) Kalman and article filters: Materials from Greg Welch and Gary Bisho, University of North Carolina at Chael Hill: The Kalman Filter, htt:// (online resource) An Introduction to the Kalman Filter, STC Lecture Series (slides) An Introduction to the Kalman Filter, SIGGRAPH 2 (slides and aer) An Introduction to the Kalman Filter, TR95-4, University of North Carolina at Chael Hill (tech re) Massimo Piccardi, UTS 47 References (2) Peter S. Maybec, Stochastic models, estimation, and control. Volume, Academic Press, 979 Siome Klein Goldenstein, A gentle Introduction to redictive filters, Revista de Informática Teórica e Alicada - RITA, Volume, vol., no., 24 M S Arulamalam, S Masell, N Gordon, and T Cla, idem Probabilistic data association: Kirubarajan, T, Bar-Shalom, Y., Probabilistic data association techniques for target tracing in clutter, Proceedings of the IEEE, vol. 92, no. 3, Mar 24, I. Co, A Review of Statistical Data Association Techniques for Motion Corresondence, Int'l J. Comuter Vision, vol., no., , 993 Rasmussen, C., Hager, G.D., Probabilistic Data Association Methods for Tracing Comle Visual Objects, PAMI(23), No. 6, June 2, Massimo Piccardi, UTS 48

25 Than you! I hoe the toics covered will rove useful for your future research Massimo Piccardi inext Research Centre, FEIT, University of Technology, Sydney massimo@it.uts.edu.au Massimo Piccardi, UTS 49