Using the Kalman filter Extended Kalman filter

Transcription

1 Using he Kalman filer Eended Kalman filer Doz. G. Bleser Prof. Sricker Compuer Vision: Objec and People Tracking SA-

2 Ouline Recap: Kalman filer algorihm Using Kalman filers Eended Kalman filer algorihm 2

3 Recap: Kalman filer Bayes updae rule + linear Gaussian models Kalman filer correc measure predic poserior likelihood moion model poserior a - N ; N z; C Q N ; A Bu R N ; 3

4 Recap: Kalman filer algorihm. Kalman_filer u z : 2. Predicion: 3. A Bu T 4. A A R 5. Correcion: T T 6. K C C C Q 7. K z C 8. I K C 9. Reurn Requiremens: Iniial belief is Gaussian Linear Gaussian sae space model 4

5 Using Kalman filers Applicaion o a simple 2D racking problem Some slides based on K. Smih 5 SA-

6 Track an aircraf in a video sequence Assumpions: Image processing provides noisy 2D posiions of aircraf Moion model is cons. pos. Linear Gaussian sysem Formulae he sae space model: ~ ~ 6

7 Sae: 2D image posiion Track an aircraf in a video sequence Measuremen: 2D image posiion 7

8 Track an aircraf in a video sequence Moion/dynamic model: ~ Measuremen model: C Q ~ Q 8

9 Track an aircraf in a video sequence. Kalman_filer u z : 2. Predicion: 3. A Bu T 4. A A R 5. Correcion: T T 6. K C C C Q 7. K z C 8. I K C 9. Reurn C Q Noe: 9

10 Track an aircraf in a video sequence Predicion: Σ Σ Prediced measuremen prediced sae Measuremen updae: Σ Σ Σ Σ Kalman gain Σ Innovaion residual

11 Discussion Predicion: Σ Σ Prediced measuremen prediced sae Measuremen updae: Σ Σ Σ Σ Kalman gain Wha happens if: Innovaion residual

12 Discussion Wha happens if he objec moion violaes our model assumpion Effecs? Big innovaions racking lags Possible soluions? Adap noise seings increase process noise/reduce meas. noise Choose a beer model

13 We can esimae velociy! predicion pas measuremens

14 Simple consan velociy model Moion/dynamic model: Measuremen model: 4 Q N y y y obs obs 2 2 Q R N y y 2 2 R

15 Consan velociy model Assuming ha he objec follows a cons. velociy model: Beer predicion Con d esimae even if we don have measuremens 5

16 Discussion Consan posiion model Consan velociy model y y y 6

17 Discussion How o rack very agile moions? Increased process noise Consan acceleraion model More reacive racking Insable if no measuremens available Alernaive soluions: Swiched model Knowledge from airplane available? 7

18 Kalman limiaions Uni modal disribuions fail for unprediced moion Problem if image processing depends on predicion racking vs. deecion Possible soluions: Swiched model ground conac as conrol inpu Muliple hypohesis racking Predicion oo far from acual locaion o recover 8

19 Kalman filer limiaions Applies o linear Gaussian models Many visual racking problems are nonlinear e.g. as soon as we move o racking in 3D Couresy of G. Panin 9

20 Humanoid robo caches flying balls Several cameras observe flying balls The 3D rajecory is esimaed A robo is supposed o cach he balls nonlinear esimaion problem Couresy of U. Frese 2

21 Eended Kalman filer Slides based on S. Thrun 2 SA-

22 Nonlinear models Gaussian noise Moion model: Nonlinear funcion mos general model A B u g u Measuremen model: Nonlinear funcion g u ~ Ofen used: model wih addiive noise z C z h ~ z h 22

23 Linear funcion X Y ~ N AX B Y ~ N A B AA T 23

24 Nonlinear funcion PDF obained from 5. Mone Carlo samples + hisogramming Then sample mean and covariance 24

25 Bayes updae rule Problem: We do no say in he Gaussian world if moion and/or measuremen models are nonlinear funcions of he sae Non-Gauss Nonlinear+Gauss Gauss Nonlinear+Gauss No general closed form soluion for Bayes filer Approimae soluions: Keep he funcions and approimae disribuions Linearize he funcions and use again he Kalman filer 25

26 EKF linearizaion Mismach = linearizaion error Mone Carlo ransform + sample saisics blue Gaussian Firs Order Taylor approimaion red Gaussian 26

27 EKF linearizaion: addiive noise Linearize and wih Firs Order Taylor Epansion Linearizaion poins: bes available esimae Predicion linearize around : Correcion linearize around : 27 G u g u g u g u g u g H h h h h h Jacobian mari Jacobian mari

28 28 Eended Kalman filer: addiive noise. Eended_Kalman_filer u z : 2. Predicion: Correcion: Reurn u g T R G G T T Q H H H K h z K H K I u g G h H u A B T R A A T T Q C C C K C z K C K I

29 Linearizaion: non addiive noise Linearizaion poins: bes esimae and assuming zero mean noise Predicion linearize around and : Correcion linearize around and : 29 W G u g u g u g u g u g u g V H h h h h h h Jacobians Jacobians

30 3 Eended Kalman filer: non addiive noise. Eended_Kalman_filer u z : 2. Predicion: Correcion: Reurn u g T T W W R G G T T T V V Q H H H K h z K H K I u g G h H h V u g W

31 Gauss approimaion formula Le: be unknown wih mean and covariance Σ a nonlinear funcion of The Gauss approimaion for mean and covariance Σ is: Σ 3

32 EKF linearizaion: problems Bigger uncerainy bigger linearizaon error 32

33 EKF linearizaion: problems 33

34 EKF linearizaion: problems Higher local nonlineariy bigger linearizaon error 34

35 EKF linearizaion: problems 35

36 Eended Kalman filer: summary Kalman Filer was opimal for linear Gaussian models Problem wih EKF: he funcions and are linearized hence approimaed No opimal we keep Gaussian represenaions bu is no Gaussian If and are highly nonlinear difficul o quanize he esimaion may become unsable Good resuls if funcions are appro. linear around mean Less cerain esimae is more affeced by linearizaion errors EKF is no an opimal Bayesian racker However i is successfully used in many applicaions 36

37 Eended Kalman filer: limiaions Quesion: is he EKF feasible for he above esimaion problem? Implicaions of using Gaussians: Gaussians are unimodal! They possess a single maimum! Typical for many racking problems: poserior focused around rue sae wih small margin of uncerainy Poor mach for mulimodal problem global esimaion 37

38 Eended Kalman filer: limiaions No non Gaussian observaion models M. Breiensein F. Reichlin B. Leibe E. Koller-Meier L. Van Gool Robus Tracking-by-Deecion using a Deecor Confidence Paricle Filer Inernaional Conference on Compuer Vision ICCV 29 38

39 More Bayes filers for he mos ineresed ones Ieraed Eended Kalman filer Unscened Kalman filer Informaion filer Alernaives for EKF Hisogram filer Paricle filer Ineracing muliple model filer Muliple hypohesis racking Nonlinear funcions muliple modes non Gaussian noise in general less efficien Miure of Gaussians fied number of modes differen model assumpions Bibliography: Books by Sebasian Thrun and Y. Bar Shalom on hp://av.dfki.de 39

40 Oulook Ne lecures: Sae space models for 3D visual racking and applicaions People racking based on RGB D daa Oral eam: Suggesed periods: Please wrie an o leivy_michelly.kaul@dfki.de wih cc o gabriele.bleser@dfki.de and ask for an appoinmen. The should conain: Full name mariculaion number faculy course of sudies Preferred daes and ime 4