1 / 63

Tracking

Tracking. Image Processing Seminar 2008 Oren Mor. What is Tracking?. Estimating pose Possible from a variety of measured sensors Electrical Mechanical Inertial Optical Acoustic magnetic. Tracking applications - Examples. Tracking missiles Tracking heads/hands/drumsticks

gellert
Télécharger la présentation

Tracking

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Tracking Image Processing Seminar 2008 Oren Mor

  2. What is Tracking? • Estimating pose • Possible from a variety of measured sensors • Electrical • Mechanical • Inertial • Optical • Acoustic • magnetic

  3. Tracking applications - Examples • Tracking missiles • Tracking heads/hands/drumsticks • Extracting lip motion from video • Lots of computer vision applications • Economics • Navigation SIGGRAPH 2001

  4. Noise • Each sensor has fundamental limitations related to the associated physical medium • Information obtained from any sensor is part of a sequence of estimates • One of the most well-known mathematical tools for stochastic estimation from noisy sensor measurement is the Kalman Filter City of Vancouver

  5. Part I – Kalman Filter

  6. Rudolf Emil Kalman • Born in 1930 in Hungary • BS and MS from MIT • PhD 1957 from Columbia • Filter developed in 1960-61 • Now retired SIGGRAPH 2001

  7. What is the Kalman Filter? • Just some applied math • A linear system: f(a+b)=f(a)+f(b) • Noisy data in → hopefully less noisy out • But delay is the price for filtering… • Predictor-corrector estimator that minimizes the estimated error covariance • We’ll talk about The Discrete Kalman filter SIGGRAPH 2001

  8. Recursive Filters • Sequential update of previous estimate • Allow on-line processing of data • Rapid adaptation to changing signals characteristics • Consists of two steps: • Prediction step: • Update step:

  9. General Idea

  10. A The process to be estimated • Discrete time controlled process governed by linear stochastic difference equation • A is the state transition model applied to the previous state • B is the control-input model applied to the control vector • W is the process noise wikipedia

  11. A The process to be estimated • With a measurement that is • H is the observation model, which maps the process space into the observed space wikipedia

  12. A The process to be estimated • and are the process and measurement noise respectively • We assume • The noise vectors are assumed to be mutually independent wikipedia

  13. Computational Origins of the filter • is the a priori state estimated at step k given knowledge of the process prior to step k • is the a posteriori state estimated at step k given measurement Kalman filter web site

  14. Computational Origins of the filter • The a priori error estimate • The a posteriori error estimate • The a priori estimate error covariance is • The a posteriori estimate error covariance is Kalman filter web site

  15. Computational Origins of the filter • Computing a posteriori state estimate as a linear combination • The residual reflects the discrepancy between predicted measurement and the actual measurement • K minimizes the a posteriori error covariance equation Kalman filter web site

  16. Interesting Observations • As the measurement error covariance R approaches zero, K weights the residual is more “trusted”, is “trusted” less • As the a priori estimate error covariance approaches zero, K weights the residual less is less “trusted”, is “trusted” more Kalman filter web site

  17. The Discrete Kalman Filter Algorithm • Time update equations are responsible for projecting forward the current state and error covariance – predictor equations • Measurement update equations are responsible for the feedback – corrector equations Kalman filter web site

  18. Predict → Correct • Predicting the new state and its uncertainty • Correcting with the new measurement Kalman filter web site

  19. The Discrete Kalman Filter Algorithm Kalman filter web site

  20. Example • A truck on a straight frictionless endless road • Starts from position 0 • Has random acceleration • Measured every ∆t, imprecisely • We’ll derive a model from which we create a Kalman filter wikipedia

  21. Example - Model • No control, so B and u is ignored • The position and velocity is described by linear state space • We assume that the acceleration ak, normally distributed, with mean 0 and STD from Newton laws of motion where wikipedia

  22. Example - measurement • A noisy measurement of the true position. Assume the noise is normally distributed with STD where wikipedia

  23. Example – initialization • We know the starting state with perfect percision and the covariance matrix is zero And we’re ready to run the KF iterations! wikipedia

  24. Extended Kalman filter • Most non-trivial systems are non-linear • In the extended Kalman filter the state transition and observation model • The EKF is not an optimal estimator • If initial state or process model is wrong, the filter quickly diverges • The de facto standard in navigation systems and GPS wikipedia

  25. Part II – The Condensation Algorithm

  26. Introduction • We talked about Kalman filter, which relied on a Gaussian model • Another family of algorithms uses nonparametric models • No restrictions to linear processes or Gaussian noise

  27. Dynamic System • State transition: • State only depends on previous state • Measurement equation: • Both functions are not necessarily linear • The state noise on both equations is usually not Gaussian

  28. Propagation in one time step Isard and Blake, IJCV

  29. General Prediction-Update Framework • Assume that is available at time k-1 • Prediction step: (using Chapman-Kolmogoroff equation) • This is the prior to state at time k without knowing the measurement • Update step: • Compute posterior from predicted prior and new measurement

  30. Particle Filter – general concept • If we cannot solve the integrals required for a Bayesian recursive filter analytically, we represent the posterior probabilities by a set of randomly chosen weighted samples Vampire-project

  31. Sequential Importance Sampling • Let set of support points (samples, particles) • Whole trajectory for each particle • Let associated weights, normalized to • Then: (discrete weighted approximation to the true posterior)

  32. Sequential Importance Sampling • Usually we cannot draw samples from directly. Assume we sample directly from a different importance function . Our approximation is still correct if • The trick: We can choose freely!

  33. Factored sampling Isard and Blake, IJCV

  34. Probability distribution example • Each sample is shown as a curve • Thickness proportional to the weight • Estimator of the distribution mean Movie Isard and Blake, IJCV

  35. Sequential Importance Sampling • If the importance function is chosen to factorize such that Then one can augment old particles by to get new particles

  36. Sequential Importance Sampling • Weight update (after some lengthly computations) • Furthermore, if (only depends on last states and observations) And we don’t need to preserve trajectories and the histories

  37. SIS algorithm – Pseudo code

  38. Problem – Degeneracy problem • Problem with SIS approach: after a few iterations, most particles have negligible weight (the weight is concentrated on a few particles only) Counter measures: • Brute force: many samples • Good choice of importance density • resampling

  39. Resampling Approaches • Whenever degeneracy rises above threshold: replace old set of samples (+weights) with new set of samples (+weights), such that sample density better reflects posterior • This eliminates particles with low weight and chooses more particles in more probable regions

  40. General Particle filter – Pseudo code Movie

  41. Problems • Particles with high weight are selected more and more often, others die out slowly • Resampling limits the ability to parallelize the algorithm

  42. Advantages of PF • Can deal with non-linears • Can deal with non-Gaussian noise • Can be implemented in O(Ns) • Mostly parallelizable • Easy to implement

  43. Questions?

  44. Part III – Multiple Object Tracking

  45. Intoduction • Difficulties for Multiple Object Tracking • Objects may interact. • The objects may present more or less the same appearances • BraMBLe: A Bayesian Multiple-Blob Tracker, M. Isard and J. MacCormick, ICCV’01. Qi Zhao

  46. Problem • The goal is to track an unknown number of blobs from static camera video. Qi Zhao

  47. Number, Positions, Shapes, Velocities, … Solution • The Bayesian Multiple-BLob (BraMBLe) tracker is a Bayesian solution. • It estimates State at frame t Image Sequence Qi Zhao

  48. Modeling idea ObservationLikelihood Posterior State Distribution Prior • Sequential Bayes ObservationLikelihood PosteriorStateDistribution Prior Instead of modeling directly, BraMBLe models and . Qi Zhao

  49. Object State Number of objects Object Model X • The blob configuration is Qi Zhao

  50. Object Model X Identity Velocity Shape Location Qi Zhao

More Related