19_Temporal_Models
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1.Computer vision: models, learning and inference Chapter 19 Temporal models
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3.Goal To track object state from frame to frame in a video Difficulties: Clutter (data association) One image may not be enough to fully define state Relationship between frames may be complicated
4.Structure 4 4 Computer vision: models, learning and inference. ©2011 Simon J.D. Prince Temporal models Kalman filter Extended Kalman filter Unscented Kalman filter Particle filters Applications
5.5 Temporal Models Consider an evolving system Represented by an unknown vector, w This is termed the state Examples: 2D Position of tracked object in image 3D Pose of tracked object in world Joint positions of articulated model OUR GOAL: To compute the marginal posterior distribution over w at time t. 5 Computer vision: models, learning and inference. ©2011 Simon J.D. Prince
6.6 Estimating State Two contributions to estimating the state: A set of measurements x t , which provide information about the state w t at time t. This is a generative model: the measurements are derived from the state using a known probability relation Pr( x t  w 1 … w T ) A time series model , which says something about the expected way that the system will evolve e.g., Pr( w t  w 1 ... w t1 , w t+1 … w T ) 6 Computer vision: models, learning and inference. ©2011 Simon J.D. Prince
7.Temporal Models 7 Computer vision: models, learning and inference. ©2011 Simon J.D. Prince
8.Only the immediate past matters (Markov) the probability of the state at time t is conditionally independent of states at times 1...t2 given the state at time t1. Measurements depend on only the current state the likelihood of the measurements at time t is conditionally independent of all of the other measurements and the states at times 1...t1, t+1..t given the state at time t. Assumptions 8 Computer vision: models, learning and inference. ©2011 Simon J.D. Prince
9.Graphical Model World states Measurements 9 Computer vision: models, learning and inference. ©2011 Simon J.D. Prince
10.Recursive Estimation Time 1 Time 2 Time t from temporal model 10 Computer vision: models, learning and inference. ©2011 Simon J.D. Prince
11.Computing the prior (time evolution) Each time, the prior is based on the Chapman Kolmogorov equation Prior at time t Temporal model Posterior at time t1 11 Computer vision: models, learning and inference. ©2011 Simon J.D. Prince
12.Summary Temporal Evolution Measurement Update Alternate between: Temporal model Measurement model 12 Computer vision: models, learning and inference. ©2011 Simon J.D. Prince
13.Structure 13 13 Computer vision: models, learning and inference. ©2011 Simon J.D. Prince Temporal models Kalman filter Extended Kalman filter Unscented Kalman filter Particle filters Applications
14.Kalman Filter The Kalman filter is just a special case of this type of recursive estimation procedure. Temporal model and measurement model carefully chosen so that if the posterior at time t1 was Gaussian then the prior at time t will be Gaussian posterior at time t will be Gaussian The Kalman filter equations are rules for updating the means and covariances of these Gaussians 14 Computer vision: models, learning and inference. ©2011 Simon J.D. Prince
15.The Kalman Filter Previous time step Prediction Measurement likelihood Combination 15 Computer vision: models, learning and inference. ©2011 Simon J.D. Prince
16.Kalman Filter Definition Time evolution equation Measurement equation State transition matrix Additive Gaussian noise Additive Gaussian noise Relates state and measurement 16 Computer vision: models, learning and inference. ©2011 Simon J.D. Prince
17.Kalman Filter Definition Time evolution equation Measurment equation State transition matrix Additive Gaussian noise Additive Gaussian noise Relates state and measurement 17 Computer vision: models, learning and inference. ©2011 Simon J.D. Prince
18.Temporal evolution 18 Computer vision: models, learning and inference. ©2011 Simon J.D. Prince
19.Measurement incorporation 19 Computer vision: models, learning and inference. ©2011 Simon J.D. Prince
20.Kalman Filter This is not the usual way these equations are presented. Part of the reason for this is the size of the inverses: f is usually landscape and so f T f is inefficient Define Kalman gain: 20 Computer vision: models, learning and inference. ©2011 Simon J.D. Prince
21.Mean Term Using Matrix inversion relations: 21 Computer vision: models, learning and inference. ©2011 Simon J.D. Prince
22.Covariance Term Kalman Filter Using Matrix inversion relations: 22 Computer vision: models, learning and inference. ©2011 Simon J.D. Prince
23.Final Kalman Filter Equation Innovation (difference between actual and predicted measurements Prior variance minus a term due to information from measurement 23 Computer vision: models, learning and inference. ©2011 Simon J.D. Prince
24.Kalman Filter Summary Time evolution equation Measurement equation Inference 24 Computer vision: models, learning and inference. ©2011 Simon J.D. Prince
25.Kalman Filter Example 1 25 Computer vision: models, learning and inference. ©2011 Simon J.D. Prince
26.Kalman Filter Example 2 Alternates: 26 Computer vision: models, learning and inference. ©2011 Simon J.D. Prince
27.27 Smoothing Estimates depend only on measurements up to the current point in time . Sometimes want to estimate state based on future measurements as well Fixed Lag Smoother : This is an online scheme in which the optimal estimate for a state at time t t is calculated based on measurements up to time t , where t is the time lag. i.e. we wish to calculate Pr ( w t  t x 1 . . . x t ). Fixed Interval Smoother : We have a fixed time interval of measurements and want to calculate the optimal state estimate based on all of these measurements. In other words , instead of calculating Pr ( w t x 1 . . . x t ) we now estimate Pr (w t x 1 . . . x T ) where T is the total length of the interval. 27 Computer vision: models, learning and inference. ©2011 Simon J.D. Prince
28.28 Fixed lag smoother 28 Computer vision: models, learning and inference. ©2011 Simon J.D. Prince State evolution equation Measurement equation Estimate delayed by t
29.Fixedlag Kalman Smoothing 29 Computer vision: models, learning and inference. ©2011 Simon J.D. Prince

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