1 Formation et Analyse d’Images Session 7 Daniela Hall 7 November 2005

2 Course Overview Session 1 (19/09/05) –Overview –Human vision –Homogenous coordinates –Camera models Session 2 (26/09/05) –Tensor notation –Image transformations –Homography computation Session 3 (3/10/05) –Camera calibration –Reflection models –Color spaces Session 4 (10/10/05) –Pixel based image analysis 17/10/05 course is replaced by Modelisation surfacique

3 Course overview Session (24/10/05) 9:45 – 12:45 –Contrast description –Hough transform Session 7 (7/11/05) –Kalman filter Session 8 (14/11/05) –Tracking of regions, pixels, and lines Session 9 (21/11/05) –Gaussian filter operators Session 10 (5/12/05) –Scale Space Session 11 (12/12/05) –Stereo vision –Epipolar geometry Session 12 (16/01/06): exercises and questions

4 Session overview 1.Kalman filter 2.Robust tracking of targets

5 Kalman filter The Kalman filter ist a optimal recursiv estimator. Kalman filtering has been applied in areas such as –aerospace, –marine navigation, –nuclear power plant instrumentation, –manufactring, and many others. The typical problem tries to estimate position and speed from the measurements. Tutorial: G. Welch, G. Bishop: An Introduction to the Kalman Filter, TR , Univ of N. Carolina, USA

6 Kalman filter The Kalman filter tries to estimate the state x of a discrete time controlled process that is governed by the equation with measurement Process noise Measurement noise Process noise covariance Q and measurement noise covariance R might in practise change over time, but are assumed constant Matrix A relates the state at time k to the state at previous time k-1, in absence of noise. In practise A might change over time, but is assumed constant. Matrix B relates the optional control input to the state x. We let it aside for the moment. Matrix H relates the state to the measurement.

7 Notations Measurement Measurement noise covariance Process noise Process noise covariance Kalman gain

8 Kalman filter notations A priori state estimate A posteriori state estimate A priori estimate error A posteriori estimate error A priori estimate error covariance A posteriori estimate error covariance

9 Kalman filter Goal: find a posteriori state estimate ^ x k as a linear combination of an a priori estimate ^ x k - and a weighted difference between the actual measurement z k and a measurement prediction H ^ x k - The difference (z k – H ^ x k - ) is called innovation or residual K is the gain or blending factor that minimizes the a posteriori error covariance.

10 Kalman gain K Matrix K is the gain that minimizes the a posteriori error covariance. –The equations that need to be minimized –How to minimize

11 Kalman gain K One form of the result is Measurement cov small weights residual heavier A priori estimate error small weights residual little

12 Kalman gain K When the error covariance R approaches 0, the actual measurement z k is trusted more, while the predicted measurement ^x k - is trusted less When the a priori estimate error covariance approaches 0, the actual measurement z k is trusted less, while the predicted measurement ^x k - is trusted more.

13 Discrete Kalman filter algorithm The Kalman filter estimates a process by using a form of feedback control. The filter estimates the process state at some time and then obtains feedback in form of noisy measurements. The Kalman filter equation fall in 2 groups –time update equations –measurement update equations Time update equations project forward in time the current state and error covariance estimates to obtain a priori estimates. The measurement update equations implement feedback. They incorporate a new measurement into the a priori estimate to form an improved a posteriori estimate.

14 Kalman filter algorithm Time Update (« Predict ») Measurement update (« Correct ») The time update projects current state estimate ahead in time. The measurement update adjusts the projected estimate by an actual measurement.

15 Kalman filter Time update equations (predict) –Project state and covariance estimates forward in time Measurement update equations (correct) –Compute Kalman gain K –Measure process zk –Compute a posteriori estimate xk –Compute a posteriori error covariance estimate Pk Initial estimates for ^x k-1 and P k-1

16 Filter parameters and tuning R: measurement noise covariance can be measured a priori to the filter operation (off-line) Q: process noise covariance. Can not be measured, because we can not directly observe the process we are measuring. If we choose Q large enough (lots of uncertainty), a poor process model can produce acceptable results. Parameter tuning: We can increase filter performance by tuning the parameters R and Q. We can even use a distinct Kalman filter for tuning. If R and Q are constant, the estimation error cov Pk and the Kalman gain Kk will stabilize quickly and stay constant. In this case, Pk and Kk can be precomputed.

17 Session overview 1.Kalman filter 2.Robust tracking of targets

18 Robust tracking of objects Trigger regions Detection New targets List of targets Predict List of predictions Correct Detection Measurements

19 Robust tracking of objects Measurement State vector State equation Source: M. Kohler: Using the Kalman Filter to track Human interactive motion, Research report No 629/Feb 1997, University of Dortmund Germany

20 Robust Tracking of objects Measurement noise error covariance Temporal matrix Process noise error covariance a affects the computation speed (large a increases uncertainty and therefore the search regions)

21 Form of the temporal matrix A Matrix A relates a posteriori state estimate ^x k-1 to the a priori state estimate ^x k - The new a priori state estimate requires the temporal derivative According to a Taylor serie we can write

22 Kalman filter notations A priori state estimate A posteriori state estimate A priori estimate error A posteriori estimate error A priori estimate error covariance A posteriori estimate error covariance

23 Kalman filter Time update equations (predict) –Project state and covariance estimates forward in time Measurement update equations (correct) –Compute Kalman gain K –Measure process z k –Compute a posteriori estimate x k –Compute a posteriori error covariance estimate P k Initial estimates for ^x k-1 and P k-1

24 Example A 1D point moves with a certain speed on a continuous scale We have a sensor that gives only integer values Compute a Kalman filter for the process.

25 Example results p true position p’ true speed z measured pos z’ measured grad ^x estimated pos ^x’ estimated grad K and P converge quickly