1、朱振峰信息科學研究所 九教B611， 計算機視覺 Computer Vision 計算機視覺 Computer Vision Lecture 9-4 Dynamitic System Filter Based (Kalman Fliter) for Object TrackingRudolf Emil KalmanBorn 1930 in HungaryBS and MS from MITPhD 1957 from ColumbiaFilter developed in 1960-61 Object Tracking Object Tracking Object Tracking Object

2、 Tracking Object Tracking Object Tracking Object Tracking Object TrackingKalman filter for tracking Model for trackingObject has internal state Capital indicates random variableSmall represents particular valueObtained measurements in frame i areValue of the measurement General Steps of TrackingPred

3、iction: What is the next state of the object given past measurements Data association: Which measures are relevant for the state?Correction: Compute representation of the state from prediction and measurements.TrackingpredictcorrectOnly immediate past mattersMeasurements depend only on current state

4、Important simplifications Fortunately it doesnt limit to much!Independence AssumptionsLinear Dynamic ModelsState is linear transformed plus Gaussian noiseRelevant measures are linearly obtained from state plus Gaussian noiseSufficient to maintain mean and standard deviationA really simple exampleWe

5、are on a boat at night and lost our positionWe know: star positionConstant Velocityp is position of boat, v is velocity of boatstate is We only measure position so Marc makes a measuremen2,Conditional Density FunctionJan makes a measurementConditional Density Functio2,Combi

6、ne measurements & variance2Conditional Density FunctionOnline weighted average!Kalman filterJust some applied math.A linear dynamic system: f(a+b) = f(a) + f(b)Noisy data in hopefully less noisy out.But delay is the price for filtering.Predict CorrectKF operates by Predicting the new st

7、ate and its uncertaintyCorrecting with the new measurementpredictcorrectWhat is it used for?Tracking missilesTracking heads/hands/drumsticksExtracting lip motion from videoFitting Bezier patches to point dataLots of computer vision applicationsEconomicsNavigationA really simple exampleWe are on a bo

8、at at night and lost our positionWe know: move with constant velocity star positionBut suppose were movingNot all the difference is error. Some may be motionKF can include a motion modelEstimate velocity and positio2Process ModelDescribes how the state changes over timeThe state for the

9、 first example was scalarThe process model was “nothing changes”A better model might be constant velocity motion Measurement Model“What you see from where you are” not“Where you are from what you see”Constant Velocityp is position of boat, v is velocity of boatstate is We only measure position so St

10、ate and Error CovarianceFirst two moments of Gaussian processError CovarianceProcess State (Mean)The Process ModelUncertainty over intervalState transitionProcess dynamicsDifficult to determineMeasurement ModelMeasurement uncertaintyMeasurementmatrixMeasurement relationship to statePredict (Time Upd

11、ate)a priori state, error covariance, measurementMeasurement Update (Correct)Kalman gaina posteriori state and error covarianceMinimizes posteriori error covarianceThe Kalman GainWeights between prediction and measurements to posteriori error covarianceFor no measurement uncertainty State is deduced only from measurementThe Kalman GainSimple univariate (scalar) examplea posteriori state and error covarianceSummaryPREDICTCORRECTEstimating a ConstantThe state transition matrix The measurem