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Berkeley COMPSCI C280 - Lecture Notes

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Slide 1Tracking scenariosThings to consider in trackingThree main issues in trackingSimplifying AssumptionsSlide 6Tracking as inductionBase casePrediction stepUpdate stepThe Kalman FilterRecall the three main issues in trackingThe Kalman FilterThe Kalman Filter in 1DThe Kalman FilterPrediction for 1D Kalman filterSlide 17The Kalman FilterMeasurement update for 1D Kalman filterSlide 20Kalman filter for computing an on-line averageSlide 22What happens if the x dynamics are given a non-zero variance?Slide 24Linear dynamic modelsExamples of linear state space modelsConstant velocitySlide 28Constant accelerationSlide 30Periodic motionPeriodic motionn-Dn-Dn-D Predictionn-D Correctionn-D correctionKalman Gain MatrixSlide 39Slide 40Slide 41Slide 42Slide 43SmoothingSlide 45Slide 46Slide 47Slide 48Slide 49ResourcesEmbellishments for trackingAbrupt changesMultiple model filtersMM estimateP likelihoodNo lagSmooth when stillEmbellishments for trackingJepson, Fleet, and El-Maraghi trackerWandering, Stable, and Lost appearance modelSlide 61Slide 62Non-toy image representationThe motion trackerJepson, Fleet, and El-Maraghi trackerAdd fleet&jepson tracking slidesEmbellishments for tracking(KF) Distribution propagationDistribution propagationRepresenting non-linear DistributionsRepresenting non-linear DistributionsDiscretize by evenly sampling over the entire state spaceRepresenting Distributions using Weighted SamplesRepresenting Distributions using Weighted SamplesSlide 76Sampled representation of a probability distributionTracking, in particle filter representationParticle filterSampled PredictionSampled Correction (Bayes rule)Naïve PF TrackingSample impoverishmentResample the priorResampling concentrates samplesA practical particle filter with resamplingPictorial viewSlide 88Slide 89Animation of condensation algorithmApplicationsContour trackingHead trackingLeaf trackingHand trackingDesired operations with a probability densityComputing an expectation using sampled representationMarginalizing a sampled densityMarginalizing a sampled densitySampled Bayes ruleC280, Computer VisionProf. Trevor [email protected] 19: TrackingTracking scenarios•Follow a point•Follow a template•Follow a changing template•Follow all the elements of a moving person, fit a model to it.23What are the dynamics of the thing being tracked?How is it observed?Things to consider in tracking4Three main issues in tracking5Simplifying Assumptions6Kalman filter graphical model and corresponding factorized joint probabilityx1x2x3y1y2y3)|()|()|()|()|()(),,,,,(33232212111321321xyPxxPxyPxxPxyPxPyyyxxxP 7Tracking as induction•Make a measurement starting in the 0th frame•Then: assume you have an estimate at the ith frame, after the measurement step.•Show that you can do prediction for the i+1th frame, and measurement for the i+1th frame.8Base case9Prediction stepgiven10Update stepgiven11The Kalman Filter•Key ideas: –Linear models interact uniquely well with Gaussian noise - make the prior Gaussian, everything else Gaussian and the calculations are easy–Gaussians are really easy to represent --- once you know the mean and covariance, you’re done12Recall the three main issues in tracking(Ignore data association for now)13The Kalman Filter[figure from http://www.cs.unc.edu/~welch/kalman/kalmanIntro.html]14The Kalman Filter in 1D•Dynamic Model•NotationPredicted meanCorrected mean15The Kalman Filter16Prediction for 1D Kalman filter•The new state is obtained by–multiplying old state by known constant–adding zero-mean noise•Therefore, predicted mean for new state is–constant times mean for old state•Old variance is normal random variable–variance is multiplied by square of constant–and variance of noise is added.1718The Kalman Filter19Measurement update for 1D Kalman filterNotice:–if measurement noise is small, we rely mainly on the measurement,–if it’s large, mainly on the prediction– s does not depend on y2021Kalman filter for computing an on-line average•What Kalman filter parameters and initial conditions should we pick so that the optimal estimate for x at each iteration is just the average of all the observations seen so far?22Iteration 0 1 2000xiiiixx01,0,1,1 iimdiimdKalman filter modelInitial conditions0y10y1210yy 21210yy 213210yyy 3123What happens if the x dynamics are given a non-zero variance?24Iteration 0 1 2000xiiiixx00y1,1,1,1 iimdiimdKalman filter modelInitial conditions10y23210yy 3235852210yyy 853210yy 25Linear dynamic models•A linear dynamic model has the form•This is much, much more general than it looks, and extremely powerfulyiN Mixi;mi xiN Di 1xi 1;di 26Examples of linear state space models•Drifting points–assume that the new position of the point is the old one, plus noise D = IdentityyiN Mixi;mi xiN Di 1xi 1;di cic.nist.gov/lipman/sciviz/images/random3.gif http://www.grunch.net/synergetics/images/random3.jpg27 Constant velocity •We have–(the Greek letters denote noise terms)•Stack (u, v) into a single state vector–which is the form we had aboveuiui 1 tvi 1ivivi 1iuvi1 t0 1uvi 1 noiseyiN Mixi;mi xiN Di 1xi 1;di Di-1xi-1xi28positionpositionConstantVelocityModelvelocitytimemeasurement,positiontime29 Constant acceleration•We have–(the Greek letters denote noise terms)•Stack (u, v) into a single state vector–which is the form we had aboveuiui 1 tvi 1ivivi 1 tai 1iaiai 1iuvai1 t 00 1 t0 0 1uvai 1 noiseyiN Mixi;mi xiN Di 1xi 1;di Di-1xi-1xi30timepositionpositionvelocityConstantAccelerationModel31Assume we have a point, moving on a line with a periodic movement defined with a differential eq: can be defined as with state defined as stacked position


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Berkeley COMPSCI C280 - Lecture Notes

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