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6.801/866Tracking with Non-linear Dynamic ModelsDistribution propogationDistribution propogationA little nonlinearity becomes very non-GaussianEKFRepresenting non-linear DistributionsRepresenting non-linear DistributionsRepresenting non-linear DistributionsRepresenting Distributions using Weighted SamplesRepresenting Distributions using Weighted SamplesExpectation over sampled densityExpectation over sampled densitySampled representation of a probability distributionMarginalizing a sampled densityMarginalizing a sampled densitySampled BayesSampled BayesSampled PredictionSampled CorrectionNaïve PF TrackingSample impoverishmentSample impoverishmentResample the priorResampling concentrates samplesA practical particle filter with resamplingA variantA variant (animation)ApplicationsContour trackingHead trackingLeaf trackingHand trackingMixed state trackingA drawing interfaceArticulated trackingInteresting ExtensionsTracking with Non-linear Dynamic Models6.801/866Tracking with Non-linear Dynamic ModelsT. DarrellTracking with Non-linear Dynamic Models• Distribution propagation• Problems with non-linearities• Sampling densities• Particle filtering• Tracking peopleDistribution propogation[Isard 1998]Distribution propogation[Isard 1998]A little nonlinearity becomes very non-GaussianNotGaussian!EKFLinearize system at each time point to form an Extended Kalman Filter (EKF)– Compute Jacobian matrix whose (l,m)’th value is evaluated at – use this for forward model at each step in KFUseful in many engineering applications, but not as successful in computer vision….Representing non-linear DistributionsRepresenting non-linear DistributionsUnimodal parametric models fail to capture real-world densities…Representing non-linear DistributionsMixture models are appealing, but very hard to propagate analytically![ but see Cham and Rehg’s MHT approach]Representing Distributions using Weighted SamplesRather than a parametric form, use a set of samples to represent a density:Representing Distributions using Weighted SamplesRather than a parametric form, use a set of samples to represent a density:1. a set of sample locations2. weights associated with those locationsfrom what distribution?how does that distribution effect the weights?[Isard 1998]Expectation over sampled densityIf = thenExpectation over sampled densitySampled representation of a probability distributionMarginalizing a sampled densityIf we have a sampled representation of a joint densityand we wish to marginalize over one variable:we can simply ignore the corresponding components of the samples (!):Marginalizing a sampled densitySampled BayesTransforming a Sampled Representation of a Prior into a Sampled Representation of a Posterior:Sampled BayesSampled Prediction= ?~=Drop elements to getSampled CorrectionPrior Æ posteriorReweight with yieldingNaïve PF Tracking• Start with samples from something simple (Gaussian)• RepeatDoesn’t work that well–Predict– CorrectSample impoverishmentTest with linear case:kf: xpf: oSample impoverishment10 of the 100 particles:Resample the priorIn a sampled density representation, the frequency of samples can be traded off against weight:These new samples are a representation of the same density.I.e., make N draws with replacement from the original set of samples, using the weights as the probability of drawing a sample.s.t.…Resampling concentrates samplesA practical particle filter with resamplingA variant[Isard 1998]A variant (animation)[Isard 1998]ApplicationsTracking– hands– bodies–leavesContour tracking[Isard 1998]Head tracking[Isard 1998]Leaf tracking[Isard 1998]Hand tracking[Isard 1998]Mixed state tracking[Isard 1998]A drawing interface[Isard 1998]Articulated trackingInteresting Extensions• Multiple people / objects– state has to model multi-body configuration– resampling is tricky!• Multiple modalities – fuse observation likelihoods– audio/visual localization and source separationTracking with Non-linear Dynamic Models• Distribution propagation• Problems with non-linearities• Sampling densities• Particle filtering• Tracking people[Figures from F&P except as


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MIT 6 801 - Lecture Notes

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