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## Non-linear tracking

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**Non-linear tracking**Marc Pollefeys COMP 256 Some slides and illustrations from D. Forsyth, M. Isard, T. Darrell …**Final project presentation**No further assignments, focus on project Final presentation: • Presentation and/or Demo (your choice, but let me know) • Short paper (Due April 22 by 23:59) (preferably Latex IEEE proc. style) • Final presentation/demo April 24 and 26**System state dynamics**Observation dynamics We are interested in: Belief or posterior density Bayes Filters Estimating system state from noisy observations**Recall “law of total probability” and “Bayes’**rule” From above, constructing two steps of Bayes Filters Predict: Update:**Assumptions: Markov Process**Predict: Update:**Bayes Filter**How to use it? What else to know? Motion Model Perceptual Model Start from:**Step 0: initialization**Step 1: updating Example 1**Step 3: updating**Step 4: predicting Step 2: predicting Example 1 (continue)**Several types of Bayes filters**• They differs in how to represent probability densities • Kalman filter • Multihypothesis filter • Grid-based approach • Topological approach • Particle filter**Recall general problem**Assumptions of Kalman Filter: Belief of Kalman Filter is actually a unimodal Gaussian Advantage: computational efficiency Disadvantage: assumptions too restrictive Kalman Filter**Multi-hypothesis Tracking**• Belief is a mixture of Gaussian • Tracking each Gaussian hypothesis using a Kalman filter • Deciding weights on the basis of how well the hypothesis predict the sensor measurements • Advantage: • can represent multimodal Gaussian • Disadvantage: • Computationally expensive • Difficult to decide on hypotheses**Grid-based Approaches**• Using discrete, piecewise constant representations of the belief • Tessellate the environment into small patches, with each patch containing the belief of object in it • Advantage: • Able to represent arbitrary distributions over the discrete state space • Disadvantage • Computational and space complexity required to keep the position grid in memory and update it**Topological approaches**• A graph representing the state space • node representing object’s location (e.g. a room) • edge representing the connectivity (e.g. hallway) • Advantage • Efficiency, because state space is small • Disadvantage • Coarseness of representation**Particle filters**• Also known as Sequential Monte Carlo Methods • Representing belief by sets of samples or particles are nonnegative weights called importance factors • Updating procedure is sequential importance sampling with re-sampling**Step 0: initialization**Each particle has the same weight Step 1: updating weights. Weights are proportional to p(z|x) Example 2: Particle Filter**Step 3: updating weights. Weights are proportional to p(z|x)**Step 4: predicting. Predict the new locations of particles. Step 2: predicting. Predict the new locations of particles. Example 2: Particle Filter Particles are more concentrated in the region where the person is more likely to be**Compare Particle Filter with Bayes Filter with Known**Distribution Updating Example 1 Example 2 Predicting Example 1 Example 2**Comments on Particle Filters**• Advantage: • Able to represent arbitrary density • Converging to true posterior even for non-Gaussian and nonlinear system • Efficient in the sense that particles tend to focus on regions with high probability • Disadvantage • Worst-case complexity grows exponentially in the dimensions**Particle Filtering in CV: Initial Particle Set**• Particles at t = 0 drawn from wide prior because of large initial uncertainty • Gaussian with large covariance • Uniform distribution from MacCormick & Blake, 1998 State includes shape & position; prior more constrained for shape**¼(1)**¼(N) ¼(2) ¼(N-1) ¼(3) courtesy of D. Fox Particle Filtering: Sampling • Normalize N particle weights so that they sum to 1 • Resample particles by picking randomly and uniformly in [0, 1] range Ntimes • Analogous to spinning a roulette wheel with arc-lengths of bins equal to particle weights • Adaptively focuses on promising areas of state space**Deterministic component (aka “drift”)**Random component (aka “diffusion”) Particle Filtering: Prediction • Update each particle using generative form of dynamics: • Drift may be nonlinear (i.e., different displacement for each particle) • Each particle diffuses independently • Typically modeled with a Gaussian**Particle Filtering: Measurement**• For each particle s(i), compute new weight ¼(i) as measurement likelihood ¼(i) =P (zjs(i)) • Enforcing plausibility: Particles that represent impossible configurations are given 0 likelihood • E.g., positions outside of image from MacCormick & Blake, 1998 A snake measurement likelihood method**Sampling occurs**here Particle Filtering Steps (aka CONDENSATION) drift diffuse measurement likelihood measure from Isard & Blake, 1998**Particle Filtering Visualization**courtesy of M. Isard 1-D system, red curve is measurement likelihood**CONDENSATION: Example State Posterior**from Isard & Blake, 1998 Note how initial distribution “sharpens”**Example: Contour-based Head Template Tracking**courtesy of A. Blake**Example: Recovering from Distraction**from Isard & Blake, 1998**Obtaining a State Estimate**• Note that there’s no explicit state estimate maintained—just a “cloud” of particles • Can obtain an estimate at a particular time by querying the current particle set • Some approaches • “Mean” particle • Weighted sum of particles • Confidence: inverse variance • Really want a mode finder—mean of tallest peak**Condensation:Estimating Target State**From Isard & Blake, 1998 State samples (thickness proportional to weight) Mean of weighted state samples**Multi-Modal Posteriors**• The MAP estimate is just the tallest one when there are multiple peaks in the posterior • This is fine when one peak dominates, but when they are of comparable heights, we might sometimes pick the wrong one • Committing to just one possibility can lead to mistracking • Want a wider sense of the posterior distribution to keep track of other good candidate states adapted from [Hong, 1995] Multiple peaks in the measurement likelihood**MCMC-based particle filter**(Khan, Balch & Dellaert PAMI05) Model interaction (higher dimensional state-space) CNN video