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Particle Filtering Robert Collins Penn State Sampling Methods: Particle Filtering CSE586 Computer Vision II CSE Dept, Penn State Univ Robert Collins Penn State Recall: Importance Sampling Procedure to estimate EP(f(x)): 1) Generate N samples xi from Q(x) 2) form importance weights 3) compute empirical estimate of EP(f(x)), the expected value of f(x) under distribution P(x), as Robert Collins Penn State Resampling Note: We thus have a set of weighted samples (xi, wi | i=1,…,N) If we really need random samples from P, we can generate them by resampling such that the likelihood of choosing value xi is proportional to its weight wi This would now involve now sampling from a discrete distribution of N possible values (the N values of xi ) Therefore, regardless of the dimensionality of vector x, we are resampling from a 1D distribution (we are essentially sampling from the indices 1...N, in proportion to the importance weights wi). So we can using the inverse transform sampling method we discussed earlier. Robert Collins Penn State Sequential Monte Carlo Methods Sequential Importance Sampling (SIS) and the closely related algorithm Sampling Importance Sampling (SIR) are known by various names in the literature: - bootstrap filtering - particle filtering - Condensation algorithm - survival of the fittest General idea: Importance sampling on time series data, with samples and weights updated as each new data term is observed. Well-suited for simulating recursive Bayes filtering! Robert Collins Penn State Recall: Bayes Filtering Two-step Iteration at Each Time t: Motion Prediction Step: Data Correction Step (Bayes rule): Robert Collins Penn State Recall: Bayes Filtering Problem: in general we get intractible integrals Motion Prediction Step: Data Correction Step (Bayes rule): Robert Collins Penn State Sequential Monte Carlo Methods Intuition: • Represent probability distributions by samples (called particles). • Each particle is a “guess” at the true state. • For each one, simulate it’s motion update and add noise to get a motion prediction. Measure the likelihood of this prediction, and weight the resulting particles proportional to their likelihoods. Robert Collins Penn State Back to Bayes Filtering This integral in the denominator of Bayes rule disappears as a consequence of representing distributions by a weighted set of samples. Since we have only a finite number of samples, the normalization constant will be the sum of the weights! Data Correction Step (Bayes rule): Robert Collins Penn State Back to Bayes Filtering Now let’s write the Bayes filter by combining motion prediction and data correction steps into one equation. new posterior data term motion term old posterior Robert Collins Penn State Monte Carlo Bayes Filtering Assume the posterior at time t-1 (which is the prior at time t) has been approximated as a set of N weighted particles: So that Where is the delta dirac function Useful property: Robert Collins Penn State Monte Carlo Bayes Filtering Then the motion prediction integral simplifies to a summation Motion prediction integral The prior had been approximated by N particles Exchange order of summation and integration Property of Dirac delta function Robert Collins Penn State Monte Carlo Bayes Filtering Our Bayes filtering equation thus simplifies as well Plugging in result from previous page Bringing term that doesn’t depend on i into the summation Robert Collins Penn State Monte Carlo Bayes Filtering Our new posterior is therefore but this is still not amenable to computation in closed-form for arbitrary motion models and likelihood functions (e.g. we would have to integrate it to compute the normalization constant c) Idea : Let’s approximate the posterior as a set of N samples! Idea 2 : Hey wait a minute, the prior was already represented as a set of N samples! Why don’t we just “update” each of those? Robert Collins Penn State Monte Carlo Bayes Filtering i i Approach: for each sample x t-1 , generate a new sample x t from by importance sampling using some convenient proposal distribution So, generate a sample and compute its importance weight Robert Collins Penn State Monte Carlo Bayes Filtering We then can approximate our posterior as where Robert Collins Penn State SIS Algorithm Robert Collins Penn State SIS Degeneracy Unfortunately, pure SIS suffers from degeneracy. In many cases, after a few iterations, all but one particle will have negligible weight. Illustration of degeneracy: Time 1 w Time 10 w Time 19 w Robert Collins Penn State Resampling to Combat Degeneracy Sampling with replacement to get N new samples, each having equal weight 1/N Samples with high weight get replicated Samples with low weight die off Concentrates particles in areas of higher probability Robert Collins Penn State Generic Particle Filter Robert Collins Penn State Sample Importance Resample (SIR) SIR is a special case of the generic particle filter where: - the prior density is used as the proposal density - resampling is done every iteration therefore and thus cancellation the old weights are all equal due to resampling Robert Collins Penn State SIR Algorithm Robert Collins Penn State Drawing from the Prior Density note, when we use the prior as the importance density, we only need to sample from the process noise distribution (typically uniform or Gaussian). Why? Recall: xk = fk (xk-1, vk-1) v is process noise Thus we can sample from the prior P(xk | xk-1) by starting with i i sample x k-1, generating a noise vector v k-1 from the noise process, and forming the noisy sample i i i x k = fk (x k-1, v k-1) If the noise is additive, this leads to a very simple interpretation: move each particle using motion prediction, then add noise. Robert Collins Penn State SIR Filtering Illustration M (m) 1 xk 2 , M M ~ (m) 1 m 1 xk 1, M m 1 (m) (m) M xk 1 ,wk 1 m 1 M (m) 1 xk 1 , M M ~ (m) 1 m 1 xk , M m 1 (m) (m) M xk ,wk m 1 M (m) 1 xk 1 , M m 1 x Robert Collins Penn State Problems with SIS/SIR Degeneracy: in SIS, after several iterations all samples except one tend to have negligible weight. Thus a lot of computational effort is spent on particles that make no contribution. Resampling is supposed to fix this, but also causes a problem... Sample Impoverishment: in SIR, after several iterations all samples tend to collapse into a single state. The ability to representation multimodal distributions is thus short-lived. Robert Collins CSE598G Particle Filter Failure Analysis References King and Forsyth, “How Does CONDENSATION Behave with a Finite Number of Samples?” ECCV 2000, 695-709. Karlin and Taylor, A First Course in Stochastic Processes, 2nd edition, Academic Press, 1975. Robert Collins CSE598G Particle Filter Failure Analysis Summary Condensation/SIR is aymptotically correct as the number of samples tends towards infinity. However, as a practical issue, it has to be run with a finite number of samples. Iterations of Condensation form a Markov chain whose state space is quantized representations of a density. This Markov chain has some undesirable properties • high variance - different runs can lead to very different answers • low apparent variance within each individual run (appears stable) • state can collapse to single peak in time roughly linear in number of samples • tracker may appear to follow peaks in the posterior even in the absence of any meaningful measurements. These properties generally known as “sample impoverishment” Robert Collins CSE598G Stationary Analysis For simplicity, we focus on tracking problems with stationary distributions (posterior should be the same at any time step). [because it is hard to really focus on what is going on when the posterior modes are deterministically moving around. Any movement of modes in our analysis will be due to behavior of the particle filter] Robert Collins CSE598G A Simple PMF State Space Consider 10 particles representing a probability mass function over 2 locations. PMF state space: {(0,10)(1,9)(2,8)(3,7)(4,6)(5,5) (4,6) (6,4)(7,3)(8,2)(9,1)(10,0)} 1 2 We will now instantiate a particular two-state filtering model that we can analyze in closed-form, and explore the Markov chain process (on the PMF state space above) that describes how particle filtering performs on that process. Robert Collins CSE598G Discrete, Stationary, No Noise Assume a stationary process model with no-noise process model: Xk+1 = F Xk + vk I 0 Identity no noise process model: Xk+1 = Xk Robert Collins CSE598G Perfect Two-State Ambiguity Let our two filtering states be {a,b}. We define both prior distribution and observation model to be ambiguous (equal belief in a and b). .5 X = a .5 X = a P(X ) = 0 P(Z|X ) = 0 0 k .5 X0 = b .5 X0 = b from process model: a b a 1 0 P(Xk+1 | Xk) = b 0 1 Robert Collins CSE598G Recall: Recursive Filtering Prediction: predicted current state state transition previous estimated state Update: measurement predicted current state estimated current state normalization term These are exact propagation equations. Robert Collins CSE598G Analytic Filter Analysis Predict 1 .5 0 .5 = .5 0 .5 1 .5 = .5 Update .5 .5 = .25/(.25+.25) = .5 .5 .5 = .25/(.25+.25) = .5 Robert Collins CSE598G Analytic Filter Analysis Therefore, for all k, the posterior distribution is .5 Xk = a P(Xk | z1:k) = .5 Xk = b which agrees with our intuition in regards to the stationarity and ambiguity of our two-state model. Now let’s see how a particle filter behaves... Robert Collins CSE598G Particle Filter Consider 10 particles representing a probability mass function over our 2 locations {a,b}.
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