Single-object tracking in clutter
4.3 The probabilistic data association filter
Algorithm 16 Nearest neighbor filter recursion equations at time k
1: Prediction:
ˆxk|k−1, Pk|k−1
= KFP
ˆxk−1|k−1, Pk−1|k−1, F, Q . 2: Measurement selection{Section 5.5.2, (5.78)}:
yk(i) = arg min
yk( j),∀ j∈{1,..., mk}
yk( j) − H ˆxk|k−1T
S−1k|k−1
yk( j) − H ˆxk|k−1 , where Sk|k−1= HPk|k−1HT + Rk.
3: Output trajectory estimate:
ˆxk|k, Pk|k
= KFE
yk(i), ˆxk|k−1, Pk|k−1, H, R .
4.3 The probabilistic data association filter
The most successful algorithm in the class of Bayesian all-neighbors filters is the probabilistic data association (PDA) filter. The algorithm updates the object state estimate using all validated measurements and their respective posterior probabil-ity weightings. The PDA algorithm is derived by making the assumption that the prediction density in the optimal Bayesain recursion given all past observations is a Gaussian density, although, strictly speaking, it is a Gaussian mixture. This PDA filter approximation is based on the following six strong assumptions:
1. The object being tracked exists and no other object exists.
2. The object motion obeys linear Gaussian statistics.
3. Only one measurement can be from the object of interest.
4. The measurement noise is white Gaussian.
5. The object may or may not be detected all the time and is detected with proba-bility of detection PD.
6. All non-object originated measurements are assumed to be originated from clutter that is uniformly distributed in space and Poisson distributed in time.
7. Only measurements that fall within a proximity of the expected measurement (i.e., that fall within the validation gate) are considered for processing
The transition and prediction density evaluation
The object dynamics, measurement and noise models are the same as those in the nearest neighbor filter in Section 4.2. Thus the transition and prediction densities are the same as (4.11).
The likelihood function
While the transition and predicted densities of NNF and the PDAF are the same, the key difference is in the way both filters extract information out of the received measurements – i.e., the likelihood function. The likelihood in the PDA is approx-imated by choosing the subset of measurements from the total measurement set yk by gating the measurements based on their statistical distance to the predicted measurements. Since both the measurement noise and model process noise are Gaussian, a chi-square test (Bar-Shalom and Fortmann, 1988) determines the sta-tistical distance and whether a given measurement lies within the minimum volume hyper-ellipsoid that contains a set percentage of the probability distribution of the predicted measurement – the validation gate. The measurements that fall within this ellipsoid are said to be gated. The validation gate is an ellipsoid defined by
G= {y ∈ Rn : [y − ˆyk|k−1]S(k)−1[y − ˆyk|k−1]T ≤ γ },
where √γ is the gate size, ˆyk|k−1 is the predicted measurement and S(k) is its covariance. A measurement falling inside the validation gate is accepted as a mea-surement and is indexed as a validated meamea-surement. That is, y becomes one of the received measurements{yk(1), yk(2), . . . , yk(mk)}. The volume of the n-dimensional gate is
Vk = πn/2 (n/2 + 1)
%|S(k)|γ1/2,
where|S(k)| is the determinant of S(k).
By assuming that the clutter measurements are uniformly distributed in space and have equal probability of being at any point within the gate we can derive the likelihood of the PDA filter. Let PG denote the probabil-ity that the correct measurement falls within the gate, then the likelihood
p(yk(1), yk(2), . . . , yk(mk)|xk, mk, θk(i)) can be evaluated as p(yk(1), yk(2), . . . , yk(mk)|xk, mk, θk(i)) =
1 Vk
mk−1
p(yk(i)|xk), (4.14) where p(yk(i)|xk) is the likelihood of the ith measurement being object-originated. It is N(yk(i); Hxk, Rk) when the complete measurement space is con-sidered. However, since the effective measurement space is truncated to the vali-dation gate, the likelihood is effectively truncated by the chi-square ellipsoid. This leads to the truncated Gaussian density and does not integrate to 1. By normalizing it with its area in the gate (i.e., the gating probability) it can be re-interpreted as a measurement probability density.
4.3 The probabilistic data association filter 113
Now the first term of (4.2) can be expanded into p(yk, mk|xk, yk−1, mk−1)
theorem as follows: let PDstand for the probability that the object is detected and PG for the probability that the object-originated measurement falls into the gate.
Using Bayes’ theorem,
whereNOTA is the complement event of event A. LetμF(mk) be the distribution of mkclutter measurements that are gated. After some rearranging,
p(θk(0)|mk) = (1 − PDPG)μF(mk)
and for i = 1, . . . , mk, assuming that all measurements gated are equally likely to The two most commonly used distributions for μF(mk) are Poisson and non-parametric (uniform) clutter densities as defined below:
μF(mk) = exp(−λVk)(λVk)mk
mk! , for Poisson clutter, where λVkis the mean of the distribution, and
μF(mk) = 1
N ∀ mk = 0, 1 . . . , N − 1, for non-parametric clutter.
For example, for the non-parametric model (Bar-Shalom and Fortmann, 1988), P(θk(i)|mk) =
1− PDPG, i = 0, PDPG/mk, i = 1, 2, · · · , mk. Whereas for the Poisson clutter distribution,
P(θk(i)|mk) Using the result from (4.17) we have
δ =
4.3 The probabilistic data association filter 115
we can concisely represent the normalization factor in the context of single-object tracking in clutter by the following equation:
δ = a0+ a1 mk
i=1
ei. (4.18)
The conditional density
Returning to the density of prime interest, the conditional density of the object state, we have
p(xk|yk, mk) = 1 δ
mk
i=0
p(yk(1), yk(2), . . . , yk(mk), mk|xk, θk(i))p(θk(i)|mk)
× p(xk|yk−1, mk−1),
whereδ = p(yk, mk|yk−1, mk−1) is the normalizing factor. Substituting the likeli-hood of (4.17), and the predicted density of (4.11), the conditional density can be rewritten as
p(xk|yk, mk) = 1 δ
1 Vk
mk
N(xk; ˆxk|k−1, Pk|k−1)p(θk(0)|mk)p(mk|yk−1, mk−1)
+1 δ
1 Vk
mk−1 mk
i=1
p(yk(i)|xk)N(xk; ˆxk|k−1, Pk|k−1)p(θk(i)|mk)
× p(mk|yk−1, mk−1),
where
p(yk(i)|xk)N(xk; ˆxk|k−1, Pk|k−1)= PG−1N(yk(i); Hxk, Rk)N(xk; ˆxk|k−1, Pk|k−1).
(4.19) The above expression can be further simplied by multiplying and dividing it by N(xk; ˆxk|k−1, Pk|k−1). Thus we have,
p(yk(i)|xk)N(xk; ˆxk|k−1, Pk|k−1)
= PG−1N(yk(i); ˆyk, Sk)N(yk(i); Hxk, Rk)N(xk; ˆxk|k−1, Pk|k−1) N(yk(i); ˆyk, Sk)
= PG−1N(yk(i); ˆyk, Sk)N(xk; ˆxik|k, Pik|k),
where, using the Gaussian distributions theorem of Appendix A,
ˆxik|k, Pik|k
= KFE
yk(i), ˆxk|k−1, Pk|k−1, H, Rk
.
4.3 The probabilistic data association filter 117 These are the update of a Kalman filter using the i th measurement yk(i), as in the PDAF solution. Substituting, we have
where, by substituting the normalization factorδ = a0+ a1
mk
Calculating the data association probabilities, β(·) is denoted by a pseudo-function
{βk(i)}mi=0k
= STDA
{pk(i)}mi=1k ,
where βk(i) are calculated for the ith validated measurement (the measurement within the validation gate Vk).
Thus the posterior density of the target state p(xk|yk, mk) is a Gaussian mixture whose mean and covariance are given as follows:
ˆxk|k, Pk|k
= GMix!
ˆxik|k, Pik|k, βk(i)"
i
.
The PDA conditional density estimate calculation is denoted by the First the a posteriori estimation mean ˆxk|k(i) and covariance Pk|k(i) are calcu-lated given each measurement possibility i ≥ 0,
[ˆxk|k(i), Pik|k] =
after which the mean and covariance of the resulting Gaussian mixture are calcu-lated by
4.3.1 The probability data association filter equations
Algorithm 17 PDA filter recursion equations at time k