lecture26-2022-12-01.tmpRpePur

Today in ECE 6555

Announcements
- 2 lectures left (including today)
- One more homework (optional, I didn't realize how close the end of the semester was…)
- One final exam take-home
Last time
- Gaussian processes
- Check office hours video for discussion of
Today
- PhD filter for multi-target tracking
Questions?

Probabilistic state-space model
- State vector $x \in R^{n}$
- Markov process or order 1 with transition density $f_{k | k - 1} (x_{k} | x_{k - 1})$ governing state evolution
- Observation $z \in R^{d}$
- Noisy observation with lieklihood $g_{k} (z_{k} | x_{k})$
Bayesian filtering
1. Initialization: start from known $p (x_{0})$
2. For $i \geq 1$
  1. Prediction: compute $p (x_{i} | z_{0 : i - 1})$ by Chapman-Kolmogorov equation $p (x_{i} | y_{0 : i - 1}) = \int f_{i} (x_{i} | x_{i - 1}) p (x_{i - 1} | y_{0 : i - 1}) d x_{i - 1}$
  2. Update: compute $p (x_{i} | z_{0 : i})$ by Bayes' rule $p (x_{i} | y_{0 : i}) = \frac{1}{Z_{i}} p (y_{i} | x_{i}) p (x_{i} | z_{0 : i - 1}) with Z_{i} = \int g_{i} (z_{i} | x_{i}) p (x_{i} | y_{0 : i - 1}) d x_{i}$

Multi-target tracking extends the problem to multiple objects
- At time $k$ , $M (k)$ targets with state vectors ${x_{i}}_{i = 1}^{M (k)}$ , each in $X \subset R^{n}$
  - Targets may die, evolve, or new targets may appear
- At time $k$ , $N (k)$ measurements ${z_{i}}_{i = 1}^{N (k)}$ , each in $Z \subset R^{d}$
  - Measurements cannot be associated to targets
No ordering on the targets and measurements (no association)
- Represented as collections $X_{k} ≜ {x_{i}}_{i = 1}^{M (k)} \in F (X)$ , $Z_{k} ≜ {z_{i}}_{i = 1}^{N (k)} \in F (Z)$
- Collections modeled as random finite sets (RFS) to model undertainty
RFS model
- Given $X_{k - 1}$ , each target $x_{k - 1} \in X_{k - 1}$ survives with probability $p_{S, k} (x_{k - 1})$ or dies
- Conditioned on existence, state in $X_{k - 1}$ evolves according to $f_{k | k - 1} (x_{k} | x_{x_{k - 1}})$
- A given state $x_{k - 1}$ can evolve either into an RFS $S_{k | k - 1} (x_{k - 1})$ equal to $x_{k}$ or $\emptyset$
- Spontaneous births of targets described by independent RFS $Γ_{k}$
- Existing target $ξ$ can spawn new targets described by independent RFS $B_{k | k - 1} (ξ)$
$X_{k} = [⋃_{ξ \in X_{k - 1}} S_{k | k - 1} (ξ)] ⋃ [⋃_{ξ \in X_{k - 1}} B_{k | k - 1} (ξ)] ⋃ Γ_{k}$

Measurement model
- Target $x_{k} \in X_{k}$ detected with probabiltiy $p_{D, k} (x_{k})$
- If detected, target generates observation $z_{k}$ according to $g_{k} (z_{k} | x_{k})$
- A given state $x_{k} \in X_{k}$ can generatesa an RFS $Θ_{k} (x_{k})$ equal to $z_{k}$ or $\emptyset$
- Sensors can also receive clutter (false measurements) $K_{k}$
$Z_{k} = [⋃_{ξ \in X_{k}} Θ_{k} (ξ)] ⋃ K_{k}$
Ideal Bayes filter to track multiple-target posterior density $p_{k | k - 1} (X_{k} | Z_{1 : k - 1}) = \int f_{k | k - 1} (X_{k} | X) p_{k - 1} (X | Z_{1 : k - 1}) μ_{s} (X)$ $p_{k} (X_{k} | Z_{1 : k}) = \frac{g_{k} (Z_{k} | X_{k}) p_{k | k - 1} (X_{k} | Z_{1 : k - 1})}{\int g_{k} (Z_{k} | X) p_{k | k - 1} (X_{k} | Z_{1 : k - 1}) μ_{s} (X)}$
- Intractable!

Probability Hypothesis Density filter is analogous to the Kalman filter for single-target tracking
- Approximation to alleviate computational burden in the multiple-target Bayes filter
- Propagates a first-order statistical moment of the posterior multiple-target state
Intensity function: first- order moment of RFS $X \in X$
- Intensity if $v : X \to R$ such that for any $S \subset X$ $\int | X \cap S | P (d X) = \int_{S} v (x) d x$
- $\hat{N} = \int v (x) d x$ is the expected number of targets in $X$
- local maxima of $v$ can be used to generated estimate of elements in $X$
Poisson RFS
- Completely characterized by intensity function
- $P_{} (| X | = n)$ is poisson with mean $\hat{N}$
- elements of $X$ are i.i.d. according to the probability density function $v (x) / N$
Clutter, Birth, Spawn RFS modeled as Poisson RFS

Intensities
- $γ_{k}$ : intensity of birth RFS $Γ_{k}$
- $β_{k | k - 1} (\cdot | ξ)$ : intensity of spawn RFS $B_{k | k - 1} (ξ)$
- $p_{S, k} (ξ)$ : probability that target survives
- $p_{D, k} (ξ)$ : probability that target is detected
- $κ_{k}$ : intensity of clutter RFS
Assumptions
- Each target evolves and generates observations independently of one another
- Clutter is Poisson and independent of target-originated measurements
- The predicted multiple-target RFS governed by $p_{k | k - 1}$ is Poisson
  - approximation but true without spawning if $X_{k - 1}$ and $Γ_{k}$ Poisson
Recursions for intensity $v_{k | k - 1} (x) = \int p_{S, k} (ξ) f_{k | k - 1} (x | ξ) v_{k - 1} (ξ) d ξ + \int β_{k | k - 1} (x | ξ) v_{k - 1} (ξ) d ξ + γ_{k} (x)$ $v_{k} (x) = (1 - p_{D, k} (x)) v_{k | k - 1} (x) + \sum_{z \in Z_{k}} \frac{p_{D, k} (x) g_{k} (z | x) v_{k | k - 1} (x)}{κ_{k} (z) + \int p_{D, k} (ξ) g_{k} (z | ξ) v_{k | k - 1} (ξ) d ξ}$

Linear Gaussian models for dynamics and measurment
- $f_{k | k - 1} (x | ξ) \sim N (x; F_{k - 1} ξ, Q_{k - 1})$
- $g_{k} (z | x) \sim N (z; H_{k} x, R_{k})$
Survival and detection probabilities state independent $p_{S, k} (x) = p_{S, k} p_{D, k} (x) = p_{D, k}$
Gaussian mixtures for birth and spawn intensities $γ_{k} (x) ≜ \sum_{i = 1}^{J_{γ, k}} w_{γ, k}^{(i)} N (x; m_{γ, k}^{(i)}, P_{γ, k}^{(i)})$ $β_{k | k - 1} (x | ξ) ≜ \sum_{i = 1}^{J_{β, k}} w_{β, k}^{(j)} N (x; F_{β, k - 1}^{(j)} ξ + d_{β, k - 1}^{(j)}, Q_{β, k - 1}^{(j)})$
Key insight: Gaussian mixtures are preserved

Assume that $v_{k - 1} (x) = \sum_{i = 1}^{J_{k - 1}} w_{k - 1}^{(i)} N (x; m_{k - 1}^{(i)}, P_{k - 1}^{(i)})$ then $v_{k - 1 | k - 1} (x) = v_{S, k | k - 1} (x) + v_{β, k | k - 1} (x) + γ_{k} (x)$
where $v_{S, k | k - 1} (x) = p_{S, k} \sum_{j = 1}^{J_{k - 1}} w_{k - 1}^{(j)} N (x; m_{S, k | k - 1}^{(j)}, P_{S, k | k - 1}^{(j)})$ $m_{S, k | k - 1}^{(j)} = F_{k - 1} m_{k - 1}^{(j)} P_{S, k | k - 1}^{(j)} = F_{k - 1} P_{k - 1}^{(j)} F_{k - 1}^{T} + Q_{k - 1}$

$v_{β, k | k - 1} (x) = \sum_{j = 1}^{J_{k - 1}} \sum_{ℓ = 1}^{J_{β, k}} w_{k - 1}^{(j)} w_{β, k}^{ℓ} N (x; m_{β, k | k - 1}^{(j, ℓ)}, P_{β, k | k - 1}^{(j, ℓ)})$
where $m_{β, k | k - 1}^{(j, ℓ)} = F_{β, k - 1}^{(ℓ)} m_{k - 1}^{(j)} + d_{β, k - 1}^{(ℓ)} P_{β, k | k - 1}^{(j, ℓ)} = F_{β, k - 1}^{(ℓ)} P_{β, k - 1}^{(j)} {F_{β, k - 1}^{(ℓ)}}^{T} + Q_{β, k - 1}^{(ℓ)}$

Assume that $v_{k | k - 1} (x) = \sum_{i = 1}^{J_{k | k - 1}} w_{k | k - 1}^{(i)} N (x; m_{k | k - 1}^{(i)}, P_{k | k - 1}^{(i)})$ then $v_{k - 1} (x) = (1 - p_{D, k}) v_{k | k - 1} (x) + \sum_{z \in Z_{k}} v_{D, k} (x; z)$ $v_{D, k} (x; z) = \sum_{j = 1}^{J_{k | k - 1}} w_{k}^{(j)} (z) N (x; m_{k | k}^{(j)}, P_{k | k}^{(j)})$ $w_{k}^{(j)} = \frac{p_{D, k} w_{k | k - 1}^{(j)} q_{k}^{(j) (z)}}{κ_{k} (z) + p_{D, k} \sum_{ℓ = 1}^{J_{k | k - 1}} w_{k | k - 1}^{(ℓ)} q_{k}^{(ℓ)} (z)}$