MultiTarget State Extraction for the SMCPHD Filter
Abstract
:1. Introduction
2. Problem Formulation
2.1. The PHD Filter
2.2. Review of the SMCPHD Filter
 Step 1 Prediction:
 For $\text{\hspace{0.17em}}i=1,\mathrm{...},{L}_{k1}\text{\hspace{0.17em}}$, sample ${\tilde{x}}_{k}^{(i)}$ from a proposal density ${q}_{k}(\cdot {x}_{k1}^{(i)},{\mathrm{Z}}_{k})$ for persistent targets and compute the predicted weights:$${\tilde{w}}_{kk1}^{(i)}=\frac{{\varphi}_{kk1}({\tilde{x}}_{k}^{(i)},{x}_{k1}^{(i)}){w}_{k1}^{(i)}}{{q}_{k}({\tilde{x}}_{k}^{(i)}{x}_{k1}^{(i)},{Z}_{k})}$$
 For $i={L}_{k1}+1,\mathrm{...},{L}_{k1}+{J}_{k}$, sample ${\tilde{x}}_{k}^{(i)}$ from a proposal density ${p}_{k}(\cdot {\mathrm{Z}}_{k})\text{\hspace{0.17em}}$ for newborn targets and compute the corresponding weights:$${\tilde{w}}_{kk1}^{(i)}=\frac{{\gamma}_{k}({\tilde{x}}_{k}^{(i)})}{{J}_{k}{p}_{k}({\tilde{x}}_{k}^{(i)}{Z}_{k})}$$
 Step 2 Update:
 For each $z\in {\mathrm{Z}}_{k}$, compute:$${C}_{k}(z)={\displaystyle \sum _{j=1}^{{L}_{k1}+{J}_{k}}{\psi}_{k,z}({\tilde{x}}_{k}^{(j)}){\tilde{w}}_{kk1}^{(j)}}$$
 For $i=1,\mathrm{...},{L}_{k1}+{J}_{k}$, update weights:$${\tilde{w}}_{k}^{(i)}=\left[(1{p}_{D,k}({\tilde{x}}_{k}^{(i)}))+{\displaystyle \sum _{z\in {\mathrm{Z}}_{k}}\frac{{\psi}_{k,z}({\tilde{x}}_{k}^{(i)})\text{\hspace{0.17em}}}{{\mathit{\kappa}}_{k}(z)+{C}_{k}(z)}}\right]{\tilde{w}}_{kk1}^{(i)}$$
 Step 3 Resampling:
 Compute the total mass ${\widehat{N}}_{kk}={\displaystyle {\sum}_{i=1}^{{L}_{k1}+{J}_{k}}{\tilde{w}}_{k}^{(i)}}$ and then resample ${\{{\tilde{w}}_{k}^{(i)}/{\widehat{N}}_{kk},{\tilde{x}}_{k}^{(i)}\}}_{i=1}^{{L}_{k1}+{J}_{k}}$ to get ${\{{w}_{k}^{(i)}/{\widehat{N}}_{kk},{\mathit{x}}_{k}^{(i)}\}}_{i=1}^{{L}_{k}}$.
 Rescale the weights by ${\widehat{N}}_{kk}$ to obtain ${\{{w}_{k}^{(i)},{x}_{k}^{(i)}\}}_{i=1}^{{L}_{k}^{}}$.
 Step 4 MultiTarget Parameter Estimation:
 Estimate the number of targets ${N}_{k}$ (by rounding ${\widehat{N}}_{kk}$).
 Extract the target state set ${\widehat{\mathrm{X}}}_{k}=\{{\widehat{x}}_{k}^{(1)},\mathrm{...},{\widehat{x}}_{k}^{({N}_{k})}\}$ from the particles that represent the posterior intensity, where ${\widehat{x}}_{k}^{(1)},\mathrm{...},\text{\hspace{0.17em}}{\widehat{x}}_{k}^{({N}_{k})}$ denote the estimated multitarget state.
3. The Proposed MultiTarget State Extraction Method
3.1. Particles and Measurements Classification
3.2. MultiTarget State Extraction
Algorithm 1. Multitarget state extraction for detected targets. 
Given: The estimated target number ${N}_{k}$, ${\mathrm{Z}}_{E,k}$ and ${\mathrm{X}}_{C,k}$.

Algorithm 2. Methods for state estimation of undetected targets. 
Given: ${\overline{\mathrm{X}}}_{k}$ and ${\widehat{\mathrm{X}}}_{k}$ 
1. Set ${\mathrm{U}}_{k}={\overline{\mathrm{X}}}_{k}$, $\mathcal{I}=\varnothing $ 
2. for $i=1,\mathrm{...},{N}_{k}$ do 
3. $\mathcal{I}=\{{\tilde{x}}_{k}^{(p)}\in {\mathrm{U}}_{k}({\tilde{x}}_{k}^{(p)}{\widehat{x}}_{k}^{(i)}){[{\widehat{P}}_{k}^{(i)}]}^{1}{({\tilde{x}}_{k}^{(p)}{\widehat{x}}_{k}^{(i)})}^{T}\le \mathsf{\lambda}\}$ 
4. ${\mathrm{U}}_{k}={\mathrm{U}}_{k}\backslash \mathcal{I}$ 
5. end for 
6. ${\mathrm{U}}_{k}={\mathrm{U}}_{k}\backslash ({\mathrm{U}}_{k}\cap {\{{\tilde{x}}_{k}^{(i)}\}}_{i={L}_{k1}+1}^{{L}_{k1}+{J}_{k}})$. 
7. Compute ${\overline{N}}_{\mathrm{U},k}$ according to (19) 
8. if ${\overline{N}}_{U,k}\ge 1$ 
9. State extraction using clustering method 
10. end 
3.3. Notes on Implementation
4. Simulation
5. Conclusions
Acknowledgments
Author Contributions
Conflicts of Interest
Abbreviations
FMM  Finite mixture models 
JPDA  Joint probabilistic data association 
MEAP  Multiexpected a posterior 
MHT  Multiple hypothesis tracking 
MTT  Multitarget tracking 
OSPA  Optimal Subpattern Assignment 
PHD  Probability hypothesis density 
RFS  Random finite sets 
SMC  Sequential Monte Carlo 
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Si, W.; Wang, L.; Qu, Z. MultiTarget State Extraction for the SMCPHD Filter. Sensors 2016, 16, 901. https://doi.org/10.3390/s16060901
Si W, Wang L, Qu Z. MultiTarget State Extraction for the SMCPHD Filter. Sensors. 2016; 16(6):901. https://doi.org/10.3390/s16060901
Chicago/Turabian StyleSi, Weijian, Liwei Wang, and Zhiyu Qu. 2016. "MultiTarget State Extraction for the SMCPHD Filter" Sensors 16, no. 6: 901. https://doi.org/10.3390/s16060901