# Parallel Computing Based Dynamic Programming Algorithm of Track-before-Detect

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Models and Method Statement

#### 2.1. Target Dynamic Model and Measurement Model

#### 2.2. Basic Dynamic Programming Track-before-Detect (TBD) Algorithm

## 3. Multi-Target Dynamic Programming for Track-before-Detect

#### 3.1. Target Cancellation

#### 3.2. Parallel Computing-Based DP-TBD

#### 3.2.1. Partition of the Target State Transition Set

#### 3.2.2. Implementation Steps Based on Parallel Computing

- Step 1:
- According to (11), partition the state transition set $\mathsf{\Gamma}\left({\mathbf{x}}_{k}\right)$ by:$$\begin{array}{cc}\hfill \mathsf{\Gamma}\left({\mathbf{x}}_{k}\right)& =\bigcup _{i=1}^{{N}_{c}}{\mathsf{\Gamma}}^{i}\left({\mathbf{x}}_{k}\right)\hfill \\ \hfill {\mathsf{\Gamma}}^{i}\left({\mathbf{x}}_{k}\right)& =\{{\mathbf{x}}_{k-1}=[{x}_{k-1},{\dot{x}}_{k-1},{y}_{k-1},{\dot{y}}_{k-1}]|{x}_{k-1}\in [{x}_{k},{x}_{k}+{\delta}_{x}^{i}]\phantom{\rule{0.166667em}{0ex}},{y}_{k-1}\in [{y}_{k},{y}_{k}+{\delta}_{y}^{i}]\}\hfill \end{array}$$
- Step 2:
- Implement MT-DP-TBD in ${N}_{c}$ computing cores with ${N}_{c}$ transition subset ${\mathsf{\Gamma}}^{i}\left({\mathbf{x}}_{k}\right)$, respectively. For $1\le i\le {N}_{c}$:
- Step 2.1:
- DP integration: For $1\le k\le K$ and all ${\mathbf{x}}_{k}\in {\mathbb{R}}^{4}$:$${I}^{i}\left({\mathbf{x}}_{k}\right)={z}_{k}\left({\mathbf{x}}_{k}\right)+\underset{{\mathbf{x}}_{k-1}\in {\mathsf{\Gamma}}^{i}\left({\mathbf{x}}_{k}\right)}{max}{I}^{i}\left({\mathbf{x}}_{k-1}\right)$$
- Step 2.2:
- Obtain a candidate target state ${\mathbf{x}}_{K}^{m}$ at the ${K}^{\mathrm{th}}$ scan:$$\begin{array}{c}\hfill {\mathbf{x}}_{K}^{m}=arg\underset{{\mathbf{x}}_{K}\in {\mathbb{R}}^{4}}{max}{I}^{i}\left({\mathbf{x}}_{K}\right)\\ \hfill s.t.\phantom{\rule{1.em}{0ex}}{I}^{i}\left({\mathbf{x}}_{K}\right)>{V}_{DT}\end{array}$$
- Step 2.3:
- Target cancellation: Backtrack the trajectory ${\hat{\mathbf{X}}}_{1:K}^{m}$ by ${\mathsf{\Phi}}_{{\mathbf{x}}_{K}^{m}}$ defined in (10), and detach the measurement information related to ${\hat{\mathbf{X}}}_{1:K}^{m}$ from the original measurement data ${Z}_{1:K}$, then go to Step 2.1. Step 2 is a recursive process and ends when ${I}^{i}\left({\mathbf{x}}_{K}\right)<{V}_{DT}$. Then, each computing core gets a track collection ${\hat{\mathbf{X}}}^{i}=[{\left({\hat{\mathbf{X}}}_{1:K}^{1}\right)}^{i},{\left({\hat{\mathbf{X}}}_{1:K}^{2}\right)}^{i},\cdots ,{\left({\hat{\mathbf{X}}}_{1:K}^{m}\right)}^{i}]$.
- Step 3:
- Merge the ${N}_{c}$ track collections. If the coincidence of every two tracks exceeds 50%, they will be merged and considered to belong to one target. Then, the final estimation of all tracks is given as $\hat{\mathbf{X}}=[{\hat{\mathbf{X}}}_{1:K}^{1},{\hat{\mathbf{X}}}_{1:K}^{2},\cdots ,{\hat{\mathbf{X}}}_{1:K}^{M}]$. This method avoids the trajectories’ repetition caused by the operation of different transition sets and eliminates the false trajectories generated due to the spread of target energy.

## 4. Simulations and Analysis

#### 4.1. Interference of Cross Targets

#### 4.2. Simulation of PC-DP-TBD

#### 4.3. Performance Analysis

#### 4.4. Computational Expense

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Davey, S.J.; Rutten, M.G.; Cheung, B. A comparison of detection performance for several track-before-detect algorithms. Eurasip J. Adv. Signal Process.
**2007**, 22–27. [Google Scholar] [CrossRef] - Larson, R.; Peschon, J. A dynamic programming approach to trajectory estimation. IEEE Trans. Autom. Control
**1966**, 11, 537–540. [Google Scholar] [CrossRef] - Huang, X.; Zhao, Y.; Hao, Y. Dynamic Programming Algorithm for Track-Before-Detect Technology. Command Inf. Syst. Technol.
**2015**, 53–71. [Google Scholar] [CrossRef] - Tonissen, S.; Evans, R. Peformance of dynamic programming techniques for Track-Before-Detect. IEEE Trans. Aerosp. Electron. Syst.
**1996**, 32, 1440–1451. [Google Scholar] [CrossRef] - Cai, F.; Fan, H.; Fu, Q. Dual-Channel Particle Filter Based Track-Before-Detect for Monopulse Radar. Math. Probl. Eng.
**2014**, 2014, 750279. [Google Scholar] [CrossRef] - Rutten, M.G.; Ristic, B.; Gordon, N.J. A comparison of particle filters for recursive track-before-detect. In Proceedings of the 2005 7th International Conference on Information Fusion, Philadelphia, PA, USA, 25–28 July 2005; Volume 1, p. 7. [Google Scholar] [CrossRef]
- Jing, C.; Lin, Z.; Li, J. Detection and tracking of an underwater target using the combination of a particle filter and track-before-detect. In Proceedings of the OCEANS 2016-Shanghai, Shanghai, China, 10–13 April 2016; pp. 1–5. [Google Scholar] [CrossRef]
- Bi, X.; Du, J.; Zhang, Q.; Wang, W. Improved multi-target radar TBD algorithm. J. Syst. Eng. Electron.
**2015**, 26, 1229–1235. [Google Scholar] [CrossRef] [Green Version] - Carlson, B.D.; Evans, E.D.; Wilson, S.L. Search radar detection and track with the Hough transform. I. system concept. IEEE Trans. Aerosp. Electron. Syst.
**1994**, 30, 102–108. [Google Scholar] [CrossRef] - Buzzi, S.; Lops, M.; Venturino, L.; Ferri, M. Detection of an Unknown Number of Targets via Track-Before-Detect Procedures. In Proceedings of the 2007 IEEE Radar Conference, Boston, MA, USA, 17–20 April 2007; pp. 180–185. [Google Scholar]
- Yi, W.; Morelande, M.R.; Kong, L.; Yang, J. An Efficient Multi-Frame Track-Before-Detect Algorithm for Multi-Target Tracking. IEEE J. Sel. Top. Signal Process.
**2013**, 7, 421–434. [Google Scholar] [CrossRef] - Grossi, E.; Lops, M.; Venturino, L. A Track-Before-Detect Algorithm With Thresholded Observations and Closely-Spaced Targets. IEEE Signal Process. Lett.
**2013**, 20, 1171–1174. [Google Scholar] [CrossRef] - Bruno, M.G.S.; Moura, J.M.F. Multiframe detector/tracker: optimal performance. IEEE Trans. Aerosp. Electron. Syst.
**2001**, 37, 925–945. [Google Scholar] [CrossRef] - Yan, B. Track-before-detect algorithm based on dynamic programming for multi-extended-targets detection. IET Signal Process.
**2017**, 11, 674–686. [Google Scholar] [CrossRef] - McDonald, M.; Balaji, B. Impact of Measurement Model Mismatch on Nonlinear Track-Before-Detect Performance for Maritime RADAR Surveillance. IEEE J. Ocean. Eng.
**2011**, 36, 602–614. [Google Scholar] [CrossRef] - Jiang, H.; Yi, W.; Cui, G.; Kong, L.; Yang, X. Knowledge-Based Track-Before-Detect Strategies for Fluctuating Targets in K-Distributed Clutter. IEEE Sens. J.
**2016**, 16, 7124–7132. [Google Scholar] [CrossRef] - Moyer, L.R.; Spak, J.; Lamanna, P. A Multi-Dimensional Hough Transform-Based Track-Before-Detect Technique for Detecting Weak Targets in Strong Clutter Backgrounds. IEEE Trans. Aerosp. Electron. Syst.
**2011**, 47, 3062–3068. [Google Scholar] [CrossRef] - Johnston, L.A.; Krishnamurthy, V. Performance analysis of a track before detect dynamic programming algorithm. In Proceedings of the 2000 IEEE International Conference on Acoustics, Speech, and Signal Processing, Istanbul, Turkey, 5–9 June 2000; Proceedings (Cat. No.00CH37100). Volume 1, pp. 49–52. [Google Scholar] [CrossRef]
- Johnston, L.A.; Krishnamurthy, V. Performance analysis of a dynamic programming track before detect algorithm. IEEE Trans. Aerosp. Electron. Syst.
**2002**, 38, 228–242. [Google Scholar] [CrossRef] - Orton, M.; Fitzgerald, W. A Bayesian approach to tracking multiple targets using sensor arrays and particle filters. IEEE Trans. Signal Process.
**2002**, 50, 216–223. [Google Scholar] [CrossRef]

**Figure 1.**Target state transition process illustrated by admissible search regions among successive scans. The target velocity is assumed to be one cell/frame, and its position in the ${k}^{\mathrm{th}}$ frame is given. The corresponding transition areas in the $k-{1}^{\mathrm{th}}$ and $k-{2}^{\mathrm{th}}$ frames are represented by the shadow cells.

**Figure 2.**Target state transition area and the corresponding partition adopted in parallel computing dynamic programming-based track-before-detect (PC-DP-TBD). MT, multi-target.

**Figure 4.**Two target tracks with a target SNR of 10 dB; the solid circles represent the position of targets among $K=6$ frames of measurement.

**Figure 7.**This figure shows the DP integration result of one PC-DP-TBD batch process from four processors respectively. (

**a**) ${\delta}_{x}^{1}$, ${\delta}_{y}^{1}$; (

**b**) ${\delta}_{x}^{2}$, ${\delta}_{y}^{2}$; (

**c**) ${\delta}_{x}^{3}$, ${\delta}_{y}^{3}$; (

**d**) ${\delta}_{x}^{4}$, ${\delta}_{y}^{4}$.

**Figure 9.**The detection probability ${P}_{d}$ of all six targets against SNR from 4 dB–15 dB for ${P}_{f}a={10}^{-1}$. The ${P}_{d}$ of PC-DP-TBD method is indicated by the solid lines, and the dotted lines indicate the ${P}_{d}$ of MT-DP-TBD for comparison.

**Figure 10.**The RMSE of all six targets against SNR from 4 dB–15 dB for ${P}_{f}a={10}^{-1}$. The RMSE of PC-DP-TBD method is indicated by the solid lines, and the dotted lines indicate the RMSE of MT-DP-TBD for comparison.

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Guo, Q.; Li, Z.; Song, W.; Fu, W.
Parallel Computing Based Dynamic Programming Algorithm of Track-before-Detect. *Symmetry* **2019**, *11*, 29.
https://doi.org/10.3390/sym11010029

**AMA Style**

Guo Q, Li Z, Song W, Fu W.
Parallel Computing Based Dynamic Programming Algorithm of Track-before-Detect. *Symmetry*. 2019; 11(1):29.
https://doi.org/10.3390/sym11010029

**Chicago/Turabian Style**

Guo, Qiang, Zhenwu Li, Wenming Song, and Wenyu Fu.
2019. "Parallel Computing Based Dynamic Programming Algorithm of Track-before-Detect" *Symmetry* 11, no. 1: 29.
https://doi.org/10.3390/sym11010029