﻿ 基于分段序列离散度的异步航迹关联算法
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Asynchronous track-to-track association algorithm based on discrete degree of segmented sequence
YI Xiao, DU Jinpeng
Naval Aviation University, Yantai 264001, China
Abstract: To solve the problem of asynchronous unequal rate track association, this paper proposes an asynchronous track-to-track association algorithm based on discrete degree of segmented sequence. A discrete information measurement of segmented and mixed track sequence is defined, and the segmentation rules of track sequence with unequal lengths are presented. The association determination is performed via calculation of the discrete degree and adoption of the classical assignment method. In view of the ambiguity problem, a secondary inspection link is established. Compared with traditional algorithms, this algorithm requires no time alignment and is not affected by noise distribution. The simulation results show that the algorithm can maintain stable association effect with high accuracy under the conditions of asynchronous track and different sensor sampling rates. Furthermore, it can effectively distinguish complex situations such as track crossing, bifurcation and merging, exhibiting clear advantages.
Keywords: asynchronous track    discrete degree    track association    segmented sequence    target tracking

1 数据集的离散度度量

 $\Delta (X,Y) = \left\{ \begin{array}{l} \mathop {{\rm{max}}}\limits_{\Pi _M^{nN}} V({X_{\Pi _M^{nN}}} \cup nY) + \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \mathop {{\rm{max}}}\limits_{\Pi _M^{nN}} V(X_{\Pi _M^n}^{\rm{c}} \cup {Y_{\Pi _N^m}}),M \ge N\\ \Delta (Y,X),M < N \end{array} \right.$ （1）

 ${X_{\Pi _p^q}} = \{ x_j^\pi |x_j^\pi \in X,j = 1,2, \cdots ,q\}$

 $\bar x = \sum\limits_{i = 1}^p {{x_i}} /p,s = \sqrt {\sum\limits_{i = 1}^p {{{({x_i} - \bar x)}^2}} /p}$
2 基于分段序列离散度的航迹关联算法 2.1 问题描述

 ${U_s} = \{ 1,2, \cdots ,{m_s}\} ,{U_w} = \{ 1,2, \cdots ,{m_w}\}$ （2）

 ${\hat \lambda _{ij}} = \Delta (\hat X_s^i,\hat X_w^j)$ （3）

 ${\lambda _{ij}} = \Delta (X_s^i,X_w^j)$ （4）

2.2 不等长航迹序列的分段划分

 $\begin{array}{*{20}{c}} {X \to \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over X} = \{ \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over X} (1),\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over X} (2), \cdots ,\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over X} (n)\} = }\\ {\{ \{ \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over x} (1)\} ,\{ \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over x} (2)\} , \cdots ,\{ \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over x} (n)\} \} } \end{array}$

 $\left\{ {\begin{array}{*{20}{l}} {M \ge n}\\ {\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over X} (i) \cap \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over X} (j) = \emptyset ,i \ne j}\\ {\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over X} (1) \cup \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over X} (2) \cup \cdots \cup \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over X} (n) = X}\\ {\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over X} (j) = \{ \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over x} (j)\} = \left( {{x_i}|i \in \left( {\frac{{j - 1}}{n}M,\frac{j}{n}M} \right]} \right)}\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} j = 1,2, \cdots ,n} \end{array}} \right.$

 ${\zeta _s}(k) = \{ \zeta _s^1(k),\zeta _s^2(k), \cdots ,\zeta _s^i(k), \cdots ,\zeta _{{s^s}}^m(k)\}$

 $\begin{array}{l} \zeta _s^i(k + b) = \{ \hat X_s^i(k + b + {f^1}),\\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \hat X_s^i(k + b + {f^2}), \cdots ,\hat X_s^i(k + b + {f^{{n^i}}})\} \end{array}$

 $\begin{array}{l} \zeta _s^i(k) = \zeta _s^i(k + 1) \cup \zeta _s^i(k + 2) \cup \cdots \cup \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \zeta _s^i(k + a) = \{ \hat X_s^i(k + 1 + {f^1}),\hat X_s^i(k + 1 - 1)\\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {f^2}), \cdots ,\hat X_s^i(k + a + {f^{{n^i}}})\} \end{array}$

 $\begin{array}{l} \zeta _s^i(k) = \{ \hat X_s^i(k + f_s^1),\hat X_s^i(k + f_s^2), \cdots ,\\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \hat X_s^i(k + f_{{s^s}}^{{n^i}})\} \end{array}$

1) 确保分段后同一组对比序列具有相同的段数，分段数目为nj=INTL[nwj/nsi]+2。

2) 序列分段时，尽可能保证每个分段航迹子序列所含航迹点数目相等。

3) 尽量避免某分段航迹子序列只包含单一航迹点。

4) 原则优先级为：$1) \succ 2) \succ 3)$

2.3 分段航迹序列离散度的计算

 ${{\mathit{\boldsymbol{{ }\!\!\varPsi\!\!\text{ } }}}_{x}}=\left[ \begin{matrix} \overset\frown{X_{w}^{1}}(1) & \cdots & \overset\frown{X_{w}^{1}}(p) & \cdots & \overset\frown{X_{w}^{1}}({{n}_{1}}) & 0 & 0 \\ \overset\frown{X_{s,1}^{i}}(1) & \cdots & \overset\frown{X_{s,1}^{i}}(p) & \cdots & \overset\frown{X_{s,1}^{i}}({{n}_{1}}) & 0 & 0 \\ \vdots & {} & \vdots & {} & \vdots & \vdots & \vdots \\ \overset\frown{X_{w}^{k}}(1) & \cdots & \overset\frown{X_{w}^{k}}(p) & \cdots & \overset\frown{X_{w}^{k}}({{n}_{1}}) & \cdots & \overset\frown{X_{w}^{k}}({{n}_{k}}) \\ \overset\frown{X_{s,k}^{i}}(1) & \cdots & \overset\frown{X_{s,k}^{i}}(p) & \cdots & \overset\frown{X_{s,k}^{i}}({{n}_{1}}) & \cdots & \overset\frown{X_{s,k}^{i}}({{n}_{k}}) \\ \vdots & {} & \vdots & {} & {} & \vdots & \vdots \\ \overset\frown{X_{w}^{j}}(1) & \cdots & \overset\frown{X_{w}^{j}}(p) & \cdots & \cdots & \overset\frown{X_{w}^{j}}({{n}_{j}}) & 0 \\ \overset\frown{X_{s,j}^{1}}(1) & \cdots & \overset\frown{X_{s,j}^{1}}(p) & \cdots & \cdots & \overset\frown{X_{s,j}^{1}}({{n}_{j}}) & 0 \\ \vdots & {} & \vdots & {} & {} & \vdots & \vdots \\ \overset\frown{X_{{{w}^{w}}}^{m}}(1) & \cdots & \overset\frown{X_{{{w}^{w}}}^{m}}(p) & \cdots & \cdots & 0 & 0 \\ \overset\frown{X_{s,{{m}_{w}}}^{i}}(1) & \cdots & \overset\frown{X_{s,{{m}_{w}}}^{i}}(p) & \cdots & \cdots & 0 & 0 \\ \end{matrix} \right]$ （5）

 $\begin{array}{*{20}{l}} {{\mathit{\boldsymbol{ \boldsymbol{\varTheta} }}_x} = {{[{\delta _{e,f}}]}_{{m_w} \times {n_k}}} = }\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \left[ {\begin{array}{*{20}{c}} {{\delta _{1,1}}}&{{\delta _{1,2}}}& \cdots &{{\delta _{1,{n_k}}}}\\ {{\delta _{2,1}}}&{{\delta _{2,2}}}& \cdots &{{\delta _{2,{n_k}}}}\\ \vdots & \vdots &{}& \vdots \\ {{\delta _{{m_w}}},1}&{{\delta _{{m_w}}},2}& \cdots &{{\delta _{{m_w}}},{n_k}} \end{array}} \right]} \end{array}$ （6）

 $\begin{array}{*{35}{l}} {{\delta }_{e,f}}=\Delta (\overset\frown{X_{w}^{e}}(f),\overset\frown{X_{s,e}^{i}}(f)) \\ e=1,2,\cdots ,{{m}_{w}};f=1,2,\cdots ,{{n}_{k}} \\ \end{array}$ （7）

 ${{\lambda }_{x}}=\lambda (\overset\frown{X_{w}^{j}},\overset\frown{X_{s,j}^{i}})=\frac{1}{{{n}_{j}}}\sum\limits_{f=1}^{{{n}_{j}}}{{{\delta }_{j,f}}}$ （8）

 ${\lambda _{ij}} = {\alpha _1}{\lambda _x} + {\alpha _2}{\lambda _y}$ （9）

 $\left\{ {\begin{array}{*{20}{l}} {{\alpha _1} = \frac{{1/|{\sigma _x}|}}{{1/|{\sigma _x}| + 1/|{\sigma _y}|}}}\\ {{\alpha _2} = \frac{{1/|{\sigma _y}|}}{{1/|{\sigma _x}| + 1/|{\sigma _y}|}}} \end{array}} \right.$ （10）
2.4 航迹关联判定

 ${\vartheta _{ij}} = \left\{ {\begin{array}{*{20}{l}} 1\\ 0 \end{array}} \right.$ （11）

 $L(k) = \sum\limits_i^{{m_s}} {\sum\limits_j^{{m_w}} {{\vartheta _{ij}}} } {\lambda _{ij}}(k)$ （12）

 $\left\{ {\begin{array}{*{20}{l}} {\mathop {{\rm{min}}}\limits_{{\vartheta _{ij}}} \sum\limits_i^{{m_s}} {\sum\limits_j^{{m_w}} {{\vartheta _{ij}}} } {\lambda _{ij}}(k)}\\ {\sum\limits_{j = 1}^{{m_w}} {{\vartheta _{ij}}} = 1\quad \forall i = 1,2, \cdots ,{m_s}}\\ {\sum\limits_{i = 1}^{{m_s}} {{\vartheta _{ij}}} = 1\quad \forall j = 1,2, \cdots ,{m_w}} \end{array}} \right.$ （13）

 $n_{ij}^{{L_1}} = n_{ij}^{{L_1}} + 1,n_{ij}^{{L_2}} = n_{ij}^{{L_2}} + 1$ （14）

 图 2 算法流程图 Fig. 2 Flowchart of algorithm
3 仿真验证与分析 3.1 仿真环境

 ${\rm{Ec}}(k) = \frac{{\sum\limits_{i = 1}^M {{C_i}} (k)}}{{MN}}$ （15）

3.2 算法性能比较与分析

 图 3 不同算法的正确关联率对比 Fig. 3 Comparison of correct associating rates of different algorithms

 图 4 关联结果随采样率之比的变化 Fig. 4 Variation of correlation results with changing ratio of sampling rate

 图 5 关联结果随信噪比的变化 Fig. 5 Variation of correlation results with changing SNR

 雷达2开机时延/s 正确关联率 T1=0.2 s, T2=0.4 s T1=0.5 s, T2=1 s T1=1.1 s, T2=2 s 0.1 0.930 1 0.914 9 0.875 7 0.2 0.928 9 0.916 6 0.890 1 0.3 0.924 1 0.915 8 0.885 4

 噪声分布形式 高斯分布 瑞利分布 指数分布 均匀分布 正确关联率 0.922 6 0.921 7 0.916 2 0.918 4
3.3 航迹分叉合并情况的可辨性分析

 图 6 航迹存在分叉合并时的正确关联率对比 Fig. 6 Comparison of correct associating rates of track bifurcation merging

 图 7 航迹分叉与合并示意图 Fig. 7 Schematic diagram of track bifurcation and merging

 图 8 分段航迹序列离散度的变化 Fig. 8 Variation of segmented track sequence dispersion

3.4 算法复杂度分析

 图 9 不同算法的耗时对比 Fig. 9 Comparison of time consumption of different algorithms

 算法类型 运算量 文献[16]中算法 本文算法 乘法运算量 2mn+m 2mn+4m 加法运算量 4mn-m 6mn-2m
4 结论

1) 本文提出一种基于分段序列离散度的异步航迹关联算法，给出离散度的具体度量指标和不等长航迹序列的分段划分规则，并针对多义性问题给出二次检验方法。

2) 本文算法无需时域配准，可在多种环境下直接对异步不等速率航迹进行准确关联，具有稳定性。算法不受噪声分布的影响，且噪声强度对算法的影响相对较小，具有良好的抗杂波干扰性。

3) 本文算法可有效分辨航迹交叉、分叉和合并等复杂情况。

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http://dx.doi.org/10.7527/S1000-6893.2020.23694

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#### 文章信息

YI Xiao, DU Jinpeng

Asynchronous track-to-track association algorithm based on discrete degree of segmented sequence

Acta Aeronautica et Astronautica Sinica, 2020, 41(7): 323694.
http://dx.doi.org/10.7527/S1000-6893.2020.23694