TracktoTrack Association for Intelligent Vehicles by Preserving Local Track Geometry
Abstract
:1. Introduction
 The mathematical formulation for T2TASB is presented. Moreover, the local track geometry with kconnected neighborhood is derived to improve the robustness and accuracy of T2TASB. The proposed method extends the CPD method by considering the geometric relationship between neighboring tracks.
 An EM algorithm is proposed for T2TASB. The optimal T2TASB correspondence matrix and transformation function between local tracks are estimated simultaneously.
 The performance of the proposed method is validated by the experiments and computer simulations using the KITTI dataset.
2. A New Method for T2TASB
3. EM Solution for the Proposed Method
 1).
 Estep: ${E}_{L}(\Theta ,{\Theta}^{\left(m\right)})={Q}_{1}$
 2).
 Mstep: ${\Theta}^{(m+1)}=max{E}_{L}(\Theta ,{\Theta}^{\left(m\right)})$,
3.1. EStep
3.2. MStep
Algorithm 1 Proposed LTGP method for T2TASB 
Require: 
Local tracks ${\mathbf{X}}_{k}^{1}$ and ${\mathbf{X}}_{k}^{2}$, parameters w, $\alpha $, $\beta $, $\gamma $, M. 

Ensure: 
Transformed local track from sensor 2 is $f\left({\mathbf{X}}_{k}^{2}\right)={\mathbf{X}}_{k}^{2}+{\mathbf{G}}_{k}{\mathbf{W}}_{k}$. 
Association matrix for T2TASB is ${\mathbf{C}}^{k}(t,l)$ as in (3.2). 
4. Computer Simulations
5. Experiments on KITTI Dataset
6. Conclusions
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
Appendix A. The Relationship of Two Local Tracks
Abbreviations
T2TA  Tracktotrack association 
T2TASB  Tracktotrack association with sensor bias 
LTGP  Local track geometry preservation 
GMM  Gaussian mixture model 
EM  Expectationmaximization 
NN  Nearest neighbor 
GNN  Global nearest neighbor 
ML  Maximum likelihood 
OSPA  Optimal subpattern assignment 
CPD  Coherent point drift 
${\mathbf{X}}_{k}^{s}$  Local tracks from sensor s at time k 
K  Total number of discrete time steps 
${N}_{k}^{s}$  Number of tracks at time k by sensor s 
${\mathbf{x}}_{t,k}^{1}$  tth data from sensor 1 at time k 
${\mathbf{x}}_{l,k}^{2}$  Centroid of the lth component from sensor 2 at time k 
$\mathcal{N}$  Gaussian distribution 
${\sigma}_{k}^{2}$  Equal isotropic covariance at time k 
f  Nonrigid transformation 
$\mathbf{I}$  Identity matrix 
D  Size of a local track vector 
${\pi}_{t,l}^{k}$  Membership probability of tth row and lth column element in ${\pi}^{k}$ at time k 
${\pi}^{k}$  Membership probability matrix at time k 
${\mathbf{Z}}^{k}$  Indicator matrix 
${\mathbf{z}}_{t}^{k}$  a $1\times {N}_{k}^{1}$ binary vector for $l=1,2,\dots {N}_{k}^{2}$ at time k 
${z}_{t,l}^{k}$  tth row and lth column element in ${\mathbf{z}}_{t}^{k}$ at time k 
${\mathbf{W}}_{k}$  an ${N}_{k}^{2}\times D$ dimensional weight matrix of the Gaussian kernel 
${\mathbf{G}}_{k}$  an ${N}_{k}^{2}\times {N}_{k}^{2}$ Gaussian kernel matrix 
${g}_{ij}$  an ith row and jth column element in ${\mathbf{G}}_{k}$ 
$\beta $  the width parameter in the smoothing Gaussian filter 
$\mathbf{Tr}(.)$  Trace of a matrix 
$\mathbf{L}$  ${N}_{k}^{2}\times {N}_{k}^{2}$ weighted matrix 
${L}_{lj}$  a lth row and jth column element in $\mathbf{L}$ 
${\mathbf{G}}_{k}(i,.)$  ith row of ${\mathbf{G}}_{k}$ 
$\gamma $  Tradeoff parameter controlling between Q and $E\left(\mathbf{L}\right)$ 
${\mathbf{R}}_{k}$  an ${N}_{k}^{1}\times {N}_{k}^{2}$ matrix 
${\mathbf{C}}^{k}$  Cost matrix of T2TASB at time k as an ${N}_{k}^{1}\times ({N}_{k}^{2}+1)$ matrix 
$[{x}_{t,k}^{1},{y}_{t,k}^{1}]$  xaxis and yaxis positions of target ${\mathbf{x}}_{t,k}^{1}$ 
$[{x}_{j,k}^{1},{y}_{j,k}^{2}]$  xaxis and yaxis positions of target ${\mathbf{x}}_{j,k}^{2}$ 
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Method  GNN without Registration (s)  Reference PatternBased (s)  CPD(s)  LTGP(s)  

Sequence  
KITTI_01  0.0004  0.0008  0.0094  0.0159  
KITTI_20  0.0004  0.0013  0.0102  0.0169  
KITTI_16  0.0008  0.0015  0.0111  0.0176  
KITTI_17  0.0004  0.0009  0.0104  0.0163 
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Zou, K.; Zhu, H.; De Freitas, A.; Li, Y.; Esmaeili Najafabadi, H. TracktoTrack Association for Intelligent Vehicles by Preserving Local Track Geometry. Sensors 2020, 20, 1412. https://doi.org/10.3390/s20051412
Zou K, Zhu H, De Freitas A, Li Y, Esmaeili Najafabadi H. TracktoTrack Association for Intelligent Vehicles by Preserving Local Track Geometry. Sensors. 2020; 20(5):1412. https://doi.org/10.3390/s20051412
Chicago/Turabian StyleZou, Ke, Hao Zhu, Allan De Freitas, Yongfu Li, and Hamid Esmaeili Najafabadi. 2020. "TracktoTrack Association for Intelligent Vehicles by Preserving Local Track Geometry" Sensors 20, no. 5: 1412. https://doi.org/10.3390/s20051412