Electromagnetics - Modelling, Simulation, Control and Industrial Applications - Abstract
Wang, Han
In this work we apply an extended Kalman filter (EKF) to track both the location and the orientation of a small mobile target from multistatic response (MSR) measurements of a two dimensional conductivity problem. A mobile target of conductivity constant $kappaneq 1$ moves in a homogeneous medium of conductivity 1. At time $t$ the target is $D_t=z_t+ R_theta_tD$ with $z_t$ the center and $theta_t$ the orientation which are changing over time. We suppose that the trajectory of the target is enclosed inside a circle on which $N$ point sources/receivers $x_s$ are positioned. The measurement recorded for each pair of sources/receivers is the MSR matrix $mathbfV=(V_sr)_s,r$, being defined as the difference between the potential field due to the presence of the inclusion and the background field for source $x_s$, measured at the receiver $x_r$. Data are acquired with a short sampling step in time while the target is moving, and the objective of tracking is to estimate $z_t, theta_t_t$ from the MSR data stream $mathbfV_t_t$ in a efficient and robust way. $D$ and the contrast of conductivity $lambda$ are fully characterized by its infinite orders of Generalized Polarization Tensors (GPT) $M_alpha,beta(lambda, D)$, whose harmonic combinations are called contracted GPTs (CGPT) citeammari_enhancement_2011-1, and can be put into a matrix form $mathbfM$. The MSR data are linearly related to the CGPT using asymptotic expansion of the Green function as $mathbfV_t = mathbfL(mathbfM_t) + mathbfW_t$, with $ mathbfL$ a linear operator determined by the acquisition system hence independent of time, $mathbfM_t$ the CGPT of the target $D_t$, and $mathbfW_t$ the white noise of measurement. We suppose that the CGPT $mathbfM(D)$ of $D$ is known (using the reconstruction and dictionary matching algorithms developed in citeammari_target_2012). We introduce the state vector of velocity, position and orientation: $X_t=(v_t, z_t, theta_t)$, and assume that $X_t = mathbfFX_t-1 + U_t$ with $mathbfF$ the system state matrix, and $U_t$ a Gaussian random vector. On the other hand, explicit relations between CGPTs of the target $D_t$ and $D$ can be expressed through an operator $T_t$ as $mathbfM_t=mathbfT_t(mathbfM_D)$ citeammari_target_2012 which depends non linearly on $z_t, theta_t$, so that $ mathbfL(mathbfT_t(mathbfM_D)) = h(z_t, theta_t; mathbfM_D)$ and $mathbfV_t = h(X_t; mathbfM_D) + mathbfW_t$. Furthermore, $h$ can be linearized, which allows us to apply the EKF to estimate $X_t$ from $mathbfV_t$. Numerical experiments show that with the first two orders of CGPT $mathbfM_D$, the path $(z_t, theta_t)$ can be estimated correctly up to $10%$ of measurement noise even with a poor initial guess. In the limited view case, the effect of noise is severe on the tracking, unless the arrays of sources/receivers offer a good directional diversity. bibliographystyleplain beginthebibliography1 bibitemammari_target_2012 H. Ammari, T. Boulier, J. Garnier, W. Jing, H. Kang, and H. Wang, Target identification using dictionary matching of generalized polarization tensors, Arxiv preprint arXiv:1204.3035, 2012. bibitemAGKLY11 H. Ammari, J. Garnier, H. Kang, M. Lim, and S. Yu, Generalized polarization tensors for shape description, submitted, 2011. bibitemammari_enhancement_2011-1 H. Ammari, H. Kang, M. Lim, and H. Lee, Enhancement of near cloaking using generalized polarization tensors vanishing structures. Part I: The conductivity problem, Comm. Math. Phys., to appear. endthebibliography