Particle Flows for Source Localization in 3-D Using TDOA Measurements

TL;DR

This work tackles 3-D localization of an unknown number of static sources using TDOA measurements through a Bayesian BP framework. It integrates particle flow, both deterministic (EDH/LEDH) and stochastic (Gromov's), to combat nonlinear measurement models and DA uncertainty, employing a Gaussian mixture representation of beliefs to capture multimodal posteriors. The proposed approach enables sequential processing across sensors and can correctly determine the number of sources while yielding accurate location estimates, with stochastic Gromov's flow achieving the best accuracy–runtime tradeoff in simulations. The results demonstrate significant improvements over bootstrap particle filtering, highlighting the practical impact for multisensor passive localization in challenging environments.

Abstract

Localization using time-difference of arrival (TDOA) has myriad applications, e.g., in passive surveillance systems and marine mammal research. In this paper, we present a Bayesian estimation method that can localize an unknown number of static sources in 3-D based on TDOA measurements. The proposed localization algorithm based on particle flow (PFL) can overcome the challenges related to the highly nonlinear TDOA measurement model, the data association (DA) uncertainty, and the uncertainty in the number of sources to be localized. Different PFL strategies are compared within a unified belief propagation (BP) framework in a challenging multisensor source localization problem. In particular, we consider PFL-based approximation of beliefs based on one or multiple Gaussian kernels with parameters computed using deterministic and stochastic flow processes. Our numerical results demonstrate that the proposed method can correctly determine the number of sources and provide accurate location estimates. The stochastic flow demonstrates greater accuracy compared to the deterministic flow when using the same number of particles.

Figures (4)

• Figure 1: Source position and hyperboloids resulting from the TDOA measurements of two sensors. Each sensor consists of two receiver pairs. A dashed green line indicates the intersection of the two hyperboloids.
• Figure 2: Factor graph for message passing of an unknown number of sources at a single sensor $s$, corresponding to the propagation of the joint PDF \ref{['eq:jointPosteriorComplete']}. Messages calculated using pfl are depicted in red. These messages are calculated based on messages depicted in blue. The following short notations are used: $n_{\mathrm{m}} \space\triangleq M_{s}$, $n_{\mathrm{p}} \space\triangleq J_{s-1}$, $\underline{\bm{y}}^{j} \space\triangleq \underline{\bm{y}}_{s}^{(j)}$, $\overline{\bm{y}}^{\space m} \space\triangleq \overline{\bm{y}}_{s}^{(m)}\space$, $\bm{a}^j \space\triangleq \bm{a}^{(j)}_s$, $\bm{b}^m \space\triangleq \bm{b}^{(m)}_s$, $f^j \space\triangleq f(\underline{\bm{y}}^{(j)}_{s} | \bm{y}^{(j)}_{s-1})$, $q^j \space\triangleq q( \underline{\bm{x}}^{(j)}_{s}\space, \underline{r}^{(j)}_{s}\space, a_{s}^{(j)};\bm{z}_{s} )$, $v^m \space\triangleq v( \overline{\bm{x}}^{(m)}_{s} \space,\overline{r}^{(m)}_{s} \space,b_{s}^{(m)};\bm{z}^{(m)}_{s} )$, $\Psi^{j,m} \space\triangleq \Psi_{j,m}(a^{(j)}_{s} \space,b^{(m)}_{s})$, $\gamma_j \triangleq \gamma_{s}^{(j)}(\underline{\bm{y}}^{(j)}_{s} )$, $\beta_j \triangleq \beta^{(j)}_{s}(a^{(j)}_{s})$, $\xi_m \triangleq \xi^{(m)}_{s}(b^{(m)}_{s})$, $\varsigma_{m} \triangleq \varsigma^{(m)}_s(\overline{\bm{y}}^{(m)}_{s} )$, $\tilde{f}^j\space\triangleq \tilde{f}(\bm{y}_{s-1}^{(j)})$, $\alpha_j \triangleq \alpha_s(\underline{\bm{y}}^{(j)}_{s})$, $\kappa_j \triangleq \kappa^{(j)}_{s}(a^{(j)}_{s} )$, $\iota_{m} \triangleq \iota_s^{(m)}(b^{(m)}_{s} )$, and $\tilde{f}^j_+\space\triangleq \tilde{f}(\bm{y}_{s}^{(j)})$.
• Figure 3: Particle-based representation after the edh flow of the spatial PDFs by gmm model. There exist three signal sources in a 3-D environment with clutters, and the state is sequentially filtered by sensor: (a) $s=1$, (b) $s=2$, (c) $s=4$, and (d) $s=9=n_{\text{s}}$. Filled diamonds indicate the locations of receivers involved in the current update step; the locations of other receivers are indicated by empty diamonds. X-marks indicate the sources' ground truth positions.
• Figure 4: Statistics of the OSPA error for different algorithms. Each column corresponds to the algorithm with the same ID as in Table I.