Set-Type Belief Propagation with Applications to Poisson Multi-Bernoulli SLAM

TL;DR

This paper extends belief propagation to set-valued random finite sets, addressing the limitation of standard BP which assumes fixed-size vector variables. By formulating set-type belief propagation (BP) on sequences of random finite sets and introducing partitioning, merging, and auxiliary-variable shifting factors, the authors derive PMB-based SLAM filters and establish their relationship to conventional vector-type BP methods. The main contributions include a full set-type BP framework, a PMB-SLAM-specific derivation with auxiliary variables, and a demonstration that the proposed approach can outperform vector-type BP-SLAM under informative birth scenarios, while remaining compatible with existing set- and vector-based formulations. The work provides a principled alternative to heuristics in multi-target tracking and SLAM, enabling more accurate handling of unknown target cardinalities and data associations in dynamic environments.

Abstract

Belief propagation (BP) is a useful probabilistic inference algorithm for efficiently computing approximate marginal probability densities of random variables. However, in its standard form, BP is only applicable to the vector-type random variables with a fixed and known number of vector elements, while certain applications rely on RFSs with an unknown number of vector elements. In this paper, we develop BP rules for factor graphs defined on sequences of RFSs where each RFS has an unknown number of elements, with the intention of deriving novel inference methods for RFSs. Furthermore, we show that vector-type BP is a special case of set-type BP, where each RFS follows the Bernoulli process. To demonstrate the validity of developed set-type BP, we apply it to the PMB filter for SLAM, which naturally leads to new set-type BP-mapping, SLAM, multi-target tracking, and simultaneous localization and tracking filters. Finally, we explore the relationships between the vector-type BP and the proposed set-type BP PMB-SLAM implementations and show a performance gain of the proposed set-type BP PMB-SLAM filter in comparison with the vector-type BP-SLAM filter.

Figures (8)

• Figure 1: Factor graph representation of the factorized vector density \ref{['eq:vector-factorized']}.
• Figure 2: Factor graph representation of set-density of \ref{['eq:exSetfac']}.
• Figure 3: Factor graph of a PMB density with auxiliary variables, see \ref{['eq:PMB_Aux']}, the product of a PPP and $n$ Bernoulli densities.
• Figure 4: Factor graph representation with a partitioning and merging factor: (left-right) one incoming message is partitioned into multiple messages; and (right-left) multiple messages are merged into one message.
• Figure 5: Concatenated factor graph of the joint densities of \ref{['eq:P_factorizedDensity']} and \ref{['eq:factorizedDensity']}. The special factors for set densities, discussed in Section \ref{['sec:SetSpecial']}, are represented by white squares. The abbreviated notations for the factor nodes are represented by dropping the arguments and $\tilde{(\cdot)}$ on sets, as follows: $f_\mathsf{u}^s \triangleq \mathtt{b}(\mathbf{s}_{k-1})$, $f_\mathsf{u}^\mathrm{U} \triangleq \mathtt{b}(\mathcal{X}_{k-1}^\mathrm{U})$, $f_\mathsf{u}^i \triangleq \mathtt{b}(\mathcal{X}_{k-1}^i)$, $f^A \triangleq f^A(\mathbf{s}_k,\mathbf{s}_{k-1})$, $f^B \triangleq f^B({\mathcal{X}}_{k}^\mathrm{S},{\mathcal{X}}_{k-1}^\mathrm{U})$, $f^C \triangleq f^C(\mathcal{X}_k^\mathrm{B})$, $f^D \triangleq f^D({\mathcal{X}}_{k}^\mathrm{S},{\mathcal{X}}_{k}^\mathrm{B},{\mathcal{P}}_{k}^\mathrm{U})$, $f_i^E \triangleq f_i^E(\mathcal{X}_{k}^i,\mathcal{X}_{k-1}^i)$, $f^F \triangleq f^F( {\mathcal{X}}_{k}^\mathrm{U}, {\mathcal{P}}_k^1,\cdots ,{\mathcal{P}}_k^{J_k})$, $f_j^G \triangleq f_j^G({\mathcal{P}}_k^j,{\mathcal{Y}}_k^j)$, $f^H \triangleq f^H(\mathbf{s}_k,\mathcal{X}_k^\mathrm{U})$, $f_j^I \triangleq f_j^I(\mathbf{s}_k,\mathcal{Y}_k^j,d_k^j)$, $f_i^J \triangleq f_i^J(\mathbf{s}_k,\mathcal{X}_k^i,c_k^i)$, and $f_{i,j}^K \triangleq f_{i,j}^K(c_{k}^i,d_{k}^j)$.

Theorems & Definitions (25)

• Example 1
• Definition 1: Factorization of Set-Density and Factor Graph
• Example 2
• Definition 2: Set-Type bp Update Rules
• Theorem 1
• proof
• Corollary 1
• proof
• Example 3
• Remark 1: Vector-Type BP is a Special Case of Set-Type BP