Sensing Method for Two-Target Detection in Time-Constrained Vector Poisson Channel

TL;DR

This work studies sensor scheduling for two Poisson sources observed under a fixed total time $T$ with a switchable configuration that yields either individual counts $(Y_1,Y_2)$ over times $(T_1,T_2)$ or joint counts $(Y_3)$ over time $T_3$. It casts the problem in both information-theoretic terms, maximizing $I(X_1,X_2;Y_1,Y_2,Y_3)$, and detection-theoretic terms, maximizing the Bayesian probability of total correct detections $P_d$, by optimizing the time allocations $(T_1,T_2,T_3)$ under $T=T_1+T_2+T_3$. Key findings show that the optimal schedule often lies on the symmetry line $T_1=T_2=(1-T_3)/2$, with the preference for joint versus individual sensing depending on the prior $p$, and that the two objective metrics do not always agree on the same optimum. The study provides a computational framework for evaluating vector Poisson channels, demonstrates a conjectured global concavity of $I(X_1,X_2;Y_1,Y_2,Y_3)$, and outlines extensions to larger sensor networks and adaptive switching strategies that are relevant for practical time-constrained counting systems.

Abstract

It is an experimental design problem in which there are two Poisson sources with two possible and known rates, and one counter. Through a switch, the counter can observe the sources individually or the counts can be combined so that the counter observes the sum of the two. The sensor scheduling problem is to determine an optimal proportion of the available time to be allocated toward individual and joint sensing, under a total time constraint. Two different metrics are used for optimization: mutual information between the sources and the observed counts, and probability of detection for the associated source detection problem. Our results, which are primarily computational, indicate similar but not identical results under the two cost functions.

Figures (16)

• Figure 1: Illustration of sensing method for Bayesian detection of $2-$long hidden random vector $X$ from $3-$long observable random vector $Y$ in vector Poisson channel under a total observation time constraint, where $\{ \mathcal{P}_i \}$ is a conditional Poisson point process given Bernoulli distributed input $X_i$.
• Figure 2: An abstract example: $Y_1$, $Y_2$ and $Y_3$ are total counts of photons accumulated in times $T_1$, $T_2$ and $T_3$, respectively. LED source 1 (and source 2) initiates a homogeneous Poisson process of intensity $\lambda_0$ if Bernoulli distributed random input signal $X_1$ (and $X_2$) takes value of $0$ with probability $(1-p)$; and $\lambda_1$ if input signal $X_1$ (and $X_2$) takes value of $1$ with probability $p$. Once the input random vector $[X_0 \: X_1]$ assumes any of the four possible states $[X_0 \: X_1: 00, 01, 10, 11]$, it doesn't change its state during the course of Photon counting for given time $T$.
• Figure 3: Mutual information $I(X;Y)$ and Bayesian probability of total correct detections $P_d$ vs. time $T_3$ where $0 \le T_3 \le T$ and $T_1=T_2=\frac{T-T_3}{2}$ such that time constraint $T=T_1+T_2+T_3$ is satisfied.
• Figure 4: Scalar Poisson channel: mutual information $I(X_1;Y_1)$ and probability of total correct detections $P_d$ vs. time $T.$
• Figure 5: Concavity and convexity of three terms in $I(X_1 X_2;Y_1 Y_2 Y_3)=I(X_1;Y_1)+I(X_2;Y_2)+I(X_1 X_2;Y_3|Y_1 Y_2)$ when $(T_1,T_2,T_3):=(\frac{1-T_3}{2},\frac{1-T_3}{2},T_3)$ such that $0 \le T_3 \le 1$.

Theorems & Definitions (4)

• Theorem 1
• Theorem 2
• Conjecture 1
• Corollary 1: Feasible region for optimal solution