[Paper Review] Maximum Likelihood Localization of Radiation Sources with unknown Source Intensity

Henry E. Baidoo‐Williams|arXiv (Cornell University)|Jul 27, 2016

Microwave Imaging and Scattering AnalysisEngineering15 references18 citations

TL;DR

This paper presents a novel maximum likelihood approach for localizing radiation sources with unknown intensity using the minimal number of sensors—N+1 in ℝᴺ—ensuring a unique global optimum within the open convex hull of sensors. By constructing a non-concave likelihood function and employing a projection-augmented gradient ascent algorithm, the method guarantees global convergence without requiring initialization within a basin of attraction, eliminating false stationary points in the region of interest.

ABSTRACT

In this paper, we consider a novel and robust maximum likelihood approach to localizing radiation sources with unknown statistics of the source signal strength. The result utilizes the smallest number of sensors required theoretically to localize the source. It is shown, that should the source lie in the open convex hull of the sensors, precisely $N+1$ are required in $\mathbb{R}^N, ~N \in \{1,\cdots,3\}$. It is further shown that the region of interest, the open convex hull of the sensors, is entirely devoid of false stationary points. An augmented gradient ascent algorithm with random projections should an estimate escape the convex hull is presented.

Motivation & Objective

• To address the challenge of localizing radiation sources when source intensity is unknown, a critical limitation in prior work requiring initialization within a basin of attraction.
• To minimize sensor count to the theoretical minimum—N+1 sensors in ℝᴺ—while ensuring unique localization.
• To eliminate false stationary points in the region of interest (the open convex hull of sensors), enabling robust and reliable localization.
• To develop an initialization-independent algorithm that achieves global uniform asymptotic convergence in probability.
• To demonstrate robustness through simulations under realistic noise and uncertainty conditions.

Proposed method

• Formulates a maximum likelihood estimation framework for radiation source localization with unknown source intensity A*, treating A* as a nuisance parameter.
• Derives a non-concave profit function based on Poisson-distributed sensor measurements, incorporating path loss and attenuation effects.
• Introduces a gradient ascent algorithm that maximizes the likelihood function under the constraint that the estimate remains within the convex hull of sensors.
• Employs random projections and a projection-based correction mechanism when the estimate escapes the convex hull, ensuring convergence.
• Uses a robust simulation framework with independent Poisson-distributed sensor readings and background noise to validate algorithm performance.
• Proves that the likelihood function has a unique global maximizer within the open convex hull, with no false stationary points.

Experimental results

Research questions

• RQ1 Can radiation source localization be achieved with the minimal number of sensors (N+1 in ℝᴺ) when source intensity is unknown?
• RQ2 Does the maximum likelihood function for unknown source intensity have false stationary points within the open convex hull of sensors?
• RQ3 Can a gradient-based algorithm achieve global convergence without requiring initialization within a basin of attraction?
• RQ4 How does the algorithm perform under realistic noise and uncertainty in sensor measurements?
• RQ5 What is the convergence behavior and robustness of the algorithm in low-SNR and high-noise regimes?

Key findings

• The maximum likelihood function has a unique global maximizer within the open convex hull of N+1 sensors in ℝᴺ, with no false stationary points.
• The algorithm achieves global uniform asymptotic convergence in probability, independent of initialization, due to the absence of local maxima in the region of interest.
• Simulations show the algorithm converges rapidly, with RMSE decreasing significantly within 500 iterations, even with 10,000 random initializations.
• In both 2D and 3D scenarios, the estimate leaves the convex hull less than 0.001% of the time, even at low SNR, demonstrating high robustness.
• For N=2, the average RMSE across SNR values is below 10 meters in favorable conditions, with convergence speed confirmed by RMSE plots over iterations.
• The algorithm remains robust under uncertainty, as sensor readings differ from those used in gradient computation, confirming resilience to measurement noise and unknown A*.