Abstract-There are many systems for counting photons such as gamma-rays emitted from radioactive sources. Many of these systems are also position-sensitive, which means that the system provides directional information about recorded events. This paper investigates whether or not the additional information provided by position-sensitive capability improves the performance of detecting a point-source in background. We analyze the asymptotic performance of the generalized likelihood ratio test (GLRT) and a test based on the maximum-likelihood (ML) estimate of the source intensity for systems with and without position-sensitive capability. When the background intensity is known and detector sensitivity is spatially uniform, we prove that position-sensitive capability increases the area under the receiver operating characteristic curve (AUC). For cases when detector sensitivity is nonuniform or background intensity is unknown, we provide numerical results to illustrate the effect of the parameters on detection performance.

Abstract-Likelihood-based test statistics for the task of detecting a radioactive source in background using a gamma-ray imaging system often have intractable distributions. This complicates the tasks of predicting detection performance and setting thresholds that ensure desired false-alarm rates. Asymptotic distributions of test statistics can aid in predicting performance and in setting detection thresholds. However, in applications with complex sensors, like gamma-ray imaging, often only approximate statistical models for the measurements are available. Standard asymptotic approximations can yield inaccurate performance predictions when based on misspecified models. This paper considers asymptotic properties of detection tests based on maximum likelihood (ML) estimates under model mismatch, i.e., when the statistical model used for detection differs from the true distribution. We provide general expressions for the asymptotic distribution of likelihood-based test statistics when the number of measurements is Poisson, and expressions specific to gamma-ray source detection that one can evaluate using a modest amount of data from a real system or Monte Carlo simulation. Considering a simulated Compton imaging system, we show that the proposed expressions yield more accurate detection performance predictions than previous expressions that ignore model mismatch. These expressions require less data and computation than conventional empirical methods.

This work investigates three penalized-likelihood expectation maximization (EM) algorithms for image reconstruction with Poisson data where the images are known a priori to be sparse in the space domain. The penalty functions considered are the 1 norm, the 0 "norm," and a penalty function based on the sum of logarithms of pixel values, R(x) = np j=1 log xj δ + 1. Our results show that the 1 penalized algorithm reconstructs scaled versions of the maximum-likelihood (ML) solution, which does not improve the sparsity over the traditional ML estimate. Due to the singularity of the Poisson log-likelihood at zero, the 0 penalized EM algorithm is equivalent to the maximum-likelihood EM algorithm. We demonstrate that the penalty based on the sum of logarithms produces sparser images than the ML solution. We evaluated these algorithms using experimental data from a position-sensitive Compton-imaging detector, where the spatial distribution of photon-emitters is known to be sparse.

We address the question of whether or not the directional or imaging information offered by a position-sensitive gamma-ray detector improves the detection accuracy when searching for a source of known shape amid a background of known intensity. We formulate the detection problem as a composite hypothesis testing problem and examine the behavior of the generalized likelihood ratio test (GLRT) in terms of the area under the receiver operating characteristic (AUC). Due to the analytical complexity of the GLRT in this case, we examine its asymptotic properties when the number of detected photons is large. We find that a detector of uniform sensitivity can more accurately detect a source when imaging information is used.

Abstract-In emission tomography, the Poisson statistics of the observations make penalized-likelihood reconstruction with an ℓ1 penalty more difficult than in the case where the observed data is Gaussian. Previously proposed methods for enforcing sparsity of the reconstructed image with respect to some transform use approximations of the ℓ1 norm. These approximations facilitate the derivation of monotonic algorithms using optimization transfer methods. Recently, augmented Lagrangian methods have been applied to restoration of images corrupted by Poisson noise without requiring approximations to the ℓ1 norm. This work extends previously derived augmented Lagrangian-based algorithms to penalized likelihood reconstruction for emission tomography with an exact ℓ1 penalty. We compare the proposed algorithm to an incremental optimization transfer algorithm that performs penalized-likelihood reconstruction with a hyperbolic approximation to the ℓ1 penalty. The results show that the proposed algorithm reduces the cost function nearly as quickly as the incremental optimization transfer algorithm. The potential advantage of the proposed method is that it solves the ℓ1 penalized-likelihood problem exactly.