Accelerated strategy for the MLEM algorithm.

The iterative maximum-likelihood expectation-maximization (ML-EM) algorithm is an excellent algorithm for image reconstruction and usually provides better images than the filtered backprojection (FBP) algorithm. However, a windowed FBP algorithm can outperform the ML-EM in certain occasions, when the least-squared difference from the true image, that is, the least-squared error (LSE), is used as the comparison criterion. Computer simulations were carried out for the two algorithms. For a given data set the best reconstruction (compared to the true image) from each algorithm was first obtained, and the two reconstructions are compared. The stopping iteration number of the ML-EM algorithm and the parameters of the windowed FBP algorithm were determined, so that they produced an image that was closest to the true image. However, to use the LSE criterion to compare algorithms, one must know the true image. How to select the optimal parameters when the true image is unknown is a practical open problem. For noisy Poisson projections, computer simulation results indicate that the ML-EM images are better than the regular FBP images, and the windowed FBP algorithm images are better than the ML-EM images. For the noiseless projections, the FBP algorithms outperform the ML-EM algorithm. The computer simulations reveal that the windowed FBP algorithm can provide a reconstruction that is closer to the true image than the ML-EM algorithm.

Three-dimensional (3D) iterative maximum likelihood expectation maximization (ML-EM) algorithms for single-photon emission tomography (SPET) are capable of correcting image-degrading effects of non-uniform attenuation, distance-dependent camera response and patient shape-dependent scatter. However, the resulting improvements in quantitation, resolution and signal-to-noise ratio (SNR) are obtained at the cost of a huge computational burden. This paper presents a new acceleration method for ML-EM: dual matrix ordered subsets (DM-OS). DM-OS combines two acceleration methods: (a) different matrices for projection and back-projection and (b) ordered subsets of projections. DM-OS was compared with ML-EM on simulated data and on physical thorax phantom data, for both 180 degrees and 360 degrees orbits. Contrast, normalized standard deviation and mean squared error were calculated for the digital phantom experiment. DM-OS resulted in similar image quality to ML-EM, even for speed-up factors of 200 compared to ML-EM in the case of 120 projections. The thorax phantom data could be reconstructed 50 times faster (60 projections) using DM-OS with preservation of image quality. ML-EM and DM-OS with scatter compensation showed significant improvement of SNR compared to ML-EM without scatter compensation. Furthermore, inclusion of complex image formation models in the computer code is simplified in the case of DM-OS. It is thus shown that DM-OS is a fast and relatively simple algorithm for 3D iterative scatter compensation, with similar results to conventional ML-EM, for both 180 degrees and 360 degrees acquired data.

We investigated the spatial resolution and quantitative properties of SPECT images reconstructed using filteredbackprojection (FB) and maximum likelihood-expectation maximization (ML-EM) algorithms. We studied ML-EM as a function of iteration number and projectorhackprojector (php) model, and FE3 as a function of filter. To quantify the image resolution for a shift-variant process we defined a local point response function (PRF). To justify the use of the local PRF to quantify resolution for nonlinear ML-EM we tested the regime over which the algorithm behaved linearly with respect to input perturbations. Using Fourier domain analogs of the local PRF we demonstrated that ML-EM with an accurate php can lead to images of better resolution than FB with a Butterworth or Metz filter or ML-EM with a php that does not fully model the imaging process. We also showed that the resolution improves with increasing MLEM iteration number, and the extent of the improvement depends on the accuracy of the php model. We conclude that the local PRF can be used to characterize spatial resolution in SPECT images, and that ML-EM offers the possibility of superior resolution, particularly if the algorithm contains a good model of the imaging process.

The maximum likelihood expectation maximisation (ML-EM) algorithm can be used to reduce reconstruction artefacts produced by filtered backprojection (FBP) methods in pinhole single-photon emission tomography (SPET). However, ML-EM suffers from noise propagation along iterations, which leads to quantitatively unpleasant reconstruction results. To avoid this increase in noise, the median root prior (MRP) algorithm for pinhole SPET was implemented. Projection data of a line source and Picker's thyroid phantom were collected using a single-head gamma camera with a pinhole collimator. MRP was added to existing pinhole ML-EM reconstruction algorithm and the phantom studies were reconstructed using MRP, ML-EM and FBP for comparison. Coefficients of variation, contrasts and full-widths at half-maximum were calculated and showed a clear reduction in noise without significant loss of resolution or decrease in contrast when MRP was applied. MRP also produced visually pleasing images even with high iteration numbers, free of the checkerboard-type noise patterns which are typical of ML-EM images.

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> Compton imaging is a gamma ray imaging technique that has many possible applications including homeland security and medical imaging. Using the Compton scattering formula the origin of a scattered <formula formulatype="inline"> <tex Notation="TeX">$\gamma$</tex></formula>-ray can be localized to a point on the surface of a cone using a minimum of two position and energy measurements. List-mode Maximum likelihood expectation maximization (MLEM) is an iterative statistical algorithm that makes successive approximations to the most probable source distribution that would have led to the observed data. Conventional MLEM is often performed in two dimensions, which still requires knowledge of the source-to-detector distance. We propose iteration over all three spatial dimensions in order to maximize the probability in three-dimensional space. This paper reports on the development of a three-dimensional MLEM (3DMLEM) algorithm to reconstruct images from a Compton imager in three dimensions for single, multiple and line sources. Several techniques for reducing convergence times are also discussed. </para>

Emission tomography, including single-photon emission computed tomography (SPECT), requires image reconstruction from noisy and incomplete projection data. The maximum-likelihood expectation maximization (MLEM) algorithm is widely used due to its statistical foundation and non-negativity preservation, but it is highly sensitive to noise, particularly in low-count conditions. Although total variation (TV) regularization can reduce noise, it often oversmooths structural details and requires careful parameter tuning. We propose a Graph-Enhanced Expectation Maximization (GREM) algorithm that incorporates graph-based neighborhood information into an MLEM-type multiplicative reconstruction scheme. The method is motivated by a penalized formulation combining a Kullback–Leibler divergence term with a graph Laplacian regularization term, promoting local structural consistency while preserving edges. The resulting update retains the multiplicative structure of MLEM and preserves the non-negativity of the image estimates. Numerical experiments using synthetic phantoms under multiple noise levels, as well as clinical 99mTc-GSA liver SPECT data, demonstrate that GREM consistently outperforms conventional MLEM and TV-regularized MLEM in terms of PSNR and MS-SSIM. These results indicate that GREM provides an effective and practical approach for edge-preserving noise suppression in emission tomography without relying on external training data.

MLEM reconstruction method applied to partial defect verification using simulated data

The purpose of this study is to compare and evaluate the performance of a multiple-scattering Compton imager (MSCI) to measure prompt gamma-rays emitted during proton therapy. Because prompt gamma-rays are generated simultaneously during the proton beam delivery, the falloff position of the Bragg peak of the proton beam can be determined from the distribution of prompt gamma-rays. The detection system was designed using three CdZnTe detector layers that can track radiation of unknown energy on the basis of effective Compton scattering events. The simple back-projection, filtered back-projection, and maximum likelihood expectation maximization (MLEM) algorithms were applied for the reconstructed Compton images. The falloff positions of the Bragg peaks determined from individual MSCIs were compared with the theoretical values calculated using the Monte Carlo N-Particle eXtended code. Moreover, the performance of the MSCI was compared with that of a previously developed system based on a slit collimator gamma camera. In summary, the MSCI with the MLEM reconstruction algorithm was better than the other reconstruction methods in terms of the falloff position of the Bragg peak, the angular resolution, and the signal-to-noise ratio.

The energy-imaging integrated deconvolution (EIID) algorithm is capable of deconvolving the source image at any specific energy, as well as the incident spectrum from any direction. This spectrum-image reconstruction method takes place in integrated spatial and energy domain using the maximum likelihood expectation maximization (MLEM) algorithm. This technique has been demonstrated for combined two-, three- and four-interaction Compton events from an array of 2 cm × 2 cm × 1.5 cm detectors in our previous work. By including Compton continuum events in the system model and using a sensitivity image to correct efficiency effects, the MLEM solution estimates the true incident gamma-ray spectrum with correct branching ratio. For each direction, we can then identify the source isotope based on the peak energy and relative peak area. After identifying the source isotope, the source activity can be estimated. From the reconstructed image and change of the relative peak areas from the branching ratio, the presence of shielding and the material and thickness of shielding can be estimated.

Tomographic absorption spectroscopy (TAS) is experiencing a surge of interest due to recent progress in laser technology and advanced imaging concepts such as nonlinear tomography and compressed sensing. Nevertheless, even though numerous algorithms have been adapted from other tomographic areas such as medical imaging and engineering process control, and applied in TAS applications, systematic comparison between those methods has not been investigated. In this work, we aim to test major inversion algorithms on both the so-called rank-deficient (RD) and discrete ill-posed (DIP) problems. Comparative studies were performed on extensive cases, and the results for three representative phantoms are reported here. Other important topics such as determination of control parameters and the semiconvergent behavior of iterative methods have also been studied. According to our study, Landweber outperformed other methods for two RD cases and is slightly inferior to the maximum likelihood expectation maximization (MLEM) algorithm for the third one. It has also been found that Landweber and MLEM feature better immunity to semiconvergence than the others; however, since MLEM is sensitive to an initial guess, it is less robust than Landweber. On the other hand, the truncated singular value decomposition (TSVD) method is recommended for DIP problems due to its superior performance as it can effectively suppress detrimental effects from noise, which will be amplified during the inversion procedure. It should be emphasized that even though this study was conducted under the context of TAS, we expect it to provide useful insights for other tomographic modalities.

Bayesian spiking neurons (BSNs) provide a probablisitic and intuitive interpretation of how spiking neurons could work and have been shown to be equivalent to leaky integrate-and-fire neurons under certain conditions [1]. The study of BSNs has been restricted mainly to small networks because online learning, which currently involves a maximum-likelihood-expectation-maximisation (ML-EM) approach [2,3], is quite slow. Here a new approach to estimating the parameters of Bayesian spiking neurons, referred to as fast learning (FL), is presented and compared to online ML-EM learning. Learning in a BSN is local to the neuron and involves estimation of the transition rate and observation rate parameters of an underlying implicit hidden Markov model (HMM), the hidden state of which the BSN output encodes [1]. Rather than estimating the parameters by maximizing the log-likelihood of the hidden states and the synaptic observations given the parameters as is done in ML-EM [2,3], the FL algorithm directly calculates statistics upon which the parameters depend. This is achieved by taking advantage of the relationship between the log-odds ratio of the hidden state computed by the BSN and the probability that the hidden state is ‘on’ given the past synaptic observations, . Online learning in a two BSN neuron hierarchy is explored, where the first neuron receives N=20 synapses driven by Poisson processes and the second neuron receives input from only the first neuron. Simulations were performed for a fixed set of transition rates and observation rates for 10 different perturbations of the initial transition and observation rate estimates: ±0-20%, ±20-40%,..., ±180-200%. Initial rates were not allowed to go below 10-6. Each perturbation condition was simulated 100 times by randomly selecting the initial parameter values. Although the FL algorithm is not as exact as the ML-EM at estimating the true parameter values for small perturbations of the initial rate estimates relative to the true rates, the FL algorithm is able to reliably estimate the parameters for initial perturbations of up to 200%, whereas the ML-EM algorithm estimates begin to deviate after perturbations of about 40-60%. Moreover, the simplicity of the FL algorithm means that it runs on the order of 25 times faster than the ML-EM implementation considered. These results hold true for both the first and second neurons in the two BSN neuron hierarchy. For the first neuron in the hierarchy, the RMS difference in the time series of the probability calculated for the estimated and the true parameter values, follows a similar pattern to the parameter estimates when the FL and ML-EM algorithms are compared, with average RMS errors of 0.2% obtained for the FL algorithm across the range of perturbations studied. Although we do not have a formal proof of convergence of the FL algorithm, we conclude that the FL algorithm can stably estimate the parameters over a large range of initial perturbations and it can do this very quickly. Thus the FL algorithm makes online learning in networks of BSNs much more tractable.

Depth-of-interaction enhanced Compton camera using pixelated LYSO(Ce) scintillator arrays and dual-ended SiPMs.

In this study, we propose a novel method for reconstructing the radiation distribution of a one-dimensional radioactive source using Machine Learning (ML) algorithms and plastic scintillation optical fiber. The wavelength spectrum unfolding technique is used to estimate the source position accurately. We compare the accuracy and time efficiency of three different algorithms, namely, Generalized reduced gradient (GRG), Maximum likelihood expectation maximization (MLEM), and ML, in the single-source and dual-source cases. Our results demonstrate that MLEM algorithm has a shorter reconstruction time with comparable accuracy of position and intensity compared to GRG algorithm for single-source case. For dual-source case, ML algorithm provides real-time estimation of position and intensity with acceptable errors, while GRG algorithm has a larger error in intensity estimation and longer computation time. Our proposed ML algorithms offer useful guidance for practical applications in radiation source location.

Comparison of reconstructed prompt gamma emissions using maximum likelihood estimation and origin ensemble algorithms for a Compton camera system tailored to proton range monitoring

One of the major problems associated with parallel hole collimators (PCs) is the trade-off between their resolution and sensitivity. To solve this problem, a novel PC - namely, extended parallel hole collimator (EPC) - was proposed, in which particular trapezoidal denticles were increased upon septa on the side of the detector. In this study, an EPC was designed and its performance was compared with that of two PCs, PC35 and PC41, with a hole size of 1.5 mm and hole lengths of 35 and 41 mm, respectively. The Monte Carlo method was used to calculate the important parameters such as resolution, sensitivity, scattering, and penetration ratio. A Jaszczak phantom was also simulated to evaluate the resolution and contrast of tomographic images, which were produced by the EPC6, PC35, and PC41 using the Monte Carlo N-particle version 5 code, and tomographic images were reconstructed by using maximum likelihood expectation maximization algorithm. Sensitivity of the EPC6 was increased by 20.3% in comparison with that of the PC41 at the identical spatial resolution and full-width at tenth of maximum here. Moreover, the penetration and scattering ratio of the EPC6 was 1.2% less than that of the PC41. The simulated phantom images show that the EPC6 increases contrast-resolution and contrast-to-noise ratio compared with those of PC41 and PC35. When compared with PC41 and PC35, EPC6 improved trade-off between resolution and sensitivity, reduced penetrating and scattering ratios, and produced images with higher quality. EPC6 can be used to increase detectability of more details in nuclear medicine images.