An Improved Maximum Likelihood Approach to Image Reconstruction Using Ordered Subsets and Data Subdivisions

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.

Negative binomial maximum likelihood expectation maximization (NB-MLEM) algorithm for reconstruction of pre-corrected PET data

Quantitative positron emission tomography imaging often requires correcting the image data for deformable motion. With cyclic motion, this is traditionally achieved by separating the coincidence data into a relatively small number of gates, and incorporating the inter-gate image transformation matrices into the reconstruction algorithm. In the presence of non-cyclic deformable motion, this approach may be impractical due to a large number of required gates. In this paper, we propose an alternative approach to iterative image reconstruction with correction for deformable motion, wherein unorganized point clouds are used to model the imaged objects in the image space, and motion is corrected for explicitly by introducing a time-dependence into the point coordinates. The image function is represented using constant basis functions with finite support determined by the boundaries of the Voronoi cells in the point cloud. We validate the quantitative accuracy and stability of the proposed approach by reconstructing noise-free and noisy projection data from digital and physical phantoms. The point-cloud-based maximum likelihood expectation maximization (MLEM) and one-pass list-mode ordered-subset expectation maximization (OSEM) algorithms are validated. The results demonstrate that images reconstructed using the proposed method are quantitatively stable, with noise and convergence properties comparable to image reconstruction based on the use of rectangular and radially-symmetric basis functions.

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

Comparison of ML–EM algorithm and ART for reconstruction of gas hold-up profile in a bubble column

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.

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.

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.

This paper presents the development and validation of a Monte Carlo-based singe photon emission computed tomography reconstruction program for parallel-hole collimation contained within the SIMIND Monte Carlo framework. The Monte Carlo code is used as an accurate forward-projector and is combined with a simplified back-projector to perform iterative tomographic reconstruction using the Maximum Likelihood Expectation Maximization and Ordered Subsets Expectation Maximization algorithms, together forming a program called SIMREC. The Monte Carlo simulation transforms the estimated source distribution directly from activity to counts in its projections. Hence, the reconstructed image is expressed in activity without reference to an external calibration. The program is tested using phantom measurements of spheres filled with 99mTc, 177Lu and 131I placed in air and centrally and peripherally in a water-filled elliptical phantom. The feasibility of applying the reconstruction to patients is also demonstrated for a range of radiopharmaceuticals. The deviation in total activity in the spheres ranged between −4.1% and 6.2% compared with the activity determined when preparing the phantom. The SIMREC program was found to be accurate with respect to activity estimation and to reconstruct visually acceptable images within a few hours when applied to patient examples.

This work explores the feasibility and impact of including both the motion correction and the image registration transformation parameters from positron emission tomography (PET) image space to magnetic resonance (MR), or stereotaxic, image space within the system matrix of PET image reconstruction. This approach is motivated by the fields of neuroscience and psychiatry, where PET is used to investigate differences in activation patterns between different groups of participants, requiring all images to be registered to a common spatial atlas. Currently, image registration is performed after image reconstruction which introduces interpolation effects into the final image. Furthermore, motion correction (also requiring registration) introduces a further level of interpolation, and the overall result of these operations can lead to resolution degradation and possibly artifacts. It is important to note that performing such operations on a post-reconstruction basis means, strictly speaking, that the final images are not ones which maximize the desired objective function (e.g. maximum likelihood (ML), or maximum a posteriori reconstruction (MAP)). To correctly seek parameter estimates in the desired spatial atlas which are in accordance with the chosen reconstruction objective function, it is necessary to include the transformation parameters for both motion correction and registration within the system modeling stage of image reconstruction. Such an approach not only respects the statistically chosen objective function (e.g. ML or MAP), but furthermore should serve to reduce the interpolation effects. To evaluate the proposed method, this work investigates registration (including motion correction) using 2D and 3D simulations based on the high resolution research tomograph (HRRT) PET scanner geometry, with and without resolution modeling, using the ML expectation maximization (MLEM) reconstruction algorithm. The quality of reconstruction was assessed using bias-variance and root mean squared error analyses, comparing the proposed method to conventional post-reconstruction registration methods. An overall reduction in bias (for a cold region: from 41% down to 31% (2D) and 97% down to 65% (3D), and for a hot region: from 11% down to 8% (2D) and from 16% down to 14% (3D)) and in root mean squared error analyses (for a cold region: from 43% to 37% (2D) and from 97% to 65% (3D), and for a hot region: from 11% to 9% (2D) and from 16% down to 14% (3D)) in reconstructed regional mean activities (full regions of interest; all with statistical significance: p < 5 × 10−10) is found when including the motion correction and registration in the system matrix of the MLEM reconstruction, with resolution modeling. However, this improvement in performance comes with an extra computational cost of about 40 min. In this context, this work constitutes an important step toward the goal of estimating parameters of interest directly from the raw Poisson-distributed PET data, and hence toward the complete elimination of post-processing steps.

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.

Magnetic induction tomography (MIT) is a kind of imaging technology, which uses the principle of electromagnetic detection to measure the conductivity distribution. In this research, we make an effort to improve the quality of image reconstruction mainly via the image reconstruction of MIT analysis, including solving the forward problem and image reconstruction. With respect to the forward problem, the variational finite element method is adopted. We transform the solution of a nonlinear partial differential equation into linear equations by using field subdividing and the appropriate interpolation function so that the voltage data of the sensing coils can be calculated. With respect to the image reconstruction, a method of modifying the iterative Newton-Raphson (NR) algorithm is presented in order to improve the quality of the image. In the iterative NR, weighting matrix and L1-norm regularization are introduced to overcome the drawbacks of large estimation errors and poor stability of the reconstruction image. On the other hand, within the incomplete-data framework of the expectation maximization (EM) algorithm, the image reconstruction can be converted to the problem of EM through the likelihood function for improving the under-determined problem. In the EM, the missing-data is introduced and the measurement data and the sensitivity matrix are compensated to overcome the drawback that the number of the measurement voltage is far less than the number of the unknown. In addition to the two aspects above, image segmentation is also used to make the lesion more flexible and adaptive to the patients’ real conditions, which provides a theoretical reference for the development of the application of the MIT technique in clinical applications. The results show that solving the forward problem with the variational finite element method can provide the measurement voltage data for image reconstruction, the improved iterative NR method and EM algorithm can enhance the image quality from different points, and the proposed image segmentation can make the lesions of the reconstruction image more likely to be identified.

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.