Multiscale Bayesian image reconstruction in positron emission tomography

Image quality of positron emission tomography (PET) reconstructions is degraded by subject motion occurring during the acquisition. Magnetic resonance (MR)-based motion correction approaches have been studied for PET/MR scanners and have been successful at capturing regular motion patterns, when used in conjunction with surrogate signals (e.g. navigators) to detect motion. However, handling irregular respiratory motion and bulk motion remains challenging. In this work, we propose an MR-based motion correction method relying on subspace-based real-time MR imaging to estimate motion fields used to correct PET reconstructions. We take advantage of the low-rank characteristics of dynamic MR images to reconstruct high-resolution MR images at high frame rates from highly undersampled k-space data. Reconstructed dynamic MR images are used to determine motion phases for PET reconstruction and estimate phase-to-phase nonrigid motion fields able to capture complex motion patterns such as irregular respiratory and bulk motion. MR-derived binning and motion fields are used for PET reconstruction to generate motion-corrected PET images. The proposed method was evaluated on in vivo data with irregular motion patterns. MR reconstructions accurately captured motion, outperforming state-of-the-art dynamic MR reconstruction techniques. Evaluation of PET reconstructions demonstrated the benefits of the proposed method in terms of motion artifacts reduction, improving the contrast-to-noise ratio by up to a factor 3 and achieveing a target-to-background ratio up to 90% superior compared to standard/uncorrected methods. The proposed method can improve the image quality of motion-corrected PET reconstructions in clinical applications.

In this article, we evaluate Parallel Level Sets (PLS) and Bowsher's method as segmentation-free anatomical priors for regularized brain positron emission tomography (PET) reconstruction. We derive the proximity operators for two PLS priors and use the EM-TV algorithm in combination with the first order primal-dual algorithm by Chambolle and Pock to solve the non-smooth optimization problem for PET reconstruction with PLS regularization. In addition, we compare the performance of two PLS versions against the symmetric and asymmetric Bowsher priors with quadratic and relative difference penalty function. For this aim, we first evaluate reconstructions of 30 noise realizations of simulated PET data derived from a real hybrid positron emission tomography/magnetic resonance imaging (PET/MR) acquisition in terms of regional bias and noise. Second, we evaluate reconstructions of a real brain PET/MR data set acquired on a GE Signa time-of-flight PET/MR in a similar way. The reconstructions of simulated and real 3D PET/MR data show that all priors were superior to post-smoothed maximum likelihood expectation maximization with ordered subsets (OSEM) in terms of bias-noise characteristics in different regions of interest where the PET uptake follows anatomical boundaries. Our implementation of the asymmetric Bowsher prior showed slightly superior performance compared with the two versions of PLS and the symmetric Bowsher prior. At very high regularization weights, all investigated anatomical priors suffer from the transfer of non-shared gradients.

Including accurate modeling of the point spread function (PSF) in positron emission tomography (PET) reconstruction algorithms results in improvements in image spatial resolution and contrast. In this work, we sampled the PSF in our first-generation radio-frequency brain dedicated PET insert for simultaneous PET/MR imaging using a 100 µCi NEMA standard 250 µm diameter Na-22 point source at 13 different positions within a subsection of the system field of view (FoV). The acquired list mode data was converted into the canonical sinogram format from which the spatial positioning of the source and standard deviations were calculated. The subset was then used to extrapolate the PSF for the full system FoV. This model was then fed as an input parameter into a graphical processing unit based ordered subset expectation maximization (OSEM) reconstruction algorithm and used to generate reconstructed images with and without spatially varying PSF modeling for the Na-22 point source and a Hoffman brain phantom. Results indicate that for point source reconstruction, the FWHM of the horizontal profile of the point source is smaller with spatially variant PSF especially closer to the edges. Effect of spatially varying PSF modeling is also presented with Hoffman phantom reconstruction and CNR value has increased with spatially varying PSF.

The authors studied certain characteristics of the expectation-maximization (EM) algorithm for image reconstruction in positron emission tomography (PET). They have attempted to speed up the algorithm by performing some time-consuming computation only once, at the initiation stage. They have studied the resulting memory size, disk space, and CPU requirements versus the number of pixels in the reconstructed image and the number of gamma -ray detectors in the tomograph. The convergence and image quality have been studied as a function of the number of iterations performed. Preliminary results are presented. >

Positron emission tomography (PET) reconstruction has become an ill-posed inverse problem due to low-count projection data, and a robust algorithm is urgently required to improve imaging quality. Recently, the deep image prior (DIP) has drawn much attention and has been successfully applied in several image restoration tasks, such as denoising and inpainting, since it does not need any labels (reference image). However, overfitting is a vital defect of this framework. Hence, many methods have been proposed to mitigate this problem, and DeepRED is a typical representation that combines DIP and regularization by denoising (RED). In this article, we leverage DeepRED from a Bayesian perspective to reconstruct PET images from a single corrupted sinogram without any supervised or auxiliary information. In contrast to the conventional denoisers customarily used in RED, a DnCNN-like denoiser, which can add an adaptive constraint to DIP and facilitate the computation of derivation, is employed. Moreover, to further enhance the regularization, Gaussian noise is injected into the gradient updates, deriving a Markov chain Monte Carlo (MCMC) sampler. Experimental studies on brain and whole-body datasets demonstrate that our proposed method can achieve better performance in terms of qualitative and quantitative results compared to several classic and state-of-the-art methods.

The promise of new PET image reconstruction

To accelerate image reconstruction of positron emission tomography (PET) data, we introduced an approach for parallel architectures by applying the message passing paradigm to an existing implementation of the ordered-subsets expectation-maximization (OSEM) algorithm for two- or three-dimensional (2D/3D) PET. To reduce the amount of time needed to complete a reconstruction, a cluster was used as well as different multi-core systems. A main focus was on the multi-core processors, as these systems are increasingly common and easy to use in medical environments.

Background. Recently, approaches have utilized the superior anatomical information provided by magnetic resonance imaging (MRI) to guide the reconstruction of positron emission tomography (PET). One of those approaches is the Bowsher’s prior, which has been accelerated lately with a convolutional neural network (CNN) to reconstruct MR-guided PET in the imaging domain in routine clinical imaging. Two differently trained Bowsher-CNN methods (B-CNN0 and B-CNN) have been trained and tested on brain PET/MR images with non-PSMA tracers, but so far, have not been evaluated in other anatomical regions yet. Methods. A NEMA phantom with five of its six spheres filled with the same, calibrated concentration of 18F-DCFPyL-PSMA, and thirty-two patients (mean age 64 ± 7 years) with biopsy-confirmed PCa were used in this study. Reconstruction with either of the two available Bowsher-CNN methods were performed on the conventional MR-based attenuation correction (MRAC) and T1-MR images in the imaging domain. Detectable volume of the spheres and tumors, relative contrast recovery (CR), and background variation (BV) were measured for the MRAC and the Bowsher-CNN images, and qualitative assessment was conducted by ranking the image sharpness and quality by two experienced readers. Results. For the phantom study, the B-CNN produced 12.7% better CR compared to conventional reconstruction. The small sphere volume (<1.8 ml) detectability improved from MRAC to B-CNN by nearly 13%, while measured activity was higher than the ground-truth by 8%. The signal-to-noise ratio, CR, and BV were significantly improved (p < 0.05) in B-CNN images of the tumor. The qualitative analysis determined that tumor sharpness was excellent in 76% of the PET images reconstructed with the B-CNN method, compared to conventional reconstruction. Conclusions. Applying the MR-guided B-CNN in clinical prostate PET/MR imaging improves some quantitative, as well as qualitative imaging measures. The measured improvements in the phantom are also clearly translated into clinical application.

MRI-guided (MRIg) positron emission tomography (PET) reconstruction can potentially reduce noise and increase spatial resolution compared to standard clinical ordered-subsets expectation-maximization (OSEM) image quality. However, to adjust for the desired image quality, the balance between measured data and prior information usually requires manual tuning. This work presents an adaptive method to automatically control the influence of the magnetic resonance imaging (MRI) information on the PET emission data using maximum a posteriori (MAP) image reconstruction, robust against a wide range of counts. The method was evaluated on different static brain PET datasets using [18F]-FDG, [18F]-Florbetapir and [11C]-PiB, acquired in a simultaneous PET/MRI scanner and a PET/CT scanner, followed by an MRI scan. Noise in gray and white matter was measured for a wide range of statistics. Furthermore, noise and quantification accuracy were analyzed in different cortical and subcortical brain regions with different levels of tracer uptake, and at different levels of counts. Results demonstrated consistent improved image quality in terms of noise and spatial resolution with MRI-guided MAP PET (MRIg-MAP) reconstruction compared to OSEM. Additionally, it was shown that the number of collected counts could be reduced by ~1.6– <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$2.3\times $ </tex-math></inline-formula> using MRIg-MAP reconstruction compared to OSEM, without increasing the noise significantly, either by reducing the scan time or injected activity. In conclusion, we presented a novel method to adaptively balance the influence of the anatomical information on the emission data for MRIg-MAP reconstruction, which showed image quality improvements compared to OSEM for different radiotracers, at different levels of counts, and applicable to simultaneous and sequential PET-MRI scans.

The objective of this research is to investigate the potential of total variation (TV) nullspace smoothing to mitigate the edge artifacts produced in expectation maximization (EM) reconstruction. Its success in reducing noise and preserving edges in image restoration, suggests that TV norm minimization could be explored in the reconstruction of positron emission tomography (PET) images. For example, the following criteria could be used for reconstruction: Find the image that has the minimum TV norm such that the measurements produced by the image are sufficiently close to the actual measurements. It is argued here, however, that this criterion may cause distortion of edges because of the Poisson nature of PET measurements. Moreover, the above criterion does not exploit the Poisson model of the data as the EM algorithm does. As a potentially better alternative, TV nullspace smoothing of EM reconstructions is proposed here. Rather than minimizing the whole EM reconstructed image with respect to the TV norm, in nullspace smoothing, only a portion of the reconstructed image is minimized; namely, the nullspace component of the reconstructed image. This may be advantageous because PET data contains no information about the nullspace component of the object being scanned. A second potential advantage stems from the fact that EM algorithm attempts to find an image that maximizes the likelihood function associated with a Poisson model of the data. The nullspace smoothed image is as optimal, in the maximum likelihood sense, as the original unsmoothed image. Methodology. To perform this investigation, noise-less projections of a mathematical phantom were simulated. Results. It was seen that the edge artifacts of the TV nullspace smoothed image were uniformly better than those in the original unsmoothed EM reconstruction. Conclusion. Although the results presented here are encouraging, more work needs to be done in the future to demonstrate the usefulness of nullspace smoothing with real data.

We present a Bayesian velocity field reconstruction algorithm that performs the reconstruction of the mass density field using only peculiar velocity data. Our method consistently accounts for the inhomogeneous Malmquist (IHM) bias using analytical integration along the line of sight. By testing our method on a simulation, we show that our method gives an unbiased reconstruction of the velocity field. We show that not accounting for the IHM bias can lead to significant biases in the Bayesian reconstructions. We applied our method to a peculiar velocity data set consisting of the SFI++ and 2MTF Tully–Fisher catalogues and the A2 supernovae compilation, thus obtaining a novel velocity reconstruction in the local Universe. Our velocity reconstructions have a cosmological power spectrum consistent with the theoretical expectation. Furthermore, we obtain a full description of the uncertainties on reconstruction through samples of the posterior distribution. We validate our velocity reconstruction of the local Universe by comparing it to an independent reconstruction using the 2M++ galaxy catalogue, obtaining good agreement between the two reconstructions. Using Bayesian model comparison, we find that our velocity model performs better than the adaptive kernel smoothed velocity with the same peculiar velocity data. However, our velocity model does not perform as well as the velocity reconstruction from the 2M++ galaxy catalogue, due to the sparse and noisy nature of the peculiar velocity tracer samples. The method presented here provides a way to include peculiar velocity data in initial condition reconstruction frameworks.

An algorithm is described for three-dimensional whole-body (3DWB) image reconstruction in positron emission tomography. For whole-body applications, improvements to the popular fixed-axial-acceptance-angle technique are achieved by combining axially adjacent projection data available with a long-axis data set. Time-consuming reprojection of unmeasured oblique lines of response is reduced or eliminated by axial overlap of bed positions, while pixel variance and reconstructed axial resolution are made more uniform by the overlap. Improvements in noise and resolution uniformity are accompanied by gains in reconstruction efficiency, and may be optimized against increased acquisition time due to overlapping acquisition segments and reduced axial coverage. An 11-detector-ring overlap improves axial uniformity in coronal images of a long, uniform cylinder from 23% to 8% with uniform axial resolution. Associated with a 37% improvement in reconstruction time is a 34% reduction in axial coverage for four bed positions. A smaller degree of overlap is found to provide the best trade-off between image uniformity, total scan duration, and reconstruction time because of a proportionally greater reduction in reprojected lines of response. Using a sample optimization scheme, we find a three-ring overlap is best for a 60 cm axial field of view and a five-ring overlap for an 80 cm axial field of view.

Several algorithms for image reconstruction in positron emission tomography (PET) have been described in the medical and statistical literature. We study a continuous idealization of the PET reconstruction problem, considered as an example of bivariate density estimation based on indirect observations. Given a large sample of indirect observations, we consider the size of the equivalent sample of observations, whose original exact positions would allow equally accurate estimation of the image of interest. Both for indirect and for direct observations, we establish exact minimax rates of convergence of estimation, for all possible estimators, over suitable smoothness classes of functions. A key technical device is a modulus of continuity appropriate to global function estimation. For indirect data and (in practice unobservable) direct data, the rates for mean integrated square error are $n^{-p/(p + 2)}$ and $(n/\\log n)^{-p/(p + 1)}$, respectively, for densities in a class corresponding to bounded square-integrable $p$th derivatives. We obtain numerical values for equivalent sample sizes for minimax linear estimators using a slightly modified error criterion. Modifications of the model to incorporate attenuation and the third dimension effect do not affect the minimax rates. The approach of the paper is applicable to a wide class of linear inverse problems.

Generalized Gibbs priors based positron emission tomography reconstruction

The maximum likelihood expectation maximization (ML-EM) is an attractive approach for image reconstruction in positron emission tomography (PET) since it considers the Poisson character of the emission data and it converges to the maximum likelihood solution. However, the algorithm possesses a very slow rate of convergence, and in practical applications it has to be terminated well before approaching the maximum likelihood solution. Recently, a row action maximum likelihood algorithm (RAMLA) was proposed for maximizing the Poisson likelihood at a considerably faster rate of convergence. We examine the performance of this new algorithm for fully 3D PET reconstruction since it enables us to study, within a reasonable amount of computer time, the images produced at much higher likelihood than would be produced by the standard ML-EM within the same period of time.