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Purpose: In spite of the general acceptance of iterative reconstruction for clinical use, analytic algorithms provide an important alternative tool due to their linearity, unbiased performance, and predictability for quantitative imaging and quality control studies. On modern time-of-flight (TOF) positron emission tomography scanners with excellent timing resolution, substantial angular compression of (histoprojection) data is possible without loss of resolution, but this also brings challenges for analytical algorithms. We propose TOF and non-TOF Fourier-based analytic approaches that appropriately handle the data sparsity on modern TOF systems. Approach: The proposed TOF algorithm (3D-DIFTOF-direct inversion Fourier transform for TOF) works directly on histoprojection data. The proposed Fourier-based approaches for histoprojection data are further extended to include non-TOF reconstruction (TOF-binned 3D-DIFT), which is particularly useful in time calibration procedures due to its insensitivity to time calibration errors. TOF information is used here to extend available histoprojection data to a larger number of views, essential for artifact-free non-TOF reconstruction. The proposed algorithms are compared with standard analytic techniques on Siemens scanners-space-based confidence-weighted TOF FBP and non-TOF DIFT. Results: 3D-DIFTOF reconstruction demonstrates both improved NEMA-based resolution and contrast versus background variability trade-offs. Similarly, the TOF-binned 3D-DIFT approach shows improved contrast-noise trade-offs over the standard non-TOF approach and is well suited for timing calibration. Conclusions: Our results demonstrate that the proposed 3D-DIFTOF technique provides an improved and more faithful characterization of image resolution compared with standard space-based analytic reconstructions. The proposed tools also provide accurate translation of sparse TOF data available on clinical scanners to upsampled data for non-TOF algorithms., (© 2020 Society of Photo-Optical Instrumentation Engineers (SPIE).)

The development of appropriate tools to quantify long-term carbon (C) budgets following forest transitions, that is, shifts from deforestation to afforestation, and to identify their drivers are key issues for forging sustainable land-based climate-change mitigation strategies. Here, we develop a new modeling approach, CRAFT (CaRbon Accumulation in ForesTs) based on widely available input data to study the C dynamics in French forests at the regional scale from 1850 to 2015. The model is composed of two interconnected modules which integrate biomass stocks and flows (Module 1) with litter and soil organic C (Module 2) and build upon previously established coupled climate-vegetation models. Our model allows to develop a comprehensive understanding of forest C dynamics by systematically depicting the integrated impact of environmental changes and land use. Model outputs were compared to empirical data of C stocks in forest biomass and soils, available for recent decades from inventories, and to a long-term simulation using a bookkeeping model. The CRAFT model reliably simulates the C dynamics during France's forest transition and reproduces C-fluxes and stocks reported in the forest and soil inventories, in contrast to a widely used bookkeeping model which strictly only depicts C-fluxes due to wood extraction. Model results show that like in several other industrialized countries, a sharp increase in forest biomass and SOC stocks resulted from forest area expansion and, especially after 1960, from tree growth resulting in vegetation thickening (on average 7.8 Mt C/year over the whole period). The difference between the bookkeeping model, 0.3 Mt C/year in 1850 and 21 Mt C/year in 2015, can be attributed to environmental and land management changes. The CRAFT model opens new grounds for better quantifying long-term forest C dynamics and investigating the relative effects of land use, land management, and environmental change., (© 2020 The Authors. Global Change Biology published by John Wiley & Sons Ltd.)

Dual-panel PET system configuration can lead to spatially variable point-spread functions (PSF) of considerable deformations due to depth-of-interaction effects and limited angular coverage. If not modelled properly, these effects result in decreased and inconsistent recovery of lesion activity across the field-of-view (FOV), as well as mispositioning of lesions in the reconstructed image caused by strong PSF asymmetries. We implemented and evaluated models of such PSF deformations with spatially-variant image-based resolution modeling (IRM) within reconstruction (varRM) using the Direct Image REConstruction for Time-of-flight (DIRECT) method and within post-reconstruction deconvolution methods. In addition, DIRECT reconstruction was performed with a spatially-invariant IRM (invRM) and without resolution modeling (noRM) for comparison. The methods were evaluated using simulated data for a realistic breast model with a set of 5 mm lesions located throughout the FOV of a dual-panel Breast-PET scanner. We simulated high-count data to focus on the ability of each method to correctly recover the PSF deformations, and a clinically realistic count level to assess the impact of low count data on the quantitative performance of the evaluated techniques. Performance of the methods evaluated herein was assessed by comparing lesion activity recovery (%BIAS), consistency (%SD) across the FOV, overall error (%RMSE), and recovery of each lesion location. As expected, all techniques using IRM provide considerable improvement over the noRM reconstruction. For the high-count cases, the overall quantitative performance of all IRM techniques, whether within reconstruction or within post-reconstruction, is similar if the lesion location misplacements are ignored. However, invRM provides less consistent performance on activity across lesions and is not able to recover accurate lesion locations. For a clinically realistic count level, varRM reconstruction consistently outperforms all compared approaches, while the post-reconstruction IRM approaches exhibit higher %SD and %RMSE values due to being more affected by the data noise than the within-reconstruction IRM approaches.

Previously, we proposed to use a coincidence collimator to achieve fractional-crystal resolution in PET imaging. We have designed and fabricated a collimator prototype for a small-animal PET scanner, A-PET. To compensate for imperfections in the fabricated collimator prototype, collimator normalization, as well as scanner normalization, is required to reconstruct quantitative and artifact-free images. In this study, we develop a normalization method for the collimator prototype based on the A-PET normalization using a uniform cylinder phantom. We performed data acquisition without the collimator for scanner normalization first, and then with the collimator from eight different rotation views for collimator normalization. After a reconstruction without correction, we extracted the cylinder parameters from which we generated expected emission sinograms. Single scatter simulation was used to generate the scattered sinograms. We used the least-squares method to generate the normalization coefficient for each LOR based on measured, expected and scattered sinograms. The scanner and collimator normalization coefficients were factorized by performing two normalizations separately. The normalization methods were also verified using experimental data acquired from A-PET with and without the collimator. In summary, we developed a model-base collimator normalization that can significantly reduce variance and produce collimator normalization with adequate statistical quality within feasible scan time.

The point-spread-functions (PSFs) of reconstructed images can be deformed due to detector effects such as resolution blurring and parallax error, data acquisition geometry such as insufficient sampling or limited angular coverage in dual-panel PET systems, or reconstruction imperfections/simplifications. PSF deformation decreases quantitative accuracy and its spatial variation lowers consistency of lesion uptake measurement across the imaging field-of-view (FOV). This can be a significant problem with dual panel PET systems even when using TOF data and image reconstruction models of the detector and data acquisition process. To correct for the spatially variant reconstructed PSF distortions we propose to use an image-based resolution model (IRM) that includes such image PSF deformation effects. Originally the IRM was mostly used for approximating data resolution effects of standard PET systems with full angular coverage in a computationally efficient way, but recently it was also used to mitigate effects of simplified geometric projectors. Our work goes beyond this by including into the IRM reconstruction imperfections caused by combination of the limited angle, parallax errors, and any other (residual) deformation effects and testing it for challenging dual panel data with strongly asymmetric and variable PSF deformations. We applied and tested these concepts using simulated data based on our design for a dedicated breast imaging geometry (B-PET) consisting of dual-panel, time-of-flight (TOF) detectors. We compared two image-based resolution models; i) a simple spatially invariant approximation to PSF deformation, which captures only the general PSF shape through an elongated 3D Gaussian function, and ii) a spatially variant model using a Gaussian mixture model (GMM) to more accurately capture the asymmetric PSF shape in images reconstructed from data acquired with the B-PET scanner geometry. Results demonstrate that while both IRMs decrease the overall uptake bias in the reconstructed image, the second one with the spatially variant and accurate PSF shape model is also able to ameliorate the spatially variant deformation effects to provide consistent uptake results independent of the lesion location within the FOV.

Iterative reconstruction algorithms are routinely used for clinical practice; however, analytic algorithms are relevant candidates for quantitative research studies due to their linear behavior. While iterative algorithms also benefit from the inclusion of accurate data and noise models the widespread use of time-of-flight (TOF) scanners with less sensitivity to noise and data imperfections make analytic algorithms even more promising. In our previous work we have developed a novel iterative reconstruction approach (DIRECT: direct image reconstruction for TOF) providing convenient TOF data partitioning framework and leading to very efficient reconstructions. In this work we have expanded DIRECT to include an analytic TOF algorithm with confidence weighting incorporating models of both TOF and spatial resolution kernels. Feasibility studies using simulated and measured data demonstrate that analytic-DIRECT with appropriate resolution and regularization filters is able to provide matched bias versus variance performance to iterative TOF reconstruction with a matched resolution model.

Fully 3D time-of-flight (TOF) PET scanners offer the potential of previously unachievable image quality in clinical PET imaging. TOF measurements add another degree of redundancy for cylindrical PET scanners and make photon-limited TOF-PET imaging more robust than non-TOF PET imaging. The data space for 3D TOF-PET data is five-dimensional with two degrees of redundancy. Previously, consistency equations were used to characterize the redundancy of TOF-PET data. In this paper, we first derive two Fourier consistency equations and Fourier-John equation for 3D TOF PET based on the generalized projection-slice theorem; the three partial differential equations (PDEs) are the dual of the sinogram consistency equations and John's equation. We then solve the three PDEs using the method of characteristics. The two degrees of entangled redundancy of the TOF-PET data can be explicitly elicited and exploited by the solutions of the PDEs along the characteristic curves, which gives a complete understanding of the rich structure of the 3D x-ray transform with TOF measurement. Fourier rebinning equations and other mapping equations among different types of PET data are special cases of the general solutions. We also obtain new Fourier rebinning and consistency equations (FORCEs) from other special cases of the general solutions, and thus we obtain a complete scheme to convert among different types of PET data: 3D TOF, 3D non-TOF, 2D TOF and 2D non-TOF data. The new FORCEs can be used as new Fourier-based rebinning algorithms for TOF-PET data reduction, inverse rebinnings for designing fast projectors, or consistency conditions for estimating missing data. Further, we give a geometric interpretation of the general solutions--the two families of characteristic curves can be obtained by respectively changing the azimuthal and co-polar angles of the biorthogonal coordinates in Fourier space. We conclude the unified Fourier theory by showing that the Fourier consistency equations are necessary and sufficient for 3D x-ray transform with TOF measurement. Finally, we give numerical examples of inverse rebinning for a 3D TOF PET and Fourier-based rebinning for a 2D TOF PET using the FORCEs to show the efficacy of the unified Fourier solutions.

Due to the unique geometry, dual-panel PET scanners have many advantages in dedicated breast imaging and on-board imaging applications since the compact scanners can be combined with other imaging and treatment modalities. The major challenges of dual-panel PET imaging are the limited-angle problem and data truncation, which can cause artifacts due to incomplete data sampling. The time-of-flight (TOF) information can be a promising solution to reduce these artifacts. The TOF planogram is the native data format for dual-panel TOF PET scanners, and the non-TOF planogram is the 3D extension of linogram. The TOF planograms is five-dimensional while the objects are three-dimensional, and there are two degrees of redundancy. In this paper, we derive consistency equations and Fourier-based rebinning algorithms to provide a complete understanding of the rich structure of the fully 3D TOF planograms. We first derive two consistency equations and John's equation for 3D TOF planograms. By taking the Fourier transforms, we obtain two Fourier consistency equations and the Fourier-John equation, which are the duals of the consistency equations and John's equation, respectively. We then solve the Fourier consistency equations and Fourier-John equation using the method of characteristics. The two degrees of entangled redundancy of the 3D TOF data can be explicitly elicited and exploited by the solutions along the characteristic curves. As the special cases of the general solutions, we obtain Fourier rebinning and consistency equations (FORCEs), and thus we obtain a complete scheme to convert among different types of PET planograms: 3D TOF, 3D non-TOF, 2D TOF and 2D non-TOF planograms. The FORCEs can be used as Fourier-based rebinning algorithms for TOF-PET data reduction, inverse rebinnings for designing fast projectors, or consistency conditions for estimating missing data. As a byproduct, we show the two consistency equations are necessary and sufficient for 3D TOF planograms. Finally, we give numerical examples of implementation of a fast 2D TOF planogram projector and Fourier-based rebinning for a 2D TOF planograms using the FORCEs to show the efficacy of the Fourier-based solutions.

In positron emission tomography (PET) imaging, attenuation correction with accurate attenuation estimation is crucial for quantitative patient studies. Recent research showed that the attenuation sinogram can be determined up to a scaling constant utilizing the time-of-flight information. The TOF-PET data can be naturally and efficiently stored in a histo-image without information loss, and the radioactive tracer distribution can be efficiently reconstructed using the DIRECT approaches. In this paper, we explore transmission-less attenuation estimation from TOF-PET histo-images. We first present the TOF-PET histo-image formation and the consistency equations in the histo-image parameterization, then we derive a least-squares solution for estimating the directional derivatives of the attenuation factors from the measured emission histo-images. Finally, we present a fast solver to estimate the attenuation factors from their directional derivatives using the discrete sine transform and fast Fourier transform while considering the boundary conditions. We find that the attenuation histo-images can be uniquely determined from the TOF-PET histo-images by considering boundary conditions. Since the estimate of the attenuation directional derivatives can be inaccurate for LORs tangent to the patient boundary, external sources, e.g. a ring or annulus source, might be needed to give an accurate estimate of the attenuation gradient for such LORs. The attenuation estimation from TOF-PET emission histo-images is demonstrated using simulated 2D TOF-PET data.

Positron emission tomography (PET) has become an important modality in medical and molecular imaging. However, in most PET applications, the resolution is still mainly limited by the physical crystal sizes or the detector's intrinsic spatial resolution. To achieve images with better spatial resolution in a central region of interest (ROI), we have previously proposed using collimation in PET scanners. The collimator is designed to partially mask detector crystals to detect lines of response (LORs) within fractional crystals. A sequence of collimator-encoded LORs is measured with different collimation configurations. This novel collimated scanner geometry makes the reconstruction problem challenging, as both detector and collimator effects need to be modeled to reconstruct high-resolution images from collimated LORs. In this paper, we present a LOR-interleaving (LORI) algorithm, which incorporates these effects and has the advantage of reusing existing reconstruction software, to reconstruct high-resolution images for PET with fractional-crystal collimation. We also develop a 3D ray-tracing model incorporating both the collimator and crystal penetration for simulations and reconstructions of the collimated PET. By registering the collimator-encoded LORs with the collimator configurations, high-resolution LORs are restored based on the modeled transfer matrices using the non-negative least-squares method and EM algorithm. The resolution-enhanced images are then reconstructed from the high-resolution LORs using the MLEM or OSEM algorithm. For validation, we applied the LORI method to a small-animal PET scanner, A-PET, with a specially designed collimator. We demonstrate through simulated reconstructions with a hot-rod phantom and MOBY phantom that the LORI reconstructions can substantially improve spatial resolution and quantification compared to the uncollimated reconstructions. The LORI algorithm is crucial to improve overall image quality of collimated PET, which can have significant implications in preclinical and clinical ROI imaging applications.

Purpose: Spatial resolution in positron emission tomography (PET) is still a limiting factor in many imaging applications. To improve the spatial resolution for an existing scanner with fixed crystal sizes, mechanical movements such as scanner wobbling and object shifting have been considered for PET systems. Multiple acquisitions from different positions can provide complementary information and increased spatial sampling. The objective of this paper is to explore an efficient and useful reconstruction framework to reconstruct super-resolution images from super-sampled low-resolution data sets., Methods: The authors introduce a super-sampling data acquisition model based on the physical processes with tomographic, downsampling, and shifting matrices as its building blocks. Based on the model, we extend the MLEM and Landweber algorithms to reconstruct images from super-sampled data sets. The authors also derive a backprojection-filtration-like (BPF-like) method for the super-sampling reconstruction. Furthermore, they explore variant methods for super-sampling reconstructions: the separate super-sampling resolution-modeling reconstruction and the reconstruction without downsampling to further improve image quality at the cost of more computation. The authors use simulated reconstruction of a resolution phantom to evaluate the three types of algorithms with different super-samplings at different count levels., Results: Contrast recovery coefficient (CRC) versus background variability, as an image-quality metric, is calculated at each iteration for all reconstructions. The authors observe that all three algorithms can significantly and consistently achieve increased CRCs at fixed background variability and reduce background artifacts with super-sampled data sets at the same count levels. For the same super-sampled data sets, the MLEM method achieves better image quality than the Landweber method, which in turn achieves better image quality than the BPF-like method. The authors also demonstrate that the reconstructions from super-sampled data sets using a fine system matrix yield improved image quality compared to the reconstructions using a coarse system matrix. Super-sampling reconstructions with different count levels showed that the more spatial-resolution improvement can be obtained with higher count at a larger iteration number., Conclusions: The authors developed a super-sampling reconstruction framework that can reconstruct super-resolution images using the super-sampling data sets simultaneously with known acquisition motion. The super-sampling PET acquisition using the proposed algorithms provides an effective and economic way to improve image quality for PET imaging, which has an important implication in preclinical and clinical region-of-interest PET imaging applications.

Collimation can improve both the spatial resolution and sampling properties compared to the same scanner without collimation. Spatial resolution improves because each original crystal can be conceptually split into two (i.e., doubling the number of in-plane crystals) by masking half the crystal with a high-density attenuator (e.g., tungsten); this reduces coincidence efficiency by 4× since both crystals comprising the line of response (LOR) are masked, but yields 4× as many resolution-enhanced (RE) LORs. All the new RE LORs can be measured by scanning with the collimator in different configurations. In this simulation study, the collimator was assumed to be ideal, neither allowing gamma penetration nor truncating the field of view. Comparisons were made in 2D between an uncollimated small-animal system with 2-mm crystals that were assumed to be perfectly absorbing and the same system with collimation that narrowed the effective crystal size to 1 mm. Digital phantoms included a hot-rod and a single-hot-spot, both in a uniform background with activity ratio of 4:1. In addition to the collimated and uncollimated configurations, angular and spatial wobbling acquisitions of the 2-mm case were also simulated. Similarly, configurations with different combinations of the RE LORs were considered including (i) all LORs, (ii) only those parallel to the 2-mm LORs; and (iii) only cross pairs that are not parallel to the 2-mm LORs. Lastly, quantitative studies were conducted for collimated and uncollimated data using contrast recovery coefficient and mean-squared error (MSE) as metrics. The reconstructions show that for most noise levels there is a substantial improvement in image quality (i.e., visual quality, resolution, and a reduction in artifacts) by using collimation even when there are 4× fewer counts or - in some cases - comparing with the noiseless uncollimated reconstruction. By comparing various configurations of sampling, the results show that it is the matched combination of both improved spatial resolution of each LOR and the increase in the number of LORs that yields improved reconstructions. Further, the quantitative studies show that for low-count scans, the collimated data give better MSE for small lesions and the uncollimated data give better MSE for larger lesions; for high-count studies, the collimated data yield better quantitative values for the entire range of lesion sizes that were evaluated.

We describe a GPU-accelerated framework that efficiently models spatially (shift) variant system response kernels and performs forward- and back-projection operations with these kernels for the DIRECT (Direct Image Reconstruction for TOF) iterative reconstruction approach. Inherent challenges arise from the poor memory cache performance at non-axis aligned TOF directions. Focusing on the GPU memory access patterns, we utilize different kinds of GPU memory according to these patterns in order to maximize the memory cache performance. We also exploit the GPU instruction-level parallelism to efficiently hide long latencies from the memory operations. Our experiments indicate that our GPU implementation of the projection operators has slightly faster or approximately comparable time performance than FFT-based approaches using state-of-the-art FFTW routines. However, most importantly, our GPU framework can also efficiently handle any generic system response kernels, such as spatially symmetric and shift-variant as well as spatially asymmetric and shift-variant, both of which an FFT-based approach cannot cope with.

Early clinical results with time-of-flight (TOF) positron emission tomography (PET) systems have demonstrated the advantages of TOF information in PET reconstruction. Reconstruction approaches in TOF-PET systems include list-mode and binned iterative algorithms as well as confidence-weighted analytic methods. List-mode iterative TOF reconstruction retains the resolutions of the data in the spatial and temporal domains without any binning approximations but is computationally intensive. We have developed an approach [DIRECT (direct image reconstruction for TOF)] to speed up TOF-PET reconstruction that takes advantage of the reduced angular sampling requirement of TOF data by grouping list-mode data into a small number of azimuthal views and co-polar tilts and depositing the grouped events into histo-images, arrays with the sampling and geometry of the final image. All physical effects are included in the system model and deposited in the same histo-image structure. Using histo-images allows efficient computation during reconstruction without ray-tracing or interpolation operations. The DIRECT approach was compared with 3-D list-mode TOF ordered subsets expectation maximization (OSEM) reconstruction for phantom and patient data taken on the University of Pennsylvania research LaBr (3) TOF-PET scanner. The total processing and reconstruction time for these studies with DIRECT without attention to code optimization is approximately 25%-30% that of list-mode TOF-OSEM to achieve comparable image quality. Furthermore, the reconstruction time for DIRECT is independent of the number of events and/or sizes of the spatial and TOF kernels, while the time for list-mode TOF-OSEM increases with more events or larger kernels. The DIRECT approach is able to reproduce the image quality of list-mode iterative TOF reconstruction both qualitatively and quantitatively in measured data with a reduced time.

The use of time-of-flight (TOF) information during positron emission tomography (PET) reconstruction has been found to improve the image quality. In this work we quantified this improvement using two existing methods: 1) a very simple analytical expression only valid for a central point in a large uniform disk source and 2) efficient analytical approximations for postfiltered maximum likelihood expectation maximization (MLEM) reconstruction with a fixed target resolution, predicting the image quality in a pixel or in a small region of interest based on the Fisher information matrix. Using this latter method the weighting function for filtered backprojection reconstruction of TOF PET data proposed by C. Watson can be derived. The image quality was investigated at different locations in various software phantoms. Simplified as well as realistic phantoms, measured both with TOF PET systems and with a conventional PET system, were simulated. Since the time resolution of the system is not always accurately known, the effect on the image quality of using an inaccurate kernel during reconstruction was also examined with the Fisher information-based method. First, we confirmed with this method that the variance improvement in the center of a large uniform disk source is proportional to the disk diameter and inversely proportional to the time resolution. Next, image quality improvement was observed in all pixels, but in eccentric and high-count regions the contrast-to-noise ratio (CNR) increased less than in central and low- or medium-count regions. Finally, the CNR was seen to decrease when the time resolution was inaccurately modeled (too narrow or too wide) during reconstruction. Although the maximum CNR is not very sensitive to the time resolution error, using an inaccurate TOF kernel tends to introduce artifacts in the reconstructed image.

For modern time-of-flight (TOF) positron emission tomography (PET) systems, in which the number of possible lines of response and TOF bins is much larger than the number of acquired events, the most appropriate reconstruction approaches are considered to be list-mode methods. However, their shortcomings are relatively high computational costs for reconstruction and for sensitivity matrix calculation. Efficient treatment of TOF data within the proposed DIRECT approach is obtained by 1) angular (azimuthal and co-polar) grouping of TOF events to a set of views as given by the angular sampling requirements for the TOF resolution, and 2) deposition (weighted-histogramming) of these grouped events, and correction data, into a set of "histo-images," one histo-image per view. The histo-images have the same geometry (voxel grid, size and orientation) as the reconstructed image. The concept is similar to the approach involving binning of the TOF data into angularly subsampled histo-projections-projections expanded in the TOF directions. However, unlike binning into histo-projections, the deposition of TOF events directly into the image voxels eliminates the need for tracing and/or interpolation operations during the reconstruction. Together with the performance of reconstruction operations directly in image space, this leads to a very efficient implementation of TOF reconstruction algorithms. Furthermore, the resolution properties are not compromised either, since events are placed into the image elements of the desired size from the beginning. Concepts and efficiency of the proposed data partitioning scheme are demonstrated in this work by using the DIRECT approach in conjunction with the row-action maximum-likelihood (RAMLA) algorithm.

Fourier-based approaches for three-dimensional (3-D) reconstruction are based on the relationship between the 3-D Fourier transform (FT) of the volume and the two-dimensional (2-D) FT of a parallel-ray projection of the volume. The critical step in the Fourier-based methods is the estimation of the samples of the 3-D transform of the image from the samples of the 2-D transforms of the projections on the planes through the origin of Fourier space, and vice versa for forward-projection (reprojection). The Fourier-based approaches have the potential for very fast reconstruction, but their straightforward implementation might lead to unsatisfactory results if careful attention is not paid to interpolation and weighting functions. In our previous work, we have investigated optimal interpolation parameters for the Fourier-based forward and back-projectors for iterative image reconstruction. The optimized interpolation kernels were shown to provide excellent quality comparable to the ideal sinc interpolator. This work presents an optimization of interpolation parameters of the 3-D direct Fourier method with Fourier reprojection (3D-FRP) for fully 3-D positron emission tomography (PET) data with incomplete oblique projections. The reprojection step is needed for the estimation (from an initial image) of the missing portions of the oblique data. In the 3D-FRP implementation, we use the gridding interpolation strategy, combined with proper weighting approaches in the transform and image domains. We have found that while the 3-D reprojection step requires similar optimal interpolation parameters as found in our previous studies on Fourier-based iterative approaches, the optimal interpolation parameters for the main 3D-FRP reconstruction stage are quite different. Our experimental results confirm that for the optimal interpolation parameters a very good image accuracy can be achieved even without any extra spectral oversampling, which is a common practice to decrease errors caused by interpolation in Fourier reconstruction.

In this paper we propose a comprehensive energy-based scatter correction approach for positron emission tomography (PET). We take advantage of the marked difference between the energy spectra of the unscattered and scattered photons, and use the detailed energy information that comes with the list-mode data for the estimation of the scattered events distribution in the data space. Also, inside the maximum-likelihood expectation maximization (ML-EM) image reconstruction algorithm, we introduce energy-dependent factors that individualize the correction terms for each event, given its position and energy information. The central piece of our approach is the two-dimensional detector energy response model represented as a linear combination of four components, each one representing a particular state a PET event can be found in: both photons unscattered, the second scattered while the first not, the first photon scattered while the second not and both photons scattered. For a set of events collected in the vicinity of a point in the projection space, the coefficient of each component is determined by applying a statistical estimator. As a result we obtain the number of scattered events that are in the given set. The model also gives us the variation of scatter fraction with the photon pair energies for that particular position in the data space. A simulation study that demonstrates the proposed methods is presented.

Introduction: Fully three-dimensional (3-D) reconstruction algorithms have the potential to better utilize PET data acquired in true 3-D acquisition modes, as compared to 2-D approaches involving data reduction through rebinning procedures followed by a 2-D or 2.5-D reconstruction. Our previous studies, involving simulated and real phantom data from PET scanners working purely in 3-D acquisition mode, showed that the fully 3-D reconstruction approaches bring appreciable improvement on physical measures and visual image quality as well. Reconstruction procedures utilized in those studies were based upon the row action maximum likelihood algorithm (RAMLA) implemented on our clinical scanners. The purpose of this study was to investigate if the fully 3-D RAMLA reconstruction software brings an appreciable improvement in clinical image quality and reader confidence., Materials and Methods: Ninety patient scans acquired on the Philips CPET scanner at our institution were reconstructed using 2.5-D RAMLA and 3-D RAMLA with differing reconstruction parameters. These scans were blindly presented to four experienced PET scan readers who graded various aspects of image quality., Results: Our study concluded that 3-D RAMLA on the CPET scanner reduces artifacts and image noise and improves clinical confidence in interpreting PET images., Conclusion: This study led to the routine use of 3-D RAMLA reconstruction on the CPET scanner at our institution.

A new algorithm for three-dimensional reconstruction of two-dimensional crystals from projections is presented, and its applicability to biological macromolecules imaged using transmission electron microscopy (TEM) is investigated. Its main departures from the traditional approach is that it works in real space, rather than in Fourier space, and it is iterative. This has the advantage of making it convenient to introduce additional constraints (such as the support of the function to be reconstructed, which may be known from alternative measurements) and has the potential of more accurately modeling the TEM image formation process. Phantom experiments indicate the superiority of the new approach even without the introduction of constraints in addition to the projection data.

Iterative image reconstruction algorithms play an increasingly important role in modern tomographic systems, especially in emission tomography. With the fast increase of the sizes of the tomographic data, reduction of the computation demands of the reconstruction algorithms is of great importance. Fourier-based forward and back-projection methods have the potential to considerably reduce the computation time in iterative reconstruction. Additional substantial speed-up of those approaches can be obtained utilizing powerful and cheap off-the-shelf fast Fourier transform (FFT) processing hardware. The Fourier reconstruction approaches are based on the relationship between the Fourier transform of the image and Fourier transformation of the parallel-ray projections. The critical two steps are the estimations of the samples of the projection transform, on the central section through the origin of Fourier space, from the samples of the transform of the image, and vice versa for back-projection. Interpolation errors are a limitation of Fourier-based reconstruction methods. We have applied min-max optimized Kaiser-Bessel interpolation within the nonuniform FFT (NUFFT) framework and devised ways of incorporation of resolution models into the Fourier-based iterative approaches. Numerical and computer simulation results show that the min-max NUFFT approach provides substantially lower approximation errors in tomographic forward and back-projection than conventional interpolation methods. Our studies have further confirmed that Fourier-based projectors using the NUFFT approach provide accurate approximations to their space-based counterparts but with about ten times faster computation, and that they are viable candidates for fast iterative image reconstruction.

The recently proposed Fourier rebinning (FORE) technique of 3D PET reconstruction is investigated over a wide range of axial acceptance angles. In this study we evaluate the performance of the FORE technique using spatial resolution, contrast and noise figures of merit and compare reconstruction performance of the FORE (followed by multislice 2D reconstruction) to the 3D-RP technique for large-acceptance-angle data (+/-26.25 degrees). Our results show that the FORE technique does not affect the transverse resolution. On the other hand the axial resolution using FORE deteriorates faster, compared with the 3D-RP, at large radii as the acceptance angle increases. Concerning the noise behaviour, we have found that filtering has better ability to suppress the noise in the FORE reconstruction, compared with the 3D-RP reconstruction, especially in the slices near the edge of the axial field of view. Overall, the combination of good performance and fast reconstruction time makes the FORE technique a practical choice for 3D PET applications.