# Image Reconstruction

Panin YV and Matej S

**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.

Li Y, Matej S, Karp JS.

Recent research showed that attenuation images can be determined from emission data, jointly with activity images, up to a scaling constant when utilizing the time-of-flight (TOF) information. We aim to develop practical CT-less joint reconstruction for clinical TOF PET scanners to obtain quantitatively accurate activity and attenuation images. In this work, we present a joint reconstruction of activity and attenuation based on MLAA (maximum likelihood reconstruction of attenuation and activity) with autonomous scaling determination and joint TOF scatter estimation from TOF PET data. Our idea for scaling is to use a selected volume of interest (VOI) in a reconstructed attenuation image with known attenuation, e.g., a liver in patient imaging. First, we construct a unit attenuation medium which has a similar, not necessarily the same, support to the imaged emission object. All detectable LORs intersecting the unit medium have an attenuation factor of e^{-1} ≈ 0.3679, i.e., the line integral of linear attenuation coefficients is one. The scaling factor can then be determined from the difference between the reconstructed attenuation image and the known attenuation within the selected VOI normalized by the unit attenuation medium. A four-step iterative joint reconstruction algorithm is developed. In each iteration, 1) first the activity is updated using TOF OSEM from TOF list-mode data; 2) then the attenuation image is updated using XMLTR—a extended MLTR from non-TOF LOR sinograms; 3) a scaling factor is determined based on the selected VOI and both activity and attenuation images are updated using the estimated scaling; 4) scatter is estimated using TOF single scatter simulation with the jointly reconstructed activity and attenuation images. The performance of joint reconstruction is studied using simulated data from a generic whole-body clinical TOF PET scanner and a long axial FOV research PET scanner as well as 3D experimental data from the PennPET Explorer scanner. We show that the proposed joint reconstruction with proper autonomous scaling provides low bias results comparable to the reference reconstruction with known attenuation.

P. Gravel, Y. Li, and S. Matej.

*IEEE Trans Radiat Plasma Med Sci*, vol. 4, pp: 603-612, 2020.

Limited-angle data, such as data obtained from a dual-panel Breast-PET scanner, result in substantial image blur in directions coinciding with the missing cone of the image spectrum. On systems with time-of-flight (TOF) capabilities, this blur is reduced as given by the TOF uncertainty, with the image spectrum being correspondingly expanded into the missing spectral cone. Modeling of the TOF uncertainty in the reconstruction is expected to deconvolve this residual TOF blurring. We have however observed that, as a tradeoff, this TOF de-blurring process also introduces ringing artifacts at the edges, analogous to the edge effects observed with line-of-response (LOR) resolution modeling, which attempts to deconvolve the blur due to detector resolution effects. However, in the former case, the ringing artifacts are much wider due to the spatial extent of the TOF uncertainty as compared to the width of typical LOR resolution blur. We illustrate and investigate the effects of using matched, as well as under-modeled and over-modeled, TOF kernels on edge artifacts in reconstruction from limited-angle data, and compare them with TOF reconstructions of complete data. Although for the conventional data with full angular coverage the reconstruction is fairly insensitive to the exact size of the TOF kernel and TOF modeling does not produce ringing artifacts, it is not the case for the limited-angle data. We show that it is important to use some form of regularization of the TOF uncertainty deconvolution process within reconstruction of the limited-angle data, such as decreasing the TOF kernel size.

Gravel P, Surti S, Krishnamoorthy S, Karp JS, Matej S

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 5mm 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.

Iriarte A, Marabini R, Matej S, Sorzano COS, Lewitt RM

Positron emission tomography (PET) is a nuclear imaging modality that provides *in vivo* quantitative measurements of the spatial and temporal distribution of compounds labeled with a positron emitting radionuclide. In the last decades, a tremendous effort has been put into the field of mathematical tomographic image reconstruction algorithms that transform the data registered by a PET camera into an image that represents slices through the scanned object. Iterative image reconstruction methods often provide higher quality images than conventional direct analytical methods. Aside from taking into account the statistical nature of the data, the key advantage of iterative reconstruction techniques is their ability to incorporate detailed models of the data acquisition process. This is mainly realized through the use of the so-called system matrix, that defines the mapping from the object space to the measurement space. The quality of the reconstructed images relies to a great extent on the accuracy with which the system matrix is estimated. Unfortunately, an accurate system matrix is often associated with high reconstruction times and huge storage requirements. Many attempts have been made to achieve realistic models without incurring excessive computational costs. As a result, a wide range of alternatives to the calculation of the system matrix exists. In this article we present a review of the different approaches used to address the problem of how to model, calculate and store the system matrix.

Li Y, Defrise M, Matej S, Metzler SD.

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 (FCEs) and the Fourier–John equation (FJE), which are the duals of the consistency equations and John's equation, respectively. We then solve the FCEs and FJE 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.

Li Y, Matej S, Metzler SD.

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.

Lyon MC, Sitek A, Metzler SD, Moore SC

**Purpose**: The authors are currently developing a dual‐resolution multiple‐pinhole microSPECT imaging system based on three large NaI(Tl) gamma cameras. Two multiple‐pinhole tungsten collimator tubes will be used sequentially for whole‐body “scout” imaging of a mouse, followed by high‐resolution (hi‐res) imaging of an organ of interest, such as the heart or brain. Ideally, the whole‐body image will be reconstructed in real time such that data need only be acquired until the area of interest can be visualized well‐enough to determine positioning for the hi‐res scan. The authors investigated the utility of the origin ensemble (OE) algorithm for online and offline reconstructions of the scout data. This algorithm operates directly in image space, and can provide estimates of image uncertainty, along with reconstructed images. Techniques for accelerating the OE reconstruction were also introduced and evaluated.

**Methods**: System matrices were calculated for our 39‐pinhole scout collimator design. SPECT projections were simulated for a range of count levels using the MOBY digital mouse phantom. Simulated data were used for a comparison of OE and maximum‐likelihood expectation maximization (MLEM) reconstructions. The OE algorithm convergence was evaluated by calculating the total‐image entropy and by measuring the counts in a volume‐of‐interest (VOI) containing the heart. Total‐image entropy was also calculated for simulated MOBY data reconstructed using OE with various levels of parallelization.

**Results**: For VOI measurements in the heart, liver, bladder, and soft‐tissue, MLEM and OE reconstructed images agreed within 6%. Image entropy converged after ∼2000 iterations of OE, while the counts in the heart converged earlier at ∼200 iterations of OE. An accelerated version of OE completed 1000 iterations in <9 min for a 6.8M count data set, with some loss of image entropy performance, whereas the same dataset required ∼79 min to complete 1000 iterations of conventional OE. A combination of the two methods showed decreased reconstruction time and no loss of performance when compared to conventional OE alone.

**Conclusions**: OE‐reconstructed images were found to be quantitatively and qualitatively similar to MLEM, yet OE also provided estimates of image uncertainty. Some acceleration of the reconstruction can be gained through the use of parallel computing. The OE algorithm is useful for reconstructing multiple‐pinhole SPECT data and can be easily modified for real‐time reconstruction.

Matej S, Li Y, Panetta J, Karp JS, Surti S.

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.

Matej S, Daube-Witherspoon ME, Karp JS.

*Phys Med Biol*, vol. 61, pp. 3365-3386, 2016. Paper selected by PMB for their Highlights of 2016.

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 timeof-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.

Li, Y Matej S, Karp JS, Metzler SD.

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.

Daube-Witherspoon ME, Matej S, Werner ME, Surti S, Karp JS.

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.

Matej S, Surti S, Jayanthi S, Daube-Witherspoon ME, Lewitt RM, Karp JS.

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 ldquohisto-images,rdquo 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.

Vandenberghe S, Daube-Witherspoon ME, Lewitt RM, Karp JS.

Faster scintillators like LaBr_{3} and LSO have sparked renewed interest in PET scanners with time-of-flight (TOF) information. The TOF information adds another dimension to the data set compared to conventional three-dimensional (3D) PET with the size of the projection data being multiplied by the number of TOF bins. Here we show by simulations and analytical reconstruction that angular sampling for two-dimensional (2D) TOF PET can be reduced significantly compared to what is required for conventional 2D PET. Fully 3D TOF PET data, however, have a wide range of oblique and transverse angles. To make use of the smaller necessary angular sampling we reduce the 3D data to a set of 2D histoprojections. This is done by rebinning the 3D data to 2D data and by mashing these 2D data into a limited number of angles. Both methods are based on the most likely point given by the TOF measurement. It is shown that the axial resolution loss associated with rebinning reduces with improved timing resolution and becomes less than 1 mm for a TOF resolution below 300 ps. The amount of angular mashing that can be applied without tangential resolution loss increases with improved TOF resolution. Even quite coarse angular mashing (18 angles out of 324 measured angles for 424 ps) does not significantly reduce image quality in terms of the contrast or noise. The advantages of the proposed methods are threefold. Data storage is reduced to a limited number of 2D histoprojections with TOF information. Compared to listmode format we have the advantage of a predetermined storage space and faster reconstruction. The method does not require the normalization of projections prior to rebinning and can be applied directly to measured listmode data.

Matej S, Fessler JA, Kazantsev IG.

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.

Popescu LM, Matej S, Lewitt RM.

In positron emission tomography (PET), the format in which the data is stored has a major influence on the image reconstruction procedure. The use of the list-mode format preserves all of the measured attributes of the detected photon pairs but the events are stored in the order that they were measured, which allows only sequential access to the data. This fact limits the number of applicable algorithms and often computing speed or memory capacity constraints require the use of algorithms that do not make full use of the original precise information in the data. In this paper we show how through a change of the format in which the data is stored one can keep all the initial information about the individual events while providing random access to subsets of events belonging to given geometrical regions, thus making possible the use of maximum likelihood ordered subsets (OSEM) type algorithms with data provided as a collection of individual events (list-mode), and facilitating the adaptation of other types of algorithms. The structured data format also allows for more compact (compressed) storage of the information compared to the simple list-mode format.

Daube-Witherspoon ME, Matej S, Karp JS, Lewitt RM.

Three-dimensional (3D) reconstructions from fully 3D positron emission tomography (PET) data can yield high-quality images but at a high computational cost. The 3D row action maximum likelihood algorithm (3D RAMLA) with spherically-symmetric basis functions (blobs) has recently been modified to reconstruct multi-slice 2D PET data after Fourier rebinning (FORE) but still using 3D basis functions (2.5D RAMLA. In this study 2.5D RAMLA and 3D RAMLA were applied to several patient and phantom PET datasets to assess their clinical performance. RAMLA performance was compared to that for the reconstruction techniques in routine clinical use on the authors' PET scanners. Torso phantom and whole-body patient scans acquired on the C-PET scanner were reconstructed after FORE with filtered back-projection (FORE+FBP), the ordered subsets expectation maximization algorithm (FORE+OSEM), and FORE+2.5D RAMLA for various reconstruction parameters. The 3D Hoffman brain phantom scanned on the HEAD Penn-PET scanner was reconstructed with the 3D reprojection algorithm (3DRP) and 3D RAMLA, as well, as FORE+FBP, FORE+OSEM, and FORE+2.5D RAMLA. The authors' results demonstrate improvement of 3D and 2.5D RAMLA with blob basis functions, compared to the reconstruction methods currently in clinical use, in terms of contrast recovery and noise, especially in regions of limited statistics.

Matej S, Lewitt RM.

The direct Fourier method (DFM) for three-dimensional (3-D) reconstruction of a 3-D volume is 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 direct Fourier method has the potential for very fast reconstruction, but a straightforward implementation of the method leads to unsatisfactory results. This paper presents an implementation of the direct Fourier method for fully 3-D positron emission tomography (PET) data with incomplete oblique projections (3D-FRP) that gives results as good as, or better than, those of a much slower 3-D filtered backprojection method (3DRP), and in the same time as a fast but less accurate method using Fourier rebinning (FORE) followed by slice-by-slice reconstruction. In common with 3DRP, 3D-FRP is based on a discretization of an inversion formula, so it is geometrically accurate for large oblique angles, and both methods involve reprojection of an initial image. The critical two steps in the 3D-FRP method are the estimations 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 reprojection. These steps use a gridding strategy, combined with new approaches for weighting in the transform and image domains. The authors' experimental results confirm that good image accuracy can be achieved together with a short reconstruction time.

Matej S, Lewitt RM.

Spherically symmetric volume elements with smooth tapering of the values near their boundaries are alternatives to the more conventional voxels for the construction of volume images in the computer. Their use, instead of voxels, introduces additional parameters which enable the user to control the shape of the volume element (blob) and consequently to control the characteristics of the images produced by iterative methods for reconstruction from projection data. For images composed of blobs, efficient algorithms have been designed for the projection and discrete back-projection operations, which are the crucial parts of iterative reconstruction methods. The authors have investigated the relationship between the values of the blob parameters and the properties of images represented by the blobs. Experiments show that using blobs in iterative reconstruction methods leads to substantial improvement in the reconstruction performance, based on visual quality and on quantitative measures, in comparison with the voxel case. The images reconstructed using appropriately chosen blobs are characterized by less image noise for both noiseless data and noisy data, without loss of image resolution.

Matej S, Lewitt RM.

Incorporation of spherically-symmetric volume elements (blobs), instead of the conventional voxels, into iterative image reconstruction algorithms, has been found in our previous studies to lead to significant improvement in the quality of the reconstructed images. Furthermore, for three-dimensional (3D) positron emission tomography the 3D algebraic reconstruction technique using blobs can reach comparable or even better quality than the 3D filtered backprojection method after only one cycle through the projection data. The only shortcoming of the blob reconstruction method is an increased computational demand, because of the overlapping nature of the blobs. In our previous studies the blobs were placed on the same 3D simple cubic grid as used for voxel basis functions. For spherically-symmetric basis functions there are more advantageous arrangements of the 3D grid, enabling a more isotropic distribution of the spherical functions in the 3D space and a better packing efficiency of the image spectrum. Our studies confirmed that, when using the body centered cubic grid, the number of grid points can be effectively reduced, decreasing the computational and memory demands while preserving the quality of the reconstructed images.

Furuie SS, Herman GT, Narayan TK, Kinahan PE, Karp JS, Lewitt RM, Matej S.

Presents a practical methodology for evaluating 3D positron emission tomography (PET) reconstruction methods. It includes generation of random samples from a statistically described ensemble of 3D images resembling those to which PET would be applied in a medical situation, generation of corresponding projection data with noise and detector point spread function simulating those of a 3D PET scanner, assignment of figures of merit appropriate for the intended medical applications, optimization of the reconstruction algorithms on a training set of data, and statistical testing of the validity of hypotheses that say that two reconstruction algorithms perform equally well (from the point of view of a particular figure of merit) as compared to the alternative hypotheses that say that one of the algorithms outperforms the other. Although the methodology was developed with the 3D PET in mind, it can be used, with minor changes, for other 3D data collection methods, such as fully 3D CT or SPECT.

Lewitt RM, Muehllehner G, Karp JS.

A fast method is described for reconstructing volume images from three-dimensional (3D) coincidence data in positron emission tomography (PET). The reconstruction method makes use of all coincidence data acquired by high-sensitivity PET systems that do not have inter-slice absorbers (septa) to restrict the axial acceptance angle. The reconstruction method requires only a small amount of storage and computation, making it well suited for dynamic and whole-body studies. The method consists of three steps: (i) rebinning of coincidence data into a stack of 2D sinograms; (ii) slice-by-slice reconstruction of the sinogram associated with each slice to produce a preliminary 3D image having strong blurring in the axial (z) direction, but with different blurring at different z positions; and (iii) spatially variant filtering of the 3D image in the axial direction (i.e. 1D filtering in z for each x-y column) to produce the final image. The first step involves a new form of the rebinning operation in which multiple sinograms are incremented for each oblique coincidence line (multi-slice rebinning). The axial filtering step is formulated and implemented using the singular value decomposition. The method has been applied successfully to simulated data and to measured data for different kinds of phantom (multiple point sources, multiple discs, a cylinder with cold spheres, and a 3D brain phantom).

Karp JS, Muehllehner G, Lewitt RM.

A method is introduced to compensate for missing projection data that can result from gas between detectors or from malfunctioning detectors. This method uses constraints in the Fourier domain to estimate the missing data, thus completing the data set so that the filtered backprojection algorithm can be used to reconstruct artifact-free images. The image reconstructed from estimates using this technique and a data set with gaps is nearly indistinguishable from an image reconstructed from a complete data set without gaps, using a simulated brain phantom.

Daube-Witherspoon ME, Muehllehner G.

Improved axial spatial resolution in positron emission tomography (PET) scanners will lead to reduced sensitivity unless the axial acceptance angle for the coincidences is kept constant. A large acceptance angle, however, violates assumptions made in most reconstruction algorithms, which reconstruct parallel independent slices, rather than a three-dimensional volume. Two methods of treating the axial information from a volume PET scanner are presented. Qualitative and quantitative errors introduced by the approximations are examined for simulated objects with sharp boundaries and for a more anatomically realistic distribution with smooth activity gradients.

Daube-Witherspoon ME, Muehllehner G.

The trend in the design of scanners for positron emission computed tomography has traditionally been to improve the transverse spatial resolution to several millimeters while maintaining relatively coarse axial resolution (1-2 cm). Several scanners are being built with fine sampling in the axial as well as transverse directions, leading to the possibility of the true volume imaging. The number of possible coincidence pairs in these scanners is quite large. The usual methods of image reconstruction cannot handle these data without making approximations. It is computationally most efficient to reduce the size of this large, sparsely populated array by back-projecting the coincidence data prior to reconstruction. While analytic reconstruction techniques exist for back-projected data, an iterative algorithm may be necessary for those cases where the point spread function is spatially variant. A modification of the maximum likelihood algorithm is proposed to reconstruct these back-projected data. The method, the iterative image space reconstruction algorithm (ISRA), is able to reconstruct data from a scanner with a spatially variant point spread function in less time than other proposed algorithms. Results are presented for single-slice data, simulated and actual, from the PENN-PET scanner.

### Section imaging by computer calculation.

Muehllehner G, Wetzel RA.