Iterative image reconstruction using inverse fourier rebinning for fully 3-D PET

We describe a fast forward and back projector pair based on inverse Fourier rebinning for use in iterative image reconstruction for fully three-dimensional (3-D) positron emission tomography (PET). The projector pair is used as part of a factored system matrix that takes into account detector-pair response by using shift-variant sinogram blur kernels, thereby combining the computational advantages of Fourier rebinning with iterative reconstruction using accurate system models. The forward projector consists of a two-dimensional (2-D) projector, which maps 3-D images into 2-D direct sinograms, followed by exact inverse rebinning which maps the 2-D into fully 3-D sinograms. The back projector is implemented as the transpose of the forward projector and differs from the true exact rebinning operator in the sense that it does not require reprojection to compute missing line of responses (LORs). We compensate for two types of inaccuracies that arise in a cylindrical PET scanner when using inverse Fourier rebinning: 1) nonuniform radial sampling and 2) nonconstant oblique angles in the radial direction in a single oblique sinogram. We examine the effects of these corrections on sinogram accuracy and reconstructed image quality. We evaluate performance of the new projector pair for maximum a posteriori (MAP) reconstruction of simulated and in vivo data. The new projector results in only a small loss in resolution towards the edge of the field-of-view when compared to the fully 3-D geometric projector and requires an order of magnitude less computation.     

Iterative image reconstruction using inverse Fourier rebinning for fully 3-D PET

We describe a fast forward and back projector pair based on inverse Fourier rebinning for use in iterative image reconstruction for fully 3-D positron emission tomography (PET). The projector pair is used as part of a factored system matrix that takes into account detector-pair response by using shift-variant sinogram blur kernels, thereby combining the computational advantages of Fourier rebinning with iterative reconstruction using accurate system models. The forward projector consists of a 2-D projector, which maps 3-D images into 2-D direct sinograms, followed by exact inverse rebinning which maps the 2-D into fully 3-D sinograms. The back projector is implemented as the transpose of the forward projector and differs from the true exact rebinning operator in the sense that it does not require reprojection to compute missing lines of response (LORs). We compensate for two types of inaccuracies that arise in a cylindrical PET scanner when using inverse Fourier rebinning: 1) nonuniform radial sampling and 2) nonconstant oblique angles in the radial direction in a single oblique sinogram. We examine the effects of these corrections on sinogram accuracy and reconstructed image quality. We evaluate performance of the new projector pair for maximum a posteriori (MAP) reconstruction of simulated and in vivo data. The new projector results in only a small loss in resolution towards the edge of the field-of-view when compared to the fully 3-D geometric projector and requires an order of magnitude less computation.     

Fourier-based reconstruction for fully 3-D PET: optimization of interpolation parameters

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.     

Resolution and noise properties of MAP reconstruction for fully 3-D PET

We derive approximate analytical expressions for the local impulse response and covariance of images reconstructed from fully three-dimensional (3-D) positron emission tomography (PET) data using maximum a posteriori (MAP) estimation. These expressions explicitly account for the spatially variant detector response and sensitivity of a 3-D tomograph. The resulting spatially variant impulse response and covariance are computed using 3-D Fourier transforms. A truncated Gaussian distribution is used to account for the effect on the variance of the nonnegativity constraint used in MAP reconstruction. Using Monte Carlo simulations and phantom data from the microPET small animal scanner, we show that the approximations provide reasonably accurate estimates of contrast recovery and covariance of MAP reconstruction for priors with quadratic energy functions. We also describe how these analytical results can be used to achieve near-uniform contrast recovery throughout the reconstructed volume.     

Analytical calculation of volumes-of-intersection for iterative, fully 3-D PET reconstruction

Use of iterative algorithms to reconstruct three-dimensional (3-D) positron emission tomography (PET) data requires the computation of the system probability matrix. The pure geometrical contribution can easily be approximated by the length-of-intersection (LOI) between lines-of-response (LOR) and individual voxels. However, more accurate geometrical projectors are desirable. Therefore, we have developed a fast method for the analytical calculation of the 3-D shape and volume of volumes-of-intersection (VOI). This method provides an alternative robust projector with a uniformly continuous sampling of the image space. The enhanced calculation effort is facilitated by using several speedup techniques. Exploiting intrinsic symmetry relations and the sparseness of the system matrix allows to create an efficiently compressed matrix which can be precomputed and completely stored in memory. In addition, a new voxel addressing scheme has been implemented. This scheme avoids time-consuming symmetry transformations of voxel addresses by using an octant-wise symmetrically ordered field of voxels. The above methods have been applied for a fully 3-D, iterative reconstruction of 3-D sinograms recorded with a Siemens/CTI ECAT HR+ PET scanner. A comparison of the performance of the reconstruction using LOI weighting and VOI weighting is presented.     

Fast spatio-temporal image reconstruction for dynamic PET

In tomographic imaging, dynamic images are typically obtained by reconstructing the frames of a time sequence independently, one by one. A disadvantage of this frame-by-frame reconstruction approach is that it fails to account. For temporal correlations in the signal. Ideally, one should treat the entire image sequence as a single spatio-temporal signal. However, the resulting reconstruction task becomes computationally intensive. Fortunately, as the authors show in this paper, the spatio-temporal reconstruction problem call be greatly simplified by first applying a temporal Karhunen-Loeve (KL) transformation to the imaging equation. The authors show that if the regularization operator is chosen to be separable into space and time components, penalized weighted least squares reconstruction of the entire image sequence is approximately equivalent to frame-by-frame reconstruction in the space-KL domain. By this approach, spatio-temporal reconstruction can be achieved at reasonable computational cost. One can achieve further computational savings by discarding high-order KL components to avoid reconstructing them. Performance of the method is demonstrated through statistical evaluations of the bias-variance tradeoff obtained by computer simulation reconstruction     

Parallelization of the EM algorithm for 3-D PET image reconstruction

The EM algorithm for PET image reconstruction has two major drawbacks that have impeded the routine use of the EM algorithm: the long computation time due to slow convergence and a large memory required for the image, projection, and probability matrix. An attempt is made to solve these two problems by parallelizing the EM algorithm on multiprocessor systems. An efficient data and task partitioning scheme, called partition-by-box, based on the message passing model is proposed. The partition-by-box scheme and its modified version have been implemented on a message passing system, Intel iPSC/2, and a shared memory system, BBN Butterfly GP1000. The implementation results show that, for the partition-by-box scheme, a message passing system of complete binary tree interconnection with fixed connectivity of three at each node can have similar performance to that with the hypercube topology, which has a connectivity of log(2) N for N PEs. It is shown that the EM algorithm can be efficiently parallelized using the (modified) partition-by-box scheme with the message passing model.     

Iterative crystal efficiency calculation in fully 3-D PET

The calculation of the intrinsic efficiency of individual crystals is one of the steps needed to obtain accurate images of the radioisotope distribution in positron emission tomography (PET). These efficiencies can be computed by comparing the number of coincidence counts obtained when the crystals are equally illuminated by the same source. However, because the number of coincidence counts acquired for one crystal also depends on the efficiency of the other crystals in coincidence, most methods of crystal efficiency calculation need to assume that the influence of the other crystals is negligible. If there are large crystal efficiency variations, this approximation may lead to systematic errors. We have recently implemented an iterative method for a single ring of detectors that does not rely on this assumption. In this paper, we describe a fully three-dimensional (3-D) iterative method that better exploits the sensitivity of the tomograph and allows reduced acquisition times or the use of narrow energy windows. We compare the performance of the iterative method (single-ring and extended to fully 3-D) with noniterative techniques for different acquisition times of a uniform cylinder. Two different energy windows were used to assess the performance of each method with different levels of variations of crystal efficiency. The results showed that the iterative methods are more accurate when large efficiency variations exist and that only the fully 3-D methods provided good efficiency estimates with very low duration scans. We, thus, conclude that iterative fully 3-D methods provide the best estimations and can be used in a larger range of situations than can the other methods tested.     

3-D model localization using high-resolution reconstruction ofmonocular image sequences

In this paper, we present a complete system for the recognition and localization of a three-dimensional (3-D) model from a sequence of monocular images with known motion. The originality of this system is twofold. First, it uses a purely 3-D approach, starting from the 3-D reconstruction of the scene and ending by the 3-D matching of the model. Second, unlike most monocular systems, we do not use token tracking to match successive images. Rather, subpixel contour matching is used to recover more precisely complete 3-D contours. In contrast with the token tracking approaches, which yield a representation of the 3-D scene based on disconnected segments or points, this approach provides us with a denser and higher level representation of the scene. The reconstructed contours are fused along successive images using a simple result derived from the Kalman filter theory. The fusion process increases the localization precision and the robustness of the 3-D reconstruction. Finally, corners are extracted from the 3-D contours. They are used to generate hypotheses of the model position, using a hypothesize-and-verify algorithm that is described in detail. This algorithm yields a robust recognition and precise localization of the model in the scene. Results are presented on infrared image sequences with different resolutions, demonstrating the precision of the localization as well as the robustness and the low computational complexity of the algorithms.     

Interior-point methodology for 3-D PET reconstruction

Interior-point methods have been successfully applied to a wide variety of linear and nonlinear programming applications. This paper presents a class of algorithms, based on path-following interior-point methodology, for performing regularized maximum-likelihood (ML) reconstructions on three-dimensional (3-D) emission tomography data. The algorithms solve a sequence of subproblems that converge to the regularized maximum likelihood solution from the interior of the feasible region (the nonnegative orthant). We propose two methods, a primal method which updates only the primal image variables and a primal-dual method which simultaneously updates the primal variables and the Lagrange multipliers. A parallel implementation permits the interior-point methods to scale to very large reconstruction problems. Termination is based on well-defined convergence measures, namely, the Karush-Kuhn-Tucker first-order necessary conditions for optimality. We demonstrate the rapid convergence of the path-following interior-point methods using both data from a small animal scanner and Monte Carlo simulated data. The proposed methods can readily be applied to solve the regularized, weighted least squares reconstruction problem.     

Investigation of time-of-flight benefit for fully 3-D PET

The purpose of this paper is to determine the benefit that can be achieved in image quality for a time-of-flight (TOF) fully three-dimensional (3-D) whole-body positron emission tomography (PET) scanner. We simulate a 3-D whole-body time-of-flight PET scanner with a complete modeling of spatial and energy resolutions. The scanner is based on LaBr3 Anger-logic detectors with which 300ps timing resolution has been achieved. Multiple simulations were performed for 70-cm long uniform cylinders with 27-cm and 35-cm diameters, containing hot spheres (22, 17, 13, and 10-mm diameter) in a central slice and 10-mm diameter hot spheres in a slice at 1/4 axial FOV. Image reconstruction was performed with a list-mode iterative TOF algorithm and data were analyzed after attenuation and scatter corrections for timing resolutions of 300, 600, 1000 ps and non-TOF for varying count levels. The results show that contrast recovery improves slightly with TOF (NEMA NU2-2001 analysis), and improved timing resolution leads to a faster convergence to the maximum contrast value. Detectability for 10-mm diameter hot spheres estimated using a nonprewhitening matched filter (NPW SNR) also improves nonlinearly with TOF. The gain in image quality using contrast and noise measures is proportional to the object diameter and inversely proportional to the timing resolution of the scanner. The gains in NPW SNR are smaller, but they also increase with increasing object diameter and improved timing resolution. The results show that scan times can be reduced in a TOF scanner to achieve images similar to those from a non-TOF scanner, or improved image quality achieved for same scan times.     

Practical considerations for 3-D image reconstruction using spherically symmetric volume elements

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.     

Guest editorial: fully 3-D reconstruction of medical images

The 11 papers in this special issue focus on 3-D reconstruction of medical images. Two main classes of reconstructive algorithms are covered in this issue: analytic and iterative. The papers are briefly summarized here.     

Fast higher-order MR image reconstruction using singular-vector separation

Medical resonance imaging (MRI) conventionally relies on spatially linear gradient fields for image encoding. However, in practice various sources of nonlinear fields can perturb the encoding process and give rise to artifacts unless they are suitably addressed at the reconstruction level. Accounting for field perturbations that are neither linear in space nor constant over time, i.e., dynamic higher-order fields, is particularly challenging. It was previously shown to be feasible with conjugate-gradient iteration. However, so far this approach has been relatively slow due to the need to carry out explicit matrix-vector multiplications in each cycle. In this work, it is proposed to accelerate higher-order reconstruction by expanding the encoding matrix such that fast Fourier transform can be employed for more efficient matrix-vector computation. The underlying principle is to represent the perturbing terms as sums of separable functions of space and time. Compact representations with this property are found by singular-vector analysis of the perturbing matrix. Guidelines for balancing the accuracy and speed of the resulting algorithm are derived by error propagation analysis. The proposed technique is demonstrated for the case of higher-order field perturbations due to eddy currents caused by diffusion weighting. In this example, image reconstruction was accelerated by two orders of magnitude.     

Ultra fast symmetry and SIMD-based projection-backprojection (SSP) algorithm for 3-D PET image reconstruction

Remarkable progress in positron emission tomography (PET) development has occurred in recent years, in hardware, software, and computer implementation of image reconstruction. Recent development in PET scanners such as the high-resolution research tomograph (HRRT) developed by CTI (now Siemens) represents such a case and is capable of greatly enhanced resolution as well as sensitivity. In these PET scanners, the amount of coincidence line data collected contains more than 4.5 x 10(9) coincidence lines of response generated by as many nuclear detectors as 120 000. This formidable amount of data and the reconstruction of this data set pose a real problem in HRRT and have also been of the major bottle neck in further developments of high resolution PET scanners as well as their applications. In these classes of PET scanners, therefore, obtaining one set of reconstructed images often requires many hours of image reconstruction. For example, in HRRT with full data collection in a normal brain scan (using SPAN 3), the image reconstruction time is close to 80 min, making it practically impossible to attempt any list-mode-based dynamic imaging since the image reconstruction time would take many days (as much as 43 h or more for 32-frame dynamic image reconstruction). To remedy this data-handling problem in image reconstruction, we developed a new algorithm based on the symmetry properties of the projection and backprojection processes, especially in the 3-D OSEM algorithm where multiples of projection and back-projection are required. In addition, the single-instruction multiple-data (SIMD) technique also allowed us to successfully incorporate the symmetry properties mentioned above, thereby effectively reducing the total image reconstruction time to a few minutes. We refer to this technique as the symmetry and SIMD-based projection-backprojection (SSP) technique or algorithm and the details of the technique will be discussed and an example of the application of the technique to the HRRT's OSEM algorithm will be presented as a demonstration.     

A fast fully 4-D incremental gradient reconstruction algorithm for list mode PET data

We describe a fast and globally convergent fully four-dimensional incremental gradient (4DIG) algorithm to estimate the continuous-time tracer density from list mode positron emission tomography (PET) data. Detection of 511-keV photon pairs produced by positron-electron annihilation is modeled as an inhomogeneous Poisson process whose rate function is parameterized using cubic B-splines. The rate functions are estimated by minimizing the cost function formed by the sum of the negative log-likelihood of arrival times, spatial and temporal roughness penalties, and a negativity penalty. We first derive a computable bound for the norm of the optimal temporal basis function coefficients. Based on this bound we then construct and prove convergence of an incremental gradient algorithm. Fully 4-D simulations demonstrate the substantially faster convergence behavior of the 4DIG algorithm relative to preconditioned conjugate gradient. Four-dimensional reconstructions of real data are also included to illustrate the performance of this method.     

Fast image reconstruction network in image stitching

Compared with the traditional feature-based image stitching algorithm, the free-view image stitching algorithm based on deep learning has the advantages of fast stitching speed and good effect. However, these algorithms still cannot achieve real-time splicing speed. For the image reconstruction stage, we redesign a new fast image reconstruction network. This network is designed based on ShuffleNet, and the new network structure and loss function will reduce the time required for image reconstruction. In addition, this network can also reduce the performance loss after the network is lightweight. It is proved by experiments that the fast image reconstruction network can realize real-time high-resolution free-view image reconstruction.     

Microwave image reconstruction from 3-D fields coupled to 2-D parameter estimation

An efficient Gauss-Newton iterative imaging technique utilizing a three-dimensional (3-D) field solution coupled to a two-dimensional (2-D) parameter estimation scheme (3-D/2-D) is presented for microwave tomographic imaging in medical applications. While electromagnetic wave propagation is described fully by a 3-D vector field, a 3-D scalar model has been applied to improve the efficiency of the iterative reconstruction process with apparently limited reduction in accuracy. In addition, the image recovery has been restricted to 2-D but is generalizable to three dimensions. Image artifacts related primarily to 3-D effects are reduced when compared with results from an entirely two-dimensional inversion (2-D/2-D). Important advances in terms of improving algorithmic efficiency include use of a block solver for computing the field solutions and application of the dual mesh scheme and adjoint approach for Jacobian construction. Methods which enhance the image quality such as the log-magnitude/unwrapped phase minimization were also applied. Results obtained from synthetic measurement data show that the new 3-D/2-D algorithm consistently outperforms its 2-D/2-D counterpart in terms of reducing the effective imaging slice thickness in both permittivity and conductivity images over a range of inclusion sizes and background medium contrasts.     

Efficient fully 3-D iterative SPECT reconstruction with Monte Carlo-based scatter compensation

Quantitative accuracy of single photon emission computed tomography (SPECT) images is highly dependent on the photon scatter model used for image reconstruction. Monte Carlo simulation (MCS) is the most general method for detailed modeling of scatter, but to date, fully three-dimensional (3-D) MCS-based statistical SPECT reconstruction approaches have not been realized, due to prohibitively long computation times and excessive computer memory requirements. MCS-based reconstruction has previously been restricted to two-dimensional approaches that are vastly inferior to fully 3-D reconstruction. Instead of MCS, scatter calculations based on simplified but less accurate models are sometimes incorporated in fully 3-D SPECT reconstruction algorithms. We developed a computationally efficient fully 3-D MCS-based reconstruction architecture by combining the following methods: 1) a dual matrix ordered subset (DM-OS) reconstruction algorithm to accelerate the reconstruction and avoid massive transition matrix precalculation and storage; 2) a stochastic photon transport calculation in MCS is combined with an analytic detector modeling step to reduce noise in the Monte Carlo (MC)-based reprojection after only a small number of photon histories have been tracked; and 3) the number of photon histories simulated is reduced by an order of magnitude in early iterations, or photon histories calculated in an early iteration are reused. For a 64 x 64 x 64 image array, the reconstruction time required for ten DM-OS iterations is approximately 30 min on a dual processor (AMD 1.4 GHz) PC, in which case the stochastic nature of MCS modeling is found to have a negligible effect on noise in reconstructions. Since MCS can calculate photon transport for any clinically used photon energy and patient attenuation distribution, the proposed methodology is expected to be useful for obtaining highly accurate quantitative SPECT images within clinically acceptable computation times.