System models for PET statistical iterative reconstruction: A review.

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.

2.5-D simultaneous multislice reconstruction by series expansion methods from Fourier-rebinned PET data.

True three-dimensional (3-D) volume reconstruction from fully 3-D data in positron emission tomography (PET) has only a limited clinical use because of its large computational burden. Fourier rebinning (FORE) of the fully 3-D data into a set of 2-D sinogram data decomposes the 3-D reconstruction process into multiple 2-D reconstructions of decoupled 2-D image slices, thus substantially decreasing the computational burden even in the case when the 2-D reconstructions are performed by an iterative reconstruction algorithm. On the other hand, the approximations involved in the rebinning combined with the decoupling of the image slices cause a certain reduction of image quality, especially when the signal-to-noise ratio of the data is low. We propose a 2.5-D Simultaneous Multislice Reconstruction approach, based on the series expansion principle, where the volume is represented by the superposition of 3-D spherically symmetric bell-shaped basis functions. It takes advantage of the time reduction due to the use of the FORE (2-D) data, instead of the original fully 3-D data, but at the same time uses a 3-D iterative reconstruction approach with 3-D basis functions. The same general approach can be applied to any reconstruction algorithm belonging to the class of series expansion methods (iterative or noniterative) using 3-D basis functions that span multiple slices, and can be used for any multislice sinogram or list mode data whether obtained by a special rebinning scheme or acquired directly by a PET scanner in the 2-D mode using septa. Our studies confirm that the proposed 2.5-D approach provides a considerable improvement in reconstruction quality, as compared to the standard 2-D reconstruction approach, while the reconstruction time is of the same order as that of the 2-D approach and is clinically practical even on a general-purpose computer.

Performance of the Fourier rebinning algorithm for PET with large acceptance angles.

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.

Three-dimensional imaging characteristics of the HEAD PENN-PET scanner.

UNLABELLED: A volume-imaging PET scanner, without interplane septa, for brain imaging has been designed and built to achieve high performance, specifically in spatial resolution and sensitivity. The scanner is unique in its use of a single annular crystal of Nal(Tl), which allows a field of view (FOV) of 25.6 cm in both the transverse and axial directions. Data are reconstructed into an image matrix of 128(3) with (2 mm)3 voxels, using three-dimensional image reconstruction algorithms. METHODS: Point-source measurements are performed to determine spatial resolution over the scanner FOV, and cylindrical phantom distributions are used to determine the sensitivity, scatter fraction and counting rate performance of the system. A three-dimensional brain phantom and 18F-FDG patient studies are used to evaluate image quality with three-dimensional reconstruction algorithms. RESULTS: The system spatial resolution is measured to be 3.5 mm in both the transverse and axial directions, in the center of the FOV. The true sensitivity, using the standard NEMA phantom (6 liter), is 660 kcps/microCi/ml, after subtracting a scatter fraction of 34%. Due to deadtime effects, we measure a peak true counting rate, after scatter and randoms subtraction, of 100 kcps at 0.7 mCi for a smaller brain-sized (1.1 liter) phantom, and 70 kcps for a head-sized (2.5 liter) phantom at the same activity. A typical 18F-FDG clinical brain study requires only 2 mCi to achieve high statistics (100 million true events) with a scan time of 30 min. CONCLUSION: The HEAD PENN-PET scanner is based on a cost-effective design using Nal(Tl) and has been shown to achieve high performance for brain studies and pediatric whole-body studies. As a full-time three-dimensional imaging scanner with a very large axial acceptance angle, high sensitivity is achieved. The system becomes counting-rate limited as the activity is increased, but we achieve high image quality with a small injected dose. This is a significant advantage for clinical imaging, particularly for pediatric patients.

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.

Fourier correction for spatially variant collimator blurring in SPECT.

In single-photon emission computed tomography (SPECT), projection data are acquired by rotating the photon detector around a patient, either in a circular orbit or in a noncircular orbit. The projection data of the desired spatial distribution of emission activity is blurred by the point-response function of the collimator that is used to define the range of directions of gamma-ray photons reaching the detector. The point-response function of the collimator is not spatially stationary, but depends on the distance from the collimator to the point. Conventional methods for deblurring collimator projection data are based on approximating the actual distance-dependent point-response function by a spatially invariant blurring function, so that deconvolution methods can be applied independently to the data at each angle of view. A method is described here for distance-dependent preprocessing of SPECT projection data prior to image reconstruction. Based on the special distance-dependent characteristics of the Fourier coefficients of the sinogram, a spatially variant inverse filter can be developed to process the projection data in all views simultaneously. The algorithm is first derived from Fourier analysis of the projection data from the circular orbit geometry. For circular orbit projection data, experimental results from both simulated data and real phantom data indicate the potential of this method. It is shown that the spatial filtering method can be extended to the projection data from the noncircular orbit geometry. Experiments on simulated projection data from an elliptical orbit demonstrate correction of the spatially variant blurring and distortion in the reconstructed image caused by the noncircular orbit geometry.

A methodology for testing for statistically significant differences between fully 3D PET reconstruction algorithms.

We present a practical methodology for evaluating 3D 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 cr or SPECT.

Evaluation of task-oriented performance of several fully 3D PET reconstruction algorithms.

The relative performance of five fully 3D PET reconstruction algorithms is evaluated. The algorithms are a filtered backprojection (FBP) method and two variants each of the EM-ML and ART iterative methods. For each of the iterative methods, one variant makes use of voxels and the other makes use of 'blobs' (spherically symmetric functions smoothly decaying to zero at their boundaries) as basis functions in its discrete reconstruction model. The methods are evaluated from the point of view of the efficacy of the reconstructions produced by them for three typical medical tasks--estimation of the average activity inside specific regions of interest, detection of hot spots, and detection of cold spots. A free parameter is allowed in the description of each of the five algorithms; the parameters are determined by a training process during which a value of the free parameter is selected which (nearly) maximizes a technical figure of merit. Such training and the actual comparative evaluation is done by making use of randomly generated phantoms and their projection data. The methodology allows assignation of levels of statistical significance to claims of the relative superiority of one algorithm over another for a particular task. We find that using blobs as basis functions in the iterative algorithms is definitely advantageous over using voxels. This result has high statistical significance. (We also include a visual illustration of it.) Comparing FBP, EM-ML using blobs, and ART using blobs, we do not find a clear difference in the overall performance of the investigated variants of the methods. If anything, our results suggest that ART using blobs may be the most efficacious of the three.

Alternatives to voxels for image representation in iterative reconstruction algorithms.

Spherically symmetric volume elements are alternatives to the more conventional voxels for the construction of volume images in the computer. The image representation, and the calculation of projections of it, are essential components of iterative algorithms for image reconstruction from projection data. A two-parameter family of spherical volume elements is described that allows control of the smoothness properties of the represented image, whereas conventional voxels are discontinuous. The rotational symmetry of the spherical elements leads to efficient calculation of projections of the represented image, as required in iterative reconstruction algorithms. For volume elements whose shape is ellipsoidal (rather than spherical) it is shown that efficient calculation of the projections is also possible by means of an image space transformation.

Multidimensional digital image representations using generalized Kaiser-Bessel window functions.

Inverse problems that require the solution of integral equations are inherent in a number of indirect imaging applications, such as computerized tomography. Numerical solutions based on discretization of the mathematical model of the imaging process, or on discretization of analytic formulas for iterative inversion of the integral equations, require a discrete representation of an underlying continuous image. This paper describes discrete image representations, in n-dimensional space, that are constructed by the superposition of shifted copies of a rotationally symmetric basis function. The basis function is constructed using a generalization of the Kaiser-Bessel window function of digital signal processing. The generalization of the window function involves going from one dimension to a rotationally symmetric function in n dimensions and going from the zero-order modified Bessel function of the standard window to a function involving the modified Bessel function of order m. Three methods are given for the construction, in n-dimensional space, of basis functions having a specified (finite) number of continuous derivatives, and formulas are derived for the Fourier transform, the x-ray transform, the gradient, and the Laplacian of these basis functions. Properties of the new image representations using these basis functions are discussed, primarily in the context of two-dimensional and three-dimensional image reconstruction from line-integral data by iterative inversion of the x-ray transform. Potential applications to three-dimensional image display are also mentioned.

Constrained Fourier space method for compensation of missing data in emission computed tomography.

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.

Accelerated iterative reconstruction for positron emission tomography based on the em algorithm for maximum likelihood estimation.

The EM method that was originally developed for maximum likelihood estimation in the context of mathematical statistics may be applied to a stochastic model of positron emission tomography (PET). The result is an iterative algorithm for image reconstruction that is finding increasing use in PET, due to its attractive theoretical and practical properties. Its major disadvantage is the large amount of computation that is often required, due to the algorithm's slow rate of convergence. This paper presents an accelerated form of the EM algorithm for PET in which the changes to the image, as calculated by the standard algorithm, are multiplied at each iteration by an overrelaxation parameter. The accelerated algorithm retains two of the important practical properties of the standard algorithm, namely the selfnormalization and nonnegativity of the reconstructed images. Experimental results are presented using measured data obtained from a hexagonal detector system for PET. The likelihood function and the norm of the data residual were monitored during the iterative process. According to both of these measures, the images reconstructed at iterations 7 and 11 of the accelerated algorithm are similar to those at iterations 15 and 30 of the standard algorithm, for two different sets of data. Important theoretical properties remain to be investigated, namely the convergence of the accelerated algorithm and its performance as a maximum likelihood estimator.

Reconstruction from Sparsely Sampled Data by ART with Interpolated Rays.

After a brief discussion of the algebraic reconstruction techniques (ART), we introduce the attenuation problem in positron emission tomography (PET). We anticipate that a generalization of ART, the so-called cyclic subgradient projection (CSP) method, may be useful for solving this problem. This, however, has not been successfully realized, due to the fact that data collected by our proposed stationary PET detector ring are too sparsely sampled. That this is, in fact, a major problem is demonstrated by showing that ordinary ART produces reconstructions with unacceptably strong artifacts even on perfect (no attenuation) data collected according to the PET geometry. We demonstrate that the source of this artifact is the sparse sampling, and we propose the use of interpolated rays to overcome the problem. This approach is successful, as is illustrated by showing reconstructions from sparsely sampled data by ART with interpolated rays.

Evaluation of a preprocessing algorithm for truncated CT projections.

A preprocessing algorithm for completing truncated computed tomography (CT) projection data (so that reconstruction methods suitable for complete projection data can be applied) is experimentally investigated. The study is motivated primarily by the problem of patients who are too obese to fit into the fan beam of a rotate-only (third generation) scanner, and secondarily by dose reduction considerations. Four sets of patient data collected by a rotate-only scanner are used, and the reconstructions from the complete and truncated projection data are compared. It is illustrated that the preprocessing algorithm leads to qualitatively good images, since local variations are nearly identical in the reconstruction from the complete and from the truncated data. Quantitatively, the results are less encouraging; the exact values of CT numbers differ in the two reconstructions by a significant (although slowly varying) amount across the two reconstructed images.

Demonstration of a software package for the reconstruction of the dynamically changing structure of the human heart from cone beam x-ray projections.

The Dynamic Spatial Reconstructor (DSR) is a device constructed at the Biodynamics Research Unit of the Mayo Clinic for (among other things) the visualization of the beating heart inside the intact thorax. The device consists of 28 rotating X-ray sources arranged on a circular arc at 6 degrees intervals (total span 162 degrees) and a matching set of 28 imaging systems. The whole thorax of the patient is projected onto the two-dimensional screen of the imaging systems by cone beams of X rays from the sources. All of the X-ray sources are switched on and off within a total period of 10 milliseconds. The Medical Image Processing Group at the State University of New York at Buffalo has developed a software package for the design and evaluation of algorithms to be used by the DSR. In this paper we illustrate the operation of the package and a particular algorithm for the reconstruction of the dynamically changing structure of the heart from data collected by the DSR.

Processing of incomplete measurement data in computed tomography.

Conventional approaches to computed tomography involve scanning the entire cross section and producing an image whose spatial and density resolution is uniform over its entire area. If the extent of each scan is restricted to the width of the lesion being investigated, then the x-ray dose is reduced, but a set of incomplete "truncated" projections is measured. Conversely, projections are "hollow" when their inner parts cannot be measured, e.g., when there is a metallic object within the body cross section. We present procedures for preprocessing incomplete projections so that images can be reconstructed from them using the convolution/back projection method.

Computed tomography with fan beam geometry.

Data collection for computed tomography (CT), using a fan beam of radiation, is considered in detail. The manner in which the required projection data set is built up as scanning proceeds is demonstrated. It is shown that by offsetting the fan beam detector from its symmetrical position by a fraction of the element spacing, and by making measurements over 360 degrees, the spatial resolution of the detector may be increased by up to a factor of two. Results of simulations relating to an actual CT device under construction are presented.

A possible conformation for double-stranded polynucleotides.

A model is presented for double-stranded polynucleotides which involves side-by-side meshing of the two strands rather than double helical intertwining. The sugar-phosphate backbone has a twisted strip-like character, yet base-pairing of the Watson-Crick type is still possible. Structural features of the basic model are described and coordinates are presented for a representative example. The structure has, on the whole, reasonable sterochemical contacts, and can be shown to produce a fiber diffraction pattern with x-rays not unlike that of the B form of DNA.