We review positron-emission tomography (PET), which has inherent advantages that avoid the shortcomings of other nuclear medicine imaging methods. PET image reconstruction methods with origins in signal and image processing are discussed, including the potential problems of these methods. A summary of statistical image reconstruction methods, which can yield improved image quality, is also presented.

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Positron-emission tomography

The authors present the results of testing an algorithm for three-dimensional image reconstruction that uses all gamma-ray coincidence events detected by a PET (positron emission tomography) volume imaging scanner. By using two iterations of an analytic filter-backprojection method, the algorithm is not constrained by the requirement of a spatially invariant detector point spread function, which limits normal analytic techniques. Removing this constraint allows the incorporation of all detected events, regardless of orientation, which improves the statistical quality of the final reconstructed image. >

Analytic 3D image reconstruction using all detected events

This paper presents two new rebinning algorithms for the reconstruction of three-dimensional (3-D) positron emission tomography (PET) data. A rebinning algorithm is one that first sorts the 3-D data into an ordinary two-dimensional (2-D) data set containing one sinogram for each transaxial slice to be reconstructed; the 3-D image is then recovered by applying to each slice a 2-D reconstruction method such as filtered-backprojection. This approach allows a significant speedup of 3-D reconstruction, which is particularly useful for applications involving dynamic acquisitions or whole-body imaging. The first new algorithm is obtained by discretizing an exact analytical inversion formula. The second algorithm, called the Fourier rebinning algorithm (FORE), is approximate but allows an efficient implementation based on taking 2-D Fourier transforms of the data. This second algorithm was implemented and applied to data acquired with the new generation of PET systems and also to simulated data for a scanner with an 18/spl deg/ axial aperture. The reconstructed images were compared to those obtained with the 3-D reprojection algorithm (3DRP) which is the standard "exact" 3-D filtered-backprojection method. Results demonstrate that FORE provides a reliable alternative to 3DRP, while at the same time achieving an order of magnitude reduction in processing time.

Exact and approximate rebinning algorithms for 3-D PET data

The recent introduction of high-resolution molecular imaging technology is considered by many experts as a major breakthrough that will potentially lead to a revolutionary paradigm shift in health care and revolutionize clinical practice. This paper intends to balance the capabilities of the two major molecular imaging modalities used in nuclear medicine, namely positron emission tomography (PET) and single photon emission computed tomography (SPECT). The motivations are many-fold: (1) to gain a better understanding of the strengths and limitations of the two imaging modalities in the context of recent and ongoing developments in hardware and software design; (2) to emphasize that certain issues, historically and commonly thought as limitations of one technology, may now instead be viewed as challenges that can be addressed; (3) to point out that current state of the art PET and SPECT scanners can (greatly) benefit from improvements in innovative image reconstruction algorithms; and (4) to identify important areas of research in PET and SPECT imaging that will be instrumental to further improvements in the two modalities. Both technologies are poised to advance molecular imaging and have a direct impact on clinical and research practice to influence the future of molecular medicine.

PET versus SPECT: strengths, limitations and challenges

A method is presented that directly calculates the mean number of scattered coincidences in data acquired with fully 3D positron emission tomography (PET). This method uses a transmission scan, an emission scan, the physics of Compton scatter, and a mathematical model of the scanner in a forward calculation of the number of events for which one photon has undergone a single Compton interaction. The distribution of events for which multiple Compton interactions have occurred is modelled as a linear transformation of the single-scatter distribution. Computational efficiency is achieved by sampling at rates no higher than those required by the scatter distribution and by implementing the algorithm using look-up tables. Evaluation studies in phantoms with large scatter fractions show that the method yields images with quantitative accuracy equivalent to that of slice-collimated PET in clinically useful times.

https://iopscience.iop.org/article/10.1088/0031-9155/41/1/012/pdf

Model-based scatter correction for fully 3D PET

An exact inversion formula written in the form of shift-variant filtered-backprojection (FBP) is given for reconstruction from cone-beam data taken from any orbit satisfying H.K. Tuy's (1983) sufficiency conditions. The method is based on a result of P. Grangeat (1987), involving the derivative of the three-dimensional (3D) Radon transform, but unlike Grangeat's algorithm, no 3D rebinning step is required. Data redundancy, which occurs when several cone-beam projections supply the same values in the Radon domain, is handled using an elegant weighting function and without discarding data. The algorithm is expressed in a convenient cone-beam detector reference frame, and a specific example for the case of a dual orthogonal circular orbit is presented. When the method is applied to a single circular orbit (even though Tuy's condition is not satisfied), it is shown to be equivalent to the well-known algorithm of L.A. Feldkamp et al. (1984). >

A cone-beam reconstruction algorithm using shift-variant filtering and cone-beam backprojection

The current trend in attenuation correction for single photon emission computed tomography (SPECT) is to measure and reconstruct the attenuation coefficient map using a transmission scan, performed either sequentially or simultaneously with the emission scan. This approach requires dedicated hardware and increases the cost (and in some cases the scanning time) required to produce a clinical SPECT image. Furthermore, if short focal-length fan-beam collimators are used for transmission imaging, the projection data may be truncated, leading to errors in the attenuation coefficient map. Our goal is to obtain information about the attenuation distribution from only the measured emission data by exploiting the fact that only certain attenuation distributions are consistent with a given emission dataset. Ultimately this consistency information will either be used directly to compensate for attenuation or combined with the incomplete information from fan-beam transmission measurements to produce a more accurate attenuation coefficient map. In this manuscript the consistency conditions (which relate the measured SPECT data to the sinogram of the attenuation distribution) are used to find the uniform elliptical attenuation object which is most consistent with the measured emission data. This object is then used for attenuation correction during the reconstruction of the emission data. The method is tested using both simulated and experimentally acquired data from uniformly and nonuniformly attenuating objects. The results show that, for uniform elliptical attenuators, the consistency conditions of the SPECT data can be used to produce an accurate estimate of the attenuation map without performing any transmission measurements. The results also show that, in certain circumstances, the consistency conditions can prove useful for attenuation compensation with nonuniform attenuators.

Toward accurate attenuation correction in SPECT without transmission measurements

A number of authors have studied the problems associated with full three-dimensional reconstruction, especially in the case of positron tomography where three-dimensional reconstruction is likely to offer the greatest benefits. While most approaches follow that of filtered backprojection, the relationship between the various filters that have been proposed is far from evident. The authors clarify this relationship by analysing and generalising the different classes of published filters and establish the properties and characteristics of a general solution to the three-dimensional reconstruction problem. Some guidelines are suggested for the choice of an appropriate filter in a given situation.

https://iopscience.iop.org/article/10.1088/0031-9155/34/5/002/pdf

Three-dimensional image reconstruction from complete projections.

Discusses two approaches towards improving the sensitivity and enlarging the useful field of view for a rotating positron camera by removing the constraint on phi . Weighted backprojection appears superior in that the camera response is more uniform and the images are less noisy than for the post-reconstruction scaling method. Fourier deconvolution reconstruction may be retained since the response function of the overall system is still shift invariant. In a typical imaging situation, the increase in sensitivity may be as much as a factor of 2.5, which represents a significant improvement when considering the poor statistics of nuclear medicine images. As a consequence of decoupling the sensitivity from the diameter of the field of view, transverse sections through internal organs with dimensions approaching those of the detectors may therefore be accurately imaged with good sensitivity. The weighted backprojection method is now applied in routine clinical imaging (Frey et al., 1984).

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Increased sensitivity and field of view for a rotating positron camera

Full three-dimensional scanning allows a significant improvement in image quality in X-ray transmission computerized tomography (CT), in single-photon emission computerized tomography (SPECT) and in positron emission tomography (PET). Increased detection efficiency is obtained by increasing the solid angle seen by the detectors and by detecting photons which are no longer confined to a set of parallel slices as in the standard 2D scanning mode. Consequently, 3D image reconstruction cannot be factored as usual into a set of independent 2D reconstructions, and hence one has to invert the 3D X-ray transform with limited data. Assuming a basic knowledge of standard 2D tomography, this paper presents a review of analytic methods for the reconstruction of a 3D image from a set of 2D parallel projections along some limited set of directions. This inverse problem is overdetermined, i.e., the projection data are redundant, and the consequences of this property are analysed. Redundancy is used to generate classes of exact filters for 3D filtered-backprojection, thereby allowing considerable versatility in the design of inversion algorithms tailored to specific applications. The review also covers the inversion of the 3D X-ray transform when the 2D parallel projections are incompletely measured (truncated), a situation which arises for example in PET, In view of data redundancy, it is possible to build convolution kernels for filtered-backprojection which have a limited support and are not, therefore, affected by truncation. A similar analysis and utilization of data redundancy could be proposed in any application where, instead of trying to define the smallest data set from which the problem can be solved, one attempts to optimize the signal-to-noise ratio by measuring and by incorporating into the reconstruction as much data as practically feasible.

https://iopscience.iop.org/article/10.1088/0266-5611/11/2/001/pdf

R. Clack

Papers

A cone-beam reconstruction algorithm using shift-variant filtering and cone-beam backprojection

Toward accurate attenuation correction in SPECT without transmission measurements

Three-dimensional image reconstruction from complete projections.

Increased sensitivity and field of view for a rotating positron camera

Image reconstruction from truncated, two-dimensional, parallel projections