Non-Gaussian space-variant resolution modelling for list-mode reconstruction

Partial volume effect is an important source of bias in PET images that can be lowered by accounting for the point spread function (PSF) of the scanner. We measured such a PSF in various points of a clinical PET scanner and modelled it as a product of matrices acting in image space, taking the asymmetrical, shift-varying and non-Gaussian character of the PSF into account (AMP modelling), and we integrated this accurate image space modelling into a conventional list-mode OSEM algorithm (EM-AMP reconstruction). We showed on the one hand that when a sufficiently high number of iterations are considered, the AMP modelling lead to better recovery coefficients at reduced background noise compared to reconstruction where no or only partial resolution modelling is performed, and on the other hand that for a small number of iterations, a Gaussian modelling gave the best recovery coefficients. Moreover, we have demonstrated that a deconvolution based on the AMP system response model leads to the same recovery coefficients as the corresponding EM-AMP reconstruction, but at the expense of an increased background noise.

Resolution improvement and noise reduction in human pinhole SPECT using a multi-ray approach and the SHINE method

PURPOSE

This work aimed at quantifying the gains in spatial resolution and noise that could be achieved when using resolution modelling based on a multi-ray approach and additionally the Statistical and Heuristic Noise Extraction (SHINE) method in human pinhole single photon emission tomography (PH-SPECT).

METHODS

PH-SPECT of two line phantoms and one homogeneous cylinder were recorded using parameters suited for studies of the human neck area. They were reconstructed using pinhole dedicated ordered subsets expectation maximisation algorithm including a resolution recovery technique based on 7 or 21 rays. Optionally, the SPECT data were SHINE pre-processed. Transverse and axial full widths at half-maximum (FWHM) were obtained from the line phantoms. The noise was quantified using the coefficient of variation (COV) derived from the uniform phantom. Two human PH-SPECT studies of the thyroid (a hot nodule and a very low uptake) were processed with the same algorithms.

RESULTS

Depending on the number of iterations, FWHM decreased by 30 to 50% when using the multi-ray approach in the reconstruction process. The SHINE method did not affect the resolution but decreased the COV by at least 20% and by 45% when combined with the multi-ray method. The two human studies illustrated the gain in spatial resolution and the decrease in noise afforded both by the multi-ray reconstruction and the SHINE method.

CONCLUSION

Iterative reconstruction with resolution modelling allows to obtain high resolution human PH-SPECT studies with reduced noise content. The SHINE method affords an additional noise reduction without compromising the resolution.

SOLVING THE INTERIOR PROBLEM OF COMPUTED TOMOGRAPHY USING A PRIORI KNOWLEDGE

The case of incomplete tomographic data for a compactly supported attenuation function is studied. When the attenuation function is a priori known in a subregion, we show that a reduced set of measurements is enough to uniquely determine the attenuation function over all the space. Furthermore, we found stability estimates showing that reconstruction can be stable near the region where the attenuation is known. These estimates also suggest that reconstruction stability collapses quickly when approaching the set of points that are viewed under less than 180 degrees. This paper may be seen as a continuation of the work "Truncated Hilbert transform and Image reconstruction from limited tomographic data" that was published in Inverse Problems in 2006. This continuation tackles new cases of incomplete data that could be of interest in applications of computed tomography.

Single and multipinhole collimator design evaluation method for small animal SPECT

High-resolution functional imaging of small animals is often obtained by single pinhole SPECT with circular orbit acquisition. Multipinhole SPECT adds information due to its improved sampling, and can improve the trade-off between resolution and sensitivity. To evaluate different pinhole collimator designs an efficient method is needed that quantifies the reconstruction image quality. In this paper, we propose a fast, approximate method that examines the quality of individual voxels of a postsmoothed maximum likelihood expectation maximization (MLEM) reconstruction by studying their linearized local impulse response (LLIR) and (co)variance for a predefined target resolution. For validation, the contrast-to-noise ratios (CNRs) in some voxels of a homogeneous sphere and of a realistic rat brain software phantom were calculated for many single and multipinhole designs. A good agreement was observed between the CNRs obtained with the approximate method and those obtained with postsmoothed MLEM reconstructions of simulated noisy projections. This good agreement was quantified by a least squares fit through these results, which yielded a line with slope 1.02 (1.00 expected) and a y-intercept close to zero (0 expected). 95.4% of the validation points lie within three standard deviations from that line. Using the approximate method, the influence on the CNR of varying a parameter in realistic single and multipinhole designs was examined. The investigated parameters were the aperture diameter, the distance between the apertures and the axis-of-rotation, the focal distance, the acceptance angle, the position of the apertures, the focusing distance, and the number of pinholes. The results can generally be explained by the change in sensitivity, the amount of postsmoothing, and the amount of overlap in the projections. The method was applied to multipinhole designs with apertures focusing at a single point, but is also applicable to other designs.

Penalized maximum-likelihood sinogram restoration for dual focal spot computed tomography

Due to various system non-idealities, the raw data generated by a computed tomography (CT) machine are not readily usable for reconstruction. Although the deterministic nature of corruption effects such as crosstalk and afterglow permits correction by deconvolution, there is a drawback because deconvolution usually amplifies noise. Methods that perform raw data correction combined with noise suppression are commonly termed sinogram restoration methods. The need for sinogram restoration arises, for example, when photon counts are low and non-statistical reconstruction algorithms such as filtered backprojection are used. Many modern CT machines offer a dual focal spot (DFS) mode, which serves the goal of increased radial sampling by alternating the focal spot between two positions on the anode plate during the scan. Although the focal spot mode does not play a role with respect to how the data are affected by the above-mentioned corruption effects, it needs to be taken into account if regularized sinogram restoration is to be applied to the data. This work points out the subtle difference in processing that sinogram restoration for DFS requires, how it is correctly employed within the penalized maximum-likelihood sinogram restoration algorithm and what impact it has on image quality.

Fast fully 3-D image reconstruction in PET using planograms

We present a method of performing fast and accurate three-dimensional (3-D) backprojection using only Fourier transform operations for line-integral data acquired by planar detector arrays in positron emission tomography. This approach is a 3-D extension of the two-dimensional (2-D) linogram technique of Edholm. By using a special choice of parameters to index a line of response (LOR) for a pair of planar detectors, rather than the conventional parameters used to index a LOR for a circular tomograph, all the LORs passing through a point in the field of view (FOV) lie on a 2-D plane in the four-dimensional (4-D) data space. Thus, backprojection of all the LORs passing through a point in the FOV corresponds to integration of a 2-D plane through the 4-D "planogram." The key step is that the integration along a set of parallel 2-D planes through the planogram, that is, backprojection of a plane of points, can be replaced by a 2-D section through the origin of the 4-D Fourier transform of the data. Backprojection can be performed as a sequence of Fourier transform operations, for faster implementation. In addition, we derive the central-section theorem for planogram format data, and also derive a reconstruction filter for both backprojection-filtering and filtered-backprojection reconstruction algorithms. With software-based Fourier transform calculations we provide preliminary comparisons of planogram backprojection to standard 3-D backprojection and demonstrate a reduction in computation time by a factor of approximately 15.

HeinzelCluster: accelerated reconstruction for FORE and OSEM3D

Using iterative three-dimensional (3D) reconstruction techniques for reconstruction of positron emission tomography (PET) is not feasible on most single-processor machines due to the excessive computing time needed, especially so for the large sinogram sizes of our high-resolution research tomograph (HRRT). In our first approach to speed up reconstruction time we transform the 3D scan into the format of a two-dimensional (2D) scan with sinograms that can be reconstructed independently using Fourier rebinning (FORE) and a fast 2D reconstruction method. On our dedicated reconstruction cluster (seven four-processor systems, Intel PIII@700 MHz, switched fast ethernet and Myrinet, Windows NT Server), we process these 2D sinograms in parallel. We have achieved a speedup > 23 using 26 processors and also compared results for different communication methods (RPC, Syngo, Myrinet GM). The other approach is to parallelize OSEM3D (implementation of C Michel), which has produced the best results for HRRT data so far and is more suitable for an adequate treatment of the sinogram gaps that result from the detector geometry of the HRRT. We have implemented two levels of parallelization for four dedicated cluster (a shared memory fine-grain level on each node utilizing all four processors and a coarse-grain level allowing for 15 nodes) reducing the time for one core iteration from over 7 h to about 35 min.

180 degree pinhole SPET with a tilted detector and OS-EM reconstruction: phantom studies and potential clinical applications

This study investigated the feasibility of ordered subsets expectation maximisation (OS-EM) reconstruction of pinhole single-photon emission tomography (SPET) acquired with a tilted detector head and a 180 degrees orbit. Phantom and patient data were recorded using a standard single-head camera. Reconstructions were performed using a dedicated OS-EM algorithm. Reconstructed images of line, uniformity and Picker's thyroid phantoms showed that the geometry, physical size and uniformity of the radioactive objects were preserved. For the range of radius corresponding to the patient studies, the measured full-widths at half-maximum lay between 4.90+/-0.25 mm and 6.05+/-0.25 mm. Finally, the gain in resolution associated with the use of the pinhole collimator instead of a parallel-hole collimator was highlighted in a parathyroid exploration and in a shoulder bone study.

Rebinning-based algorithms for helical cone-beam CT

Several image reconstruction algorithms based on rebinning have been proposed recently for helical cone-beam CT. These algorithms separate the 3D reconstruction into a set of independent 2D reconstructions for a set of surfaces: planar or non-planar surfaces are defined and then reconstructed using 2D filtered backprojection from a 2D fan-beam or parallel-beam set of data estimated from the cone-beam (CB) measurements. The first part of this paper presents a unified derivation of rebinning algorithms for planar and non-planar surfaces. An integral equation is derived for the surface allowing the best rebinning and an iterative algorithm converging to the solution of that equation is given. The second part presents an efficient method to correct the residual reconstruction artefacts observed with rebinning algorithms when the cone-angle is too large for the required accuracy. This correction algorithm involves a CB backprojection and the reconstruction time is slightly longer than for the zero-boundary (ZB) method.

Comparison of 3-D reconstruction with 3D-OSEM and with FORE+OSEM for PET

The combination of Fourier rebinning (FORE) and the ordered subsets expectation-maximization (OSEM), a fast statistical algorithm, appears as a promising alternative to the fully three-dimensional (3-D) iterative approach for clinical positron emission tomography (PET) data. In this paper, we evaluated the properties of FORE+OSEM and compared it with fully 3-D OSEM using both simulations and data acquired by commercial scanners. The aim is to determine to what extent the speed advantage of FORE+OSEM is paid for by a possible degradation of image quality in the case of noisy clinical PET data. A forward- and back-projection pair based on a line integral model was used in two-dimensional OSEM and 3-D OSEM (3D-OSEM) instead of a system matrix. Different variants of both approaches have been studied with simulations in terms of contrast-noise tradeoff. Two variants--FORE+OSEM with attenuation weighting (AW) [FORE+OSEM(AW)] and 3D-OSEM with attenuation-normalization weighting (ANSP) and a shifted-Poisson (SP) model [3D-OSEM(ANSP)]--were compared with measured phantom data and patient data. Based on the results from both simulations and measured data, we conclude that: 1) both attenuation (-normalization) weighting and the SP model improve the image quality but slow down the convergence and 2) despite its approximate nature, FORE+OSEM does not show apparent image degradation compared with 3D-OSEM for data with a noise level typical of a whole-body FDG scan.

Efficient cardiac diffusion tensor MRI by three-dimensional reconstruction of solenoidal tensor fields

Tensor tomography is being investigated as a technique for reconstruction of in vivo diffusion tensor fields that can potentially be used to reduce the number of magnetic resonance imaging (MRI) measurements. Specifically, assessments are being made of the reconstruction of cardiac diffusion tensor fields from 3D Radon planar projections using a filtered backprojection algorithm in order to specify the helical fiber structure of myocardial tissue. Helmholtz type decomposition is proposed for 3D second order tensor fields. Using this decomposition a Fourier projection theorem is formulated in terms of the solenoidal and irrotational components of the tensor field. From the Fourier projection theorem, two sets of Radon directional measurements, one that reconstructs the solenoidal component and one that reconstructs the irrotational component of the tensor field, are prescribed. Based on these observations filtered backprojection reconstruction formulae are given for the reconstruction of a 3D second order tensor field and its solenoidal and irrotational components from Radon projection measurements. Computer simulations demonstrate the validity of the mathematical formulations and demonstrate that a realistic model of the helical fiber structure of the myocardial tissue specifies a diffusion tensor field for which the first principal vector (the vector associated with the maximum eigenvalue) of the solenoidal component accurately approximates the first principal vector of the diffusion tensor. A priori knowledge of this allows the orientation of the myocardial fiber structure to be specified utilizing one half of the number of MRI measurements of a normal diffusion tensor field study.

Automatic determination of left ventricular ejection fraction from gated blood-pool tomography

The aim of this study was to develop and validate a new algorithm to automatically compute left ventricular ejection fraction (LVEF) from gated blood-pool tomography (GBPT). The results were compared with those of conventional planar radionuclide angiocardiography (PRNA).

METHODS

Fifty-three consecutive patients received an injection of 740 MBq (99m)Tc-labeled human serum albumin. PRNA and GBPT were performed consecutively in a random sequence. PRNA served as the reference, and GBPT images were processed using a new edge detection algorithm. The algorithm is fast (<45 s), fully automatic, and works in three-dimensional space. The method includes identification of the valve plane and the septum. The left ventricular cavity at end-diastole is delineated by segmentation using an iterative threshold technique. An optimal threshold is reached when the corresponding isocontour best fits the first derivative of the end-diastolic count distribution in three dimensions. This optimal threshold is then applied to delineate the left ventricular cavity on the other time bins. The data are corrected for the partial-volume effect. Left ventricular volumes are determined using a geometry-based method and are used to calculate the ejection fraction.

RESULTS

The success rate of the new algorithm was 94%. LVEFs calculated from GBPT agreed well with those calculated from PRNA (r = 0.78; GBPT = 0.94 PRNA + 6.33). The systematic error was 2.8%, and the random error was 8.8%. Excellent inter- and intraobserver reproducibility was found, with average differences of 1.1% +/- 4.6% and 1.1% +/- 5.0%, respectively, between the two measurements.

CONCLUSION

This new algorithm provides a fast, automated, and objective method to calculate LVEF from GBPT.

A short reader's guide to 3D tomographic reconstruction

This text summarizes the main technical problems related to 3D image reconstruction in PET, SPECT and CT, and provides references to a selection of key papers and to review papers.

Quasi-exact filtered backprojection algorithm for long-object problem in helical cone-beam tomography

Exact reconstruction from axially truncated cone-beam projections acquired with a helical vertex path is a challenging problem for which solutions are currently under investigation by some researchers. This paper deals with a difficult class of this problem called the long-object problem. Its purpose is to reconstruct a central region of interest (ROI) of a long object when the helical path extends only a little bit above and below the ROI. By extending the authors' recent approach based on the triangular decomposition of the Grangeat formula, we derive quasi-exact reconstruction algorithms whose overall structure is of filtered backprojection (FBP) style. Unlike the previous similar approaches to the long-object problem, the proposed FBP algorithms do not require additional two circular scans at the ends of the helical path. Furthermore, the algorithms require a significantly smaller detector area and achieve improved image quality even for a large pitch compared with the approximate Feldkamp algorithms. One drawback of the proposed algorithms is the computational time, which is much longer than for the Feldkamp algorithms. We show some simulation results to demonstrate the performances of the proposed algorithms.

A solution to the long-object problem in helical cone-beam tomography

This paper presents a new algorithm for the long-object problem in helical cone-beam (CB) computerized tomography (CT). This problem consists in reconstructing a region-of-interest (ROI) bounded by two given transaxial slices, using axially truncated CB projections corresponding to a helix segment long enough to cover the ROI, but not long enough to cover the whole axial extent of the object. The new algorithm is based on a previously published method, referred to as CB-FBP (Kudo et al 1998 Phys. Med. Biol. 43 2885-909), which is suitable for quasi-exact reconstruction when the helix extends well beyond the support of the object. We first show that the CB-FBP algorithm simplifies dramatically, and furthermore constitutes a solution to the long-object problem, when the object under study has line integrals which vanish along all PI-lines. (A PI line is a line which connects two points of the helix separated by less than one pitch.) Exploiting a geometric property of the helix, we then show how the image can be expressed as the sum of two images, where the first image can be reconstructed from the measured CB projections by a simple backprojection procedure, and the second image has zero PI-line integrals and hence can be reconstructed using the simplified CB-FBP algorithm. The resulting method is a quasi-exact solution to the long-object problem, called the ZB method. We present its implementation and illustrate its performance using simulated CB data of the 3D Shepp phantom and of a more challenging head-like phantom.

Interest of the ordered subsets expectation maximization (OS-EM) algorithm in pinhole single-photon emission tomography reconstruction: a phantom study

Pinhole single-photon emission tomography (SPET) has been proposed to improve the trade-off between sensitivity and resolution for small organs located in close proximity to the pinhole aperture. This technique is hampered by artefacts in the non-central slices. These artefacts are caused by truncation and by the fact that the pinhole SPET data collected in a circular orbit do not contain sufficient information for exact reconstruction. The ordered subsets expectation maximization (OS-EM) algorithm is a potential solution to these problems. In this study a three-dimensional OS-EM algorithm was implemented for data acquired on a single-head gamma camera equipped with a pinhole collimator (PH OS-EM). The aim of this study was to compare the PH OS-EM algorithm with the filtered back-projection algorithm of Feldkamp, Davis and Kress (FDK) and with the conventional parallel-hole geometry as a whole, using a line source phantom, Picker's thyroid phantom and a phantom mimicking the human cervical column. Correction for the angular dependency of the sensitivity in the pinhole geometry was based on a uniform flood acquisition. The projection data were shifted according to the measured centre of rotation. No correction was made for attenuation, scatter or distance-dependent camera resolution. The resolution measured with the line source phantom showed a significant improvement with PH OS-EM as compared with FDK, especially in the axial direction. Using Picker's thyroid phantom, one iteration with eight subsets was sufficient to obtain images with similar noise levels in uniform regions of interest to those obtained with the FDK algorithm. With these parameters the reconstruction time was 2.5 times longer than for the FDK method. Furthermore, there was a reduction in the artefacts caused by the circular orbit SPET acquisition. The images obtained from the phantom mimicking the human cervical column indicated that the improvement in image quality with PH OS-EM is relevant for future clinical use and that the improvements obtained using the OS-EM algorithm are more significant for the pinhole geometry than for the conventional parallel-hole geometry. We conclude that PH OS-EM is a practical and promising alternative for pinhole SPET reconstruction.

Exact rebinning methods for three-dimensional PET

The high computational cost of data processing in volume PET imaging is still hindering the routine application of this successful technique, especially in the case of dynamic studies. This paper describes two new algorithms based on an exact rebinning equation, which can be applied to accelerate the processing of three-dimensional (3-D) PET data. The first algorithm, FOREPROJ, is a fast-forward projection algorithm that allows calculation of the 3-D attenuation correction factors (ACF's) directly from a two-dimensional (2-D) transmission scan, without first reconstructing the attenuation map and then performing a 3-D forward projection. The use of FOREPROJ speeds up the estimation of the 3-D ACF's by more than a factor five. The second algorithm, FOREX, is a rebinning algorithm that is also more than five times faster, compared to the standard reprojection algorithm (3DRP) and does not suffer from the image distortions generated by the even faster approximate Fourier rebinning (FORE) method at large axial apertures. However, FOREX is probably not required by most existing scanners, as the axial apertures are not large enough to show improvements over FORE with clinical data. Both algorithms have been implemented and applied to data simulated for a scanner with a large axial aperture (30 degrees), and also to data acquired with the ECAT HR and the ECAT HR+ scanners. Results demonstrate the excellent accuracy achieved by these algorithms and the important speedup when the sinogram sizes are powers of two.

Attenuation correction in whole-body FDG oncological studies: the role of statistical reconstruction

Whole-body fluorine-18 fluoro-2-d-deoxyglucose positron emission tomography (FDG-PET) is widely used in clinical centres for diagnosis, staging and therapy monitoring in oncology. Images are usually not corrected for attenuation since filtered backprojection (FBP) reconstruction methods require a 10 to 15-min transmission scan per bed position on most current PET devices equipped with germanium-68 rod transmission sources. Such an acquisition protocol would increase the total scanning time beyond acceptable limits. The aim of this work is to validate the use of iterative reconstruction methods, on both transmission and emission scans, in order to obtain a fully corrected whole-body study within a reasonable scanning time of 60 min. Five minute emission and 3-min transmission scans are acquired at each of the seven bed positions. The transmission data are reconstructed with OSEM (ordered subsets expectation maximization) and the last iteration is reprojected to obtain consistent attenuation correction factors (ACFs). The emission image is then also reconstructed with OSEM, using the emission scan corrected for normalization, scatter and decay together with the set of consistent ACFs as inputs. The total processing time is about 35 min, which is acceptable in a clinical environment. The image quality, readability and accuracy of uptake quantification were assessed in 38 patients scanned for various malignancies. The sensitivity for tumour detection was the same for the non-attenuation-corrected (NAC-FBP) and the attenuation-corrected (AC-OSEM) images. The AC-OSEM images were less noisy and easier to interpret. The interobserver reproducibility was significantly increased when compared with non-corrected images (96.1% vs 81.1%, P<0.01). Standardized uptake values (SUVs) measured on images reconstructed with OSEM (AC-OSEM) and filtered backprojection (AC-FBP) were similar in all body regions except in the pelvic area, where SUVs were higher on AC-FBP images (mean increase 7.74%, P<0. 01). Our results show that, when statistical reconstruction is applied to both transmission and emission data, high quality quantitative whole-body images are obtained within a reasonable scanning (60 min) and processing time, making it applicable in clinical practice.

Single-slice rebinning method for helical cone-beam CT

In this paper, we present reconstruction results from helical cone-beam CT data, obtained using a simple and fast algorithm, which we call the CB-SSRB algorithm. This algorithm combines the single-slice rebinning method of PET imaging with the weighting schemes of spiral CT algorithms. The reconstruction is approximate but can be performed using 2D multislice fan-beam filtered backprojection. The quality of the results is surprisingly good, and far exceeds what one might expect, even when the pitch of the helix is large. In particular, with this algorithm comparable quality is obtained using helical cone-beam data with a normalized pitch of 10 to that obtained using standard spiral CT reconstruction with a normalized pitch of 2.

A performance study of 3D reconstruction algorithms for positron emission tomography

This paper investigates the statistical and systematic accuracy of five three-dimensional reconstruction algorithms for multi-ring PET scanners operated without septa: the reprojection method, the direct Fourier reconstruction, the FAVOR algorithm, and the single-slice and multi-slice rebinning algorithms. Simulated data of a uniform cylinder, of Gaussian sources, and of spherical sources are used to compare respectively the noise properties, the modulation transfer function, and the recovery coefficients of the algorithms. Brain scans reconstructed with the different algorithms are compared by calculating the linear regression of the mean values within regions of interest. The most significant observations are a slight loss of transaxial resolution with the reprojection algorithm in the external slices of the scanner, and increased noise in the images reconstructed using multi-slice rebinning.

Cone-beam filtered-backprojection algorithm for truncated helical data

This paper investigates 3D image reconstruction from truncated cone-beam (CB) projections acquired with a helical vertex path. First, we show that a rigorous derivation of Grangeat's formula for truncated projections leads to a small additional term compared with previously published similar formulations. This correction term is called the boundary term. Next, this result is used to develop a CB filtered-backprojection (FBP) algorithm for truncated helical projections. This new algorithm only requires the CB projections to be measured within the region that is bounded, in the detector, by the projections of the upper and lower turns of the helix. Finally, simulations with mathematical phantoms demonstrate that: (i) the boundary term is necessary to obtain high-quality imaging of low-contrast structures and (ii) good image quality is obtained even with large values of the pitch of the helix.

Iterative reconstruction for helical CT: a simulation study

Iterative reconstruction algorithms for helical CT are presented. The algorithms are derived from two-dimensional reconstruction algorithms, by adapting the projector/backprojector to the helical orbit of the source, and by constraining the axial frequencies with a Gaussian sieve. Simulations have been carried out and the performance of the iterative algorithms is compared to that of filtered backprojection of synthetic (interpolated) two-dimensional sinograms. The iterative algorithms produce superior bias-noise curves. Axial resolution is superior, but disturbing edge-artefacts are introduced.

Iterative reconstruction for helical CT: a simulation study

Iterative reconstruction algorithms for helical CT are presented. The algorithms are derived from two-dimensional reconstruction algorithms, by adapting the projector/backprojector to the helical orbit of the source, and by constraining the axial frequencies with a Gaussian sieve. Simulations have been carried out and the performance of the iterative algorithms is compared to that of filtered backprojection of synthetic (interpolated) two-dimensional sinograms. The iterative algorithms produce superior bias-noise curves. Axial resolution is superior, but disturbing edge-artefacts are introduced.

Direct reconstruction of cone-beam data acquired with a vertex path containing a circle

Cone-beam data acquired with a vertex path satisfying the data sufficiency condition of Tuy can be reconstructed using exact filtered backprojection algorithms. These algorithms are based on the application to each cone-beam projection of a two-dimensional (2-D) filter that is nonstationary, and therefore more complex than the one-dimensional (1-D) ramp filter used in the approximate algorithm of Feldkamp, Davis, and Kress (1984) (FDK). We determine in this paper the general conditions under which the 2-D nonstationary filter reduces to a 2-D stationary filter, and also give the explicit expression of the corresponding convolution kernel. Using this result and the redundancy of the cone-beam data, a composite algorithm is derived for the class of vertex paths that consist of one circle and some complementary subpath designed to guarantee data sufficiency. In this algorithm the projections corresponding to vertex points along the circle are filtered using a 2-D stationary filter, whereas the other projections are handled with a 2-D nonstationary filter. The composite algorithm generalizes the method proposed by Kudo and Saito (1990), in which the circle data are processed with a 1-D ramp filter as in the FDK algorithm. The advantage of the 2-D filter introduced in this paper is to guarantee that the filtered cone-beam projections do not contain singularities in smooth regions of the object. Tests of the composite algorithm on simulated data are presented.

Exact and approximate rebinning algorithms for 3-D PET data

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 degrees 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.

Optimal collimator choice for sequential iodine-123 and technetium-99m imaging

Dual-isotope studies with technetium-99m and iodine-123 may be useful for various organs, including brain and myocardium. For the images obtained with each of the tracers to be comparable, it is important that activity ratios (activity in one part of the image/reference activity in the image) are preserved by the imaging method. We have used a Rollo phantom to study how collimator response affects such ratios. All investigations were performed with 123I(p,5n) and on a Siemens Orbiter 3700 camera fitted with either a low-energy high-resolution (LEHR) or a medium-energy (ME) collimator. Images were made of a Rollo phantom filled with an aqueous solution of either 99Tc or 123I, and placed on the collimator surface with 8 cm of methyl-methacrylate interposed. Count densities were measured in ROIs drawn in each cell of the phantom, and normalised to the maximal ROI value in the image. The mean square error (MSE) was used to assess how well the ratios of count densities approximated the known activity ratios based on the dimensions of the cells of the phantom. For 99mTc, regardless of the collimator used, the count density ratios approximated the activity ratios fairly well (LEHR: MSE=0.008; ME: MSE=0.020). For 123I, count density ratios obtained with the LEHR were consistently higher than activity ratios (MSE=0. 235), whereas the differences between the measured and the theoretical values were less with the ME collimator (MSE=0.013). Contrast fidelity of the 123I images obtained with the LEHR collimator could be improved with Jaszczak scatter correction with k=1, but this led to unfavourable signal-to-noise ratios. For sequential 99mTc/123I studies with extended sources, ME is to be preferred because of its higher contrast accuracy. Spatial resolution is less for the ME than for the LEHR collimator (FWHM with scatter: LEHR/99mTc=6.9 mm, LEHR/123I=7.4 mm, ME/99mTc= 10.1 mm, ME/123I=11.1 mm), but remains similar for both tracers when the ME is used.

Three-dimensional multimodality medical image registration using a parameter accumulation approach

A parameter accumulation method based on the Hough transformation is proposed to register three-dimensional (3-D) multimodality medical images. The estimation of registration parameters is decomposed into separate estimations of rotation, using directional vectors, and translation, using positional vectors. Similarly, the rotation parameters are decomposed into the rotation axis and angle, which are then estimated separately. This kind of decomposition reduces the parametric dimension and improves the computing efficiency which has been a major concern in implementing the Hough transformation. When 3-D rotation is involved, evaluating registration error is not straightforward. This paper introduces an equivalent error angle as a criterion to evaluate the performance of 3-D registration methods. Experimental results indicate that a least-squares fitting is superior to the parameter accumulation with data contaminated by additive noise only. When mismatched feature points (outliers) exist in the data set, however, the parameter accumulation approach is more accurate. The application of the proposed approach to the registration of 3-D PET and CT images is demonstrated.

Filtered backprojection reconstruction of combined parallel beam and cone beam SPECT data

The reconstruction problem for a combined parallel beam (PB) and cone beam (CB) imaging geometry has been addressed. The general algorithm (CB-FBP) of Defrise and Clack has been applied to this geometry and shown to provide accurate images as expected. A second algorithm specifically tailored to the PB-CB geometry was developed. It uses the general principles of the CB-FBP method to combine a shift-variant filtering of the PB data with a standard reconstruction of the CB data using the algorithm of Feldkamp, Davis and Kress (FDK). This 'mixed' algorithm has the advantage of fewer interpolation steps, thereby reducing reconstruction time and providing more accurate reconstructions. The algorithms were applied to noiseless data from a computer-generated 3D Shepp phantom. Both the CB-FBP algorithm and the mixed algorithm successfully combine the CB and the PB data to correct the well known artefacts observed when reconstructing CB data acquired with a circular orbit. The mixed algorithm is about twice as fast as CB-FBP and results in better image quality, due to decreased discretization errors. However, the two algorithms yield comparable image quality when applied to a disc phantom measured with a two-headed SPECT system.

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

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 rotating PET scanner using BGO block detectors: design, performance and applications

Recent advances in fully three-dimensional reconstruction for multi-ring PET scanners have led us to explore the potential of a prototype scanner based on the rotation of two opposing arrays of BGO block detectors. The prototype contains only one-third of the number of detectors in the equivalent full ring scanner, resulting in reduced cost. With a lower energy threshold at 250 keV, the absolute efficiency of the scanner is 0.5% and the scatter fraction is 35% for a 20-cm cylinder. Transaxial and axial spatial resolution is about 6 mm. The maximum noise equivalent count rate estimated for a 15-cm diameter cylinder is 36,000 cps at a concentration of 26 kBq/ml. The minimum scan time for a 18F-fluoro-2-deoxyglucose (FDG) brain study is 55 sec. The camera has been validated for clinical applications using both FDG and 82Rb.

A normalization technique for 3D PET data

Prior to reconstruction, emission data from a multi-ring PET camera must be corrected (normalized) for variations in detector sensitivity. The appropriate correction coefficients are obtained by measuring the response of all coincidence lines to a calibrated source of activity (a blank scan). State-of-the-art cameras may contain up to a million such lines of response (LORs), and therefore around 400 million counts will be required to calibrate each LOR to a statistical accuracy of 5%. Alternatively, by modelling the LOR sensitivity as the product of the individual detector efficiencies and a geometrical factor, a calibration procedure has been proposed which requires the determination of only 6000 parameters from this same data set. A significant improvement in the statistical accuracy of the coefficients can therefore be expected. Recently, multi-ring scanners have been operated with the septa retracted, increasing the number of measured LORs by a factor of eight. The acquisition of the calibration data necessary to achieve adequate statistical accuracy then becomes prohibitive. We show that, by modelling the LOR sensitivity, it is possible, with certain approximations, to normalize a septa-retracted emission data set with good accuracy. The input to the model is a high statistics blank scan acquired with the septa extended, which offers a number of practical advantages.

Fully three-dimensional reconstruction for a PET camera with retractable septa

A fully 3-D reconstruction algorithm has been developed to reconstruct data from a 16 ring PET camera (a Siemens/CTI 953B) with automatically retractable septa. The tomograph is able to acquire coincidences between any pair of detector rings and septa retraction increases the total system count rate by a factor of 7.8 (including scatter) and 4.7 (scatter subtracted) for a uniform, 20 cm diameter cylinder. The reconstruction algorithm is based on 3-D filtered backprojection, expressed in a form suitable for the multi-angle sinogram data. Sinograms which are not measured due to the truncated cylindrical geometry of the tomograph, but which are required for a spatially invariant response function, are obtained by forward projection. After filtering, the complete set of sinograms is backprojected into a 3-D volume of 128x128x31 voxels using a voxel-driven procedure. The algorithm has been validated with simulation, and tested with both phantom and clinical data from the 953B.

Implementation of three-dimensional image reconstruction for multi-ring positron tomographs

In view of the number of PET studies involving low count rate acquisitions, there has been increasing interest recently in the development of positron cameras capable of fully three-dimensional acquisition and reconstruction. This interest has given impetus to the study of algorithms for 3D reconstruction, including those algorithms suitable for application to multi-ring PET scanners. While 2D reconstruction methods can often be generalised to 3D, a number of implementation problems arise which are unique to the 3D approach. This paper examines some of the difficulties associated with the generalisation of the filtered backprojection algorithm to 3D, paying particular attention to the approximations and variable transformations required for application to data from a multi-ring scanner.

Three-dimensional image reconstruction from complete projections

Three-dimensional medical image reconstruction for both transmission and emission tomography has traditionally decomposed the problem into a set of two-dimensional reconstructions on parallel transverse sections. There is, however, increasing interest in reconstructing projection data directly in three dimensions. For emission tomography in particular, such a reconstruction procedure would clearly make more efficient use of the available photon flux. In the past few years, 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. This paper clarifies this relationship by analysing and generalising the different classes of published filters and establishes 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.

Aspects of three dimensional reconstruction for a multi ring positron tomograph

An important feature of multi ring positron tomographs is the inter plane septa, the purpose of which is to reduce random and scattered coincidences. In general, such septa also eliminate the coincidence lines of response between pairs of detectors more than one ring apart. The operation of a camera without septa must result in an increase not only in the true coincidence rate, but also in the singles, and therefore in the dead time and randoms rate, and in the scattered coincidences. A configuration option in the coincidence hardware of the 8 ring, 15 slice ECAT 931/08-12 enables a full set of 64 sinograms to be acquired when the septa are removed. The detector normalisation and transmission data for studies with the septa out can be obtained using a rotating pin source. To take maximum advantage of the additional signal, the emission data must be reconstructed using a fully three dimensional reconstruction algorithm. This paper presents an analysis of some phantom studies acquired without septa and reconstructed in three dimensions. The results are compared with data acquired with septa for the same phantoms imaged under similar conditions. It is found that, with the septa removed, the signal to noise for a uniform, 20 cm diameter cylinder improves by a factor of 2.8 in the centre of the field of view, whereas in regions distant from the centre in the axial direction, the signal to noise decreases due to the increase in scatter and randoms. An improvement in signal to noise is observed in 6 cm of the 10 cm axial length of the tomograph.

An algorithm for three-dimensional reconstruction incorporating cross-plane rays

Conventional multislice positron cameras reconstruct a three-dimensional distribution of a positron-emitting radioscope as a set of two-dimensional transverse sections. Consequently, annihilation photons which cross two or more planes are eliminated from the data. Such an approach makes efficient use of the emitted photon flux. A method is proposed which makes more efficient use of the available photons by including both oblique and transverse section in the reconstruction. The implementation of the method consists of centering a scaled convolution filter on each detected coincidence event line and backprojecting the filter values through the three-dimensional reconstruction volume. The final image is normalized to allow for the different number of oblique and transverse sections that contribute to each point in the imaging volume. The method has been evaluated using both simulated data and measured data obtained with a routing area detector positron camera.