Table of contents

A new reconstruction scheme for multi-row spiral CT is described and results are presented. The spiral path is decomposed into small, overlapping segments which are used for a separate convolution and backprojection yielding a stack of segment images which contain only projection data of a partial scan (typically in the range of 20°). These segment image stacks are, in a second step, reformatted to the requested image planes. In a third step, the reformatted segment images are added to obtain full images. The main benefit of the proposed algorithm is superior image quality. A 64-row dataset with a cone angle of 6.4° and a table feed of 80 mm per spiral turn has been reconstructed with excellent image quality. A filter direction for three-dimensional (3D) backprojection algorithms is suggested by investigating the limit where the partial scan size goes to zero.

In this paper we continue studying a theoretically exact filtered backprojection inversion formula for cone beam spiral CT proposed earlier by the author. Our results show that if the phantom f is constant along the axial direction, the formula is equivalent to the 2D Radon transform inversion. Also, the inversion formula remains exact as spiral pitch goes to zero and in the limit becomes again the 2D Radon transform inversion formula. Finally, we show that according to the formula the processed cone beam projections should be backprojected using both the inverse distance squared law and the inverse distance law.

Based on the duality of nonlinear programming, this paper proposes an accurate row-action type iterative algorithm which is appropriate to reconstruct sparse objects from a limited number of projections. The cost function we use is the L_p norm with p ≈ 1.1. This norm allows us to pick up a sparse solution from a set of feasible solutions to the measurement equation. Furthermore, since it is both strictly convex and differentiable, we can use the duality of nonlinear programming to construct a row-action type iterative algorithm to find a solution. We also impose the bound constraint on pixel values to pick up a better solution. We demonstrate that this method works well in three-dimensional blood vessel reconstruction from a limited number of cone beam projections.

Dynamic cone-beam reconstruction algorithms are required to reconstruct three-dimensional (3D) image sequences on dynamic 3D CT combining multi-row two-dimensional (2D) detectors and sub-second scanners. The speed-up of the rotating gantry allows one to improve the temporal resolution of the image sequence, but at the same time, it implies increase in the dose delivered during a given time period to keep constant the signal-to-noise ratio associated with each frame. The alternative solution proposed in this paper is to process data acquisition on several half-turns in order to reduce the dose delivered per rotation with the same signal-to-noise ratio. In order to compensate for time evolution and motion artefacts, we propose to use a dynamic particle model to describe the object evolution during the scan. In this article, we first introduce the dynamic particle model and the dynamic CT acquisition model. Then, we explain the principle of the proposed dynamic cone-beam reconstruction algorithm. Lastly, we present preliminary results on simulated data.

A fast iterative method is described for processing clinical PET scans acquired in three dimensions, that is, with no inter-plane septa, using standard computers to replace dedicated processors used until the late 1990s. The method is based on sinogram resampling, Fourier rebinning, Monte Carlo scatter simulation and iterative reconstruction using the attenuation-weighted OSEM method and a projector based on a Gaussian pixel model. Resampling of measured sinogram values occurs before Fourier rebinning, to minimize parallax and geometric distortions due to the circular geometry, and also to reduce the size of the sinogram. We analyse the geometrical and statistical effects of resampling, showing that the lines of response are positioned correctly and that resampling is equivalent to about 4 mm of post-reconstruction filtering. We also present phantom and patient results. In this approach, multi-bed clinical oncology scans can be ready for diagnosis within minutes.

Source trajectories and reconstruction algorithms for clinical volumetric computerized tomography (VCT) will require optimization for efficiency and image quality. VCT data is highly overdetermined, satisfying an ultrahyperbolic partial differential equation. Characteristic boundary value problems for the hyperbolic wave equation and ultrahyperbolic equation are compared, focusing in this paper on a mathematically instructive open-gantry VCT geometry. This example provides physical insight into consistency conditions upon VCT data, clearly showing which information about the object can and cannot be recovered from a set of measured projections. Furthermore, this example demonstrates that efficient numerical solvers for the ultrahyperbolic equation can provide tremendous flexibility in the choice of reconstruction algorithm.

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 our 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.

Some recent medical imaging applications such as functional imaging (PET and SPECT) or interventional imaging (CT fluoroscopy) involve increasing amounts of data. In order to reduce the image reconstruction time, we develop a new fast 3D reconstruction algorithm based on a divide and conquer approach.

The proposed multichannel algorithm performs an indirect frequential subband decomposition of the image f to be reconstructed (f = ∑f_j) through the filtering of the projections Rf. The subband images f_j are reconstructed on a downsampled grid without information suppression. In order to reduce the computation time, we do not backproject the null filtered projections and we downsample the number of projections according to the Shannon conditions associated with the subband image. Our algorithm is based on filtering and backprojection operators. Using the same algorithms for these basic operators, our approach is three and a half times faster than a classical FBP algorithm for a 2D image 512 × 512 and six times faster for a 3D image 32 × 512 × 512.

Artefacts can result when reconstructing a dynamic image sequence from inconsistent single photon emission computed tomography (SPECT) projection data acquired by a slowly rotating gantry. The artefacts can lead to biases in kinetic parameters estimated from time–activity curves generated by overlaying volumes of interest on the images. Insufficient sampling and truncation of projections by cone-beam collimators can cause additional artefacts. To overcome these sources of bias in conventional image based dynamic data analysis, we have been investigating the estimation of time–activity curves and kinetic model parameters directly from dynamic SPECT projection data by modelling the spatial and temporal distribution of the radiopharmaceutical throughout the projected field of view. In the present work, we perform Monte Carlo simulations to study the effects of the temporal modelling on the statistical variability of the reconstructed spatiotemporal distributions. The simulations utilize fast methods for fully four-dimensional (4D) direct estimation of spatiotemporal distributions and their statistical uncertainties, using a spatial segmentation and temporal B-splines. The simulation results suggest that there is benefit in modelling higher orders of temporal spline continuity. In addition, the accuracy of the time modelling can be increased substantially without unduly increasing the statistical uncertainty, by using relatively fine initial time sampling to capture rapidly changing activity distributions.

In backprojection cone-beam CT the cone-beam projection images are first filtered, then 3D backprojected into the object space. In this paper the point spread function (PSF) for the filtering operation is studied. For the cases where the normalization matrix is a constant, ie all integration planes intersect the scan path the same number of times, the derivation of the PSF is extended to the general case of limited angular range for the Radon line integrals. It is found that the 2D component of the PSF can be reduced to the form of space-variant 1D Hilbert transforms. The application of the PSF to a number of aspects in long object imaging will be discussed.

The continuous scanning mode in three-dimensional (3D) whole-body PET studies has the advantage of axial sensitivity uniformity over the majority of the axial FOV. However, this scan mode requires large data handling compared to conventional discrete scans. In this work, we have implemented and evaluated a new continuous 3D scan method using 'on-the-fly' Fourier rebinning. In this method, sinograms for the pair of rings are added in real-time into the sinograms of the incremented ring pairs by moving the bed axially one detector width at a time. For an N-ring scanner, 2N − 1 sinograms are transferred to a host computer at each bed position and rebinned into direct two-dimensional (2D) sinograms using Fourier rebinning. Phantom and human studies showed that the axial image uniformity is achieved without degrading the axial resolution. This method can minimize the time for off-line data processing and makes the continuous 3D scan more feasible in clinical whole-body studies.

The RSH SPECT scanner provides parallel-beam attenuated projections for a fully 3D acquisition geometry. The geometry can be represented by circles on the unit sphere of projection directions, one circle for each position of the detector head. Unlike most other fully 3D geometries this one is particularly challenging because there are no 2D subsets in the data. When no attenuation is present, it is well known that an unmeasured projection can be synthesized if it lies inside one of the measured circles. The main result of this work is that under some assumptions on the attenuation distribution, attenuated projections within a circle can be synthesized from available attenuated projections. One consequence is that RSH SPECT projections can be rebinned into a conventional SPECT geometry for which analytic attenuation correction techniques are available.

This work presents new mathematical results on the inversion of the exponential x-ray transform. It is shown that a reconstruction formula can be obtained for any dataset whose projection directions consist of a union of half great circles on the unit sphere. A basic example of such a dataset is the semi-equatorial band. The discussion in the paper is mostly focused on this example. The reconstruction formula takes the form of a Neumann (geometric) series and is both exact and stable. The exponential x-ray transform has been mainly studied in SPECT imaging. In this context, our results demonstrate mathematically that fully 3D image reconstruction in SPECT with non-zero attenuation does not always require symmetric datasets (opposing views).

A novel approach to reconstructing the principal directions of a diffusion tensor field directly from magnetic resonance imaging (MRI) data using a tensor tomography data acquisition approach was developed. If tensor eigenvalues are assumed to be known, the reconstruction of principal directions requires fewer measurements than the reconstruction of the full tensor field. The tensor tomography data acquisition method (rotating diffusion gradients) leads to a unique reconstruction of principal directions, whereas the conventional MRI acquisition technique (stationary diffusion gradients) leads to an ambiguous reconstruction of principal directions when the same number of measurements are used. A computer-generated phantom was used to simulate the diffusion tensor field in the mid-ventricular region of the myocardium. The principal directions of the diffusion tensor field were assumed to align with the fibre structure of the myocardium. An iterative algorithm was used to reconstruct the principal directions. Computer simulations verify that the proposed method provides accurate reconstruction of the principal directions of a diffusion tensor field.

In this paper we present a scatter correction method for a regularized list mode maximum likelihood reconstruction algorithm for the positron emission mammograph (PEM) that is being developed at our laboratory. The scatter events inside the object are modelled as additive Poisson random variables in the forward model of the reconstruction algorithm. The mean scatter sinogram is estimated using a Monte Carlo simulation program. With the assumption that the background activity is nearly uniform, the Monte Carlo scatter simulation only needs to run once for each PEM configuration. This saves computation time. The crystal scatters are modelled as a shift-invariant blurring in image domain because they are more localized. Thus, the useful information in the crystal scatters can be deconvolved in high-resolution reconstructions. The propagation of the noise from the estimated scatter sinogram into the reconstruction is analysed theoretically. The results provide an easy way to calculate the required number of events in the Monte Carlo scatter simulation for a given noise level in the image. The analysis is also applicable to other scatter estimation methods, provided that the covariance of the estimated scatter sinogram is available.

We describe a method for normalization in 3D PET for use with maximum a posteriori (MAP) or other iterative model-based image reconstruction methods. This approach is an extension of previous factored normalization methods in which we include separate factors for detector sensitivity, geometric response, block effects and deadtime. Since our MAP reconstruction approach already models some of the geometric factors in the forward projection, the normalization factors must be modified to account only for effects not already included in the model. We describe a maximum likelihood approach to joint estimation of the count-rate independent normalization factors, which we apply to data from a uniform cylindrical source. We then compute block-wise and block-profile deadtime correction factors using singles and coincidence data, respectively, from a multiframe cylindrical source. We have applied this method for reconstruction of data from the Concorde microPET P4 scanner. Quantitative evaluation of this method using well-counter measurements of activity in a multicompartment phantom compares favourably with normalization based directly on cylindrical source measurements.

We describe an approach to fast iterative reconstruction from fully three-dimensional (3D) PET data using a network of PentiumIII PCs configured as a Beowulf cluster. To facilitate the use of this system, we have developed a browser-based interface using Java. The system compresses PET data on the user's machine, sends these data over a network, and instructs the PC cluster to reconstruct the image. The cluster implements a parallelized version of our preconditioned conjugate gradient method for fully 3D MAP image reconstruction. We report on the speed-up factors using the Beowulf approach and the impacts of communication latencies in the local cluster network and the network connection between the user's machine and our PC cluster.

In this work, we investigate longitudinal sampling and aliasing effects in multi-slice helical CT. We demonstrate that longitudinal aliasing can be a significant, complicated, and potentially detrimental effect in multi-slice helical CT reconstructions. Multi-slice helical CT scans are generally undersampled longitudinally for all pitches of clinical interest, and the resulting aliasing effects are spatially variant. As in the single-slice case, aliasing is shown to be negligible at the isocentre for circularly symmetric objects due to a fortuitous aliasing cancellation phenomenon. However, away from the isocentre, aliasing effects can be significant, spatially variant, and highly pitch dependent. This implies that measures more sophisticated than isocentre slice sensitivity profiles are needed to characterize longitudinal properties of multi-slice helical CT systems. Such measures are particularly important in assessing the question of whether there are preferred pitches in helical CT. Previous analyses have generally focused only on isocentre sampling patterns, and our more global analysis leads to somewhat different conclusions than have been reached before, suggesting that pitches 3, 4, 5, and 6 are favourable, and that half-integer pitches are somewhat suboptimal.

Image quality and quantitative accuracy in single-photon emission computed tomography (SPECT) can be degraded by, e.g., the effects of photon attenuation and finite spatial resolution. It is generally considered that adequate compensation for such effects on SPECT images requires data acquired over 2π. Recently, using the existing consistency condition on the data function, Noo and Wagner (2001 Inverse Problems17 1357–72) have shown analytically that data acquired over only π can be used to correct completely for the effect of uniform attenuation in SPECT. It remains unknown, however, whether data acquired only over π in SPECT with non-uniform attenuation and/or 3D distance-dependent spatial resolution (DDSR) contain complete information for accurate image reconstruction. In this work, we develop a heuristic perspective, which is referred to as the potato peeler perspective, to show conceptually that data in SPECT with non-uniform attenuation and/or 3D DDSR acquired over 2π contain redundant information and that such information can be used to reduce the scanning angle in SPECT. Specifically, we show heuristically that, in SPECT with only non-uniform attenuation, the scanning angle can be reduced from 2π to π and that, in SPECT with both non-uniform attenuation and DDSR with a physically realistic form, the scanning angle can be reduced from 2π to π in a practical sense. We conduct computer simulation studies, and the results from these studies corroborate the observations obtained based upon the heuristic potato peeler perspective.

The papers in this special issue were invited from contributors to theSixth International Meeting on Fully Three-Dimensional Image Reconstruction in Radiology and Nuclear Medicine held at Pacific Grove, California, USA (30 October - 2 November 2001). This was the sixth in a series of biennial meetings with the aim of bringing together people actively researching problems related to fully three-dimensional tomography in radiology and nuclear medicine. The previous five meetings have been held at Corsendonk, Belgium (1991); Snowbird, Utah, USA (1993); Aix-les-Bains, France (1995); Laurel Highlands, Pennsylvania, USA (1997); and Egmond aan Zee, the Netherlands (1999).

To encourage discussions and the exchange of ideas and techniques, the Asilomar Conference Center with its relaxed setting was chosen for 2001. Approximately 110 participants were in attendance. Tuition support was provided for all students who applied.

There were 84 abstracts submitted to the meeting. Of these, 27 were chosen for oral sessions (35 minute presentations, including discussion) and 26 were chosen for poster sessions. A keynote address entitled `Why Do Patients and Their Care-Givers Need 3D and 4D Imaging?' was delivered by Thomas F Budinger.

Authors of the 53 presentations were invited to submit manuscripts for this special issue of Physics in Medicine and Biology. Of these, 35 were submitted of which 19 were accepted for publication.

Shortly before the 2001 meeting we learned of the passing of our dear friend and colleague, Paul Edholm. A short remembrance of the man and his contributions follows this editorial.

Finally, I would like to thank those, in addition to the participants, who made this meeting a success. Our International Scientific Committee, Supplementary Manuscript Reviewers, and Sponsors are listed below, as is the Local Organizing Committee. Of special note are Martin S Boswell and Michelle K Huesman, whose tireless efforts made my job significantly easier.

R H Huesman Special Issue Editor

Local Organizing Committee at the Center for Functional Imaging, LBL

Ronald H Huesman Gregory J Klein Jinyi Qi Bryan W Reutter Martin S Boswell Michelle K Huesman Hilma C Johnsen Thomas F Budinger

International Scientific Committee

Dale Bailey, Guy's Hospital, London, England Harrison H Barrett, University of Arizona, USA Freek Beekman, University Hospital Utrecht, The Netherlands Thomas F Budinger, Lawrence Berkeley Lab, USA Irene Buvat, U494 INSERM, Paris, France Anna Celler, Vancouver Hospital, Canada Margaret E Daube-Witherspoon, University of Pennsylvania, USA Michel Defrise, Free University of Brussels, Belgium Jeffrey A Fessler, University of Michigan, USA Pierre Grangeat, LETI/CEA, Grenoble, France Grant T Gullberg, University of Utah, USA Ronald H Huesman, Lawrence Berkeley Lab, USA Ronald J Jaszczak, Duke University, USA Paul E Kinahan, University of Pittsburgh, USA Gregory J Klein, Lawrence Berkeley Lab, USA Hiroyuki Kudo, University of Tsukuba, Japan Christian Michel, Universite Catholique de Louvain, Belgium Steve R Meikle, Royal Prince Alfred Hospital, Australia Jinyi Qi, Lawrence Berkeley Lab, USA Bryan W Reutter, Lawrence Berkeley Lab, USA Vesna Sossi, UBC/TRIUMF, Vancouver, Canada David W Townsend, University of Pittsburgh, USA Benjamin M W Tsui, University of North Carolina, USA Gengsheng L Zeng, University of Utah, USA

Supplementary Manuscript Reviewers

James E Bowsher, Duke University, USA David Brasse, University of Pittsburgh, USA Rolf Clackdoyle, University of Utah, USA Richard M Leahy, University of Southern California, USA Scott D Metzler, Duke University, USA Frederic Noo, University of Utah, USA Katsuyuki Taguchi, Toshiba Corp, Japan

Sponsors

CTI, Inc. GE Medical Systems ADAC Laboratories Apple Computer, Inc. US Department of Energy Office of Science

Host

Center for Functional Imaging, EO Lawrence Berkeley National Laboratory

Paul Edholm in memoriam

Our friend and colleague Professor Paul (Palle) Edholm passed away on 9 October 2001. He was born 1926, got his MD from the Karolinska Institue in Stockholm in 1960, and was appointed Professor of Radiology at Linköping University in 1970.

In hindsight many of us can't say when and why we chose a specific branch or problem as our research interest. Not so with Paul. Some time in the 1950s he had his first encounter with classical linear tomo-synthesis and after that his mind was set. He became absolutely fascinated by the problem of ridding these tomograms of the shadows cast onto the plane in focus by neighbouring planes. He constantly pursued possible solutions presented in 1960 in his thesis `The tomogram, its formation and content'. Not many colleagues in his native country, MDs and PhDs alike, could follow or appreciate his work at this time. Paul pushed ahead single-handedly, keeping an eye on the international developments. He began to consider another data set, which he later coined the sinogram. Always aware of the practical side of things, he felt that the processing of these data, including the necessary filtering step, had to be done by optical means. From his manuscripts in the late 1960s to early 1970s it is clear that he was quite close to a complete understanding of what is now known as the filtered back-projection reconstruction method.

While struggling with his mechanical-optical designs he received the news from EMI in England about the first computer-aided tomography scanner, the importance of which he understood immediately. For many years he became the most wanted lecturer in Sweden on radiology and medical imaging. And he was more than pleased when his alma mater Karolinska in 1979 decided to give the Nobel prize in medicine and physiology to the inventors of CT.

After his arrival in Linköping in the early 1970s his creative mind started to generate several new ideas. One of these was coined ectomography. Instead of the linear movement in classical tomography, the x-ray source in ectomography moves in a plane above the patient, under which 2D-projections are captured on film. This scanning geometry was later carried over to a SPECT camera design with a slanted and rotating collimator. In the mid 1980s he conceived the linogram, which can be seen as an unconventional sampling of the Radon space. A useful feature of the linogram is that the samples in the Fourier domain appear along concentric squares instead of circles as in the sinogram case. Much of this work came about during his sabbaticals with the Medical Image Processing Group at the University of Pennsylvania and in collaboration with members of that group. The frequency-distance transform also stems from this time period. During rotation, points of a flat object that reside on a line parallel to the detector, all have the same velocity vector projected onto the detector. Hence, the sinusoids of these points in the sinogram will at this instant have the same slope. Therefore, in the 2D Fourier transform of the sinogram, called the F2S-data space, the contributions originating from object points lined up at a certain distance from the detector will appear along a line through origin in F2S!

The 1990s were Paul's last decade as a scientist. He spent this period mostly with the EE-department at Linköping University to the benefit of many PhD-students and colleagues alike. His lectures on CT, MRI, optical illusions and many other subjects used to keep the audience spellbound. His modest personality, his sense of humour, and the consistency he showed in pursuing a problem made him a wonderful team-player in this new environment. In spite of an eye defect that prevented him from using true stereovision, his ability to see and think in three dimensions was outstanding. His approach was always to reduce as much as possible to pure geometry.

To travel with Paul was an intellectual joy. He could recite long passages of Ulysses by heart. Digressions on technology in the Roman Empire went along with explanations on why the sunlight under tall trees comes in circular patches but appears as waving lines on a sandy water bed. More than anything else, however, the living kingdom and its astounding diversity kept him fascinated throughout his life. He lived and died as a genuine Darwinist. To him it was self-evident that no man is anything but a small part of Nature's great cycle.

Per-Erik Danielsson