Analysis of an exact inversion algorithm for spiral cone-beam CT

Evaluation of the spatial resolution characteristics of a cone-beam breast CT scanner

The purpose of this study was to examine the spatial resolution of a prototype pendant-geometry cone-beam breast computed tomography (CT) system. Modulation transfer functions (MTFs) of the reconstructed image in the coronal (x and y) plane were computed as a function of the cone angle, the radial distance from the axis of rotation, the size of the reconstruction matrix, the back-projection filter used, and the number of projections acquired for the reconstruction. The results show that the cone angle and size of the reconstruction matrix have minimal impact on the MTF, while the MTF degraded radially from the axis of rotation (from 0.76 at 2.6 mm from axis of rotation down to 0.37 at 76.9 mm from axis of rotation at f=0.5 mm(-1)). The Ramp reconstruction filter increases the MTF near the axis of rotation relative to the Shepp-Logan filter, while an increase in the number of projections from 500 to 1000 increased the MTF near the periphery of the reconstructed image. The MTF in the z direction (anterior-posterior direction) was also evaluated. The z-direction MTF values tend to be higher when compared to the coronal MTF (0.85 at f =0.5 mm(-1)), and tend to be very constant throughout the coronal plane direction. The results suggest that an increase in the MTF for the prototype breast CT system is possible by optimizing various scanning and reconstruction parameters.

Tilted cone-beam reconstruction with row-wise fan-to-parallel rebinning

Reconstruction algorithms for cone-beam CT have been the focus of many studies. Several exact and approximate reconstruction algorithms were proposed for step-and-shoot and helical scanning trajectories to combat cone-beam related artefacts. In this paper, we present a new closed-form cone-beam reconstruction formula for tilted gantry data acquisition. Although several algorithms were proposed in the past to combat errors induced by the gantry tilt, none of the algorithms addresses the scenario in which the cone-beam geometry is first rebinned to a set of parallel beams prior to the filtered backprojection. We show that the image quality advantages of the rebinned parallel-beam reconstruction are significant, which makes the development of such an algorithm necessary. Because of the rebinning process, the reconstruction algorithm becomes more complex and the amount of iso-centre adjustment depends not only on the projection and tilt angles, but also on the reconstructed pixel location. In this paper, we first demonstrate the advantages of the row-wise fan-to-parallel rebinning and derive a closed-form solution for the reconstruction algorithm for the step-and-shoot and constant-pitch helical scans. The proposed algorithm requires the 'warping' of the reconstruction matrix on a view-by-view basis prior to the backprojection step. We further extend the algorithm to the variable-pitch helical scans in which the patient table travels at non-constant speeds. The algorithm was tested extensively on both the 16- and 64-slice CT scanners. The efficacy of the algorithm is clearly demonstrated by multiple experiments.

Extending Three-Dimensional Weighted Cone Beam Filtered Backprojection (CB-FBP) Algorithm for Image Reconstruction in Volumetric CT at Low Helical Pitches

A three-dimensional (3D) weighted helical cone beam filtered backprojection (CB-FBP) algorithm (namely, original 3D weighted helical CB-FBP algorithm) has already been proposed to reconstruct images from the projection data acquired along a helical trajectory in angular ranges up to [0, 2 π]. However, an overscan is usually employed in the clinic to reconstruct tomographic images with superior noise characteristics at the most challenging anatomic structures, such as head and spine, extremity imaging, and CT angiography as well. To obtain the most achievable noise characteristics or dose efficiency in a helical overscan, we extended the 3D weighted helical CB-FBP algorithm to handle helical pitches that are smaller than 1: 1 (namely extended 3D weighted helical CB-FBP algorithm). By decomposing a helical over scan with an angular range of [0, 2π + Δβ] into a union of full scans corresponding to an angular range of [0, 2π], the extended 3D weighted function is a summation of all 3D weighting functions corresponding to each full scan. An experimental evaluation shows that the extended 3D weighted helical CB-FBP algorithm can improve noise characteristics or dose efficiency of the 3D weighted helical CB-FBP algorithm at a helical pitch smaller than 1: 1, while its reconstruction accuracy and computational efficiency are maintained. It is believed that, such an efficient CB reconstruction algorithm that can provide superior noise characteristics or dose efficiency at low helical pitches may find its extensive applications in CT medical imaging.

A combination-weighted Feldkamp-based reconstruction algorithm for cone-beam CT

The combination-weighted Feldkamp algorithm (CW-FDK) was developed and tested in a phantom in order to reduce cone-beam artefacts and enhance cranio-caudal reconstruction coverage in an attempt to improve image quality when utilizing cone-beam computed tomography (CBCT). Using a 256-slice cone-beam CT (256CBCT), image quality (CT-number uniformity and geometrical accuracy) was quantitatively evaluated in phantom and clinical studies, and the results were compared to those obtained with the original Feldkamp algorithm. A clinical study was done in lung cancer patients under breath holding and free breathing. Image quality for the original Feldkamp algorithm is degraded at the edge of the scan region due to the missing volume, commensurate with the cranio-caudal distance between the reconstruction and central planes. The CW-FDK extended the reconstruction coverage to equal the scan coverage and improved reconstruction accuracy, unaffected by the cranio-caudal distance. The extended reconstruction coverage with good image quality provided by the CW-FDK will be clinically investigated for improving diagnostic and radiotherapy applications. In addition, this algorithm can also be adapted for use in relatively wide cone-angle CBCT such as with a flat-panel detector CBCT.

Formulation of four Katsevich algorithms in native geometry

We derive formulations of the four exact helical Katsevich algorithms in the native cylindrical detector geometry, which allow efficient implementation in modern computed tomography scanners with wide cone beam aperture. Also, we discuss some aspects of numerical implementation.

Handling of long objects in iterative improvement of nonexact reconstruction in helical cone-beam CT

In medical helical cone-beam CT, it is common that the region-of-interest (ROI) is contained inside the helix cylinder, while the complete object is long and extends outside the top and the bottom of the cylinder. This is the Long Object Problem. Analytical reconstruction methods for helical cone-beam CT have been designed to handle this problem. It has been shown that a moderate amount of over-scanning is sufficient for reconstruction of a certain ROI. The over-scanning projection rays travel both through the ROI, as well as outside the ROI. This is unfortunate for iterative methods since it seems impossible to compute accurate values for the projection rays which travel partly inside and partly outside the ROI. Therefore, it seems that the useful ROI will diminish for every iteration step. We propose the following solution to the problem. First, we reconstruct volume regions also outside the ROI. These volume regions will certainly be incompletely reconstructed, but our experimental results show that they serve well for projection generation. This is rather counter-intuitive and contradictory to our initial assumptions. Second, we use careful extrapolation and masking of projection data. This is not a general necessity, but needed for the chosen iterative algorithm, which includes rebinning and iterative filtered backprojection. Our idea here was to use an approximate reconstruction method which gives cone-beam artifacts and then improve the reconstructed result by iterative filtered backprojection. The experimental results seem very encouraging. The cone-beam artifacts can indeed be removed. Even voxels close to the boundary of the ROI are as well enhanced by the iterative loop as those in the middle of the ROI.

The radon-split method for helical cone-beam CT and its application to nongated reconstruction

The mathematical analysis of exact filtered back-projection algorithms is strictly related to Radon inversion. We show how filter-lines can be defined for the helical trajectory, which serve for the extraction of contributions of particular kinds of Radon-planes. Due to the Fourier-slice theorem, Radon-planes with few intersections with the helix are associated with low-frequency contributions to transversal slices. This insight leads to different applications of the new method. The application presented here enables the incorporation of an arbitrary amount of redundant data in an approximate way. This means that the back-projection is not restricted to an n-Pi interval. A detailed mathematical analysis, in which we demonstrate how the defined filter-lines work, concludes this paper.

Image reconstruction

We give an overview of the role of Physics in Medicine and Biology in the development of tomographic reconstruction algorithms. We focus on imaging modalities involving ionizing radiation, CT, PET and SPECT, and cover a wide spectrum of reconstruction problems, starting with classical 2D tomography in the 1970s up to 4D and 5D problems involving dynamic imaging of moving organs.

A three-dimensional-weighted cone beam filtered backprojection (CB-FBP) algorithm for image reconstruction in volumetric CT—helical scanning

Based on the structure of the original helical FDK algorithm, a three-dimensional (3D)-weighted cone beam filtered backprojection (CB-FBP) algorithm is proposed for image reconstruction in volumetric CT under helical source trajectory. In addition to its dependence on view and fan angles, the 3D weighting utilizes the cone angle dependency of a ray to improve reconstruction accuracy. The 3D weighting is ray-dependent and the underlying mechanism is to give a favourable weight to the ray with the smaller cone angle out of a pair of conjugate rays but an unfavourable weight to the ray with the larger cone angle out of the conjugate ray pair. The proposed 3D-weighted helical CB-FBP reconstruction algorithm is implemented in the cone-parallel geometry that can improve noise uniformity and image generation speed significantly. Under the cone-parallel geometry, the filtering is naturally carried out along the tangential direction of the helical source trajectory. By exploring the 3D weighting's dependence on cone angle, the proposed helical 3D-weighted CB-FBP reconstruction algorithm can provide significantly improved reconstruction accuracy at moderate cone angle and high helical pitches. The 3D-weighted CB-FBP algorithm is experimentally evaluated by computer-simulated phantoms and phantoms scanned by a diagnostic volumetric CT system with a detector dimension of 64 x 0.625 mm over various helical pitches. The computer simulation study shows that the 3D weighting enables the proposed algorithm to reach reconstruction accuracy comparable to that of exact CB reconstruction algorithms, such as the Katsevich algorithm, under a moderate cone angle (4 degrees) and various helical pitches. Meanwhile, the experimental evaluation using the phantoms scanned by a volumetric CT system shows that the spatial resolution along the z-direction and noise characteristics of the proposed 3D-weighted helical CB-FBP reconstruction algorithm are maintained very well in comparison to the FDK-type algorithms. Moreover, the experimental evaluation by clinical data verifies that the proposed 3D-weighted CB-FBP algorithm for image reconstruction in volumetric CT under helical source trajectory meets the challenges posed by diagnostic applications of volumetric CT imaging.

Parallel Implementation of Katsevich's FBP Algorithm

For spiral cone-beam CT, parallel computing is an effective approach to resolving the problem of heavy computation burden. It is well known that the major computation time is spent in the backprojection step for either filtered-backprojection (FBP) or backprojected-filtration (BPF) algorithms. By the cone-beam cover method [1], the backprojection procedure is driven by cone-beam projections, and every cone-beam projection can be backprojected independently. Basing on this fact, we develop a parallel implementation of Katsevich's FBP algorithm. We do all the numerical experiments on a Linux cluster. In one typical experiment, the sequential reconstruction time is 781.3 seconds, while the parallel reconstruction time is 25.7 seconds with 32 processors.

Towards an inline reconstruction architecture for micro-CT systems

Recent developments in micro-CT have revolutionized the ability to examine in vivo living experimental animal models such as mouse with a spatial resolution less than 50 microm. The main requirements of in vivo imaging for biological researchers are a good spatial resolution, a low dose induced to the animal during the full examination and a reduced acquisition and reconstruction time for screening purposes. We introduce inline acquisition and reconstruction architecture to obtain in real time the 3D attenuation map of the animal fulfilling the three previous requirements. The micro-CT system is based on commercially available x-ray detector and micro-focus x-ray source. The reconstruction architecture is based on a cluster of PCs where a dedicated communication scheme combining serial and parallel treatments is implemented. In order to obtain high performance transmission rate between the detector and the reconstruction architecture, a dedicated data acquisition system is also developed. With the proposed solution, the time required to filter and backproject a projection of 2048 x 2048 pixels inside a volume of 140 mega voxels using the Feldkamp algorithm is similar to 500 ms, the time needed to acquire the same projection.

A three-dimensional reconstruction algorithm for an inverse-geometry volumetric CT system

An inverse-geometry volumetric computed tomography (IGCT) system has been proposed capable of rapidly acquiring sufficient data to reconstruct a thick volume in one circular scan. The system uses a large-area scanned source opposite a smaller detector. The source and detector have the same extent in the axial, or slice, direction, thus providing sufficient volumetric sampling and avoiding cone-beam artifacts. This paper describes a reconstruction algorithm for the IGCT system. The algorithm first rebins the acquired data into two-dimensional (2D) parallel-ray projections at multiple tilt and azimuthal angles, followed by a 3D filtered backprojection. The rebinning step is performed by gridding the data onto a Cartesian grid in a 4D projection space. We present a new method for correcting the gridding error caused by the finite and asymmetric sampling in the neighborhood of each output grid point in the projection space. The reconstruction algorithm was implemented and tested on simulated IGCT data. Results show that the gridding correction reduces the gridding errors to below one Hounsfield unit. With this correction, the reconstruction algorithm does not introduce significant artifacts or blurring when compared to images reconstructed from simulated 2D parallel-ray projections. We also present an investigation of the noise behavior of the method which verifies that the proposed reconstruction algorithm utilizes cross-plane rays as efficiently as in-plane rays and can provide noise comparable to an in-plane parallel-ray geometry for the same number of photons. Simulations of a resolution test pattern and the modulation transfer function demonstrate that the IGCT system, using the proposed algorithm, is capable of 0.4 mm isotropic resolution. The successful implementation of the reconstruction algorithm is an important step in establishing feasibility of the IGCT system.

Evaluation of x-ray scatter properties in a dedicated cone-beam breast CT scanner

The magnitude of scatter contamination on a first-generation prototype breast computed tomography (CT) scanner was evaluated using the scatter-to-primary ratio (SPR) metric. The SPR was measured and characterized over a wide range of parameters relevant to breast CT imaging, including x-ray beam energy, breast diameter, breast composition, isocenter-to-detector distance, collimated slot thickness, and grid ratio. The results demonstrated that in the absence of scatter reduction techniques, the SPR levels for the average breast (e.g., 14 cm diameter 50/50 composition cylindrical phantom) are quite high (approximately 0.5 at the center of the phantom for 80 kVp in true cone-beam CT geometry), and increases as the diameter of the phantom is increased (to approximately 1.0 at the center of a 18 cm diameter 50/50 phantom). The x-ray beam energy and the phantom compositions had only minimal impact on the measured SPR. When an ideal bowtie filter was used, the SPRs at the central axis of the 14 and 18 cm cylindrical phantoms were reduced while the SPRs at the edge of the phantoms were increased. Lastly, collimation in the vertical direction had a significant impact on the SPRs at the central axis of the phantoms. These high SPR levels might lead to cupping artifacts and increased noise in the reconstructed CT images, and this suggests that efficient scatter rejection and/or correction techniques may be required to improve the quality and accuracy of cone beam CT images.

Cardiac cone-beam CT volume reconstruction using ART

Modern computed tomography systems allow volume imaging of the heart. Up to now, approximately two-dimensional (2D) and 3D algorithms based on filtered backprojection are used for the reconstruction. These algorithms become more sensitive to artifacts when the cone angle of the x-ray beam increases as it is the current trend of computed tomography (CT) technology. In this paper, we investigate the potential of iterative reconstruction based on the algebraic reconstruction technique (ART) for helical cardiac cone-beam CT. Iterative reconstruction has the advantages that it takes the cone angle into account exactly and that it can be combined with retrospective cardiac gating fairly easily. We introduce a modified ART algorithm for cardiac CT reconstruction. We apply it to clinical cardiac data from a 16-slice CT scanner and compare the images to those obtained with a current analytical reconstruction method. In a second part, we investigate the potential of iterative reconstruction for a large area detector with 256 slices. For the clinical cases, iterative reconstruction produces excellent images of diagnostic quality. For the large area detector, iterative reconstruction produces images superior to analytical reconstruction in terms of cone-beam artifacts.

A three-dimensional weighted cone beam filtered backprojection (CB-FBP) algorithm for image reconstruction in volumetric CT under a circular source trajectory

The original FDK algorithm proposed for cone beam (CB) image reconstruction under a circular source trajectory has been extensively employed in medical and industrial imaging applications. With increasing cone angle, CB artefacts in images reconstructed by the original FDK algorithm deteriorate, since the circular trajectory does not satisfy the so-called data sufficiency condition (DSC). A few 'circular plus' trajectories have been proposed in the past to help the original FDK algorithm to reduce CB artefacts by meeting the DSC. However, the circular trajectory has distinct advantages over other scanning trajectories in practical CT imaging, such as head imaging, breast imaging, cardiac, vascular and perfusion applications. In addition to looking into the DSC, another insight into the CB artefacts existing in the original FDK algorithm is the inconsistency between conjugate rays that are 180 degrees apart in view angle (namely conjugate ray inconsistency). The conjugate ray inconsistency is pixel dependent, varying dramatically over pixels within the image plane to be reconstructed. However, the original FDK algorithm treats all conjugate rays equally, resulting in CB artefacts that can be avoided if appropriate weighting strategies are exercised. Along with an experimental evaluation and verification, a three-dimensional (3D) weighted axial cone beam filtered backprojection (CB-FBP) algorithm is proposed in this paper for image reconstruction in volumetric CT under a circular source trajectory. Without extra trajectories supplemental to the circular trajectory, the proposed algorithm applies 3D weighting on projection data before 3D backprojection to reduce conjugate ray inconsistency by suppressing the contribution from one of the conjugate rays with a larger cone angle. Furthermore, the 3D weighting is dependent on the distance between the reconstruction plane and the central plane determined by the circular trajectory. The proposed 3D weighted axial CB-FBP algorithm can be implemented in either the native CB geometry or the so-called cone-parallel geometry. By taking the cone-parallel geometry as an example, the experimental evaluation shows that, up to a moderate cone angle corresponding to a detector dimension of 64 x 0.625 mm, the CB artefacts can be substantially suppressed by the proposed algorithm, while advantages of the original FDK algorithm, such as the filtered backprojection algorithm structure, 1D ramp filtering and data manipulation efficiency, are maintained.

EnPiT: filtered back-projection algorithm for helical CT using an n-Pi acquisition

In this paper, we formulate a reconstruction algorithm for an n-Pi acquisition, where n can be any positive odd integer. The algorithm is a generalization of the method presented in (Bontus et al. 2003). It is based on the results obtained by Katsevich (2004). For the algorithm, different sets of filter-lines have to be defined. We describe the variation of these lines along the detector in some detail, before we discuss, how the method gives all Radon-plane contributions the correct weighting. The different sets of filter-lines are all contained within the n-Pi window, such that a practical realization is possible. Reconstruction results, which we present in the final section, show convincing image quality.

PI-line-based image reconstruction in helical cone-beam computed tomography with a variable pitch

Current applications of helical cone-beam computed tomography (CT) involve primarily a constant pitch where the translating speed of the table and the rotation speed of the source-detector remain constant. However, situations do exist where it may be more desirable to use a helical scan with a variable translating speed of the table, leading a variable pitch. One of such applications could arise in helical cone-beam CT fluoroscopy for the determination of vascular structures through real-time imaging of contrast bolus arrival. Most of the existing reconstruction algorithms have been developed only for helical cone-beam CT with constant pitch, including the backprojection-filtration (BPF) and filtered-backprojection (FBP) algorithms that we proposed previously. It is possible to generalize some of these algorithms to reconstruct images exactly for helical cone-beam CT with a variable pitch. In this work, we generalize our BPF and FBP algorithms to reconstruct images directly from data acquired in helical cone-beam CT with a variable pitch. We have also performed a preliminary numerical study to demonstrate and verify the generalization of the two algorithms. The results of the study confirm that our generalized BPF and FBP algorithms can yield exact reconstruction in helical cone-beam CT with a variable pitch. It should be pointed out that our generalized BPF algorithm is the only algorithm that is capable of reconstructing exactly region-of-interest image from data containing transverse truncations.

Fan-beam and cone-beam image reconstruction via filtering the backprojection image of differentiated projection data

In this paper, a new image reconstruction scheme is presented based on Tuy's cone-beam inversion scheme and its fan-beam counterpart. It is demonstrated that Tuy's inversion scheme may be used to derive a new framework for fanbeam and cone-beam image reconstruction. In this new framework, images are reconstructed via filtering the backprojection image of differentiated projection data. The new framework is mathematically exact and is applicable to a general source trajectory provided the Tuy data sufficiency condition is satisfied. By choosing a piece-wise constant function for one of the components in the factorized weighting function, the filtering kernel is one dimensional, viz. the filtering process is along a straight line. Thus, the derived image reconstruction algorithm is mathematically exact and efficient. In the cone-beam case, the derived reconstruction algorithm is applicable to a large class of source trajectories where the pi-lines or the generalized pi-lines exist. In addition, the new reconstruction scheme survives the super-short scan mode in both the fan-beam and cone-beam cases provided the data are not transversely truncated. Numerical simulations were conducted to validate the new reconstruction scheme for the fan-beam case.

A unified framework for exact cone-beam reconstruction formulas

In this paper, we present concise proofs of several recently developed exact cone-beam reconstruction methods in the Tuy inversion framework, including both filtered-backprojection and backprojection-filtration formulas in the cases of standard spiral, nonstandard spiral, and more general scanning loci. While a similar proof of the Katsevich formula was previously reported, we present a new proof of the Zou and Pan backprojection-filtration formula. Our proof combines both odd and even data extensions so that only the cone-beam transform itself is utilized in the backprojection-filtration inversion. More importantly, our formulation is valid for general smooth scanning curves, in agreement with an earlier paper from our group [Ye, Zhao, Yu, and Wang, Proc. SPIE 5535, 293-300 (Aug. 6 2004)]. As a consequence of that proof, we obtain a new inversion formula, which is in a two-dimensional filtering backprojection format. A possibility for generalization of the Katsevich filtered-backprojection reconstruction method is also discussed from the viewpoint of this framework.

Image reconstruction for the circle-and-arc trajectory

Proposed is an exact shift-invariant filtered backprojection algorithm for the circle-and-arc trajectory. The algorithm has several important features. First, it allows for the circle to be incomplete. Second, axial truncation of the cone beam data is allowed. Third, the length of the arc is determined only by the region of interest and is independent of the size of the entire object. The algorithm is quite flexible and can be used for even more general trajectories that consist of several circular segments and arcs. The algorithm applies also in the case when the circle (or, circles) is complete. A numerical experiment with the clock phantom demonstrated good image quality.

Exact fan-beam image reconstruction algorithm for truncated projection data acquired from an asymmetric half-size detector

In this paper, we present a new algorithm designed for a specific data truncation problem in fan-beam CT. We consider a scanning configuration in which the fan-beam projection data are acquired from an asymmetrically positioned half-sized detector. Namely, the asymmetric detector only covers one half of the scanning field of view. Thus, the acquired fan-beam projection data are truncated at every view angle. If an explicit data rebinning process is not invoked, this data acquisition configuration will reek havoc on many known fan-beam image reconstruction schemes including the standard filtered backprojection (FBP) algorithm and the super-short-scan FBP reconstruction algorithms. However, we demonstrate that a recently developed fan-beam image reconstruction algorithm which reconstructs an image via filtering a backprojection image of differentiated projection data (FBPD) survives the above fan-beam data truncation problem. Namely, we may exactly reconstruct the whole image object using the truncated data acquired in a full scan mode (2pi angular range). We may also exactly reconstruct a small region of interest (ROI) using the truncated projection data acquired in a short-scan mode (less than 2pi angular range). The most important characteristic of the proposed reconstruction scheme is that an explicit data rebinning process is not introduced. Numerical simulations were conducted to validate the new reconstruction algorithm.

Helical cardiac cone beam CT reconstruction with large area detectors: a simulation study

Retrospectively gated cardiac volume CT imaging has become feasible with the introduction of heart rate adaptive cardiac CT reconstruction algorithms. The development in detector technology and the rapid introduction of multi-row detectors has demanded reconstruction schemes which account for the cone geometry. With the extended cardiac reconstruction (ECR) framework, the idea of approximate helical cone beam CT has been extended to be used with retrospective gating, enabling heart rate adaptive cardiac cone beam reconstruction. In this contribution, the ECR technique is evaluated for systems with an increased number of detector rows, which leads to larger cone angles. A simulation study has been carried out based on a 4D cardiac phantom consisting of a thorax model and a dynamic heart insert. Images have been reconstructed for different detector set-ups. Reconstruction assessment functions have been calculated for the detector set-ups employing different rotation times, relative pitches and heart rates. With the increased volume coverage of large area detector systems, low-pitch scans become feasible without resulting in extensive scan times, inhibiting single breath hold acquisitions. ECR delivers promising image results when being applied to systems with larger cone angles.

The frequency split method for helical cone-beam reconstruction

A new approximate method for the utilization of redundant data in helical cone-beam CT is presented. It is based on the observation that the original WEDGE method provides excellent image quality if only little more than 180 degrees data are used for back-projection, and that significant low-frequency artifacts appear if a larger amount of redundant data are used. This degradation is compensated by the frequency split method: The low-frequency part of the image is reconstructed using little more than 180 degrees of data, while the high frequency part is reconstructed using all data. The resulting algorithm shows no cone-beam artifacts in a simulation of a 64-row scanner. It is further shown that the frequency split method hardly degrades the signal-to-noise ratio of the reconstructed images and that it behaves robustly in the presence of motion.

Exact and approximate algorithms for helical cone-beam CT

This paper concerns image reconstruction for helical x-ray transmission tomography (CT) with multi-row detectors. We introduce two approximate cone-beam (CB) filtered-backprojection (FBP) algorithms of the Feldkamp type, obtained by extending to three dimensions (3D) two recently proposed exact FBP algorithms for 2D fan-beam reconstruction. The new algorithms are similar to the standard Feldkamp-type FBP for helical CT. In particular, they can reconstruct each transaxial slice from data acquired along an arbitrary segment of helix, thereby efficiently exploiting the available data. In contrast to the standard Feldkamp-type algorithm, however, the redundancy weight is applied after filtering, allowing a more efficient numerical implementation. To partially alleviate the CB artefacts, which increase with increasing values of the helical pitch, a frequency-mixing method is proposed. This method reconstructs the high frequency components of the image using the longest possible segment of helix, whereas the low frequencies are reconstructed using a minimal, short-scan, segment of helix to minimize CB artefacts. The performance of the algorithms is illustrated using simulated data.

Image reconstruction for the circle and line trajectory

We propose an exact shift-invariant filtered backprojection algorithm for inversion of the cone beam data in the case when the source trajectory consists of an incomplete circle and a line segment. The algorithm allows for axial truncation of the cone beam data. The length of the line scan is determined only by the region of interest and is independent of the size of the entire object. The algorithm is quite flexible and can also be used for more general trajectories consisting of several (complete or incomplete) circles and line segments. Results of numerical experiments demonstrate good image quality.

A filtered backprojection algorithm for cone beam reconstructionusing rotational filtering under helical source trajectory

With the evolution from multi-detector-row CT to cone beam (CB) volumetric CT, maintaining reconstruction accuracy becomes more challenging. To combat the severe artifacts caused by a large cone angle in CB volumetric CT, three-dimensional reconstruction algorithms have to be utilized. In practice, filtered backprojection (FBP) reconstruction algorithms are more desirable due to their computational structure and image generation efficiency. One of the CB-FBP reconstruction algorithms is the well-known FDK algorithm that was originally derived for a circular x-ray source trajectory by heuristically extending its two-dimensional (2-D) counterpart. Later on, a general CB-FBP reconstruction algorithm was derived for noncircular, such as helical, source trajectories. It has been recognized that a filtering operation in the projection data along the tangential direction of a helical x-ray source trajectory can significantly improve the reconstruction accuracy of helical CB volumetric CT. However, the tangential filtering encounters latitudinal data truncation, resulting in degraded noise characteristics or data manipulation inefficiency. A CB-FBP reconstruction algorithm using one-dimensional rotational filtering across detector rows (namely CB-RFBP) is proposed in this paper. Although the proposed CB-RFBP reconstruction algorithm is approximate, it approaches the reconstruction accuracy that can be achieved by exact helical CB-FBP reconstruction algorithms for moderate cone angles. Unlike most exact CB-FBP reconstruction algorithms in which the redundant data are usually discarded, the proposed CB-RFBP reconstruction algorithm make use of all available projection data, resulting in significantly improved noise characteristics and dose efficiency. Moreover, the rotational filtering across detector rows not only survives the so-called long object problem, but also avoids latitudinal data truncation existing in other helical CB-FBP reconstruction algorithm in which a tangential filtering is carried out, providing better noise characteristics, dose efficiency and data manipulation efficiency.

An alternative derivation of Katsevich's cone-beam reconstruction formula

In this paper an alternative derivation of Katsevich's cone-beam image reconstruction algorithm is presented. The starting point is the classical Tuy's inversion formula. After (i) using the hidden symmetries of the intermediate functions, (ii) handling the redundant data by weighting them, (iii) changing the weighted average into an integral over the source trajectory parameter, and (iv) imposing an additional constraint on the weighting function, a filtered backprojection reconstruction formula from cone beam projections is derived. The following features are emphasized in the present paper: First, the nontangential condition in Tuy's original data sufficiency conditions has been relaxed. Second, a practical regularization scheme to handle the singularity is proposed. Third, the derivation in the cone beam case is in the same fashion as that in the fan-beam case. Our final cone-beam reconstruction formula is the same as the one discovered by Katsevich in his most recent paper. However, the data sufficiency conditions and the regularization scheme of singularities are different. A detailed comparison between these two methods is presented.

Exact filtered backprojection reconstruction for dynamic pitch helical cone beam computed tomography

We present an exact filtered backprojection reconstruction formula for helical cone beam computed tomography in which the pitch of the helix varies with time. We prove that the resulting algorithm, which is functionally identical to the constant pitch case, provides exact reconstruction provided that the projection of the helix onto the detector forms convex boundaries and that PI lines are unique. Furthermore, we demonstrate that both of these conditions are satisfied provided the sum of the translational velocity and the derivative of the translational acceleration does not change sign. As a special case, we show that gantry tilt can also be handled by our dynamic pitch formula. Simulation results demonstrate the resulting algorithm.

Spiral scan long object reconstruction through PI line reconstruction

The response of a point object in a cone beam (CB) spiral scan is analysed. Based on the result, a reconstruction algorithm for long object imaging in spiral scan cone beam CT is developed. A region-of-interest (ROI) of the long object is scanned with a detector smaller than the ROI, and a portion of it can be reconstructed without contamination from overlaying materials. The top and bottom surfaces of the ROI are defined by two sets of PI lines near the two ends of the spiral path. With this novel definition of the top and bottom ROI surfaces and through the use of projective geometry, it is straightforward to partition the cone beam image into regions corresponding to projections of the ROI, the overlaying objects or both. This also simplifies computation at source positions near the spiral ends, and makes it possible to reduce radiation exposure near the spiral ends substantially through simple hardware collimation. Simulation results to validate the algorithm are presented.

Extended parallel backprojection for standard three-dimensional and phase-correlated four-dimensional axial and spiral cone-beam CT with arbitrary pitch, arbitrary cone-angle, and 100% dose usage

We have developed a new approximate Feldkamp-type algorithm that we call the extended parallel backprojection (EPBP). Its main features are a phase-weighted backprojection and a voxel-by-voxel 180 degrees normalization. The first feature ensures three-dimensional (3-D) and 4-D capabilities with one and the same algorithm; the second ensures 100% detector usage (each ray is accounted for). The algorithm was evaluated using simulated data of a thorax phantom and a cardiac motion phantom for scanners with up to 256 slices. Axial (circle and sequence) and spiral scan trajectories were investigated. The standard reconstructions (EPBPStd) are of high quality, even for as many as 256 slices. The cardiac reconstructions (EPBPCI) are of high quality as well and show no significant deterioration of objects even far off the center of rotation. Since EPBPCI uses the cardio interpolation (CI) phase weighting the temporal resolution is equivalent to that of the well-established single-slice and multislice cardiac approaches 180 degrees CI, 180 degrees MCI, and ASSRCI, respectively, and lies in the order of 50 to 100 ms for rotation times between 0.4 and 0.5 s. EPBP appears to fulfill all required demands. Especially the phase-correlated EPBP reconstruction of cardiac multiple circle scan data is of high interest, e.g., for dynamic perfusion studies of the heart.

Weighted FBP—a simple approximate 3D FBP algorithm for multislice spiral CT with good dose usage for arbitrary pitch

A new 3D reconstruction scheme, weighted filtered backprojection (WFBP) for multirow spiral CT based on an extension of the two-dimensional SMPR algorithm is described and results are presented. In contrast to other 3D algorithms available, the algorithm makes use of all available data for all pitch values. The algorithm is a FBP algorithm: linear convolution of the parallel data along the row direction followed by a 3D backprojection. Data usage for arbitrary pitch values is maintained through a weighting scheme which takes into account redundant data. If proper row weighting is applied, the image quality is superior to the image quality of the SMPR algorithm.

On two versions of a 3 algorithm for spiral CT

A 3pi algorithm is obtained in which all the derivatives are confined to a detector array. Distance weighting of backprojection coefficients of the algorithm is studied. A numerical experiment indicates that avoiding differentiation along the source trajectory improves spatial resolution. Another numerical experiment shows that the terms depending on the non-standard distance weighting l/[x - y (s)] can no longer be ignored.

A quasiexact reconstruction algorithm for helical CT using a 3-Pi acquisition

Recently, an exact reconstruction method for helical CT was published by A. Katsevich. The algorithm is of the filtered backprojection type and is, therefore, computationally efficient. Moreover, during backprojection, only data are used which correspond to an illumination interval of 180 degrees as seen from the object-point. We propose a new reconstruction method, which is applicable to data obtained with a 3-Pi acquisition [IEEE Trans. Med. Imaging 19, 848-863 (2000)]. The method uses the same filter types as the Katsevich algorithm, but the directions and the number of the filter lines are chosen differently. For the derivation of the new algorithm, we analyze the relationship of the Katsevich method and radon inversion. A certain radon plane can intersect with the backprojection interval related to a 3-Pi acquisition either once, three, or five times. In analogy to the definition of quasiexactness introduced by Kudo et al. for a 1-Pi acquisition, we use the term quasiexactness for algorithms on a 3-Pi acquisition, if radon planes with one or three intersections within the backprojection interval are treated correctly. Using the results on the relationship with radon inversion, we can prove that our algorithm is quasiexact in this sense. We use simulation results in order to demonstrate that the algorithm yields excellent image quality.

Future generation CT imaging

X-ray CT technology has been available for more than 30 years, yet continued technological advances have kept CT imaging at the forefront of medical imaging innovation. Consequently, the number of clinical CT applications has increased steadily. Other imaging modalities might be superior to CT imaging for some specific applications, but no other single modality is more often used in chest imaging today. Future technological developments in the area of high-resolution detectors, high-capacity x-ray tubes, advanced reconstruction algorithms, and improved visualization techniques will continue to expand the imaging capability. Future CT imaging technology will combine improved imaging capability with advanced and specific computer-assisted tools, which will expand the usefulness of CT imaging in many areas.

A simple derivation and analysis of a helical cone beam tomographic algorithm for long object imaging via a novel definition of region of interest

We derive and analyse a simple algorithm first proposed by Kudo et al (2001 Proc. 2001 Meeting on Fully 3D Image Reconstruction in Radiology and Nuclear Medicine (Pacific Grove, CA) pp 7-10) for long object imaging from truncated helical cone beam data via a novel definition of region of interest (ROI). Our approach is based on the theory of short object imaging by Kudo et al (1998 Phys. Med. Biol. 43 2885-909). One of the key findings in their work is that filtering of the truncated projection can be divided into two parts: one, finite in the axial direction, results from ramp filtering the data within the Tam window. The other, infinite in the z direction, results from unbounded filtering of ray sums over PI lines only. We show that for an ROI defined by PI lines emanating from the initial and final source positions on a helical segment, the boundary data which would otherwise contaminate the reconstruction of the ROI can be completely excluded. This novel definition of the ROI leads to a simple algorithm for long object imaging. The overscan of the algorithm is analytically calculated and it is the same as that of the zero boundary method. The reconstructed ROI can be divided into two regions: one is minimally contaminated by the portion outside the ROI, while the other is reconstructed free of contamination. We validate the algorithm with a 3D Shepp-Logan phantom and a disc phantom.

Adaptive temporal resolution optimization in helical cardiac cone beam CT reconstruction

Cone beam computed tomography scanners in combination with heart rate adaptive reconstruction schemes have the potential to enable cardiac volumetric computed tomography (CT) imaging for a larger number of patients and applications. In this publication, an adaptive scheme for the automatic and patient-specific reconstruction optimization is introduced to improve the temporal resolution and image quality. The optimization method permits the automatic determination of the required amount of gated helical cone beam projection data for the reconstruction volume. It furthermore allows one to optimize subvolume reconstruction yielding an increased temporal resolution. In addition, methods for the assessment of the temporal resolution are given which enable a quantitative documentation of the reconstruction improvements. Results are presented for patient data sets acquired in low pitch helical mode using a 16-slice cone beam CT system with parallel ECG recording.

Exact helical reconstruction using native cone-beam geometries

This paper is about helical cone-beam reconstruction using the exact filtered backprojection formula recently suggested by Katsevich (2002a Phys. Med. Biol. 47 2583-97). We investigate how to efficiently and accurately implement Katsevich's formula for direct reconstruction from helical cone-beam data measured in two native geometries. The first geometry is the curved detector geometry of third-generation multi-slice CT scanners, and the second geometry is the flat detector geometry of C-arms systems and of most industrial cone-beam CT scanners. For each of these two geometries, we determine processing steps to be applied to the measured data such that the final outcome is an implementation of the Katsevich formula. These steps are first described using continuous-form equations, disregarding the finite detector resolution and the source position sampling. Next, techniques are presented for implementation of these steps with finite data sampling. The performance of these techniques is illustrated for the curved detector geometry of third-generation CT scanners, with 32, 64 and 128 detector rows. In each case, resolution and noise measurements are given along with reconstructions of the FORBILD thorax phantom.

General surface reconstruction for cone-beam multislice spiral computed tomography

A new family of cone-beam reconstruction algorithm, the General Surface Reconstruction (GSR), is proposed and formulated in this paper for multislice spiral computed tomography (CT) reconstructions. It provides a general framework to allow the reconstruction of planar or nonplanar surfaces on a set of rebinned short-scan parallel beam projection data. An iterative surface formation method is proposed as an example to show the possibility to form nonplanar reconstruction surfaces to minimize the adverse effect between the collected cone-beam projection data and the reconstruction surfaces. The improvement in accuracy of the nonplanar surfaces over planar surfaces in the two-dimensional approximate cone-beam reconstructions is mathematically proved and demonstrated using numerical simulations. The proposed GSR algorithm is evaluated by the computer simulation of cone-beam spiral scanning geometry and various mathematical phantoms. The results demonstrate that the GSR algorithm generates much better image quality compared to conventional multislice reconstruction algorithms. For a table speed up to 100 mm per rotation, GSR demonstrates good image quality for both the low-contrast ball phantom and thorax phantom. All other performance parameters are comparable to the single-slice 180 degrees LI (linear interpolation) algorithm, which is considered the "gold standard." GSR also achieves high computing efficiency and good temporal resolution, making it an attractive alternative for the reconstruction of next generation multislice spiral CT data.

Helical cardiac cone beam reconstruction using retrospective ECG gating

In modern computer tomography (CT) systems, the fast rotating gantry and the increased detector width enable 3D imaging of the heart. Cardiac volume CT has a high potential for non-invasive coronary angiography with high spatial resolution and short scan time. Due to the increased detector width, true cone beam reconstruction methods are needed instead of adapted 2D reconstruction schemes. In this paper, the extended cardiac reconstruction method is introduced. It integrates the idea of retrospectively gated cardiac reconstruction for helical data acquisition into a cone beam reconstruction framework. It leads to an efficient and flexible algorithmic scheme for the reconstruction of single- and multi-phase cardiac volume datasets. The method automatically adapts the number of cardiac cycles used for the reconstruction. The cone beam geometry is fully taken into account during the reconstruction process. Within this paper, results are presented on patient datasets which have been acquired using a 16-slice cone beam CT system.

A new framework of image reconstruction from fan beam projections

In this paper, we present a unified framework to reconstruct images for both fan beam and cone beam projections. The important feature of our theoretical framework is that it does not depend on the classical concept of the Radon transform at all. This property allows us to directly generalize the ideas and techniques developed in this paper to the cone beam reconstruction problem. In this paper, we extract such a framework from developing a new image reconstruction scheme from fan beam projections. Our new scheme also provides us new understanding of fan beam reconstruction problem. Our main results for the fan beam reconstruction are the following: First, we derive a general reconstruction scheme, in which the data sufficiency condition is transparently revealed by the reconstruction formula. Specifically, the data sufficiency condition for an accurate reconstruction of a region of interest (ROI) is that all the lines passing through the ROI must intersect the source trajectory at least once. Second, we further simplify the general reconstruction scheme by following three major steps of our new framework: (i) using symmetries of intermediate function; (ii) handling the data redundancy; (iii) changing discrete summation over the possible focal points into an integral along the source trajectory. After these steps, we obtain a new filtered backprojection algorithm. The key characteristic of this new algorithm is to take derivative of measured data with respect to the trajectory parameter. In practice, we can trade this derivative to some other continuous functions. In the configuration of a circular source trajectory with a third generation arc/collinear detector, we demonstrate how to remove the undesirable derivative of measured projection data. It results in a new algorithm for the sequential reconstruction of a ROI with a general normalized weighting function.

A Grangeat-type half-scan algorithm for cone-beam CT

Modern CT and micro-CT scanners are rapidly moving from fan-beam toward cone-beam geometry. Half-scan CT algorithms are advantageous in terms of temporal resolution, and widely used in fan-beam and cone-beam geometry. While existing half-scan algorithms for cone-beam CT are in the Feldkamp framework, in this paper we compensate missing data explicitly in the Grangeat framework, and formulate a half-scan algorithm in the circular scanning case. The half-scan spans 180 degrees plus two cone angles that guarantee sufficient data for reconstruction of the midplane defined by the source trajectory. The smooth half-scan weighting functions are designed for the suppression of data inconsistency. Numerical simulation results are reported for verification of our formulas and programs. This Grangeat-type half-scan algorithm produces excellent image quality, without off-mid-plane artifacts associated with Feldkamp-type half-scan algorithms. The Grangeat-type half-scan algorithm seems promising for quantitative and dynamic biomedical applications of CT and micro-CT.