Cone-beam filtered-backprojection algorithm for truncated helical data

A Computationally Efficient Reconstruction Algorithm for Circular Cone-Beam Computed Tomography Using Shallow Neural Networks

Circular cone-beam (CCB) Computed Tomography (CT) has become an integral part of industrial quality control, materials science and medical imaging. The need to acquire and process each scan in a short time naturally leads to trade-offs between speed and reconstruction quality, creating a need for fast reconstruction algorithms capable of creating accurate reconstructions from limited data. In this paper, we introduce the Neural Network Feldkamp-Davis-Kress (NN-FDK) algorithm. This algorithm adds a machine learning component to the FDK algorithm to improve its reconstruction accuracy while maintaining its computational efficiency. Moreover, the NN-FDK algorithm is designed such that it has low training data requirements and is fast to train. This ensures that the proposed algorithm can be used to improve image quality in high-throughput CT scanning settings, where FDK is currently used to keep pace with the acquisition speed using readily available computational resources. We compare the NN-FDK algorithm to two standard CT reconstruction algorithms and to two popular deep neural networks trained to remove reconstruction artifacts from the 2D slices of an FDK reconstruction. We show that the NN-FDK reconstruction algorithm is substantially faster in computing a reconstruction than all the tested alternative methods except for the standard FDK algorithm and we show it can compute accurate CCB CT reconstructions in cases of high noise, a low number of projection angles or large cone angles. Moreover, we show that the training time of an NN-FDK network is orders of magnitude lower than the considered deep neural networks, with only a slight reduction in reconstruction accuracy.

Three-Dimensional Weighting in Cone Beam FBP Reconstruction and Its Transformation Over Geometries

GOALS

With substantially increased number of detector rows in multidetector CT (MDCT), axial scan with projection data acquired along a circular source trajectory has become the method-of-choice in increasing clinical applications. Recognizing the practical relevance of image reconstruction directly from the projection data acquired in the native cone beam (CB) geometry, especially in scenarios wherein the most achievable in-plane resolution is desirable, we present a three-dimensional (3-D) weighted CB-FBP algorithm in such geometry in this paper.

METHODS

We start the algorithm's derivation in the cone-parallel geometry. Via changing of variables, taking the Jacobian into account and making heuristic and empirical assumptions, we arrive at the formulas for 3-D weighted image reconstruction in the native CB geometry.

RESULTS

Using the projection data simulated by computer and acquired by an MDCT scanner, we evaluate and verify performance of the proposed algorithm for image reconstruction directly from projection data acquired in the native CB geometry.

CONCLUSION

The preliminary data show that the proposed algorithm performs as well as the 3-D weighted CB-FBP algorithm in the cone-parallel geometry.

SIGNIFICANCE

The proposed algorithm is anticipated to find its utility in extensive clinical and preclinical applications wherein the reconstruction of images in the native CB geometry, i.e., the geometry for data acquisition, is of relevance.

Dual scan CT image recovery from truncated projections

There are computerized tomography (CT) scanners available commercially for imaging small objects and they are often categorized as mini-CT X-ray machines. One major limitation of these machines is their inability to scan large objects with good image quality because of the truncation of projection data. An algorithm is proposed in this work which enables such machines to scan large objects while maintaining the quality of the recovered image.

Reconstruction of 3D Regions-of-Interest from Data in Reduced Helical Cone-beam Scans

The suffciency conditions are derived for exact image reconstruction of a 3D ROI from projections acquired with a reduced helical scan over an angular range considerably smaller than that required by image reconstruction in, e.g., the conventional long object problem, for which the scanned angular range is often more than 2π. ROI reconstruction is investigated by a recently developed filtered-backprojection algorithm that can make use of data acquired with a reduced helical scan. Preliminary numerical studies demonstrate and validate the ROI reconstruction. This work may have significant practical implications because a reduced scan in CT often translates to reduced motion artifacts and reduced radiation dose delivered to the subject.

Axial Cone-Beam Reconstruction by Weighted BPF/DBPF and Orthogonal Butterfly Filtering

GOAL

The backprojection-filtration (BPF) and the derivative backprojection filtered (DBPF) algorithms, in which Hilbert filtering is the common algorithmic feature, are originally derived for exact helical reconstruction from cone-beam (CB) scan data and axial reconstruction from fan beam data, respectively. These two algorithms can be heuristically extended for image reconstruction from axial CB scan data, but induce severe artifacts in images located away from the central plane, determined by the circular source trajectory. We propose an algorithmic solution herein to eliminate the artifacts.

METHODS

The solution is an integration of three-dimensional (3-D) weighted axial CB-BPF/DBPF algorithm with orthogonal butterfly filtering, namely axial CB-BPF/DBPF cascaded with orthogonal butterfly filtering. Using the computer simulated Forbild head and thoracic phantoms that are rigorous in inspecting the reconstruction accuracy, and an anthropomorphic thoracic phantom with projection data acquired by a CT scanner, we evaluate the performance of the proposed algorithm.

RESULTS

Preliminary results show that the orthogonal butterfly filtering can eliminate the severe streak artifacts existing in the images reconstructed by the 3-D weighted axial CB-BPF/DBPF algorithm located at off-central planes.

CONCLUSION

Integrated with orthogonal butterfly filtering, the 3-D weighted CB-BPF/DBPF algorithm can perform at least as well as the 3-D weighted CB-FBP algorithm in image reconstruction from axial CB scan data.

SIGNIFICANCE

The proposed 3-D weighted axial CB-BPF/DBPF cascaded with orthogonal butterfly filtering can be an algorithmic solution for CT imaging in extensive clinical and preclinical applications.

3D algebraic iterative reconstruction for cone-beam x-ray differential phase-contrast computed tomography

Due to the potential of compact imaging systems with magnified spatial resolution and contrast, cone-beam x-ray differential phase-contrast computed tomography (DPC-CT) has attracted significant interest. The current proposed FDK reconstruction algorithm with the Hilbert imaginary filter will induce severe cone-beam artifacts when the cone-beam angle becomes large. In this paper, we propose an algebraic iterative reconstruction (AIR) method for cone-beam DPC-CT and report its experiment results. This approach considers the reconstruction process as the optimization of a discrete representation of the object function to satisfy a system of equations that describes the cone-beam DPC-CT imaging modality. Unlike the conventional iterative algorithms for absorption-based CT, it involves the derivative operation to the forward projections of the reconstructed intermediate image to take into account the differential nature of the DPC projections. This method is based on the algebraic reconstruction technique, reconstructs the image ray by ray, and is expected to provide better derivative estimates in iterations. This work comprises a numerical study of the algorithm and its experimental verification using a dataset measured with a three-grating interferometer and a mini-focus x-ray tube source. It is shown that the proposed method can reduce the cone-beam artifacts and performs better than FDK under large cone-beam angles. This algorithm is of interest for future cone-beam DPC-CT applications.

Generation of voxelized breast phantoms from surgical mastectomy specimens

PURPOSE

In the research and development of dedicated tomographic breast imaging systems, digital breast object models, also known as digital phantoms, are useful tools. While various digital breast phantoms do exist, the purpose of this study was to develop a realistic high-resolution model suitable for simulating three-dimensional (3D) breast imaging modalities. The primary goal was to design a model capable of producing simulations with realistic breast tissue structure.

METHODS

The methodology for generating an ensemble of digital breast phantoms was based on imaging surgical mastectomy specimens using a benchtop, cone-beam computed tomography system. This approach allowed low-noise, high-resolution projection views of the mastectomy specimens at each angular position. Reconstructions of these projection sets were processed using correction techniques and diffusion filtering prior to segmentation into breast tissue types in order to generate phantoms.

RESULTS

Eight compressed digital phantoms and 20 uncompressed phantoms from which an additional 96 pseudocompressed digital phantoms with voxel dimensions of 0.2 mm(3) were generated. Two distinct tissue classification models were used in forming breast phantoms. The binary model classified each tissue voxel as either adipose or fibroglandular. A multivalue scaled model classified each tissue voxel as percentage of adipose tissue (range 1%-99%). Power spectral analysis was performed to compare simulated reconstructions using the breast phantoms to the original breast specimen reconstruction, and fits were observed to be similar.

CONCLUSIONS

The digital breast phantoms developed herein provide a high-resolution anthropomorphic model of the 3D uncompressed and compressed breast that are suitable for use in evaluating and optimizing tomographic breast imaging modalities. The authors believe that other research groups might find the phantoms useful, and therefore they offer to make them available for wider use.

Evaluation of Algebraic Iterative Image Reconstruction Methods for Tetrahedron Beam Computed Tomography Systems

Tetrahedron beam computed tomography (TBCT) performs volumetric imaging using a stack of fan beams generated by a multiple pixel X-ray source. While the TBCT system was designed to overcome the scatter and detector issues faced by cone beam computed tomography (CBCT), it still suffers the same large cone angle artifacts as CBCT due to the use of approximate reconstruction algorithms. It has been shown that iterative reconstruction algorithms are better able to model irregular system geometries and that algebraic iterative algorithms in particular have been able to reduce cone artifacts appearing at large cone angles. In this paper, the SART algorithm is modified for the use with the different TBCT geometries and is tested using both simulated projection data and data acquired using the TBCT benchtop system. The modified SART reconstruction algorithms were able to mitigate the effects of using data generated at large cone angles and were also able to reconstruct CT images without the introduction of artifacts due to either the longitudinal or transverse truncation in the data sets. Algebraic iterative reconstruction can be especially useful for dual-source dual-detector TBCT, wherein the cone angle is the largest in the center of the field of view.

Combination of converging collimators for high-sensitivity brain SPECT

The objective of this study, which is related to human brain SPECT, was to increase the sensitivity of a triple-camera SPECT system and reduce statistical noise in reconstructed images using a combination of converging collimators. The reason for combining collimators is to ensure both high sensitivity and sufficient sampling without trading off spatial resolution.

METHODS

A high-sensitivity half-cone-beam (HCB) collimator, designed specifically for brain imaging, was combined with other collimators and compared with conventional parallel-beam and fanbeam circular orbit acquisitions. For comparison, previously studied HCB collimation with a circle-and-helix data acquisition trajectory was also included in this study. Simulations of the Hoffman 3-dimensional brain phantom were performed to calculate the efficiencies of collimators and their combinations and to quantitatively evaluate reconstruction bias, statistical noise, and signal-to-noise ratios in the reconstructed images. Experimental brain phantom data were also acquired and compared for different acquisition types. Finally, a patient brain scan was obtained with a combination of HCB and fanbeam collimators and compared with a triple-fanbeam circular orbit acquisition.

RESULTS

A combination of 2 HCB collimators and 1 fanbeam collimator, compared with a triple-fanbeam collimator, can increase the photon detection efficiency by 27% and by more than a factor of 2, compared with triple-parallel-hole collimation, with equal spatial resolution measured on the axis of rotation. Quantitative analysis of reconstruction bias and visual analysis of the images showed no signs of sampling artifacts. Reconstructed images in the simulations, experimental brain phantom, and patient brain scans showed improved quality with this collimator combination due to increased sensitivity and reduced noise. Lesion visibility was also improved, as confirmed by signal-to-noise ratios. Alternatively, triple-HCB circle-and-helix acquisition has also shown competitive results, with a slight disadvantage in axial sampling and implementation procedure.

CONCLUSION

Combined HCB and fanbeam collimation is a promising approach for high-sensitivity brain SPECT.

Motion-compensated iterative cone-beam CT image reconstruction with adapted blobs as basis functions

This paper presents a three-dimensional method to reconstruct moving objects from cone-beam X-ray projections using an iterative reconstruction algorithm and a given motion vector field. For the image representation, adapted blobs are used, which can be implemented efficiently as basis functions. Iterative reconstruction requires the calculation of line integrals (forward projections) through the image volume, which are compared with the actual measurements to update the image volume. In the existence of a divergent motion vector field, a change in the volumes of the blobs has to be taken into account in the forward and backprojections. An efficient method to calculate the line integral through the adapted blobs is proposed. It solves the problem, how to compensate for the divergence in the motion vector field on a grid of basis functions. The method is evaluated on two phantoms, which are subject to three different known motions. Moreover, a motion-compensated filtered back-projection reconstruction method is used, and the reconstructed images are compared. Using the correct motion vector field with the iterative motion-compensated reconstruction, sharp images are obtained, with a quality that is significantly better than gated reconstructions.

Development of a non-linear weighted hybrid cone-beam CT reconstruction for circular trajectories

We investigated an image reconstruction algorithm to reduce cone-beam artifacts in cone-beam CT. Our new algorithm to reduce such artifacts features: (1) a change in weighting with respect to projection data obtained at different projection angles; (2) distribution of correction coefficients so that they are larger near the center of the detector, while taking individual channel data for the detector into account, and smaller near the edges; (3) three-dimensional back-projection of corrected projection data. These findings confirmed that this algorithm reduces cone-beam artifacts and generates high-quality reconstruction images.

Approximate and exact cone-beam reconstruction with standard and non-standard spiral scanning

The long object problem is practically important and theoretically challenging. To solve the long object problem, spiral cone-beam CT was first proposed in 1991, and has been extensively studied since then. As a main feature of the next generation medical CT, spiral cone-beam CT has been greatly improved over the past several years, especially in terms of exact image reconstruction methods. Now, it is well established that volumetric images can be exactly and efficiently reconstructed from longitudinally truncated data collected along a rather general scanning trajectory. Here we present an overview of some key results in this area.

Chord-based image reconstruction in cone-beam CT with a curved detector

Modern computed tomography (CT) scanners use cone-beam configurations for increasing volume coverage, improving x-ray-tube utilization, and yielding isotropic spatial resolution. Recently, there have been significant developments in theory and algorithms for exact image reconstruction from cone-beam projections. In particular, algorithms have been proposed for image reconstruction on chords; and advantages over the existing algorithms offered by the chord-based algorithms include the high flexibility of exact image reconstruction for general scanning trajectories and the capability of exact reconstruction of images within a region of interest from truncated data. These chord-based algorithms have been developed only for flat-panel detectors. Many cone-beam CT scanners employ curved detectors for important practical considerations. Therefore, in this work, we have derived chord-based algorithms for a curved detector so that they can be applied to reconstructing images directly from data acquired by use of a CT scanner with a curved detector. We have also conducted preliminary numerical studies to demonstrate and evaluate the reconstruction properties of the derived chord-based algorithms for curved detectors.

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.

Exact BPF and FBP algorithms for nonstandard saddle curves

A hot topic in cone-beam CT research is exact cone-beam reconstruction from a general scanning trajectory. Particularly, a nonstandard saddle curve attracts attention, as this construct allows the continuous periodic scanning of a volume-of-interest (VOI). Here we evaluate two algorithms for reconstruction from data collected along a nonstandard saddle curve, which are in the filtered backprojection (FBP) and backprojection filtration (BPF) formats, respectively. Both the algorithms are implemented in a chord-based coordinate system. Then, a rebinning procedure is utilized to transform the reconstructed results into the natural coordinate system. The simulation results demonstrate that the FBP algorithm produces better image quality than the BPF algorithm, while both the algorithms exhibit similar noise characteristics.

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.

A differentiable Shepp–Logan phantom and its applications in exact cone-beam CT

Recently, several exact cone-beam reconstruction algorithms, such as the generalized filtered-backprojection (FBP) and backprojection-filtration (BPF) methods, have been developed to solve the long object problem. Although the well-known 3D Shepp-Logan phantom (SLP) is often used to validate these algorithms, it is deficient due to the discontinuity of the SLP. In this paper, we first construct a differentiable polynomial function to approximate the unit rectangular function on [-1, 1]. Then, we use this function to obtain a differentiable ellipsoid phantom, whose x-ray transform is differentiable for any smooth scanning trajectory. Finally, we propose a differentiable Shepp-Logan phantom (DSLP) for numerical simulation of the exact cone-beam CT algorithms. Our numerical simulation shows that the reconstructed DSLP has a better image quality than the reconstructed SLP, and is complementary to the traditional SLP for evaluation of the exact cone-beam CT algorithms.

Tilted plane Feldkamp type reconstruction algorithm for spiral cone beam CT

An approximate image reconstruction method for spiral cone beam computed tomography (CT), called tilted plane Feldkamp type reconstruction algorithm (TPFR), is presented in this paper, which extends Feldkamp cone beam reconstruction algorithm to deal with its inaccuracy and artifact problems caused by large cone angle. This is done by tilting the reconstructing planes to minimize the cone angle and optimally fit the spiral segment of the source. The tilted plane image reconstruction requires reforming the three-dimensional projection data set for the tilted plane and application of Feldkamp algorithm to the reformed data set. Analytical and computational results can show that the image reconstruction performance of the proposed TPFR algorithm is superior to that of the Feldkamp reconstruction algorithm in the image quality, volume coverage speed, maximum achievable pitch value, and slice sensitivity profiles. Moreover, it provides more accurate image reconstruction than the existing two-dimensional reconstruction algorithms.

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.

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.

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.

Feldkamp-type VOI reconstruction from super-short-scan cone-beam data

Based on the fan-beam reconstruction formula recently developed by Noo et al. [Phys. Med. Biol. 47, 2525-2546 (2002)] we develop a Feldkamp-type algorithm for the reconstruction of a volume of interest (VOI) from super-short-scan data. With either a circular or spiral scanning locus in our VOI reconstruction scheme, we first estimate fan-beam data from cone-beam data using the popular "cosine correction" scheme, and perform reconstruction based on Noo's FBP-type fan-beam reconstruction. Our proposed algorithm is tested using the three-dimensional (3-D) Shepp-Logan phantom. The experimental results show that the new algorithm can be applied to multi-source 4-D CT with significantly superior temporal resolution and temporal consistency relative to the Katsevich algorithm, which is the state of the art for exact helical cone-beam reconstruction.

Partial volume and aliasing artefacts in helical cone-beam CT

A generalization of the quasi-exact algorithms of Kudo et al (2000 IEEE Trans. Med. Imaging 19 902-21) is developed that allows for data acquisition in a 'practical' frame for clinical diagnostic helical, cone-beam computed tomography (CT). The algorithm is investigated using data that model nonlinear partial volume averaging. This investigation leads to an understanding of aliasing artefacts in helical, cone-beam CT image reconstruction. An ad hoc scheme is proposed to mitigate artefacts due to the nonlinear partial volume and aliasing artefacts.

Impact of polychromatic x-ray sources on helical, cone-beam computed tomography and dual-energy methods

Recently, there has been much work devoted to developing accurate and efficient algorithms for image reconstruction in helical, cone-beam computed tomography (CT). Little attention, however, has been directed to the effect of physical factors on helical, cone-beam CT image reconstruction. This work investigates the effect of polychromatic x-rays on image reconstruction in helical, cone-beam computed tomography. A pre-reconstruction dual-energy technique is developed to reduce beam-hardening artefacts and enhance contrast in soft tissue.

Exact image reconstruction on PI-lines from minimum data in helical cone-beam CT

The development of accurate and efficient algorithms for image reconstruction from helical cone-beam projections remains a subject of active research. In the last few years, a number of quasi-exact and exact algorithms have been developed. Among them, the Katsevich algorithms are of filtered backprojection type and thus possess computational advantages over other existing exact algorithms. In this work, we propose an alternative approach to reconstructing exactly an image from helical cone-beam projections. Based on this approach, we develop an algorithm that requires less data than do the existing quasi-exact and exact algorithms, including the Katsevich algorithms. Our proposed algorithm is also of filtered backprojection type with one-dimensional filtering performed along a PI-line in image space. Therefore, it is (at least) computationally as efficient as the Katsevich algorithms. We have performed a preliminary numerical study to demonstrate and validate the proposed algorithm using computer-simulation data. The implication of the proposed approach and algorithm appears to be significant in that they can naturally address the long object problem as well as the super-short scan problem and, most importantly, in that they provide the opportunity to reconstruct images within any selected region of interest from minimum data, allowing the use of detector with a reduced size, the selection of a minimum number of rotation angles and thus the reduction of radiation dose delivered to the imaged subject.

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.

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.

Image reconstruction on PI-lines by use of filtered backprojection in helical cone-beam CT

Recently, we have derived a general formula for image reconstruction from helical cone-beam projections. Based upon this formula, we have also developed an exact algorithm for image reconstruction on PI-line segments from minimum data within the Tam-Danielsson window. This previous algorithm can be referred to as a backprojection-filtration algorithm because it reconstructs an image by first backprojection of the data derivatives and then filtration of the backprojections on PI-line segments. In this work, we propose an alternative algorithm, which reconstructs an image by first filtering the modified data along the cone-beam projections of the PI-lines onto the detector plane and then backprojecting the filtered data onto PI-line segments. Therefore, we refer to this alternative algorithm as the filtered-backprojection algorithm. A preliminary computer-simulation study was performed for validating and demonstrating this new algorithm. Furthermore, we derive a practically useful expression to accurately compute the derivative of the data function for image reconstruction. The proposed filtered-backprojection algorithm can reconstruct the image within any selected ROI inside the helix and thus can handle naturally the long object problem and the super-short scan problem. It can also be generalized to reconstruct images from data acquired with other scanning configurations such as the helical scan with a varying pitch.

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.

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.

Experiments with the nonlinear and chaotic behaviour of the multiplicative algebraic reconstruction technique (MART) algorithm for computed tomography

Among the iterative reconstruction algorithms for tomography, the multiplicative algebraic reconstruction technique (MART) has two advantages that make it stand out from other algorithms: it confines the image (and therefore the projection data) to the convex hull of the patient, and it maximizes entropy. In this paper, we have undertaken a series of experiments to determine the importance of MART nonlinearity to image quality. Variants of MART were implemented aiming to exploit and exaggerate the nonlinear properties of the algorithm. We introduce the Power MART, Boxcar Averaging MART and Bouncing MART algorithms. Power MART is linked to the relaxation concept. Its behaviour is similar to that of the chaos of a logistic equation. There appears to be an antagonism between increasing nonlinearity and noise in the projection data. The experiments confirm our general observation that regularization as a means of solving simultaneous linear equations that are underdetermined is suboptimal: it does not necessarily select the correct image from the hyperplane of solutions, and so does not maximize the image quality:x-ray dose ratio. Our investigations prove that there is scope to optimize CT algorithms and thereby achieve greater dose reduction.

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.

Grangeat-type helical half-scan computerized tomography algorithm for reconstruction of a short object

Currently, cone-beam computerized tomography (CT) and micro-CT scanners are under rapid development for major biomedical applications. Half-scan cone-beam image reconstruction algorithms assume only part of a scanning turn, and are advantageous in terms of temporal resolution and image artifacts. While the existing half-scan cone-beam algorithms are in the Feldkamp framework, we have published a half-scan algorithm in the Grangeat framework for a circular trajectory [Med. Phys. 30, 689-700 (2003)]. In this paper, we extend our previous work to a helical case without data truncation. We modify the Grangeat's formula for utilization and estimation of Radon data. Specifically, we categorize each characteristic point in the Radon space into singly, doubly, triply sampled, and shadow regions, respectively. A smooth weighting strategy is designed to compensate for data redundancy and inconsistency. In the helical half-scan case, the concepts of projected trajectories and transition points on meridian planes are introduced to guide the design of weighting functions. Then, the shadow region is recovered via linear interpolation after smooth weighting. The Shepp-Logan phantom is used to verify the correctness of the formulation, and demonstrate the merits of the Grangeat-type half-scan algorithm. Our Grangeat-type helical half-scan algorithm is not only valuable for quantitative and/or dynamic biomedical applications of CT and micro-CT, but also serves as an intermediate step towards solving the long object problem.

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

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.

Filtering point spread function in backprojection cone-beam CT and its applications in long object imaging

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, i.e. 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.

Segmented multiple plane reconstruction: a novel approximate reconstruction scheme for multi-slice spiral CT

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 images 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.

Computation of unmeasured third-generation VCT views from measured views

We compute unmeasured cone-beam projections from projections measured by a third-generation helical volumetric computed tomography system by solving a characteristic problem for an ultrahyperbolic differential equation [John (1938)]. By working in the Fourier domain, we convert the second-order PDE into a family of first-order ordinary differential equations. A simple first-order integration is used to solve the ODEs.

Artifact analysis of approximate helical cone-beam CT reconstruction algorithms

In this paper, four approximate cone-beam CT reconstruction algorithms are compared: Advanced single slice rebinning (ASSR) as a representative of algorithms employing a two dimensional approximation, PI, PI-SLANT, and 3-PI which all use a proper three dimensional back-projection. A detailed analysis of the image artifacts produced by these techniques shows that aliasing in the z-direction is the predominant source of artifacts for a 16-row scanner with 1.25 mm nominal slice thickness. For a detector with isotropic resolution of 0.5 mm, we found that ASSR and PI produce different kinds of artifacts which are almost at the same level, while PI-SLANT produces none of these artifacts. It is shown that the use of redundant data in the 3-PI method suppresses aliasing artifacts efficiently for both scanners.

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.

Cone-beam image reconstruction using spherical harmonics

Image reconstruction from cone-beam projections is required for both x-ray computed tomography (CT) and single photon emission computed tomography (SPECT). Grangeat's algorithm accurately performs cone-beam reconstruction provided that Tuy's data sufficiency condition is satisfied and projections are complete. The algorithm consists of three stages: (a) Forming weighted plane integrals by calculating the line integrals on the cone-beam detector, and obtaining the first derivative of the plane integrals (3D Radon transform) by taking the derivative of the weighted plane integrals. (b) Rebinning the data and calculating the second derivative with respect to the normal to the plane. (c) Reconstructing the image using the 3D Radon backprojection. A new method for implementing the first stage of Grangeat's algorithm was developed using spherical harmonics. The method assumes that the detector is large enough to image the whole object without truncation. Computer simulations show that if the trajectory of the cone vertex satisfies Tuy's data sufficiency condition, the proposed algorithm provides an exact reconstruction.

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.

Single-slice rebinning reconstruction in spiral cone-beam computed tomography

At the advent of multislice computed tomography ICT) a variety of approximate cone-beam algorithms have been proposed suited for reconstruction of small cone-angle CT data in a spiral mode of operation. The goal of this study is to identify a practical and efficient approximate cone-beam method, extend its potential for medical use, and demonstrate its performance at medium cone-angles required for area detector CT. We will investigate two different approximate single-slice rebinning algorithms for cone-beam CT: the multirow Fourier reconstruction (MFR) and an extension of the advanced single-slice rebinning method (ASSR), which combines the idea of ASSR with a z-filtering approach. Thus, both algorithms, MFR and ASSR, are formulated in the framework of z-filtering using optimized spiral interpolation algorithms. In each view, X-ray samples to be used for reconstruction are identified, which describe an approximation to a virtual reconstruction plane. The performance of approximate reconstruction should improve as the virtual reconstruction plane better fits the spiral focus path. The image quality of the respective reconstruction will be assessed with respect to image artifacts, spatial resolution, contrast resolution, and image noise. It turns out that the ASSR method using tilted reconstruction planes is a practical and efficient algorithm, providing image quality comparable to that of a single-row scanning system even with a 46-row detector at a table feed of 64 mm. Both algorithms tolerate any table feed below the maximum value associated to the detector height. Due to the z-filter approach, all detector data sampled can be used for image reconstruction.

The n-PI-method for helical cone-beam CT

A new class of acquisition schemes for helical cone-beam computed tomography (CB-CT) scanning is introduced, and their effect on the reconstruction methods is analyzed. These acquisition schemes are based on a new detector shape that is bounded by the helix. It will be shown that the data acquired with these schemes are compatible with exact reconstruction methods, and the adaptation of exact reconstruction algorithms to the new acquisition geometry is described. At the same time, the so-called PI-sufficiency condition is fulfilled. Moreover, a good fit to the acquisition requirements of the various medical applications of cone-beam CT is achieved. In contrast to other helical cone-beam acquisition and reconstruction methods, the n-PI-method introduced in this publication allows for variable pitches of the acquisition helix. This additional feature will introduce a higher flexibility into the acquisition protocols of future medical cone-beam scanners. An approximative n-PI-filtered backprojection (n-PI-FBP) reconstruction method is presented and verified. It yields convincing image quality.