Feldkamp-based cone-beam reconstruction for gantry-tilted helical multislice CT

Depending on the clinical application, it is frequently necessary to tilt the gantry of an x-ray CT system with respect to the patient and couch. For single-slice fan-beam systems, tilting the gantry introduces no errors or artifacts. Most current systems, however, are helical multislice systems with up to 16 slices. The multislice helical reconstruction algorithms used to create CT images must be modified to account for tilting of the gantry. If they are not, the quality of reconstructed images will be poor with the presence of significant artifacts, such as smearing and double-imaging of anatomical structures. Current CT systems employ three primary types of reconstruction algorithms: helical fan-beam approximation, advanced single-slice rebinning, and Feldkamp-based algorithms. This paper presents a generalized helical cone-beam Feldkamp-based algorithm that is valid for both tilted and nontilted orientations of the gantry. Unlike some of the other algorithms, generalization of the Feldkamp algorithm to include gantry tilt is simple and straightforward with no significant increase in computational complexity. The effect of gantry tilt for helical Feldkamp reconstruction is to introduce a lateral shift in the isocenter of the reconstructed slice of interest, which is a function of the tilt, couch speed, and view angle. The lateral shift is easily calculated and incorporated into the helical Feldkamp backprojection algorithm. A tilt-generalized helical Feldkamp algorithm has been developed and incorporated into Aquilion 16-slice CT (Toshiba, Japan) scanners. This paper describes modifications necessary for the tilt generalization and its verification.     

A rebinned backprojection-filtration algorithm for image reconstruction in helical cone-beam CT

In the last few years, mathematically exact algorithms, including the backprojection-filtration (BPF) algorithm, have been developed for accurate image reconstruction in helical cone-beam CT. The BPF algorithm requires minimum data, and can reconstruct region-of-interest (ROI) images from data containing truncations. However, similar to other existing reconstruction algorithms for helical cone-beam CT, the BPF algorithm involves a backprojection with a spatially varying weighting factor, which is computationally demanding and, more importantly, can lead to undesirable numerical properties in reconstructed images. In this work, we develop a rebinned BPF algorithm in which the backprojection invokes no spatially varying weighting factor for accurate image reconstruction from helical cone-beam projections. This rebinned BPF algorithm is computationally more efficient and numerically more stable than the original BPF algorithm, while it also retains the nice properties of the original BPF algorithm such as minimum data requirement and ROI-image reconstruction from truncated data. We have also performed simulation studies to validate and evaluate the rebinned BPF algorithm.     

Aperture weighted cardiac reconstruction for cone-beam CT

Multi-row detectors together with fast rotating gantries made cardiac imaging possible for CT. Due to the cardiac motion, ECG gating has to be integrated into the reconstruction of the data measured on a low pitch helical trajectory. Since the first multi-row scanners were introduced, it has been shown that approximative true cone-beam reconstruction methods are most suitable for the task of retrospectively gated cardiac volume CT. In this paper, we present the aperture weighted cardiac reconstruction (AWCR), which is a three-dimensional reconstruction algorithm of the filtered back-projection type. It is capable of handling all illumination intervals of an object point, which occur as a consequence of a low pitch helical cone-beam acquisition. Therefore, this method is able to use as much redundant data as possible, resulting in an improvement of the image homogeneity, the signal to noise ratio and the temporal resolution. Different optimization techniques like the heart rate adaptive cardiac weighting or the automatic phase determination can be adopted to AWCR. The excellent image quality achieved by AWCR is presented for medical datasets acquired with both a 40-slice and a 64-slice cone-beam CT scanner.     

A new weighting scheme for cone-beam helical CT to reduce the image noise

Reducing the patient dose while keeping the image noise at the same level is desired for x-ray CT examinations. In order to achieve the goal, we propose a new weighting scheme taking the validity of the data and redundant data samples into account. The method is evaluated with a new generalized version of the Feldkamp helical reconstruction algorithm. It allows us to enlarge the projection angular range used in reconstruction, and thus, to reduce the image noise by increasing the detector utilization rate to 100% without sacrificing the image quality or z-resolution. This concept can be adapted to other exact or approximate algorithms as far as they use redundant data samples.     

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.     

Exact reconstruction of volumetric images in reverse helical cone-beam CT

Helical scanning configuration has been used widely in diagnostic cone-beam computed tomography (CBCT) for acquiring data sufficient for exact image reconstruction over an extended volume. In image-guided radiation therapy (IGRT) and other applications of CBCT, it can be difficult, if not impossible, to implement mechanically a multiple-turn helical trajectory on the imaging systems due to hardware constraints. However, imaging systems in these applications often allow for the implementation of a reverse helical trajectory in which the rotation direction changes between two consecutive turns. Because the reverse helical trajectory satisfies Tuy's condition, when projections of the imaged object are nontruncated, it yields data sufficient for exact image reconstruction within the reverse helix volume. The recently developed chord-based algorithms such as the backprojection filtration (BPF) algorithm can readily be applied to reconstructing images on chords of a reverse helical trajectory, and they can thus reconstruct an image within a volume covered by the chords. Conversely, the chord-based algorithms cannot reconstruct images within regions that are not intersected by chords. In a reverse helix volume, as shown below, chordless regions exist in which no images can thus be reconstructed by use of the chord-based algorithms. In this work, based upon Pack-Noo's formula, a shift-invariant filtered backprojection (FBP) algorithm is derived for exact image reconstruction within the reverse helix volume, including the chordless region. Numerical studies have also been conducted to demonstrate the chordless region in a reverse helix volume and to validate the FBP algorithm for image reconstruction within the chordless region. Results of the numerical studies confirm that the FBP algorithm can exactly reconstruct an image within the entire reverse helix volume, including the chordless region. It is relatively straightforward to extend the FBP algorithm to reconstruct images for general trajectories, including reverse helical trajectories with variable pitch, tilted axis, and/or additional segments between turns.     

CEnPiT: helical cardiac CT reconstruction

Computer tomography (CT) scanners with an increasing number of detector rows offer the potential of shorter scanning times. Nevertheless, the reconstruction problem becomes more challenging, since cone beam artifacts are likely to enter. Here, we consider helical cardiac CT. We analyze how a relationship can be established between exact reconstruction algorithms and the demand to perform a cardiac gating. Utilizing the redundancies requires the consideration of all kinds of Radon planes. For the reconstruction algorithm proposed here, we separate the data into two parts. The first part contains contributions of Radon planes, which are measured with a large number of redundancies. The second part contains the remaining contributions. As it turns out, the second part contributes rather to the low-frequency contents of trans-axial slices. Therefore, we propose to perform a gated back-projection only for the first part, while the second part is back-projected in an ungated way. Data from the complete source trajectory are employed in the reconstruction process in contrary to conventional helical cardiac reconstruction methods. Moreover, all different types of Radon planes are taken into account in the reconstruction, though an ECG-dependent cardiac gating is applied. The reconstruction results, which we present for clinical and simulated data, demonstrate the high potential of CEnPiT for helical cardiac CT with large cone angle systems.     

Minimum data image reconstruction algorithms with shift-invariant filtering for helical, cone-beam CT

We derive accurate and efficient reconstruction algorithms for helical, cone-beam CT that employ shift-invariant filtering. Specifically, a new backprojection-filtration algorithm is developed, and a minimum data filtered-backprojection algorithm is derived. These reconstruction algorithms with shift-invariant filtering can accept data with transverse truncation, and hence allow for minimum data image reconstruction.     

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.     

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.     

Single-slice rebinning method for helical cone-beam CT

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

A 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.     

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.     

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.     

An extended data function and its generalized backprojection for image reconstruction in helical cone-beam CT

We have recently proposed a general formula (i.e., equations (9) to (11) in Zou and Pan (2004a Phys. Med. Biol. 49 941-59)) for image reconstruction from helical cone-beam data. On the basis of the formula, we have also developed two reconstruction algorithms, which are referred to as the backprojection filtration (BPF) algorithm (Zou and Pan 2004a) and the filtered backprojection (FBP) algorithm (Zou and Pan 2004b Phys. Med. Biol. 49 2717-31), respectively. The two algorithms have been implemented and evaluated in numerical studies. In this note, however, we point out that the data function previously used for proving the general formula in Zou and Pan (2004a) is incomplete and that, instead, an extended data function and its generalized backprojection, which are described in this note, should be used to complete the proof of the general formula. On the other hand, we also demonstrate in this note that the additional term in the extended data function has no effect on the previously developed BPF and FBP algorithms. The results can also be extended to general, smooth trajectories.     

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.     

Iterative image reconstruction in helical cone-beam x-ray CT using a stored system matrix approach

We present a stored system matrix (SM) approach for iterative x-ray CT image reconstruction with helical cone-beam geometry. Because of the symmetry of a helical source trajectory, it is sufficient to calculate and store the SM entries for one transaxial slice only and for all source positions illuminating the slice. This is made possible by (1) selecting the reconstruction slice thickness to be an integer multiple of the source translation per projection view, and (2) discretizing the 3D reconstruction volume on a rotated stack of slices. Using the proposed method, the memory requirement for reconstructing a full field-of-view of clinical scanners is manageable on current computing platforms. The same storage principle can be generalized and applied to volume-of-interest (VOI) image reconstruction for helical cone-beam CT. In this case, the stored SM entries correspond to a partial- or full-ring region on one transaxial slice, and for all source positions illuminating the ring. The size and location of the ring depend on the size and the location of the VOI and the scan geometry. We demonstrate by both computer simulations and clinical patient data the speed and efficacy of iterative image reconstruction using the stored SM approach.     

Redundant data and exact helical cone-beam reconstruction

This paper is about helical cone-beam reconstruction and the use of redundant data in the framework of two reconstruction methods. The first method is the approximate wedge reconstruction formula introduced by Tuy at the 3D meeting in 1999. The second method is a (exact) hybrid implementation of the exact filtered backprojection formula of Katsevich (2004 Adv. Appl. Math. at press) that combines filtering in the native cone-beam geometry with backprojection in the wedge geometry. The similarity of the two methods is explored and their image quality performance is compared for geometries with up to 112 detector rows. Furthermore, the concept of aperture weighting is introduced to allow the handling of variable amounts of redundant data. A reduction of motion artefacts using redundant data is demonstrated for geometries with 16, 32 and 112 detector rows using a pitch factor of 1.25. For scans with up to 100 rows, utilizing 50% of the redundant data provided excellent results without any introduction of cone-beam artefacts. For larger cone angles, an alternative approach that utilizes all available redundant data, even at reduced pitch factors, is suggested.     

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.