Regularized iterative weighted filtered backprojection for helical cone-beam CT

Contemporary reconstruction methods employed for clinical helical cone-beam computed tomography (CT) are analytical (noniterative) but mathematically nonexact, i.e., the reconstructed image contains so called cone-beam artifacts, especially for higher cone angles. Besides cone artifacts, these methods also suffer from windmill artifacts: alternating dark and bright regions creating spiral-like patterns occurring in the vicinity of high z-direction derivatives. In this article, the authors examine the possibility to suppress cone and windmill artifacts by means of iterative application of nonexact three-dimensional filtered backprojection, where the analytical part of the reconstruction brings about accelerated convergence. Specifically, they base their investigations on the weighted filtered backprojection method [Stierstorfer et al., Phys. Med. Biol. 49, 2209-2218 (2004)]. Enhancement of high frequencies and amplification of noise is a common but unwanted side effect in many acceleration attempts. They have employed linear regularization to avoid these effects and to improve the convergence properties of the iterative scheme. Artifacts and noise, as well as spatial resolution in terms of modulation transfer functions and slice sensitivity profiles have been measured. The results show that for cone angles up to +/-2.78 degrees, cone artifacts are suppressed and windmill artifacts are alleviated within three iterations. Furthermore, regularization parameters controlling spatial resolution can be tuned so that image quality in terms of spatial resolution and noise is preserved. Simulations with higher number of iterations and long objects (exceeding the measured region) verify that the size of the reconstructible region is not reduced, and that the regularization greatly improves the convergence properties of the iterative scheme. Taking these results into account, and the possibilities to extend the proposed method with more accurate modeling of the acquisition process, the authors believe that iterative improvement with non-exact methods is a promising technique for medical CT applications.     

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.     

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 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.     

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.     

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.     

A filtered backprojection algorithm for cone beam reconstruction using 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.     

A motion-compensated scheme for helical cone-beam reconstruction in cardiac CT angiography

Since coronary heart disease is one of the main causes of death all over the world, cardiac computed tomography (CT) imaging is an application of very high interest in order to verify indications timely. Due to the cardiac motion, electrocardiogram (ECG) gating has to be implemented into the reconstruction of the measured projection data. However, the temporal and spatial resolution is limited due to the mechanical movement of the gantry and due to the fact that a finite angular span of projections has to be acquired for the reconstruction of each voxel. In this article, a motion-compensated reconstruction method for cardiac CT is described, which can be used to increase the signal-to-noise ratio or to suppress motion blurring. Alternatively, it can be translated into an improvement of the temporal and spatial resolution. It can be applied to the entire heart in common and to high contrast objects moving with the heart in particular, such as calcified plaques or devices like stents. The method is based on three subsequent steps: As a first step, the projection data acquired in low pitch helical acquisition mode together with the ECG are reconstructed at multiple phase points. As a second step, the motion-vector field is calculated from the reconstructed images in relation to the image in a reference phase. Finally, a motion-compensated reconstruction is carried out for the reference phase using those projections, which cover the cardiac phases for which the motion-vector field has been determined.     

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.     

A model for filtered backprojection reconstruction artifacts due to time-varying attenuation values in perfusion C-arm CT

Filtered backprojection is the basis for many CT reconstruction tasks. It assumes constant attenuation values of the object during the acquisition of the projection data. Reconstruction artifacts can arise if this assumption is violated. For example, contrast flow in perfusion imaging with C-arm CT systems, which have acquisition times of several seconds per C-arm rotation, can cause this violation. In this paper, we derived and validated a novel spatio-temporal model to describe these kinds of artifacts. The model separates the temporal dynamics due to contrast flow from the scan and reconstruction parameters. We introduced derivative-weighted point spread functions to describe the spatial spread of the artifacts. The model allows prediction of reconstruction artifacts for given temporal dynamics of the attenuation values. Furthermore, it can be used to systematically investigate the influence of different reconstruction parameters on the artifacts. We have shown that with optimized redundancy weighting function parameters the spatial spread of the artifacts around a typical arterial vessel can be reduced by about 70%. Finally, an inversion of our model could be used as the basis for novel dynamic reconstruction algorithms that further minimize these artifacts.     

A general exact method for synthesizing parallel-beam projections from cone-beam projections via filtered backprojection

In recent years, image reconstruction methods for cone-beam computed tomography (CT) have been extensively studied. However, few of these studies discussed computing parallel-beam projections from cone-beam projections. In this paper, we focus on the exact synthesis of complete or incomplete parallel-beam projections from cone-beam projections. First, an extended central slice theorem is described to establish a relationship between the Radon space and the Fourier space. Then, data sufficiency conditions are proposed for computing parallel-beam projection data from cone-beam data. Using these results, a general filtered backprojection algorithm is formulated that can exactly synthesize parallel-beam projection data from cone-beam projection data. As an example, we prove that parallel-beam projections can be exactly synthesized in an angular range in the case of circular cone-beam scanning. Interestingly, this angular range is larger than that derived in the Feldkamp reconstruction framework. Numerical experiments are performed in the circular scanning case to verify our method.     

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.     

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.     

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.     

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.     

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.     

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 shift-invariant filtered backprojection (FBP) cone-beam reconstruction algorithm for the source trajectory of two concentric circles using an equal weighting scheme

In this paper, a shift-invariant filtered backprojection cone-beam image reconstruction algorithm is derived, based upon Katsevich's general inversion scheme, and validated for the source trajectory of two concentric circles. The source trajectory is complete according to Tuy's data sufficiency condition and is used as the basis for an exact image reconstruction algorithm. The algorithm proceeds according to the following steps. First, differentiate the cone-beam projection data with respect to the detector coordinates and with respect to the source trajectory parameter. The data are then separately filtered along three different orientations in the detector plane with a shift-invariant Hilbert kernel. Eight different filtration groups are obtained via linear combinations of weighted filtered data. Voxel-based backprojection is then carried out from eight sets of view angles, where separate filtered data are backprojected from each set according to the backprojection sets' associated filtration group. The algorithm is first derived for a scanning configuration consisting of two concentric and orthogonal circles. By performing an affine transformation on the image object, the developed image reconstruction algorithm has been generalized to the case where the two concentric circles are not orthogonal. Numerical simulations are presented to validate the reconstruction algorithm and demonstrate the dose advantage of the equal weighting scheme.