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

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

Region-of-interest reconstruction of motion-contaminated data using a weighted backprojection filtration algorithm

The recently developed weighted backprojection filtration (WBPF) algorithm using data redundancy has capabilities that make this algorithm an attractive candidate for reconstructing images from motion-contaminated projection data. First, the WBPF algorithm is capable of reconstructing region-of-interest (ROI) images from reduced-scan fan-beam data, which have less data than the short-scan data required to reconstruct the entire field of view (FOV). Second, this algorithm can reconstruct ROI images from truncated data. Using phantom simulation studies, we demonstrate how these unique capabilities can be exploited to reduce the amount of motion-contaminated data used for reconstruction. In particular, we use examples from cardiac imaging to illustrate how off-center phantom positioning combined with phase-interval ROI reconstruction can result in the suppression of motion artifacts. In terms of temporal resolution, reduced-scan reconstruction with 45% of a full-scan dataset can be used to improve the temporal resolution of a short-scan reconstruction by 25.8% if ungated data are used. For data gated at 66 beats per minute, reduced-scan reconstruction with 45% of a full-scan dataset can be used to improve the temporal resolution of a short-scan reconstruction by 7.9%. As a result of our studies, we believe that the WBPF algorithm demonstrates the potential for reconstructing quality ROI images from motion-contaminated fan-beam data.     

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.     

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.     

Image reconstruction via the finite Hilbert transform of the derivative of the backprojection

An exact analytical image reconstruction method is presented for two-dimensional imaging. The method performs backprojection, the derivative and finite Hilbert transforms. This method can be applied to many imaging geometries. The backprojection procedure is imaging-geometry dependent, while the differentiation and the finite Hilbert transform procedures are identical for all imaging geometries. This algorithm is applicable to list-mode data in nuclear medicine, while other filtered backprojection algorithms cannot be applied directly to the list-mode data.     

Fast and exact 2D image reconstruction by means of Chebyshev decomposition and backprojection

A new algorithm for the reconstruction of two-dimensional (2D) images from projections is described. The algorithm is based on the decomposition of the projections into Chebyshev polynomials of the second kind, which are the ideal basis functions for this application. The Chebyshev decomposition is done via the fast discrete sine transform. A discrete reconstruction filter is applied that corresponds to the ramp filter used in standard filtered backprojection (FBP) reconstruction. In contrast to FBP, the filter is applied to the Chebyshev coefficients and not to the Fourier coefficients of the projections. Then the reconstructed image is simply obtained by means of backprojection. Consequently, the method can be considered as a Chebyshev-domain filtered backprojection (CD-FBP). The total calculation time is dominated by the backprojection step only and is comparable to FBP. The merits of CD-FBP as compared with standard FBP are that: (a) The result is exact if the 2D function to be reconstructed can be decomposed into polynomials of finite degree, and if the sampling is adequate. Otherwise a polynomial approximation results. (b) The algorithm is inherently discrete. (c) It is particularly well suited for reconstructions from projections with non-equidistant samples that occur for instance in 2D PET (positron emission tomography) imaging and in a special form of fan beam scanning. Examples of applications comprise reconstructions of the Shepp and Logan head phantom in various sampling geometries, and a real PET test object. In the PET example an increased resolution is observed in comparison with standard FBP.     

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 cone beam filtered backprojection (CB-FBP) reconstruction algorithm for a circle-plus-two-arc orbit.

The circle-plus-arc orbit possesses advantages over other "circle-plus" orbits for the application of x-ray cone beam (CB) volume CT in image-guided interventional procedures requiring intraoperative imaging, in which movement of the patient table is to be avoided. A CB circle-plus-two-arc orbit satisfying the data sufficiency condition and a filtered backprojection (FBP) algorithm to reconstruct longitudinally unbounded objects is presented here. In the circle suborbit, the algorithm employs Feldkamp's formula and another FBP implementation. In the arc suborbits, an FBP solution is obtained originating from Grangeat's formula, and the reconstruction computation is significantly reduced using a window function to exclude redundancy in Radon domain. The performance of the algorithm has been thoroughly evaluated through computer-simulated phantoms and preliminarily evaluated through experimental data, revealing that the algorithm can regionally reconstruct longitudinally unbounded objects exactly and efficiently, is insensitive to the variation of the angle sampling interval along the arc suborbits, and is robust over practical x-ray quantum noise. The algorithm's merits include: only 1D filtering is implemented even in a 3D reconstruction, only separable 2D interpolation is required to accomplish the CB backprojection, and the algorithm structure is appropriate for parallel computation.     

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.     

Filtered backprojection formula for exact image reconstruction from cone-beam data along a general scanning curve

Recently, Katsevich proved a filtered backprojection formula for exact image reconstruction from cone-beam data along a helical scanning locus, which is an important breakthrough since 1991 when the spiral cone-beam scanning mode was proposed. In this paper, we prove a generalized Katsevich's formula for exact image reconstruction from cone-beam data collected along a rather flexible curve. We will also give a general condition on filtering directions. Based on this condition, we suggest a natural choice of filtering directions, which is more convenient than Katsevich's choice and can be applied to general scanning curves. In the derivation, we use analytical techniques instead of geometric arguments. As a result, we do not need the uniqueness of the PI lines. In fact, our formula can be used to reconstruct images on any chord as long as a scanning curve runs from one endpoint of the chord to the other endpoint. This can be considered as a generalization of Orlov's classical theorem. Specifically, our formula can be applied to (i) nonstandard spirals of variable radii and pitches (with PI- or n-PI-windows), and (ii) saddlelike curves.     

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 local region of interest image reconstruction via filtered backprojection for fan-beam differential phase-contrast computed tomography

Recently, x-ray differential phase contrast computed tomography (DPC-CT) has been experimentally implemented using a conventional source combined with several gratings. Images were reconstructed using a parallel-beam reconstruction formula. However, parallel-beam reconstruction formulae are not directly applicable for a large image object where the parallel-beam approximation fails. In this note, we present a new image reconstruction formula for fan-beam DPC-CT. There are two major features in this algorithm: (1) it enables the reconstruction of a local region of interest (ROI) using data acquired from an angular interval shorter than 180 degrees + fan angle and (2) it still preserves the filtered backprojection structure. Numerical simulations have been conducted to validate the image reconstruction algorithm.     

Antialiasing backprojection for helical MDCT

Helical CTs are well known to suffer from aliasing artifacts because of their finite longitudinal sampling pitch. The artifact pattern is typically strong streaks from bone edges in clinical images. Especially in the case of multidetector row CT, the artifact resulting from longitudinal aliasing is often called a windmill artifact because the visible streaks form a windmill pattern when the object is of a particular shape. The scan must be performed using a very thin slice thickness, i.e., fine sampling in the longitudinal direction, with a longer scan time to mitigate this aliasing artifact. Some elaborate longitudinal interpolation methods to remediate longitudinal aliasing have been proposed, but they have not been successful in practice despite their theoretical importance. A periodic swing of the focal spot in the longitudinal direction, a so-called z-flying focal spot, was introduced recently to achieve finer sampling. Although it is a useful technique, some important deficiencies exist: It is sufficiently effective only near the isocenter and is difficult to apply to a scan using a thick slice thickness, even though longitudinal aliasing is more serious at the thicker scan. In this paper, the author addresses the nature of interlaced (or unequally spaced) sampling and derives a new principle of data treatment that can suppress the aliased spectra selectively. According to this principle, the common practice of image reconstruction, which backprojects data along the original sampling ray path, is never the best choice. The author proposes a new scheme of backprojection, which involves the longitudinal shift of projection data. A proper choice of longitudinal shift for backprojection provides effective and selective suppression of aliased spectra, with retention of the original frequency spectrum depending on the level of focus swing. With this shifted backprojection, the swing of focus can be made much smaller than for a conventional z-flying focal spot. The required amount of shift for backprojection is position dependent. Nevertheless, its implementation in the reconstruction process can be achieved simply by relocating the x-ray source and detector assembly from positions of actual scanning. Through simulation, the combination of shifted backprojection and the small swing of focus is evaluated. Results confirm that the artifact attributable to longitudinal aliasing is well suppressed in the entire field of view, whereas the penalty on the slice sensitivity profile (or longitudinal resolution) can be kept minimal. Moreover, this method solves other deficiencies of z-flying focus, such as inapplicability to scans with a thicker slice thickness.     

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.     

Cone beam collimation for single photon emission computed tomography: analysis, simulation, and image reconstruction using filtered backprojection

This paper presents an analysis of two cone beam configurations (having focal lengths of 40 and 60 cm) for the acquisition of single photon emission computed tomography (SPECT) projection data. A three-dimensional filtered backprojection algorithm is used to reconstruct SPECT images of cone beam projection data obtained using Monte Carlo simulations. The mathematical analysis resulted in on-axis point source sensitivities (calculated for a distance of 15 cm from the collimator surface) for cone beam configurations that were 1.4-3 times the sensitivities of parallel-hole and fan beam geometries having similar geometric resolutions. Cone beam collimation offers the potential for improved sensitivity for SPECT devices using large-field-of-view scintillation cameras.     

The backprojection method applied to classical tomography

A method based upon backprojection of projection images which allows six degrees of freedom in the selection of the plane to be tomosynthesized is described. Data can be collected with any type of classical tomographic apparatus, including linear, circular, or other complex motion unit, which has been modified by the addition of a digital imaging chain. A set of projection images can be acquired in a single sweep, although, in principle, they could be acquired simultaneously. In addition, C-arms can also be used with no limitations of the selection of tomosynthesized planes.     

扇形束CT超短扫描优质重建算法研究

本研究提出一种基于π线的扇形束CT超短扫描优质重建算法，新算法仍采用经典的FBP重建算法框架，当且仅当满足通过感兴趣区域的任一直线均与扫描轨道相交时，就可以有效地进行该区域的精确重建。新重建公式中回避了投影数据的求导运算，将加权Hilbert滤波转化为Hilbert滤波与Ramp滤波的组合形式。由于实际重建过程中要对离散数据进行处理，新算法中导数的回避将增加数值计算稳定性，提高重建图像质量。为验证新算法理论分析的有效性，通过与经典的超短扫描重建算法作对比实验，结果印证了新算法可完成超短扫描优质重建，在数值计算稳定性与重建图像质量提高两方面均有尚佳表现。     

Distance-driven projection and backprojection in three dimensions

Projection and backprojection are operations that arise frequently in tomographic imaging. Recently, we proposed a new method for projection and backprojection, which we call distance-driven, and that offers low arithmetic cost and a highly sequential memory access pattern. Furthermore, distance-driven projection and backprojection avoid several artefact-inducing approximations characteristic of some other methods. We have previously demonstrated the application of this method to parallel and fan beam geometries. In this paper, we extend the distance-driven framework to three dimensions and demonstrate its application to cone beam reconstruction. We also present experimental results to demonstrate the computational performance, the artefact characteristics and the noise-resolution characteristics of the distance-driven method in three dimensions.     

Hypertrophy without increased isometric strength after weight training

Summary Eight men (20–23 years) weight trained 3 days week^–1 for 19 weeks. Training sessions consisted of six sets of a leg press exercise (simultaneous hip and knee extension and ankle plantar flexion) on a weight machine, the last three sets with the heaviest weight that could be used for 7–20 repetitions. In comparison to a control group (n = 6) only the trained group increased (P<0.01) weight lifting performance (heaviest weight lifted for one repetition, 29%), and left and right knee extensor cross-sectional area (CAT scanning and computerized planimetry, 11%, P<0.05). In contrast, training caused no increase in maximal voluntary isometric knee extension strength, electrically evoked knee extensor peak twitch torque, and knee extensor motor unit activation (interpolated twitch method). These data indicate that a moderate but significant amount of hypertrophy induced by weight training does not necessarily increase performance in an isometric strength task different from the training task but involving the same muscle group. The failure of evoked twitch torque to increase despite hypertrophy may further indicate that moderate hypertrophy in the early stage of strength training may not necessarily cause an increase in intrinsic muscle force generating capacity.     

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.