Image reconstruction from fan-beam projections on less than a short scan

This work is concerned with 2D image reconstruction from fan-beam projections. It is shown that exact and stable reconstruction of a given region-of-interest in the object does not require all lines passing through the object to be measured. Complete (non-truncated) fan-beam projections provide sufficient information for reconstruction when 'every line passing through the region-of-interest intersects the vertex path in a non-tangential way'. The practical implications of this condition are discussed and a new filtered-backprojection algorithm is derived for reconstruction. Experiments with computer-simulated data are performed to support the mathematical results.     

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

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

A unified analysis of FBP-based algorithms in helical cone-beam and circular cone- and fan-beam scans

A circular scanning trajectory is and will likely remain a popular choice of trajectory in computed tomography (CT) imaging because it is easy to implement and control. Filtered-backprojection (FBP)-based algorithms have been developed previously for approximate and exact reconstruction of the entire image or a region of interest within the image in circular cone-beam and fan-beam cases. Recently, we have developed a 3D FBP-based algorithm for image reconstruction on PI-line segments in a helical cone-beam scan. In this work, we demonstrated that the 3D FBP-based algorithm indeed provided a rather general formulation for image reconstruction from divergent projections (such as cone-beam and fan-beam projections). On the basis of this formulation we derived new approximate or exact algorithms for image reconstruction in circular cone-beam or fan-beam scans, which can be interpreted as special cases of the helical scan. Existing algorithms corresponding to the derived algorithms were identified. We also performed a preliminary numerical study to verify our theoretical results in each of the cases. The results in the work can readily be generalized to other non-circular trajectories.     

Analytical fan-beam and cone-beam reconstruction algorithms with uniform attenuation correction for SPECT

In this paper, we developed an analytical fan-beam reconstruction algorithm that compensates for uniform attenuation in SPECT. The new fan-beam algorithm is in the form of backprojection first, then filtering, and is mathematically exact. The algorithm is based on three components. The first one is the established generalized central-slice theorem, which relates the 1D Fourier transform of a set of arbitrary data and the 2D Fourier transform of the backprojected image. The second one is the fact that the backprojection of the fan-beam measurements is identical to the backprojection of the parallel measurements of the same object with the same attenuator. The third one is the stable analytical reconstruction algorithm for uniformly attenuated Radon data, developed by Metz and Pan. The fan-beam algorithm is then extended into a cone-beam reconstruction algorithm, where the orbit of the focal point of the cone-beam imaging geometry is a circle. This orbit geometry does not satisfy Tuy's condition and the obtained cone-beam algorithm is an approximation. In the cone-beam algorithm, the cone-beam data are first backprojected into the 3D image volume; then a slice-by-slice filtering is performed. This slice-by-slice filtering procedure is identical to that of the fan-beam algorithm. Both the fan-beam and cone-beam algorithms are efficient, and computer simulations are presented. The new cone-beam algorithm is compared with Bronnikov's cone-beam algorithm, and it is shown to have better performance with noisy projections.     

Exact fan-beam and 4pi-acquisition cone-beam SPECT algorithms with uniform attenuation correction

This paper presents analytical fan-beam and cone-beam reconstruction algorithms that compensate for uniform attenuation in single photon emission computed tomography. First, a fan-beam algorithm is developed by obtaining a relationship between the two-dimensional (2D) Fourier transform of parallel-beam projections and fan-beam projections. Using this relationship, 2D Fourier transforms of equivalent parallel-beam projection data are obtained from the fan-beam projection data. Then a quasioptimal analytical reconstruction algorithm for uniformly attenuated Radon data, developed by Metz and Pan, is used to reconstruct the image. A cone-beam algorithm is developed by extending the fan-beam algorithm to 4pi solid angle geometry. The cone-beam algorithm is also an exact algorithm.     

An FDK-like cone-beam SPECT reconstruction algorithm for non-uniform attenuated projections acquired using a circular trajectory

In this paper, Novikov's inversion formula of the attenuated two-dimensional (2D) Radon transform is applied to the reconstruction of attenuated fan-beam projections acquired with equal detector spacing and of attenuated cone-beam projections acquired with a flat planar detector and circular trajectory. The derivation of the fan-beam algorithm is obtained by transformation from parallel-beam coordinates to fan-beam coordinates. The cone-beam reconstruction algorithm is an extension of the fan-beam reconstruction algorithm using Feldkamp-Davis-Kress's (FDK) method. Computer simulations indicate that the algorithm is efficient and is accurate in reconstructing slices close to the central slice of the cone-beam orbit plane. When the attenuation map is set to zero the implementation is equivalent to the FDK method. Reconstructed images are also shown for noise corrupted projections.     

A novel extension of the parallel-beam projection-slice theorem to divergent fan-beam and cone-beam projections

The general goal of this paper is to extend the parallel-beam projection-slice theorem to divergent fan-beam and cone-beam projections without rebinning the divergent fan-beam and cone-beam projections into parallel-beam projections directly. The basic idea is to establish a novel link between the local Fourier transform of the projection data and the Fourier transform of the image object. Analogous to the two- and three-dimensional parallel-beam cases, the measured projection data are backprojected along the projection direction and then a local Fourier transform is taken for the backprojected data array. However, due to the loss of the shift invariance of the image object in a single view of the divergent-beam projections, the measured projection data is weighted by a distance dependent weight w(r) before the local Fourier transform is performed. The variable r in the weighting function w(r) is the distance from the backprojected point to the x-ray source position. It is shown that a special choice of the weighting function, w(r)=1/r, will facilitate the calculations and a simple relation can be established between the Fourier transform of the image function and the local Fourier transform of the 1/r-weighted backprojection data array. Unlike the parallel-beam cases, a one-to-one correspondence does not exist for a local Fourier transform of the backprojected data array and a single line in the two-dimensional (2D) case or a single slice in the 3D case of the Fourier transform of the image function. However, the Fourier space of the image object can be built up after the local Fourier transforms of the 1/r-weighted backprojection data arrays are shifted and then summed in a laboratory frame. Thus the established relations Eq. (27) and Eq. (29) between the Fourier space of the image object and the Fourier transforms of the backprojected data arrays can be viewed as a generalized projection-slice theorem for divergent fan-beam and cone-beam projections. Once the Fourier space of the image function is built up, an inverse Fourier transform could be performed to reconstruct tomographic images from the divergent beam projections. Due to the linearity of the Fourier transform, an image reconstruction step can be performed either when the complete Fourier space is available or in parallel with the building of the Fourier space. Numerical simulations are performed to verify the generalized projection-slice theorem by using a disc phantom in the fan-beam case.     

Respiratory motion estimation from slowly rotating x-ray projections: theory and simulation

Understanding the movement of tumors caused by respiratory motion is very important for conformal radiatherapy. However, respiratory motion is very difficult to study by conventional x-ray CT imaging since object motion causes inconsistent projection views, leading to artifacts in reconstructed images. We propose to estimate the parameters of a nonrigid, free breathing motion model from a set of projection views of the thorax that are acquired using a slowly rotating cone-beam CT scanner. This approach involves deforming a motion-free reference thorax volume according to the estimated parameters and comparing its projections to the corresponding measured projection views. The parameters are optimized by minimizing a regularized squared error cost function. Simulation results with a fan-beam geometry show good agreement between the estimated motion and the true motion, which supports the potential of this approach for estimating four-dimensional (three-dimensional spatial + temporal) respiratory motion.     

Weighted expectation maximization reconstruction algorithms with application to gated megavoltage tomography

We propose and investigate weighted expectation maximization (EM) algorithms for image reconstruction in x-ray tomography. The development of the algorithms is motivated by the respiratory-gated megavoltage tomography problem, in which the acquired asymmetric cone-beam projections are limited in number and unevenly sampled over view angle. In these cases, images reconstructed by use of the conventional EM algorithm can contain ring- and streak-like artefacts that are attributable to a combination of data inconsistencies and truncation of the projection data. By use of computer-simulated and clinical gated fan-beam megavoltage projection data, we demonstrate that the proposed weighted EM algorithms effectively mitigate such image artefacts.     

A new data consistency condition for fan-beam projection data

The sum of all attenuation data acquired in one view of parallel-beam projections is a view angle independent constant. This fact is known as a data consistency condition on the two-dimensional Radon transforms. It plays an important role in tomographic image reconstruction and artifact correction. In this paper, a novel fan-beam data consistency condition (FDCC) is derived and presented. Using the FDCC, individual projection data in one view of fan-beam projections can be estimated from filtering all other projection data measured from different view angles. Numerical simulations are performed to validate the new FDCC in correcting ring artifacts caused by malfunctioning detector cells.     

Optimal noise control in and fast reconstruction of fan-beam computed tomography image.

We proposed a linear approach that exploits statistically complementary information inherent in the projection data of fan-beam computed tomography (CT) for achieving a bias-free image-variance reduction in fan-beam CT. This linear approach leads to the development of infinite classes of hybrid algorithms for image reconstruction in fan-beam CT. These hybrid algorithms are computationally more efficient and numerically less susceptible to data noise and to the effect of finite sampling than the conventional fan-beam filtered back-projection (FFBP) algorithm. We also developed infinite classes of generalized fan-beam filtered back-projection (GFFBP) algorithms, which include the conventional FFBP algorithm as a special member. We demonstrated theoretically and quantitatively that the hybrid and GFFBP algorithms are identical (or different) in the absence (or presence) of data noise and of the effect of finite sampling. More importantly, we identified the statistically optimal hybrid algorithm that may have potentially significant implication to image reconstruction in fan-beam CT. Extensive numerical results of computer-simulation studies validated our theoretical results.     

Image reconstruction in peripheral and central regions-of-interest and data redundancy

Algorithms have been developed for image reconstruction within a region-of-interest (ROI) from fan-beam data less than that required for reconstructing the entire image. However, these algorithms do not admit truncated data. In this work, we investigate exact ROI-image reconstruction from fan-beam data containing truncations by use of the so-called fan-beam backprojection-filtration (BPF) algorithm. We also generalize the fan-beam BPF algorithm to exploit redundant information inherent in the truncated fan-beam data. Because the parallel-beam scan can be interpreted as a special case of the fan-beam scan, based upon the fan-beam BPF algorithm, we derive a parallel-beam BPF algorithm for exactly reconstructing ROI images from truncated parallel-beam data. Furthermore, we investigate image reconstruction within two types of distinctive ROIs, which are referred to as the peripheral and central ROIs, respectively, from fan-beam data containing truncations and discuss their potential clinical applications. The results can readily be generalized to reconstructing 3D ROI images from data acquired in circular and helical cone-beam scan. They can also be extended to address ROI-image-reconstruction problems in parallel-, fan-, and cone-beam scans with general trajectories. The work not only has significant implications for clinical and animal-imaging applications of CT, but also may find applications in other imaging modalities.     

Fan-beam filtered-backprojection reconstruction without backprojection weight

In this paper, we address the problem of two-dimensional image reconstruction from fan-beam data acquired along a full 2pi scan. Conventional approaches that follow the filtered-backprojection (FBP) structure require a weighted backprojection with the weight depending on the point to be reconstructed and also on the source position; this weight appears only in the case of divergent beam geometries. Compared to reconstruction from parallel-beam data, the backprojection weight implies an increase in computational effort and is also thought to have some negative impacts on noise properties of the reconstructed images. We demonstrate here that direct FBP reconstruction from full-scan fan-beam data is possible with no backprojection weight. Using computer-simulated, realistic fan-beam data, we compared our novel FBP formula with no backprojection weight to the use of an FBP formula based on equal weighting of all data. Comparisons in terms of signal-to-noise ratio, spatial resolution and computational efficiency are presented. These studies show that the formula we suggest yields images with a reduced noise level, at almost identical spatial resolution. This effect increases quickly with the distance from the center of the field of view, from 0% at the center to 20% less noise at 20 cm, and to 40% less noise at 25 cm. Furthermore, the suggested method is computationally less demanding and reduces computation time with a gain that was found to vary between 12% and 43% on the computers used for evaluation.     

Image reconstruction in regions-of-interest from truncated projections in a reduced fan-beam scan

In a reduced fan-beam scan, the scanned angular range is smaller than that in a short scan (i.e., a half-scan). In this work, we have developed a new algorithm, which is referred to as the backprojection-filtration (BPF) algorithm, for exact image reconstruction within ROIs from reduced-scan data containing truncations. Explicit conditions on data acquisition have also been derived for exact image reconstruction within an ROI. We have performed a preliminary quantitative study whose results demonstrated and verified the proposed fan-beam BPF algorithm and the derived conditions on data acquisition. The proposed BPF algorithm can have significant implications for clinical and animal CT imaging, therapy imaging, electron paramagnetic resonance imaging and other tomographic imaging because it allows for reconstruction from truncated data and for a potentially drastic reduction of radiation dose and/or of imaging time.     

An alternative proof of Bukhgeim and Kazantsev's inversion formula for attenuated fan-beam projections

An inversion formula was developed by Bukhgeim and Kazantsev for attenuated fan-beam projections [Russian Academy of Science Siberian Branch: The Sobolev Institute of Mathematics (2002)]. The inversion formula was obtained by relating the attenuated fan-beam projections to unattenuated fan-beam projections and by trickily processing the unattenuated fan-beam projections. We show in this paper that the inversion formula can be readily obtained from Novikov's inversion formula for the two-dimensional (2D) attenuated radon transform. The derivation provides an alternative proof of Bukhgeim and Kazantsev's inversion formula by the use of transformation between parallel-beam coordinates and fan-beam coordinates and thus is quite elementary.     

Use of 3D reconstruction to correct for patient motion in SPECT

Patient motion occurring during data acquisition in single photon emission computed tomography (SPET) can cause serious reconstruction artefacts. We have developed a new approach to correct for head motion in brain SPECT. Prior to motion, projections are assigned to conventional projections. When head motion occurs, it is measured by a motion monitoring system, and subsequent projection data are mapped to 'virtual' projections. The appropriate position of each virtual projection is determined by applying the converse of the patient's accumulated motion to the actual camera projection. Conventional and virtual projections, taken together, form a consistent set that can be reconstructed using a three-dimensional (3D) algorithm. The technique has been tested on a range of simulated rotational movements, both within and out of the transaxial plane. For all simulated movements, the motion corrected images exhibited better agreement with a motion free reconstruction than did the uncorrected images. This technique may help to overcome one of the major remaining limitations on image quality and quantitative accuracy in SPECT.     

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.     

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

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

Projection-based motion compensation for gated coronary artery reconstruction from rotational x-ray angiograms

Three-dimensional reconstruction of coronary arteries can be performed during x-ray-guided interventions by gated reconstruction from a rotational coronary angiography sequence. Due to imperfect gating and cardiac or breathing motion, the heart's motion state might not be the same in all projections used for the reconstruction of one cardiac phase. The motion state inconsistency causes motion artefacts and degrades the reconstruction quality. These effects can be reduced by a projection-based 2D motion compensation method. Using maximum-intensity forward projections of an initial uncompensated reconstruction as reference, the projection data are transformed elastically to improve the consistency with respect to the heart's motion state. A fast iterative closest-point algorithm working on vessel centrelines is employed for estimating the optimum transformation. Motion compensation is carried out prior to and independently from a final reconstruction. The motion compensation improves the accuracy of reconstructed vessel radii and the image contrast in a software phantom study. Reconstructions of human clinical cases are presented, in which the motion compensation substantially reduces motion blur and improves contrast and visibility of the coronary arteries.     

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.