Image reconstruction with shift-variant filtration and its implication for noise and resolution properties in fan-beam computed tomography

In computed tomography (CT), the fan-beam filtered backprojection (FFBP) algorithm is used widely for image reconstruction. It is known that the FFBP algorithm can significantly amplify data noise and aliasing artifacts in situations where the focal lengths are comparable to or smaller than the size of the field of measurement (FOM). In this work, we propose an algorithm that is less susceptible to data noise, aliasing, and other data inconsistencies than is the FFBP algorithm while retaining the favorable resolution properties of the FFBP algorithm. In an attempt to evaluate the noise properties in reconstructed images, we derive analytic expressions for image variances obtained by use of the FFBP algorithm and the proposed algorithm. Computer simulation studies are conducted for quantitative evaluation of the spatial resolution and noise properties of images reconstructed by use of the algorithms. Numerical results of these studies confirm the favorable spatial resolution and noise properties of the proposed algorithm and verify the validity of the theoretically predicted image variances. The proposed algorithm and the derived analytic expressions for image variances can have practical implications for both estimation and detection/classification tasks making use of CT images, and they can readily be generalized to other fan-beam geometries.     

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.     

Fast reconstruction with uniform noise properties in halfscan computed tomography

The hybrid algorithms developed recently for the reconstruction of fan-beam images possess computational and noise properties superior to those of the fan-beam filtered backprojection (FFBP) algorithm. However, the hybrid algorithms cannot be applied directly to a halfscan fan-beam sinogram because they require knowledge of a fullscan fan-beam sinogram. In this work, we developed halfscan-hybrid algorithms for image reconstruction in halfscan computed tomography (CT). Numerical evaluation indicates that the proposed halfscan-hybrid algorithms are computationally more efficient than are the widely used halfscan-FFBP algorithms. Also, the results of quantitative studies demonstrated clearly that the noise levels in images reconstructed by use of the halfscan-hybrid algorithm are generally lower and spatially more uniform than are those in images reconstructed by use of the halfscan-FFBP algorithm. Such reduced and uniform image noise levels may be translated into improvement of the accuracy and precision of lesion detection and parameter estimation in noisy CT images without increasing the radiation dose to the patient. Therefore, the halfscan-hybrid algorithms may have significant implication for image reconstruction in conventional and helical CT.     

Image reconstruction 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 tube combined with gratings. Images were reconstructed using a parallel-beam reconstruction formula. However, parallel-beam reconstruction formulae are not applicable when the parallel-beam approximation fails. In this paper, we present a new image reconstruction formula for fan-beam DPC-CT. There are several novel features of the new image reconstruction formula: (i) when the scanning angular range of data acquisition is larger than pi + gamma(m) (gamma(m) is the full fan angle), the entire field of view can be exactly reconstructed; (ii) when the scanning angular range is smaller than pi + gamma(m), a local region of interest (ROI) can be exactly reconstructed; (iii) it enables an exact reconstruction for a local ROI when the projection data are truncated at some view angles; (iv) it enlarges the imaging field of view when the detector is asymmetrically placed. In this last case, the data are truncated from every view angle. Numerical simulations have been conducted to validate the new reconstruction formula.     

Image covariance and lesion detectability in direct fan-beam x-ray computed tomography

We consider noise in computed tomography images that are reconstructed using the classical direct fan-beam filtered backprojection algorithm, from both full- and short-scan data. A new, accurate method for computing image covariance is presented. The utility of the new covariance method is demonstrated by its application to the implementation of a channelized Hotelling observer for a lesion detection task. Results from the new covariance method and its application to the channelized Hotelling observer are compared with results from Monte Carlo simulations. In addition, the impact of a bowtie filter and x-ray tube current modulation on reconstruction noise and lesion detectability are explored for full-scan reconstruction.     

A new scheme for view-dependent data differentiation in fan-beam and cone-beam computed tomography

In computed tomography, analytical fan-beam (FB) and cone-beam (CB) image reconstruction often involves a view-dependent data differentiation. The implementation of this differentiation step is critical in terms of resolution and image quality. In this work, we present a new differentiation scheme that is robust to changes in the data acquisition geometry and to coarse view sampling. Our scheme was compared to two previously suggested methods, which we call the direct scheme and the chain-rule scheme. Image reconstructions were performed from computer-simulated data of the Shepp-Logan phantom, the FORBILD thorax phantom and a modified FORBILD head phantom. For FB reconstruction, we investigated three acquisition geometries: a circular, an ellipse-shaped and a square-shaped trajectory. For CB reconstruction, the circle-plus-line trajectory was considered. Image comparison showed that the new scheme performs consistently well when varying the scenario, in both FB and CB geometry, unlike the other two schemes.     

Density resolution of proton computed tomography

Conformal proton radiation therapy requires accurate prediction of the Bragg peak position. Protons may be more suitable than conventional x-rays for this task since the relative electron density distribution can be measured directly with proton computed tomography (CT). However, proton CT has its own limitations, which need to be carefully studied before this technique can be introduced into routine clinical practice. In this work, we have used analytical relationships as well as the Monte Carlo simulation tool GEANT4 to study the principal resolution limits of proton CT. The noise level observed in proton CT images of a cylindrical water phantom with embedded tissue-equivalent density inhomogeneities, which were generated based on GEANT4 simulations, compared well with predictions based on Tschalar's theory of energy loss straggling. The relationship between phantom thickness, initial energy, and the relative electron density resolution was systematically investigated to estimate the proton dose needed to obtain a given density resolution. We show that a reasonable density resolution can be achieved with a relatively small dose, which is comparable to or even lower than that of x-ray CT.     

New low-contrast resolution phantoms for computed tomography

Computed tomography (CT) has been established as a major imaging modality in diagnostic radiology. Accordingly, acceptance testing and quality control of CT scanners is of great importance. While most procedures and phantoms for testing are widely accepted, there is still discussion and uncertainty about low-contrast (LC) sensitivity. In our opinion this unsatisfactory situation is caused at least in part by the lack of suitable phantoms for LC resolution measurements. We investigated the commonly used phantoms for LC detectability, the Catphan, and for LC resolution, the ATS phantom. While the Catphan showed stable object contrasts, the ATS phantom's measured contrast exhibited a strong dependence on temperature and x-ray quality. Based on newly developed polyurethane resin materials, we designed and tested a LC resolution phantom with several different contrast steps. The object contrasts showed no dependence on temperature and beam quality. The new LC resolution phantom proved to be very suitable for measuring a scanner's low-contrast sensitivity in the image plane, one of the most important image quality parameters. To assess LC resolution in three dimensions we designed an additional phantom with rows of spherical objects. A first prototype was evaluated in a multicenter study. The setup proved to be very helpful to quantify the in-plane and axial LC sensitivity of spiral CT scan modes.     

Interactively variable isotropic resolution in computed tomography

An individual balancing between spatial resolution and image noise is necessary to fulfil the diagnostic requirements in medical CT imaging. In order to change influencing parameters, such as reconstruction kernel or effective slice thickness, additional raw-data-dependent image reconstructions have to be performed. Therefore, the noise versus resolution trade-off is time consuming and not interactively applicable. Furthermore, isotropic resolution, expressed by an equivalent point spread function (PSF) in every spatial direction, is important for the undistorted visualization and quantitative evaluation of small structures independent of the viewing plane. Theoretically, isotropic resolution can be obtained by matching the in-plane and through-plane resolution with the aforementioned parameters. Practically, however, the user is not assisted in doing so by current reconstruction systems and therefore isotropic resolution is not commonly achieved, in particular not at the desired resolution level. In this paper, an integrated approach is presented for equalizing the in-plane and through-plane spatial resolution by image filtering. The required filter kernels are calculated from previously measured PSFs in x/y- and z-direction. The concepts derived are combined with a variable resolution filtering technique. Both approaches are independent of CT raw data and operate only on reconstructed images which allows for their application in real time. Thereby, the aim of interactively variable, isotropic resolution is achieved. Results were evaluated quantitatively by measuring PSFs and image noise, and qualitatively by comparing the images to direct reconstructions regarded as the gold standard. Filtered images matched direct reconstructions with arbitrary reconstruction kernels with standard deviations in difference images of typically between 1 and 17 HU. Isotropic resolution was achieved within 5% of the selected resolution level. Processing times of 20-100 ms per frame allow for interactive use.     

Half-scan and single-plane intensity diffraction tomography for phase objects

A reconstruction theory for intensity diffraction tomography (I-DT) has been proposed that permits reconstruction of a weakly scattering object without explicit knowledge of phase information. In this work, we examine the application of I-DT, using either planar- or spherical-wave incident wavefields, for imaging three-dimensional (3D) phase objects. We develop and investigate two algorithms for reconstructing phase objects that utilize only half of the measurements that would be needed to reconstruct a complex-valued object function. Each reconstruction algorithm reconstructs the phase object by use of different sets of intensity measurements. Although the developed reconstruction algorithms are equivalent mathematically, we demonstrate that their numerical and noise propagation properties differ considerably. We implement numerically the reconstruction algorithms and present reconstructed images to demonstrate their use and to corroborate our theoretical assertions.     

High temporal resolution and streak-free four-dimensional cone-beam computed tomography

Cone-beam computed tomography (CBCT) has been clinically used to verify patient position and to localize the target of treatment in image-guided radiation therapy (IGRT). However, when the chest and the upper abdomen are scanned, respiratory-induced motion blurring limits the utility of CBCT. In order to mitigate this blurring, respiratory-gated CBCT, i.e. 4D CBCT, was introduced. In 4D CBCT, the cone-beam projection data sets acquired during a gantry rotation are sorted into several respiratory phases. In these gated reconstructions, the number of projections for each respiratory phase is significantly reduced. Consequently, undersampling streaking artifacts are present in the reconstructed images, and the image contrast resolution is also significantly compromised. In this paper, we present a new method to simultaneously achieve both high temporal resolution ( approximately 100 ms) and streaking artifact-free image volumes in 4D CBCT. The enabling technique is a newly proposed image reconstruction method, i.e. prior image constrained compressed sensing (PICCS), which enables accurate image reconstruction using vastly undersampled cone-beam projections and a fully sampled prior image. Using PICCS, a streak-free image can be reconstructed from 10-20 cone-beam projections while the signal-to-noise ratio is determined by a denoising feature of the selected objective function and by the prior image, which is reconstructed using all of the acquired cone-beam projections. This feature of PICCS breaks the connection between the temporal resolution and streaking artifacts' level in 4D CBCT. Numerical simulations and experimental phantom studies have been conducted to validate the method.     

The noise power spectrum in computed X-ray tomography

An expression is derived showing that the two-dimensional noise power spectrum of computed X-ray tomography is proportional to [G(k)]2/k where k is the radial spatial frequency and G(k) is the one-dimensional corrective filter used in the filtered back-projection reconstuction technique. It is shown that predicted noise power spectra compare well with those estimated from CT reconstructions of simulated noise for both the ramp filter and the Hanning-weighted ramp filter. A consequence of the non-uniform shape of the noise power spectrum is that statistical noise in CT reconstructions is correlated from point to point. Because of this correlation when the reconstructed CT values are averaged over some region, the uncertainty of the average depends on the shape of the region as well as its area. This dependence is confirmed by computer simulations.     

Favorable noise uniformity properties of Fourier-based interpolation and reconstruction approaches in single-slice helical computed tomography

Volumes reconstructed by standard methods from single-slice helical computed tomography (CT) data have been shown to have noise levels that are highly nonuniform relative to those in conventional CT. These noise nonuniformities can affect low-contrast object detectability and have also been identified as the cause of the zebra artifacts that plague maximum intensity projection (MIP) images of such volumes. While these spatially variant noise levels have their root in the peculiarities of the helical scan geometry, there is also a strong dependence on the interpolation and reconstruction algorithms employed. In this paper, we seek to develop image reconstruction strategies that eliminate or reduce, at its source, the nonuniformity of noise levels in helical CT relative to that in conventional CT. We pursue two approaches, independently and in concert. We argue, and verify, that Fourier-based longitudinal interpolation approaches lead to more uniform noise ratios than do the standard 360LI and 180LI approaches. We also demonstrate that a Fourier-based fan-to-parallel rebinning algorithm, used as an alternative to fanbeam filtered backprojection for slice reconstruction, also leads to more uniform noise ratios, even when making use of the 180LI and 360LI interpolation approaches.     

High temporal resolution for multislice helical computed tomography

Multislice helical computed tomography (CT) substantially reduces scanning time. However, the temporal resolution of individual images is still insufficient for imaging rapidly moving organs such as the heart and adjacent pulmonary vessels. It may, in some cases, be worse than with current single-slice helical CT. The purpose of this study is to describe a novel image reconstruction algorithm to improve temporal resolution in multislice helical CT, and to evaluate its performance against existing algorithms. The proposed image reconstruction algorithm uses helical interpolation followed by data weighting based on the acquisition time. The temporal resolution, the longitudinal (z-axis) spatial resolution, the image noise, and the in-plane image artifacts created by a moving phantom were compared with those from the basic multislice helical reconstruction (helical filter interpolation, HFI) algorithm and the basic single-slice helical reconstruction algorithm (180 degrees linear interpolation, 180LI) using computer simulations. Computer simulation results were verified with CT examinations of the heart and lung vasculature using a 0.5 second multislice scanner. The temporal resolution of HFI algorithm varies from 0.28 and 0.86 s, depending on helical pitch. The proposed method improves the resolution to a constant value of 0.29 s, independent of pitch, allowing moving objects to be imaged with reduced blurring or motion artifacts. The spatial (z) resolution was slightly worse than with the HFI algorithm; the image noise was worse than with the HFI algorithm but was comparable to axial (step-and-shoot) CT. The proposed method provided sharp images of the moving objects, portraying the anatomy accurately. The proposed algorithm for multislice helical CT allowed us to obtain CT images with high temporal resolution. It may improve the image quality of clinical cardiac, lung, and vascular CT imaging.     

Spatially varying longitudinal aliasing and resolution in spiral computed tomography

Spiral computed tomography (CT) has revolutionized conventional CT as a truly three-dimensional imaging modality. A number of studies aimed at evaluating the longitudinal resolution in spiral CT have been presented, but the spatially varying nature of the longitudinal resolution in spiral CT has been largely left undiscussed. In this paper, we investigate the longitudinal resolution in spiral CT as affected by the spatially varying longitudinal aliasing. We propose the treatment of aliasing as a signal dependent, additive noise, and define a new image quality parameter, the contrast-to-aliased-noise ratio (CNaR), that relates to possible image degradation or loss of resolution caused by aliasing. We performed CT simulations and actual phantom scans using a resolution phantom consisting of sequences of spherical beads of different diameters, extending along the longitudinal axis. Our results show that the off-isocenter longitudinal resolution differs significantly from the longitudinal resolution at the isocenter and that the CNaR decreases with distance from the isocenter, and is a function of pitch and the helical interpolation algorithm used. The longitudinal resolution was observed to worsen with decreasing CNaR. We conclude that the longitudinal resolution in spiral CT is spatially varying, and can be characterized by the CNaR measured at the transaxial location of interest.     

Motion correction for improved target localization with on-board cone-beam computed tomography

On-board imager (OBI) based cone-beam computed tomography (CBCT) has become available in radiotherapy clinics to accurately identify the target in the treatment position. However, due to the relatively slow gantry rotation (typically about 60 s for a full 360 degrees scan) in acquiring the CBCT projection data, the patient's respiratory motion causes serious problems such as blurring, doubling, streaking and distortion in the reconstructed images, which heavily degrade the image quality and the target localization. In this work, we present a motion compensation method for slow-rotating CBCT scans by incorporating into image reconstruction a patient-specific motion model, which is derived from previously obtained four-dimensional (4D) treatment planning CT images of the same patient via deformable registration. The registration of the 4D CT phases results in transformations representing a temporal sequence of three-dimensional (3D) deformation fields, or in other words, a 4D model of organ motion. The algorithm was developed heuristically in two-dimensional (2D) parallel-beam geometry and extended to 3D cone-beam geometry. By simulations with digital phantoms capable of translational motion and other complex motion, we demonstrated that the algorithm can reduce the motion artefacts locally, and restore the tumour size and shape, which may thereby improve the accuracy of target localization and patient positioning when CBCT is used as the treatment guidance.     

The effect of computed tomography noise and tissue heterogeneity on cerebral blood flow determination by xenon-enhanced computed tomography

The errors associated with derivation of cerebral blood flow values by the xenon-enhanced computed tomography (CT) method have been evaluated as a function of tissue heterogeneity and CT noise. The results of this study indicate that CT noise introduces large errors in the derived flow value when data for a single, unprocessed voxel are used for this purpose. CT noise increases the derived flow values in a systematic way. Tissue heterogeneity results in a systematic error which lowers the derived flow values. Errors due to both parameters are computed for typical and extreme conditions.     

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.     

A simple theorem relating noise and patient dose in computed tomography

Presented is a simple model describing the dependence of image noise in computed tomography on the x-ray beam profile. The model is used to derive the x-ray profile which minimizes total image noise at constant integral patient dose. The profile may be produced with a bow-tie-type beam shaping filter. Results of the analysis are validated using a computer simulation of computed tomography (CT) acquisition and reconstruction.