Calibration-free beam hardening reduction in x-ray CBCT using the epipolar consistency condition and physical constraints.

BACKGROUND: The beam hardening effect is a typical source of artifacts in x-ray cone beam computed tomography (CBCT). It causes streaks in reconstructions and corrupted Hounsfield units toward the center of objects, widely known as cupping artifacts. PURPOSE: We present a novel efficient projection data-based method for reduction of beam-hardening artifacts and incorporate physical constraints on the shape of the compensation functions. The method is calibration-free and requires no additional knowledge of the scanning setup. METHOD: The mathematical model of the beam hardening effect caused by a single material is analyzed. We show that the effect of beam hardening on the resulting functions on the line integral measurements are monotonous and concave functions of the ideal data. This holds irrespective of any limiting assumptions on the energy dependency of the material, the detector response or properties of the x-ray source. A regression model for the beam hardening effect respecting these theoretical restrictions is proposed. Subsequently, we present an efficient method to estimate the parameters of this model directly in projection domain using an epipolar consistency condition. Computational efficiency is achieved by exploiting the linearity of an intermediate function in the formulation of our optimization problem. RESULTS: Our evaluation shows that the proposed physically constrained ECC 2 algorithm is effective even in challenging measured data scenarios with additional sources of inconsistency. CONCLUSIONS: The combination of mathematical consistency condition and a compensation model that is based on the properties of x-ray physics enables us to improve image quality of measured data retrospectively and to decrease the need for calibration in a data-driven manner.

Extended ellipse-line-ellipse trajectory for long-object cone-beam imaging with a mounted C-arm system.

Recent reports show that three-dimensional cone-beam (CB) imaging with a floor-mounted (or ceiling-mounted) C-arm system has become a valuable tool in interventional radiology. Currently, a circular short scan is used for data acquisition, which inevitably yields CB artifacts and a short coverage in the direction of the patient table. To overcome these two limitations, a more sophisticated data acquisition geometry is needed. This geometry should be complete in terms of Tuy's condition and should allow continuous scanning, while being compatible with the mechanical constraints of mounted C-arm systems. Additionally, the geometry should allow accurate image reconstruction from truncated data. One way to ensure such a feature is to adopt a trajectory that provides full R-line coverage within the field-of-view (FOV). An R-line is any segment of line that connects two points on a source trajectory, and the R-line coverage is the set of points that belong to an R-line. In this work, we propose a novel geometry called the extended ellipse-line-ellipse (ELE) for long-object imaging with a mounted C-arm system. This trajectory is built from modules consisting of two elliptical arcs connected by a line. We demonstrate that the extended ELE can be configured in many ways so that full R-line coverage is guaranteed. Both tight and relaxed parametric settings are presented. All results are supported by extensive mathematical proofs provided in appendices. Our findings make the extended ELE trajectory attractive for axially-extended FOV imaging in interventional radiology.

Towards clinical application of a Laplace operator-based region of interest reconstruction algorithm in C-arm CT.

It is known that a reduction of the field-of-view in 3-D X-ray imaging is proportional to a reduction in radiation dose. The resulting truncation, however, is incompatible with conventional reconstruction algorithms. Recently, a novel method for region of interest reconstruction that uses neither prior knowledge nor extrapolation has been published, named approximated truncation robust algorithm for computed tomography (ATRACT). It is based on a decomposition of the standard ramp filter into a 2-D Laplace filtering and a 2-D Radon-based residual filtering step. In this paper, we present two variants of the original ATRACT. One is based on expressing the residual filter as an efficient 2-D convolution with an analytically derived kernel. The second variant is to apply ATRACT in 1-D to further reduce computational complexity. The proposed algorithms were evaluated by using a reconstruction benchmark, as well as two clinical data sets. The results are encouraging since the proposed algorithms achieve a speed-up factor of up to 245 compared to the 2-D Radon-based ATRACT. Reconstructions of high accuracy are obtained, e.g., even real-data reconstruction in the presence of severe truncation achieve a relative root mean square error of as little as 0.92% with respect to nontruncated data.

Scaling calibration in region of interest reconstruction with the 1D and 2D ATRACT algorithm.

PURPOSE: Recently, a reconstruction algorithm for region of interest (ROI) imaging in C-arm CT was published, named Approximate Truncation Robust Algorithm for Computed Tomography (ATRACT). Even in the presence of substantial data truncation, the algorithm is able to reconstruct images without the use of explicit extrapolation or prior knowledge. However, the method suffers from a scaling and offset artifact in the reconstruction. Hence, the reconstruction results are not quantitative. It is our goal to reduce the scaling and offset artifact so that Hounsfield unit (HU) values can be used for diagnosis. METHODS: In this paper, we investigate two variants of the ATRACT method and present the analytical derivations of these algorithms in the Fourier domain. Then, we propose an empirical correction measure that can be applied to the ATRACT algorithm, to effectively compensate the scaling and offset issue. The proposed method is evaluated on ten clinical datasets in the presence of different degrees of artificial truncation. RESULTS: With the proposed correction approach, we achieved an average relative root-mean-square error (rRMSE) of 2.81% with respect to non-truncated Feldkamp, Davis, and Kress reconstruction, even for severely truncated data. The rRMSE is reduced to as little as 10% of the image reconstructed without the scaling calibration. CONCLUSIONS: The reconstruction results show that ROI reconstruction of high accuracy can be achieved since the scaling and offset artifact are effectively eliminated by the proposed method. With this improvement, the HU values may be used for post-processing operations such as bone or soft tissue segmentation if some tolerance is accepted.

Approximate truncation robust computed tomography--ATRACT.

We present an approximate truncation robust algorithm to compute tomographic images (ATRACT). This algorithm targets at reconstructing volumetric images from cone-beam projections in scenarios where these projections are highly truncated in each dimension. It thus facilitates reconstructions of small subvolumes of interest, without involving prior knowledge about the object. Our method is readily applicable to medical C-arm imaging, where it may contribute to new clinical workflows together with a considerable reduction of x-ray dose. We give a detailed derivation of ATRACT that starts from the conventional Feldkamp filtered-backprojection algorithm and that involves, as one component, a novel original formula for the inversion of the two-dimensional Radon transform. Discretization and numerical implementation are discussed and reconstruction results from both, simulated projections and first clinical data sets are presented.

Simulation tools for two-dimensional experiments in x-ray computed tomography using the FORBILD head phantom.

Mathematical phantoms are essential for the development and early stage evaluation of image reconstruction algorithms in x-ray computed tomography (CT). This note offers tools for computer simulations using a two-dimensional (2D) phantom that models the central axial slice through the FORBILD head phantom. Introduced in 1999, in response to a need for a more robust test, the FORBILD head phantom is now seen by many as the gold standard. However, the simple Shepp-Logan phantom is still heavily used by researchers working on 2D image reconstruction. Universal acceptance of the FORBILD head phantom may have been prevented by its significantly higher complexity: software that allows computer simulations with the Shepp-Logan phantom is not readily applicable to the FORBILD head phantom. The tools offered here address this problem. They are designed for use with Matlab®, as well as open-source variants, such as FreeMat and Octave, which are all widely used in both academia and industry. To get started, the interested user can simply copy and paste the codes from this PDF document into Matlab® M-files.

Accurate image reconstruction using real C-arm data from a Circle-plus-arc trajectory.

OBJECTIVE: Developing an efficient tool for accurate three-dimensional imaging from projections measured with C-arm systems. MATERIAL AND METHODS: A circle-plus-arc trajectory, which is complete and thus amenable to accurate reconstruction, is used. This trajectory is particularly attractive as its implementation does not require moving the patient. For reconstruction, we use the "M-line method", which allows processing the data in the efficient filtered backprojection mode. This method also offers the advantage of not requiring an ideal data acquisition geometry, i.e., the M-line algorithm can account for known deviations in the scanning geometry, which is important given that sizeable deviations are generally encountered in C-arm imaging. RESULTS: A robust implementation scheme of the "M-line method" that applies straightforwardly to real C-arm data is presented. In particular, a numerically stable technique to compute the view-dependent derivative with respect to the source trajectory parameter is applied, and an efficient way to compute the π-line backprojection intervals via a polygonal weighting mask is presented. Projection data of an anthropomorphic thorax phantom were acquired on a medical C-arm scanner and used to demonstrate the benefit of using a complete data acquisition geometry with an accurate reconstruction algorithm versus using a state-of-the-art implementation of the conventional Feldkamp algorithm with a circular short scan of cone-beam data. A significant image quality improvement based on visual assessment is shown in terms of cone-beam artifacts.

Cone-beam artifact evaluation of the factorization method.

PURPOSE: The authors investigate the CB artifact behavior of the factorization approach recently suggested for image reconstruction in circular cone-beam computed tomography. This investigation is carried out in a typical C-arm geometry and involves simulated data and for the first time also phantom and clinical CB data acquired with a commercially available angiographic system. METHODS: The CB artifact level is first measured using quantitative figures-of-merit that are computed from the reconstructions of the mathematical FORBILD head phantom and of a modified disk phantom. The authors then show reconstructions from a physical thorax phantom and clinical head data sets for a visual assessment of image quality. The performance of the factorization method is primarily compared to that of short-scan FDK, but the authors also show the results obtained with the full-scan FDK and the virtual PI-line BPF method for the simulation studies, as a benchmark. RESULTS: Quantitatively, the FORBILD head phantom reconstructions of both FDK methods show a spatially averaged bias of up to 1.2% in the axial slices about 9 cm away from the plane of the scan, which is placed 4 cm below the central slice through the phantom. The artifact level for the short-scan FDK method and the virtual PI-line BPF method noticeably depends on the scan orientation. The factorization approach can significantly reduce both, this dependency as well as the reconstruction bias. It also shows visually an improved quality of the clinical images compared to short-scan FDK, particularly close to the spine and in the subcranial regions of the clinical data sets. CONCLUSIONS: The factorization approach comes with noticeably lower reconstruction bias than the FDK methods and is least sensitive to the scan orientation among all considered short-scan methods. The data inconsistencies contained in the real data sets, such as scatter, beam hardening, or data truncation, show only little impact on the factorization results. Hence, in both, reconstructions from real and simulated data, the factorization method yields better image quality than short-scan FDK, albeit at the cost of some slight, directed high-frequency artifacts that are mostly visible in axial slices.

Line plus arc source trajectories and their R-line coverage for long-object cone-beam imaging with a C-arm system.

Cone-beam imaging with C-arm systems has become a valuable tool in interventional radiology. Currently, a simple circular trajectory is used, but future applications should use more sophisticated source trajectories, not only to avoid cone-beam artifacts but also to allow extended volume imaging. One attractive strategy to achieve these two goals is to use a source trajectory that consists of two parallel circular arcs connected by a line segment, possibly with repetition. In this work, we address the question of R-line coverage for such a trajectory. More specifically, we examine to what extent R-lines for such a trajectory cover a central cylindrical region of interest (ROI). An R-line is a line segment connecting any two points on the source trajectory. Knowledge of R-line coverage is crucial because a general theory for theoretically exact and stable image reconstruction from axially truncated data is only known for the points in the scanned object that lie on R-lines. Our analysis starts by examining the R-line coverage for the elemental trajectories consisting of (i) two parallel circular arcs and (ii) a circular arc connected orthogonally to a line segment. Next, we utilize our understanding of the R-lines for the aforementioned elemental trajectories to determine the R-line coverage for the trajectory consisting of two parallel circular arcs connected by a tightly fit line segment. For this trajectory, we find that the R-line coverage is insufficient to completely cover any central ROI. Because extension of the line segment beyond the circular arcs helps to increase the R-line coverage, we subsequently propose a trajectory composed of two parallel circular arcs connected by an extended line. We show that the R-lines for this trajectory can fully cover a central ROI if the line extension is long enough. Our presentation includes a formula for the minimum line extension needed to achieve full R-line coverage of an ROI with a specified size, and also includes a preliminary study on the required detector size, showing that the R-lines added by the line extension are not constraining.

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.

Filtered Backprojection Reconstruction with Depth-Dependent Filtering.

A direct filtered-backprojection (FBP) reconstruction algorithm is presented for circular cone-beam computed tomography (CB-CT) that allows the filter operation to be applied efficiently with shift-variant band-pass characteristics on the kernel function. Our algorithm is derived from the ramp-filter based FBP method of Feldkamp et al. and obtained by decomposing the ramp filtering into a convolution involving the Hilbert kernel (global operation) and a subsequent differentiation operation (local operation). The differentiation is implemented as a finite difference of two (Hilbert filtered) data samples and carried out as part of the backprojection step. The spacing between the two samples, which defines the low-pass characteristics of the filter operation, can thus be selected individually for each point in the image volume. We here define the sample spacing to follow the magnification of the divergent-beam geometry and thus obtain a novel, depth-dependent filtering algorithm for circular CB-CT. We evaluate this resulting algorithm using computer-simulated CB data and demonstrate that our algorithm yields results where spatial resolution and image noise are distributed much more uniformly over the field-of-view, compared to Feldkamp's approach.

A factorization approach for cone-beam reconstruction on a circular short-scan.

In this paper, we introduce a new algorithm for 3-D image reconstruction from cone-beam (CB) projections acquired along a partial circular scan. Our algorithm is based on a novel, exact factorization of the initial 3-D reconstruction problem into a set of independent 2-D inversion problems, each of which corresponds to finding the object density on one, single plane. Any such 2-D inversion problem is solved numerically using a projected steepest descent iteration scheme. We present a numerical evaluation of our factorization algorithm using computer-simulated CB data, without and with noise, of the FORBILD head phantom and of a disk phantom. First, we study quantitatively the impact of the reconstruction parameters on the algorithm performance. Next, we present reconstruction results for visual assessment of the achievable image quality and provide, for comparison, results obtained with two other state-of-the-art reconstruction algorithms for the circular short-scan.

Truncation correction for oblique filtering lines.

State-of-the-art filtered backprojection (FBP) algorithms often define the filtering operation to be performed along oblique filtering lines in the detector. A limited scan field of view leads to the truncation of those filtering lines, which causes artifacts in the final reconstructed volume. In contrast to the case where filtering is performed solely along the detector rows, no methods are available for the case of oblique filtering lines. In this work, the authors present two novel truncation correction methods which effectively handle data truncation in this case. Method 1 (basic approach) handles data truncation in two successive preprocessing steps by applying a hybrid data extrapolation method, which is a combination of a water cylinder extrapolation and a Gaussian extrapolation. It is independent of any specific reconstruction algorithm. Method 2 (kink approach) uses similar concepts for data extrapolation as the basic approach but needs to be integrated into the reconstruction algorithm. Experiments are presented from simulated data of the FORBILD head phantom, acquired along a partial-circle-plus-arc trajectory. The theoretically exact M-line algorithm is used for reconstruction. Although the discussion is focused on theoretically exact algorithms, the proposed truncation correction methods can be applied to any FBP algorithm that exposes oblique filtering lines.

Geometric calibration of the circle-plus-arc trajectory.

In this paper, a novel geometric calibration method for C-arm cone-beam scanners is presented which allows the calibration of the circle-plus-arc trajectory. The main idea is the separation of the trajectory into two circular segments (circle segment and arc segment) which are calibrated independently. This separation makes it possible to reuse a calibration phantom which has been successfully applied in clinical environments to calibrate numerous routinely used C-arm systems. For each trajectory segment, the phantom is placed in an optimal position. The two calibration results are then combined by computing the transformation the phantom underwent between the independent calibration runs. This combination can be done in a post-processing step by using standard linear algebra. The method is not limited to circle-plus-arc trajectories and works for any calibration procedure in which the phantom has a preferred orientation with respect to a trajectory segment. Results are presented for both simulated as well as real data acquired with a C-arm system. We also present the first image reconstruction results for the circle-plus-arc trajectory using real C-arm data.

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.

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.