Hewei Gao

Scatter correction method for x-ray CT using primary modulation: Phantom studies

PURPOSE Scatter correction is a major challenge in x-ray imaging using large area detectors. Recently, the authors proposed a promising scatter correction method for x-ray computed tomography (CT) using primary modulation. Proof of concept was previously illustrated by Monte Carlo simulations and physical experiments on a small phantom with a simple geometry. In this work, the authors provide a quantitative evaluation of the primary modulation technique and demonstrate its performance in applications where scatter correction is more challenging. METHODS The authors first analyze the potential errors of the estimated scatter in the primary modulation method. On two tabletop CT systems, the method is investigated using three phantoms: A Catphan 600 phantom, an anthropomorphic chest phantom, and the Catphan 600 phantom with two annuli. Two different primary modulators are also designed to show the impact of the modulator parameters on the scatter correction efficiency. The first is an aluminum modulator with a weak modulation and a low modulation frequency, and the second is a copper modulator with a strong modulation and a high modulation frequency. RESULTS On the Catphan 600 phantom in the first study, the method reduces the error of the CT number in the selected regions of interest (ROIs) from 371.4 to 21.9 Hounsfield units (HU); the contrast to noise ratio also increases from 10.9 to 19.2. On the anthropomorphic chest phantom in the second study, which represents a more difficult case due to the high scatter signals and object heterogeneity, the method reduces the error of the CT number from 327 to 19 HU in the selected ROIs and from 31.4% to 5.7% on the overall average. The third study is to investigate the impact of object size on the efficiency of our method. The scatter-to-primary ratio estimation error on the Catphan 600 phantom without any annulus (20 cm in diameter) is at the level of 0.04, it rises to 0.07 and 0.1 on the phantom with an elliptical annulus (30 cm in the minor axis and 38 cm in the major axis) and with a circular annulus (38 cm in diameter). CONCLUSIONS On the three phantom studies, good scatter correction performance of the proposed method has been demonstrated using both image comparisons and quantitative analysis. The theory and experiments demonstrate that a strong primary modulation that possesses a low transmission factor and a high modulation frequency is preferred for high scatter correction accuracy.

Modulator design for x‐ray scatter correction using primary modulation: Material selection

PURPOSE An optimal material selection for primary modulator is proposed in order to minimize beam hardening of the modulator in x-ray cone-beam computed tomography (CBCT). Recently, a measurement-based scatter correction method using primary modulation has been developed and experimentally verified. In the practical implementation, beam hardening of the modulator blocker is a limiting factor because it causes inconsistency in the primary signal and therefore degrades the accuracy of scatter correction. METHODS This inconsistency can be purposely assigned to the effective transmission factor of the modulator whose variation as a function of object filtration represents the magnitude of beam hardening of the modulator. In this work, the authors show that the variation reaches a minimum when the K-edge of the modulator material is near the mean energy of the system spectrum. Accordingly, an optimal material selection can be carried out in three steps. First, estimate and evaluate the polychromatic spectrum for a given x-ray system including both source and detector; second, calculate the mean energy of the spectrum and decide the candidate materials whose K-edge energies are near the mean energy; third, select the optimal material from the candidates after considering both the magnitude of beam hardening and the physical and chemical properties. RESULTS A tabletop x-ray CBCT system operated at 120 kVp is used to validate the material selection method in both simulations and experiments, from which the optimal material for this x-ray system is then chosen. With the transmission factor initially being 0.905 and 0.818, simulations show that erbium provides the least amount of variation as a function of object filtrations (maximum variations are 2.2% and 4.3%, respectively, only one-third of that for copper). With different combinations of aluminum and copper filtrations (simulating a range of object thicknesses), measured overall variations are 2.5%, 1.0%, and 8.6% for 25.4 microm of copper, erbium, and tungsten, respectively. With and without 300 microm of copper in the beam, the measured variations for 25.4 microm of copper, erbium, and tungsten, 1 mm of aluminum, as well as 406 microm of copper, are 1.8%, 0.2%, 5.5%, 1.9%, and 7.5%, respectively. CONCLUSIONS The spatial variation in the effective transmission factor of the modulator blocker due to beam hardening caused by the modulator itself reaches a minimum when the K-edge of the modulator material is near the mean energy of the spectrum. An optimal modulator material selection using the K-edge discontinuity is therefore proposed.

Beam Hardening Correction for Middle-Energy Industrial Computerized Tomography

In this paper, a new beam hardening correction (BHC) method for middle-energy industrial computerized tomography (CT) is presented. Our method is derived from linearization and is straightforward without iteration involved. The linearization is commonly used as a preprocessing method in BHC. Conventionally, only one-material objects can be conveniently corrected by linearization. In industrial CT, two-material objects, especially cylinders with high-Z material outside and low-Z material inside are frequently encountered. Our approach focuses on this kind of objects. The new method works well as long as the two-material object meets the conditions that the thickness of the outer material (usually wall) is thick enough and the second-order item of the Taylor expansion of the linearization is relatively small. We pointed out and proved that there is an approximately constant scaling factor difference between our linearization step and an ideal correction based on prior knowledge of objects. The scaling factor magnifies the attenuation coefficient of the inner material after reconstruction. Therefore, a weighting function is introduced into our algorithm as a restoration. To sum up, there are three steps in our method: 1) correct raw projections by the mapping function of the outer material; 2) reconstruct the cross-section image from the modified projections; 3) scale the image by a weighting function. With this method, the beam hardening artifacts are greatly reduced and the overall attenuation coefficients are accurately obtained. We also presented a compensation step to remove the countercupping artifacts in case that the conditions are not fully met. Our method is well verified in both numerical simulations and practical experiments on a 450-KeV CT system

Direct filtered-backprojection-type reconstruction from a straight-line trajectory

A computed tomography (CT) imaging configuration with a straight-line trajectory is investigated, and a direct filtered backprojection (FBP) algorithm is presented. This kind of system may be useful for industrial applications and security inspections. Projections from a straight-line trajectory have a special property where data from each detector element correspond to a parallel-beam projection of a certain view angle. However, the sampling steps of parallel beams differ from view to view. Rebinning raw projections into uniformly sampled parallel-beam projections is a common choice for this type of reconstruction problem. However, the rebinning procedure suffers a loss of spatial resolution because of interpolations. Our reconstruction method is first derived from the Fourier slice theorem, where a coordinate transform and geometrical relations in projection and backprojection are used. It is then extended to 3-D scanning geometry. Finally, data-shift preprocessing is proposed to reduce computation and memory requirements by removing useless projections in raw data. In this method, the spatial resolution is better preserved and the reconstruction is less sensitive to data truncation than in the rebinning-to-parallel-beam method. To deal with limited angle problem, an iterative reconstruction reprojection method is introduced to estimate missing data and improve the image quality.

Volumetric imaging from a multisegment straight-line trajectory and a practical reconstruction algorithm

A new imaging configuration whose trajectory is a multisegment straight line is investigated, and a practical reconstruction algorithm is proposed. It is a natural extension of an imaging configuration with a straight-line trajectory. These kinds of scanning systems may be useful in industry and security inspections. As is known, projection data from a single straight-line trajectory are incomplete and their reconstruction suffers from a limited-angle problem. A multisegment straight-line trajectory can be used to compensate for this deficiency. To reconstruct images, a practical reconstruction algorithm is derived. It is of the Feldkamp-Davis-Kress (FDK) type, and is efficient and straightforward. Like the FDK algorithm, our reconstruction is exact in the midplane and can be exact everywhere if the density of the scanned object is independent of the direction z, though the integral of the reconstructed image along z is no longer preserved. Numerical simulations validate our method.

Empirical Cupping Correction for CT Scanners with Primary Modulation (ECCP)

X-ray CT measures the attenuation of polychromatic x-rays through an object. The rawdata acquired, which are the negative logarithm of the relative x-ray intensity behind the patient, must undergo water precorrection to linearize the measurement and to convert them into line integrals that are ready for reconstruction. The function to linearize the measured projection data depends on the detected spectrum of the ray. This spectrum may vary as a function of the detector position, e.g. in cases where the heel effect becomes relevant, or where a bow-tie filter introduces channel-dependent beam hardening, or in cases where a primary modulator is used to modulate the primary intensity of the spectrum. We propose a new approach that allows to handle these effects. Our empirical cupping correction for primary modulation (ECCP) corrects for artifacts, such as cupping artifacts or ring artifacts, that are induced by non-linearities in the projection data due to spatially varying pre- or post filtration of the x-rays. To do so, ECCP requires nothing but a simple scan of a homogeneous phantom of nearly arbitrary shape. Based on this information, coefficients of a polynomial series are calculated and stored for later use. Numerical examples and physical measurements are shown to demonstrate the quality of the precorrection. ECCP achieves to remove the cupping artifacts and to obtain well-calibrated CT-values even in cases of strong primary modulation. A combination of ECCP with analytical techniques yielding a hybrid cupping correction method is possible and allows for channel-dependent correction functions.

An Improved Form of Linogram Algorithm for Image Reconstruction

The implementation of the discrete ramp filter in the filtered backprojection (FBP) algorithms has been carefully investigated by Kak and Slane in Principles of Computerized Tomographic Imaging. In the linogram algorithms, however, it was rarely used in a correct way. Instead, an oversampling (zero-padding) factor of four is usually taken to reduce the dishing artifacts. We here improve the linogram algorithm by using a new strategy of weighting instead of in its original implementation. The new weighting is produced via the Fourier transform of the discrete ramp filter similar to that in Kak and Slaney. We explicitly derive the connection between the oversampling processing and the implementation of the discrete ramp filter in the spatial domain: if projection data are zero-padded to double length, with the discrete ramp filter, the effect is theoretically equivalent to zero-padding infinitely long in the original implementation; without zero-padding, the modified algorithm can obtain almost accurate reconstruction in the central part of an image. Our theoretical analysis also gives the optimal way of implementing the linogram algorithm, leading to savings in computational time and memory space. Results of theoretical analysis are validated by numerical simulations.

An Extrapolation Method for Image Reconstruction from a Straight-line Trajectory

In this paper, we present an extrapolation method for compensation of missing data in image reconstruction from a straight-line trajectory (LCT scan). We adopt the idea for band-limited signal extrapolation. The missing data are estimated and updated during each iteration by a reprojection from the reconstructed image, in which the reconstruction and reprojection steps are implemented by the linogram technique and its inverse, respectively. The convergence condition of this extrapolation is studied. Our method has some benefits. First, the linogram reconstruction does not need any interpolation at all. Second, only O(N2 log N) arithmetic operations are required in each iteration when the projection and the reconstruction are both of size NtimesN. Third, prior knowledge on the image and the Fourier spaces such as non-negativity can be introduced to improve the convergence efficiency. The performances of this approach are compared with the iterative reconstruction reprojection (IRR) method.

Straight-Line-Trajectory-Based X-Ray Tomographic Imaging for Security Inspections: System Design, Image Reconstruction and Preliminary Results

With the source and detector kept stationary while an inspected object moves between them, straight-line-trajectory-based tomographic imaging (SLTTI) has the capability to provide both digital radiologic images and tomographic images simultaneously, at high speed and with low-cost equipment. It has a promising future for fast security inspection for air cargo examinations. In order to fully investigate the feasibility of this approach, an experimental SLTTI platform has been developed. In this work, the system design is first presented. Then, a hybrid image reconstruction algorithm is proposed to obtain high quality tomographic images from the system to solve the limited-angle problem it represents. In the proposed algorithm, an initial reconstruction using either a direct filtered-backprojection (FBP) -type algorithm or the linogram technique is followed by a Gerchberg-Papoulis (GP) extrapolation to compensate for missing data. The GP extrapolation is implemented by the linogram technique, and constrained by a total variation (TV) minimization. By taking advantage of the FBP, the linogram and the TV techniques, we establish a fast and stable iterative reconstruction for the SLTTI. Finally, numerical simulations and realistic experiments are given, through which our proposals are validated to be feasible and effective. In the numerical examples, our algorithm can reduce the reconstruction inaccuracies for both high contrast and low contrast objects by factors of more than 15 and can generate nearly artifact free images as if complete projection data were available. A realistic experiment on a CatPhan©600 phantom suggests that high quality tomographic images can be achieved from 120-deg of source fan-beam angle with undersampled data. For a group of explosive simulacra using 90 projection views over 120-deg of source fan-beam angle, the reconstruction inaccuracy can be reduced to lower than 4%.

Optimization of system parameters for modulator design in x-ray scatter correction using primary modulation

The impact of the system parameters of the modulator on X-ray scatter correction using primary modulation is studied and an optimization of the modulator design is presented. Recently, a promising scatter correction method for X-ray computed tomography (CT) that uses a checkerboard pattern of attenuating blockers (primary modulator) placed between the X-ray source and the object has been developed and experimentally verified. The blocker size, d, and the blocker transmission factor, α, are critical to the performance of the primary modulation method. In this work, an error caused by aliasing of primary whose magnitude depends on the choices of d and α, and the scanned object, is set as the object function to be minimized, with constraints including the X-ray focal spot, the physical size of the detector element, and the noise level. The optimization is carried out in two steps. In the first step, d is chosen as small as possible but should meet a lower-bound condition. In the second step, α should be selected to balance the error level in the scatter estimation and the noise level in the reconstructed image. The lower bound of d on our tabletop CT system is 0.83 mm. Numerical simulations suggest 0.6 < α < 0.8 is appropriate. Using a Catphan 600 phantom, a copper modulator (d = 0.89 mm, α = 0.70) expectedly outperforms an aluminum modulator (d = 2.83 mm, α = 0.90). With the aluminum modulator, our method reduces the average error of CT number in selected contrast rods from 371.4 to 25.4 Hounsfield units (HU) and enhances the contrast to noise ratio (CNR) from 10.9 to 17.2; when the copper modulator is used, the error is further reduced to 21.9 HU and the CNR is further increased to 19.2.