Fourier properties of the fan-beam sinogram

A Novel Low-Dose Dual-Energy Imaging Method for a Fast-Rotating Gantry-Type CT Scanner

CT scan by use of a beam-filter placed between the x-ray source and the patient allows a single-scan low-dose dual-energy imaging with a minimal hardware modification to the existing CT systems. We have earlier demonstrated the feasibility of such imaging method with a multi-slit beam-filter reciprocating along the direction perpendicular to the CT rotation axis in a cone-beam CT system. However, such method would face mechanical challenges when the beam-filter is supposed to cooperate with a fast-rotating gantry in a diagnostic CT system. In this work, we propose a new scanning method and associated image reconstruction algorithm that can overcome these challenges. We propose to slide a beam-filter that has multi-slit structure with its slits being at a slanted angle with the CT gantry rotation axis during a scan. A streaky pattern would show up in the sinogram domain as a result. Using a notch filter in the Fourier domain of the sinogram, we removed the streaks and reconstructed an image by use of the filtered-backprojection algorithm. The remaining image artifacts were suppressed by applying l₀ norm based smoothing. Using this image as a prior, we have reconstructed low- and high-energy CT images in the iterative reconstruction framework. An image-based material decomposition then followed. We conducted a simulation study to test its feasibility using the XCAT phantom and also an experimental study using the Catphan phantom, a head phantom, an iodine-solution phantom, and a monkey in anesthesia, and showed its successful performance in image reconstruction and in material decomposition.

Fourier Properties of Symmetric-Geometry Computed Tomography and Its Linogram Reconstruction With Neural Network

In this work, we investigate the Fourier properties of a symmetric-geometry computed tomography (SGCT) with linearly distributed source and detector in a stationary configuration. A linkage between the 1D Fourier Transform of a weighted projection from SGCT and the 2D Fourier Transform of a deformed object is established in a simple mathematical form (i.e., the Fourier slice theorem for SGCT). Based on its Fourier slice theorem and its unique data sampling in the Fourier space, a Linogram-based Fourier reconstruction method is derived for SGCT. We demonstrate that the entire Linogram reconstruction process can be embedded as known operators into an end-to-end neural network. As a learning-based approach, the proposed Linogram-Net has capability of improving CT image quality for non-ideal imaging scenarios, a limited-angle SGCT for instance, through combining weights learning in the projection domain and loss minimization in the image domain. Numerical simulations and physical experiments on an SGCT prototype platform showed that our proposed Linogram-based method can achieve accurate reconstruction from a dual-SGCT scan and can greatly reduce computational complexity when compared with the filtered backprojection type reconstruction. The Linogram-Net achieved accurate reconstruction when projection data are complete and significantly suppressed image artifacts from a limited-angle SGCT scan mimicked by using a clinical CT dataset, with the average CT number error in the selected regions of interest reduced from 67.7 Hounsfield Units (HU) to 28.7 HU, and the average normalized mean square error of overall images reduced from 4.21e-3 to 2.65e-3.

Geometry and energy constrained projection extension

BACKGROUND

In clinical computed tomography (CT) applications, when a patient is obese or improperly positioned, the final tomographic scan is often partially truncated. Images directly reconstructed by the conventional reconstruction algorithms suffer from severe cupping and direct current bias artifacts. Moreover, the current methods for projection extension have limitations that preclude incorporation from clinical workflows, such as prohibitive computational time for iterative reconstruction, extra radiation dose, hardware modification, etc.METHOD:In this study, we first established a geometrical constraint and estimated the patient habitus using a modified scout configuration. Then, we established an energy constraint using the integral invariance of fan-beam projections. Two constraints were extracted from the existing CT scan process with minimal modification to the clinical workflows. Finally, we developed a novel dual-constraint based optimization model that can be rapidly solved for projection extrapolation and accurate local reconstruction.

RESULTS

Both numerical phantom and realistic patient image simulations were performed, and the results confirmed the effectiveness of our proposed approach.

CONCLUSION

We establish a dual-constraint-based optimization model and correspondingly develop an accurate extrapolation method for partially truncated projections. The proposed method can be readily integrated into the clinical workflow and efficiently solved by using a one-dimensional optimization algorithm. Moreover, it is robust for noisy cases with various truncations and can be further accelerated by GPU based parallel computing.

Motion compensation for cone-beam CT using Fourier consistency conditions

In cone-beam CT, involuntary patient motion and inaccurate or irreproducible scanner motion substantially degrades image quality. To avoid artifacts this motion needs to be estimated and compensated during image reconstruction. In previous work we showed that Fourier consistency conditions (FCC) can be used in fan-beam CT to estimate motion in the sinogram domain. This work extends the FCC to [Formula: see text] cone-beam CT. We derive an efficient cost function to compensate for [Formula: see text] motion using [Formula: see text] detector translations. The extended FCC method have been tested with five translational motion patterns, using a challenging numerical phantom. We evaluated the root-mean-square-error and the structural-similarity-index between motion corrected and motion-free reconstructions. Additionally, we computed the mean-absolute-difference (MAD) between the estimated and the ground-truth motion. The practical applicability of the method is demonstrated by application to respiratory motion estimation in rotational angiography, but also to motion correction for weight-bearing imaging of knees. Where the latter makes use of a specifically modified FCC version which is robust to axial truncation. The results show a great reduction of motion artifacts. Accurate estimation results were achieved with a maximum MAD value of 708 μm and 1184 μm for motion along the vertical and horizontal detector direction, respectively. The image quality of reconstructions obtained with the proposed method is close to that of motion corrected reconstructions based on the ground-truth motion. Simulations using noise-free and noisy data demonstrate that FCC are robust to noise. Even high-frequency motion was accurately estimated leading to a considerable reduction of streaking artifacts. The method is purely image-based and therefore independent of any auxiliary data.

An Improved Extrapolation Scheme for Truncated CT Data Using 2D Fourier-Based Helgason-Ludwig Consistency Conditions

We improve data extrapolation for truncated computed tomography (CT) projections by using Helgason-Ludwig (HL) consistency conditions that mathematically describe the overlap of information between projections. First, we theoretically derive a 2D Fourier representation of the HL consistency conditions from their original formulation (projection moment theorem), for both parallel-beam and fan-beam imaging geometry. The derivation result indicates that there is a zero energy region forming a double-wedge shape in 2D Fourier domain. This observation is also referred to as the Fourier property of a sinogram in the previous literature. The major benefit of this representation is that the consistency conditions can be efficiently evaluated via 2D fast Fourier transform (FFT). Then, we suggest a method that extrapolates the truncated projections with data from a uniform ellipse of which the parameters are determined by optimizing these consistency conditions. The forward projection of the optimized ellipse can be used to complete the truncation data. The proposed algorithm is evaluated using simulated data and reprojections of clinical data. Results show that the root mean square error (RMSE) is reduced substantially, compared to a state-of-the-art extrapolation method.

Data correlation based noise level estimation for cone beam projection data

BACKGROUND

In regularized iterative reconstruction algorithms, the selection of regularization parameter depends on the noise level of cone beam projection data.

OBJECTIVE

Our aim is to propose an algorithm to estimate the noise level of cone beam projection data.

METHODS

We first derived the data correlation of cone beam projection data in the Fourier domain, based on which, the signal and the noise were decoupled. Then the noise was extracted and averaged for estimation. An adaptive regularization parameter selection strategy was introduced based on the estimated noise level. Simulation and real data studies were conducted for performance validation.

RESULTS

There exists an approximately zero-energy double-wedge area in the 3D Fourier domain of cone beam projection data. As for the noise level estimation results, the averaged relative errors of the proposed algorithm in the analytical/MC/spotlight-mode simulation experiments were 0.8%, 0.14% and 0.24%, respectively, and outperformed the homogeneous area based as well as the transformation based algorithms. Real studies indicated that the estimated noise levels were inversely proportional to the exposure levels, i.e., the slopes in the log-log plot were -1.0197 and -1.049 with respect to the short-scan and half-fan modes. The introduced regularization parameter selection strategy could deliver promising reconstructed image qualities.

CONCLUSIONS

Based on the data correlation of cone beam projection data in Fourier domain, the proposed algorithm could estimate the noise level of cone beam projection data accurately and robustly. The estimated noise level could be used to adaptively select the regularization parameter.

Cone beam computed tomography scanning and diagnosis for dental implants

Cone beam computed tomography (CBCT) has become an important new technology for oral and maxillofacial surgery practitioners. CBCT provides improved office-based diagnostic capability and applications for surgical procedures, such as CT guidance through the use of computer-generated drill guides. A thorough knowledge of the basic science of CBCT as well as the ability to interpret the images correctly and thoroughly is essential to current practice.

Full data consistency conditions for cone-beam projections with sources on a plane

Cone-beam consistency conditions (also known as range conditions) are mathematical relationships between different cone-beam projections, and they therefore describe the redundancy or overlap of information between projections. These redundancies have often been exploited for applications in image reconstruction. In this work we describe new consistency conditions for cone-beam projections whose source positions lie on a plane. A further restriction is that the target object must not intersect this plane. The conditions require that moments of the cone-beam projections be polynomial functions of the source positions, with some additional constraints on the coefficients of the polynomials. A precise description of the consistency conditions is that the four parameters of the cone-beam projections (two for the detector, two for the source position) can be expressed with just three variables, using a certain formulation involving homogeneous polynomials. The main contribution of this work is our demonstration that these conditions are not only necessary, but also sufficient. Thus the consistency conditions completely characterize all redundancies, so no other independent conditions are possible and in this sense the conditions are full. The idea of the proof is to use the known consistency conditions for 3D parallel projections, and to then apply a 1996 theorem of Edholm and Danielsson that links parallel to cone-beam projections. The consistency conditions are illustrated with a simulation example.