A general exact reconstruction for cone-beam CT via backprojection-filtration

In this paper, we prove a generalized backprojection-filtration formula for exact cone-beam image reconstruction with an arbitrary scanning locus. Our proof is independent of the shape of the scanning locus, as long as the object is contained in a region where there is a chord through any interior point. As special cases, this generalized formula can be applied with cone-beam scanning along nonstandard spiral and saddle curves, as well as in an n-PI window setting. The algorithmic implementation and numerical results are described to support the correctness of our general claim.     

A cone-beam reconstruction algorithm for circle-plus-arc data-acquisition geometry

In cone-beam computerized tomography (CT), projections acquired with the focal spot constrained on a planar orbit cannot provide a complete set of data to reconstruct the object function exactly. There are severe distortions in the reconstructed noncentral transverse planes when the cone angle is large. In this work, a new method is proposed which can obtain a complete set of data by acquiring cone-beam projections along a circle-plus-arc orbit. A reconstruction algorithm using this circle-plus-arc orbit is developed, based on the Radon transform and Grangeat's formula. This algorithm first transforms the cone-beam projection data of an object to the first derivative of the three-dimensional (3-D) Radon transform, using Grangeat's formula, and then reconstructs the object using the inverse Radon transform. In order to reduce interpolation errors, new rebinning equations have been derived accurately, which allows one-dimensional (1-D) interpolation to be used in the rebinning process instead of 3-D interpolation. A noise-free Defrise phantom and a Poisson noise-added Shepp-Logan phantom were simulated and reconstructed for algorithm validation. The results from the computer simulation indicate that the new cone-beam data-acquisition scheme can provide a complete set of projection data and the image reconstruction algorithm can achieve exact reconstruction. Potentially, the algorithm can be applied in practice for both a standard CT gantry-based volume tomographic imaging system and a C-arm-based cone-beam tomographic imaging system, with little mechanical modification required.     

The n-PI-method for helical cone-beam CT

A new class of acquisition schemes for helical cone-beam computed tomography (CB-CT) scanning is introduced, and their effect on the reconstruction methods is analyzed. These acquisition schemes are based on a new detector shape that is bounded by the helix. It will be shown that the data acquired with these schemes are compatible with exact reconstruction methods, and the adaptation of exact reconstruction algorithms to the new acquisition geometry is described. At the same time, the so-called PI-sufficiency condition is fulfilled. Moreover, a good fit to the acquisition requirements of the various medical applications of cone-beam CT is achieved. In contrast to other helical cone-beam acquisition and reconstruction methods, the n-PI-method introduced in this publication allows for variable pitches of the acquisition helix. This additional feature will introduce a higher flexibility into the acquisition protocols of future medical cone-beam scanners. An approximative n-PI-filtered backprojection (n-PI-FBP) reconstruction method is presented and verified. It yields convincing image quality.     

Image reconstruction from cone-beam projections: necessary and sufficient conditions and reconstruction methods

Previously unknown sufficient conditions, a necessary condition, and reconstruction methods for image reconstruction from cone-beam projections are developed. A sufficient condition developed is contained in the following statement. Statement 5: If one every plane that intersects the object, there exists at least one cone-beam source point, then the object can be reconstructed. Reconstruction methods for an arbitrary configuration of source points that satisfy Statement 5 are derived. By requiring additional conditions on the configuration of source points, a more efficient reconstruction method is developed. It is shown that when the configuration of source points is a circle, Statement 5 is not satisfied. In spite of this, several suggestions are made for reconstruction from a circle of source points.     

Comments on ;A cone-beam filtered backprojection reconstruction algorithm for cardiac single photon emission computed tomography' by G. T. Gullberg and G. L. Zeng

In the above-titled work by G. T. Gullberg and G. L. Zeng (ibid., vol.11, no.1, p.91-101, 1992), a fan-beam reconstruction formula of a noncircular scanning locus was derived and extended for half-scan cone-beam reconstruction. However, the reconstruction formula is not exact mathematically unless a necessary condition is satisfied. In this correspondence, the commenters derive this necessary condition and provide a geometrical explanation of the condition.     

Noise properties of chord-image reconstruction

Recently, there has been much progress in algorithm development for image reconstruction in cone-beam computed tomography (CT). Current algorithms, including the chord-based algorithms, now accept minimal data sets for obtaining images on volume regions-of-interest (ROIs) thereby potentially allowing for reduction of X-ray dose in diagnostic CT. As these developments are relatively new, little effort has been directed at investigating the response of the resulting algorithm implementations to physical factors such as data noise. In this paper, we perform an investigation on the noise properties of ROI images reconstructed by using chord-based algorithms for different scanning configurations. We find that, for the cases under study, the chord-based algorithms yield images with comparable quality. Additionally, it is observed that, in many situations, large data sets contain extraneous data that may not reduce the ROI-image variances.     

Fourier rebinning applied to multiplanar circular-orbit cone-beam SPECT

We study the application of Fourier rebinning methods to dual-planar cone-beam SPECT. Dual-planar cone-beam SPECT involves the use of a pair of dissimilar cone-beam collimators on a dual-camera SPECT system. Each collimator has its focus in a different axial plane. While dual-planar data is best reconstructed with fully three-dimensional (3-D) iterative methods, these methods are slow and have prompted a search for faster reconstruction techniques. Fourier rebinning was developed to estimate equivalent parallel projections from 3-D PET data, but it simply expresses a relationship between oblique projections taken in planes not perpendicular to the axis of rotation and direct projections taken in those that are. We find that it is possible to put cone-beam data in this context as well. The rebinned data can then be reconstructed using either filtered backprojection (FBP) or parallel iterative algorithms such as OS-EM. We compare the Feldkamp algorithm and fully 3-D OSEM reconstruction with Fourier-rebinned reconstructions on realistically-simulated Tc-99m HMPAO brain SPECT data. We find that the Fourier-rebinned reconstructions exhibit much less image noise and lower variance in region-of-interest (ROI) estimates than Feldkamp. Also, Fourier-rebinning followed by OSEM with nonuniform attenuation correction exhibits less bias in ROI estimates than Feldkamp with Chang attenuation correction. The Fourier-rebinned ROI estimates exhibit bias and variance comparable to those from fully 3-D OSEM and require considerably less processing time. However, in areas off the axis of rotation, the axial-direction resolution of FORE-reconstructed images is poorer than that of images reconstructed with 3-D OSEM. We conclude that Fourier rebinning is a practical and potentially useful approach to reconstructing data from dual-planar circular-orbit cone-beam systems.     

VOIR: a volumetric image reconstruction algorithm based on Fouriertechniques for inversion of the 3-D Radon transform

A novel volumetric image reconstruction algorithm known as VOIR is presented for inversion of the 3-D Radon transform or its radial derivative. The algorithm is a direct implementation of the projection slice theorem for plane integrals. It generalizes one of the most successful methods in 2-D Fourier image reconstruction involving concentric-square rasters to 3-D; in VOIR, the spectral data, which is calculated by fast Fourier techniques, lie on concentric cubes and are interpolated by a bilinear method on the sides of these concentric cubes. The algorithm has great computational advantages over filtered-backprojection algorithms; for images of side dimension N, the numerical complexity of VOIR is O(N(3) log N) instead of O(N (4)) for backprojection techniques. An evaluation of the image processing performance is reported by comparison of reconstructed images from simulated cone-beam scans of a contrast and resolution test object. The image processing performance is also characterized by an analysis of the edge response from the reconstructed images.     

低剂量X射线CT重建算法仿真分析

对Shepp-Logan头部图像模型提供的原始图像进行低剂量X射线计算机断层成像（CT,Computed Tomography）平移/旋转扫描,利用所得数据对图像的统计重建算法进行仿真。将仿真结果与传统的滤波反投影重建算法的重建结果进行比较,并分别将两种算法的重建图像与原图像进行比较。通过比较得出,滤波反投影重建算法不能有效重建低剂量CT图像,而统计重建算法重建效果优于前者,能够较好地恢复原图像,从而能够适用于低剂量条件下的CT图像重建。     

A cone-beam filtered backprojection reconstruction algorithm for cardiac single photon emission computed tomography

A filtered backprojection reconstruction algorithm was developed for cardiac single photon emission computed tomography with cone-beam geometry. The algorithm reconstructs cone-beam projections collected from ;short scan' acquisitions of a detector traversing a noncircular planar orbit. Since the algorithm does not correct for photon attenuation, it is designed to reconstruct data collected over an angular range of slightly more than 180 degrees with the assumption that the range of angles is oriented so as not to acquire the highly attenuated posterior projections of emissions from cardiac radiopharmaceuticals. This sampling scheme is performed to minimize the attenuation artifacts that result from reconstructing posterior projections. From computer simulations, it is found that reconstruction of attenuated projections has a greater effect upon quantitation and image quality than any potential cone-beam reconstruction artifacts resulting from insufficient sampling of cone-beam projections. With nonattenuated projection data, cone beam reconstruction errors in the heart are shown to be small (errors of at most 2%).     

Reduction of artifacts induced by misaligned geometry in cone-beam CT

The quality of computed tomography (CT) images frequently suffers from artifacts caused by scanner misalignments. In this paper, we discuss the results of our improved approximate cone-beam reconstruction formula, which corrects for the mechanical misalignment of the scanner. Based on the general filtered back-projection (FBP) algorithm proposed by Feldkamp, the new formula reduces the artifacts in reconstructed images. Six parameters are employed to describe the scanner misalignment. Experimental results show that the images reconstructed by using the new formula are clearer than those reconstructed by using the general FBP algorithm in a misaligned system. Through the new formula, we have reduced the artifacts caused by misalignment.     

Time-resolved interventional cardiac C-arm cone-beam CT: an application of the PICCS algorithm

Time-resolved cardiac imaging is particularly interesting in the interventional setting since it would provide both image guidance for accurate procedural planning and cardiac functional evaluations directly in the operating room. Imaging the heart in vivo using a slowly rotating C-arm system is extremely challenging due to the limitations of the data acquisition system and the high temporal resolution required to avoid motion artifacts. In this paper, a data acquisition scheme and an image reconstruction method are proposed to achieve time-resolved cardiac cone-beam computed tomography imaging with isotropic spatial resolution and high temporal resolution using a slowly rotating C-arm system. The data are acquired within 14 s using a single gantry rotation with a short scan angular range. The enabling image reconstruction method is the prior image constrained compressed sensing (PICCS) algorithm. The prior image is reconstructed from data acquired over all cardiac phases. Each cardiac phase is then reconstructed from the retrospectively gated cardiac data using the PICCS algorithm. To validate the method, several studies were performed. Both numerical simulations using a hybrid motion phantom with static background anatomy as well as physical phantom studies have been used to demonstrate that the proposed method enables accurate reconstruction of image objects with a high isotropic spatial resolution. A canine animal model scanned in vivo was used to further validate the method.     

基于变螺距螺旋FDK重建算法的研究

目前广泛应用的锥束螺旋CT主要采用螺旋FDK算法进行图像重建,但为了使算法能更应用于实际,许多学者对标准的螺旋轨迹进行了拓展.本文主要研究了变螺距螺旋扫描轨迹.变螺距螺旋扫描可以提高扫描速度,有更高的时间、空间分辨率.本文详细介绍了变螺距螺旋扫描轨迹,给出了具体的变螺距螺旋扫描轨迹方程.针对图像重建,本文主要从FDK重...     

Handling of long objects in iterative improvement of nonexact reconstruction in helical cone-beam CT

In medical helical cone-beam CT, it is common that the region-of-interest (ROI) is contained inside the helix cylinder, while the complete object is long and extends outside the top and the bottom of the cylinder. This is the Long Object Problem. Analytical reconstruction methods for helical cone-beam CT have been designed to handle this problem. It has been shown that a moderate amount of over-scanning is sufficient for reconstruction of a certain ROI. The over-scanning projection rays travel both through the ROI, as well as outside the ROI. This is unfortunate for iterative methods since it seems impossible to compute accurate values for the projection rays which travel partly inside and partly outside the ROI. Therefore, it seems that the useful ROI will diminish for every iteration step. We propose the following solution to the problem. First, we reconstruct volume regions also outside the ROI. These volume regions will certainly be incompletely reconstructed, but our experimental results show that they serve well for projection generation. This is rather counter-intuitive and contradictory to our initial assumptions. Second, we use careful extrapolation and masking of projection data. This is not a general necessity, but needed for the chosen iterative algorithm, which includes rebinning and iterative filtered backprojection. Our idea here was to use an approximate reconstruction method which gives cone-beam artifacts and then improve the reconstructed result by iterative filtered backprojection. The experimental results seem very encouraging. The cone-beam artifacts can indeed be removed. Even voxels close to the boundary of the ROI are as well enhanced by the iterative loop as those in the middle of the ROI.     

采用高频能量的CT几何参数自标定方法

针对锥束CT系统中几何参数失配引起几何伪影的问题,提出了一种采用空域高频能量的几何伪影自校正算法。该算法以重建图像的空域高频能量为目标函数建立优化模型,通过单纯形法迭代求解使空域高频能量最大的几何参数最优解。利用投影图像的特性提取部分参数作为输入初值,减小算法搜索范围。并采用图形处理器（Graphic Processing Unit,GPU）对自校正过程中的图像重建并行加速,减少重建时间,提高校正速度。实验结果表明:该算法具有较高的求解精度,最大相对误差不超过5%,对重建图像中的几何伪影有较好的校正效果。同时,在不影响精度的情况下减少了迭代次数,算法执行效率提高了18.8%。     

Tomographic reconstruction using an adaptive tetrahedral mesh defined by a point cloud

Medical images in nuclear medicine are commonly represented in three dimensions as a stack of two-dimensional images that are reconstructed from tomographic projections. Although natural and straightforward, this may not be an optimal visual representation for performing various diagnostic tasks. A method for three-dimensional (3-D) tomographic reconstruction is developed using a point cloud image representation. A point cloud is a set of points (nodes) in space, where each node of the point cloud is characterized by its position and intensity. The density of the nodes determines the local resolution allowing for the modeling of different parts of the image with different resolution. The reconstructed volume, which in general could be of any resolution, size, shape, and topology, is represented by a set of nonoverlapping tetrahedra defined by the nodes. The intensity at any point within the volume is defined by linearly interpolating inside a tetrahedron from the values at the four nodes that define the tetrahedron. This approach creates a continuous piecewise linear intensity over the reconstruction domain. The reconstruction provides a distinct multiresolution representation, which is designed to accurately and efficiently represent the 3-D image. The method is applicable to the acquisition of any tomographic geometry, such as parallel-, fan-, and cone-beam; and the reconstruction procedure can also model the physics of the image detection process. An efficient method for evaluating the system projection matrix is presented. The system matrix is used in an iterative algorithm to reconstruct both the intensity and location of the distribution of points in the point cloud. Examples of the reconstruction of projection data generated by computer simulations and projection data experimentally acquired using a Jaszczak cardiac torso phantom are presented. This work creates a framework for voxel-less multiresolution representation of images in nuclear medicine.     

A Filtered Backprojection Algorithm for Triple-Source Helical Cone-Beam CT

Multisource cone-beam computed tomography (CT) is an attractive approach of choice for superior temporal resolution, which is critically important for cardiac imaging and contrast enhanced studies. In this paper, we present a filtered-backprojection (FBP) algorithm for triple-source helical cone-beam CT. The algorithm is both exact and efficient. It utilizes data from three inter-helix PI-arcs associated with the inter-helix PI-lines and the minimum detection windows defined for the triple-source configuration. The proof of the formula is based on the geometric relations specific to triple-source helical cone-beam scanning. Simulation results demonstrate the validity of the reconstruction algorithm. This algorithm is also extended to a multisource version for $( 2N + 1 )$ -source helical cone-beam CT. With parallel computing, the proposed FBP algorithms can be significantly faster than our previously published multisource backprojection-filtration algorithms. Thus, the FBP algorithms are promising in applications of triple-source helical cone-beam CT.      

Efficient Monte Carlo based scatter artifact reduction in cone-beam micro-CT

Cupping and streak artifacts caused by the detection of scattered photons may severely degrade the quantitative accuracy of cone-beam X-ray computed tomography (CT) images. In order to overcome this problem, we propose and validate the following iterative scatter artifact reduction scheme. First, an initial image is reconstructed from the scatter-contaminated projections. Next, the scatter component of the projections is estimated from the initial reconstruction by a Monte Carlo (MC) simulation. The estimate obtained is then utilized during the reconstruction of a scatter-corrected image. The last two steps are repeated until an adequate correction is obtained. The estimation of the noise-free scatter projections in this scheme is accelerated in the following way: first, a rapid (i.e., based on a low number of simulated photon tracks) MC simulation is executed. The noisy result of this simulation is de-noised by a three-dimensional fitting of Gaussian basis functions. We demonstrate that, compared to plain MC, this method shortens the required simulation time by three to four orders of magnitude. Using simulated projections of a small animal phantom, we show that one cycle of the scatter correction scheme is sufficient to produce reconstructed images that barely differ from the reconstructions of scatter-free projections. The reconstructions of data acquired with a charge-coupled device based micro-CT scanner demonstrate a nearly complete removal of the scatter-induced cupping artifact. Quantitative errors in a water phantom are reduced from around 12% for reconstructions without the scatter correction to 1% after the proposed scatter correction has been applied. In conclusion, a general, accurate, and efficient scatter correction algorithm is developed that requires no mechanical modifications of the scanning equipment and results in only a moderate increase in the total reconstruction time.     

Motion-compensated and gated cone beam filtered back-projection for 3-D rotational X-ray angiography

This paper presents a method to reconstruct moving objects from cone beam X-ray projections acquired during a single rotational run using a given motion vector field. The method is applicable to voxel driven cone-beam filtered back-projection reconstruction approaches. Here, a formulation based on the algorithm of Feldkamp, Davis, and Kress (FDK) is presented. The motion correction is applied during the back-projection step by shifting the voxel to be reconstructed according to the motion vector field. The method is applied to three-dimensional (3-D) rotational X-ray angiography. Projections from a beating coronary heart phantom are simulated. Motion-compensated reconstructions with varying accuracy of the applied motion field are carried out for a late diastolic heart phase and compared to the reconstruction obtained with the standard FDK-method from projections of the corresponding motion-free model in the same heart phase. Furthermore, gated reconstructions are calculated by weighting the projections according to their cardiac phase without using a motion vector field. Different gating window widths are applied, and the reconstructions are compared. Using the correct motion field with the motion-compensated reconstruction, the image quality of the standard reconstruction from the corresponding motion-free coronary model can almost be recovered. The reconstructed image quality stays acceptable if the accuracy of the motion field sampling points is better than 1 mm. The gated reconstructions with a window width of 15%-20% of the cardiac cycle lead to superior results compared to nearest neighbor gating, especially for histogram based visualization and analysis. The motion-compensated reconstructions provide sharp images of the coronaries far surpassing the image quality of gated reconstructions.     

Super-resolution reconstruction of images based on uncontrollable microscanning and genetic algorithm

Aiming at these disadvantages like lack of details, poor contrast and blurry edges of infrared images reconstructed by traditional controllable microscanning super-resolution reconstruction （SRR）, this paper proposes a novel algorithm, which samples multiple low-resolution images （LRIs） by uncontrollable microscanning, and then uses LRIs as chro- mosomes of genetic algorithm （GA）. After several generations of evolution, optimal LRIs are got to reconstruct the high-resolution image （HRI）. The experimental results show that the average gradient of the image reconstructed by the proposed algorithm is increased to 1.5 times of that of the traditional SRR algorithm, and the amounts of information, the contrast and the visual effect of the reconstructed image are improved.