CT filtration aliasing artifacts

Two methods for filtering computerized tomography (CT) projections for the filtered backprojection reconstruction algorithm are evaluated. The methods are based on derivations made in the spatial and Fourier domains. The Fourier method of filtration produces images with DC shifts and low-frequency shading. The spatial method does not generate similar artifacts. It is shown that the artifacts result because of aliasing artifacts that arise when a spatial waveform with infinite extent is sampled in the Fourier domain. It is also shown that it is possible to connect the artifacts generated with the Fourier method by replacing the DC and the first two frequency components with the corresponding terms from the discrete Fourier transforms of the filter used in the spatial method.     

On intermediate view estimation in computed tomography

In many applications in computed tomography, practical limitations in data acquisition restrict the number of projections (views). The use of the standard convolution backprojection algorithm for reconstruction from an inadequate number of projections results in view aliasing artifacts. One approach to alleviating the effects of such artifacts consists of artificially increasing the number of views, by estimating a set of intermediate views. Two possible methods of estimating the intermediate views are interpolation and reprojection. In this paper, a study of the two is considered. Based on the merits and demerits of the two methods, a combination of the two methods is investigated. Specifically, a reconstruction from the available sinogram augmented by intermediate view reprojections, and the projections interpolated from the original views and the reprojections, provide an additional improvement with respect to view aliasing artifacts. The advantage of computing reprojections over smaller regions of interest is discussed. When the number of available projections is reasonably high but not adequate to produce an artifact-free reconstruction, estimating the intermediate views by interpolation provides an improvement without much additional degradation, at minimal computational cost.This work was supported by Siemens Medical Systems and by the Medical Research Council of Canada (through grant no. MT-13356). The work of Dr. Holdsworth was supported in part by a Research Scholarship from the Heart and Stroke Foundation of Canada.     

Partial Radon transforms

This article formally defines partial Radon transforms for functions of more than two dimensions. It shows that a generalized projection-slice theorem exists which connects planar and hyperplanar projections of a function to its Fourier transform. In addition, a general theoretical framework is provided for carrying out n-dimensional backprojection reconstruction in a multistage fashion through the use of the partial Radon transform.     

A formulation in concordance with the sampling theorem for band-limited images reconstruction from projections

A rigorous method for band-limited images reconstruction from sampled projections is presented. The problem is formulated as the minimization of a quadratic criterion expressed in the frequency domain; consequences of spectral support limitation and sampling are taken into account. Then, the optimal reconstruction is obtained in two steps: a classical 1D-convolution/backprojection followed by the computation of the solution of a 2D convolution equation. The proposed solution for this second step is based on a conjugate gradient method. In both steps, the computations are performed in the frequency domain. This reduces the number of computations and allows the execution in a reasonable amount of time. A particular choice of the weights establishes a link between this approach and Radon's classical backprojection method: the classical method is optimal when the number of projections is infinite. In practice, the method proposed here improves the results obtained when computing an estimate of the solution by backprojections from a finite number of samples and projections when the projections are noisefree.     

The gridding method for image reconstruction by Fourier transformation

The authors explore a computational method for reconstructing an n-dimensional signal f from a sampled version of its Fourier transform f;. The method involves a window function w; and proceeds in three steps. First, the convolution g;=w;*f; is computed numerically on a Cartesian grid, using the available samples of f;. Then, g=wf is computed via the inverse discrete Fourier transform, and finally f is obtained as g/w. Due to the smoothing effect of the convolution, evaluating w;*f; is much less error prone than merely interpolating f;. The method was originally devised for image reconstruction in radio astronomy, but is actually applicable to a broad range of reconstructive imaging methods, including magnetic resonance imaging and computed tomography. In particular, it provides a fast and accurate alternative to the filtered backprojection. The basic method has several variants with other applications, such as the equidistant resampling of arbitrarily sampled signals or the fast computation of the Radon (Hough) transform.     

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.     

Respiratory compensation in projection imaging using a magnification and displacement model

Respiratory motion during the collection of computed tomography (CT) projections generates structured artifacts and a loss of resolution that can render the scans unusable. This motion is problematic in scans of those patients who cannot suspend respiration, such as the very young or intubated patients. Here, the authors present an algorithm that can be used to reduce motion artifacts in CT scans caused by respiration. An approximate model for the effect of respiration is that the object cross section under interrogation experiences time-varying magnification and displacement along two axes. Using this model an exact filtered backprojection algorithm is derived for the case of parallel projections. The result is extended to generate an approximate reconstruction formula for fan-beam projections. Computer simulations and scans of phantoms on a commercial CT scanner validate the new reconstruction algorithms for parallel and fan-beam projections. Significant reduction in respiratory artifacts is demonstrated clinically when the motion model is satisfied. The method can be applied to projection data used in CT, single photon emission computed tomography (SPECT), positron emission tomography (PET), and magnetic resonance imaging (MRI).     

O(N3 log N) backprojection algorithm for the 3-D radon transform

We present a novel backprojection algorithm for three-dimensional (3-D) radon transform data that requires O(N3 log2 N) operations for reconstruction of an N x N x N volume from O(N2) plane-integral projections. Our algorithm uses a hierarchical decomposition of the 3-D radon transform to recursively decompose the backprojection operation. Simulations are presented demonstrating reconstruction quality comparable to the standard filtered backprojection, which requires O(N5) computations under the same circumstances.     

A harmonic decomposition reconstruction algorithm for spatially varying focal length collimators

Spatially varying focal length fan-beam collimators can be used in single photon emission computed tomography to improve detection efficiency and to reduce reconstruction artifacts resulting from the truncation of projection data. It has been proven that there exists no convolution backprojection algorithm for this type of collimator, so a complicated interpolation between two nonparallel projection rays is necessary for existing algorithms. The interpolation may generate blurring and artifacts in the reconstructed images. Based on a harmonic decomposition technique and the translation property of Fourier series, a semifrequency resampling technique is proposed to avoid the above mentioned interpolations. By this technique, the harmonic decomposition of projection data for spatially varying focal length fan beam collimators has the same form as that for parallel-beam collimators in the semifrequency domain (Fourier transform with respect to angular variables only). An alternative version of the inverse Cormack transform is then proposed to reconstruct the images. The derived reconstruction algorithm was implemented in a Pentium II/266 PC computer. Numerical simulations demonstrated its efficiency (3 s for 128×128 reconstruction arrays) and its robust performance (compared to the existing algorithms)     

基于光束偏转层析技术的三种Radon变换迭代法

本文通过计算机模拟研究,利用气体温度场的先验知识,考查了用光束偏转层析技术重建三维温度场的基于卷积反投影的三种Radon变换迭代法。作为一种应用实例,测试了某截面上的气体温度分布,并与热电侧测量的值进行了比较。     

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%).     

A Reconstruction Algorithm for Photoacoustic Imaging Based on the Nonuniform FFT

Fourier reconstruction algorithms significantly outperform conventional backprojection algorithms in terms of computation time. In photoacoustic imaging, these methods require interpolation in the Fourier space domain, which creates artifacts in reconstructed images. We propose a novel reconstruction algorithm that applies the one-dimensional nonuniform fast Fourier transform to photoacoustic imaging. It is shown theoretically and numerically that our algorithm avoids artifacts while preserving the computational effectiveness of Fourier reconstruction.      

A fast and accurate Fourier algorithm for iterative parallel-beamtomography

We use a series-expansion approach and an operator framework to derive a new, fast, and accurate Fourier algorithm for iterative tomographic reconstruction. This algorithm is applicable for parallel-ray projections collected at a finite number of arbitrary view angles and radially sampled at a rate high enough that aliasing errors are small. The conjugate gradient (CG) algorithm is used to minimize a regularized, spectrally weighted least-squares criterion, and we prove that the main step in each iteration is equivalent to a 2-D discrete convolution, which can be cheaply and exactly implemented via the fast Fourier transform (FFT). The proposed algorithm requires O(N²logN) floating-point operations per iteration to reconstruct an N×N image from P view angles, as compared to O(N ²P) floating-point operations per iteration for iterative convolution-backprojection algorithms or general algebraic algorithms that are based on a matrix formulation of the tomography problem. Numerical examples using simulated data demonstrate the effectiveness of the algorithm for sparse- and limited-angle tomography under realistic sampling scenarios. Although the proposed algorithm cannot explicitly account for noise with nonstationary statistics, additional simulations demonstrate that for low to moderate levels of nonstationary noise, the quality of reconstruction is almost unaffected by assuming that the noise is stationary     

X-ray computed tomography for medical imaging

The article looks at reconstruction in 2-D and 3-D tomography. We have not dealt with some of the issues in reconstruction such as sampling and aliasing artifacts, finite detector aperture artifacts, beam hardening artifacts, etc., in greater detail since these are beyond the scope of an introductory tutorial. We examine the physical and mathematical concepts of the Radon (1917) transform, and the basic parallel beam reconstruction algorithms are discussed. We also develop the algorithms for fan-beam CT, and discuss the mathematical principles of cone-beam CT     

CT reconstruction by using the MLS-ART technique and the KCDimaging system. I. Low-energy X-ray studies

We investigated the use of the kinestatic charge detector (KCD) combined with the multilevel scheme algebraic reconstruction technique (MLS-ART) for X-ray computer tomography (CT) reconstruction. The KCD offers excellent detective quantum efficiency and contrast resolution. These characteristics are especially helpful for applications in which a limited number of projections are used. In addition, the MLS-ART algorithm offers better contrast resolution than does the conventional convolution backprojection (CBP) technique when the number of projections is limited. Here we present images of a Rando-head phantom that was reconstructed by using the KCD and MLS-ART. We also present, for comparison, the images reconstructed by using the CBP technique. The combination of MLS-ART and the KCD yielded satisfactory images after just one or two iterations.     

A Kalman filtering approach to stochastic global andregion-of-interest tomography

We define two forms of stochastic tomography. In global tomography, the goal is to reconstruct an object from noisy observations of all of its projections. In region-of-interest (ROI) tomography, the goal is to reconstruct a small portion of an object (an ROI) from noisy observations of its projections densely sampled in and near the ROI and sparsely sampled away from the ROI. We solve both problems by expanding the object and its projections in a circular harmonic (Fourier) series in the angular variable so that the Radon transform becomes Abel transforms of integer orders applied to the harmonics. The algorithm has three major components. First, we fit state-space models to each order of Abel transform and thus represent the Radon transform operation as a parallel bank of systems, each of which computes the appropriate Abel transform of a circular harmonic. A variable transformation here allows either the global or ROI problem to be solved. Second, the object harmonics are modeled as a Brownian branch. This is a two-point boundary value system, which is Markovianized into a form suitable for the Kalman filter. Finally, a parallel bank of Kalman smoothing filters independently estimates each circular harmonic from the noisy projection data. Numerical examples illustrate the proposed procedure.     

Backprojection by upsampled Fourier series expansion andinterpolated FFT

A fast backprojection method through the use of interpolated fast Fourier transform (FFT) is presented. The computerized tomography (CT) reconstruction by the convolution backprojection (CBP) method has produced precise images. However, the backprojection part of the conventional CBP method is not very efficient. The authors propose an alternative approach to interpolating and backprojecting the convolved projections onto the image frame. First, the upsampled Fourier series expansion of the convolved projection is calculated. Then, using a Gaussian function, it is projected by the aliasing-free interpolation of FFT bins onto a rectangular grid in the frequency domain. The total amount of computation in this procedure for a 512x512 image is 1/5 of the conventional backprojection method with linear interpolation. This technique also allows the arbitrary control of the frequency characteristics.     

Hilbert transform based FBP algorithm for fan-beam CT full and partial scans

This paper presents a new type of filtered backprojection (FBP) algorithm for fan-beam full- and partial-scans. The filtering is shift-invariant with respect to the angular variable. The backprojection does not include position-dependent weights through the Hilbert transform and the one-dimensional transformation between the fan- and parallel-beam coordinates. The strong symmetry of the filtered projections directly leads to an exact reconstruction for partial data. The use of the Hilbert transform avoids the approximation introduced by the nonuniform cutoff frequency required in the ramp filter-based FBP algorithm. Variance analysis indicates that the algorithm might lead to a better uniformity of resolution and noise in the reconstructed image. Numerical simulations are provided to evaluate the algorithm with noise-free and noisy projections. Our simulation results indicate that the algorithm does have better stability over the ramp-filter-based FBP and circular harmonic reconstruction algorithms. This may help improve the image quality for in place computed tomography scanners with single-row detectors.     

Pixel-based reconstruction (PBR) promising simultaneous techniques for CT reconstructions

Algorithms belonging to the class of pixel-based reconstruction (PBR) algorithms, which are similar to simultaneous iterative reconstruction techniques (SIRTs) for reconstruction of objects from their fan beam projections in X-ray transmission tomography, are discussed. The general logic of these algorithms is discussed. Simulation studies indicate that, contrary to previous results with parallel beam projections, the iterative algebraic algorithms do not diverge when a more logical technique of obtaining the pseudoprojections is used. These simulations were carried out under conditions in which the number of object pixels exceeded (double) the number of detector pixel readings, i.e., the equations were highly underdetermined. The effect of the number of projections on the reconstruction and the convergence (empirical) to the exact solution is shown. For comparison, the reconstructions obtained by convolution backprojection are also given.     

Comments on the filtered backprojection algorithm, rangeconditions, and the pseudoinverse solution

The filtered backprojection (FBP) algorithm is widely used in computed tomography for inverting the two-dimensional Radon transform. In this paper, we analyze the processing of an inconsistent data function by the FBP algorithm (in its continuous form). Specifically, we demonstrate that an image reconstructed using the FBP algorithm can be represented as the sum of a pseudoinverse solution and a residual image generated from an inconsistent component of the measured data. This reveals that, when the original data function is in the range of the Radon transform, the image reconstructed using the FBP algorithm corresponds to the pseudoinverse solution. When the data function is inconsistent, we demonstrate that the FBP algorithm makes use of a nonorthogonal projection of the data function to the range of the Radon transform.