Convolution Backprojection Research Articles

Hamming Signature of Three Composite Specimens

Three samples having composite structure are examined by an X-ray tomographic scanner for pixel sizes of 20 to 160 μm. The data are processed by the convolution backprojection algorithm and the resulting tomographic images are analyzed for the Hamming class of filters. The characteristic signature of the three specimens is interpreted with the help of two error theorems.

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 x 128 reconstruction arrays) and its robust performance (compared to the existing algorithms).

High-speed parallel implementation of a modified PBR algorithm on DSP-based EH topology

Algebraic Reconstruction Technique (ART) is an age-old method used for solving the problem of three-dimensional (3-D) reconstruction from projections in electron microscopy and radiology. In medical applications, direct 3-D reconstruction is at the forefront of investigation. The simultaneous iterative reconstruction technique (SIRT) is an ART-type algorithm with the potential of generating in a few iterations tomographic images of a quality comparable to that of convolution backprojection (CBP) methods. Pixel-based reconstruction (PBR) is similar to SIRT reconstruction, and it has been shown that PBR algorithms give better quality pictures compared to those produced by SIRT algorithms. In this work, we propose a few modifications to the PBR algorithms. The modified algorithms are shown to give better quality pictures compared to PBR algorithms. The PBR algorithm and the modified PBR algorithms are highly compute intensive, Not many attempts have been made to reconstruct objects in the true 3-D sense because of the high computational overhead. In this study, we have developed parallel two-dimensional (2-D) and 3-D reconstruction algorithms based on modified PBR. We attempt to solve the two problems encountered by the PBR and modified PBR algorithms, i.e., the long computational time and the large memory requirements, by parallelizing the algorithm on a multiprocessor system. We investigate the possible task and data partitioning schemes by exploiting the potential parallelism in the PBR algorithm subject to minimizing the memory requirement. We have implemented an extended hypercube (EH) architecture for the high-speed execution of the 3-D reconstruction algorithm using the commercially available fast floating point digital signal processor (DSP) chips as the processing elements (PEs) and dual-port random access memories (DPR) as channels between the PEs. We discuss and compare the performances of the PBR algorithm on an IBM 6000 RISC workstation, on a Silicon Graphics Indigo 2 workstation, and on an EH system. The results show that an EH(3,1) using DSP chips as PEs executes the modified PBR algorithm about 100 times faster than an LBM 6000 RISC workstation. We have executed the algorithms on a 4-node IBM SP2 parallel computer. The results show that execution time of the algorithm on an EH(3,1) is better than that of a 4-node IBM SP2 system. The speed-up of an EH(3,1) system with eight PEs and one network controller is approximately 7.85.

Analytical inversion formula for uniformly attenuated fan-beam projections

A convolution backprojection algorithm was derived by Tretiak and Metz (198) to reconstruct two-dimensional (2-D) transaxial slices from uniformly attenuated parallel-beam projections. Using transformation of coordinates, this algorithm can be modified to obtain a formulation useful to reconstruct uniformly attenuated fan-beam projections. Unlike that for parallel-beam projections, this formulation does not produce a filtered backprojection reconstruction algorithm but instead has a formulation that is an inverse integral operator with a spatially varying kernel. This algorithm thus requires more computation time than does the filtered backprojection reconstruction algorithm for the uniformly attenuated parallel-beam case. However, the fan-beam reconstructions demonstrate the same image quality as that of parallel-beam reconstructions.

Multilevel algebraic reconstruction technique for x-ray computed tomography

The performance of an iterative image reconstruction technique, the Algebraic Reconstruction Technique (ART), was improved to be basically a one-iteration noniterative technique producing even better image quality for x-rayCT. In this thesis, the factors affecting the performance of ART were first analyzed. Then a novel projection access order, the multilevel scheme (MLS) for ART was proposed. MLS is exactly the sequence of 1D FFT if the number of projections is a power of 2. Experimental tests using real CT data demonstrated that the new MLS-ART technique outperforms the conventional Convolution Backprojection (CBP) method, by producing higher spatial resolution when the number of projections is taken satisfying the sampling criterion and smaller noise when the number of projections is relatively small. A computer simulation study which matches the real CT dosage and noise conditions further quantifies that MLS produces a larger MTF when the number of projections is taken above half of that required by the sampling criterion and a larger SNR when the number of projections is taken below that half. It also improves the performance of ART itself, in both the computational speed (by more than 10 times) and the physical image quality (both the high and low contrast detectabilities). This work provides a thorough physical comparison among different CTreconstruction techniques and it confirms that MLS ART will find applications in reconstructions of different CT models, especially in situations where the projection data are limited.

Fast direct Fourier methods, based on one- and two-pass coordinate transformations, yield accurate reconstructions of x-ray CT clinical images

The conversion from polar to Cartesian coordinates can be carried out with two-pass algorithms. The paper describes two different methods based on concentric square frames and octagonal frames and their results, obtained with accurate interpolations based on the `moving window Shannon reconstruction' (MWSR). The embedding of these algorithms in direct Fourier methods (DFMs) of tomographic reconstruction is discussed. With respect to one-pass methods and to the use of octagonal frames, the square frame method makes it possible to carry out the first pass, a radial resampling, in the direct space, before computing 1D Fourier transforms (FTs) of projections. Reconstructions of clinical images from the raw data of a third-generation x-ray tomograph are presented and compared with those obtained with one-pass DFMs and with the convolution back-projection method (CBPM) performed by the instrument. The simple algorithm using square frames yields results in complete agreement with other DFM protocols and the CBPM. On a general-purpose computer, the execution of DFM protocols based on one-pass and two-pass coordinate transformations is 35 to 55 times faster than the CBPM and make the algorithms attractive for modern instrumentation.

Simulation Study of Noise Influence in 3-Dimensional Reconstruction using High-Angle Hollow-Cone Dark-Field Transmission Electron Microscope Images

Three-dimensional reconstruction of crystal structures by filtered convolution back projection of high-angle hollow-cone dark-field transmission electron microscope images (HADF images) of a specimen is investigated. To determine the distribution of Cu particles in Si crystal from reconstructed images, the necessary conditions are estimated. Undistorted distribution of particles is obtained from a simulated reconstruction when the specimen is inclined between ±50°, the inclination axis direction error is ±2° and Poisson noise is 30 db. Under these conditions, the distribution of Cu particles in a Si crystal is experimentally reconstructed.

A study of errors due to reconstruction filters in computed tomography

In the summation convolution backprojection method of image reconstruction in computed tomography, the final image accuracy depends on the convolution filter used. Filters are designed to attenuate high spatial frequencies when noisy projection data are used. This paper explores the differences between the images reconstructed using a range of filters, and compares the results with the case of the ramp filter that provides the “best” image for ideal, noise-free, projection data. It is shown that systematic errors between these images and the best image exist, and that these errors are related to the second differential of the reconstruction filter with respect to spatial frequency. This error determination may be used to correct computed tomography images that have been reconstructed using inappropriate filters, and this theory is tested using noise-free projection data from two computer simulated images. It is shown that the corrected images are far closer to the original images.

Reconstruction from incomplete data in cone‐beam tomography

A method for reconstruction of an object f(x) x=(x,y,z) from a limited set of cone-beam projection data has been developed. This method uses a modified form of convolution back-projection and projection onto convex sets (POCS) for handling the limited (or incomplete) data problem. In cone-beam tomography, one needs to have a complete geometry to completely reconstruct the original three-dimensional object. While complete geometries do exist, they are of little use in practical implementations. The most common trajectory used in practical scanners is circular, which is incomplete. It is, however, possible to recover some of the information of the original signal f(x) based on a priori knowledge of the nature of f(x). If this knowledge can be posed in a convex set framework, then POCS can be utilized. In this report, we utilize this a priori knowledge as convex set constraints to reconstruct f(x) using POCS. While we demonstrate the effectiveness of our algorithm for circular trajectories, it is essentially geometry independent and will be useful in any limited-view cone-beam reconstruction.

Truncation artifacts in tomographic reconstructions from projections

The artifacts in tomographic reconstructions from truncated sets of projections are analyzed. The shift-variant impulse response of the tomographic system for parallel-beam geometry is derived. A number of propositions are made describing the observed artifacts. A graphical scheme for the prediction of the location and shape of the truncation artifacts is presented and applied to reconstructions from simulated projections. The artifact analysis is applied to images obtained with the commonly used convolution backprojection reconstruction algorithms, and it is extended to reconstructions from fan-beam projections. The analysis is performed for the continuous imaging domain so as to separate the clipping artifacts clearly from those attributed to the digital implementation of the reconstruction algorithms.

An Analysis of Biological Hard Tissues Using the Tomographic Reconstruction Error Formula

A variety of teeth and bone specimens have been scanned using the constant statistics tomographic scanner developed at the London Hospital Medical College. Each data set has been reconstructed with several Hamming filters, and for a geometric resolution ranging from 20 to 50 micrometers. The reconstructions have been studied using the approximate error formula for the convolution backprojection algorithm. Some inferences are made on the relative structure of the cross sections.

Model-based reconstruction of multiple circular and elliptical objects from a limited number of projections

We consider tomographic image reconstruction from a limited number of noisy projections. An efficient algorithm based on maximum likelihood estimation (MLE) is developed to reconstruct images of multiple discs with unknown locations and radii. The algorithm is successfully applied to images with signal-to-noise ratio (SNR) as low as 0 dB, using as few as 16 projections, and containing as many as twelve discs with widely varying radii. Experimental results show that our approach significantly outperforms conventional convolution back projection. The algorithm is successfully extended to the multiple ellipse case.

Convolution backprojection formulas for 3-D vector tomography with application to MRI

Vector tomography is the reconstruction of vector fields from measurements of their projections. In previous work, it has been shown that the reconstruction of a general three-dimensional (3-D) vector field is possible from the so-called inner product measurements. It has also been shown how the reconstruction of either the irrotational or solenoidal component of a vector field can be accomplished with fewer measurements than that required for the full field. The present paper makes three contributions. First, in analogy to the two-dimensional (2-D) approach of Norton (1988), several 3-D projection theorems are developed. These lead directly to new vector field reconstruction formulas that are convolution backprojection formulas. It is shown how the local reconstruction property of these 3-D reconstruction formulas permits reconstruction of point flow or of regional flow from a limited data set. Second, simulations demonstrating 3-D reconstructions, both local and nonlocal, are presented. Using the formulas derived herein and those derived in previous work, these results demonstrate the reconstruction of the irrotational and solenoidal components, their potential functions, and the field itself from simulated inner product measurement data. Finally, it is shown how 3-D inner product measurements can be acquired using a magnetic resonance scanner.

A new method of characterizing patterns in X-ray tomographic images of a composite specimen

A 5 × 3 mm composite specimen consisting of 50 micron thick aluminum foils and 50 micron thick adhesive tape was scanned by the Fraunhofer-Institute X-ray tomographic scanner for a pixel size of 20 microns. This data was further processed by the convolution backprojection algorithm and several Hamming and exponential filters were employed. The results indicate that different regions of this specimen exhibit a distinct “fractal signature” for a certain type of filter. Characterization guidelines for composite materials have also been suggested using this combination of tomographic filters and fractals.

A Study of Errors due to Reconstruction Filters in Computed Tomography

Abstract In the summation convolution backprojection method of image reconstruction in computed tomography, the final image accuracy depends on the convolution filter used. Filters are designed to attenuate high spatial frequencies when noisy projection data are used. This paper explores the differences between the images reconstructed using a range of filters, and compares the results with the case of the ramp filter that provides the “best” image for ideal, noise-free, projection data. It is shown that systematic errors between these images and the best image exist, and that these errors are related to the second differential of the reconstruction filter with respect to spatial frequency. This error determination may be used to correct computed tomography images that have been reconstructed using inappropriate filters, and this theory is tested using noise-free projection data from two computer simulated images. It is shown that the corrected images are far closer to the original images.

An Analysis of Biological Hard Tissues Using the Tomographic Reconstruction Error Formula

A variety of teeth and bone specimens have been scanned using the constant statistics tomographic scanner developed at the London Hospital Medical College. Each data set has been reconstructed with several Hamming filters, and for a geometric resolution ranging from 20 to 50 micrometers. The reconstructions have been studied using the approximate error formula for the convolution backprojection algorithm. Some inferences are made on the relative structure of the cross sections.

Linear array implementation of the EM algorithm for PET image reconstruction

The PET image reconstruction based on the EM algorithm has several attractive advantages over the conventional convolution backprojection algorithms. However, the PET image reconstruction based on the EM algorithm is computationally burdensome for today's single processor systems. In addition, a large memory is required for the storage of the image, projection data, and the probability matrix. Since the computations are easily divided into tasks executable in parallel, multiprocessor configurations are the ideal choice for fast execution of the EM algorithms. In this study, we attempt to overcome these two problems by parallelizing the EM algorithm on a multiprocessor system. The parallel EM algorithm on a linear array topology using the commercially available fast floating point digital signal processor (DSP) chips as the processing elements (PE's) has been implemented. The performance of the EM algorithm on a 386/387 machine, IBM 6000 RISC workstation, and on the linear array system is discussed and compared. The results show that the computational speed performance of a linear array using 8 DSP chips as PE's executing the EM image reconstruction algorithm is about 15.5 times better than that of the IBM 6000 RISC workstation. The novelty of the scheme is its simplicity. The linear array topology is expandable with a larger number of PE's. The architecture is not dependent on the DSP chip chosen, and the substitution of the latest DSP chip is straightforward and could yield better speed performance.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Asymmetric fan transmission CT on SPECT systems

For proper attenuation correction of SPECT images, a set of 3D attenuation maps specific to the imaging slices is needed. Among the many different approaches for deriving the attenuation maps, fan beam transmission CT (FBTCT), performed on the same SPECT system as emission imaging, has many promising and clinically practical features. The major problem of FBTCT is that the current SPECT systems do not have a large enough field of view (FOV) to cover the typical cross-sectional size of patients. To address this problem, the authors have developed a novel asymmetric fan (AsF) sampling scheme to extend the FOV to practical sizes for clinical TCT imaging on existing SPECT systems. This AsF scheme samples only half of the intended FOV in each projection; the other half would be sampled in an opposing projection after detector rotation. The authors have implemented the AsF sampling on a three-head SPECT system through a specially designed source-collimator assembly. The authors have modified the conventional convolution backprojection algorithm to facilitate simple and fast image reconstruction. The feasibility of the approach is confirmed by the quality of the derived TCT images of various phantoms and human subjects. The AsF sampling scheme could also have applications in other general transmission CT systems.

A reconstruction algorithm using singular value decomposition of a discrete representation of the exponential radon transform using natural pixels

An algorithm to correct for constant attenuation in SPECT is derived from the singular value decomposition (SVD) of a discrete representation of the exponential Radon transform using natural pixels. The algorithm is based on the assumption that a continuous image can be obtained by backprojecting the discrete array q, which is the least squares solution to Mq=p, where p is the array of discrete measurements, and the matrix M represents the composite operator of the backprojection operator A/sub /spl mu//* followed by the projection operator A/sub /spl mu//. A singular value decomposition of M is used to solve the equation Mq=p, and the final image is obtained by sampling the backprojection of the solution q at a discrete array of points. Analytical expressions are given to calculate the matrix elements of M that are integrals of exponential factors over the overlapped area of two projection strip functions (natural pixels). A spectral analysis of the exponential Radon transform is compared with that of the Radon transform. The condition number of the spectrum increases with increased attenuation coefficient, which correlates with the increase in statistical error propagation seen in clinical images obtained with low-energy radionuclides. Computer simulations using 32 projections sampled over 360 degrees show an improvement in the SVD reconstruction over the convolution backprojection reconstruction, especially when the projection data is corrupted with noise.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

High-speed computation of the EM algorithm for PET image reconstruction

The PET image reconstruction based on the EM algorithm has several attractive advantages over the conventional convolution backprojection algorithms. However, two major drawbacks have impeded the routine use of the EM algorithm, namely, the long computational time due to slow convergence and the large memory required for the storage of the image, projection data and the probability matrix. Here, the authors attempt to solve these two problems by parallelizing the EM algorithm on a multiprocessor system. They have implemented an extended hypercube (EH) architecture for the high-speed computation of the EM algorithm using the commercially available fast floating point digital signal processor (DSP) chips as the processing elements (PEs). The authors discuss and compare the performance of the EM algorithm on a 386/387 machine, CD 4360 mainframe, and on the EH system. The results show that the computational speed performance of an EH(3,1) using DSP chips as PEs executing the EM image reconstruction algorithm is about 130 times better than that of the CD 4360 mainframe. The EH topology is expandable with more number of PEs.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>