A truncated best-path algorithm

A variant of the best-path (BP) algorithm that can be used for deducing a posteriori symbol probabilities for input sequences of unlimited length is proposed. Decoders using the proposed algorithm have fixed memory requirements and fixed decoding delays regardless of the length of the transmitted sequence. This is made possible by utilizing the Viterbi algorithm's ability to self-initialize itself and by segmenting the decoding process     

Linograms in image reconstruction from projections

The notion of a linogram is introduced. It corresponds to the notion of a sinogram in the conventional representation of projection data in image reconstruction. In the sinogram, points which correspond to rays which go through a fixed point in the cross section to be reconstructed all fall on a sinusoidal curve. In the linogram, however, these points fall on a straight line. Thus, backprojection corresponds to integration along straight lines in the linogram. A theorem is proved expressing the backprojection operator in terms of the Radon transform and simple changes of variables. Consequences of this theorem are discussed. A novel image reconstruction method based on the theorem is presented.     

Multicriterion maximum entropy image reconstruction from projections

A solution algorithm for the image reconstruction problem with three criteria, maximum entropy, minimum nonuniformity and peakedness, and least square error between the original projection data and projection due to reconstruction is presented. Theoretical results of precedence properties which are respected by all noninferior solutions are first derived. These precedence properties are then incorporated into a multiple-criteria optimization framework to improve the computational efficiency. Comparisons of the new algorithm to the MART and MENT algorithms are carried out using computer-generated noise-free and Gaussian noisy projections. Results of the computational experiment and the efficiency of the multiobjective entropy optimization algorithm (MEOA) are reported.     

Block-iterative methods for image reconstruction from projections

The simultaneous MART algorithm (SMART) and the expectation maximization method for likelihood maximization (EMML) are extended to block-iterative versions, BI-SMART and BI-EMML, that converge to a solution in the feasible case, for any choice of subsets. The BI-EMML reduces to the "ordered subset" EMML of Hudson et al. (1992, 1994) when their "subset balanced" property holds.     

Tomographic reconstruction from experimentally obtained cone-beam projections

Direct reconstruction in three dimensions for two-dimensional projection data has been achieved by cone-beam reconstruction techniques. In this paper explicit formulas for a cone-beam convolution and back-projection reconstruction algorithm are given in a form which can be easily coded for a computer. The algorithm is justified by analyzing tomographic reconstructions of a uniformly attenuating sphere from simulated noisy projection data. A particular feature of this algorithm is the use of a one-dimensional rather than two-dimensional convolution function, greatly speeding up the reconstruction. The technique is applicable however large the cone angle of data capture and correctly reduces to the pure fan-beam reconstruction technique in the central section of the cone. The method has been applied to data captured on a cone-beam CT scanner designed for bone mineral densitometry.     

Digital reconstruction of multidimensional signals from their projections

In a wide variety of applications it is necessary to infer the structure of a multidimensional object from a set of its projections. There has been a long-standing interest in this problem and a number of different techniques have been proposed. In this paper, we present a tutorial review of the reconstruction problem and some of the algorithms which have been proposed for its solution. In addition, we present a number of new algorithms that appear to have some advantages over previous algorithms. Some comparisons of these algorithms applied to reconstructions of two-dimensional pictures are given. Furthermore, a number of new theoretical results are presented relating to the minimum number of projections necessary for exact reconstruction.     

A note on exact image reconstruction from a limited number of projections

In a recent paper in this journal by Kesidis and Papamarkos “A new method for the exact reconstruction of any gray-scale image from its projections is proposed.” In this note we point out that this method is a special case of a well-known approach (peeling) and that it can produce exact reconstructions only under assumptions that are not realistic for practical methods of data collection. Further, we point out that some statements made in the paper regarding disadvantages of the algebraic reconstruction techniques (ART) as compared to the method of the paper are false.     

Exact image reconstruction from a limited number of projections

A new method for the exact reconstruction of any gray-scale image from its projections is proposed. The original image is projected into several view angles and the projection samples are stored in an accumulator array. In order to reconstruct the image, the accumulator array is considered as an accumulation of sinusoidal contributions each one corresponding to a certain pixel of the original image. The proposed method defines conditions for the necessary number of projections and the density of ray samples on the projection axis. These conditions insure that, for each pixel, there is at least one sample in the accumulator array where only this particular pixel contributes. This characteristic projection sample is used during the reconstruction phase to determine the coordinates and the gray-scale value of the corresponding image pixel. A variation of the method is also proposed where the reconstruction is performed using a limited number of projection samples in certain view angles. Specifically, the number of necessary samples equals at most the overall number of pixels in the original image. This approach leads to a significant reduction of memory and processing time requirements since it provides exact image reconstruction using one projection sample per pixel.     

Real-valued Hankel transform approach to image reconstruction from projections

A Hankel transform relation based on the real-valued Hartley transform and its application in image reconstuction from projections are proposed. In comparison with its counterpart based on the Fourier transform, the proposed method has evident advantages in computational and memory requirements because all operations are performed in the real-number space.<>     

Tomographic image reconstruction from a limited number of projections

This paper addresses the reconstruction of a function from its rectangularly sampled Radon transform. After reviewing the Radon transform sampling requirements and describing a convolution/backprojection reconstruction algorithm, two basic issues are addressed. First it is shown that even when the Radon transform of an image is adequately sampled (in the Nyquist sense), the standard convolution/backprojection algorithm may lead to a poor reconstruction unless the Radon transform is first reconstructed to a required set of projections. Secondly, when the Radon transform is undersampled in projections, reconstruction approaches are illustrated which span the range from those which achieve a high uniform-resolution reconstruction with large amounts of aliasing artifacts to those which achieve a low nonuniform-resolution reconstruction with virtually no aliasing artifacts. Reconstruction filters which seek a compromise between those two extremes are defined.     

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.     

A high-speed reconstruction from projections using direct Fourier method with optimized parameters-an experimental analysis

Some issues of the direct Fourier method (DFM) implementation are discussed. A hybrid spline-linear interpolation for the DFM is proposed. The results of comprehensive simulation research are presented. The following reconstruction problems and parameters are emphasized: interpolation, increasing the radial density of the polar raster, filtering, the 2-D inverse Fourier transformation dimension, and considering the cases of noiseless and noisy input data. For the a priori prescribed resolution of the reconstructed image, values of reconstruction parameters have been determined which are optimal with regard to reconstruction quality and computation cost. The computational requirements of the DFM algorithm which correspond to distinct interpolation schemes are compared to one another for CT and MR tomography, respectively. The estimations obtained are compared to computational characteristics of the convolution backprojection method.     

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.     

On the possibility of direct Fourier reconstruction from divergent-beam projections

A great number of tomographic systems, especially those equipped with fast data acquisition techniques, scan the objects investigated by divergent (fan) X-ray beams. Fan-beam projection data require special reconstruction techniques to be implemented. Among reconstruction techniques from parallel projection data, the direct Fourier method (DFM) proved to be one of the most promising ones, especially for high-speed image reconstruction in the high-end 3-D and dynamic tomographic systems. The goal of this work is to answer the topical question: how would direct use of the DFM influence the quality of image reconstruction from the fan-beam data? The formula describing the error caused by such an approximation is derived. The conclusions deduced from the formula are confirmed by computer simulations. The boundary values of data acquisition geometry parameters have been estimated for the case of using the DFM without recalculating the fan-beam data.     

Digital reconstruction of images from their projections in polar coordinates

In a wide variety of applications, it is necessary to infer the structure of a three-dimensional object from its projections. Usually, the problem is reduced to the reconstruction of a series of two-dimensional (planar) sections from a set of one-dimensional (linear) projections. In this paper, starting from a polar representation of the density function, we present some closed form formulas, which can be successfully used in the reconstruction of images. The results of some numerical tests are also presented.     

Region of interest reconstruction from truncated data in circular cone-beam CT

The circular scanning trajectory is one of the most widely adopted data-acquisition configurations in computed tomography (CT). The Feldkamp, Davis, Kress (FDK) algorithm and its various modifications have been developed for reconstructing approximately three-dimensional images from circular cone-beam data. When data contain transverse truncations, however, these algorithms may reconstruct images with significant truncation artifacts. It is of practical significance to develop algorithms that can reconstruct region-of-interest (ROI) images from truncated circular cone-beam data that are free of truncation artifacts and that have an accuracy comparable to that obtained from nontruncated cone-beam data. In this work, we have investigated and developed a backprojection-filtration (BPF)-based algorithm for ROI-image reconstruction from circular cone-beam data containing transverse truncations. Furthermore, we have developed a weighted BPF algorithm to exploit "redundant" information in data for improving image quality. In an effort to validate and evaluate the proposed BPF algorithms for circular cone-beam CT, we have performed numerical studies by using both computer-simulation data and experimental data acquired with a radiotherapy cone-beam CT system. Quantitative results in these studies demonstrate that the proposed BPF algorithms for circular cone-beam CT can reconstruct ROI images free of truncation artifacts.     

A general cone-beam reconstruction algorithm

Considering the characteristics of the X-ray microscope system being developed at SUNY at Buffalo and the limitations of available cone-beam reconstruction algorithms, a general cone-beam reconstruction algorithm and several special versions of it are proposed and validated by simulation. The cone-beam algorithm allows various scanning loci, handles reconstruction of rod-shaped specimens which are common in practice, and facilitates near real-time reconstruction by providing the same computational efficiency and parallelism as L.A. Feldkamp et al.'s (1984) algorithm. Although the present cone-beam algorithm is not exact, it consistently gives satisfactory reconstructed images. Furthermore, it has several nice properties if the scanning locus meets some conditions. First, reconstruction within a midplane is exact using a planar scanning locus. Second, the vertical integral of a reconstructed image is equal to that of the actual image. Third, reconstruction is exact if an actual image is independent of rotation axis coordinate z. Also, the general algorithm can uniformize and reduce z-axis artifacts, if a helix-like scanning locus is used.     

A phase reconstruction algorithm from bispectrum [seismicreflection data]

A computationally efficient procedure for the reconstruction of the impulse response of a (minimum- or nonminimum-phase) linear time-invariant system from its bispectrum is presented. This method is based on computing cepstrum of the impulse response sequence from the ω₁=ω₂ slice of the bispectrum. The algorithm can be implemented by using only the one-dimensional fast Fourier transform algorithm     

有限角CT少数投影重建图像技术

叙述了有限角CT少数投影重建图像技术研究所取得的进展：正交全息光路、干涉条纹处理系统、修改的联合代数重建技术、频谱分析图像重建技术和神经网络重建技术。     

On radar target shape estimation using algorithms for reconstruction from projections

In radar identification of perfectly conducting targets, a need arises to determine a convex body from its cross-sectional areas normal to the lines of sight for a number of view angles. An approach, based on the theory of the Radon transform is used here to show that this problem is equivalent to the problem of image reconstruction from projections, which has been solved in many diverse disciplines. Two of the inversion algorithms developed in these fields are used to estimate object shapes from their cross-sectional areas, obtainable from the objects' far-field, backscattered ramp responses.