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Showing papers on "Iterative reconstruction published in 1985"


Journal ArticleDOI
TL;DR: In this article, a mathematical model tailored to the physics of positron emissions is presented, and the model is used to describe the image reconstruction problem of PET as a standard problem in statistical estimation from incomplete data.
Abstract: Positron emission tomography (PET)—still in its research stages—is a technique that promises to open new medical frontiers by enabling physicians to study the metabolic activity of the body in a pictorial manner. Much as in X-ray transmission tomography and other modes of computerized tomography, the quality of the reconstructed image in PET is very sensitive to the mathematical algorithm to be used for reconstruction. In this article, we tailor a mathematical model to the physics of positron emissions, and we use the model to describe the basic image reconstruction problem of PET as a standard problem in statistical estimation from incomplete data. We describe various estimation procedures, such as the maximum likelihood (ML) method (using the EM algorithm), the method of moments, and the least squares method. A computer simulation of a PET experiment is then used to demonstrate the ML and the least squares reconstructions. The main purposes of this article are to report on what we believe is an...

804 citations


Book ChapterDOI
01 Jan 1985
TL;DR: Maximum entropy, using the Shannon/Jaynes form -Σ p log p, is an enormously powerful tool for reconstructing positive, additive images from a wide variety of types of data.
Abstract: Maximum entropy, using the Shannon/Jaynes form -Σ p log p, is an enormously powerful tool for reconstructing positive, additive images from a wide variety of types of data. The alternative form Σ log f, due to Burg, is shown to be inappropriate for image reconstruction in general, including radio astronomy, and also for the reconstruction of the profiles of power spectra.

566 citations


Journal ArticleDOI
TL;DR: This paper presents a computationally efficient gridding algorithm which can be used with direct Fourier transformation to achieve arbitrarily small artifact levels.
Abstract: The Fourier inversion method for reconstruction of images in computerized tomography has not been widely used owing to the perceived difficulty of interpolating from polar or other measurement grids to the Cartesian grid required for fast numerical Fourier inversion. Although the Fourier inversion method is recognized as being computationally faster than the back-projection method for parallel ray projection data, the artifacts resulting from inaccurate interpolation have generally limited application of the method. This paper presents a computationally efficient gridding algorithm which can be used with direct Fourier transformation to achieve arbitrarily small artifact levels. The method has potential for application to other measurement geometries such as fan-beam projections and diffraction tomography and NMR imaging.

536 citations


Journal ArticleDOI
TL;DR: Previously unknown sufficient conditions, a necessary condition, and reconstruction methods for image reconstruction from cone-beam projections are developed, and by requiring additional conditions on the configuration of source points a more efficient reconstruction method is developed.
Abstract: 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.

466 citations


Journal ArticleDOI
01 Apr 1985
TL;DR: An overview of the theory of sampling and reconstruction of multidimensional signals, including the role of the camera and display apertures, and the human visual system is presented and a class of nonlinear interpolation algorithms which adapt to the motion in the scene is presented.
Abstract: Sampling is a fundamental operation in all image communication systems A time-varying image, which is a function of three independent variables, must be sampled in at least two dimensions for transmission over a one-dimensional analog communication channel, and in three dimensions for digital processing and transmission At the receiver, the sampled image must be interpolated to reconstruct a continuous function of space and time In imagery destined for human viewing, the visual system forms an integral part of the reconstruction process This paper presents an overview of the theory of sampling and reconstruction of multidimensional signals The concept of sampling structures based on lattices is introduced The important problem of conversion between different sampling structures is also treated This theory is then applied to the sampling of time-varying imagery, including the role of the camera and display apertures, and the human visual system Finally, a class of nonlinear interpolation algorithms which adapt to the motion in the scene is presented

301 citations


Journal ArticleDOI
TL;DR: Inverse Monte Carlo (IMOC) as discussed by the authors is used to find an inverse solution to the photon transport equation (an integral equation for photon flux from a specified source) for a parameterized source and specific boundary conditions.
Abstract: Inverse Monte Carlo (IMOC) is presented as a unified reconstruction algorithm for Emission Computed Tomography (ECT) providing simultaneous compensation for scatter, attenuation, and the variation of collimator resolution with depth. The technique of inverse Monte Carlo is used to find an inverse solution to the photon transport equation (an integral equation for photon flux from a specified source) for a parameterized source and specific boundary conditions. The system of linear equations so formed is solved to yield the source activity distribution for a set of acquired projections. For the studies presented here, the equations are solved using the EM (Maximum Likelihood) algorithm although other solution algorithms, such as Least Squares, could be employed. While the present results specifically consider the reconstruction of camera-based Single Photon Emission Computed Tomographic (SPECT) images, the technique is equally valid for Positron Emission Tomography (PET) if a Monte Carlo model of such a system is used. As a preliminary evaluation, experimentally acquired SPECT phantom studies for imaging Tc-99m (140 keV) are presented which demonstrate the quantitative compensation for scatter and attenuation for a two dimensional (single slice) reconstruction. The algorithm may be expanded in a straight forward manner to full three dimensional reconstruction including compensation for out of plane scatter.

190 citations


Journal ArticleDOI
TL;DR: A new ray-driven projector-backprojector which can easily be adapted for hardware implementation is described and simulated in software and discretely models the attenuated Radon transform of a source distributed within an attenuating medium as line integrals of discrete pixels.
Abstract: A new ray-driven projector-backprojector which can easily be adapted for hardware implementation is described and simulated in software. The projector-backprojector discretely models the attenuated Radon transform of a source distributed within an attenuating medium as line integrals of discrete pixels, obtained using the standard sampling technique of averaging the emission source or attenuation distribution over small square regions. Attenuation factors are calculated for each pixel during the projection and backprojection operations instead of using precalculated values. The calculation of the factors requires a specification of the attenuation distribution, estimated either from an assumed constant distribution and an approximate body outline or from transmission measurements. The distribution of attenuation coefficients is stored in memory for efficient access during the projection and backprojection operations. The reconstruction of the source distribution is obtained by using a conjugate gradient or SIRT type iterative algorithm which requires one projection and one backprojection operation for each iteration.

179 citations


Journal ArticleDOI
TL;DR: An overview of techniques proposed to tackle the problem of limited-view computed tomography, employing diverse theories such as signal recovery, image restoration, constrained deconvolution, and constrained optimization, as well as novel schemes such as iterative object-dependent algorithms incorporating a priori knowledge and use of multispectral radiation are presented.
Abstract: In many applications of computed tomography, it may not be possible to acquire projection data at all angles, as required by the most commonly used algorithm of convolution backprojection. In such a limited-data situation, we face an ill-posed problem in attempting to reconstruct an image from an incomplete set of projections. Many techniques have been proposed to tackle this situation, employing diverse theories such as signal recovery, image restoration, constrained deconvolution, and constrained optimization, as well as novel schemes such as iterative object-dependent algorithms incorporating a priori knowledge and use of multispectral radiation. We present an overview of such techniques and offer a challenge to all readers to reconstruct images from a set of limited-view data provided here.

153 citations


Journal ArticleDOI
TL;DR: A reconstruction procedure based on linear system theory has been developed for 3-D light-microscopic images that allows a significant improvement in spatial resolution in the image planes perpendicular to the optical axis.
Abstract: A reconstruction procedure based on linear system theory has been developed for 3-D light-microscopic images. Inverse filtering with the 3-D optical transfer function was used for image reconstruction. The procedure allows a significant improvement in spatial resolution in the image planes perpendicular to the optical axis.

151 citations


Journal ArticleDOI
TL;DR: It is found that basis functions based on cubic B-splines offer significant improvements in the calculational accuracy that can be achieved with iterative tomographic reconstruction algorithms.
Abstract: In the local basis-function approach, a reconstruction is represented as a linear expansion of basis functions, which are arranged on a rectangular grid and possess a local region of support. The basis functions considered here are positive and may overlap. It is found that basis functions based on cubic B-splines offer significant improvements in the calculational accuracy that can be achieved with iterative tomographic reconstruction algorithms. By employing repetitive basis functions, the computational effort involved in these algorithms can be minimized through the use of tabulated values for the line or strip integrals over a single-basis function. The local nature of the basis functions reduces the difficulties associated with applying local constraints on reconstruction values, such as upper and lower limits. Since a reconstruction is specified everywhere by a set of coefficients, display of a coarsely represented image does not require an arbitrary choice of an interpolation function.

137 citations


Journal ArticleDOI
TL;DR: The conjugate gradient method incorporating the object-extent constraining provides the fastest convergence and the least error in image reconstruction of a three-dimensional object using an incomplete projection-data set.
Abstract: The conjugate gradient method incorporating the object-extent constraint is applied to image reconstruction of a three-dimensional object using an incomplete projection-data set. The missing information is recovered by constraining the solution with the knowledge of the outer boundary of the object-extent which may be a priori measured or known. The algorithm is derived from the least-squares criterion as an advanced version of conventional iterative reconstruction algorithms such as SIRT (Simultaneous Iterative Reconstruction Technique) and ILST (Iterative Least Squares Technique). In the case of reconstruction from noisy projection data, a method based on the minimum mean-square error criterion is also proposed. Computer simulated reconstruction images of a phantom using limited angle and number of views are presented. The result shows that the conjugate gradient method incorporating the object-extent constraining provides the fastest convergence and the least error.

Journal ArticleDOI
TL;DR: Along with a review of some of the mathematical foundations of computed tomography, the article contains new results on derivation of reconstruction formulas in a general setting encompassing all standard formulas.
Abstract: Along with a review of some of the mathematical foundations of computed tomography, the article contains new results on derivation of reconstruction formulas in a general setting encompassing all standard formulas; discussion and examples of the role of the point spread function with recipes for producing suitable ones; formulas for, and examples of, the reconstruction of certain functions of the attenuation coefficient, e.g., sharpened versions of it, some of them with the property that reconstruction at a point requires only the attenuation along rays meeting a small neighborhood of the point.

Journal ArticleDOI
TL;DR: This work demonstrates that conventional spin-echo Fourier transform image acquisitions naturally encode a component of flow velocity that lies within the image plane, and requires neither special pulse sequences nor image reconstruction and format software for its implementation.
Abstract: The ability of the nuclear magnetic resonance signal to encode information about macroscopic motion has been recognized since the works of Hahn and Carr and Purcell. In the medical imaging setting this ability has led to a variety of ingenious magnetic resonance flow imaging schemes that ultimately may become competitive with X-ray angiography in sensitivity and specificity while remaining radically noninvasive. This work demonstrates that conventional spin-echo Fourier transform image acquisitions naturally encode a component of flow velocity that lies within the image plane. By displacing just the real part of the complex image data (phase display), the velocity distribution within the subject is revealed in image form. This method of flow imaging requires neither special pulse sequences nor image reconstruction and format software for its implementation. Further, images that intersect a flow channel longitudinally, demonstrating in-plane flow, yield an unusually large quantity of physiologic information per image. Phantom and in vivo flow images are presented. Also described is a phantom based on a rotating disk that enables calibration of the velocity/phase-shift constant for an untested pulse sequence.

Journal ArticleDOI
TL;DR: In this paper, a method of estimating the propagation velocity and also the depth of an object from a pulse-echo image based on the minimum squared error is proposed, and the result of underground object image reconstruction from real pulseecho data is reported to demonstrate high estimation accuracy of the propagation velocities and fine resolution of the reconstructed image.
Abstract: The resolving capability of underground imaging radar employing a multifrequency holographic approach is characterized in terms of controlling parameters such as the synthetic aperture length, soil conductivity and dielectric constant, and antenna beamwidth. The propagation velocity of electromagnetic wave in soil, which varies from soil to soil, is an essential parameter for reconstructing object images using the holographic approach. Hence, a method of estimating the propagation velocity and also the depth of an object from a pulse-echo image based on the minimum squared error is proposed. The result of underground object image reconstruction from real pulse-echo data is reported to demonstrate high estimation accuracy of the propagation velocity and fine resolution of the reconstructed image.

Journal ArticleDOI
TL;DR: New theoretical results are developed which state conditions under which two-dimensional signals are uniquely specified to within a scale factor with this information and show that these conditions include a broad class of signals.
Abstract: In this paper, we present new results on the reconstruction of signals from only the sign of the real part of the Fourier transform. Specifically, we develop new theoretical results which state conditions under which two-dimensional signals are uniquely specified to within a scale factor with this information and show that these conditions include a broad class of signals. Furthermore, we apply this result to the problem of reconstructing two-dimensional signals from their zero crossings. We also present two algorithms for reconstructing a signal from sign information in either the time or frequency domain.

Journal ArticleDOI
TL;DR: A new technique that uses photogrammetry and computer graphics for the 3D display of cerebral blood vessels provides anatomic images for diagnosis and treatment planning.
Abstract: A new technique that uses photogrammetry and computer graphics for the 3D display of cerebral blood vessels provides anatomic images for diagnosis and treatment planning.

Journal ArticleDOI
TL;DR: A new isocentric two-film reconstruction algorithm for brachytherapy seed and needle implants that has no requirements that the two films be orthogonal, symmetric, or even be taken in a transverse plane is developed.
Abstract: We have developed a new isocentric two‐film reconstruction algorithm for brachytherapy seed and needle implants. The algorithm has no requirements that the two films be orthogonal, symmetric, or even be taken in a transverse plane. In addition, there is no requirement that the two films even have the same number of images. We have found removal of these usual constraints useful for head and neck implants where images are often obscured by patient anatomy. The inherent image matching ambiguities associated with traditional two‐film techniques are minimized by considering the image end points, rather than just the image centroids. For two films, the new algorithm, which considers all image combinations at o n e time, matches all the end‐point images on one film with those on the other, and then reconstructs the end‐point positions of the seeds. The algorithm minimizes the difference between the actual images and the projected images from the reconstructed seeds. The new two‐film image matching problem is shown to be equivalent to the well‐known assignment problem. For an implant of N seeds, this equivalence allows the two‐film problem to be solved by an algorithm (ACM algorithm 548) that scales with a polynomial power of N, rather than N! as is usually assumed. An implant of N seeds can be matched and reconstructed in approximately (N/20)2 s on a VAX 11/780.

Journal ArticleDOI
TL;DR: In this paper, an iterative solution that employs alternating projections is presented, which is similar to the Gerchberg-Papoulis algorithm, and a method of overrelaxing the projections to improve convergence is studied.
Abstract: It is desired to preprocess an input image so that, when it is distorted by an imaging system, a prescribed output image is produced. The system of interest is a linear, shift-invariant, band-limited system followed by a hard limiter. Such a system is found, for example, in microphotography, when a camera that is band limited by diffraction effects is used to print on very-high-contrast film. In this paper, an iterative solution that employs alternating projections is presented. Two variations of the procedure, which is similar to the Gerchberg–Papoulis algorithm, are applied to several examples having different space–bandwidth products. Also, a method of overrelaxing the projections to improve convergence is studied.

Journal ArticleDOI
William J. Dallas1
TL;DR: A method for constructing visualizations of current distributions with magnetic field measurements of biological current is presented and two uses are discussed: SQUID tomography and longitudinal region-of-interest reconstruction.
Abstract: A method for constructing visualizations of current distributions with magnetic field measurements of biological current is presented Two uses are discussed: SQUID tomography and longitudinal region-of-interest reconstruction

Journal ArticleDOI
TL;DR: A novel approach to 360° measurement and reconstruction of the surface topography of 3-D diffuse objects is presented, based on the principle of phase-measuring profilometry with a projected sinusoidal grating.
Abstract: A novel approach to 360° measurement and reconstruction of the surface topography of 3-D diffuse objects is presented. The method is fully automated and based on the principle of phase-measuring profilometry with a projected sinusoidal grating. A complete 3-D shape is reconstructed from a series of line-section profiles corresponding to discrete angular positions of the object. The system consists simply of a slide projector with a translatable grating, a linear detector array, and a microcomputer for control and processing. Experimental results for a general 3-D object and a performance analysis are presented.

Journal ArticleDOI
TL;DR: A projection space iterative reconstruction-reprojection (PSIRR) algorithm based on backprojection-re projection in the projection space is proposed and image quality of the PSIRR algorithm shows a substantial improvement compared to the original IRR algorithm.
Abstract: Recently, an iterative reconstruction-reprojection (IRR) algorithm has been suggested for application to limited view computed tomography (CT). In the IRR, the interpolation operation is performed in the object space during backprojection-reprojection. The errors associated with the interpolation degrade the reconstructed image and may cause divergence unless a large number of rays is used. In this paper, we propose a projection space iterative reconstruction-reprojection (PSIRR) algorithm based on backprojection-reprojection in the projection space. In the process of the backprojection-reprojection, iteration is performed with a single equation instead of multiple equations and interpolation is eliminated. Computer simulation results are presented, and image quality of the PSIRR algorithm shows a substantial improvement compared to the original IRR algorithm. In addition, the new algorithm was applied to ultrasonic attenuation CT using a sponge phantom.

Journal ArticleDOI
TL;DR: When tomography is performed with electromagnetic or acoustical radiation, refraction may cause sufficient bending of the probing rays that ordinary reconstruction algorithms, which are based on the assumption of straight rays, do not yield accurate results.
Abstract: When tomography is performed with electromagnetic or acoustical radiation, refraction may cause sufficient bending of the probing rays that ordinary reconstruction algorithms, which are based on the assumption of straight rays, do not yield accurate results. The resulting problem of reconstructing the refractive-index distribution of an object from time of flight or optical path length data is nonlinear. Various approaches to solving this problem approximately have been proposed and subjected to modest numerical studies. These include iterative algorithms and techniques based on linearized inverse scattering theory. One exception is the case of axisymmetric objects for which an exact solution is known.

Journal ArticleDOI
TL;DR: In this paper, a Monte Carlo approach was used to investigate the quality of the reconstruction of two-and three-dimensional objects from simulated coded-image data with respect to viewing geometry and multiplexing (mixing) of the data.
Abstract: Two algorithms have been developed for reconstructing objects from their coded images and a priori knowledge of the object class. Reconstructions from both algorithms are presented, but the results appear to be largely independent of the algorithm used. One of the algorithms, a Monte Carlo approach, is used to investigate the quality of the reconstruction of two- and three-dimensional objects from simulated coded-image data with respect to viewing geometry and multiplexing (mixing) of the data. The cases examined include reconstructions from data with and without signal-dependent photon noise. It is found that reconstructing from multiplexed data is not so serious a problem as reconstructing from data obtained with a limited viewing angle. Also, when photon noise is included in the data, reconstructions obtained from multiplexed data are better than those obtained from unmultiplexed data because of the higher photon count made available by multiplexing. It appears that the fidelity of a reconstruction depends much more strongly on the design of the data-taking system (the coded apertures) than on the reconstruction algorithm.

Journal ArticleDOI
TL;DR: This paper shows that systematic geometric distortion and other artifacts introduced in the reconstruction process can be reduced substantially by deconvolution performed via Wiener filtering using a priori knowledge derived from the given projections.
Abstract: In many applications of computed tomography, we cannot acquire the projection data at all angles evenly spaced over 360°. In such cases, the computed tomography images reconstructed using a limited number of projections, measured over a narrow angle range, are characterized by approximately elliptical distortion along the view angles used and poor contrast at angles not used (anisotropic resolution). This systematic geometric distortion is caused by the 2-D point spread function of the reconstruction process. In this paper, we show that such geometric distortion and other artifacts introduced in the reconstruction process can be reduced substantially by deconvolution performed via Wiener filtering using a priori knowledge derived from the given projections. The 2-D system transfer function used in the deconvolution is obtained from the reconstruction of a test image by the same reconstruction algorithm which has been used for reconstructing the unknown object.

Proceedings ArticleDOI
01 Apr 1985
TL;DR: This paper introduces a general formulation of constrained iterative restoration algorithms in which deterministic and/or statistical information about the undistorted signal and statistical Information about the noise are directly incorporated into the iterative procedure.
Abstract: This paper introduces a general formulation of constrained iterative restoration algorithms in which deterministic and/or statistical information about the undistorted signal and statistical information about the noise are directly incorporated into the iterative procedure. This a priori information is incorporated into the restoration algorithm by what we call "soft" or statistical constraints. Their effect on the solution depends on the amount of noise on the data; that is, the constraint operator is "turned off" for noiseless data. The development of the new iterative algorithm is based on results from regularization techniques for stabilizing ill-posed problems.

Journal ArticleDOI
TL;DR: A reconstruction algorithm is presented which possesses a simple scanning geometry and promises higher resolution than conventional transmission mode diffraction tomography algorithms, but the resolution will be shown to be limited by the bandwidth and physical size of the single plane wave transducer.

Journal ArticleDOI
TL;DR: Maximum Likelihood Estimator methods of reconstruction, which use the system matrices tailored to specific instruments and do not need matrix inversion, are shown to result in good preliminary images.
Abstract: Matrix methods of image reconstruction have not been used, in general, because of the large size of practical matrices, ill condition upon inversion and the success of Fourier-based techniques. An exception is the work that has been done at the Lawrence Berkeley Laboratory for imaging with accelerated radioactive ions. An extension of that work into more general imaging problems shows that, with a correct formulation of the problem, positron tomography with ring geometries results in well behaved matrices which can be used for image reconstruction with no distortion of the point response in the field of view and flexibility in the design of the instrument. Maximum Likelihood Estimator methods of reconstruction, which use the system matrices tailored to specific instruments and do not need matrix inversion, are shown to result in good preliminary images. A parallel processing computer structure based on multiple inexpensive microprocessors is proposed as a system to implement the matrix-MLE methods.

Journal ArticleDOI
TL;DR: In this article, a unique 4×4 germanium detector array has been fabricated to image single photon emitters, which is achieved electronically from a sequential interaction of the emitted radiation between the detector and a scintillation camera.
Abstract: A unique 4×4 germanium detector array has been fabricated to image single photon emitters. Collimation is achieved electronically from a sequential interaction of the emitted radiation between the germanium detector and a scintillation camera. An iterative reconstruction algorithm has been used to remove multiplexing in the acquired counts, thereby producing a conical projection image from multiple views. Computer simulation studies indicate that even though the degradation in signal to noise ratio during the demultiplexing step would reduce the sensitivity by about a factor of two, a large gain in sensitivity over conventional SPECT systems would result from electronic collimation.

Journal ArticleDOI
TL;DR: It is shown that the back projection theorem can be generalized to cone beam projections and this allows to state a new inversion formula suitable for both the 4 ¿ parallel and divergent geometries.
Abstract: The use of cone beam scanners raises the problem of three dimensional reconstruction from divergent projections. After a survey on bidimensional analytical reconstruction methods we examine their application to the 3D problem. Finally, it is shown that the back projection theorem can be generalized to cone beam projections. This allows to state a new inversion formula suitable for both the 4 ? parallel and divergent geometries. It leads to the generalization of the "rho-filtered back projection " algorithm which is outlined.

Journal ArticleDOI
TL;DR: A method of improving a planar image in limited angle tomography by removing blurred image information from selected out-of-focus planes is discussed and results of a phantom experiment illustrating the technique are presented.
Abstract: A method of improving a planar image in limited angle tomography by removing blurred image information from selected out-of-focus planes is discussed. Focused tomosynthesized images rather than individual projections are used. The necessary equations for removing the information from either two or four adjacent planes, produced with a symmetrical, but otherwise arbitrary blurring function, are developed and specialized to the geometry of circular tomography. Results of a phantom experiment illustrating the technique are presented.