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Purpose: To develop a tomosynthesis cone beam reconstruction algorithm for LINAC cone beam CT with the goal to achieve a dose reduction better than 1/5 in current cone beam imaging protocols for patient positioning and portal beam verification while maintaining the required image quality for patient positioning and beam delivery verification. Methods: Tomosynthesis is the process of reconstructing tomographic images with fewer acquired projection images and additional synthetically generated projections to preserve image resolution and contrast. We developed a tomosynthesis cone beam reconstruction algorithm based on combining Feldkamp' s cone beam reconstruction algorithm and an iterative maximum likelihood reconstruction algorithm. The Feldkamp method is used to reconstruct the first estimate of the tomographic data and the iterative maximum likelihood algorithm improves the resolution, contrast and considerably eliminates ghost images from the first estimate. Results: The algorithm tests show image reconstruction with the adequate image quality, contrast and resolution for patient positioning and beam delivery verification with a 1/8 dose delivered reduction using current cone beam imaging set‐up protocols. Imaging time is reduced with the same proportion allowing higher patient throughput. The algorithm has been tested using patient data for head and neck and brain treatment sites. On both treatment sites, portal beam and patient localization can be effectively performed within 3 mm accuracy using digitally reconstructed radiographs generated from the tomosynthesisreconstructed patient anatomical image. The reconstruction is executed on a desktop computer with 2GHz CPU and 4 Gb RAM in less than 5 minutes. Conclusions: We developed a tomosynthesis algorithm capable of reconstructingimages with adequate resolution to perform beam delivery verification and patient positioning. This dose reduction will allow the daily use of cone beam CT in patients that require daily set‐up verification which currently is impractical because of dose and time constraints.

Purpose: Tetrahedron Beam Computed Tomography (TBCT) circumvents some problems of cone beam CT(CBCT) but still shares the same approximate reconstruction artifact with FDK algorithm. Recently, iterative image reconstruction has been widely employed in diagnostic CT scanners in order to suppress noise and reduce imaging dose. In this study we developed an iterative TBCT image reconstruction algorithm which is able to mitigate the cone beam reconstruction artifact as well as reduce imagenoise. Methods: Simultaneous algebraic reconstruction technique (SART) image reconstruction algorithms were developed based on our TBCT benchtop geometry. The algorithms were tested using both numerical and physical phantoms. The projection matrix was calculated using a distance driven method. Noise levels in reconstructed images were quantified. The approximate image reconstruction artifact at larger cone angle was evaluated. Results: With the same numerical and real scanned projection data, SART algorithm produces similar spatial resolution as the FDK algorithm. The cone artifact was significantly reduced as compared with FDK algorithm. The noise levels were reduced 16–20% by the iterative algorithm at different regions of the images. Conclusions: Iterative reconstruction algorithm further improves the image quality of the TBCT system. TBCT system with SART image reconstruction algorithm potentially may be used as a mobile CT scanner for diagnostics. This work is partially supported by NIH grant 1R21CA130330‐01A1 and DOD Prostate Cancer Research Program W81XWH‐07‐0083.

For distributions, that are the Fourier transforms of cone-beam x-ray data, there are given formulas that allow to create numerical algorithms on base of the Kirillov-Tuy inversion formulas.

Since the appearance of high flux neutron beams, scientists experimented with neutron radiography. This high beam flux combined with modern neutron to visible light converters leads to the possibility of performing fast neutron micro-tomography. The first results of cold neutron tomography with a neutron lens are presented in this article. Samples are rotated in the beam and the projections are recorded with a neutron camera. The 3D reconstruction is performed with cone beam reconstruction software.

Purpose: We are developing a novel Tetrahedron Beam Computed Tomography (TBCT) benchtop system using a multiple pixel x‐ray tube with 75 carbon nanotubefield emissioncathodes and an in‐house curved CTdetector array. In this paper, we describe the development of multi‐slot collimators and image reconstruction algorithms.Material and Methods: The 75 x‐ray sources are arranged in a linear array with 4 mm cathode spacing. The 5‐row detector array is curved with a 145 cm radius. The linear scanning x‐ray source array and a curved detector array are aligned perpendicular to and within the rotation plane, respectively, with the source array positioned precisely on the central axis of the detector array. A group of multi‐slot collimators were designed to collimate the beam from each source exactly to the detector array. An FDK TBCT reconstruction algorithm was developed. Image reconstruction was performed using numerical and actual projection data. Results: The shapes of each slot opening are partial ellipses. The slot geometries were derived and drawn using CAD software. Actual collimators were made using a chemical milling technique. The FDK TBCT image reconstruction algorithm was adapted from the cylindrical, equiangular detector cone beam reconstruction algorithms. The TBCT reconstruction was evaluated using numerical Shepp‐Logan phantom as well as real phantom scans.Conclusions: Multi‐slot collimators were designed and built. The TBCT image reconstruction algorithm was able to accurately reconstruct 3D volume with both numerical and real projection data. This work is partially supported by NIH grant 1R21CA130330‐01A1 and DOD Prostate Cancer Research Program W81XWH‐07‐0083.

An x-ray microscope system for microtomography is under development at SUNY/Buffalo, New York. Considering the characteristics of the x-ray microscope system and the limitations of current cone-beam reconstruction algorithms, a general cone-beam image reconstruction algorithm has been developed at AMIL-ARTS. In order to study the reconstruction error characteristics of the general cone-beam algorithm, a preliminary error analysis on the algorithm is performed in this paper. The most important error source in cone-beam reconstruction is the theoretical precision limitation. Like many cone-beam reconstruction algorithms, the general cone-beam algorithm is not exact in nature. Thus, an analytic reconstruction error formula is derived which relates the error to the specimen structure and various imaging parameters. Approximately, the reconstruction error is proportional to either the distance from a voxel to the midplane or the pitch of a helix-like scanning locus, and inversely proportional to the size of the scanning locus. The reconstruction error also depends on the specimen structure. The faster the structure varies along the z direction, the larger the reconstruction error will be. Specimens are modeled as stochastic fields. Typical simulation results are then depicted and discussed.

An x-ray shadow projection microscope using a scannable point source of x-rays is under development at SUNY. The point source is generated by a focused electron beam that can be steered electromagnetically in a plane perpendicular to the optical axis of the microscope. The specimen is mounted on a rotatable mechanical stage for microtomography. An elaborate feedback system is being implemented to measure and correct the motion error of the mechanical stage. The conventional cone-beam image reconstruction algorithm suffers from two constraints. Firstly, the specimen must be contained in a sphere-like reconstruction region. Secondly, the x-ray source must be moved along a circle in the specimen coordinate system. Considering the characteristics of the x-ray microscope system of ARTS-AMIL and the limitations of the conventional cone-beam reconstruction algorithm, a general cone-beam image reconstruction algorithm is presented. The general algorithm can be applied to various scanning geometries, such as polygonal, helical, or random scanning patterns, for different reasons. In order to study the general cone-beam reconstruction algorithm, a computer simulation has been performed. With various scanning parameters, projection data were generated mathematically and then the general cone-beam reconstruction algorithm tested with the simulated data. The experimental results shows that the different kinds of scanning loci of the x-ray source consistently resulted in satisfactory 3-D reconstructed images.

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 three-dimensional reconstruction method which uses cone-beam data was implemented. Issues concerning implementation are investigated and discussed in detail. Computer simulations were used to determine the amount of image degradation occurring in each of the three steps that comprise the reconstruction method. In addition, computer simulations were used to investigate issues concerning sampling. Improvements in the implementation were made. Additionally, suggestions for future efforts to improve the implementation are made. >

The laboratory scale X-ray shadow projection imaging system currently under development at SUNY has the capability of making multiple cone-beam projections from physically separated X-ray sources. To take advantage of such a hardware design, we have developed a multiple cone-beam reconstruction algorithm. The reconstruction errors of the single cone-beam and the multiple cone-beam algorithm are simulated and compared using mathematical phantoms. It is shown that the multiple cone-beam algorithm significantly improves the reconstruction accuracy for off-mid-plane structures.

The authors evaluated cone beam single-photon emission-computed tomography (SPECT) of the heart. A new cone beam reconstruction algorithm was used to reconstruct data collected from "short scan" acquisitions (of slightly more than 180 degrees) of a detector anteriorally traversing a noncircular orbit. The less than 360 degrees acquisition was used to minimize the attenuation artifacts that result from reconstructing posterior projections of 201T1 emissions from the heart. The algorithm includes a new method for reconstructing truncated projections of background tissue activity that eliminates reconstruction ring artifacts. Phantom and patient results are presented which compare a high-resolution cone beam collimator (50-cm focal length; 6.0-mm full width at half maximum [FWHM] at 10 cm) to a low-energy general purpose (LEGP) parallel hole collimator (8.2-mm FWHM at 10 cm) which is 1.33 times more sensitive. The cone beam tomographic results are free of reconstruction artifacts and show improved spatial and contrast resolution over that obtained with the LEGP parallel hole collimator. The limited angular sampling restrictions and truncation problems associated with cone beam tomography do not deter from obtaining diagnostic information. However, even though these preliminary results are encouraging, a thorough clinical study is still needed to investigate the specificity and sensitivity of cone beam tomography.

The cone beam X-ray transform modelizes the measurements on new 3D medical imaging devices using 2D detectors, for instance X-ray transmission tomographs using image intensifiers or gamma-ray emission tomographs using convergent collimators. The most commonly used reconstruction algorithm performs cone beam back projection (FELDKAMP 1984). But it induces some distortions for objects far from the plane of the cone apex. We have established an exact formula between the cone beam X-ray transform and the first derivative of the 3D Radon transform (GRANGEAT 1987). It shows that the distortions are induced by the shadow zone in the Radon domain related to the planes which intersect the object but not the apex trajectory. In the Radon domain, it becomes possible to restore the missing information by interpolation. Then the reconstruction principle is to compute and to invert the first derivative of the Radon transform. In this communication, we compare these two algorithms on reconstructions performed on simulated acquisitions. We study the shape and level distortions along lines parallel to the rotation axis. We present an analysis of the axial and radial variations of the Modulation Transfer Function (MTF) and of their distortions. We conclude that the Radon transform algorithm provides a regularized solution to the distortions, with optimized computing time on modern scientific computers.© (1991) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.

A laboratory sized X-ray projection microscope and microtomographic imaging system is being developed at SUNYAB (AMIL-ARTS). High resolution two dimensional images can be recorded rapidly using a high resolution phosphor and a cooled slow scan CCD camera. The effective source size is kept small to maximize the resolution. For this purpose the electron beam is focused onto a thin film target. Three dimensional information can be derived from the high resolution two dimensional images by generating stereo pairs or by reconstructing the complete tomographic image. Real time stereo pairs can be produced by rapidly moving the position of the electron spot between two locations on the thin film target thus irradiating the specimen with X-rays from two spatially distinct directions. For high resolution tomography or microtomography the two dimensional images are recorded for a large number of different orientations of the specimen. A cone beam reconstruction scheme based on a convolution and back projection algorithm is being developed. To optimize the experimental parameters in microtomography a variety of mathematical phantoms have been examined using computer simulation technique. The experimental results reveal the quantitative relationship between the accuracy of the reconstructed image and the experimental parameters such as object shape orientation and position. 1.© (1991) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.

Either for medical imaging or for non destructive testing, X-ray provides a very accurate mean to investigate internal structures. The object is described by a 3D map f of the local density. The use of a 2D X-ray detector like an image intensifier in front of the ponctual X-ray source defines a cone beam geometry. When the source moves along a curve, the acquisition measurements are modelized by the cone beam X-ray transform of the function f. This same model can be applied to emission tomography when cone beam collimators are used.

Purpose: In order to cover entire organs, multislice CT requires helical acquisition. This need for table motion in multislice CT causes image volumes acquired for whole organ motion and perfusion studies to lack temporal uniformity. Dynamic volume CT mitigates this limitation by providing the ability to acquire an entire organ with isotropic resolution in a single gantry rotation with no table movement. The purpose of this study is to examine the technical aspects of the Aquilion ONE, the first dynamic volume CT scanner, and the effect on image quality versus a standard 64‐slice system. Method and Materials: An Aquilion ONE with a detector configuration of 320× 0.5mm channels is examined and compared to an Aquilion 64 with a 64×0.5mm detector configuration. Studies performed include low contrast resolution, high contrast resolution, uniformity, slice sensitivity profile, and CT number accuracy. These studies were performed using a CATPHAN 500. Results: The technical challenges associated with dynamic volume, such as cone beam reconstruction, and their physics implications, such as image clipping, will be presented. It will be shown that low contrast, high contrast, slice sensitivity profile, and CT number accuracy are equivalent to a standard 64‐slice system. Some differences in uniformity were found and will be discussed. Conclusion: Dynamic volume CT allows clinicians to visualizecontrast flow, motion dynamics, and entire volumes at a single instant in time while maintaining the image quality found on current systems. Dynamic volume CT promises to transform the way medicine approaches stroke patients, myocardial perfusion studies, and imaging of other moving body parts such as the lung and joints.

Three novel reconstruction methods have been discussed along with the necessary background theory. Using computer simulated data, a comparison was made between a standard cone-beam algorithm and an algorithm which is based upon the three novel reconstruction methods presented here. It was seen that the novel algorithm eliminates the z directional blurring and required the same number of computer operations.

The paper describes our 3D X-ray CT algorithm “RADON” using attenuation measurements acquired with a bidimensional detector. Our inversion diagram uses the first derivative of the Radon transform synthesis then its inversion. The potentiality of that new method, particularly for the large aperture, prompted us to develop an optimized software offering convenience and high performances on a modem scientific computer. After a brief recall of the basic principle of X-ray imaging processing, we will introduce the theoretical developments resulting in the present inversion diagram. A general algorithm structure will be proposed afterwards. As a conclusion we will present the performances and the results obtained with ceramic rotors examination.

Backprojection is an essential step of the cone-beam reconstruction algorithm for computed tomography. Conventional backprojection maps a reconstructed voxel to the projection space by interpolating across adjacent detector channels and rows of a single projection. Although this approach is computationally efficient, both theoretical analysis and phantom experiments have shown that a significant degradation in z resolution can result. To overcome this shortcoming, we propose a conjugate cone-beam backprojection approach in which two projections that are 180 deg apart are backprojected simultaneously. By carefully selecting the helical pitch and designing the interpolation function, the z resolution of the reconstructed images can be significantly improved. The proposed conjugate backprojection algorithm has the potential to reduce data extrapolation artifacts due to limited detector size in step-and-shoot data acquisition. Extensive computer simulations and phantom experiments are used to demonstrate the efficacy of our approach.

Purpose: Hounsfield Units calibration with adaptive compensation of beam hardening for a dose limited breast CT system. Method and Materials: Following a complete cone beam CT scan of the target object (a human breast), geometrical parameters of the object, including the mass center location and maximum radius from the mass center were calculated promptly. These parameters were used to compute X‐ray projection images of a water cylinder based upon photon attenuation in the cylinder and detector response. Projection images of the target object were corrected prior to reconstruction by logarithmic subtraction of the water cylinder projection images.CTimages relative to the Hounsfield Units (HU) scale were produced after cone beam reconstruction of the corrected projection images. Custom built water phantoms were tested with various inserts, including polyethylene, polystyrene, PMMA, nylon, polycarbonate and Teflon, with a density range from 0.92g/cm3 to 2.2g/cm3. Results: A wide range of breast diameters (10cm–18cm) and compositions (0%–100% glandular) were evaluated, and reconstructed and scaled HU values demonstrated excellent uniformity, linearity and consistency. Typical HU values were within 5% of theoretical values. The proposed method was applied to clinical breast scans. The “cupping” artifact caused by beam hardening in the original image was corrected as expected. Conclusion: Conventional methods of Hounsfield Units conversion are based on the scan of few fix‐sized water phantoms and lack the flexibility to compensate for the beam hardening from objects with various sizes. The introduced noise from water phantom scans is also not negligible, especially for a dose limited breast CT system with much lower mAs than whole body CT systems. The proposed method can compensate for beam hardening over a wide range of breast diameters and compositions without increasing noise contents in the image.

Purpose: There are few limitations in conventional CT for localizing the dummy sources due to the slice image which has a certain distance between images. The use of the cone‐beam CT in brachytherapy also has limitations because of the economical efficiency in brachytherapy fields. To overcome former problems, we performed the CBR using x‐ray simulator. Method: The home‐made anthropomorphic pelvic phantom with the localizer and the HDR applicator set was simulated on the digital x‐ray simulator. The images were obtained with 10 degree intervals, and the CBR performed in the limited projections (36, 18, 12, 9 projection). The coronal planes (anterior‐posterior views) of CBR were mainly used for evaluating a CBR results. Results: We could obtain the CBR images with dummy source and artificial anatomic structure in the phantom. The peak pixel intensity and size differences of the dummy sources, which were important factor to distinguish the dummy sources from the backgrounds in coronal planes, were compared with each CBR results of the limited projections. Conclusions: We present the method for accurate source localization in a substitution for cone‐beam CT. This CBR approach provides the exact position of the sources and the delineation of approximate organ structure in brachytherapy planning, thus it could be a useful and economical technique for improving the planning results.

As CT has evolved from axial to helical to multislice helical, the advanced applications have been focused on volumetric imaging. Improvements in coverage and temporal resolution have opened the doors to challenging imaging tasks such as coronary angiography. However, even with today's most advanced systems, there is still a several second time difference between the imaging of the superior and inferior portions of the heart. During this time period, small differences in heart rate and contrast medium can lead to discontinuities within the volume data as seen in some MPR images. Such artifacts can lead to inaccessibility or, worse, misdiagnosis. Toshiba has developed its 256 slice system to cover over 12 cm of anatomy in a single rotation with 0.5 mm slices to allow the complete coverage of the heart at a single, instantaneous time point. Such a system will enable seamless coverage free from banding artifacts and discontinuities. Furthermore, wide volume coverage will allow dynamic evaluation of the flow of contrast material and perfusion over an entire organ rather than in just 3–4 cm, opening the doors to new clinical applications. No new system is without technological challenges. The wide coverage of the 256 slice system requires a cone angle that is four times larger than a 64 slice system and advancements in cone beam reconstruction have been necessary to meet this need. This lecture will detail the design of Toshiba's 256 slice system and provide an overview of the system's image quality, dosimetry, and clinical applications. It will show early clinical examples of the advantages of whole organ coverage. It will explore some of the technological challenges inherent in such a system and the solutions to these challenges. Educational Objectives 1. Understand the development and design of Toshiba's 256 slice system. 2. Understand the clinical utility of seamless, wide volume coverage. 3. Understand the technical challenges of the 256. 4. Understand the system's image quality.

Over the past decade, x‐ray computed tomography has experienced tremendously technological advancements: the introduction of helical/spiral and multi‐slice/volumetric acquisition. These advancements not only allow improved image quality and enable new clinical applications, but also significantly increase the technical challenges associated with image reconstruction. The first part of this lecture will cover the fundamentals of image reconstruction. For the ease of understanding, we start with an explanation of the central slice theorem (Fourier slice theorem). Both theoretical and intuitive approaches are used to illustrate the concept. The reconstruction algorithm is then extended to fan beam geometry by mathematical derivation and graphic description. Using the central slice theorem as the foundation, reconstruction algorithms for helical acquisition are discussed in the second part of the lecture. We analyze, for single slice, the major difference between helical and step‐andshoot acquisitions. Implications of different reconstruction approaches on image quality and computational complexity are also discussed. Cone beam reconstruction discussion will start with one of the most popular algorithms: FDK algorithm. The derivation of the algorithm from the fan‐beam case is first described and its extension to helical/spiral acquisition is then presented. The lecture ends with a discussion on some of the most recent advances in cone beam reconstruction, including both approximate and exact methods. Jiang Hsieh is an employee of GE Healthcare Technologies. Educational Objectives: 1. Learn the fundamentals of x‐ray CTreconstruction. 2. Understand recent advancements in reconstruction algorithms.

Over the past decade, x-ray computed tomography has experienced tremendously technological advancements: the introduction of helical/spiral and multi-slice/volumetric acquisition. These advancements not only allow improved image quality and enable new clinical applications, but also significantly increase the technical challenges associated with image reconstruction. The first part of this lecture will cover the fundamentals of image reconstruction. For the ease of understanding, we start with an explanation of the central slice theorem (Fourier slice theorem) for a 2D parallel beam geometry. Both theoretical and intuitive approaches are used to illustrate the concept. The reconstruction algorithm is then extended to fan beam geometry by mathematical derivation and graphic description. Using the central slice theorem as the foundation, reconstruction algorithms for helical acquisition are discussed in the second part of the lecture. We analyze, for single slice, the major difference between helical and step-and-shoot acquisitions. Implications of different reconstruction approaches on image quality and computational complexity are also discussed. Cone beam reconstruction discussion will start with one of the most popular algorithms: FDK algorithm. The derivation of the algorithm from the fan-beam case is first described and its extension to helical/spiral acquisition is then presented. The lecture ends with a discussion on some of the most recent advances in cone beam reconstruction, including both approximate and exact methods. Jiang Hsieh is an employee of GE Healthcare Technologies. Educational Objectives: 1. Learn the fundamentals of x-ray CT reconstruction. 2. Understand recent advancements in reconstruction algorithms.

Purpose: System calibration of a micro CT scanner for anatomic imaging of small animals. Method and Materials: A phantom was constructed with lead ball bearings (200 μm in diameter) and scanned with a micro CT scanner designed and built in our laboratory. A general mathematical algorithm was developed to locate the X-ray focal spot and measure the source to detector distance (SID) and the azimuthal rotation of the detector with information acquired from the projection image of the phantom. A wire phantom (0.070mm Ni-Cd) was scanned to calibrate the angle measurement of the stepping motor which drives the rotating stage of the scanner, and also to measure the source to iso-center distance (SIC). An array of different objects was scanned and reconstructed with these calibrated parameters to evaluate the calibration result. Results: With the calibration procedure described, critical system parameters required for cone beam reconstruction including SIC, SID, x-ray focal spot location, detector rotation angle, and stage rotation angle were measured with high precision. Subsequent phantom testing demonstrated the accuracy of this calibration method by qualitative artifact assessment, and spatial resolution metrics (MTF) also demonstrated excellent performance. Conclusion: A simple but precise method for system calibration of a cone beam micro CT system using flat panel detector was developed. This method is likely useful for CT systems across different scales (from micro CT to human CT).

Purpose: With the rapid demographic shift towards urbanization worldwide, increasing numbers of children are exposed to ozone and particulate pollutants as a result of closer proximity to transportation corridors. To quantify the effect that ozone has on mammals, a rat‐lung cast model was developed and micro‐CT techniques were used to image the lung casts at high spatial resolution.Method and Materials: A normal rat lung was characterized. At sacrifice, the trachea was cannulated, and a RTV silicone compound was injected by syringe into the airways and allowed to harden. The rat tissues were decomposed using acid bath, so that a cast of the respiratory tree of each rat remained. A custom designed micro‐CT scanner was used to evaluate the lung casts. The system used a 70 micrometer focal spot tungsten anode, a rotating stage, and a 2048 × 1024 CMOS based indirect ( Gd 2 O 2 S ) detector.Results: CT scans were acquired using the full resolution of the detector, and using between 300 and 1000 projection images. A cone beam reconstruction developed in house was used to generate high resolution volume data sets (eg. 1024 × 1024 × 512) of the lung casts. Conclusion: The micro‐CT scanner was able to accurately characterize the pulmonary structure of the rat from the lungs casts, and this approach was necessary to characterize the complex three dimensional pulmonary structures. This application of small animal imaging technology has enabled computer modeling and quantitative analysis of the biological issues surrounding ozone exposure using the rat model.

An integrated approach is developed for three-dimensional (3D) cone beam reconstruction of cerebral vasculature (both morphology and gray scale) from a limited number (4–6) of digital subtraction angiographic (DSA) projections, obtained using a standard biplane C-arm x-ray system, using the Algebraic Reconstruction Technique (ART) with non-negativity constraint. The accuracy of ART is improved by determining weights based on the intersection volume between a pyramidal ray and cubic voxel. The input data are segmented and corrected for spatial image distortion. The reconstruction (volume 512 3 ) is accelerated by retaining only the voxels containing vasculature (masking), multiscaling, and partitioning the algorithm to run in parallel. Problems in using ART are evaluated. Limited view reconstruction is demonstrated to be unstable. In addition, errors arising from geometric uncertainty, noise, and partial voluming are assessed with simulations. Corrective measures in reconstructions are described and demonstrated with experimental data. Reconstruction of a phantom containing fiducial markers at known locations demonstrated that the reconstructed geometry is accurate to less than a pixel width (0.23 mm). The reconstruction algorithm with corrective measures is tested using a thin wire phantom, and anatomic skull phantom, a human carotid artery, and human cerebral vasculature with an aneurysm. The use of weight norm cutoff and averaging ART for different terminal conditions is found to be effective. The reconstruction results are also validated with reprojections of the 3D reconstructions.

A mathematical framework is presented for cone-beam reconstruction by intermediate functions that are related to the three-dimensional Radon transform of the object being imaged. Cone-beam projection data are processed with a filter to form an intermediate function. The filter is a linear combination ah1(l) + bh2(l) of the ramp kernel h1(l) and the derivative functional h2(l). From the intermediate function, the reconstruction is completed with a convolution and backprojection, where the convolution filter is another linear combination ch1(l) + dh2(l) subject to the restriction ac + bd = 1. This formulation unifies and generalizes the important cone-beam formulas of Tuy [ SIAM J. Appl. Math.43, 546 ( 1983)], Smith [ IEEE Trans. Med. ImagingMI-4, 14 ( 1985)], and Grangeat [ Ph.D. dissertation ( Ecole Nationale Superieure des Telecommunications, Paris, 1987)]. The appropriate values of a, b, c, d for these methods are derived.

International audience; Cone beam x-ray tomography is based on the aquisition of two-dimensional projections for different positions of a cone beam x-ray source. Generally the x-ray source describes a single circle around the volume. We consider three different scanning modes: a sphere, a circle, and two orthogonal circles. We study the response of a cone beam reconstruction system defined as the weighted backprojection of the cone beam projections of an arbitrary impulse point. The spatial invariance of the system is discussed for the three acquisition geometries. Exact and approximate reconstruction methods based on three-dimensional convolution are then proposed. Some results obtained from both simulated and experimental data are presented and analyzed.

We compare two algorithms of three-dimensional (3D) cone-beam tomography: a 3D backprojection algorithm and a Radon algorithm, with a circular source trajectory. We recall the principle of these algorithms and show that when the source trajectory is circular, an exact reconstruction cannot be performed. We evaluate structures that cannot be detected and show that, for each algorithm, assumptions must be made about the object to estimate the missing information. Using simulated data, we measured the modulation transfer function of the two algorithms, evaluated the density resolution, and measured the artifacts due to missing information. We show that the 3D backprojection algorithm has good geometrical resolution along the axis of rotation and that, from the standpoint of density resolution, its useful limit is ±3°. We also show that it is sensitive to the lack of information in the horizontal planes. The useful aperture of the Radon algorithm is ±12°; for apertures lower than this limit, the algorithm is much less sensitive to the lack of information in the horizontal planes than the 3D backprojection algorithm. We present two examples of reconstructions that illustrate the limitations of these algorithms when applied to realistic samples and discuss the limits of using each algorithm when the source trajectory is circular.

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.

An inversion procedure is developed for reconstructing a function of compact support in $\mathbb{R}^3 $ from its divergent beam x-ray transform with sources on a curve. The results apply to many curves not satisfying the conditions of Tuy [SIAM J. Appl. Math., 43 (1983), pp. 546–552), for example, planar circles of sufficiently large radius. In some cases, estimates in Sobolev norms are established for the inversion operator.

A true three-dimensional cone-beam reconstruction (TTCR) algorithm for direct volume image reconstruction from 2-D cone-beam projections is developed for the complete sphere geometry. The algorithm is derived from the parallel-beam true three-dimensional reconstruction (TTR) algorithm and is based on the modified filtered backprojection technique, which uses a set of 2-D space-invariant filters. The proposed algorithm proved to be superior in spatial resolution to the parallel-beam TTR algorithm and to offer better computational efficiency. >

An analytic inversion formula allowing the reconstruction of a three-dimensional object from x-ray cone-beams is given. The formula is valid for the case where the source of the beams describes a bounded curve satisfying a set of weak conditions.

A fan beam convolution backprojection algorithm has been developed for use with any spherically-shaped detector. This work represents the first step in the development of a cone beam reconstruction algorithm to be used with a prototype volume CT device that employs a conventional image intensifier as its detector. This algorithm has been tested for a high contrast vessel phantom reconstruction using real x-ray projection data acquired with our device.

Feldkamp has developed an approximate cone-beam tomography reconstruction algorithm. [1] This paper presents an analysis of some of the limitations of this algorithm and proposes a post-processing algorithm, which has been studied in simulations. Preliminary results suggest that by incorporating a priori constraints on the object reconstruction, the reconstruction volume within which satisfactory results are achieved can be expanded beyond that produced by the Feldkamp algorithm.

A prototype volume CT system for use in angiography has been modified and tested using three different phantoms. This system consists of a fixed x-ray tube, a conventional image intensifier coupled to a charge-coupled device (CCD) camera, a computer-controlled turntable on which phantoms were placed. In order to explore the imaging performance of the system for reconstructing a three dimension (3D) vascular structure, two sets of projection images of a vascular phantom, acquired over 250 projection angles with two different sizes of image intensifier were digitized and used for a direct 3D cone beam reconstruction. The spatial resolution limits of the system were measured from the 3D reconstructed images of a specially-designed resolution phantom. The direct 3D reconstructions of a Humanoid chest phantom were obtained using this system to show the perspective of the system for a general medical application with 3D imaging. The quality of the reconstructed images indicates that the system can be used for achieving a direct 3D reconstruction of a vascular structure as well as a general 3D object. The measured spatial resolution of the system is nearly half of the nominal resolution of the image intensifier and is reduced around the edge of the image intensifier, primarily because of the pincushion distortion.

International audience; 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 ..pi.. parallel and divergent geometries. It leads to the generalization of the ''rho-filtered back projection'' algorithm which is outlined.

In airport baggage scanning it is desirable to have a system that can scan baggage moving at standard conveyor belt speeds. One way to achieve this is to use multiple electronically switched sources rather than a single source on a mechanically rotated gantry. In such a system placing the detectors opposite the sources would obstruct the beam, so they have to be offset (hence offset multi-source geometry). This results in asymmetrical axial truncation of the cone beam projections. As such projections do not constitute complete data in the sense of integral geometry, the standard cone beam reconstruction algorithms do not apply. In this series of papers we introduce a new family of rebinning methods for reconstruction from axially asymmetrically truncated cone beam projections. In the first paper we discussed the approximation of the data on the multi-sheet surface with the truncated projection data obtained from offset multi-source geometries. In this second paper we focus on the recovery of the volumetric image from the reconstruction of data rebinned to multi-sheet surfaces. Multi-sheet rebinning effects an implicit relation between the fan beam transforms on the individual sheets and the rebinned data. This relation in conjunction with the linearity of the ray transform allows us to formulate the deconvolution problem for the recovery of the volume from a stack of reconstructed images on multi-sheet surfaces. We discuss the errors in the right-hand side of the deconvolution problem (reconstruction on multi-sheet surfaces) resulting from rebinning approximation. We introduce convolution matrix models based on the distribution of the distances of the rays from the multi-sheet surface, which considerably improve the data model fit and in turn lead to a superior solution. Multiple strategies for solution of the deconvolution problem are discussed and an efficient and robust implementation is presented, which makes the method capable of real time reconstruction. We conclude with some reconstruction results from simulated as well as real data collected with a Rapiscan RTT80 scanner.

This article is concerned with the use of iterative methods for image reconstruction from projections in 3D cone beam emission tomography. It is aimed at the analysis of some of the parameters which likely play an important role in the accuracy of the results. These parameters are related to three stages of the global acquisition/reconstruction procedure; the tomographic data acquisition geometry, the sampling frequency of the projection data and the reconstruction algorithm.

We extend Grangeat's formula (1991) so that partially measured cone-beam projections can be incorporated into image reconstruction in a successful way. The extension is based on a simple fact that multiple partially measured projections may be combined to compute the 3-D Radon data inaccessible from any single orbit point. Furthermore, we show a few examples for which Grangeat's formula does not provide a method for exact reconstruction but the extended formula does.

A convolution-backprojection formula is deduced for direct reconstruction of a three-dimensional density function from a set of two-dimensional projections. The formula is approximate but has useful properties, including errors that are relatively small in many practical instances and a form that leads to convenient computation. It reduces to the standard fan-beam formula in the plane that is perpendicular to the axis of rotation and contains the point source. The algorithm is applied to a mathematical phantom as an example of its performance.