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Rectification for cone-beam projection and backprojection
Abstract
The purpose of this paper is to derive a technique for accelerating the computation of cone-beam forward and backward projections that are the basic steps of tomographic reconstruction. The cone-beam geometry of C-arm systems is commonly described with projection matrices. Such matrices provide a continuous framework for analyzing the flow of operations needed to compute backprojection for analytical reconstruction, as well as the combination of forward and backward projections for iterative reconstruction. The proposed rectification technique resampies the original data to planes that are aligned with two of the reconstructed volume main axes, so that the original cone-beam geometry can be replaced by a simpler geometry, where succession of plane magnifications are involved only. Rectification generalizes previous independent results to the cone-beam backprojection of preprocessed data as well as to cone-beam iterative reconstruction. The memory access pattern of simple magnifications provides superior predictability and is, therefore, easier to optimize, independently of the choice of the interpolation technique. Rectification is also shown to provide control over interpolation errors through oversampling, allowing tradeoffs between computation speed and precision to be set. Experimental results are provided for linear and nearest neighbor interpolations, based on simulations, as well as phantom and patient data acquired on a digital C-arm system.
... A variety of projection methods for 3D cone-beam geometries have been proposed [9,14,26,38,93,120,167]. All methods provide some compromise between computational complexity and accuracy. ...
... Rectification techniques [120] were introduced to accelerate the computation of cone-beam forward and backward projections. Riddell et al. [120] resampled the original data to planes that are aligned with two of the reconstructed volume main axes, so that the original cone-beam geometry can be replaced by a simpler geometry that involves only a succession of plane magnifications. ...
... Rectification techniques [120] were introduced to accelerate the computation of cone-beam forward and backward projections. Riddell et al. [120] resampled the original data to planes that are aligned with two of the reconstructed volume main axes, so that the original cone-beam geometry can be replaced by a simpler geometry that involves only a succession of plane magnifications. ...
Image reconstruction and motion estimation are very important for image-guided radiotherapy
(IGRT). Three-dimensional reconstruction of patient anatomy using X-ray computed tomography
(CT) allows identification of the location of a tumor prior to treatment. The locations of tumorsmay change during actual treatment due to movement such as respiratory motion. Motion estimation helps optimize the accuracy and precision of radiotherapy so that more of the normal surrounding tissue can be spared. This dissertation addresses several important issues related to these two core components of IGRT.
Firstly, we developed two new separable footprint (SF) projector methods for X-ray conebeam
CT. The SF projectors approximate the voxel footprint functions as 2D separable functions.
The SF-TR projector uses trapezoid functions in the transaxial direction and rectangular functions
in the axial direction, whereas the SF-TT projector uses trapezoid functions in both directions. Both SF projector methods are more accurate than the distance-driven (DD) projector, which is a current state-of-the-art method in the field. The SF-TT projector is more accurate than the SF-TR projector for rays associated with large cone angles. In addition, the SF-TR projector has similar computation speed with the DD projector and the SF-TT projector is about two times slower.
Secondly, we proposed a statistical penalized weighted least-squares (PWLS) method with
edge-preserving regularization to reconstruct two basis materials from a single-energy CT scan
acquired with differential filtration, such as a split filter or a bow-tie filter. It requires only the use of suitable filters between the X-ray tube and the patient. For both filtration methods, the proposed PWLS method reconstructed soft tissue and bone images with lower RMS errors, reduced the beam-hardening artifacts much more effectively and produced lower noise, as compared with the traditional non-iterative Joseph and Spital method.
Thirdly, we conducted an objective characterization of the influence of rotational arc length on accuracy of motion estimation for projection-to-volume targeting during rotational therapy. Simulations illustrate the potential accuracy of limited-angle projection-to-volume alignment. Registration accuracy can be sensitive to angular center, tends to be lower along direction of the projection set, and tends to decrease away from the rotation center.
... The majority of the bibliography can be enclosed in one of the two main research trends: Cone Beam Projection and Filtered Back-projection. Related to Cone Beam Projection one interesting paper is 'Rectification for Cone Beam Projection and Back-projection' [1], where the authors try to find a technique that decreases computational time and increases the accuracy by providing a more efficient memory access. In July 2009 appeared the paper 'Directional View Interpolation for Compensation of Sparse Angular Sampling' [2], which gives a solution to increase the accuracy in the reconstruction by generating interpolated views. ...
... • Steps: It is a slider that lets the user adjust the value of a scalar in the range [1,20] that modifies the separation between the initial angles. ...
Actually, the use of CT scanners is not restricted to a medical environment and its use has been in-corporated into industrial facilities. One of the biggest drawbacks related to the use of CT scan-ners in these environments is the cost (in memory and time) associated. There are many approaches to simulate the functioning of CT scanners with less expensive devices such as cameras. Out of all this approaches, a Filtered Back-projection method is presented in this article. Based on the Radon transform it outperforms the classical Euclidean Geometry Approach by offering a fast and accu-rate solution to the task of reconstructing images via projection methods.
... It achieves its high performance by using a hierarchical memory layout and loop unrolling techniques. It further replaces the linear interpolation by a hybrid technique that first performs a detector upsampling based on linear interpolation (similar to [21]). The upsampled detector has very fine pixels and for the subsequent backprojection step it is sufficient to carry out a nearest neighbor interpolation without impairing image quality. ...
... Riddell and Trousset implemented a rectification-based perspective backprojection on a 3.4 GHz Pentium 4 CPU [21]. Their "rectification" is similar to our real-to-ideal rebinning and therefore their algorithm is a hybrid approach. ...
Tomographic image reconstruction, such as the reconstruction of CT projection values, of tomosynthesis data, PET or SPECT events, is computational very demanding. The most time-consuming step is the backprojection which is often limited by the memory bandwidth. Recently, a novel general purpose architecture optimized for distributed computing became available: the Cell Broadband Engine (CBE). Its eight synergistic processing elements (SPEs) currently allow for a theoretical performance of 192 GFlops (3 GHz, 8 units, 4 floats per vector, 2 instructions, multiply and add, per clock). To maximize image reconstruction speed we modified our parallel-beam backprojection algorithm that is highly optimized for standard PCs, and optimized the code for the cell processor. Data mining techniques and double buffering of source data were extensively used to optimally utilize both the memory bandwidth and the available local store of each SPE. The pixel-driven backprojection code uses floating point arithmetic and either linear interpolation (LI) or nearest neighbor (NN) interpolation between neighboring detector channels. Performance was measured using simulated data with 512 parallel beam projections per half rotation and 1024 detector elements. The data were backprojected into an image of 512 by 512 pixels using our PC-based approach and the new cell-based algorithm. Both the PC and the CBE were clocked at 3 GHz. Images obtained were found to be identical with both approaches. A throughput of 11 fps (LI) and 15 fps (NN) was measured on the PC whereas the CBE achieved 126 fps (LI) and 165 fps (NN). Thereby, the cell greatly outperforms today's top-notch backprojections based on graphical processing units (GPU). Using both CBEs of our dual cell-based blade (Mercury Computer Systems) one can backproject 252 images per second with LI and and 330 images per second with NN.
... Other algorithms, such as the algorithm of Katsevich [2], require a different determination of the filtering lines, which may impose additional strategies for a robust implementation, as, for example, a trajectory fitting; see Dennerlein et al. [3]. Note that other implementations of C-arm-based cone-beam reconstruction techniques do exist; see, for example, references [4][5][6], ...
... Let us denote the azimuth and polar angle with λ 1 and λ 2 . A source position may be expressed as follows (5) The parameter r denotes the source-iso-center distance, i.e., the radius of the scan. The typical source-iso-center distance is r = 78.5 cm. ...
Developing an efficient tool for accurate three-dimensional imaging from projections measured with C-arm systems.
A circle-plus-arc trajectory, which is complete and thus amenable to accurate reconstruction, is used. This trajectory is particularly attractive as its implementation does not require moving the patient. For reconstruction, we use the "M-line method", which allows processing the data in the efficient filtered backprojection mode. This method also offers the advantage of not requiring an ideal data acquisition geometry, i.e., the M-line algorithm can account for known deviations in the scanning geometry, which is important given that sizeable deviations are generally encountered in C-arm imaging.
A robust implementation scheme of the "M-line method" that applies straightforwardly to real C-arm data is presented. In particular, a numerically stable technique to compute the view-dependent derivative with respect to the source trajectory parameter is applied, and an efficient way to compute the π-line backprojection intervals via a polygonal weighting mask is presented. Projection data of an anthropomorphic thorax phantom were acquired on a medical C-arm scanner and used to demonstrate the benefit of using a complete data acquisition geometry with an accurate reconstruction algorithm versus using a state-of-the-art implementation of the conventional Feldkamp algorithm with a circular short scan of cone-beam data. A significant image quality improvement based on visual assessment is shown in terms of cone-beam artifacts.
... Rectification techniques [24] were introduced to accelerate the computation of cone-beam forward and backward projections. Riddell et al. [24] resampled the original data to planes that are aligned with two of the reconstructed volume main axes, so that the original cone-beam geometry can be replaced by a simpler geometry that involves only a succession of plane magnifications. ...
... Rectification techniques [24] were introduced to accelerate the computation of cone-beam forward and backward projections. Riddell et al. [24] resampled the original data to planes that are aligned with two of the reconstructed volume main axes, so that the original cone-beam geometry can be replaced by a simpler geometry that involves only a succession of plane magnifications. In iterative methods, resampled measurements can simplify forward and back-projection each iteration. ...
Iterative methods for 3D image reconstruction have the potential to improve image quality over conventional filtered back projection (FBP) in X-ray computed tomography (CT). However, the computation burden of 3D cone-beam forward and back-projectors is one of the greatest challenges facing practical adoption of iterative methods for X-ray CT. Moreover, projector accuracy is also important for iterative methods. This paper describes two new separable footprint (SF) projector methods that approximate the voxel footprint functions as 2D separable functions. Because of the separability of these footprint functions, calculating their integrals over a detector cell is greatly simplified and can be implemented efficiently. The SF-TR projector uses trapezoid functions in the transaxial direction and rectangular functions in the axial direction, whereas the SF-TT projector uses trapezoid functions in both directions. Simulations and experiments showed that both SF projector methods are more accurate than the distance-driven (DD) projector, which is a current state-of-the-art method in the field. The SF-TT projector is more accurate than the SF-TR projector for rays associated with large cone angles. The SF-TR projector has similar computation speed with the DD projector and the SF-TT projector is about two times slower.
... The image quality of the direct and the hybrid approach is nearly equivalent as shown in Ref. 17. To give additional evidence, Fig. 4 shows an example of a transversal section that was reconstructed with the direct approach and with the hybrid backprojection. ...
... Riddell and Trousset implemented a rectification-based perspective backprojection on a 3.4 GHz Pentium 4 CPU. 17 Their code uses the decomposition given in Appendix B and therefore is a hybrid approach. In contrast to our cell-based hybrid algorithm that first performs the alignment A followed by the backprojection B · C Riddell and Trousset perform the "rectification" A · B followed by the backprojection C. The authors state that backprojecting 148 projections into a cylinder of 512 voxels height and diameter takes 110 s. ...
Tomographic image reconstruction, such as the reconstruction of computed tomography projection values, of tomosynthesis data, positron emission tomography or SPECT events, and of magnetic resonance imaging data is computationally very demanding. One of the most time‐consuming steps is the backprojection. Recently, a novel general purpose architecture optimized for distributed computing became available: the cell broadband engine (CBE). To maximize image reconstruction speed we modified our parallel‐beam backprojection algorithm [two dimensional (2D)] and our perspective backprojection algorithm [three dimensional (3D), cone beam for flat–panel detectors] and optimized the code for the CBE. The algorithms are pixel or voxel driven, run with floating point accuracy and use linear (LI) or nearest neighbor (NN) interpolation between detector elements. For the parallel‐beam case, 512 projections per half rotation, 1024 detector channels, and an image of size was used. The cone‐beam backprojection performance was assessed by backprojecting a full circle scan of 512 projections of size into a volume of size voxels. The field of view was chosen to completely lie within the field of measurement and the pixel or voxel size was set to correspond to the detector element size projected to the center of rotation divided by . Both the PC and the CBE were clocked at . For the parallel backprojection of 512 projections into a image, a throughput of (LI) and (NN) was measured on the PC, whereas the CBE achieved (LI) and (NN), respectively. The cone‐beam backprojection of 512 projections into the volume took on the PC and is as fast as on the cell. Thereby, the cell greatly outperforms today's top‐notch backprojections based on graphical processing units. Using both CBEs of our dual cell‐based blade (Mercury Computer Systems) allows to 2D backproject 330 images/s and one can complete the 3D cone‐beam backprojection in .
... The most time consuming operation in the inner loop of reconstruction is the ratio computation due to the perspective projection model. Our Cell-based approach avoids image rectification as suggested [4] that leads to the elimination of the homogeneous division [5] but introduces an additional lowpass operation on the projection. ...
... In this regard, a rectification-based approach has theoretically the potential to further improve the reconstruction speed [5]. During our experiments we, however, observed that this is not true for the used graphics hardware. ...
The Common Unified Device Architecture (CUDA) is a fundamentally new programming approach making use of the unified shader design of the most current Graphics Processing Units (CPUs) from NVIDIA. The programming interface allows to implement an algorithm using standard C language and a few extensions without any knowledge about graphics programming using OpenGL, DirectX, and shading languages. We apply this revolutionary new technology to the FDK method, which solves the three-dimensional reconstruction task in cone-beam CT. The computational complexity of this algorithm prohibits its use for many medical applications without hardware acceleration. Today's CPUs with their high level of parallelism are cost-efficient processors for performing the FDK reconstruction according to medical requirements. In this paper, we present an innovative implementation of the most time-consuming parts of the FDK algorithm: filtering and back-projection. We also explain the required transformations to parallelize the algorithm for the CUDA architecture. Our implementation approach further allows to do an on-the-fly- reconstruction, which means that the reconstruction is completed right after the end of data acquisition. This enables us to present the reconstructed volume to the physician in real-time, immediately after the last projection image has been acquired by the scanning device. Finally, we compare our results to our highly optimized FDK implementation on the Cell Broadband Engine Architecture (CBEA), both with respect to reconstruction speed and implementation effort.
... Another resampling transform that may be used for reconstruction is "rectification." 25 Rectification relies on the observation that, for any paired homographies (H y 0 , H y 1 ) derived from projection matrix P, each one can be deduced from the other by a magnification. The decomposition of the projection into a composition of homographies can thus be simplified into computing a single "rectification" homography derived from, for example, H y 0 , to which 2D magnification H y 1 H −1 y 0 is applied to obtain H y 1 . ...
Purpose
Discretizing tomographic forward and backward operations is a crucial step in the design of model‐based reconstruction algorithms. Standard projectors rely on linear interpolation, whose adjoint introduces discretization errors during backprojection. More advanced techniques are obtained through geometric footprint models that may present a high computational cost and an inner logic that is not suitable for implementation on massively parallel computing architectures. In this work, we take a fresh look at the discretization of resampling transforms and focus on the issue of magnification‐induced local sampling variations by introducing a new magnification‐driven interpolation approach for tomography.
Methods
Starting from the existing literature on spline interpolation for magnification purposes, we provide a mathematical formulation for discretizing a one‐dimensional homography. We then extend our approach to two‐dimensional representations in order to account for the geometry of cone‐beam computed tomography with a flat panel detector. Our new method relies on the decomposition of signals onto a space generated by nonuniform B‐splines so as to capture the spatially varying magnification that locally affects sampling. We propose various degrees of approximations for a rapid implementation of the proposed approach. Our framework allows us to define a novel family of projector/backprojector pairs parameterized by the order of the employed B‐splines. The state‐of‐the‐art distance‐driven interpolation appears to fit into this family thus providing new insight and computational layout for this scheme. The question of data resampling at the detector level is handled and integrated with reconstruction in a single framework.
Results
Results on both synthetic data and real data using a quality assurance phantom, were performed to validate our approach. We show experimentally that our approximate implementations are associated with reduced complexity while achieving a near‐optimal performance. In contrast with linear interpolation, B‐splines guarantee full usage of all data samples, and thus the X‐ray dose, leading to more uniform noise properties. In addition, higher‐order B‐splines allow analytical and iterative reconstruction to reach higher resolution. These benefits appear more significant when downsampling frames acquired by X‐ray flat‐panel detectors with small pixels.
Conclusions
Magnification‐driven B‐spline interpolation is shown to provide high‐accuracy projection operators with good‐quality adjoints for iterative reconstruction. It equally applies to backprojection for analytical reconstruction and detector data downsampling.
... After generating the ternary CT, the forward projection of the ternary CT is used as ternary sinogram p tern (see Fig. 3(c) for an example). The forward projection with rectification with p-matrix in [11] is adopted here. Then the initial sinogram p is normalized using the ternary sinogram. ...
Metal Artifact Reduction (MAR) plays an important role in Computed Tomography (CT) research and application because severe artifacts degrade the image quality and diagnosis value if metal objects are present in the field of measurement. Although there are already many works for MAR, these works are for fan beam CT, not for cone beam CT, which is the trend and receiving much research attention. In this paper, we extend the Normalized Metal Artifact Reduction (NMAR) for fan beam CT to NMAR3 for cone beam CT, by replacing the linear interpolation in the NMAR with bi-linear interpolation. Experiments are carried out on 17 sets of spine phantom CT. 15 of them have reference CT as ground truth and 2 ones not. Both quantitative and qualitative results verified that NMAR3 outperforms the baseline method, i.e., bi-linear interpolation based method.
... Dans la suite, par souci de simplicité, et par analogie avec les formules de reconstruction en géométrie rectifiée (Riddell & Trousset, 2006), nous prenons 166 C.4. Une reconstruction directe pour le bow-tie virtuel K = 2 et nous coupons le spin entre les vues frontales, Θ FRT = π 4 , 3π 4 ∪ 5π 4 , 7π 4 , et les vues latérales, Θ LAT = Θ \ Θ FRT . ...
L'arceau interventionnel est un système d'imagerie rayons X temps réel. Il dispose d'une option tomographique qui, grâce à une rotation de l'arceau autour du patient, permet d'acquérir des images en coupes dont la résolution en contraste est plus faible que celle des tomodensitomètres diagnostiques, rendant l'information clinique des tissus mous du cerveau inexploitable. Nous proposons un nouveau mode d'acquisition et de reconstruction tomographiques sur arceau interventionnel pour l'amélioration de la détection des faibles contrastes en imagerie interventionnelle des tissus mous de la tête. Afin d'émuler un filtre "bow-tie" (en noeud papillon), une double acquisition est envisagée. Les spécificités de la double acquisition imposent la conception d'un algorithme de reconstruction itérative dédié, incluant le filtre rampe dans l'énergie de minimisation. En bifurquant des approches par rétro-projection filtrée vers celles par filtration des rétro-projections, une méthode de reconstruction directe, alternative à la précédente, est proposée pour les acquisitions doubles. Pour une acquisition simple, la méthode est assurée de faire aussi bien que l'algorithme de rétro-projection filtrée quel que soit l'échantillonnage angulaire en géométrie planaire, et offre une approximation alternative à l'algorithme de Feldkamp-Davis-Kress en géométrie conique. Nous montrons qu'avec peu ou pas de modifications aux schémas précédents, les deux méthodes de reconstruction (itérative et directe) s'adaptent bien à la reconstruction de régions d'intérêt, à laquelle l'acquisition double reste étroitement liée à travers son acquisition tronquée.
... Once the vasculature in space is generated, X-ray angiograms can be simulated by imitating the principle of X-ray passing through a space object with predefined absorption coefficients. Generally, the simulation of X-ray propagation can be conducted by two methods: Voxel-driven and pixel-driven methods [29,30]. Voxel-driven approaches usually calculate the projection of every voxel in a predefined image plane. ...
This study proposes a novel geometrical force constraint method for 3-D vasculature modeling and angiographic image simulation. For this method, space filling force, gravitational force, and topological preserving force are proposed and combined for the optimization of the topology of the vascular structure. The surface covering force and surface adhesion force are constructed to drive the growth of the vasculature on any surface. According to the combination effects of the topological and surface adhering forces, a realistic vasculature can be effectively simulated on any surface. The image projection of the generated 3-D vascular structures is simulated according to the perspective projection and energy attenuation principles of X-rays. Finally, the simulated projection vasculature is fused with a predefined angiographic mask image to generate a realistic angiogram. The proposed method is evaluated on a CT image and three generally utilized surfaces. The results fully demonstrate the effectiveness and robustness of the proposed method.
... Since the trajectory of the C-arm deviates from the ideal circular orbit, coefficients p ij are directly estimated. The computation of a CB reprojection or backprojection with these matrices is described in [109]. ...
Medical imaging has known great advances over the past decades to become a powerful tool for the clinical practice. It has led to the tremendous growth of interventional radiology, in which medical devices are inserted and manipulated under image guidance through the vascular system to the pathology location and then used to deliver the therapy. In these minimally-invasive procedures, X-ray guidance is carried out with C-arm systems through two-dimensional real-time projective low-dose images. More recently, three-dimensional visualization via tomographic acquisition has also become available. This work tackles tomographic reconstruction in the aforementioned context. More specifically, it deals with the correction of motion artifacts that originate from the temporal variations of the contrast-enhanced vessels and thus tackles a central aspect of tomography: data (angular) sampling. The compressed sensing theory identifies conditions under which subsampled data can be recovered through the minimization of a least-square data fidelity term combined with sparse constraints. Relying on this theory, an original reconstruction framework is proposed based on iterative filtered backprojection, proximal splitting, '1-minimization and homotopy. This framework is derived for integrating several spatial and temporal penalties. Such a strategy is shown to outperform the analytical filtered backprojection algorithm that is used in the current clinical practice by reducing motion and sampling artifacts in well-identified clinical cases, with focus on cerebral and abdominal imaging. The obtained results emphasize one of the key contributions of this work that is the importance of homotopy in addition to regularization, to provide much needed image quality improvement in the suggested domain of applicability.
... We intentionally avoided to use detector re-binning techniques that align the virtual detector to one of the volume axis since this technique influences the achieved image quality and requires additional computations for the re-binning step. 12 ...
In most of today's commercially available cone-beam CT scanners, the well known FDK method is used for solving the 3D reconstruction task. The computational complexity of this algorithm prohibits its use for many medical applications without hardware acceleration. The brand-new Cell Broadband Engine Architecture (CBEA) with its high level of parallelism is a cost-efficient processor for performing the FDK reconstruction according to the medical requirements. The programming scheme, however, is quite different to any standard personal computer hardware. In this paper, we present an innovative implementation of the most time-consuming parts of the FDK algorithm: filtering and back-projection. We also explain the required transformations to parallelize the algorithm for the CBEA. Our software framework allows to compute the filtering and back-projection in parallel, making it possible to do an on-the-fly-reconstruction. The achieved results demonstrate that a complete FDK reconstruction is computed with the CBEA in less than seven seconds for a standard clinical scenario. Given the fact that scan times are usually much higher, we conclude that reconstruction is finished right after the end of data acquisition. This enables us to present the reconstructed volume to the physician in real-time, immediately after the last projection image has been acquired by the scanning device.
... The key is to define an overlapping length between each image pixel and each detector element, which is the state-of-the art method. These techniques were extensively studied for fast implementation and quality optimization [7][8][9][10]. Given the finite detector elemental and source focal sizes, more precise models were studied to allow higher spatial resolution [11] or generate more realistic projections [12]. ...
For finite detector and focal spot sizes, here we propose a projection model for high spatial resolution. First, for a given x-ray source point, a projection datum is modeled as an area integral over a narrow fan-beam connecting the detector elemental borders and the x-ray source point. Then, the final projection value is expressed as the integral obtained in the first step over the whole focal spot support. An ordered-subset simultaneous algebraic reconstruction technique (OS-SART) is developed using the proposed projection model. In the numerical simulation, our method produces improved spatial resolution and suppresses high-frequency artifacts.
... In non-corrected CBCT, the key points of the optimization are the use of the preservation of the straightness with the x-ray projection transform to minimize the computational operations and the reordering in memory of image data to optimize the cache memory flow. 29,37 Warping the backprojection potentially eliminates these two optimizations. However, we proposed to split the warped CT image in piecewise linear segments based on two user parameters l and m which respectively represent the minimal length of the segment pieces and a threshold for merging them based on the actual motion of the respiratory cycle ͑Fig. ...
Respiratory motion causes artifacts in cone‐beam (CB) CT images acquired on slow rotating scanners integrated with linear accelerators. Respiration‐correlated CBCT has been proposed to correct for the respiratory motion but only a subset of the CB projections is used to reconstruct each frame of the 4D CBCT image and, therefore, adequate image quality requires long acquisition times. In this article, the authors develop an on‐the‐fly solution to estimate and compensate for the respiratory motion in the reconstruction of a 3D CBCT image from all the CB projections. An a priori motion model of the patient respiratory cycle is estimated from the 4D planning CT. During the acquisition, the model is correlated with the respiration using a respiratory signal extracted from the CB projections. The estimated motion is next compensated for in an optimized reconstruction algorithm. The motion compensated for is forced to be null on average over the acquisition time to ensure that the compensation results in a CBCT image which describes the mean position of each organ, even if the a priori motion model is inaccurate. Results were assessed on simulated, phantom, and patient data. In all experiments, blur was visually reduced by motion‐compensated CBCT. Simulations showed robustness to inaccuracies of the motion model observed on patient data such as amplitude variations, phase shifts, and setup errors, thus proving the efficiency of the compensation using an a priori motion model. Noise and view‐aliasing artifacts were lower on motion‐compensated CBCT images with 1 min scan than on respiration‐correlated CBCT images with 4 min scan. Finally, on‐the‐fly motion estimation and motion‐compensated reconstruction were within the acquisition time of the CB projections and the CBCT image available a few seconds after the end of the acquisition. In conclusion, the authors developed and implemented a method for correcting the respiratory motion during the treatment fractions which can replace respiration‐correlated CBCT for improving image quality while decreasing acquisition time.
... We used the shear-warp algorithm [34] which is a two-step decomposition of the projection transform in a 3-D shear followed by a 2-D warp (Fig. 2). In this case, we modified the method as proposed by [35] to apply the warp part to the measured CB projections instead of the intermediate image of the decomposition. This allows to reduce so-called edge and aliasing artifacts [36], [37] because warp resampling acts as a low pass filter on the measured CB projections. ...
Respiratory motion is a major concern in cone-beam (CB) computed tomography (CT) of the thorax. It causes artifacts such as blur, streaks, and bands, in particular when using slow-rotating scanners mounted on the gantry of linear accelerators. In this paper, we compare two approaches for motion-compensated CBCT reconstruction of the thorax. The first one is analytic; it is heuristically adapted from the method of Feldkamp, Davis, and Kress (FDK). The second one is algebraic: the system of linear equations is generated using a new algorithm for the projection of deformable volumes and solved using the Simultaneous Algebraic Reconstruction Technique (SART). For both methods, we propose to estimate the motion on patient data using a previously acquired 4-D CT image. The methods were tested on two digital and one mechanical motion-controlled phantoms and on a patient dataset. Our results indicate that the two methods correct most motion artifacts. However, the analytic method does not fully correct streaks and bands even if the motion is perfectly estimated due to the underlying approximation. In contrast, the algebraic method allows us full correction of respiratory-induced artifacts.
... Our approach is based on the use of projection matrices. [19][20][21][22] A cone beam projection matrix A has a 4ϫ 3 dimension and relates the mapping of a point in three-dimensions ͑3D͒ ͑x , y , z͒ to its projection ͑u , v͒ on a twodimensional detector in homogeneous coordinates ΄ uw vw ...
The authors describe a dual tube/detector micro-computed tomography (micro-CT) system that has the potential to improve temporal resolution and material contrast in small animal imaging studies. To realize this potential, it is necessary to precisely calibrate the geometry of a dual micro-CT system to allow the combination of projection data acquired with each individual tube/detector in a single reconstructed image. The authors present a geometric calibration technique that uses multiple projection images acquired with the two imaging chains while rotating a phantom containing a vertical array of regularly spaced metallic beads. The individual geometries of the imaging chains are estimated from the phantom projection images using analytical methods followed by a refinement procedure based on nonlinear optimization. The geometric parameters are used to create the cone beam projection matrices required by the reconstruction process for each imaging chain. Next, a transformation between the two projection matrices is found that allows the combination of projection data in a single reconstructed image. The authors describe this technique, test it with a series of computer simulations, and then apply it to data collected from their dual tube/detector micro-CT system. The results demonstrate that the proposed technique is accurate, robust, and produces images free of misalignment artifacts.
Modern interventional x-ray systems are often equipped with flat-panel detector-based cone-beam CT (FPD-CBCT) to provide tomographic, volumetric, and high spatial resolution imaging of interventional devices, iodinated vessels, and other objects. The purpose of this work was to bring an interchangeable strip photon-counting detector (PCD) to C-arm systems to supplement (instead of retiring) the existing FPD-CBCT with a high quality, spectral, and affordable PCD-CT imaging option. With minimal modification to the existing C-arm, a
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PCD with a 0.75 mm CdTe layer, two energy thresholds, and 0.1 mm pixels was integrated with a Siemens Artis Zee interventional imaging system. The PCD can be translated in and out of the field-of-view to allow the system to switch between FPD and PCD-CT imaging modes. A dedicated phantom and a new algorithm were developed to calibrate the projection geometry of the narrow-beam PCD-CT system and correct the gantry wobbling-induced geometric distortion artifacts. In addition, a detector response calibration procedure was performed for each PCD pixel using materials with known radiological pathlengths to address concentric artifacts in PCD-CT images. Both phantom and human cadaver experiments were performed at a high gantry rotation speed and clinically relevant radiation dose level to evaluate the spectral and non-spectral imaging performance of the prototype system. Results show that the PCD-CT system has excellent image quality with negligible artifacts after the proposed corrections. Compared with FPD-CBCT images acquired at the same dose level, PCD-CT images demonstrated a 53% reduction in noise variance and additional quantitative imaging capability.
Commonly used in medical imaging for diagnostic purposes, in luggage scanning, as well as in industrial non-destructive testing applications, Computed Tomography (CT) is an imaging technique that provides cross sections of an object from measurements taken from different angular positions around the object. CT, also referred to as Image Reconstruction (IR), is known to be a very compute-intensive problem. In its simplest form, the computational load is a function of O(M × N3), where M represents the number of measurements taken around the object and N is the dimension of the object. Furthermore, research institutes report that the increase in processing power required by CT is consistently above Moore's Law. On the other hand, the changing work flow in hospital requires obtaining CT images faster with better quality from lower dose. In some cases, real time is needed. High Performance Image Reconstruction (HPIR) has to be used to match the performance requirements involved by the use of modern CT reconstruction algorithms in hospitals. Traditionally, this problem had been solved by the design of specific hardware. Nowadays, the evolution of technology makes it possible to use Components of the Shelf (COTS). Typical HPIR platforms can be built around multicore processors such as the Cell Broadband Engine (CBE), General-Purpose Graphics Processing Units (GPGPU) or Field Programmable Gate Arrays (FPGA). These platforms exhibit different level in the parallelism required to implement CT reconstruction algorithms. They also have different properties in the way the computation can be carried out, potentially requiring drastic changes in the way an algorithm can be implemented. Furthermore, because of their COTS nature, it is not always easy to take the best advantages of a given platform and compromises have to be made. Finally, a fully fleshed reconstruction platform also includes the data acquisition interface as well as the vizualisation of the reconstructed slices. These parts are the area of excellence of FPGAs and GPGPUs. However, more often then not, the processing power available in those units exceeds the requirement of a given pipeline and the remaining real estate and processing power can be used for the core of the reconstruction pipeline. Indeed, several design options can be considered for a given algorithm with yet another set of compromises.
There is a growing interest in image reconstruction from a small number of projections for computed tomography. Most of the available algorithms require a large number of iterations to reconstruct a high-quality image and they include parameters that need careful tuning. In this paper, we present a new algorithm that aims at reducing these problems. We formulate the reconstruction as an unconstrained optimization problem that consists of a measurement consistency term and a total variation regularization. The algorithm that we propose is based on the class of proximal gradient methods. Since the basic proximal gradient method is slow, we propose three modifications to improve its convergence speed. First, instead of proximal gradient iterations, we use a variance-reduced stochastic proximal gradient descent updates. Second, we apply the proximal operator with a locally adaptive regularization parameter; specifically, we partition the image into small blocks and denoise each block with a regularization parameter that depends on the probability of the presence of important image features in that block. Thirdly, at each iteration of the algorithm, we minimize the objective function over the subspace spanned by the current proximal gradient update and several previous update directions. The step size in the stochastic proximal gradient descent can be set equal to one and we suggest an easy method to find a small range that contains the acceptable values for the regularization parameter. Our experiments show that the proposed algorithm can recover a high-quality image from undersampled projections in a small number of iterations.
Tomographic image reconstruction is computationally very demanding. In all cases the backprojection represents the performance bottleneck due to the high operational count and due to the high demand put on the memory subsystem. In the past, solving this problem has lead to the implementation of specific architectures, connecting Application Specific Integrated Circuits (ASICs) or Field Programmable Gate Arrays (FPGAs) to memory through dedicated high speed busses. More recently, there have also been attempt to use Graphic Processing Units (GPUs) to perform the backprojection step. Originally aimed at the gaming market, IBM, Toshiba and Sony have introduced the Cell Broadband Engine (CBE) processor, often considered as a multicomputer on a chip. Clocked at 3 GHz, the Cell allows for a theoretical performance of 192 GFlops and a peak data transfer rate over the internal bus of 200 GB/s. This performance indeed makes the Cell a very attractive architecture for implementing tomographic image reconstruction algorithms. In this study, we investigate the relative performance of a perspective backprojection algorithm when implemented on a standard PC and on the Cell processor. We compare these results to the performance achievable with FPGAs based boards and high end GPUs. The cone-beam backprojection performance was assessed by backprojecting a full circle scan of 512 projections of 1024×1024 pixels into a volume of size 512×512×512 voxels. It took 3.2 minutes on the PC (single CPU) and is as fast as 13.6 seconds on the Cell.
Algebraic reconstruction based on incomplete projection data is a hot issue in CT application. An improved algebraic reconstruction technique (ART) is proposed based on the analysis of relationships between images with mutual perpendicular projection angles. The projection coefficient matrix is calculated by recording the indices of ray-cross grids and the lengths of grid-ray intersections. In the process of reverse projection, POCS restriction is used to reconstruct the image from the incomplete data. The experimental results show that compared with the ART algorithm, the proposed algorithm greatly improves the speed and the quality of image reconstruction.
The Common Unified Device Architecture (CUDA) introduced in 2007 by NVIDIA is a recent programming model making use of the unified shader design of the most recent graphics processing units (GPUs). The programming interface allows algorithm implementation using standard C language along with a few extensions without any knowledge about graphics programming using OpenGL, DirectX, and shading languages. We apply this novel technology to the Simultaneous Algebraic Reconstruction Technique (SART), which is an advanced iterative image reconstruction method in cone-beam CT. So far, the computational complexity of this algorithm has prohibited its use in most medical applications. However, since today's GPUs provide a high level of parallelism and are highly cost-efficient processors, they are predestinated for performing the iterative reconstruction according to medical requirements. In this paper we present an efficient implementation of the most time-consuming parts of the iterative reconstruction algorithm: forward- and back-projection. We also explain the required strategy to parallelize the algorithm for the CUDA 1.1 and CUDA 2.0 architecture. Furthermore, our implementation introduces an acceleration technique for the reconstruction compared to a standard SART implementation on the GPU using CUDA. Thus, we present an implementation that can be used in a time-critical clinical environment. Finally, we compare our results to the current applications on multi-core workstations, with respect to both reconstruction speed and (dis-)advantages. Our implementation exhibits a speed-up of more than 64 compared to a state-of-the-art CPU using hardware-accelerated texture interpolation.
Tomographic image reconstruction, such as the reconstruction of CT projection values, of tomosynthesis data, PET or SPECT events, is computational very demanding. In filtered backprojection as well as in iterative reconstruction schemes, the most time–consuming steps are forward-and backprojection which are often limited by the memory bandwidth. Recently, a novel general purpose architecture optimized for distributed computing became available: the Cell Broadband Engine (CBE). Its eight synergistic processing elements (SPEs) currently allow for a theoretical performance of 192 GFlops (3 GHz, 8 units, 4 floats per vector, 2 instructions, multiply and add, per clock). To maximize image reconstruction speed we modified our parallel–beam and perspective backprojection algorithms which are highly optimized for standard PCs, and optimized the code for the CBE processor. 1–3 In addition, we implemented an optimized perspective forwardprojection on the CBE which allows us to perform statistical image reconstructions like the ordered subset convex (OSC) algorithm. 4 Performance was measured using simulated data with 512 projections per rotation and 512 2 detector elements. The data were backprojected into an image of 512 3 voxels using our PC–based approaches and the new CBE– based algorithms. Both the PC and the CBE timings were scaled to a 3 GHz clock frequency. On the CBE, we obtain total reconstruction times of 4.04 s for the parallel backprojection, 13.6 s for the perspective backprojection and 192 s for a complete OSC reconstruction, consisting of one initial Feldkamp reconstruction, followed by 4 OSC iterations.
Tomographic image reconstruction is computationally very demanding. In all cases the backprojection represents the performance bottleneck due to the high operational count and resulting high demand put on the memory subsystem. In this study, we present the implementation of a cone beam reconstruction algorithm on the Cell Broadband Engine (CBE) processor aimed at real-time applications. The cone-beam backprojection performance was assessed by backprojecting a half-circle scan of 512 projections of 10242 pixels into a volume of size 5123 voxels. The projections are acquired on a C-Arm scanner and directed in real time to a CBE-based platform for real-time reconstruction. The acquisition speed typically ranges between 17 and 35 projections per second. On a CBE processor clocked at 3.2 GHz, our implementation performs this task in ~13 seconds, allowing for real time reconstruction.
We present an evaluation of state-of-the-art computer hardware architectures for implementing the FDK method, which solves the 3-D image reconstruction task in cone-beam computed tomography (CT). The computational complexity of the FDK method prohibits its use for many clinical applications unless appropriate hardware acceleration is employed. Today’s most powerful hardware architectures for high-performance computing applications are based on standard multi-core processors, off-the-shelf graphics boards, the Cell Broadband Engine Architecture (CBEA), or customized accelerator platforms (e.g., FPGA-based computer components).For each hardware platform under consideration, we describe a thoroughly optimized implementation of the most time-consuming parts of the FDK algorithm; the filtering step as well as the subsequent back-projection step. We further explain the required code transformations to parallelize the algorithm for the respective target architecture. We compare both the implementation complexity and the resulting performance of all architectures under consideration using the same two medical datasets which have been acquired using a standard C-arm device.Our optimized back-projection implementations achieve at least a speedup of 6.5 (CBEA, two processors), 22.0 (GPU, single board), and 35.8 (FPGA, 9 chips) compared to a standard workstation equipped with a quad-core processor.
The authors are currently developing an alternative means by which
to deliver isolated power to serialised solid-state switches that float
at high voltage. This power transmission system utilizes a novel
application of piezoceramic devices to completely isolate the power
signal. The system consists of a FET driven pulser which excites a lead
zirconate titanate (PZT) small diameter, thin disc transducer. By means
of the piezoelectric effect, the pulsed voltage excitation to the disc
generates a corresponding mechanical impulse. These impulses, in the
form of an ultrasonic acoustic signal, travel along the length of a
solid borosilicate waveguide which is acoustic impedance-matched to the
piezoceramic transducer. At the opposite end of the waveguide, a
matching transducer converts the acoustic signal back into an electric
power signal which can be conditioned as needed. This configuration is
capable of continuously delivering power through an acoustic waveguide
with a length of up to one meter. This system provides power isolation
without the voltage grading effects of transformer isolation or the high
cost of other isolation technologies. The operation of this system and
the experimental results of its application are demonstrated in this
paper
Iterative maximum likelihood (ML) transmission computed tomography algorithms have distinct advantages over Fourier-based reconstruction, but unfortunately require increased computation time. The convex algorithm [1] is a relatively fast iterative ML algorithm but it is nevertheless too slow for many applications. Therefore, an acceleration of this algorithm by using ordered subsets of projections is proposed [ordered subsets convex algorithm (OSC)]. OSC applies the convex algorithm sequentially to subsets of projections. OSC was compared with the convex algorithm using simulated and physical thorax phantom data. Reconstructions were performed for OSC using eight and 16 subsets (eight and four projections/subset, respectively). Global errors, image noise, contrast recovery, and likelihood increase were calculated. Results show that OSC is faster than the convex algorithm, the amount of acceleration being approximately proportional to the number of subsets in OSC, and it causes only a slight increase of noise and global errors in the reconstructions. Images and image profiles of the reconstructions were in good agreement. In conclusion, OSC and the convex algorithm result in similar image quality but OSC is more than an order of magnitude faster. Index Terms—Maximum likelihood (ML) reconstruction, ordered subsets , transmission computed tomography (TCT).
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.
This monograph by one of the world's leading vision researchers provides a thorough, mathematically rigorous exposition of a broad and vital area in computer vision: the problems and techniques related to three-dimensional (stereo) vision and motion. The emphasis is on using geometry to solve problems in stereo and motion, with examples from navigation and object recognition.
Iterative maximum likelihood (ML) transmission computed tomography algorithms have distinct advantages over Fourier-based reconstruction, but unfortunately require increased computation time. The convex algorithm [1] is a relatively fast iterative ML algorithm but it is nevertheless too slow for many applications. Therefore, an acceleration of this algorithm by using ordered subsets of projections is proposed [ordered subsets convex algorithm (OSC)]. OSC applies the convex algorithm sequentially to subsets of projections. OSC was compared with the convex algorithm using simulated and physical thorax phantom data. Reconstructions were performed for OSC using eight and 16 subsets (eight and four projections/subset, respectively). Global errors, image noise, contrast recovery, and likelihood increase were calculated. Results show that OSC is faster than the convex algorithm, the amount of acceleration being approximately proportional to the number of subsets in OSC, and it causes only a slight increase of noise and global errors in the reconstructions. Images and image profiles of the reconstructions were in good agreement. In conclusion, OSC and the convex algorithm result in similar image quality but OSC is more than an order of magnitude faster.
The prime motivation of this work is to devise techniques that make the algebraic reconstruction technique (ART) and related methods more efficient for routine clinical use, while not compromising their accuracy. Since most of the computational effort of ART is spent for projection/backprojection operations, we first seek to optimize the projection algorithm. Existing projection algorithms are surveyed and it is found that these algorithms either lack accuracy or speed, or are not suitable for cone-beam reconstruction. We hence devise a new and more accurate extension to the splatting algorithm, a well-known voxel-driven projection method. We also describe a new three-dimensional (3-D) ray-driven projector that is considerably faster than the voxel-driven projector and, at the same time, more accurate and perfectly suited for the demands of cone beam. We then devise caching schemes for both ART and simultaneous ART (SART), which minimize the number of redundant computations for projection and backprojection and, at the same time, are very memory conscious. We find that with caching, the cost for an ART projection/backprojection operation can be reduced to the equivalent cost of 1.12 projections. We also find that SART, due to its image-based volume correction scheme, is considerably harder to accelerate with caching. Implementations of the algorithms yield run-time ratios TSART/TART between 1.5 and 1.15, depending on the amount of caching used.
The ordered subsets EM (OSEM) algorithm has enjoyed considerable interest for emission image reconstruction due to its acceleration of the original EM algorithm and ease of programming. The transmission EM reconstruction algorithm converges very slowly and is not used in practice. In this paper, we introduce a simultaneous update algorithm called separable paraboloidal surrogates (SPS) that converges much faster than the transmission EM algorithm. Furthermore, unlike the 'convex algorithm' for transmission tomography, the proposed algorithm is monotonic even with nonzero background counts. We demonstrate that the ordered subsets principle can also be applied to the new SPS algorithm for transmission tomography to accelerate 'convergence', albeit with similar sacrifice of global convergence properties as for OSEM. We implemented and evaluated this ordered subsets transmission (OSTR) algorithm. The results indicate that the OSTR algorithm speeds up the increase in the objective function by roughly the number of subsets in the early iterates when compared to the ordinary SPS algorithm. We compute mean square errors and segmentation errors for different methods and show that OSTR is superior to OSEM applied to the logarithm of the transmission data. However, penalized-likelihood reconstructions yield the best quality images among all other methods tested.
Computational burden is a major concern when an iterative algorithm is used to reconstruct a three-dimensional (3-D) image with attenuation, detector response, and scatter corrections. Most of the computation time is spent executing the projector and backprojector of an iterative algorithm. Usually, the projector and the backprojector are transposed operators of each other. The projector should model the imaging geometry and physics as accurately as possible. Some researchers have used backprojectors that are computationally less expensive than the projectors to reduce computation time. This paper points out that valid backprojectors should satisfy a condition that the projector/backprojector matrix must not contain negative eigen-values. This paper also investigates the effects when unmatched projector/backprojector pairs are used.
Increasingly, three-dimensional (3-D) imaging technologies are used in medical diagnosis, for therapy planning, and during interventional procedures. We describe the possibilities of fast 3-D-reconstruction of high-contrast objects with high spatial resolution from only a small series of two-dimensional (2-D) planar radiographs. The special problems arising from the intended use of an open, mechanically unstable C-arm system are discussed. For the description of the irregular sampling geometry, homogeneous coordinates are used thoroughly. The well-known Feldkamp algorithm is modified to incorporate corresponding projection matrices without any decomposition into intrinsic and extrinsic parameters. Some approximations to speed up the whole reconstruction procedure and the tradeoff between image quality and computation time are also considered. Using standard hardware the reconstruction of a 256(3) cube is now possible within a few minutes, a time that is acceptable during interventions. Examples for cranial vessel imaging from some clinical test installations will be shown as well as promising results for bone imaging with a laboratory C-arm system.
Projection and backprojection are operations that arise frequently in tomographic imaging. Recently, we proposed a new method for projection and backprojection, which we call distance-driven, and that offers low arithmetic cost and a highly sequential memory access pattern. Furthermore, distance-driven projection and backprojection avoid several artefact-inducing approximations characteristic of some other methods. We have previously demonstrated the application of this method to parallel and fan beam geometries. In this paper, we extend the distance-driven framework to three dimensions and demonstrate its application to cone beam reconstruction. We also present experimental results to demonstrate the computational performance, the artefact characteristics and the noise-resolution characteristics of the distance-driven method in three dimensions.
Tis paper focuses on the design of fast algorithms
for rotating images and preserving high quality. The basis for
the approach is a decomposition of a rotation into a sequence of
one-dimensional translations. As the accuracy of these operations
is critical, we introduce a general theoretical framework that
addresses their design and performance. We also investigate
the issue of optimality and present an improved least-square
formulation of the problem. This approach leads to a separable
three-pass implementation of a rotation using one-dimensional
convolutions only. We provide explicit filter formulas for several
continuous signal models including spline and band-limited representations. Finally, we present rotation experiments and compare
the currently standard techniques with the various versions of
our algorithm. Our results indicate that the present algorithm in
its higher-order versions outperforms all standard high-accuracy
metbods of which we are aware, both in terms of speed and
quality. lts computational complexity increases linearly with the
order of accuracy. The best-qaulity results are obtained with the
sinc-based algorithm, which cab be implemented using simple
one-dimensional FFT's.
The purpose of this paper is to derive optimal spline algorithms
for the enlargement or reduction of digital images by arbitrary
(noninteger) scaling factors. In our formulation, the original and
rescaled signals are each represented by an interpolating polynomial
spline of degree n with step size one and Δ, respectively. The
change of scale is achieved by determining the spline with step size
Δ that provides the closest approximation of the original signal
in the L<sub>2</sub>-norm. We show that this approximation can be
computed in three steps: (i) a digital prefilter that provides the
B-spline coefficients of the input signal, (ii) a resampling using an
expansion formula with a modified sampling kernel that depends
explicitly on Δ, and (iii) a digital postfilter that maps the
result back into the signal domain. We provide explicit formulas for
n=0, 1, and 3 and propose solutions for the efficient implementation of
these algorithms. We consider image processing examples and show that
the present method compares favorably with standard interpolation
techniques. Finally, we discuss some properties of this approach and its
connection with the classical technique of bandlimiting a signal, which
provides the asymptotic limit of our algorithm as the order of the
spline tends to infinity
The task of reconstructing an object from its projections via tomographic methods is a time-consuming process due to the vast complexity of the data. For this reason, manufacturers of equipment for medical computed tomography (CT) rely mostly on special application specified integrated circuits (ASICs) to obtain the fast reconstruction times required in clinical settings. Although modern CPUs have gained sufficient power in recent years to be competitive for two-dimensional (2D) reconstruction, this is not the case for three-dimensional (3D) reconstructions, especially not when iterative algorithms must be applied. The recent evolution of commodity PC computer graphics boards (GPUs) has the potential to change this picture in a very dramatic way. In this paper we will show how the new floating point GPUs can be exploited to perform both analytical and iterative reconstruction from X-ray and functional imaging data. For this purpose, we decompose three popular three-dimensional (3D) reconstruction algorithms (Feldkamp filtered backprojection, the simultaneous algebraic reconstruction technique, and expectation maximization) into a common set of base modules, which all can be executed on the GPU and their output linked internally. Visualization of the reconstructed object is easily achieved since the object already resides in the graphics hardware, allowing one to run a visualization module at any time to view the reconstruction results. Our implementation allows speedups of over an order of magnitude with respect to CPU implementations, at comparable image quality.
Cone-beam reconstruction (CBR) is growing in importance, but current computer systems are slower than desirable for clinical use. We have built a high-speed system for high-quality, 3D imaging. We partitioned the problem into input, filtering, backprojection, postprocessing, and output components. We mapped most of the components to standard RACE++ processing nodes. The backprojection component is very compute-intensive; we mapped it to a field-programmable gate array (FPGA)-based adjunct processor. We built a prototype FPGA card, optimized for flexibility, and implemented the backprojection in that FPGA. This strategy allows for redesigning the backprojection function when necessary, and keeps the other details of the CBR algorithm in easily programmable processors. We present a system that performs Feldkamp CBR of 300 projections into a 5123 cubical image in 38.7 seconds. The system is designed to be scalable, so that Feldkamp CBR of 21.4 seconds can be performed with two adjunct processors, and Feldkamp CBR of other regions of interest or dimensions could be performed in proportionately shorter times. Further optimization and faster-processing parts will also contribute to continual speed improvements. This system is flexible and can be extended to perform other imaging functions, such as real-time planar angiography, with the same hardware.
Several existing volume rendering algorithms operate by factor-ing the viewing transformation into a 3D shear parallel to the data slices, a projection to form an intermediate but distorted image, and a 2D warp to form an undistorted final image. We extend this class of algorithms in three ways. First, we describe a new object-order rendering algorithm based on the factorization that is significan tly faster than published algorithms with minimal loss of image quality. Shear-warp factorizations have the property that rows of voxels in the volume are aligned with rows of pixels in the intermediate image. We use this fact to construct a scanline-based algorithm that traverses the volume and the intermediate image in synchrony, taking advantage of the spatial coherence present in both. We use spatial data structures based on run-length encoding for both the volume and the intermediate image. Our implemen-tation running on an SGI Indigo workstation renders a 256, voxel medical data set in one second. Our second extension is a shear-warp factorization for perspective viewing transformations, and we show how our rendering algorithm can support this extension. Third, we introduce a data structure for encoding spatial coherence in unclassifie d volumes (i. e. scalar fields with no precomputed opacity). When combined with our shear-warp rendering algo-rithm this data structure allows us to classify and render a 256, voxel volume in three seconds. The method extends to support mixed volumes and geometry and is parallelizable. CR Categories: I. 3.7 [Computer Graphics]: Three-Dimensional Graphics and Realism; I. 3.3 [Computer Graphics]: Picture/Image Generation, Display Algorithms.
Hybrid methods have been known for a long time as very efficient algorithms for attenuation correction in single-photon emission computed tomography, but only recently have efforts been made to formulate them with more rigorous mathematics. This has allowed us to explain their efficiency in terms of approximate inversion, and to establish a convergence condition. The present study focuses on the convergence problem and emphasizes the question of symmetry. Hybrid method operators are not symmetrical; therefore the convergence condition is not easily verified. New schemes based on a modified conjugate gradient method are presented. Convergence is proved and performances are shown to be at least as good as the standard hybrid schemes on perfect and noisy simulated data.
Noise propagation in iterative reconstruction can be reduced by exact data projection. This can be done by area-weighted projection using the convolution method. Large arrays have to be convolved in order to achieve satisfactory image quality. Two procedures are described which improve the convolution method used so far. Variable binning helps to reduce the size of the convolution arrays without loss of image quality. Computation time is further reduced by abbreviated convolution. The effects of the procedures are illustrated by means of phantom measurements.
Reconstructing a three-dimensional (3D) object from a set of its two-dimensional (2D) X-ray projections requires that the source position and image plane orientation in 3D space be obtained with high accuracy. We present a method for estimating the geometrical parameters of an X-ray imaging chain, based on the minimization of the reprojection mean quadratic error measured on reference points of a calibration phantom. This error is explicitly calculated with respect to the geometrical parameters of the conic projection, and a conjugate gradient technique is used for its minimization. By comparison to the classical unconstrained method, better results were obtained in simulation with our method, specially when only a few reference points are available. This method may be adapted to different X-ray systems and may also be extended to the estimation of the geometrical parameters of the imaging chain trajectory in the case of dynamic acquisitions.
To evaluate three-dimensional (3D) digital subtraction angiography (DSA) as a supplement to two-dimensional (2D) DSA in the endovascular treatment (EVT) of intracranial aneurysms.
In 22 ruptured aneurysms, neck visualization, aneurysm shape, and EVT feasibility were analyzed at 2D DSA (anteroposterior, lateral, and rotational views) and at maximum intensity projection (MIP) and surface shaded display (SSD) 3D DSA. The possibility of obtaining a working view for EVT at 3D DSA and the relevance of measurements in choosing the first coil also were assessed.
Two-dimensional DSA images clearly depicted the aneurysm neck in four of 22 aneurysms; MIP images, in 10; and SSD images, in 21, but SSD led to overestimation of the neck size in one aneurysm. Aneurysm shape was precisely demonstrated in five of 22 aneurysms at 2D DSA, in eight at MIP, and in all cases at SSD. In two of 22 aneurysms, EVT seemed to be nonfeasible at 2D DSA; however, SSD demonstrated feasibility and EVT was successfully performed. In one aneurysm, only SSD demonstrated the extension of the neck to a parent vessel, which was proved at surgery. Working views for EVT were deduced from 3D DSA findings in 20 of 21 aneurysms. The choice of the first coil was correct in 19 of 21 aneurysms.
Three-dimensional DSA is valuable for evaluating the potential for EVT, finding a working view, and performing accurate measurements.
A new system has been designed and built to validate the concept of 3D computerized angiography (CA). This system can acquire a set of 2D digital subtracted angiography images while rotating around a patient and then, using these images, reconstruct a 3D representation of the opacified vasculature. The design principles and main characteristics of the system are described, with special attention paid to data processing aspects. An initial in vivo evaluation of this system performed on anaesthetized animals and human volunteers is presented. The influence on the quality of the 3D reconstruction of different factors such as volume resolution, estimation method, source trajectory and number of projections is discussed.
The authors define ordered subset processing for standard algorithms (such as expectation maximization, EM) for image restoration from projections. Ordered subsets methods group projection data into an ordered sequence of subsets (or blocks). An iteration of ordered subsets EM is defined as a single pass through all the subsets, in each subset using the current estimate to initialize application of EM with that data subset. This approach is similar in concept to block-Kaczmarz methods introduced by Eggermont et al. (1981) for iterative reconstruction. Simultaneous iterative reconstruction (SIRT) and multiplicative algebraic reconstruction (MART) techniques are well known special cases. Ordered subsets EM (OS-EM) provides a restoration imposing a natural positivity condition and with close links to the EM algorithm. OS-EM is applicable in both single photon (SPECT) and positron emission tomography (PET). In simulation studies in SPECT, the OS-EM algorithm provides an order-of-magnitude acceleration over EM, with restoration quality maintained.
Computed tomography (CT) has been extensively studied for years and widely used in the modern society. Although the filtered back-projection algorithm is the method of choice by manufacturers, efforts are being made to revisit iterative methods due to their unique advantages, such as superior performance with incomplete noisy data. In 1984, the simultaneous algebraic reconstruction technique (SART) was developed as a major refinement of the algebraic reconstruction technique (ART). However, the convergence of the SART has never been established since then. In this paper, the convergence is proved under the condition that coefficients of the linear imaging system are nonnegative. It is shown that from any initial guess the sequence generated by the SART converges to a weighted least square solution.
Iterative reconstruction algorithms require the repetitive
application of reprojection and backprojection operations. Reprojection
of an image at angle θ can be decomposed into rotating the image
by angle θ and summing it at angle 0. Here, the authors show that
this “rotate and sum” (RS) technique can be reduced to
applying one-dimensional translations of the image rows, followed by an
interpolation-free summation of the image columns. This sequence
actually yields the proper reprojection magnified by a factor that
depends on the angle. By pre-processing the data to match this angular
variation in scale, most iterative reconstructions can be performed with
this variable scale reprojection scheme (VS). The main routine is a
“translate and sum” function that accesses the memory row by
row. The OSEM algorithm was implemented with the VS, the RS and the
line-length model, and OSEM reconstructions were performed on real PET
data. The VS scheme was demonstrated to be significantly faster than RS
and twice as fast as LL, without compromising image quality, and even
slightly improving it
This paper reviews and compares three maximum likelihood
algorithms for transmission tomography. One of these algorithms is the
EM algorithm, one is based on a convexity argument devised by De Pierro
(see IEEE Trans. Med. Imaging, vol.12, p.328-333, 1993) in the context
of emission tomography, and one is an ad hoc gradient algorithm. The
algorithms enjoy desirable local and global convergence properties and
combine gracefully with Bayesian smoothing priors. Preliminary numerical
testing of the algorithms on simulated data suggest that the convex
algorithm and the ad hoc gradient algorithm are computationally superior
to the EM algorithm. This superiority stems from the larger number of
exponentiations required by the EM algorithm. The convex and gradient
algorithms are well adapted to parallel computing
This paper has the dual purpose of introducing some new algorithms for emission and transmission tomography and proving mathematically that these algorithms and related antecedent algorithms converge. Like the EM algorithms for positron, single-photon, and transmission tomography, the algorithms provide maximum likelihood estimates of pixel concentration or linear attenuation parameters. One particular innovation we discuss is a computationally practical scheme for modifying the EM algorithms to include a Bayesian prior. The Bayesian versions of the EM algorithms are shown to have superior convergence properties in a vicinity of the maximum. We anticipate that some of the other algorithms will also converge faster than the EM algorithms.
This paper addresses reprojection of three-dimensional (3-D) reconstructions obtained from cone-beam scans using a C-arm imaging equipment assisted by a pose-determining system. The emphasis is on reprojecting without decomposing the estimated projection matrix (P-matrix) associated with a pose. Both voxel- and ray-driven methods are considered. The voxel-driven reprojector follows the algorithm for backprojection using a P-matrix. The ray-driven reprojector is derived by extracting from the P-matrix the equation of the line joining a detector-pixel and the X-ray source position. This reprojector can be modified to a ray-driven backprojector. When the geometry is specified explicitly in terms of the physical parameters of the imaging system, the projection matrices can be constructed. The resulting "projection-matrix method" is advantageous, especially when the scanning trajectory is irregular. The algorithms presented are useful in iterative methods of image reconstruction and enhancement procedures, apart from their well-known role in visualization and volume rendering. Reprojections of 3-D patient data compare favorably with the original X-ray projections obtained from a prototype C-arm system. The algorithms for reprojection can be modified to compute perspective maximum intensity projection.
While the computation time for reconstructing images in C.T. is not a problem in commercial systems, there are many experimental and developmental applications where resources are limited and image reconstruction places a heavy burden on the computer system. This paper describes a very efficient back-projection algorithm which results in large time savings when implemented in machine code. Also described is a minor modification to this algorithm which converts it to a re-projection procedure with comparable efficiency.
A least squares technique for reconstructing truncated data is
presented. The method involves including the ramp filter into the system
matrix. The sinogram is extrapolated with zeros for filtering but
extrapolated values are not considered during backward and forward
projections. The new matrix leads to a frequency weighted least squares
criterion. Tikhonov regularization and a priori information are
considered. For the O-order regularization, the resulting images are
biased but it is shown that this bias is recovered by a simple scaling
process. First and second order regularizations provide true
restoration. The technique is applied to simulated and real transmission
SPECT data. Results show that the frequency least squares criterion is a
valuable approach for efficiently reconstructing truncated sinograms
Transform methods for image reconstruction from projections are based on analytic inversion formulas. In this tutorial paper, the inversion formula for the case of two-dimensional (2-D) reconstruction from line integrals is manipulated into a number of different forms, each of which may be discretized to obtain different algorithms for reconstruction from sampled data. For the convolution-backprojection algorithm and the direct Fourier algorithm the emphasis is placed on understanding the relationship between the discrete operations specified by the algorithm and the functional operations expressed by the inversion formula. The performance of the Fourier algorithm may be improved, with negligible extra computation, by interleaving two polar sampling grids in Fourier space. The convolution-backprojection formulas are adapted for the fan-beam geometry, and other reconstruction methods are summarized, including the rho-filtered layergram method, and methods involving expansions in angular harmonics. A standard mathematical process leads to a known formula for iterative reconstruction from projections at a finite number of angles. A new iterative reconstruction algorithm is obtained from this formula by introducing one-dimensional (1-D) and 2-D interpolating functions, applied to sampled projections and images, respectively. These interpolating functions are derived by the same Fourier approach which aids in the development and understanding of the more conventional transform methods.
Series-expansion reconstruction methods made their first appearance in the scientific literature and in the CT scanner industry around 1970. Great research efforts have gone into them since but many questions still wait to be answered. These methods, synonymously known as algebraic methods, iterative algorithms, or optimization theory techniques, are based on the discretization of the image domain prior to any mathematical analysis and thus are rooted in a completely different branch of mathematics than the transform methods which are discussed in this issue by Lewitt [51]. How is the model set up? What is the methodology of the approach? Where does mathematical optimization theory enter? What do these reconstruction algorithms look like? How are quadratic optimization, entropy optimization, and Bayesian analysis used in image reconstruction? Finally, why study series expansion methods if transform methods are so much faster? These are some of the questions that are answered in this paper.
3-D X-ray angiography: From numerical simulations to clinical routine (invited paper)
- Y Trousset
- C Picard
- C Ponchut
- R Romeas
- R Campagnolo
- S Croci
- J M Scarabin
- M Amiel
Y. Trousset, C. Picard, C. Ponchut, R. Romeas, R. Campagnolo, S. Croci, J. M. Scarabin, and M. Amiel, "3-D X-ray angiography: From numerical simulations to clinical routine (invited paper)," in Proc. 1995 Int. Meeting Fully Three-Dimensional Image Reconstruct. Radiol. Nucl. Med., Aix-les-Bains, France, 1995, pp. 3–9.
Method of reconstruction of a three-dimensional image of an object
- R Vaillant
- Y Trousset
- R Boucherie
- R Roméas
R. Vaillant, Y. Trousset, R. Boucherie, and R. Roméas, "Method of reconstruction of a three-dimensional image of an object," U.S. Patent 6 320 928, B1, 2001.
Improved iterative image reconstruction using variable projection binning and abbreviated convolutionVariable scale reprojection for iterative reconstruction: Application to PET reconstruction with OSEM
- P Shmidlin
P. Shmidlin, "Improved iterative image reconstruction using variable projection binning and abbreviated convolution," Eur. J. Nucl. Med., vol. 21, pp. 930–936, 1994. [9] C. Riddell and S. L. Bacharach, "Variable scale reprojection for iterative reconstruction: Application to PET reconstruction with OSEM," in Nucl. Sci. Symp. 1999 Conf. Rec., Seattle, WA, 1999, pp. 1647–1651.