Estimation of geometrical parameters for fan beam tomography

A nonconvex model-based combined geometric calibration scheme for micro cone-beam CT with irregular trajectories

BACKGROUND

Many dedicated cone-beam CT (CBCT) systems have irregular scanning trajectories. Compared with the standard CBCT calibration, accurate calibration for CBCT systems with irregular trajectories is a more complex task, since the geometric parameters for each scanning view are variable. Most of the existing calibration methods assume that the intrinsic geometric relationship of the fiducials in the phantom is precisely known, and rarely delve deeper into the issue of whether the phantom accuracy is adapted to the calibration model.

PURPOSE

A high-precision phantom and a highly robust calibration model are interdependent and mutually supportive, and they are both important for calibration accuracy, especially for the high-resolution CBCT. Therefore, we propose a calibration scheme that considers both accurate phantom measurement and robust geometric calibration.

METHODS

Our proposed scheme consists of two parts: (1) introducing a measurement model to acquire the accurate intrinsic geometric relationship of the fiducials in the phantom; (2) developing a highly noise-robust nonconvex model-based calibration method. The measurement model in the first part is achieved by extending our previous high-precision geometric calibration model suitable for CBCT with circular trajectories. In the second part, a novel iterative method with optimization constraints based on a back-projection model is developed to solve the geometric parameters of each view.

RESULTS

The simulations and real experiments show that the measurement errors of the fiducial ball bearings (BBs) are within the subpixel level. With the help of the geometric relationship of the BBs obtained by our measurement method, the classic calibration method can achieve good calibration based on far fewer BBs. All metrics obtained in simulated experiments as well as in real experiments on Micro CT systems with resolutions of 9 and 4.5 μm show that the proposed calibration method has higher calibration accuracy than the competing classic method. It is particularly worth noting that although our measurement model proves to be very accurate, the classic calibration method based on this measurement model can only achieve good calibration results when the resolution of the measurement system is close to that of the system to be calibrated, but our calibration scheme enables high-accuracy calibration even when the resolution of the system to be calibrated is twice that of the measurement system.

CONCLUSIONS

The proposed combined geometrical calibration scheme does not rely on a phantom with an intricate pattern of fiducials, so it is applicable in Micro CT with high resolution. The two parts of the scheme, the "measurement model" and the "calibration model," prove to be of high accuracy. The combination of these two models can effectively improve the calibration accuracy, especially in some extreme cases.

Data sustained misalignment correction in microscopic cone beam CT via optimization under the Grangeat Epipolar consistency condition

PURPOSE

The misalignment correction in cone beam computed tomography (CBCT), which is usually carried out in an offline manner, is a difficult and tedious process. It becomes even more challenging in microscopic CBCT due to the much higher requirements on spatial resolution. In practice, however, an offline approach for misalignment correction may not be readily implementable, especially in the situations where either time is of the essence or the process needs to be carried out repetitively. Thus, an online self-calibration (i.e., data sustained misalignment correction without the involvement of specific alignment phantom) would be more practical. In this work, we investigate the data sustained misalignment correction in microscopic CBCT via optimization under the Grangeat Epipolar Consistence Condition and evaluate its performance via phantom and specimen studies.

METHODS

With the cost function defined according to the Grangeat Epipolar Consistency Condition (G-ECC) and by minimizing the cost function using the simplex-simulated annealing algorithm (SIMPSA), we evaluate and verify the G-ECC optimization-based online self-calibration method's performance. Performance is measured in sensitivity, robustness, and accuracy using the projection data of phantoms generated by computer simulation and botanical specimens acquired by a prototype microscopic CBCT.

RESULTS

The online data sustained misalignment correction in microscopic CBCT via G-ECC optimization works very well in sensitivity and robustness, in addition to its accuracy of 0.27%, 0.48%, and 0.34% relative errors, respectively, in obtaining the three geometric parameters that are the most critical to image reconstruction in CBCT. Quantitatively, the performance in meeting the requirements on spatial resolution is comparable to, if not better than, that of the offline misalignment correction method, in which a specific alignment phantom has to be used.

CONCLUSIONS

The G-ECC optimization-based online self-calibration approach provides a practical solution (as long as no latitudinal (lateral) data truncation occurs) for misalignment correction in microscopic CBCT, an application that demands high accuracy in geometric alignment for biological (cellular) imaging at super high spatial resolutions in the order of micrometers (2.1 µm).

Jitter correction for transmission X-ray microscopy via measurement of geometric moments

Transmission X-ray microscopes (TXMs) have become one of the most powerful tools for imaging 3D structures of nano-scale samples using the computed tomography (CT) principle. As a major error source, sample jitter caused by mechanical instability of the rotation stage produces shifted 2D projections, from which reconstructed images contain severe motion artifacts. In this paper, a jitter correction algorithm is proposed, that has high accuracy and computational efficiency for TXM experiments with or without nano-particle markers. Geometric moments (GMs) are measured on segmented projections for each angle and fitted to sinusoidal curves in the angular direction. Sample jitter is estimated from the difference between the measured and the fitted GMs for image correction. On a digital phantom, the proposed method removes jitter errors at different noise levels. Physical experiments on chlorella cells show that the proposed GM method achieves better spatial resolution and higher computational efficiency than the re-projection method, a state-of-the-art algorithm using iterative correction. It even outperforms the approach of manual alignment, the current gold standard, on faithfully maintaining fine structures on the CT images. Our method is practically attractive in that it is computationally efficient and lowers experimental costs in current TXM studies without using expensive nano-particles markers.

A method of determining geometry of cone beam CT scanner

The task of determining the geometry of a cone-beam CT scanner with flat panel detector and circular/spiral source trajectory is considered. Accomplishing this task implies analyzing projections of a set of points referred to as calibrating set or calibrating phantom. We take advantage of the fact that observed coordinates of a point’s projection are rational functions of the point’s location. Unknown coefficients of these functions can be recovered exactly from six projections of the point. Location of the source as well as position and orientation of the detector are determined in the scanner reference frame, which is constituted by rotation axis and central plane of the scanner. Two different projections of a calibrating set are enough to solve the task if the source trajectory is a circle. In applications where a shift of an object transversally to the central plane is required, two additional projections have to be collected in order to identify the direction of the shift. The developed formalism becomes especially simple when the detector is aligned with the rotation axis. In this case four projections of a single calibrating point rotated successfully about the rotation axis are sufficient. The error analysis carried out in the paper shows that the magnitude of deviation from the true values is of the order of the magnitude of measurement errors.

Geometric calibration of a stationary digital breast tomosynthesis system based on distributed carbon nanotube X-ray source arrays

Stationary digital breast tomosynthesis (sDBT) with distributed X-ray sources based on carbon nanotube (CNT) field emission cathodes has been recently proposed as an approach that can prevent motion blur produced by traditional DBT systems. In this paper, we simulate a geometric calibration method based on a proposed multi-source CNT X-ray sDBT system. This method is a projection matrix-based approach with seven geometric parameters, all of which can be obtained from only one projection datum of the phantom. To our knowledge, this study reports the first application of this approach in a CNT-based multi-beam X-ray sDBT system. The simulation results showed that the extracted geometric parameters from the calculated projection matrix are extremely close to the input values and that the proposed method is effective and reliable for a square sDBT system. In addition, a traditional cone-beam computed tomography (CT) system was also simulated, and the uncalibrated and calibrated geometric parameters were used in image reconstruction based on the filtered back-projection (FBP) method. The results indicated that the images reconstructed with calibrated geometric parameters have fewer artifacts and are closer to the reference image. All the simulation tests showed that this geometric calibration method is optimized for sDBT systems but can also be applied to other application-specific CT imaging systems.

A method to determine the detector locations of the cone-beam projection of the balls' centers

In geometric calibration of cone-beam computed tomography (CBCT), sphere-like objects such as balls are widely imaged, the positioning information of which is obtained to determine the unknown geometric parameters. In this process, the accuracy of the detector location of CB projection of the center of the ball, which we call the center projection, is very important, since geometric calibration is sensitive to errors in the positioning information. Currently in almost all the geometric calibration using balls, the center projection is invariably estimated by the center of the support of the projection or the centroid of the intensity values inside the support approximately. Clackdoyle's work indicates that the center projection is not always at the center of the support or the centroid of the intensity values inside, and has given a quantitative analysis of the maximum errors in evaluating the center projection by the centroid. In this paper, an exact method is proposed to calculate the center projection, utilizing both the detector location of the ellipse center and the two axis lengths of the ellipse. Numerical simulation results have demonstrated the precision and the robustness of the proposed method. Finally there are some comments on this work with non-uniform density balls, as well as the effect by the error occurred in the evaluation for the location of the orthogonal projection of the cone vertex onto the detector.

Geometric Parameters Estimation and Calibration in Cone-Beam Micro-CT

The quality of Computed Tomography (CT) images crucially depends on the precise knowledge of the scanner geometry. Therefore, it is necessary to estimate and calibrate the misalignments before image acquisition. In this paper, a Two-Piece-Ball (TPB) phantom is used to estimate a set of parameters that describe the geometry of a cone-beam CT system. Only multiple projections of the TPB phantom at one position are required, which can avoid the rotation errors when acquiring multi-angle projections. Also, a corresponding algorithm is derived. The performance of the method is evaluated through simulation and experimental data. The results demonstrated that the proposed method is valid and easy to implement. Furthermore, the experimental results from the Micro-CT system demonstrate the ability to reduce artifacts and improve image quality through geometric parameter calibration.

An analytical geometric calibration method for circular cone-beam geometry

This work is a continuation of our previous work on geometric calibration in the circular cone-beam geometry. It is well known that seven parameters completely describe such a geometry in either flat-panel X-ray computed tomography or single pinhole SPECT imaging. Previously we developed a graphical procedure to determine the detector in-plane rotation angle independently of the other six parameters. Using the discovered geometrical relationships, in this paper we determine the remaining six parameters using the cone-beam projections of a minimum of three point objects. Our method is analytical. It makes use of the parameters of the fitted ellipse from the calibration data. The parameter estimation is accurate in the noise-free case or when there is moderate projection data truncation or shorter calibration scan range ( ≤ 360°). We perform numerical evaluations to study the robustness of the proposed method under different projection noise levels and using different data acquisition ranges. Using a full 360° scan range, the estimation accuracy and precision of our method are comparable or superior to previous methods. Using a shorter acquisition range, there may be bias in the ellipse parameters obtained by simple algebraic fitting methods. This bias will propagate to the estimated geometric parameters. Such bias can be mostly eliminated by using a more sophisticated fitting algorithm. At the same noise level, the geometric parameter estimation accuracies are comparable, but the estimation precision degrades, as the acquisition range becomes shorter.

A robust geometry estimation method for spiral, sequential and circular cone-beam micro-CT

PURPOSE

The authors propose a novel method for misalignment estimation of micro-CT scanners using an adaptive genetic algorithm.

METHODS

The proposed algorithm is able to estimate the rotational geometry, the direction vector of table movement and the displacement between different imaging threads of a dual source or even multisource scanner. The calibration procedure does not rely on dedicated calibration phantoms and a sequence scan of a single metal bead is sufficient to geometrically calibrate the whole imaging system for spiral, sequential, and circular scan protocols. Dual source spiral and sequential scan protocols in micro-computed tomography result in projection data that-besides the source and detector positions and orientations-also require a precise knowledge of the table direction vector to be reconstructed properly. If those geometric parameters are not known accurately severe artifacts and a loss in spatial resolution appear in the reconstructed images as long as no geometry calibration is performed. The table direction vector is further required to ensure that consecutive volumes of a sequence scan can be stitched together and to allow the reconstruction of spiral data at all.

RESULTS

The algorithm's performance is evaluated using simulations of a micro-CT system with known geometry and misalignment. To assess the quality of the algorithm in a real world scenario the calibration of a micro-CT scanner is performed and several reconstructions with and without geometry estimation are presented.

CONCLUSIONS

The results indicate that the algorithm successfully estimates all geometry parameters, misalignment artifacts in the reconstructed volumes vanish, and the spatial resolution is increased as can be shown by the evaluation of modulation transfer function measurements.

Correction of error motion in a line-scanning tomographic optical microscope

A line-scanning tomographic optical microscope system requires precise rotation of the scanning line. Center of rotation error introduced by both the imprecision of optical and mechanical components is studied experimentally and via simulations. It was shown that a practical tolerance limit can be chosen where the influence of the investigated error on the reconstructed image quality remains insignificant. An effective and simply practical solution was presented to keep the center of rotation error below this tolerance limit and the spatial resolution of the reconstructed image close to the diffraction limit.

A graphical method for determining the in-plane rotation angle in geometric calibration of circular cone-beam CT systems

It is well known that seven parameters completely describe a circular cone-beam geometry in either flat-panel X-ray computed tomography (CT) or single pinhole SPECT imaging. This paper considers the problem of determining one of the seven parameters only, the detector in-plane rotation or twist angle η. We describe a graphical procedure that can determine η independently of all other six parameters from a geometric calibration scan of point objects. Our method is exact in the ideal noise-free case and is general in that the other two out-of-plane detector rotation angles θ and φ can be nonzero. The calibration scan typically needs at least two point objects and an even number of projection views over a full 360° data acquisition. Under certain conditions, projection data truncation or a short scan acquisition of 180° + fan angle can be accommodated without affecting the accuracy of the calibration result. The graphical method is equally applicable to rotational multipinhole SPECT geometry. In this case, the final result is averaged from the individual estimates considering each pinhole separately. We use computer simulations and a multipinhole SPECT experiment to demonstrate the accuracy and precision of the proposed method.

Geometric calibration of a micro-CT system and performance for insect imaging

Micro-CT with a high spatial resolution in combination with computer-based-reconstruction techniques is considered a powerful tool for morphological study of insects. The quality of CT images crucially depends on the precise knowledge of the scan geometry of the micro-CT system. In this paper, we have proposed a method to calculate the deviation of rotating axis for compensating deficiency of existing methods. A practical application of this geometric calibration method of the micro-CT system for insect imaging is presented. We have performed the computer-simulation study and experimental study with our prototype micro-CT system. The results demonstrate that the proposed technique is accurate and robust. In addition, we have evaluated the imaging characteristics of the detector in terms of modulation-transfer function (MTF). Finally, insect imaging performance and image reconstruction from data acquired with different energies are presented.

Direct determination of geometric alignment parameters for cone-beam scanners

This paper describes a comprehensive method for determining the geometric alignment parameters for cone-beam scanners (often called calibrating the scanners or performing geometric calibration). The method is applicable to x-ray scanners using area detectors, or to SPECT systems using pinholes or cone-beam converging collimators. Images of an alignment test object (calibration phantom) fixed in the field of view of the scanner are processed to determine the nine geometric parameters for each view. The parameter values are found directly using formulae applied to the projected positions of the test object marker points onto the detector. Each view is treated independently, and no restrictions are made on the position of the cone vertex, or on the position or orientation of the detector. The proposed test object consists of 14 small point-like objects arranged with four points on each of three orthogonal lines, and two points on a diagonal line. This test object is shown to provide unique solutions for all possible scanner geometries, even when partial measurement information is lost by points superimposing in the calibration scan. For the many situations where the cone vertex stays reasonably close to a central plane (for circular, planar, or near-planar trajectories), a simpler version of the test object is appropriate. The simpler object consists of six points, two per orthogonal line, but with some restrictions on the positioning of the test object. This paper focuses on the principles and mathematical justifications for the method. Numerical simulations of the calibration process and reconstructions using estimated parameters are also presented to validate the method and to provide evidence of the robustness of the technique.

Backprojection-based imaging geometric parameter estimation

In SPECT (single photon emission computed tomography) and X-ray CT, point sources are commonly used to acquire data for imaging parameter estimation. The parameters are estimated by fitting the point source projection data to a mathematical model of the imaging system. This paper introduces two backprojection-based parameter estimation methods. In the first method a point source is attached to the object; the second method is data-based without using a point source at all.

Brain single-photon emission CT physics principles

The basic principles of scintigraphy are reviewed and extended to 3D imaging. Single-photon emission computed tomography (SPECT) is a sensitive and specific 3D technique to monitor in vivo functional processes in both clinical and preclinical studies. SPECT/CT systems are becoming increasingly common and can provide accurately registered anatomic information as well. In general, SPECT is affected by low photon-collection efficiency, but in brain imaging, not all of the large FOV of clinical gamma cameras is needed: The use of fan- and cone-beam collimation trades off the unused FOV for increased sensitivity and resolution. The design of dedicated cameras aims at increased angular coverage and resolution by minimizing the distance from the patient. The corrections needed for quantitative imaging are challenging but can take advantage of the relative spatial uniformity of attenuation and scatter. Preclinical systems can provide submillimeter resolution in small animal brain imaging with workable sensitivity.

Geometric calibration for a dual tube/detector micro-CT system

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.

An optimization-based method for geometrical calibration in cone-beam CT without dedicated phantoms

In this paper we present a new method for the determination of geometrical misalignments in cone-beam CT scanners, from the analysis of the projection data of a generic object. No a priori knowledge of the object shape and positioning is required. We show that a cost function, which depends on the misalignment parameters, can be defined using the projection data and that such a cost function has a local minimum in correspondence to the actual parameters of the system. Hence, the calibration of the scanner can be carried out by minimizing the cost function using standard optimization techniques. The method is developed for a particular class of 3D object functions, for which the redundancy of the fan beam sinogram in the transaxial midplane can be extended to cone-beam projection data, even at wide cone angles. The method has an approximated validity for objects which do not belong to that class; in that case, a suitable subset of the projection data can be selected in order to compute the cost function. We show by numerical simulations that our method is capable to determine with high accuracy the most critical misalignment parameters of the scanner, i.e., the transversal shift and the skew of the detector. Additionally, the detector slant can be determined. Other parameters such as the detector tilt, the longitudinal shift and the error in the source-detector distance cannot be determined with our method, as the proposed cost function has a very weak dependence on them. However, due to the negligible influence of these latter parameters in the reconstructed image quality, they can be kept fixed at estimated values in both calibration and reconstruction processes without compromising the final result. A trade-off between computational cost and calibration accuracy must be considered when choosing the data subset used for the computation of the cost function. Results on real data of a mouse femur as obtained with a small animal micro-CT are shown as well, proving the capability of the proposed calibration method. In principle, the method can be adapted to other cone-beam imaging modalities (e.g., single photon emission computed tomography).

A geometric calibration method for cone beam CT systems

Cone beam CT systems are being deployed in large numbers for small animal imaging, dental imaging, and other specialty applications. A new high-precision method for cone beam CT system calibration is presented in this paper. It uses multiple projection images acquired from rotating point-like objects (metal ball bearings) and the angle information generated from the rotating gantry system is also used. It is assumed that the whole system has a mechanically stable rotation center and that the detector does not have severe out-of-plane rotation (<2 degrees). Simple geometrical relationships between the orbital paths of individual BBs and five system parameters were derived. Computer simulations were employed to validate the accuracy of this method in the presence of noise. Equal or higher accuracy was achieved compared with previous methods. This method was implemented for the geometrical calibration of both a micro CT scanner and a breast CT scanner. The reconstructed tomographic images demonstrated that the proposed method is robust and easy to implement with high precision.

Accurate technique for complete geometric calibration of cone-beam computed tomography systems

Cone-beam computed tomography systems have been developed to provide in situ imaging for the purpose of guiding radiation therapy. Clinical systems have been constructed using this approach, a clinical linear accelerator (Elekta Synergy RP) and an iso-centric C-arm. Geometric calibration involves the estimation of a set of parameters that describes the geometry of such systems, and is essential for accurate image reconstruction. We have developed a general analytic algorithm and corresponding calibration phantom for estimating these geometric parameters in cone-beam computed tomography (CT) systems. The performance of the calibration algorithm is evaluated and its application is discussed. The algorithm makes use of a calibration phantom to estimate the geometric parameters of the system. The phantom consists of 24 steel ball bearings (BBs) in a known geometry. Twelve BBs are spaced evenly at 30 deg in two plane-parallel circles separated by a given distance along the tube axis. The detector (e.g., a flat panel detector) is assumed to have no spatial distortion. The method estimates geometric parameters including the position of the x-ray source, position, and rotation of the detector, and gantry angle, and can describe complex source-detector trajectories. The accuracy and sensitivity of the calibration algorithm was analyzed. The calibration algorithm estimates geometric parameters in a high level of accuracy such that the quality of CT reconstruction is not degraded by the error of estimation. Sensitivity analysis shows uncertainty of 0.01 degrees (around beam direction) to 0.3 degrees (normal to the beam direction) in rotation, and 0.2 mm (orthogonal to the beam direction) to 4.9 mm (beam direction) in position for the medical linear accelerator geometry. Experimental measurements using a laboratory bench Cone-beam CT system of known geometry demonstrate the sensitivity of the method in detecting small changes in the imaging geometry with an uncertainty of 0.1 mm in transverse and vertical (perpendicular to the beam direction) and 1.0 mm in the longitudinal (beam axis) directions. The calibration algorithm was compared to a previously reported method, which uses one ball bearing at the isocenter of the system, to investigate the impact of more precise calibration on the image quality of cone-beam CT reconstruction. A thin steel wire located inside the calibration phantom was imaged on the conebeam CT lab bench with and without perturbations in source and detector position during the scan. The described calibration method improved the quality of the image and the geometric accuracy of the object reconstructed, improving the full width at half maximum of the wire by 27.5% and increasing contrast of the wire by 52.8%. The proposed method is not limited to the geometric calibration of cone-beam CT systems but can be used for many other systems, which consist of one or more point sources and area detectors such as calibration of megavoltage (MV) treatment system (focal spot movement during the beam delivery, MV source trajectory versus gantry angle, the axis of collimator rotation, and couch motion), cross calibration between Kilovolt imaging and MV treatment system, and cross calibration between multiple imaging systems. Using the complete information of the system geometry, it was demonstrated that high image quality in CT reconstructions is possible even in systems with large geometric nonidealities.

Geometric misalignment and calibration in cone-beam tomography

We present a new high-precision method for the geometric calibration in cone-beam computed tomography. It is based on a Fourier analysis of the projection-orbit data, recorded with a flat-panel area detector, of individual point-like objects. For circular scan trajectories the complete set of misalignment parameters which determine the deviation of the detector alignment from the ideal scan geometry are obtained from explicit analytic expressions. To derive these expressions we show how to disentangle the problems of calculating misalignment parameters and point coordinates. The calculation of the coordinates of the point objects inside the scanned volume, in units of the distance from the focal spot to the center of rotation, is then possible analytically likewise. We simulate point-projection data on a misaligned detector with various amounts of randomness added to mimic measurement uncertainties. This data is then employed in our calibration to validate the method by comparing the resulting misalignment parameters and point coordinates to the known true ones. We also present our implementation and results for the geometric calibration of micro-CT systems. The effectiveness of the corresponding misalignment correction in reducing image artifacts is exemplified by reconstructed micro-CT images.

Determination of mechanical and electronic shifts for pinhole SPECT using a single point source

The effects of uncompensated electronic and mechanical shifts may compromise the resolution of pinhole single photon emission computed tomography. The resolution degradation due to uncompensated shifts is estimated through simulated data. A method for determining the transverse mechanical and axial electronic shifts is described and evaluated. This method assumes that the tilt of the detector and the radius of rotation (ROR) are previously determined using another method. When this assumption is made, it is possible to determine the rest of the calibration parameters using a single point source. A method that determines the electronic and mechanical shifts as well as the tilt has been previously described; this method requires three point sources. It may be reasonable in most circumstances to calibrate tilt much less frequently than the mechanical shifts since the tilt is a property of the scanner whereas the mechanical shift may change every time the collimator is replaced. An alternative method for determining the ROR may also be used. Lastly, we take the view that the transverse electronic shift and the focal length change slowly and find these parameters independently.

Analytic method based on identification of ellipse parameters for scanner calibration in cone-beam tomography

This paper is about calibration of cone-beam (CB) scanners for both x-ray computed tomography and single-photon emission computed tomography. Scanner calibration refers here to the estimation of a set of parameters which fully describe the geometry of data acquisition. Such parameters are needed for the tomographic reconstruction step. The discussion is limited to the usual case where the cone vertex and planar detector move along a circular path relative to the object. It is also assumed that the detector does not have spatial distortions. We propose a new method which requires a small set of measurements of a simple calibration object consisting of two spherical objects, that can be considered as 'point' objects. This object traces two ellipses on the detector and from the parametric description of these ellipses, the calibration geometry can be determined analytically using explicit formulae. The method is robust and easy to implement. However, it is not fully general as it is assumed that the detector is parallel to the rotation axis of the scanner. Implementation details are given for an experimental x-ray CB scanner.

Three-dimensional computed tomographic reconstruction using a C-arm mounted XRII: image-based correction of gantry motion nonidealities

The image quality of 3D reconstructions produced using a C-arm mounted XRII depends on precise determination of the geometric parameters that describe the detector system in the laboratory frame of reference. We have designed a simplified calibration system that depends on images of a metal sphere, acquired during rotation of the gantry through 200 degrees. Angle-dependent shift corrections are obtained, accounting for nonideal motion in two directions: perpendicular to the axis of rotation and tangential to the circular trajectory (tau), and parallel to the axis of rotation (xi). Projection images are corrected prior to reconstruction using a simple shift-interpolation algorithm. We show that the motion of the gantry is highly reproducible during acquisitions within one day (mean standard deviation in tau and xi is 0.11 mm and 0.08 mm, respectively), and over 21 months (mean standard deviation in tau and xi is 0.10 mm and 0.06 mm, respectively). Reconstruction of a small-bead phantom demonstrates uniformity of the correction algorithm over the full volume of the reconstruction [standard deviation of full-width-half-maximum of the beads is approximately 0.25 pixels (0.13 mm) over the volume of reconstruction]. Our approach provides a simple correction technique that can be applied when trajectory deviations are significant relative to the pixel size of the detector but small relative to the detector field of view, and when the fan angle of the acquisition geometry is small (<20 degrees). A comparison with other calibration techniques in the literature is provided.

Astigmatic single photon emission computed tomography imaging with a displaced center of rotation

A filtered backprojection algorithm is developed for single photon emission computed tomography (SPECT) imaging with an astigmatic collimator having a displaced center of rotation. The astigmatic collimator has two perpendicular focal lines, one that is parallel to the axis of rotation of the gamma camera and one that is perpendicular to this axis. Using SPECT simulations of projection data from a hot rod phantom and point source arrays, it is found that a lack of incorporation of the mechanical shift in the reconstruction algorithm causes errors and artifacts in reconstructed SPECT images. The collimator and acquisition parameters in the astigmatic reconstruction formula, which include focal lengths, radius of rotation, and mechanical shifts, are often partly unknown and can be determined using the projections of a point source at various projection angles. The accurate determination of these parameters by a least squares fitting technique using projection data from numerically simulated SPECT acquisitions is studied. These studies show that the accuracy of parameter determination is improved as the distance between the point source and the axis of rotation of the gamma camera is increased. The focal length of the focal line perpendicular to the axis of rotation is determined more accurately than the focal length to the focal line parallel to this axis.

Application of convergent-beam collimation and simultaneous transmission emission tomography to cardiac single-photon emission computed tomography

Single-photon emission computed tomography (SPECT) is the most commonly performed imaging technique for perfusion studies of the heart and brain. However, these organs are much smaller than the crystal surface of gamma cameras. SPECT sensitivity and resolution can be improved by using fan- and cone-beam collimators to magnify the image of these organs over a larger portion of the crystal face. Special orbits and reconstruction algorithms must be used with convergent-beam acquisitions to prevent image distortion. Differential attenuation of source activity in the chest is one of the most vexing problems in cardiac SPECT, especially with Thallium-201. Multi-headed cameras equipped with convergent-beam collimators allow a transmission image to be obtained at the same time as emission images. Applying a transmission map of the chest attenuation values to the emission images produces a truer picture of source distribution in the heart. This article reviews the technical problems associated with convergent-beam geometry and simultaneous transmission emission tomography SPECT imaging of the heart.

Determination of both mechanical and electronic shifts in cone beam SPECT

The difference between the displacement of the centre of rotation (mechanical shift, MS) and the electronic centring misalignment (electronic shift, ES) in cone beam SPECT is evaluated. A method is proposed to determine both MS and ES using the centroid of a projected point source sampled over 360 degrees and the Marquardt non-linear fitting algorithm. Both shifts are characterized by two orthogonal components. This method is verified using Monte Carlo simulated point source data with different combinations of mechanical and electronic shifts. Both shifts can be determined correctly. We have also applied the proposed method to our cone beam SPECT system to determine both shifts as well as the focal length. The determined ES parameters are then used to correct the projections and the MS parameters are incorporated into a reconstruction algorithm. The point source images are reconstructed and the image resolutions with and without the shift corrections are measured. The experimental results demonstrate that the image resolution is improved after shift corrections. The experimental results also indicate that the shift parameters determined in the same experiment with the point source located at different places are consistent but change from time to time, suggesting that calibration of the system is needed on a periodic basis.

Review of convergent beam tomography in single photon emission computed tomography

Investigation of convergent-beam single photon emission computed tomography (SPECT) is actively being pursued to evaluate its clinical potentials. Fan-beam, cone-beam, pin-hole and astigmatic collimators are being used with rotating gamma cameras having large crystal areas, to increase the sensitivity for emission and transmission computed tomography of small organs such as the thyroid, brain or heart. With new multi-detector SPECT systems, convergent-beam geometry offers the ability to simultaneously obtain emission and transmission data necessary to quantify uptake of radiopharmaceutical distributions in the heart. The development of convergent-beam geometry in SPECT requires the integration of hardware and software. In considering hardware, the optimum detector system for cone-beam tomography is a system that satisfies the data sufficiency condition for which the scanning trajectory intersects any plane passing through the reconstructed region of interest. However, the major development of algorithms has been for the data insufficient case of single planar orbit acquisitions. The development of these algorithms have made possible the preliminary evaluation of this technology and the imaging of brain and heart are showing significant potential for the clinical application of cone-beam tomography. Presently, significant research activity is pursuing the development of algorithms for data acquisitions that satisfy the data sufficiency condition and that can be implemented easily and inexpensively on clinical SPECT systems.