Image reconstruction from truncated fan-beam projections along a circular trajectory using the virtual fan-beam method.

Objective.In this paper, we investigate how the virtual fan-beam (VFB) method can be used to perform mathematically correct 2D reconstruction in a region-of-interest (ROI), using truncated fan-beam projections acquired on a circular scan, for truncation that only occurs on one side of the object.Approach.We start by choosing a virtual fan-beam trajectory and specifying how to obtain the corresponding virtual projections. Then, three VFB formulas are obtained by applying known super-short-scan (SSS) formulas to this virtual trajectory. Two of them perform the backprojection in a virtual parallel geometry and the third in the virtual fan-beam geometry. Next, we develop two VFB formulas that perform the backprojection step in the fan-beam acquisition geometry.Main results.We present five VFB reconstruction formulas for this truncation setting. To our knowledge, the two VFB formulas performing the backprojection in the fan-beam acquisition geometry are new. Moreover, the five VFB formulas presented here obtain accurate reconstruction in a larger ROI than what has been previously reported in the literature in the same setting. A complete mathematical derivation of these five VFB formulas is given, and their implementation is described step by step. Numerical simulations, using the Forbild head and thorax phantoms, demonstrate the efficacy of these formulas. A spatial resolution analysis and a variance study indicate minor differences between these five VFB formulas.Significance.This work shows that many different VFB formulas can be applied to perform mathematically correct 2D reconstruction in a ROI, in case of truncated fan-beam projections acquired on a circular scan. Moreover, the two new VFB formulas, with backprojection in the acquisition geometry, may open the path for an extension of the VFB method to 3D reconstruction from transversely truncated cone-beam projection acquired on a circular scan.

Equal parallel and cone-beam projections: a curious property of D-symmetric object functions.

In tomographic image reconstruction, the object density function is the unknown quantity whose projections are measured by the scanner. In the three-dimensional case, we define the D-reflection of such a density function as the object obtained by a particular weighted reflection about the planez=D, and a D-symmetric function as one whose D-reflection is equal to itself. D-symmetric object functions have the curious property that their parallel projection onto the detector planez=Dis equal to their cone-beam projection onto the same detector with x-ray source location at the origin. Much more remarkable is the additional fact that for any fixed D-symmetric object,everyoblique parallel projection onto this same detector plane equals the cone-beam projection for a corresponding source location. The mathematical proof is straight forward but not particularly enlightening, and we also provide here an alternative physical demonstration that explains the various weighting terms in the context of classical tomosynthesis. Furthermore, we clarify the distinction between the new formulation presented here, and the original formulation of Edholm and co-workers who obtained similar properties but for a pair of objects whose divergent and parallel projections matched, but with no D-symmetry. We do not claim any immediate imaging application or useful physics from these notions, but we briefly comment on consequences for methods that apply data consistency conditions in image reconstruction.

Cone-beam CT sampling incompleteness: analytical and empirical studies of emerging systems and source-detector orbits.

Purpose: Motivated by emerging cone-beam computed tomography (CBCT) systems and scan orbits, we aim to quantitatively assess the completeness of data for 3D image reconstruction-in turn, related to "cone-beam artifacts." Fundamental principles of cone-beam sampling incompleteness are considered with respect to an analytical figure-of-merit [FOM, denoted tan(ψmin)] and related to an empirical FOM (denoted zmod) for measurement of cone-beam artifact magnitude in a test phantom. Approach: A previously proposed analytical FOM [tan(ψmin), defined as the minimum angle between a point in the 3D image reconstruction and the x-ray source over the scan orbit] was analyzed for a variety of CBCT geometries. A physical test phantom was configured with parallel disk pairs (perpendicular to the z-axis) at various locations throughout the field of view, quantifying cone-beam artifact magnitude in terms of zmod (the relative signal modulation between the disks). Two CBCT systems were considered: an interventional C-arm (Cios Spin 3D; Siemens Healthineers, Forcheim Germany) and a musculoskeletal extremity scanner; Onsight3D, Carestream Health, Rochester, United States)]. Simulations and physical experiments were conducted for various source-detector orbits: (a) a conventional 360 deg circular orbit, (b) tilted and untilted semi-circular (196 deg) orbits, (c) multi-source (three x-ray sources distributed along the z axis) semi-circular orbits, and (d) a non-circular (sine-on-sphere, SoS) orbit. The incompleteness of sampling [tan(ψmin)] and magnitude of cone-beam artifacts (zmod) were evaluated for each system and orbit. Results: The results show visually and quantitatively the effect of system geometry and scan orbit on cone-beam sampling effects, demonstrating the relationship between analytical tan(ψmin) and empirical zmod. Advanced source-detector orbits (e.g., three-source and SoS orbits) exhibited superior sampling completeness as quantified by both the analytical and the empirical FOMs. The test phantom and zmod metric were sensitive to variations in CBCT system geometry and scan orbit and provided a surrogate measure of underlying sampling completeness. Conclusion: For a given system geometry and source-detector orbit, cone-beam sampling completeness can be quantified analytically (in terms arising from Tuy's condition) and/or empirically (using a test phantom for quantification of cone-beam artifacts). Such analysis provides theoretical and practical insight on sampling effects and the completeness of data for emerging CBCT systems and scan trajectories.

A tomographic reconstruction algorithm for cross-sectional imaging of IMRT beams from six projections.

Objective. Patient-specific Quality Assurance (QA) measurements are of key importance in radiotherapy for safe and efficient treatment delivery and allow early detection of clinically relevant errors. Such QA processes remain challenging to implement for complex Intensity Modulated Radiation Therapy (IMRT) radiotherapy fields delivered using a multileaf collimator (MLC) which often feature small open segments and raise QA issues similar to those encountered in small field dosimetry. Recently, detectors based on long scintillating fibers have been proposed to measure a few parallel projections of the irradiation field with good performance for small field dosimetry. The purpose of this work is to develop and validate a novel approach to reconstruct MLC-shaped small irradiation fields from six projections.Approach. The proposed field reconstruction method uses a limited number of geometric parameters to model the irradiation field. These parameters are iteratively estimated with a steepest descent algorithm. The reconstruction method was first validated on simulated data. Real data were measured with a water-equivalent slab phantom equipped with a detector made of 6 scintillating-fiber ribbons placed at 1 m from the source. A radiochromic film was used to acquire a reference measurement of a first dose distribution in the slab phantom at the same source-to-detector distance and the treatment planning system (TPS) provided the reference for another dose distribution. In addition, simulated errors introduced on the delivered dose, field location and field shape were used to evaluate the ability of the proposed method to efficiently identify a deviation between the planned and delivered treatments.Main results. For a first small IMRT segment, 3%/3 mm, 2%/2 mm and 2%/1 mm gamma analysis conducted between the reconstructed dose distribution and the dose measured with radiochromic film exhibited pass rates of 100%, 99.9% and 95.7%, respectively. For a second and smaller IMRT segment, the same gamma analysis performed between the reconstructed dose distribution and the reference provided by the TPS showed pass rates of 100%, 99.4% and 92.6% for the 3%/3 mm, 2%/2 mm and 2%/1 mm gamma criteria, respectively. Gamma analysis of the simulated treatment delivery errors showed the ability of the reconstruction algorithm to detect a 3% deviation between the planned and delivered doses, as well as shifts lower than 7 mm and 3 mm when considering an individual leaf and a whole field shift, respectively.Significance. The proposed method allows accurate tomographic reconstruction of IMRT segments by processing projections measured with six scintillating-fiber ribbons and is suitable for water-equivalent real-time small IMRT segments QA.

Feasibility of attenuation map alignment in pinhole cardiac SPECT using exponential data consistency conditions.

PURPOSE: Dedicated cardiac SPECT systems do not typically include an integrated CT scanner and thus attenuation correction requires registration of separately acquired transmission scans. Data consistency conditions are equations that express the redundancy between projections while taking into account the attenuation effects. This study assessed the feasibility of applying exponential data consistency conditions to rebinned pinhole projections for attenuation-map registration in pinhole cardiac SPECT. METHODS: Simulations of an anthropomorphic computer phantom with three different tracer activity distributions were performed with and without clinical levels of noise in the projection data. The first activity distribution contained activity only within the myocardium which satisfied the assumptions of the data consistency conditions. The other two distributions violated these assumptions by adding background activity and uptake in the liver. Simulations included acquisitions with 360, 31, and 9 pinhole projections and detector pixel sizes of 0.75 and 2.5 mm. A metric based on the average difference between pairs of exponential projections was used to evaluate registration accuracy. RESULTS: When activity is restricted to the myocardium, the registration error was 3.0 mm for 31 noisy pinhole projections with a detector size of 2.5 mm. When activity is added to the background and the liver, a correction for the extra-cardiac activity is needed but when applied, a registration error of 6.0 mm was achieved. CONCLUSION: These results suggest that it may be feasible to use exponential data consistency conditions to register pinhole cardiac SPECT and CT transmission data. Taxonomy: 8-6 (IM-SPECT/Registration).

Quantification of Tomographic Incompleteness in Cone-Beam Reconstruction.

For situations of cone-beam scanning where the measurements are incomplete, we propose a method to quantify the severity of the missing information at each voxel. This incompleteness metric is geometric; it uses only the relative locations of all cone-beam vertices with respect to the voxel in question, and does not apply global information such as the object extent or the pattern of incompleteness of other voxels. The values are non-negative, with zero indicating "least incompleteness," i.e. minimal danger of incompleteness artifacts. The incompleteness value can be related to the severity of the potential reconstruction artifact at the voxel location, independent of reconstruction algorithm. We performed a computer simulation of x-ray sources along a circular trajectory, and used small multi-disk test-objects to examine the local effects of data incompleteness. The observed behavior of the reconstructed test-objects quantitatively matched the precalculated incompleteness values. A second simulation of a hypothetical SPECT breast imaging system used only 12 pinholes. Reconstructions were performed using analytic and iterative methods, and five reconstructed test-objects matched the behavior predicted by the incompleteness model. The model is based on known sufficiency conditions for data incompleteness, and provides strong predictive guidance for what can go wrong with incomplete cone-beam data.

Filtered-backprojection reconstruction for a cone-beam computed tomography scanner with independent source and detector rotations.

PURPOSE: A new cone-beam CT scanner for image-guided radiotherapy (IGRT) can independently rotate the source and the detector along circular trajectories. Existing reconstruction algorithms are not suitable for this scanning geometry. The authors propose and evaluate a three-dimensional (3D) filtered-backprojection reconstruction for this situation. METHODS: The source and the detector trajectories are tuned to image a field-of-view (FOV) that is offset with respect to the center-of-rotation. The new reconstruction formula is derived from the Feldkamp algorithm and results in a similar three-step algorithm: projection weighting, ramp filtering, and weighted backprojection. Simulations of a Shepp Logan digital phantom were used to evaluate the new algorithm with a 10 cm-offset FOV. A real cone-beam CT image with an 8.5 cm-offset FOV was also obtained from projections of an anthropomorphic head phantom. RESULTS: The quality of the cone-beam CT images reconstructed using the new algorithm was similar to those using the Feldkamp algorithm which is used in conventional cone-beam CT. The real image of the head phantom exhibited comparable image quality to that of existing systems. CONCLUSIONS: The authors have proposed a 3D filtered-backprojection reconstruction for scanners with independent source and detector rotations that is practical and effective. This algorithm forms the basis for exploiting the scanner's unique capabilities in IGRT protocols.

Data consistency conditions for truncated fanbeam and parallel projections.

PURPOSE: In image reconstruction from projections, data consistency conditions (DCCs) are mathematical relationships that express the overlap of information between ideal projections. DCCs have been incorporated in image reconstruction procedures for positron emission tomography, single photon emission computed tomography, and x-ray computed tomography (CT). Building on published fanbeam DCCs for nontruncated projections along a line, the authors recently announced new DCCs that can be applied to truncated parallel projections in classical (two-dimensional) image reconstruction. These DCCs take the form of polynomial expressions for a weighted backprojection of the projections. The purpose of this work was to present the new DCCs for truncated parallel projections, to extend these conditions to truncated fanbeam projections on a circular trajectory, to verify the conditions with numerical examples, and to present a model of how DCCs could be applied with a toy problem in patient motion estimation with truncated projections. METHODS: A mathematical derivation of the new parallel DCCs was performed by substituting the underlying imaging equation into the mathematical expression for the weighted backprojection and demonstrating the resulting polynomial form. This DCC result was extended to fanbeam projections by a substitution of parallel to fanbeam variables. Ideal fanbeam projections of a simple mathematical phantom were simulated and the DCCs for these projections were evaluated by fitting polynomials to the weighted backprojection. For the motion estimation problem, a parametrized motion was simulated using a dynamic version of the mathematical phantom, and both noiseless and noisy fanbeam projections were simulated for a full circular trajectory. The fanbeam DCCs were applied to extract the motion parameters, which allowed the motion contamination to be removed from the projections. A reconstruction was performed from the corrected projections. RESULTS: The mathematical derivation revealed the anticipated polynomial behavior. The conversion to fanbeam variables led to a straight-forward weighted fanbeam backprojection which yielded the same function and therefore the same polynomial behavior as occurred in the parallel case. Plots of the numerically calculated DCCs showed polynomial behavior visually indistinguishable from the fitted polynomials. For the motion estimation problem, the motion parameters were satisfactorily recovered and ten times more accurately for the noise-free case. The reconstructed images showed that only a faint trace of the motion blur was still visible after correction from the noisy motion-contaminated projections. CONCLUSIONS: New DCCs have been established for fanbeam and parallel projections, and these conditions have been validated using numerical experiments with truncated projections. It has been shown how these DCCs could be applied to extract parameters of unwanted physical effects in tomographic imaging, even with truncated projections.

Full data consistency conditions for cone-beam projections with sources on a plane.

Cone-beam consistency conditions (also known as range conditions) are mathematical relationships between different cone-beam projections, and they therefore describe the redundancy or overlap of information between projections. These redundancies have often been exploited for applications in image reconstruction. In this work we describe new consistency conditions for cone-beam projections whose source positions lie on a plane. A further restriction is that the target object must not intersect this plane. The conditions require that moments of the cone-beam projections be polynomial functions of the source positions, with some additional constraints on the coefficients of the polynomials. A precise description of the consistency conditions is that the four parameters of the cone-beam projections (two for the detector, two for the source position) can be expressed with just three variables, using a certain formulation involving homogeneous polynomials. The main contribution of this work is our demonstration that these conditions are not only necessary, but also sufficient. Thus the consistency conditions completely characterize all redundancies, so no other independent conditions are possible and in this sense the conditions are full. The idea of the proof is to use the known consistency conditions for 3D parallel projections, and to then apply a 1996 theorem of Edholm and Danielsson that links parallel to cone-beam projections. The consistency conditions are illustrated with a simulation example.

Planogram rebinning with the frequency-distance relationship.

We present an efficient rebinning algorithm for positron emission tomography (PET) systems with panel detectors. The rebinning algorithm is derived in the planogram coordinate system which is the native data format for PET systems with panel detectors and is the 3-D extension of the 2-D linogram transform developed by Edholm. Theoretical error bounds and numerical results are included.

Cone-beam reconstruction using the backprojection of locally filtered projections.

This paper describes a flexible new methodology for accurate cone beam reconstruction with source positions on a curve (or set of curves). The inversion formulas employed by this methodology are based on first backprojecting a simple derivative in the projection space and then applying a Hilbert transform inversion in the image space. The local nature of the projection space filtering distinguishes this approach from conventional filtered-backprojection methods. This characteristic together with a degree of flexibility in choosing the direction of the Hilbert transform used for inversion offers two important features for the design of data acquisition geometries and reconstruction algorithms. First, the size of the detector necessary to acquire sufficient data for accurate reconstruction of a given region is often smaller than that required by previously documented approaches. In other words, more data truncation is allowed. Second, redundant data can be incorporated for the purpose of noise reduction. The validity of the inversion formulas along with the application of these two properties are illustrated with reconstructions from computer simulated data. In particular, in the helical cone beam geometry, it is shown that 1) intermittent transaxial truncation has no effect on the reconstruction in a central region which means that wider patients can be accommodated on existing scanners, and more importantly that radiation exposure can be reduced for region of interest imaging and 2) at maximum pitch the data outside the Tam-Danielsson window can be used to reduce image noise and thereby improve dose utilization. Furthermore, the degree of axial truncation tolerated by our approach for saddle trajectories is shown to be larger than that of previous methods.

A two-step Hilbert transform method for 2D image reconstruction.

The paper describes a new accurate two-dimensional (2D) image reconstruction method consisting of two steps. In the first step, the backprojected image is formed after taking the derivative of the parallel projection data. In the second step, a Hilbert filtering is applied along certain lines in the differentiated backprojection (DBP) image. Formulae for performing the DBP step in fanbeam geometry are also presented. The advantage of this two-step Hilbert transform approach is that in certain situations, regions of interest (ROIs) can be reconstructed from truncated projection data. Simulation results are presented that illustrate very similar reconstructed image quality using the new method compared to standard filtered backprojection, and that show the capability to correctly handle truncated projections. In particular, a simulation is presented of a wide patient whose projections are truncated laterally yet for which highly accurate ROI reconstruction is obtained.

Inversion of the 3D exponential x-ray transform for a half equatorial band and other semi-circular geometries.

This work presents new mathematical results on the inversion of the exponential x-ray transform. It is shown that a reconstruction formula can be obtained for any dataset whose projection directions consist of a union of half great circles on the unit sphere. A basic example of such a dataset is the semi-equatorial band. The discussion in the paper is mostly focused on this example. The reconstruction formula takes the form of a Neumann (geometric) series and is both exact and stable. The exponential x-ray transform has been mainly studied in SPECT imaging. In this context, our results demonstrate mathematically that fully 3D image reconstruction in SPECT with non-zero attenuation does not always require symmetric datasets (opposing views).

Image reconstruction from fan-beam projections on less than a short scan.

This work is concerned with 2D image reconstruction from fan-beam projections. It is shown that exact and stable reconstruction of a given region-of-interest in the object does not require all lines passing through the object to be measured. Complete (non-truncated) fan-beam projections provide sufficient information for reconstruction when 'every line passing through the region-of-interest intersects the vertex path in a non-tangential way'. The practical implications of this condition are discussed and a new filtered-backprojection algorithm is derived for reconstruction. Experiments with computer-simulated data are performed to support the mathematical results.