Filtered backprojection formula for exact image reconstruction from cone-beam data along a general scanning curve

Simple quality assurance based on filtered back projection for geometrical/irradiation accuracy in single-isocenter multiple-target stereotactic radiotherapy

In single-isocenter multiple-target stereotactic radiotherapy (SIMT-SRT), it is difficult to evaluate both the geometrical accuracy and absorbed dose measurement when irradiating off-isocenter targets. This study aimed to develop a simple quality assurance (QA) method to evaluate off-isocenter irradiation position accuracy in SIMT-SRT and compare its feasibility with that of a commercial device. First, we created two types of inserts and metallic balls with a diameter of 5 mm to be inserted into a commercially available phantom (SIMT phantom). Second, we developed a dedicated analysis software using Python for the Winston-Lutz test (WLT). Third, an image processing software, including the filtered back-projection algorithm, was developed to analyze the images obtained using an electronic portal imaging device (EPID). Fourth, the feasibility of our method was evaluated by comparing it with the results of WLT using two commercially available phantoms: WL-QA and MultiMet-WL cubes. Notably, 92% of the results in one-dimensional deviations were within 0.26 mm (EPID pixel width). The correlation coefficients were 0.52, 0.92, and 0.96 in the left-right, superior-inferior, and anterior-posterior directions, respectively. In the WLT, a maximum two-dimensional deviation of 0.70 mm was detected in our method, while the deviation in the other method was within 0.5 mm. The advantage of our method is that it can evaluate the geometrical accuracy at any gantry angle during dynamic rotation irradiation using a filtered back-projection algorithm, even if the target is located off the isocenter. Our method can perform WLT at arbitrary positions and is suitable for the QA of dynamic rotation irradiation using an EPID.

Tensor framelet based iterative image reconstruction algorithm for low-dose multislice helical CT

In this study, we investigate the feasibility of improving the imaging quality for low-dose multislice helical computed tomography (CT) via iterative reconstruction with tensor framelet (TF) regularization. TF based algorithm is a high-order generalization of isotropic total variation regularization. It is implemented on a GPU platform for a fast parallel algorithm of X-ray forward band backward projections, with the flying focal spot into account. The solution algorithm for image reconstruction is based on the alternating direction method of multipliers or the so-called split Bregman method. The proposed method is validated using the experimental data from a Siemens SOMATOM Definition 64-slice helical CT scanner, in comparison with FDK, the Katsevich and the total variation (TV) algorithm. To test the algorithm performance with low-dose data, ACR and Rando phantoms were scanned with different dosages and the data was equally undersampled with various factors. The proposed method is robust for the low-dose data with 25% undersampling factor. Quantitative metrics have demonstrated that the proposed algorithm achieves superior results over other existing methods.

CT Scanning Imaging Method Based on a Spherical Trajectory

In industrial computed tomography (CT), the mismatch between the X-ray energy and the effective thickness makes it difficult to ensure the integrity of projection data using the traditional scanning model, because of the limitations of the object's complex structure. So, we have developed a CT imaging method that is based on a spherical trajectory. Considering an unrestrained trajectory for iterative reconstruction, an iterative algorithm can be used to realise the CT reconstruction of a spherical trajectory for complete projection data only. Also, an inclined circle trajectory is used as an example of a spherical trajectory to illustrate the accuracy and feasibility of this new scanning method. The simulation results indicate that the new method produces superior results for a larger cone-beam angle, a limited angle and tabular objects compared with traditional circle trajectory scanning.

Extended ellipse-line-ellipse trajectory for long-object cone-beam imaging with a mounted C-arm system

Recent reports show that three-dimensional cone-beam (CB) imaging with a floor-mounted (or ceiling-mounted) C-arm system has become a valuable tool in interventional radiology. Currently, a circular short scan is used for data acquisition, which inevitably yields CB artifacts and a short coverage in the direction of the patient table. To overcome these two limitations, a more sophisticated data acquisition geometry is needed. This geometry should be complete in terms of Tuy's condition and should allow continuous scanning, while being compatible with the mechanical constraints of mounted C-arm systems. Additionally, the geometry should allow accurate image reconstruction from truncated data. One way to ensure such a feature is to adopt a trajectory that provides full R-line coverage within the field-of-view (FOV). An R-line is any segment of line that connects two points on a source trajectory, and the R-line coverage is the set of points that belong to an R-line. In this work, we propose a novel geometry called the extended ellipse-line-ellipse (ELE) for long-object imaging with a mounted C-arm system. This trajectory is built from modules consisting of two elliptical arcs connected by a line. We demonstrate that the extended ELE can be configured in many ways so that full R-line coverage is guaranteed. Both tight and relaxed parametric settings are presented. All results are supported by extensive mathematical proofs provided in appendices. Our findings make the extended ELE trajectory attractive for axially-extended FOV imaging in interventional radiology.

A Novel Iterative CT Reconstruction Approach Based on FBP Algorithm

The Filtered Back-Projection (FBP) algorithm and its modified versions are the most important techniques for CT (Computerized tomography) reconstruction, however, it may produce aliasing degradation in the reconstructed images due to projection discretization. The general iterative reconstruction (IR) algorithms suffer from their heavy calculation burden and other drawbacks. In this paper, an iterative FBP approach is proposed to reduce the aliasing degradation. In the approach, the image reconstructed by FBP algorithm is treated as the intermediate image and projected along the original projection directions to produce the reprojection data. The difference between the original and reprojection data is filtered by a special digital filter, and then is reconstructed by FBP to produce a correction term. The correction term is added to the intermediate image to update it. This procedure can be performed iteratively to improve the reconstruction performance gradually until certain stopping criterion is satisfied. Some simulations and tests on real data show the proposed approach is better than FBP algorithm or some IR algorithms in term of some general image criteria. The calculation burden is several times that of FBP, which is much less than that of general IR algorithms and acceptable in the most situations. Therefore, the proposed algorithm has the potential applications in practical CT systems.

A novel scheme to design the filter for CT reconstruction using FBP algorithm

Background

The Filtered Back-Projection (FBP) algorithm is the most important technique for computerized tomographic (CT) imaging, in which the ramp filter plays a key role. FBP algorithm had been derived using the continuous system model. However, it has to be discretized in practical applications, which necessarily produces distortion in the reconstructed images.

Methods

A novel scheme is proposed to design the filters to substitute the standard ramp filter to improve the reconstruction performance for parallel beam tomography. The design scheme is presented under the discrete image model and discrete projection environment. The designs are achieved by constrained optimization procedures. The designed filter can be regarded as the optimal filter for the corresponding parameters in some ways.

Results

Some filters under given parameters (such as image size and scanning angles) have been designed. The performance evaluation of CT reconstruction shows that the designed filters are better than the ramp filter in term of some general criteria.

Conclusions

The 2-D or 3-D FBP algorithms for fan beam tomography used in most CT systems, are obtained by modifying the FBP algorithm for parallel beam tomography. Therefore, the designed filters can be used for fan beam tomography and have potential applications in practical CT systems.

Cone-beam reconstruction for the two-circles-plus-one-line trajectory

The Kodak Image Station In-Vivo FX has an x-ray module with cone-beam configuration for radiographic imaging but lacks the functionality of tomography. To introduce x-ray tomography into the system, we choose the two-circles-plus-one-line trajectory by mounting one translation motor and one rotation motor. We establish a reconstruction algorithm by applying the M-line reconstruction method. Numerical studies and preliminary physical phantom experiment demonstrate the feasibility of the proposed design and reconstruction algorithm.

A general local reconstruction approach based on a truncated hilbert transform

Exact image reconstruction from limited projection data has been a central topic in the computed tomography (CT) field. In this paper, we present a general region-of-interest/volume-of-interest (ROI/VOI) reconstruction approach using a truly truncated Hilbert transform on a line-segment inside a compactly supported object aided by partial knowledge on one or both neighboring intervals of that segment. Our approach and associated new data sufficient condition allows the most flexible ROI/VOI image reconstruction from the minimum account of data in both the fan-beam and cone-beam geometry. We also report primary numerical simulation results to demonstrate the correctness and merits of our finding. Our work has major theoretical potentials and innovative practical applications.

Exact interior reconstruction with cone-beam CT

Using the backprojection filtration (BPF) and filtered backprojection (FBP) approaches, respectively, we prove that with cone-beam CT the interior problem can be exactly solved by analytic continuation. The prior knowledge we assume is that a volume of interest (VOI) in an object to be reconstructed is known in a subregion of the VOI. Our derivations are based on the so-called generalized PI-segment (chord). The available projection onto convex set (POCS) algorithm and singular value decomposition (SVD) method can be applied to perform the exact interior reconstruction. These results have many implications in the CT field and can be extended to other tomographic modalities, such as SPECT/PET, MRI.

Cone-beam composite-circling scan and exact image reconstruction for a quasi-short object

Here we propose a cone-beam composite-circling mode to solve the quasi-short object problem, which is to reconstruct a short portion of a long object from longitudinally truncated cone-beam data involving the short object. In contrast to the saddle curve cone-beam scanning, the proposed scanning mode requires that the X-ray focal spot undergoes a circular motion in a plane facing the short object, while the X-ray source is rotated in the gantry main plane. Because of the symmetry of the proposed mechanical rotations and the compatibility with the physiological conditions, this new mode has significant advantages over the saddle curve from perspectives of both engineering implementation and clinical applications. As a feasibility study, a backprojection filtration (BPF) algorithm is developed to reconstruct images from data collected along a composite-circling trajectory. The initial simulation results demonstrate the correctness of the proposed exact reconstruction method and the merits of the proposed mode.

Inverse Fourier Transform in the Gamma Coordinate System

This paper provides auxiliary results for our general scheme of computed tomography. In 3D parallel-beam geometry, we first demonstrate that the inverse Fourier transform in different coordinate systems leads to different reconstruction formulas and explain why the Radon formula cannot directly work with truncated projection data. Also, we introduce a gamma coordinate system, analyze its properties, compute the Jacobian of the coordinate transform, and define weight functions for the inverse Fourier transform assuming a simple scanning model. Then, we generate Orlov's theorem and a weighted Radon formula from the inverse Fourier transform in the new system. Furthermore, we present the motion equation of the frequency plane and the conditions for sharp points of the instantaneous rotation axis. Our analysis on the motion of the frequency plane is related to the Frenet-Serret theorem in the differential geometry.

Exact interior reconstruction from truncated limited-angle projection data

Using filtered backprojection (FBP) and an analytic continuation approach, we prove that exact interior reconstruction is possible and unique from truncated limited-angle projection data, if we assume a prior knowledge on a subregion or subvolume within an object to be reconstructed. Our results show that (i) the interior region-of-interest (ROI) problem and interior volume-of-interest (VOI) problem can be exactly reconstructed from a limited-angle scan of the ROI/VOI and a 180 degree PI-scan of the subregion or subvolume and (ii) the whole object function can be exactly reconstructed from nontruncated projections from a limited-angle scan. These results improve the classical theory of Hamaker et al. (1980).

A BPF-FBP tandem algorithm for image reconstruction in reverse helical cone-beam CT

PURPOSE

Reverse helical cone-beam computed tomography (CBCT) is a scanning configuration for potential applications in image-guided radiation therapy in which an accurate anatomic image of the patient is needed for image-guidance procedures. The authors previously developed an algorithm for image reconstruction from nontruncated data of an object that is completely within the reverse helix. The purpose of this work is to develop an image reconstruction approach for reverse helical CBCT of a long object that extends out of the reverse helix and therefore constitutes data truncation.

METHODS

The proposed approach comprises of two reconstruction steps. In the first step, a chord-based backprojection-filtration (BPF) algorithm reconstructs a volumetric image of an object from the original cone-beam data. Because there exists a chordless region in the middle of the reverse helix, the image obtained in the first step contains an unreconstructed central-gap region. In the second step, the gap region is reconstructed by use of a Pack-Noo-formula-based filteredback-projection (FBP) algorithm from the modified cone-beam data obtained by subtracting from the original cone-beam data the reprojection of the image reconstructed in the first step.

RESULTS

The authors have performed numerical studies to validate the proposed approach in image reconstruction from reverse helical cone-beam data. The results confirm that the proposed approach can reconstruct accurate images of a long object without suffering from data-truncation artifacts or cone-angle artifacts.

CONCLUSIONS

They developed and validated a BPF-FBP tandem algorithm to reconstruct images of a long object from reverse helical cone-beam data. The chord-based BPF algorithm was utilized for converting the long-object problem into a short-object problem. The proposed approach is applicable to other scanning configurations such as reduced circular sinusoidal trajectories.

Exact image reconstruction with triple-source saddle-curve cone-beam scanning

In this paper, we propose an exact shift-invariant filtered backprojection (FBP) algorithm for triple-source saddle-curve cone-beam CT. In this imaging geometry, the x-ray sources are symmetrically positioned along a circle, and the trajectory of each source is a saddle curve. Then, we extend Yang's formula from the single-source case to the triple-source case. The saddle curves can be divided into four parts to yield four datasets. Each of them contains three data segments associated with different saddle curves, respectively. Images can be reconstructed on the planes orthogonal to the z-axis. Each plane intersects the trajectories at six points (or three points at the two ends) which can be used to define the filtering directions. Then, we discuss the properties of these curves and study the case of 2N+1 sources (N>or=2). A necessary condition and a sufficient condition are given to find efficient curves. Finally, we perform numerical simulations to demonstrate the feasibility of our triple-source saddle-curve approach. The results show that the triple-source geometry is advantageous for high temporal resolution imaging, especially important for cardiac imaging and small animal imaging.

Exact image reconstruction with triple-source saddle-curve cone-beam scanning

In this paper, we propose an exact shift-invariant filtered backprojection (FBP) algorithm for triple-source saddle-curve cone-beam CT. In this imaging geometry, the x-ray sources are symmetrically positioned along a circle, and the trajectory of each source is a saddle curve. Then, we extend Yang's formula from the single-source case to the triple-source case. The saddle curves can be divided into four parts to yield four datasets. Each of them contains three data segments associated with different saddle curves, respectively. Images can be reconstructed on the planes orthogonal to the z-axis. Each plane intersects the trajectories at six points (or three points at the two ends) which can be used to define the filtering directions. Then, we discuss the properties of these curves and study the case of 2N+1 sources (N>or=2). A necessary condition and a sufficient condition are given to find efficient curves. Finally, we perform numerical simulations to demonstrate the feasibility of our triple-source saddle-curve approach. The results show that the triple-source geometry is advantageous for high temporal resolution imaging, especially important for cardiac imaging and small animal imaging.

Knowledge-based dynamic volumetric cardiac computed tomography with saddle curve trajectory

Motion artifact is still a major issue in cardiac computed tomography because the current motion correction and electrocardiogram gating techniques have not fully addressed this problem. The image quality can be significantly improved by using information about the actual state of the heart and an exact reconstruction algorithm. We propose to extend a cardiac computed tomographic technique, using the knowledge of the volume and the relation between the state and the phase of the heart, to a saddle curve trajectory. This will optimize the image quality by reducing the artifacts resulting from approximate reconstruction and solve the long-object problem. Necessary background is provided, and the effectiveness of the algorithms is demonstrated in numerical simulations with the dynamic thorax phantom.

Exact image reconstruction for a circle and line trajectory with a gantry tilt

We investigate image reconstruction with a circle and line trajectory with a tilted gantry. We derive new equations for reconstruction from the line data, such as equations of filtering lines, range of filtering lines and range of the line scan. We analyze the detector requirements and show that the line scan does not impose extra requirements on the cylindrical detector size with our algorithm, that is, the axial truncation of the filtering lines does not occur. We discuss full-scan and short-scan versions of the algorithm. Evaluation of our algorithm uses simulated and real 256-slice data.

Exact reconstruction of volumetric images in reverse helical cone-beam CT

Helical scanning configuration has been used widely in diagnostic cone-beam computed tomography (CBCT) for acquiring data sufficient for exact image reconstruction over an extended volume. In image-guided radiation therapy (IGRT) and other applications of CBCT, it can be difficult, if not impossible, to implement mechanically a multiple-turn helical trajectory on the imaging systems due to hardware constraints. However, imaging systems in these applications often allow for the implementation of a reverse helical trajectory in which the rotation direction changes between two consecutive turns. Because the reverse helical trajectory satisfies Tuy's condition, when projections of the imaged object are nontruncated, it yields data sufficient for exact image reconstruction within the reverse helix volume. The recently developed chord-based algorithms such as the backprojection filtration (BPF) algorithm can readily be applied to reconstructing images on chords of a reverse helical trajectory, and they can thus reconstruct an image within a volume covered by the chords. Conversely, the chord-based algorithms cannot reconstruct images within regions that are not intersected by chords. In a reverse helix volume, as shown below, chordless regions exist in which no images can thus be reconstructed by use of the chord-based algorithms. In this work, based upon Pack-Noo's formula, a shift-invariant filtered backprojection (FBP) algorithm is derived for exact image reconstruction within the reverse helix volume, including the chordless region. Numerical studies have also been conducted to demonstrate the chordless region in a reverse helix volume and to validate the FBP algorithm for image reconstruction within the chordless region. Results of the numerical studies confirm that the FBP algorithm can exactly reconstruct an image within the entire reverse helix volume, including the chordless region. It is relatively straightforward to extend the FBP algorithm to reconstruct images for general trajectories, including reverse helical trajectories with variable pitch, tilted axis, and/or additional segments between turns.

An outlook on x-ray CT research and development

Over the past decade, computed tomography (CT) theory, techniques and applications have undergone a rapid development. Since CT is so practical and useful, undoubtedly CT technology will continue advancing biomedical and non-biomedical applications. In this outlook article, we share our opinions on the research and development in this field, emphasizing 12 topics we expect to be critical in the next decade: analytic reconstruction, iterative reconstruction, local/interior reconstruction, flat-panel based CT, dual-source CT, multi-source CT, novel scanning modes, energy-sensitive CT, nano-CT, artifact reduction, modality fusion, and phase-contrast CT. We also sketch several representative biomedical applications.

A General Scheme for Velocity Tomography

With the rapid development of x-ray source and detector technologies, multi-source scanners become a hot topic in the CT field, which can acquire several projections simultaneously. Aided with the ECG-gating technique, the multi-source scanner can collect sufficient projections to reconstruct one or more specific phases of a beating heart. Hence, we are motivated to develop velocity tomography as a new dynamic imaging mode to recover the velocity field from the projections. First, we derive a velocity field constraint equation subject to the mass conservation. Then, we present a two-step general scheme to estimate the velocity field. The first step directly or indirectly computes partial derivatives. The second step iteratively determines the velocity field subject to the constraint equation and other conditions. Finally, we describe numerical experiments in the fan-beam geometry to demonstrate the correctness and utility of our scheme.

Exact and approximate cone-beam reconstruction algorithms for C-arm based cone-beam CT using a two-concentric-arc source trajectory

In this paper, we present shift-invariant filtered backprojection (FBP) cone-beam image reconstruction algorithms for a cone-beam CT system based on a clinical C-arm gantry. The source trajectory consists of two concentric arcs which is complete in the sense that the Tuy data sufficiency condition is satisfied. This scanning geometry is referred to here as a CC geometry (each arc is shaped like the letter "C"). The challenge for image reconstruction for the CC geometry is that the image volume is not well populated by the familiar doubly measured (DM) lines. Thus, the well-known DM-line based image reconstruction schemes are not appropriate for the CC geometry. Our starting point is a general reconstruction formula developed by Pack and Noo which is not dependent on the existence of DM-lines. For a specific scanning geometry, the filtering lines must be carefully selected to satisfy the Pack-Noo condition for mathematically exact reconstruction. The new points in this paper are summarized here. (1) A mathematically exact cone-beam reconstruction algorithm was formulated for the CC geometry by utilizing the Pack-Noo image reconstruction scheme. One drawback of the developed exact algorithm is that it does not solve the long-object problem. (2) We developed an approximate image reconstruction algorithm by deforming the filtering lines so that the long object problem is solved while the reconstruction accuracy is maintained. (3) In addition to numerical phantom experiments to validate the developed image reconstruction algorithms, we also validate our algorithms using physical phantom experiments on a clinical C-arm system.

Noise properties of chord-image reconstruction

Recently, there has been much progress in algorithm development for image reconstruction in cone-beam computed tomography (CT). Current algorithms, including the chord-based algorithms, now accept minimal data sets for obtaining images on volume regions-of-interest (ROIs) thereby potentially allowing for reduction of X-ray dose in diagnostic CT. As these developments are relatively new, little effort has been directed at investigating the response of the resulting algorithm implementations to physical factors such as data noise. In this paper, we perform an investigation on the noise properties of ROI images reconstructed by using chord-based algorithms for different scanning configurations. We find that, for the cases under study, the chord-based algorithms yield images with comparable quality. Additionally, it is observed that, in many situations, large data sets contain extraneous data that may not reduce the ROI-image variances.

A new scheme for view-dependent data differentiation in fan-beam and cone-beam computed tomography

In computed tomography, analytical fan-beam (FB) and cone-beam (CB) image reconstruction often involves a view-dependent data differentiation. The implementation of this differentiation step is critical in terms of resolution and image quality. In this work, we present a new differentiation scheme that is robust to changes in the data acquisition geometry and to coarse view sampling. Our scheme was compared to two previously suggested methods, which we call the direct scheme and the chain-rule scheme. Image reconstructions were performed from computer-simulated data of the Shepp-Logan phantom, the FORBILD thorax phantom and a modified FORBILD head phantom. For FB reconstruction, we investigated three acquisition geometries: a circular, an ellipse-shaped and a square-shaped trajectory. For CB reconstruction, the circle-plus-line trajectory was considered. Image comparison showed that the new scheme performs consistently well when varying the scenario, in both FB and CB geometry, unlike the other two schemes.

Lambda tomography with discontinuous scanning trajectories

Lambda tomography (LT) is a well-known local reconstruction technology to reduce the radiation dose or accommodate a limited imaging geometry. After a theoretical analysis of the so-called Calderon operator (CO), the necessary conditions for exact LT reconstruction are presented in terms of the 2D and 3D COs. Based on our previous results on LT, a general scheme is proposed to construct exact LT formulae in terms of the 2D CO with multiple segment trajectories. Every 2D formula has a corresponding 3D cone-beam formula in the Feldkamp framework in terms of the 2D CO which was illustrated in a triple-segment case. Our simulation results verify the correctness and demonstrate the merits of the proposed scheme.

Local cone-beam tomography image reconstruction on chords

We develop reconstruction algorithms for local cone-beam tomography for use with generalized scanning trajectories. The algorithms are grounded theoretically in a recently developed chord-based theory for exact image reconstruction and principles of lambda tomography. Being chord based, they are distinct mathematically and conceptually from conventional local tomography reconstruction algorithms. The salient feature of our algorithms is that they permit reconstruction of discontinuities in the profiles of the object function along chords. By consideration of all possible chords, a 3D image that describes the locations of object discontinuities can be reconstructed. Results from microlocal analysis are applied for understanding the object features that can be reconstructed stably by use of the algorithms. A computer-simulation study is conducted to demonstrate the algorithms and compare their performance with an existing algorithm.

Handling data redundancy in helical cone beam reconstruction with a cone-angle-based window function and its asymptotic approximation

A cone-angle-based window function is defined in this manuscript for image reconstruction using helical cone beam filtered backprojection (CB-FBP) algorithms. Rather than defining the window boundaries in a two-dimensional detector acquiring projection data for computed tomographic imaging, the cone-angle-based window function deals with data redundancy by selecting rays with the smallest cone angle relative to the reconstruction plane. To be computationally efficient, an asymptotic approximation of the cone-angle-based window function is also given and analyzed in this paper. The benefit of using such an asymptotic approximation also includes the avoidance of functional discontinuities that cause artifacts in reconstructed tomographic images. The cone-angle-based window function and its asymptotic approximation provide a way, equivalent to the Tam-Danielsson-window, for helical CB-FBP reconstruction algorithms to deal with data redundancy, regardless of where the helical pitch is constant or dynamically variable during a scan. By taking the cone-parallel geometry as an example, a computer simulation study is conducted to evaluate the proposed window function and its asymptotic approximation for helical CB-FBP reconstruction algorithm to handle data redundancy. The computer simulated Forbild head and thorax phantoms are utilized in the performance evaluation, showing that the proposed cone-angle-based window function and its asymptotic approximation can deal with data redundancy very well in cone beam image reconstruction from projection data acquired along helical source trajectories. Moreover, a numerical study carried out in this paper reveals that the proposed cone-angle-based window function is actually equivalent to the Tam-Danielsson-window, and rigorous mathematical proofs are being investigated.

Approximate and exact cone-beam reconstruction with standard and non-standard spiral scanning

The long object problem is practically important and theoretically challenging. To solve the long object problem, spiral cone-beam CT was first proposed in 1991, and has been extensively studied since then. As a main feature of the next generation medical CT, spiral cone-beam CT has been greatly improved over the past several years, especially in terms of exact image reconstruction methods. Now, it is well established that volumetric images can be exactly and efficiently reconstructed from longitudinally truncated data collected along a rather general scanning trajectory. Here we present an overview of some key results in this area.

Practical cone-beam lambda tomography

As a potentially important technology for medical x-ray computed tomography (CT), lambda tomography (LT) is to reconstruct a gradient-like image only from local projection data. Based on our recently derived exact fan-beam LT formula, [H. Y. Gu and G. Wang, Int. J. Biomed. Imaging 2006(1), 1-9 (2006)] here we propose a practical cone-beam LT algorithm for LT reconstruction from local data collected along an arbitrary smooth three-dimensional curve. A key step in our algorithm is to determine an appropriate vector perpendicular to the line connecting the x-ray source and an image point. The algorithm is implemented assuming an equispatial planar detector and a nonstandard spiral trajectory. The numerical simulation results demonstrate the merits of our method.

Cone-beam mammo-computed tomography from data along two tilting arcs

Over the past several years there has been an increasing interest in cone-beam computed tomography (CT) for breast imaging. In this article, we propose a new scheme for theoretically exact cone-beam mammo-CT and develop a corresponding Katsevich-type reconstruction algorithm. In our scheme, cone-beam scans are performed along two tilting arcs to collect a sufficient amount of information for exact reconstruction. In our algorithm, cone-beam data are filtered in a shift-invariant fashion and then weighted backprojected into the three-dimensional space for the final reconstruction. Our approach has several desirable features, including tolerance of axial data truncation, efficiency in sequential/parallel implementation, and accuracy for quantitative analysis. We also demonstrate the system performance and clinical utility of the proposed technique in numerical simulations.

View-independent reconstruction algorithms for cone beam CT with general saddle trajectory

In Yang et al (2006 Phys. Med. Biol. 51 1157-72), an exact filtered backprojection (FBP) reconstruction algorithm was proposed for cone beam tomography with saddle trajectory based on the seminal works of Pack and Noo (2005a Inverse Problems 21 1105-20; 2005b 8th Int. Meeting on Fully 3D Reconstruction in Radiology and Nuclear Medicine (Salt Lake City) ed F Noo, H Kudo and L G Zeng pp 287-90). However, the artefacts due to discretization and/or sampling errors in the reconstructed images by this method were still visible, especially when the pitch is large. In this paper, two view-independent (VI) algorithms, which are similar to the FDK-type algorithms (Feldkamp et al 1984 J. Opt. Soc. Am. A 1 612-19), are proposed for planar detector geometry. The first VI algorithm involves two filtered projections and a small additional term (two-dimensional (2D) Radon transform term). One of the filtered projections is obtained by ramp filtering (as in the FDK algorithm for circular trajectory) and the other one is obtained by Hilbert transform. The 2D Radon transform term is just like the term which was first derived by Hu (1996 Scanning 18 572-81) for a circular trajectory. The second VI algorithm involves only one filtered projection term, which is obtained by differentiation followed by Hilbert transform and the 2D Radon transform term. Both algorithms involve only one backprojection step with a weighting factor as in the FDK algorithm. The simulation studies show that the pixel values of the reconstructed images by the VI algorithms are more accurate than those by the original view differencing (VD) algorithm, the streak artefacts are also reduced, and their computational times are comparable to that of the original VD algorithm. We also generalize the concept of saddle trajectory and the corresponding reconstruction algorithm. The generalized algorithm is also theoretically exact, has a shift-invariant FBP structure, and does not depend on the concept of pi-line.

Data consistency based translational motion artifact reduction in fan-beam CT

A basic assumption in the classic computed tomography (CT) theory is that an object remains stationary in an entire scan. In biomedical CT/micro-CT, this assumption is often violated. To produce high-resolution images, such as for our recently proposed clinical micro-CT (CMCT) prototype, it is desirable to develop a precise motion estimation and image reconstruction scheme. In this paper, we first extend the Helgason-Ludwig consistency condition (HLCC) from parallel-beam to fan-beam geometry when an object is subject to a translation. Then, we propose a novel method to estimate the motion parameters only from sinograms based on the HLCC. To reconstruct the moving object, we formulate two generalized fan-beam reconstruction methods, which are in filtered backprojection and backprojection filtering formats, respectively. Furthermore, we present numerical simulation results to show that our approach is accurate and robust.

Transform-based backprojection for volume reconstruction of large format electron microscope tilt series

Alignment of the individual images of a tilt series is a critical step in obtaining high-quality electron microscope reconstructions. We report on general methods for producing good alignments, and utilizing the alignment data in subsequent reconstruction steps. Our alignment techniques utilize bundle adjustment. Bundle adjustment is the simultaneous calculation of the position of distinguished markers in the object space and the transforms of these markers to their positions in the observed images, along the bundle of particle trajectories along which the object is projected to each EM image. Bundle adjustment techniques are general enough to encompass the computation of linear, projective or nonlinear transforms for backprojection, and can compensate for curvilinear trajectories through the object, sample warping, and optical aberration. We will also report on new reconstruction codes and describe our results using these codes.

Exact BPF and FBP algorithms for nonstandard saddle curves

A hot topic in cone-beam CT research is exact cone-beam reconstruction from a general scanning trajectory. Particularly, a nonstandard saddle curve attracts attention, as this construct allows the continuous periodic scanning of a volume-of-interest (VOI). Here we evaluate two algorithms for reconstruction from data collected along a nonstandard saddle curve, which are in the filtered backprojection (FBP) and backprojection filtration (BPF) formats, respectively. Both the algorithms are implemented in a chord-based coordinate system. Then, a rebinning procedure is utilized to transform the reconstructed results into the natural coordinate system. The simulation results demonstrate that the FBP algorithm produces better image quality than the BPF algorithm, while both the algorithms exhibit similar noise characteristics.

A differentiable Shepp–Logan phantom and its applications in exact cone-beam CT

Recently, several exact cone-beam reconstruction algorithms, such as the generalized filtered-backprojection (FBP) and backprojection-filtration (BPF) methods, have been developed to solve the long object problem. Although the well-known 3D Shepp-Logan phantom (SLP) is often used to validate these algorithms, it is deficient due to the discontinuity of the SLP. In this paper, we first construct a differentiable polynomial function to approximate the unit rectangular function on [-1, 1]. Then, we use this function to obtain a differentiable ellipsoid phantom, whose x-ray transform is differentiable for any smooth scanning trajectory. Finally, we propose a differentiable Shepp-Logan phantom (DSLP) for numerical simulation of the exact cone-beam CT algorithms. Our numerical simulation shows that the reconstructed DSLP has a better image quality than the reconstructed SLP, and is complementary to the traditional SLP for evaluation of the exact cone-beam CT algorithms.

A unified framework for exact cone-beam reconstruction formulas

In this paper, we present concise proofs of several recently developed exact cone-beam reconstruction methods in the Tuy inversion framework, including both filtered-backprojection and backprojection-filtration formulas in the cases of standard spiral, nonstandard spiral, and more general scanning loci. While a similar proof of the Katsevich formula was previously reported, we present a new proof of the Zou and Pan backprojection-filtration formula. Our proof combines both odd and even data extensions so that only the cone-beam transform itself is utilized in the backprojection-filtration inversion. More importantly, our formulation is valid for general smooth scanning curves, in agreement with an earlier paper from our group [Ye, Zhao, Yu, and Wang, Proc. SPIE 5535, 293-300 (Aug. 6 2004)]. As a consequence of that proof, we obtain a new inversion formula, which is in a two-dimensional filtering backprojection format. A possibility for generalization of the Katsevich filtered-backprojection reconstruction method is also discussed from the viewpoint of this framework.

Image reconstruction for the circle-and-arc trajectory

Proposed is an exact shift-invariant filtered backprojection algorithm for the circle-and-arc trajectory. The algorithm has several important features. First, it allows for the circle to be incomplete. Second, axial truncation of the cone beam data is allowed. Third, the length of the arc is determined only by the region of interest and is independent of the size of the entire object. The algorithm is quite flexible and can be used for even more general trajectories that consist of several circular segments and arcs. The algorithm applies also in the case when the circle (or, circles) is complete. A numerical experiment with the clock phantom demonstrated good image quality.