Derivation and implementation of a cone-beam reconstruction algorithm for nonplanar orbits

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.

On the data acquisition, image reconstruction, cone beam artifacts, and their suppression in axial MDCT and CBCT - A review

PURPOSE

In the clinic, computed tomography (CT) has evolved into an essential modality for diagnostic imaging by multidetector row CT (MDCT) and image guided intervention by cone beam CT (CBCT). Recognizing the increasing importance of axial MDCT/CBCT in clinical and preclinical applications, and the existence of CB artifacts in MDCT/CBCT images, we provide a review of CB artifacts' root causes, rendering mechanisms and morphology, and possible solutions for elimination and/or reduction of the artifacts.

METHODS

By examining the null space in Radon and Fourier domain, the root cause of CB artifacts (i.e., data insufficiency) in axial MDCT/CBCT is analytically investigated, followed by a review of the data sufficiency conditions and the "circle +" source trajectories. The rendering mechanisms and morphology of CB artifacts in axial MDCT/CBCT and their special cases (e.g., half/short scan and full scan with latitudinally displaced detector) are then analyzed, followed by a survey of the potential solutions to suppress the artifacts. The phenomenon of imaged zone indention and its variation over FBP, BPF/DBPF, two-pass and iterative CB reconstruction algorithms and/or schemes are discussed in detail.

RESULTS

An interdomain examination of the null space provides an insightful understanding of the root cause of CB artifacts in axial MDCT/CBCT. The decomposition of CB artifacts rendering mechanisms facilitates understanding of the artifacts' behavior under different conditions and the potential solutions to suppress them. An inspection of the imaged zone intention phenomenon provides guidance on the design and implementation of CB image reconstruction algorithms and schemes for CB artifacts suppression in axial MDCT/CBCT.

CONCLUSIONS

With increasing importance of axial MDCT/CBCT in clinical and preclinical applications, this review article can update the community with in-depth information and clarification on the latest progress in dealing with CB artifacts and thus increase clinical/preclinical confidence.

Recent Advances in X-ray Cone-beam Computed Laminography

X-ray computed tomography is an established volume imaging technique used routinely in medical diagnosis, industrial non-destructive testing, and a wide range of scientific fields. Traditionally, computed tomography uses scanning geometries with a single axis of rotation together with reconstruction algorithms specifically designed for this setup. Recently there has however been increasing interest in more complex scanning geometries. These include so called X-ray computed laminography systems capable of imaging specimens with large lateral dimensions or large aspect ratios, neither of which are well suited to conventional CT scanning procedures. Developments throughout this field have thus been rapid, including the introduction of novel system trajectories, the application and refinement of various reconstruction methods, and the use of recently developed computational hardware and software techniques to accelerate reconstruction times. Here we examine the advances made in the last several years and consider their impact on the state of the art.

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.

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 local shift-variant Fourier model and experimental validation of circular cone-beam computed tomography artifacts

Large field of view cone-beam computed tomography (CBCT) is being achieved using circular source and detector trajectories. These circular trajectories are known to collect insufficient data for accurate image reconstruction. Although various descriptions of the missing information exist, the manifestation of this lack of data in reconstructed images is generally nonintuitive. One model predicts that the missing information corresponds to a shift-variant cone of missing frequency components. This description implies that artifacts depend on the imaging geometry, as well as the frequency content of the imaged object. In particular, objects with a large proportion of energy distributed over frequency bands that coincide with the missing cone will be most compromised. These predictions were experimentally verified by imaging small, localized objects (acrylic spheres, stacked disks) at varying positions in the object space and observing the frequency spectrums of the reconstructions. Measurements of the internal angle of the missing cone agreed well with theory, indicating a right circular cone for points on the rotation axis, and an oblique, circular cone elsewhere. In the former case, the largest internal angle with respect to the vertical axis corresponds to the (half) cone angle of the CBCT system (typically approximately 5 degrees - 7.5 degrees in IGRT). Object recovery was also found to be strongly dependent on the distribution of the object's frequency spectrum relative to the missing cone, as expected. The observed artifacts were also reproducible via removal of local frequency components, further supporting the theoretical model. Larger objects with differing internal structures (cellular polyurethane, solid acrylic) were also imaged and interpreted with respect to the previous results. Finally, small animal data obtained using a clinical CBCT scanner were observed for evidence of the missing cone. This study provides insight into the influence of incomplete data collection on the appearance of objects imaged in large field of view CBCT.

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.

An efficient estimation method for reducing the axial intensity drop in circular cone-beam CT

Reconstruction algorithms for circular cone-beam (CB) scans have been extensively studied in the literature. Since insufficient data are measured, an exact reconstruction is impossible for such a geometry. If the reconstruction algorithm assumes zeros for the missing data, such as the standard FDK algorithm, a major type of resulting CB artifacts is the intensity drop along the axial direction. Many algorithms have been proposed to improve image quality when faced with this problem of data missing; however, development of an effective and computationally efficient algorithm remains a major challenge. In this work, we propose a novel method for estimating the unmeasured data and reducing the intensity drop artifacts. Each CB projection is analyzed in the Radon space via Grangeat's first derivative. Assuming the CB projection is taken from a parallel beam geometry, we extract those data that reside in the unmeasured region of the Radon space. These data are then used as in a parallel beam geometry to calculate a correction term, which is added together with Hu's correction term to the FDK result to form a final reconstruction. More approximations are then made on the calculation of the additional term, and the final formula is implemented very efficiently. The algorithm performance is evaluated using computer simulations on analytical phantoms. The reconstruction comparison with results using other existing algorithms shows that the proposed algorithm achieves a superior performance on the reduction of axial intensity drop artifacts with a high computation efficiency.

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.

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.

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.

A shift-invariant filtered backprojection (FBP) cone-beam reconstruction algorithm for the source trajectory of two concentric circles using an equal weighting scheme

In this paper, a shift-invariant filtered backprojection cone-beam image reconstruction algorithm is derived, based upon Katsevich's general inversion scheme, and validated for the source trajectory of two concentric circles. The source trajectory is complete according to Tuy's data sufficiency condition and is used as the basis for an exact image reconstruction algorithm. The algorithm proceeds according to the following steps. First, differentiate the cone-beam projection data with respect to the detector coordinates and with respect to the source trajectory parameter. The data are then separately filtered along three different orientations in the detector plane with a shift-invariant Hilbert kernel. Eight different filtration groups are obtained via linear combinations of weighted filtered data. Voxel-based backprojection is then carried out from eight sets of view angles, where separate filtered data are backprojected from each set according to the backprojection sets' associated filtration group. The algorithm is first derived for a scanning configuration consisting of two concentric and orthogonal circles. By performing an affine transformation on the image object, the developed image reconstruction algorithm has been generalized to the case where the two concentric circles are not orthogonal. Numerical simulations are presented to validate the reconstruction algorithm and demonstrate the dose advantage of the equal weighting scheme.

Physical evaluation of the weighted Feldkamp algorithms applied to the 256-detector row CT scanner for volumetric cine imaging

RATIONALE AND OBJECTIVES

To improve effective scan time and image quality in cone-beam computed tomography (CT), Parker's weighting function (half scan [HS]-Feldkamp-Davis-Kress [FDK]) extended to a larger range up to 2pi was proposed as new half-scan algorithm (NHS-FDK). We conducted a practical physical evaluation of NHS-FDK and HS-FDK using 256-detector row CT.

MATERIALS AND METHODS

Three types of weighting function (full-scan [FS-FDK], HS-FDK, and NHS-FDK) were evaluated by using 256-detector row CT for five variables, ie, point spread function, image noise, CT number uniformity, Feldkamp artifact, temporal resolution, and clinical evaluation.

RESULTS

Image noise, Feldkamp artifact, and temporal resolution were dependent on weighting function. Image noise magnitude was independent of projection angle for all regions of interest with FS-FDK, but showed a symmetric pattern with projection angle with HS-FDK and NHS-FDK. With regard to temporal resolution, NHS-FDK did not remove the motion artifact in the heart except in such slower motion organs as the pulmonary vessels, whereas HS-FDK reduced the motion artifact in the heart. HS-FDK had an even more incomplete data region in the Radon space than FS-FDK, suggesting that it would provide poor image quality distant to the midplane in the longitudinal direction. In practical testing in human subjects, HS-FDK showed inferior performance in all variables except temporal resolution.

CONCLUSION

Despite its inferiority to FS-FDK for static objects, HS-FDK may be useful in chest imaging. Contrary to previous findings using static images, NHS-FDK failed to show advantages over HS-FDK or FS-FDK in a moving phantom and human subjects.

System design and development of a pinhole SPECT system for quantitative functional imaging of small animals

Recently, small animal imaging by pinhole SPECT has been widely investigated by several researchers. We developed a pinhole SPECT system specially designed for small animal imaging. The system consists of a rotation unit for a small animal and a SPECT camera attached with a pinhole collimator. In order to acquire complete data of the projections, the system has two orbits with angles of 90 degrees and 45 degrees with respect to the object. In this system, the position of the SPECT camera is kept fixed, and the animal is rotated in order to avoid misalignment of the center of rotation (COR). We implemented a three dimensional OSEM algorithm for the reconstruction of data acquired by the system from both the orbitals. A point source experiment revealed no significant COR misalignment using the proposed system. Experiments with a line phantom clearly indicated that our system succeeded in minimizing the misalignment of the COR. We performed a study with a rat and 99mTc-HMDP, an agent for bone scan, and demonstrated a dramatic improvement in the spatial resolution and uniformity achieved by our system in comparison with the conventional Feldkamp algorithm with one set of orbital data.

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 three-dimensional-weighted cone beam filtered backprojection (CB-FBP) algorithm for image reconstruction in volumetric CT—helical scanning

Based on the structure of the original helical FDK algorithm, a three-dimensional (3D)-weighted cone beam filtered backprojection (CB-FBP) algorithm is proposed for image reconstruction in volumetric CT under helical source trajectory. In addition to its dependence on view and fan angles, the 3D weighting utilizes the cone angle dependency of a ray to improve reconstruction accuracy. The 3D weighting is ray-dependent and the underlying mechanism is to give a favourable weight to the ray with the smaller cone angle out of a pair of conjugate rays but an unfavourable weight to the ray with the larger cone angle out of the conjugate ray pair. The proposed 3D-weighted helical CB-FBP reconstruction algorithm is implemented in the cone-parallel geometry that can improve noise uniformity and image generation speed significantly. Under the cone-parallel geometry, the filtering is naturally carried out along the tangential direction of the helical source trajectory. By exploring the 3D weighting's dependence on cone angle, the proposed helical 3D-weighted CB-FBP reconstruction algorithm can provide significantly improved reconstruction accuracy at moderate cone angle and high helical pitches. The 3D-weighted CB-FBP algorithm is experimentally evaluated by computer-simulated phantoms and phantoms scanned by a diagnostic volumetric CT system with a detector dimension of 64 x 0.625 mm over various helical pitches. The computer simulation study shows that the 3D weighting enables the proposed algorithm to reach reconstruction accuracy comparable to that of exact CB reconstruction algorithms, such as the Katsevich algorithm, under a moderate cone angle (4 degrees) and various helical pitches. Meanwhile, the experimental evaluation using the phantoms scanned by a volumetric CT system shows that the spatial resolution along the z-direction and noise characteristics of the proposed 3D-weighted helical CB-FBP reconstruction algorithm are maintained very well in comparison to the FDK-type algorithms. Moreover, the experimental evaluation by clinical data verifies that the proposed 3D-weighted CB-FBP algorithm for image reconstruction in volumetric CT under helical source trajectory meets the challenges posed by diagnostic applications of volumetric CT imaging.

Physical performance evaluation of a 256-slice CT-scanner for four-dimensional imaging

We have developed a prototype 256-slice CT-scanner for four-dimensional (4D) imaging that employs continuous rotations of a cone-beam. Since a cone-beam scan along a circular orbit does not collect a complete set of data to make an exact reconstruction of a volume [three-dimensional (3D) image], it might cause disadvantages or artifacts. To examine effects of the cone-beam data collection on image quality, we have evaluated physical performance of the prototype 256-slice CT-scanner with 0.5 mm slices and compared it to that of a 16-slice CT-scanner with 0.75 mm slices. As a result, we found that image noise, uniformity, and high contrast detectability were independent of z coordinate. A Feldkamp artifact was observed in distortion measurements. Full width at half maximum (FWHM) of slice sensitivity profiles (SSP) increased with z coordinate though it seemed to be caused by other reasons than incompleteness of data. With regard to low contrast detectability, smaller objects were detected more clearly at the midplane (z = 0 mm) than at z = 40 mm, though circular-band like artifacts affected detection. The comparison between the 16-slice and the 256-slice scanners showed better performance for the 16-slice scanner regarding the SSP, low contrast detectability, and distortion. The inferiorities of the 256-slice scanner in other than distortion measurement (Feldkamp artifact) seemed to be partly caused by the prototype nature of the scanner and should be improved in the future scanner. The image noise, uniformity, and high contrast detectability were almost identical for both CTs. The 256-slice scanner was superior to the 16-slice scanner regarding the PSF, though it was caused by the smaller transverse beam width of the 256-slice scanner. In order to compare both scanners comprehensively in terms of exposure dose, noise, slice thickness, and transverse spatial resolution, K=Dsigma2ha3 was calculated, where D was exposure dose (CT dose index), sigma was magnitude of noise, h was slice thickness (FWHM of SSP), and a was transverse spatial resolution (FWHM of PSF). The results showed that the K value was 25% larger for the 16-slice scanner, and that the 256-slice scanner was 1.25 times more effective than the 16-slice scanner at the midplane. The superiority in K value for the 256-slice scanner might be partly caused by decrease of wasted exposure with a wide-angle cone-beam scan. In spite of the several problems of the 256-slice scanner, it took a volume data approximately 1.0 mm (transverse) x 1.3 mm (longitudinal) resolution for a wide field of view (approximately 100 mm long) along the zeta axis in a 1 s scan if resolution was defined by the FWHM of the PSF or the SSP, which should be very useful to take dynamic 3D (4D) images of moving organs.

Image reconstruction for the circle and line trajectory

We propose an exact shift-invariant filtered backprojection algorithm for inversion of the cone beam data in the case when the source trajectory consists of an incomplete circle and a line segment. The algorithm allows for axial truncation of the cone beam data. The length of the line scan 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 also be used for more general trajectories consisting of several (complete or incomplete) circles and line segments. Results of numerical experiments demonstrate good image quality.

An alternative derivation of Katsevich's cone-beam reconstruction formula

In this paper an alternative derivation of Katsevich's cone-beam image reconstruction algorithm is presented. The starting point is the classical Tuy's inversion formula. After (i) using the hidden symmetries of the intermediate functions, (ii) handling the redundant data by weighting them, (iii) changing the weighted average into an integral over the source trajectory parameter, and (iv) imposing an additional constraint on the weighting function, a filtered backprojection reconstruction formula from cone beam projections is derived. The following features are emphasized in the present paper: First, the nontangential condition in Tuy's original data sufficiency conditions has been relaxed. Second, a practical regularization scheme to handle the singularity is proposed. Third, the derivation in the cone beam case is in the same fashion as that in the fan-beam case. Our final cone-beam reconstruction formula is the same as the one discovered by Katsevich in his most recent paper. However, the data sufficiency conditions and the regularization scheme of singularities are different. A detailed comparison between these two methods is presented.

A new weighting scheme for cone-beam helical CT to reduce the image noise

Reducing the patient dose while keeping the image noise at the same level is desired for x-ray CT examinations. In order to achieve the goal, we propose a new weighting scheme taking the validity of the data and redundant data samples into account. The method is evaluated with a new generalized version of the Feldkamp helical reconstruction algorithm. It allows us to enlarge the projection angular range used in reconstruction, and thus, to reduce the image noise by increasing the detector utilization rate to 100% without sacrificing the image quality or z-resolution. This concept can be adapted to other exact or approximate algorithms as far as they use redundant data samples.

A simple derivation and analysis of a helical cone beam tomographic algorithm for long object imaging via a novel definition of region of interest

We derive and analyse a simple algorithm first proposed by Kudo et al (2001 Proc. 2001 Meeting on Fully 3D Image Reconstruction in Radiology and Nuclear Medicine (Pacific Grove, CA) pp 7-10) for long object imaging from truncated helical cone beam data via a novel definition of region of interest (ROI). Our approach is based on the theory of short object imaging by Kudo et al (1998 Phys. Med. Biol. 43 2885-909). One of the key findings in their work is that filtering of the truncated projection can be divided into two parts: one, finite in the axial direction, results from ramp filtering the data within the Tam window. The other, infinite in the z direction, results from unbounded filtering of ray sums over PI lines only. We show that for an ROI defined by PI lines emanating from the initial and final source positions on a helical segment, the boundary data which would otherwise contaminate the reconstruction of the ROI can be completely excluded. This novel definition of the ROI leads to a simple algorithm for long object imaging. The overscan of the algorithm is analytically calculated and it is the same as that of the zero boundary method. The reconstructed ROI can be divided into two regions: one is minimally contaminated by the portion outside the ROI, while the other is reconstructed free of contamination. We validate the algorithm with a 3D Shepp-Logan phantom and a disc phantom.

Temporal resolution and the evaluation of candidate algorithms for four-dimensional CT

The four-dimensional computed tomography ("4D-CT") with area detector has been developed for dynamic volumetric imaging with large longitudinal coverage. In this paper one of the key technologies for 4D-CT development is discussed: Image reconstruction algorithm with high temporal resolution. All of the cone-beam algorithms investigated previously assume that the object is stationary. In this paper a new class of cone-beam problem is addressed: a dynamic volumetric (4-D) imaging. A continuously rotating circular (stationary couch) scanning is employed, and then, a generalized version of the well-known Feldkamp algorithm with the following three steps is performed: (1) applying a weighting function (along the time axis) to projection data, (2) filtering the weighted data along the detector row direction, (3) cone-beam backprojecting of the filtered data along the corresponding x-ray path. The weighting function controls the time center, the temporal resolution, and the image quality. Four weighting functions developed for fan-beam reconstruction were applied to the first step: (a) a constant weight fixed at 0.5 (FS-FDK), (b) feathering both edges of the (time) window (OS-FDK), (c) Parker's weight for a half-scan (HF-FDK), and (d) an extended Parker's weight, which allows us to use a larger range of projection data up to one rotation (NHS-FDK). We evaluated them in terms of temporal resolution, image noise, and image quality. Also, the cause of the artifact has been investigated. The temporal resolution of NHF-FDK equals that of HS-FDK, which is half of the one rotation period. For the moving object, NHS-FDK offers the best image quality. The images with FS-FDK are degraded by streak artifacts; HS-FDK provides poor image quality with good temporal resolution; and images by OS-FDK are blurred due to insufficient temporal resolution. The cause of the artifact was found as an inconsistency of projection data due to object motion (in FS-FDK) and lost 3-D-Radon data caused by applying Parker's weight (in HS-FDK). A hand toy was employed for the preliminary evaluation of dynamic volumetric imaging with the real 256-slice scanner. In an overall evaluation, NHS-FDK provides the stable and the sufficient image quality both with moving and stationary objects.

Filtering point spread function in backprojection cone-beam CT and its applications in long object imaging

In backprojection cone-beam CT the cone-beam projection images are first filtered, then 3D backprojected into the object space. In this paper the point spread function (PSF) for the filtering operation is studied. For the cases where the normalization matrix is a constant, i.e. all integration planes intersect the scan path the same number of times, the derivation of the PSF is extended to the general case of limited angular range for the Radon line integrals. It is found that the 2D component of the PSF can be reduced to the form of space-variant 1D Hilbert transforms. The application of the PSF to a number of aspects in long object imaging will be discussed.

Consistency conditions upon 3D CT data and the wave equation

Source trajectories and reconstruction algorithms for clinical volumetric computerized tomography (VCT) will require optimization for efficiency and image quality. VCT data is highly overdetermined, satisfying an ultrahyperbolic partial differential equation. Characteristic boundary value problems for the hyperbolic wave equation and ultrahyperbolic equation are compared, focusing in this paper on a mathematically instructive open-gantry VCT geometry. This example provides physical insight into consistency conditions upon VCT data, clearly showing which information about the object can and cannot be recovered from a set of measured projections. Furthermore, this example demonstrates that efficient numerical solvers for the ultrahyperbolic equation can provide tremendous flexibility in the choice of reconstruction algorithm.

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.

Cone-beam volume CT breast imaging: feasibility study

X-ray projection mammography, using a film/screen combination, or digital techniques, has proven to be the most effective imaging modality currently available for early detection of breast cancer. However, the inherent superimposition of structures makes a small carcinoma (a few millimeters in size) difficult to detect when it is occult or in dense breasts, leading to a high false-positive biopsy rate. Cone-beam x-ray-projection-based volume imaging using flat panel detectors (FPDs) may allow obtaining three-dimensional breast images, resulting in more accurate diagnosis of structures and patterns of lesions while eliminating the hard compression of breasts. This article presents a novel cone-beam volume computed tomographic breast imaging (CBVCTBI) technique based on the above techniques. Through a variety of computer simulations, the key issues of the system and imaging techniques were addressed, including the x-ray imaging geometry and corresponding reconstruction algorithms, x-ray characteristics of breast tissue and lesions, x-ray setting techniques, the absorbed dose estimation, and the quantitative effect of x-ray scattering on image quality. The preliminary simulation results support the proposed CVBCTBI modality for breast imaging in respect to its feasibility and practicability. The absorbed dose level is comparable to that of current mammography and will not be a prominent problem for this imaging technique. Compared to conventional mammography, the proposed imaging technique with isotropic spatial resolution will potentially provide significantly better low-contrast detectability of breast tumors and more accurate location of breast lesions.

A cone beam filtered backprojection (CB-FBP) reconstruction algorithm for a circle-plus-two-arc orbit

The circle-plus-arc orbit possesses advantages over other "circle-plus" orbits for the application of x-ray cone beam (CB) volume CT in image-guided interventional procedures requiring intraoperative imaging, in which movement of the patient table is to be avoided. A CB circle-plus-two-arc orbit satisfying the data sufficiency condition and a filtered backprojection (FBP) algorithm to reconstruct longitudinally unbounded objects is presented here. In the circle suborbit, the algorithm employs Feldkamp's formula and another FBP implementation. In the arc suborbits, an FBP solution is obtained originating from Grangeat's formula, and the reconstruction computation is significantly reduced using a window function to exclude redundancy in Radon domain. The performance of the algorithm has been thoroughly evaluated through computer-simulated phantoms and preliminarily evaluated through experimental data, revealing that the algorithm can regionally reconstruct longitudinally unbounded objects exactly and efficiently, is insensitive to the variation of the angle sampling interval along the arc suborbits, and is robust over practical x-ray quantum noise. The algorithm's merits include: only 1D filtering is implemented even in a 3D reconstruction, only separable 2D interpolation is required to accomplish the CB backprojection, and the algorithm structure is appropriate for parallel computation.

Cone-beam image reconstruction using spherical harmonics

Image reconstruction from cone-beam projections is required for both x-ray computed tomography (CT) and single photon emission computed tomography (SPECT). Grangeat's algorithm accurately performs cone-beam reconstruction provided that Tuy's data sufficiency condition is satisfied and projections are complete. The algorithm consists of three stages: (a) Forming weighted plane integrals by calculating the line integrals on the cone-beam detector, and obtaining the first derivative of the plane integrals (3D Radon transform) by taking the derivative of the weighted plane integrals. (b) Rebinning the data and calculating the second derivative with respect to the normal to the plane. (c) Reconstructing the image using the 3D Radon backprojection. A new method for implementing the first stage of Grangeat's algorithm was developed using spherical harmonics. The method assumes that the detector is large enough to image the whole object without truncation. Computer simulations show that if the trajectory of the cone vertex satisfies Tuy's data sufficiency condition, the proposed algorithm provides an exact reconstruction.

Cone beam volume CT image artifacts caused by defective cells in x-ray flat panel imagers and the artifact removal using a wavelet-analysis-based algorithm

The application of x-ray flat panel imagers (FPIs) in cone beam volume CT (CBVCT) has attracted increasing attention. However, due to a deficient semiconductor array manufacturing process, defective cells unavoidably exist in x-ray FPIs. These defective cells cause their corresponding image pixels in a projection image to behave abnormally in signal gray level, and result in severe streak and ring artifacts in a CBVCT image reconstructed from the projection images. Since a three-dimensional (3-D) back-projection is involved in CBVCT, the formation of the streak and ring artifacts is different from that in the two-dimensional (2-D) fan beam CT. In this paper, a geometric analysis of the abnormality propagation in the 3D back-projection is presented, and the morphology of the streak and ring artifacts caused by the abnormality propagation is investigated through both computer simulation and phantom studies. In order to calibrate those artifacts, a 2D wavelet-analysis-based statistical approach to correct the abnormal pixels is proposed. The approach consists of three steps: (1) the location-invariant defective cells in an x-ray FPI are recognized by applying 2-D wavelet analysis on flat-field images, and a comprehensive defective cell template is acquired; (2) based upon the template, the abnormal signal gray level of the projection image pixels corresponding to the location-invariant defective cells is replaced with the interpolation of that of their normal neighbor pixels; (3) that corresponding to the isolated location-variant defective cells are corrected using a narrow-windowed median filter. The CBVCT images of a CT low-contrast phantom are employed to evaluate this proposed approach, showing that the streak and ring artifacts can be reliably eliminated. The novelty and merit of the approach are the incorporation of the wavelet analysis whose intrinsic multi-resolution analysis and localizability make the recognition algorithm robust under variable x-ray exposure levels between 30% and 70% of the dynamic range of an x-ray FPI.

Cone-beam reconstruction by backprojection and filtering

A new analytical method for tomographic image reconstruction from cone-beam projections acquired on the source orbits lying on a cylinder is presented. By application of a weighted cone-beam backprojection, the reconstruction problem is reduced to an image-restoration problem characterized by a shift-variant point-spread function that is given analytically. Assuming that the source is relatively far from the imaged object, a formula for an approximate shift-invariant inverse filter is derived; the filter is presented in the Fourier domain. Results of numerical experiments with circular and helical orbits are considered.

A filtering approach to image reconstruction in 3D SPECT

We present a new approach to three-dimensional (3D) image reconstruction using analytical inversion of the exponential divergent beam transform, which can serve as a mathematical model for cone-beam 3D SPECT imaging. We apply a circular cone-beam scan and assume constant attenuation inside a convex area with a known boundary, which is satisfactory in brain imaging. The reconstruction problem is reduced to an image restoration problem characterized by a shift-variant point spread function which is given analytically. The method requires two computation steps: backprojection and filtering. The modulation transfer function (MTF) of the filter is derived by means of an original methodology using the 2D Laplace transform. The filter is implemented in the frequency domain and requires 2D Fourier transform of transverse slices. In order to obtain a shift-invariant cone-beam projection-backprojection operator we resort to an approximation, assuming that the collimator has a relatively large focal length. Nevertheless, numerical experiments demonstrate surprisingly good results for detectors with relatively short focal lengths. The use of a wavelet-based filtering algorithm greatly improves the stability to Poisson noise.

The n-PI-method for helical cone-beam CT

A new class of acquisition schemes for helical cone-beam computed tomography (CB-CT) scanning is introduced, and their effect on the reconstruction methods is analyzed. These acquisition schemes are based on a new detector shape that is bounded by the helix. It will be shown that the data acquired with these schemes are compatible with exact reconstruction methods, and the adaptation of exact reconstruction algorithms to the new acquisition geometry is described. At the same time, the so-called PI-sufficiency condition is fulfilled. Moreover, a good fit to the acquisition requirements of the various medical applications of cone-beam CT is achieved. In contrast to other helical cone-beam acquisition and reconstruction methods, the n-PI-method introduced in this publication allows for variable pitches of the acquisition helix. This additional feature will introduce a higher flexibility into the acquisition protocols of future medical cone-beam scanners. An approximative n-PI-filtered backprojection (n-PI-FBP) reconstruction method is presented and verified. It yields convincing image quality.

Quasi-exact filtered backprojection algorithm for long-object problem in helical cone-beam tomography

Exact reconstruction from axially truncated cone-beam projections acquired with a helical vertex path is a challenging problem for which solutions are currently under investigation by some researchers. This paper deals with a difficult class of this problem called the long-object problem. Its purpose is to reconstruct a central region of interest (ROI) of a long object when the helical path extends only a little bit above and below the ROI. By extending the authors' recent approach based on the triangular decomposition of the Grangeat formula, we derive quasi-exact reconstruction algorithms whose overall structure is of filtered backprojection (FBP) style. Unlike the previous similar approaches to the long-object problem, the proposed FBP algorithms do not require additional two circular scans at the ends of the helical path. Furthermore, the algorithms require a significantly smaller detector area and achieve improved image quality even for a large pitch compared with the approximate Feldkamp algorithms. One drawback of the proposed algorithms is the computational time, which is much longer than for the Feldkamp algorithms. We show some simulation results to demonstrate the performances of the proposed algorithms.

Exact (spiral + circles) scan region-of-interest cone beam reconstruction via backprojection

We present a (spiral + circles) scan cone beam reconstruction algorithm in which image reconstruction proceeds via backprojection in the object space. In principle, the algorithm can reconstruct sectional region-of-interest (ROI) in a long object. The approach is a generalization of the cone beam backprojection technique developed by Kudo and Saito in two aspects: the resource-demanding normalization step in the Kudo and Saito's algorithm is eliminated through the technique of data combination that we published earlier, and the elimination of the restriction that the detector be big enough to capture the entire cone beam projection of the ROI. Restricting the projection data to the appropriate angular range required by data combination can be accomplished by a masking process. Because of the simplification resulting from the elimination of the normalization step, the most time-consuming operations of the algorithm can be approximated by the efficient step of line-by-line ramp filtering the cone beam image in the direction of the scan path, plus a correction image. The correction image, which can be computed exactly, is needed because data combination is not properly matched at the mask boundary when ramp filtering is involved. Empirical two-dimensional (2-D) point spread function (PSF) is developed to improve matching with the correction image which is computed with finite samplings. The use of transition region to further improve matching is introduced. The results of testing the algorithm on simulated phantoms are presented.

Advanced single-slice rebinning in cone-beam spiral CT

To achieve higher volume coverage at improved z-resolution in computed tomography (CT), systems with a large number of detector rows are demanded. However, handling an increased number of detector rows, as compared to today's four-slice scanners, requires to accounting for the cone geometry of the beams. Many so-called cone-beam reconstruction algorithms have been proposed during the last decade. None met all the requirements of the medical spiral cone-beam CT in regard to the need for high image quality, low patient dose and low reconstruction times. We therefore propose an approximate cone-beam algorithm which uses virtual reconstruction planes tilted to optimally fit 180 degrees spiral segments, i.e., the advanced single-slice rebinning (ASSR) algorithm. Our algorithm is a modification of the single-slice rebinning algorithm proposed by Noo et al. [Phys. Med. Biol. 44, 561-570 (1999)] since we use tilted reconstruction slices instead of transaxial slices to approximate the spiral path. Theoretical considerations as well as the reconstruction of simulated phantom data in comparison to the gold standard 180 degrees LI (single-slice spiral CT) were carried out. Image artifacts, z-resolution as well as noise levels were evaluated for all simulated scanners. Even for a high number of detector rows the artifact level in the reconstructed images remains comparable to that of 180 degrees LI. Multiplanar reformations of the Defrise phantom show none of the typical cone-beam artifacts usually appearing when going to larger cone angles. Image noise as well as the shape of the respective slice sensitivity profiles are equivalent to the single-slice spiral reconstruction, z-resolution is slightly decreased. The ASSR has the potential to become a practical tool for medical spiral cone-beam CT. Its computational complexity lies in the order of standard single-slice CT and it allows to use available 2D backprojection hardware.

A cone-beam reconstruction algorithm for circle-plus-arc data-acquisition geometry

In cone-beam computerized tomography (CT), projections acquired with the focal spot constrained on a planar orbit cannot provide a complete set of data to reconstruct the object function exactly. There are severe distortions in the reconstructed noncentral transverse planes when the cone angle is large. In this work, a new method is proposed which can obtain a complete set of data by acquiring cone-beam projections along a circle-plus-arc orbit. A reconstruction algorithm using this circle-plus-arc orbit is developed, based on the Radon transform and Grangeat's formula. This algorithm first transforms the cone-beam projection data of an object to the first derivative of the three-dimensional (3-D) Radon transform, using Grangeat's formula, and then reconstructs the object using the inverse Radon transform. In order to reduce interpolation errors, new rebinning equations have been derived accurately, which allows one-dimensional (1-D) interpolation to be used in the rebinning process instead of 3-D interpolation. A noise-free Defrise phantom and a Poisson noise-added Shepp-Logan phantom were simulated and reconstructed for algorithm validation. The results from the computer simulation indicate that the new cone-beam data-acquisition scheme can provide a complete set of projection data and the image reconstruction algorithm can achieve exact reconstruction. Potentially, the algorithm can be applied in practice for both a standard CT gantry-based volume tomographic imaging system and a C-arm-based cone-beam tomographic imaging system, with little mechanical modification required.

Cone-beam filtered-backprojection algorithm for truncated helical data

This paper investigates 3D image reconstruction from truncated cone-beam (CB) projections acquired with a helical vertex path. First, we show that a rigorous derivation of Grangeat's formula for truncated projections leads to a small additional term compared with previously published similar formulations. This correction term is called the boundary term. Next, this result is used to develop a CB filtered-backprojection (FBP) algorithm for truncated helical projections. This new algorithm only requires the CB projections to be measured within the region that is bounded, in the detector, by the projections of the upper and lower turns of the helix. Finally, simulations with mathematical phantoms demonstrate that: (i) the boundary term is necessary to obtain high-quality imaging of low-contrast structures and (ii) good image quality is obtained even with large values of the pitch of the helix.

Wavelet-based image enhancement in x-ray imaging and tomography

We consider an application of the wavelet transform to image processing in x-ray imaging and three-dimensional (3-D) tomography aimed at industrial inspection. Our experimental setup works in two operational modes-digital radiography and 3-D cone-beam tomographic data acquisition. Although the x-ray images measured have a large dynamic range and good spatial resolution, their noise properties and contrast are often not optimal. To enhance the images, we suggest applying digital image processing by using wavelet-based algorithms and consider the wavelet-based multiscale edge representation in the framework of the Mallat and Zhong approach [IEEE Trans. Pattern Anal. Mach. Intell. 14, 710 (1992)]. A contrast-enhancement method by use of equalization of the multiscale edges is suggested. Several denoising algorithms based on modifying the modulus and the phase of the multiscale gradients and several contrast-enhancement techniques applying linear and nonlinear multiscale edge stretching are described and compared by use of experimental data. We propose the use of a filter bank of wavelet-based reconstruction filters for the filtered-backprojection reconstruction algorithm. Experimental results show a considerable increase in the performance of the whole x-ray imaging system for both radiographic and tomographic modes in the case of the application of the wavelet-based image-processing algorithms.

Fast and stable cone-beam filtered backprojection method for non-planar orbits

We propose a new cone-beam shift-variant filtered backprojection (FBP) method for non-planar orbit data acquisitions. The new method is exact, fast and stable to discretization errors. It is based on the hybrid FBP framework whose key feature is to combine the shift-variant filtering with the ramp filtering to produce the filtered projections. We show that an adequate combination of the two filterings allows us to minimize contributions of the time-consuming and numerically unstable shift-variant filtering. We demonstrate the performances of the new method by simulation studies. We also apply the new method to real x-ray data acquired from our experimental system of monochromatic computed tomography.

The dual-ellipse cross vertex path for exact reconstruction of long objects in cone-beam tomography

We investigate the way data are used in the algorithm proposed by Kudo and Saito for the exact reconstruction of long objects from axially truncated cone-beam projections. Specifically, we show that the algorithm wastes a large part of the data. To overcome the problem, we propose to use a vertex path consisting of two crossing ellipses, for which we devised a new reconstruction algorithm, called the cross algorithm, which does not waste data and is still suitable to exactly handle axial truncation. Results of reconstruction are presented on simulated data and real data from an experimental scanner.

Feldkamp and circle-and-line cone-beam reconstruction for 3D micro-CT of vascular networks

Detailed morphometric knowledge of the microvascular network is needed for studies relating structure to haemodynamic function in organs like the lung. Clinical volumetric CT is limited to millimetre-order spatial resolution. Since evidence suggests that small arterioles (50 to 300 micrometres) dominate pulmonary haemodynamics, we built a micro-CT scanner, capable of imaging excised lungs in 3D with 100 microm resolution, for basic physiology research. The scanner incorporates a micro-focal (3 microm) x-ray source, an xyz theta stage and a CCD-coupled image intensifier detector. We imaged phantoms and contrast-enhanced rat lungs, reconstructing the data with either the Feldkamp or the circle-and-line cone-beam reconstruction algorithm. We present reconstructions using 180 views over 360 degrees for the circular trajectory, augmented with views from a linear scan for the circle-and-line algorithm. Especially for platelike features perpendicular to the rotation axis and remote from the midplane, the circle-and-line algorithm produces superior reconstructions compared with Feldkamp's algorithm. We conclude that the use of nonplanar source trajectories to perform micro-CT on contrast-enhanced, excised lungs can provide data useful for morphometric analysis of vascular trees, currently down to the 130 microm level.

Direct reconstruction of cone-beam data acquired with a vertex path containing a circle

Cone-beam data acquired with a vertex path satisfying the data sufficiency condition of Tuy can be reconstructed using exact filtered backprojection algorithms. These algorithms are based on the application to each cone-beam projection of a two-dimensional (2-D) filter that is nonstationary, and therefore more complex than the one-dimensional (1-D) ramp filter used in the approximate algorithm of Feldkamp, Davis, and Kress (1984) (FDK). We determine in this paper the general conditions under which the 2-D nonstationary filter reduces to a 2-D stationary filter, and also give the explicit expression of the corresponding convolution kernel. Using this result and the redundancy of the cone-beam data, a composite algorithm is derived for the class of vertex paths that consist of one circle and some complementary subpath designed to guarantee data sufficiency. In this algorithm the projections corresponding to vertex points along the circle are filtered using a 2-D stationary filter, whereas the other projections are handled with a 2-D nonstationary filter. The composite algorithm generalizes the method proposed by Kudo and Saito (1990), in which the circle data are processed with a 1-D ramp filter as in the FDK algorithm. The advantage of the 2-D filter introduced in this paper is to guarantee that the filtered cone-beam projections do not contain singularities in smooth regions of the object. Tests of the composite algorithm on simulated data are presented.

Cone-beam CT from width-truncated projections

In this paper we report cone-beam CT techniques that permit reconstruction from width-truncated projections. These techniques are variants of Feldkamp's filtered backprojection algorithm and assume quasi-redundancy of ray integrals. Two methods are derived and compared. The first method involves the use of preconvolution weighting of the truncated data. The second technique performs post-convolution weighting preceded by non-zero estimation of the missing information. The algorithms were tested using the three-dimensional Shepp-Logan head phantom. The results indicate that given an appropriate amount of overscan, satisfactory reconstruction can be achieved. These techniques can be used to solve the problem of undersized detectors.

Filtered backprojection reconstruction of combined parallel beam and cone beam SPECT data

The reconstruction problem for a combined parallel beam (PB) and cone beam (CB) imaging geometry has been addressed. The general algorithm (CB-FBP) of Defrise and Clack has been applied to this geometry and shown to provide accurate images as expected. A second algorithm specifically tailored to the PB-CB geometry was developed. It uses the general principles of the CB-FBP method to combine a shift-variant filtering of the PB data with a standard reconstruction of the CB data using the algorithm of Feldkamp, Davis and Kress (FDK). This 'mixed' algorithm has the advantage of fewer interpolation steps, thereby reducing reconstruction time and providing more accurate reconstructions. The algorithms were applied to noiseless data from a computer-generated 3D Shepp phantom. Both the CB-FBP algorithm and the mixed algorithm successfully combine the CB and the PB data to correct the well known artefacts observed when reconstructing CB data acquired with a circular orbit. The mixed algorithm is about twice as fast as CB-FBP and results in better image quality, due to decreased discretization errors. However, the two algorithms yield comparable image quality when applied to a disc phantom measured with a two-headed SPECT system.