Exact cone beam reconstruction for a saddle trajectory

Cone-beam imaging with tilted rotation axis: Method and performance evaluation

PURPOSE

The recently introduced robotic x-ray systems provide the flexibility to acquire cone-beam computed tomography (CBCT) data using customized, application-specific source-detector trajectories. We exploit this capability to mitigate the effects of x-ray scatter and noise in CBCT imaging of weight-bearing foot and cervical spine (C-spine) using scan orbits with a tilted rotation axis.

METHODS

We used an advanced CBCT simulator implementing accurate models of x-ray scatter, primary attenuation, and noise to investigate the effects of the orbital tilt angle in upright foot and C-spine imaging. The system model was parameterized using a laboratory version of a three-dimensional (3D) robotic x-ray system (Multitom RAX, Siemens Healthineers). We considered a generalized tilted axis scan configuration, where the detector remained parallel to patient's long body axis during the acquisition, but the elevation of source and detector was changing. A modified Feldkamp-Davis-Kress (FDK) algorithm was developed for reconstruction in this configuration, which departs from the FDK assumption of a detector that is perpendicular to the scan plane. The simulated foot scans involved source-detector distance (SDD) of 1386 mm, orbital tilt angles ranging 10° to 40°, and 400 views at 1 mAs/view and 0.5° increment; the C-spine scans involved -25° to -45° tilt angles, SDD of 1090 mm, and 202 views at 1.3 mAs and 1° increment The imaging performance was assessed by projection-domain measurements of the scatter-to-primary ratio (SPR) and by reconstruction-domain measurements of contrast, noise and generalized contrast-to-noise ratio (gCNR, accounting for both image noise and background nonuniformity) of the metatarsals (foot imaging) and cervical vertebrae (spine imaging). The effects of scatter correction were also compared for horizontal and tilted scans using an ideal Monte Carlo (MC)-based scatter correction and a frame-by-frame mean scatter correction.

RESULTS

The proposed modified FDK, involving projection resampling, mitigated streak artifacts caused by the misalignment between the filtering direction and the detector rows. For foot imaging (no grids), an optimized 20° tilted orbit reduced the maximum SPR from ~1.5 in a horizontal scan to <0.5. The gCNR of the second metatarsal was enhanced twofold compared to a horizontal orbit. For the C-spine (with vertical grids), imaging with a tilted orbit avoided highly attenuating x-ray paths through the lower cervical vertebrae and shoulders. A -35° tilted orbit yielded improved image quality and visualization of the lower cervical spine: the SPR of lower cervical vertebrae was reduced from ~10 (horizontal orbit) to <6 (tilted orbit), and the gCNR for C5-C7 increased by a factor of 2. Furthermore, tilted orbits showed potential benefits over horizontal orbits by enabling scatter correction with a simple frame-by-frame mean correction without substantial increase in noise-induced artifacts after the correction.

CONCLUSIONS

Tilted scan trajectories, enabled by the emerging robotic x-ray system technology, were optimized for CBCT imaging of foot and cervical spine using an advanced simulation framework. The results demonstrated the potential advantages of tilted axis orbits in mitigation of scatter artifacts and improving contrast-to-noise ratio in CBCT reconstructions.

A new method to combine 3D reconstruction volumes for multiple parallel circular cone beam orbits

PURPOSE

This article presents a new reconstruction method for 3D imaging using a multiple 360 degrees circular orbit cone beam CT system, specifically a way to combine 3D volumes reconstructed with each orbit. The main goal is to improve the noise performance in the combined image while avoiding cone beam artifacts.

METHODS

The cone beam projection data of each orbit are reconstructed using the FDK algorithm. When at least a portion of the total volume can be reconstructed by more than one source, the proposed combination method combines these overlap regions using weighted averaging in frequency space. The local exactness and the noise performance of the combination method were tested with computer simulations of a Defrise phantom, a FORBILD head phantom, and uniform noise in the raw data.

RESULTS

A noiseless simulation showed that the local exactness of the reconstructed volume from the source with the smallest tilt angle was preserved in the combined image. A noise simulation demonstrated that the combination method improved the noise performance compared to a single orbit reconstruction.

CONCLUSIONS

In CT systems which have overlap volumes that can be reconstructed with data from more than one orbit and in which the spatial frequency content of each reconstruction can be calculated, the proposed method offers improved noise performance while keeping the local exactness of data from the source with the smallest tilt angle.

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.

Why do commercial CT scanners still employ traditional, filtered back-projection for image reconstruction?

Despite major advances in x-ray sources, detector arrays, gantry mechanical design and especially computer performance, one component of computed tomography (CT) scanners has remained virtually constant for the past 25 years-the reconstruction algorithm. Fundamental advances have been made in the solution of inverse problems, especially tomographic reconstruction, but these works have not been translated into clinical and related practice. The reasons are not obvious and seldom discussed. This review seeks to examine the reasons for this discrepancy and provides recommendations on how it can be resolved. We take the example of field of compressive sensing (CS), summarizing this new area of research from the eyes of practical medical physicists and explaining the disconnection between theoretical and application-oriented research. Using a few issues specific to CT, which engineers have addressed in very specific ways, we try to distill the mathematical problem underlying each of these issues with the hope of demonstrating that there are interesting mathematical problems of general importance that can result from in depth analysis of specific issues. We then sketch some unconventional CT-imaging designs that have the potential to impact on CT applications, if the link between applied mathematicians and engineers/physicists were stronger. Finally, we close with some observations on how the link could be strengthened. There is, we believe, an important opportunity to rapidly improve the performance of CT and related tomographic imaging techniques by addressing these issues.

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.

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.

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.

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.