Consistency conditions upon 3D CT data and the wave equation

Optimization based beam-hardening correction in CT under data integral invariant constraint

In computed tomography (CT), the polychromatic characteristics of x-ray photons, which are emitted from a source, interact with materials and are absorbed by a detector, may lead to beam-hardening effect in signal detection and image formation, especially in situations where materials of high attenuation (e.g. the bone or metal implants) are in the x-ray beam. Usually, a beam-hardening correction (BHC) method is used to suppress the artifacts induced by bone or other objects of high attenuation, in which a calibration-oriented iterative operation is carried out to determine a set of parameters for all situations. Based on the Helgasson-Ludwig consistency condition (HLCC), an optimization based method has been proposed by turning the calibration-oriented iterative operation of BHC into solving an optimization problem sustained by projection data. However, the optimization based HLCC-BHC method demands the engagement of a large number of neighboring projection views acquired at relatively high and uniform angular sampling rate, hindering its application in situations where the angular sampling in projection view is sparse or non-uniform. By defining an objective function based on the data integral invariant constraint (DIIC), we again turn BHC into solving an optimization problem sustained by projection data. As it only needs a pair of projection views at any view angle, the proposed BHC method can be applicable in the challenging scenarios mentioned above. Using the projection data simulated by computer, we evaluate and verify the proposed optimization based DIIC-BHC method's performance. Moreover, with the projection data of a head scan by a multi-detector row MDCT, we show the proposed DIIC-BHC method's utility in clinical applications.

Geometry and energy constrained projection extension

BACKGROUND

In clinical computed tomography (CT) applications, when a patient is obese or improperly positioned, the final tomographic scan is often partially truncated. Images directly reconstructed by the conventional reconstruction algorithms suffer from severe cupping and direct current bias artifacts. Moreover, the current methods for projection extension have limitations that preclude incorporation from clinical workflows, such as prohibitive computational time for iterative reconstruction, extra radiation dose, hardware modification, etc.METHOD:In this study, we first established a geometrical constraint and estimated the patient habitus using a modified scout configuration. Then, we established an energy constraint using the integral invariance of fan-beam projections. Two constraints were extracted from the existing CT scan process with minimal modification to the clinical workflows. Finally, we developed a novel dual-constraint based optimization model that can be rapidly solved for projection extrapolation and accurate local reconstruction.

RESULTS

Both numerical phantom and realistic patient image simulations were performed, and the results confirmed the effectiveness of our proposed approach.

CONCLUSION

We establish a dual-constraint-based optimization model and correspondingly develop an accurate extrapolation method for partially truncated projections. The proposed method can be readily integrated into the clinical workflow and efficiently solved by using a one-dimensional optimization algorithm. Moreover, it is robust for noisy cases with various truncations and can be further accelerated by GPU based parallel computing.

Motion compensation for cone-beam CT using Fourier consistency conditions

In cone-beam CT, involuntary patient motion and inaccurate or irreproducible scanner motion substantially degrades image quality. To avoid artifacts this motion needs to be estimated and compensated during image reconstruction. In previous work we showed that Fourier consistency conditions (FCC) can be used in fan-beam CT to estimate motion in the sinogram domain. This work extends the FCC to [Formula: see text] cone-beam CT. We derive an efficient cost function to compensate for [Formula: see text] motion using [Formula: see text] detector translations. The extended FCC method have been tested with five translational motion patterns, using a challenging numerical phantom. We evaluated the root-mean-square-error and the structural-similarity-index between motion corrected and motion-free reconstructions. Additionally, we computed the mean-absolute-difference (MAD) between the estimated and the ground-truth motion. The practical applicability of the method is demonstrated by application to respiratory motion estimation in rotational angiography, but also to motion correction for weight-bearing imaging of knees. Where the latter makes use of a specifically modified FCC version which is robust to axial truncation. The results show a great reduction of motion artifacts. Accurate estimation results were achieved with a maximum MAD value of 708 μm and 1184 μm for motion along the vertical and horizontal detector direction, respectively. The image quality of reconstructions obtained with the proposed method is close to that of motion corrected reconstructions based on the ground-truth motion. Simulations using noise-free and noisy data demonstrate that FCC are robust to noise. Even high-frequency motion was accurately estimated leading to a considerable reduction of streaking artifacts. The method is purely image-based and therefore independent of any auxiliary data.

John's Equation-based Consistency Condition and Corrupted Projection Restoration in Circular Trajectory Cone Beam CT

In transmitted X-ray tomography imaging, the acquired projections may be corrupted for various reasons, such as defective detector cells and beam-stop array scatter correction problems. In this study, we derive a consistency condition for cone-beam projections and propose a method to restore lost data in corrupted projections. In particular, the relationship of the geometry parameters in circular trajectory cone-beam computed tomography (CBCT) is utilized to convert an ultra-hyperbolic partial differential equation (PDE) into a second-order PDE. The second-order PDE is then transformed into a first-order ordinary differential equation in the frequency domain. The left side of the equation for the newly derived consistency condition is the projection derivative of the current and adjacent views, whereas the right side is the projection derivative of the geometry parameters. A projection restoration method is established based on the newly derived equation to restore corrupted data in projections in circular trajectory CBCT. The proposed method is tested in beam-stop array scatter correction, metal artifact reduction, and abnormal pixel correction cases to evaluate the performance of the consistency condition and corrupted projection restoration method. Qualitative and quantitative results demonstrate that the present method has considerable potential in restoring lost data in corrupted projections.

Fourier rebinning and consistency equations for time-of-flight PET planograms

Due to the unique geometry, dual-panel PET scanners have many advantages in dedicated breast imaging and on-board imaging applications since the compact scanners can be combined with other imaging and treatment modalities. The major challenges of dual-panel PET imaging are the limited-angle problem and data truncation, which can cause artifacts due to incomplete data sampling. The time-of-flight (TOF) information can be a promising solution to reduce these artifacts. The TOF planogram is the native data format for dual-panel TOF PET scanners, and the non-TOF planogram is the 3D extension of linogram. The TOF planograms is five-dimensional while the objects are three-dimensional, and there are two degrees of redundancy. In this paper, we derive consistency equations and Fourier-based rebinning algorithms to provide a complete understanding of the rich structure of the fully 3D TOF planograms. We first derive two consistency equations and John's equation for 3D TOF planograms. By taking the Fourier transforms, we obtain two Fourier consistency equations and the Fourier-John equation, which are the duals of the consistency equations and John's equation, respectively. We then solve the Fourier consistency equations and Fourier-John equation using the method of characteristics. The two degrees of entangled redundancy of the 3D TOF data can be explicitly elicited and exploited by the solutions along the characteristic curves. As the special cases of the general solutions, we obtain Fourier rebinning and consistency equations (FORCEs), and thus we obtain a complete scheme to convert among different types of PET planograms: 3D TOF, 3D non-TOF, 2D TOF and 2D non-TOF planograms. The FORCEs can be used as Fourier-based rebinning algorithms for TOF-PET data reduction, inverse rebinnings for designing fast projectors, or consistency conditions for estimating missing data. As a byproduct, we show the two consistency equations are necessary and sufficient for 3D TOF planograms. Finally, we give numerical examples of implementation of a fast 2D TOF planogram projector and Fourier-based rebinning for a 2D TOF planograms using the FORCEs to show the efficacy of the Fourier-based solutions.

Attenuation correction in emission tomography using the emission data--A review

The problem of attenuation correction (AC) for quantitative positron emission tomography (PET) had been considered solved to a large extent after the commercial availability of devices combining PET with computed tomography (CT) in 2001; single photon emission computed tomography (SPECT) has seen a similar development. However, stimulated in particular by technical advances toward clinical systems combining PET and magnetic resonance imaging (MRI), research interest in alternative approaches for PET AC has grown substantially in the last years. In this comprehensive literature review, the authors first present theoretical results with relevance to simultaneous reconstruction of attenuation and activity. The authors then look back at the early history of this research area especially in PET; since this history is closely interwoven with that of similar approaches in SPECT, these will also be covered. We then review algorithmic advances in PET, including analytic and iterative algorithms. The analytic approaches are either based on the Helgason-Ludwig data consistency conditions of the Radon transform, or generalizations of John's partial differential equation; with respect to iterative methods, we discuss maximum likelihood reconstruction of attenuation and activity (MLAA), the maximum likelihood attenuation correction factors (MLACF) algorithm, and their offspring. The description of methods is followed by a structured account of applications for simultaneous reconstruction techniques: this discussion covers organ-specific applications, applications specific to PET/MRI, applications using supplemental transmission information, and motion-aware applications. After briefly summarizing SPECT applications, we consider recent developments using emission data other than unscattered photons. In summary, developments using time-of-flight (TOF) PET emission data for AC have shown promising advances and open a wide range of applications. These techniques may both remedy deficiencies of purely MRI-based AC approaches in PET/MRI and improve standalone PET imaging.

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.

Fourier properties of the fan-beam sinogram

PURPOSE

P. R. Edholm, R. M. Lewitt, and B. Lindholm, "Novel properties of the Fourier decomposition of the sinogram," in Proceedings of the International Workshop on Physics and Engineering of Computerized Multidimensional Imaging and Processing [Proc. SPIE 671, 8-18 (1986)] described properties of a parallel beam projection sinogram with respect to its radial and angular frequencies. The purpose is to perform a similar derivation to arrive at corresponding properties of a fan-beam projection sinogram for both the equal-angle and equal-spaced detector sampling scenarios.

METHODS

One of the derived properties is an approximately zero-energy region in the two-dimensional Fourier transform of the full fan-beam sinogram. This region is in the form of a double-wedge, similar to the parallel beam case, but different in that it is asymmetric with respect to the frequency axes. The authors characterize this region for a point object and validate the derived properties in both a simulation and a head CT data set. The authors apply these results in an application using algebraic reconstruction.

RESULTS

In the equal-angle case, the domain of the zero region is (q,k) for which / k/(k-q) / > R/L, where q and k are the frequency variables associated with the detector and view angular positions, respectively, R is the radial support of the object, and L is the source-to-isocenter distance. A filter was designed to retain only sinogram frequencies corresponding to a specified radial support. The filtered sinogram was used to reconstruct the same radial support of the head CT data. As an example application of this concept, the double-wedge filter was used to computationally improve region of interest iterative reconstruction.

CONCLUSIONS

Interesting properties of the fan-beam sinogram exist and may be exploited in some applications.

Consistency Conditions for Cone-Beam CT Data Acquired with a Straight-Line Source Trajectory

A consistency condition is developed for computed tomography (CT) projection data acquired from a straight-line X-ray source trajectory. The condition states that integrals of normalized projection data along detector lines parallel to the X-ray path must be equal. The projection data is required to be untruncated only along the detector lines parallel to the X-ray path, a less restrictive requirement compared to Fourier conditions that necessitate completely untruncated data. The condition is implemented numerically on simple image functions, a discretization error bound is estimated, and detection of motion inconsistencies is demonstrated. The results show that the consistency condition may be used to quantitatively compare the quality of projection data sets obtained from different scans of the same image object.

Alignment of cryo-electron tomography datasets

Data acquisition of cryo-electron tomography (CET) samples described in previous chapters involves relatively imprecise mechanical motions: the tilt series has shifts, rotations, and several other distortions between projections. Alignment is the procedure of correcting for these effects in each image and requires the estimation of a projection model that describes how points from the sample in three-dimensions are projected to generate two-dimensional images. This estimation is enabled by finding corresponding common features between images. This chapter reviews several software packages that perform alignment and reconstruction tasks completely automatically (or with minimal user intervention) in two main scenarios: using gold fiducial markers as high contrast features or using relevant biological structures present in the image (marker-free). In particular, we emphasize the key decision points in the process that users should focus on in order to obtain high-resolution reconstructions.

Serial reconstruction and montaging from large-field electron microscope tomograms

Electron microscope tomography [1] has been proven as an essential technique for imaging the structure of cells beyond the range of the light microscope down to the molecular level. However, because of the extreme difference in spatial scales, there is a large gap to be bridged between light and electron microscopy. Various techniques have been developed, including increasing size of the sensor arrays, serial sectioning and montaging. Data sets and reconstructions obtained by the latter techniques generate many 3D reconstructions that need to be glued together to provide information at a larger spatial scale. However, during the course of data acquisition, thin slices may become warped in optical and electron microscope preparations. We review some procedures for de-warping sections and reassembling them into larger reconstructions, and present some data from electron microscopy.

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.

Variable pitch reconstruction using John's equation

We present an algorithm to reconstruct helical cone beam computed tomography (CT) data acquired at variable pitch. The algorithm extracts a halfscan segment of projections using an extended version of the advanced single slice rebinning (ASSR) algorithm. ASSR rebins constant pitch cone beam data to fan beam projections that approximately lie on a plane that is tilted to optimally fit the source helix. For variable pitch, the error between the tilted plane chosen by ASSR and the source helix increases, resulting in increased image artifacts. To reduce the artifacts, we choose a reconstruction plane, which is tilted and shifted relative to the source trajectory. We then correct rebinned fan beam data using John's equation to virtually move the source into the tilted and shifted reconstruction plane. Results obtained from simulated phantom images and scanner images demonstrate the applicability of the proposed algorithm.

Statistical reconstruction for x-ray CT systems with non-continuous detectors

We analyse the performance of statistical reconstruction (SR) methods when applied to non-continuous x-ray detectors. Robustness to projection gaps is required in x-ray CT systems with multiple detector modules or with defective detector pixels. In such situations, the advantage of statistical reconstruction is that it is able to ignore missing or faulty pixels and that it makes optimal use of the remaining line integrals. This potentially obviates the need to fill the sinogram discontinuities by interpolation or any other approximative pre-processing techniques. In this paper, we apply SR to cone beam projections of (i) a hypothetical modular detector micro-CT scanner and of (ii) a system with randomly located defective detector elements. For the modular-detector system, SR produces reconstruction volumes free of noticeable gap-induced artefacts as long as the location of detector gaps and selection of the scanning range provide complete object sampling in the central imaging plane. When applied to randomly located faulty detector elements, SR produces images free of substantial ring artefacts even for cases where defective pixels cover as much as 3% of the detector area.

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.

Minimum data image reconstruction algorithms with shift-invariant filtering for helical, cone-beam CT

We derive accurate and efficient reconstruction algorithms for helical, cone-beam CT that employ shift-invariant filtering. Specifically, a new backprojection-filtration algorithm is developed, and a minimum data filtered-backprojection algorithm is derived. These reconstruction algorithms with shift-invariant filtering can accept data with transverse truncation, and hence allow for minimum data image reconstruction.

A new data consistency condition for fan-beam projection data

The sum of all attenuation data acquired in one view of parallel-beam projections is a view angle independent constant. This fact is known as a data consistency condition on the two-dimensional Radon transforms. It plays an important role in tomographic image reconstruction and artifact correction. In this paper, a novel fan-beam data consistency condition (FDCC) is derived and presented. Using the FDCC, individual projection data in one view of fan-beam projections can be estimated from filtering all other projection data measured from different view angles. Numerical simulations are performed to validate the new FDCC in correcting ring artifacts caused by malfunctioning detector cells.

The cone-beam algorithm of Feldkamp, Davis, and Kress preserves oblique line integrals

The algorithm of Feldkamp, Davis, and Kress [J. Opt. Soc. Am. A 1, 612-619 (1984)] is a widely used filtered-backprojection algorithm for three-dimensional image reconstruction from cone-beam (CB) projections measured with a circular orbit of the x-ray source. A well-known property of this approximate algorithm is that the integral of the reconstructed image along any axial line orthogonal to the plane of the orbit is exact when the cone-beam projections are not truncated. We generalize this result to oblique line integrals, thus providing an efficient method to compute synthetic radiographs from cone-beam projections. Our generalized result is obtained by showing that the FDK algorithm is invariant under transformations that map oblique lines onto axial lines.