Computation of unmeasured third-generation VCT views from measured views

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

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

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.

Discriminative Feature Representation to Improve Projection Data Inconsistency for Low Dose CT Imaging

In low dose computed tomography (LDCT) imaging, the data inconsistency of measured noisy projections can significantly deteriorate reconstruction images. To deal with this problem, we propose here a new sinogram restoration approach, the sinogram- discriminative feature representation (S-DFR) method. Different from other sinogram restoration methods, the proposed method works through a 3-D representation-based feature decomposition of the projected attenuation component and the noise component using a well-designed composite dictionary containing atoms with discriminative features. This method can be easily implemented with good robustness in parameter setting. Its comparison to other competing methods through experiments on simulated and real data demonstrated that the S-DFR method offers a sound alternative in LDCT.

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.

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.

Projection correlation based view interpolation for cone beam CT: primary fluence restoration in scatter measurement with a moving beam stop array

Scatter correction is an open problem in x-ray cone beam (CB) CT. The measurement of scatter intensity with a moving beam stop array (BSA) is a promising technique that offers a low patient dose and accurate scatter measurement. However, when restoring the blocked primary fluence behind the BSA, spatial interpolation cannot well restore the high-frequency part, causing streaks in the reconstructed image. To address this problem, we deduce a projection correlation (PC) to utilize the redundancy (over-determined information) in neighbouring CB views. PC indicates that the main high-frequency information is contained in neighbouring angular projections, instead of the current projection itself, which provides a guiding principle that applies to high-frequency information restoration. On this basis, we present the projection correlation based view interpolation (PC-VI) algorithm; that it outperforms the use of only spatial interpolation is validated. The PC-VI based moving BSA method is developed. In this method, PC-VI is employed instead of spatial interpolation, and new moving modes are designed, which greatly improve the performance of the moving BSA method in terms of reliability and practicability. Evaluation is made on a high-resolution voxel-based human phantom realistically including the entire procedure of scatter measurement with a moving BSA, which is simulated by analytical ray-tracing plus Monte Carlo simulation with EGSnrc. With the proposed method, we get visually artefact-free images approaching the ideal correction. Compared with the spatial interpolation based method, the relative mean square error is reduced by a factor of 6.05-15.94 for different slices. PC-VI does well in CB redundancy mining; therefore, it has further potential in CBCT studies.

Continuous and discrete data rebinning in time-of-flight PET

This paper investigates data compression methods for time-of-flight (TOF) positron emission tomography (PET), which rebin the 3-D TOF measurements into a set of 2-D TOF data for a stack of transaxial slices. The goal of this work is to develop rebinning algorithms that are more accurate than the TOF single-slice-rebinning (TOF-SSRB) method proposed by Mullani in 1982. Two approaches are explored. The first one is based on a partial differential equation, which expresses a consistency condition for TOF-PET data with a Gaussian TOF profile. From this equation we derive an analytical rebinning algorithm, which is unbiased in the limit of continuous sampling. The second approach is discrete: each 2-D rebinned data sample is calculated as a linear combination of the 3-D TOF samples in the same axial plane parallel to the axis of the scanner. The coefficients of the linear combination are precomputed by optimizing a cost function which enforces both accuracy and good variance reduction, models the TOF profile, the axial PSF of the LORs, and the specific sampling scheme of the scanner. Measurements of a thorax phantom on a prototype TOF-PET scanner with a resolution of 550 ps show that the proposed discrete method improves the bias-variance trade-off and is a promising alternative to TOF-SSRB when data compression is required to achieve clinically acceptable reconstruction time.

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.

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.

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.

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.