Helgason-Ludwig Consistency Conditions Research Articles

Statistical Interior Tomography via L1 Norm Dictionary Learning without Assuming an Object Support.

Interior tomography of X-ray computed tomography (CT) has many advantages, such as a lower radiation dose and lower detector hardware cost compared to traditional CT. However, this imaging technique only uses the projection data passing through the region of interest (ROI) for imaging; accordingly, the projection data are truncated at both ends of the detector, so the traditional analytical reconstruction algorithm cannot satisfy the demand of clinical diagnosis. To solve the above limitations, in this paper we propose a high-quality statistical iterative reconstruction algorithm that uses the zeroth-order image moment as novel prior knowledge; the zeroth-order image moment can be estimated in the projection domain using the Helgason–Ludwig consistency condition. Then, the L1norm of sparse representation, in terms of dictionary learning, and the zeroth-order image moment constraints are incorporated into the statistical iterative reconstruction framework to construct an objective function. Finally, the objective function is minimized using an alternating minimization iterative algorithm. The chest CT image simulated and CT real data experimental results demonstrate that the proposed approach can remove shift artifacts effectively and has superior performance in removing noise and persevering fine structures than the total variation (TV)-based approach.

Virtual multi-alignment theory of parallel-beam CT image reconstruction for elastic objects

In parallel-beam tomography, the virtual alignment method plays an important role in obtaining an ideally aligned reconstruction of a rigid specimen. Furthermore, the method has been developed for elastic specimens with specific motions such as periodic motion, regular expansion or contraction, and elliptical expansion or contraction to obtain a sinogram with an ideal sinogram pattern by transforming an elastic-type projection image set into a rigid-type projection image set satisfying the Helgason-Ludwig consistency condition. In this article, we present a method to convert a combined elastic specimen to a rigid specimen using the virtual multi-alignment method that allows us to obtain an ideally multi-aligned reconstruction of a combined elastic specimen.

Rigid motion artifact reduction in CT using extended difference function.

In computed tomography (CT), a patient motion would result in degraded spatial resolution and image artifacts. To eliminate motion artifacts, this study presents a method to estimate the motion parameters from sinograms based on extended difference function. Based on our previous work, we first divide the projection data into two parts according to view angles and take Radon transform. Then, we calculate the extended difference functions and search for the minimum points. The relative displacements can be determined by the minimum points, and the motion can be estimated by the relationships between the relative displacements and motion parameters. Finally, we introduce the estimated parameters into the reconstruction process to compensate for the motion effects. The simulation results show that the running times can reduce by about 30% than our previous work. In phantom experiments, the relative mean rotation excursion (RMRE) and relative mean translation excursion (RMTE) of the new method are lower than the conventional Helgason-Ludwig consistency condition (HLCC) based method and comparable to our previous work. Compare with the HLCC method, the root mean square error (RMSE) of the new method also reduces, while the Pearson correlation coefficient (CC) and mean structural similarity index (MSSIM) increase. The proposed new method yields the improved performance on accuracy of motion estimation with higher computational efficiency, and thus it can produce high-quality images.

Patient body motion correction for dynamic cardiac PET-CT by attenuation-emission alignment according to projection consistency conditions.

Patient body motion is known to cause large deviations in the determination of myocardial blood flow (MBF) with errors exceeding 300%. Accurate correction for patient whole-body motion is still a largely unsolved problem in cardiac positron emission tomography (PET) imaging. This study evaluated the efficacy of using Natterer's formulation of the Helgason-Ludwig consistency conditions on the two-dimensional Radon transform to align computed tomography to PET projection data in multiple time frames of a dynamic sequence for the purpose of frame-by-frame correction of rigid whole-body motion. The correction algorithm was evaluated with digital NCAT phantoms using realistic noise added by the analytical simulator. Count rates used in the simulation were derived from clinical patient data. In addition, a proof of concept test using measured data with a cardiac torso phantom was conducted. Motion correction resulted in significant improvement in the accuracy of MBF estimates, especially for high count-rate acquisitions. Maximum errors for 2cm of motion dropped from 325% to 25% and from 250% to 25% using global and regional partial-volume correction, respectively. Median MBF errors dropped from 33% to 4.5% and 27% to 3.8%, respectively. Importantly, the correction algorithm performed equally well to compensate for body motion in both early and late time frames. Cardiac PET-CT data used for attenuation correction (CTAC) alignment using projection consistency conditions was effective for reducing errors in MBF measurements due to simulated patient motion, and can be integrated into the image reconstruction workflow.

Rigid motion artifact reduction in CT using frequency domain analysis.

It is often unrealistic to assume that the subject remains stationary during a computed tomography (CT) imaging scan. A patient rigid motion can be decomposed into a translation and a rotation around an origin. How to minimize the motion impact on image quality is important. To eliminate artifacts caused by patient rigid motion during a CT scan, this study investigated a new method based on frequency domain analysis to estimate and compensate motion impact. Motion parameters was first determined by the magnitude correlation of projections in frequency domain. Then, the estimated parameters were applied to compensate for the motion effects in the reconstruction process. Finally, this method was extended to helical CT. In fan-beam CT experiments, the simulation results showed that the proposed method was more accurate and faster on the performance of motion estimation than using Helgason-Ludwig consistency condition method (HLCC). Furthermore, the reconstructed images on both simulated and human head experiments indicated that the proposed method yielded superior results in artifact reduction. The proposed method is a new tool for patient motion compensation, with a potential for practical application. It is not only applicable to motion correction in fan-beam CT imaging, but also to helical CT.

Improving Analytical Tomographic Reconstructions Through Consistency Conditions

This work introduces and characterizes a fast parameterless filter based on the Helgason-Ludwig consistency conditions, used to improve the accuracy of analytical reconstructions of tomographic undersampled datasets. The filter, acting in the Radon domain, extrapolates intermediate projections between those existing. The resulting sinogram, doubled in views, is then reconstructed by a standard analytical method. Experiments with simulated and real data prove that the peak-signal-to-noise ratio of the results computed by filtered backprojection is improved up to 5–6 dB, if the filter is used prior to reconstruction.

Restoration of missing data in limited angle tomography based on Helgason–Ludwig consistency conditions

In limited angle tomography, missing data in an insufficient angular scan will cause streak artifacts in the reconstructed images. Correspondingly, in the frequency domain representation of the imaged object, a double wedge-shaped region is missing. In this paper, we perform a regression in sinogram domain and an image fusion in frequency domain to restore the missing data. We first convert the sinogram restoration problem into a regression problem based on the Helgason–Ludwig consistency conditions. Due to its severe ill-posedness, regression only partially recovers the correct frequency components, especially lower frequency components, and will introduce erroneous ones, particularly higher frequencies. Bilateral filtering is utilized to retain the most prominent high frequency components and suppress erroneous ones. Afterwards, a fusion in the frequency domain utilizes the restored frequency components to fill the missing double wedge region. The proposed method is evaluated in a parallel-beam study on both numerical and clinical phantoms. The root-mean-square errors of the reconstructed images decrease from 302 to 78 HU for the noise-free Shepp–Logan phantom, from 355 to 175 HU for the noisy Shepp–Logan phantom, and from 187 to 56 HU for the clinical data. The results show that our method is promising in streak reduction and intensity offset compensation in both noise-free and noisy situations.

An Improved Extrapolation Scheme for Truncated CT Data Using 2D Fourier-Based Helgason-Ludwig Consistency Conditions.

We improve data extrapolation for truncated computed tomography (CT) projections by using Helgason-Ludwig (HL) consistency conditions that mathematically describe the overlap of information between projections. First, we theoretically derive a 2D Fourier representation of the HL consistency conditions from their original formulation (projection moment theorem), for both parallel-beam and fan-beam imaging geometry. The derivation result indicates that there is a zero energy region forming a double-wedge shape in 2D Fourier domain. This observation is also referred to as the Fourier property of a sinogram in the previous literature. The major benefit of this representation is that the consistency conditions can be efficiently evaluated via 2D fast Fourier transform (FFT). Then, we suggest a method that extrapolates the truncated projections with data from a uniform ellipse of which the parameters are determined by optimizing these consistency conditions. The forward projection of the optimized ellipse can be used to complete the truncation data. The proposed algorithm is evaluated using simulated data and reprojections of clinical data. Results show that the root mean square error (RMSE) is reduced substantially, compared to a state-of-the-art extrapolation method.

WE‐AB‐207A‐12: HLCC Based Quantitative Evaluation Method of Image Artifact in Dental CBCT

Purpose:Image artifacts are usually evaluated qualitatively via visual observation of the reconstructed images, which is susceptible to subjective factors due to the lack of an objective evaluation criterion. In this work, we propose a Helgason‐Ludwig consistency condition (HLCC) based evaluation method to quantify the severity level of different image artifacts in dental CBCT.Methods:Our evaluation method consists of four step: 1) Acquire Cone beam CT(CBCT) projection; 2) Convert 3D CBCT projection to fan‐beam projection by extracting its central plane projection; 3) Convert fan‐beam projection to parallel‐beam projection utilizing sinogram‐based rebinning algorithm or detail‐based rebinning algorithm; 4) Obtain HLCC profile by integrating parallel‐beam projection per view and calculate wave percentage and variance of the HLCC profile, which can be used to describe the severity level of image artifacts.Results:Several sets of dental CBCT projections containing only one type of artifact (i.e. geometry, scatter, beam hardening, lag and noise artifact), were simulated using gDRR, a GPU tool developed for efficient, accurate, and realistic simulation of CBCT Projections. These simulated CBCT projections were used to test our proposed method. HLCC profile wave percentage and variance induced by geometry distortion are about 3∼21 times and 16∼393 times as large as that of the artifact‐free projection, respectively. The increase factor of wave percentage and variance are 6 and133 times for beam hardening, 19 and 1184 times for scatter, and 4 and16 times for lag artifacts, respectively. In contrast, for noisy projection the wave percentage, variance and inconsistency level are almost the same with those of the noise‐free one.Conclusion:We have proposed a quantitative evaluation method of image artifact based on HLCC theory. According to our simulation results, the severity of different artifact types is found to be in a following order: Scatter>Geometry>Beam hardening>Lag>Noise>Artifact‐free in dental CBCT

Automated Motion Correction for In Vivo Optical Projection Tomography

In in vivo optical projection tomography (OPT), object motion will significantly reduce the quality and resolution of the reconstructed image. Based on the well-known Helgason-Ludwig consistency condition (HLCC), we propose a novel method for motion correction in OPT under parallel beam illumination. The method estimates object motion from projection data directly and does not require any other additional information, which results in a straightforward implementation. We decompose object movement into translation and rotation, and discuss how to correct for both translation and general motion simultaneously. Since finding the center of rotation accurately is critical in OPT, we also point out that the system's geometrical offset can be considered as object translation and therefore also calibrated through the translation estimation method. In order to verify the algorithm effectiveness, both simulated and in vivo OPT experiments are performed. Our results demonstrate that the proposed approach is capable of decreasing movement artifacts significantly thus providing high quality reconstructed images in the presence of object motion.

Statistical Projection Completion in X-ray CT Using Consistency Conditions

Projection data incompleteness arises in many situations relevant to X-ray computed tomography (CT) imaging. We propose a penalized maximum likelihood statistical sinogram restoration approach that incorporates the Helgason-Ludwig (HL) consistency conditions to accommodate projection data incompleteness. Image reconstruction is performed by the filtered-backprojection (FBP) in a second step. In our problem formulation, the objective function consists of the log-likelihood of the X-ray CT data and a penalty term; the HL condition poses a linear constraint on the restored sinogram and can be implemented efficiently via fast Fourier transform (FFT) and inverse FFT. We derive an iterative algorithm that increases the objective function monotonically. The proposed algorithm is applied to both computer simulated data and real patient data. We study different factors in the problem formulation that affect the properties of the final FBP reconstructed images, including the data truncation level, the amount of prior knowledge on the object support, as well as different approximations of the statistical distribution of the available projection data. We also compare its performance with an analytical truncation artifacts reduction method. The proposed method greatly improves both the accuracy and the precision of the reconstructed images within the scan field-of-view, and to a certain extent recovers the truncated peripheral region of the object. The proposed method may also be applied in areas such as limited angle tomography, metal artifacts reduction, and sparse sampling imaging.

An Algebraic Framework for Discrete Tomography: Revealing the Structure of Dependencies

Discrete tomography is concerned with the reconstruction of images that are defined on a discrete set of lattice points from their projections in several directions. The range of values that can be assigned to each lattice point is typically a small discrete set. In this paper we present a framework for studying these problems from an algebraic perspective, based on ring theory and commutative algebra. A principal advantage of this abstract setting is that a vast body of existing theory becomes accessible for solving discrete tomography problems. We provide proofs of several new results on the structure of dependencies between projections, including a discrete analogon of the well-known Helgason-Ludwig consistency conditions from continuous tomography.

Data Consistency Based Rigid Motion Artifact Reduction in Fan-Beam CT

It is well known that a rigid in-plane motion can be decomposed into a translation and a rotation around an origin. Based on our previous work, we first extend the Helgason-Ludwig consistency condition (HLCC) to cover a general rigid motion in fan-beam geometry. Then, we model the general motion by several parameters, and develop an iterative scheme for estimation of the in-plane motion parameters. This scheme determines the motion parameters by numerically minimizing an objective function constructed based on the HLCC. After the motion parameters are estimated, image reconstruction can be performed to compensate for the motion effects. Finally, we implement the algorithm and evaluate its performance in numerical simulations.

Two-Dimensional tomography with unknown view angles

We consider uniqueness of two-dimensional parallel beam tomography with unknown view angles. We show that infinitely many projections at unknown view angles of a sufficiently asymmetric object determine the object uniquely. An explicit expression for the required asymmetry is given in terms of the object's geometric moments. We also show that under certain assumptions finitely many projections guarantee uniqueness for the unknown view angles. Compared to previous results about uniqueness of view angles, our result reduces the minimum number of required projections to approximately half and is applicable to a larger set of objects. Our analysis is based on algebraic geometric properties of a certain system of homogeneous polynomials determined by the Helgason-Ludwig consistency conditions.

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.

Sinogram recovery with the method of convex projections for limited-data reconstruction in computed tomography

Image reconstruction from incomplete projection data is strongly required in widespread applications of computed tomography. This problem can be formulated as a sinogram–recovery problem. The sinogram–recovery problem is to find a complete sinogram that is compatible with the Helgason–Ludwig consistency condition, the measured incomplete sinogram, and other a priori knowledge about the sinogram in question. The direct use of the Helgason–Ludwig consistency condition considerably reduces computational requirements and the accumulation of digital-processing errors over the conventional iterative reconstruction–reprojection method. Most research for solving the sinogram–recovery problem is based on directly inverting systems of linear equations associated with the Helgason–Ludwig consistency condition. However, these noniterative techniques cannot be applied to various different types of limited-data situations in a unified way. Moreover, nonlinear a priori constraints such as the nonnegativity and the amplitude limit are not easily incorporated. We solve the sinogram–recovery problem by using an iterative signal-recovery technique known as the method of projection onto convex sets. Once an estimation of the complete sinogram is obtained, the conventional convolution–backprojection method can be utilized to reconstruct an image. The performance of the proposed method is investigated both with numerical phantoms and with actual x-ray data.

Sinogram recovery with the method of convex projections for limited-data reconstruction in computed tomography

Image reconstruction from incomplete projection data is strongly required in widespread applications of computed tomography. This problem can be formulated as a sinogram–recovery problem. The sinogram–recovery problem is to find a complete sinogram that is compatible with the Helgason–Ludwig consistency condition, the measured incomplete sinogram, and other a priori knowledge about the sinogram in question. The direct use of the Helgason–Ludwig consistency condition considerably reduces computational requirements and the accumulation of digital-processing errors over the conventional iterative reconstruction–reprojection method. Most research for solving the sinogram–recovery problem is based on directly inverting systems of linear equations associated with the Helgason–Ludwig consistency condition. However, these noniterative techniques cannot be applied to various different types of limited-data situations in a unified way. Moreover, nonlinear a priori constraints such as the nonnegativity and the amplitude limit are not easily incorporated. We solve the sinogram–recovery problem by using an iterative signal-recovery technique known as the method of projection onto convex sets. Once an estimation of the complete sinogram is obtained, the conventional convolution–backprojection method can be utilized to reconstruct an image. The performance of the proposed method is investigated both with numerical phantoms and with actual x-ray data.