Data consistency conditions for truncated fanbeam and parallel projections

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.

Motion estimation and correction in SPECT, PET and CT

Patient motion impacts single photon emission computed tomography (SPECT), positron emission tomography (PET) and X-ray computed tomography (CT) by giving rise to projection data inconsistencies that can manifest as reconstruction artifacts, thereby degrading image quality and compromising accurate image interpretation and quantification. Methods to estimate and correct for patient motion in SPECT, PET and CT have attracted considerable research effort over several decades. The aims of this effort have been two-fold: to estimate relevant motion fields characterizing the various forms of voluntary and involuntary motion; and to apply these motion fields within a modified reconstruction framework to obtain motion-corrected images. The aims of this review are to outline the motion problem in medical imaging and to critically review published methods for estimating and correcting for the relevant motion fields in clinical and preclinical SPECT, PET and CT. Despite many similarities in how motion is handled between these modalities, utility and applications vary based on differences in temporal and spatial resolution. Technical feasibility has been demonstrated in each modality for both rigid and non-rigid motion, but clinical feasibility remains an important target. There is considerable scope for further developments in motion estimation and correction, and particularly in data-driven methods that will aid clinical utility. State-of-the-art machine learning methods may have a unique role to play in this context.

Deep Learning-based Inaccuracy Compensation in Reconstruction of High Resolution XCT Data

While X-ray computed tomography (XCT) is pushed further into the micro- and nanoscale, the limitations of various tool components and object motion become more apparent. For high-resolution XCT, it is necessary but practically difficult to align these tool components with sub-micron precision. The aim is to develop a novel reconstruction methodology that considers unavoidable misalignment and object motion during the data acquisition in order to obtain high-quality three-dimensional images and that is applicable for data recovery from incomplete datasets. A reconstruction software empowered by sophisticated correction modules that autonomously estimates and compensates artefacts using gradient descent and deep learning algorithms has been developed and applied. For motion estimation, a novel computer vision methodology coupled with a deep convolutional neural network approach provides estimates for the object motion by tracking features throughout the adjacent projections. The model is trained using the forward projections of simulated phantoms that consist of several simple geometrical features such as sphere, triangle and rectangular. The feature maps extracted by a neural network are used to detect and to classify features done by a support vector machine. For missing data recovery, a novel deep convolutional neural network is used to infer high-quality reconstruction data from incomplete sets of projections. The forward and back projections of simulated geometric shapes from a range of angular ranges are used to train the model. The model is able to learn the angular dependency based on a limited angle coverage and to propose a new set of projections to suppress artefacts. High-quality three-dimensional images demonstrate that it is possible to effectively suppress artefacts caused by thermomechanical instability of tool components and objects resulting in motion, by center of rotation misalignment and by inaccuracy in the detector position without additional computational efforts. Data recovery from incomplete sets of projections result in directly corrected projections instead of suppressing artefacts in the final reconstructed images. The proposed methodology has been proven and is demonstrated for a ball bearing sample. The reconstruction results are compared to prior corrections and benchmarked with a commercially available reconstruction software. Compared to conventional approaches in XCT imaging and data analysis, the proposed methodology for the generation of high-quality three-dimensional X-ray images is fully autonomous. The methodology presented here has been proven for high-resolution micro-XCT and nano-XCT, however, is applicable for all length scales.

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.

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.

Data-Driven Affine Deformation Estimation and Correction in Cone Beam Computed Tomography

In computed tomography (CT), motion and deformation during the acquisition lead to streak artefacts and blurring in the reconstructed images. To remedy these artefacts, we introduce an efficient algorithm to estimate and correct for global affine deformations directly on the cone beam projections. The proposed technique is data driven and thus removes the need for markers and/or a tracking system. A relationship between affine transformations and the cone beam transform is proved and used to correct the projections. The deformation parameters that describe deformation perpendicular to the projection direction are estimated for each projection by minimizing a plane-based inconsistency criterion. The criterion compares each projection of the main scan with all projections of a fast reference scan, which is acquired prior or posterior to the main scan. Experiments with simulated and experimental data show that the proposed affine deformation estimation method is able to substantially reduce motion artefacts in cone beam CT images.

Translational motion correction algorithm for truncated cone-beam CT using opposite projections

BACKGROUND

Cone-beam computed tomography (CBCT) is widely used in various medical imaging applications, including dental examinations. Dental CBCT images often suffer from motion artifacts caused by involuntary rigid motion of patients. However, earlier motion compensation studies are not applicable for dental CBCT systems using truncated detectors.

OBJECTIVE

This study proposes a novel motion correction algorithm that can be applied for truncated dental CBCT images.

METHODS

We propose a two-step method for motion correction. First, we estimate the relative displacement of each pair of opposite projections by finding the motion vector that maximizes the two-dimensional correlation coefficients of the opposite projections. Second, we convert the relative displacement into the absolute coordinate motion that yields the highest image sharpness of the reconstruction image. Using the motion vectors in the absolute coordinate system, motion artifacts are then compensated by modifying the trajectory of the source and detector during the back-projection step of the image reconstruction process.

RESULTS

In simulation, the proposed method successfully estimated the true relative displacement. After converting to the absolute coordinate motions, the motion-compensated image was close to the ground-truth image and exhibited a lower mean-square-error than that of the uncompensated image. The results from the real data experiment also confirmed that the proposed method successfully compensated for the motion artifacts.

CONCLUSIONS

The experimental results confirmed that the proposed method was applicable to most dental CBCT systems using a truncated detector without any use of an additional motion tracking system nor prior knowledge.

Marker-free motion correction in weight-bearing cone-beam CT of the knee joint

PURPOSE

To allow for a purely image-based motion estimation and compensation in weight-bearing cone-beam computed tomography of the knee joint.

METHODS

Weight-bearing imaging of the knee joint in a standing position poses additional requirements for the image reconstruction algorithm. In contrast to supine scans, patient motion needs to be estimated and compensated. The authors propose a method that is based on 2D/3D registration of left and right femur and tibia segmented from a prior, motion-free reconstruction acquired in supine position. Each segmented bone is first roughly aligned to the motion-corrupted reconstruction of a scan in standing or squatting position. Subsequently, a rigid 2D/3D registration is performed for each bone to each of K projection images, estimating 6 × 4 × K motion parameters. The motion of individual bones is combined into global motion fields using thin-plate-spline extrapolation. These can be incorporated into a motion-compensated reconstruction in the backprojection step. The authors performed visual and quantitative comparisons between a state-of-the-art marker-based (MB) method and two variants of the proposed method using gradient correlation (GC) and normalized gradient information (NGI) as similarity measure for the 2D/3D registration.

RESULTS

The authors evaluated their method on four acquisitions under different squatting positions of the same patient. All methods showed substantial improvement in image quality compared to the uncorrected reconstructions. Compared to NGI and MB, the GC method showed increased streaking artifacts due to misregistrations in lateral projection images. NGI and MB showed comparable image quality at the bone regions. Because the markers are attached to the skin, the MB method performed better at the surface of the legs where the authors observed slight streaking of the NGI and GC methods. For a quantitative evaluation, the authors computed the universal quality index (UQI) for all bone regions with respect to the motion-free reconstruction. The authors quantitative evaluation over regions around the bones yielded a mean UQI of 18.4 for no correction, 53.3 and 56.1 for the proposed method using GC and NGI, respectively, and 53.7 for the MB reference approach. In contrast to the authors registration-based corrections, the MB reference method caused slight nonrigid deformations at bone outlines when compared to a motion-free reference scan.

CONCLUSIONS

The authors showed that their method based on the NGI similarity measure yields reconstruction quality close to the MB reference method. In contrast to the MB method, the proposed method does not require any preparation prior to the examination which will improve the clinical workflow and patient comfort. Further, the authors found that the MB method causes small, nonrigid deformations at the bone outline which indicates that markers may not accurately reflect the internal motion close to the knee joint. Therefore, the authors believe that the proposed method is a promising alternative to MB motion management.

Data consistency conditions for truncated fanbeam and parallel projections

PURPOSE

In image reconstruction from projections, data consistency conditions (DCCs) are mathematical relationships that express the overlap of information between ideal projections. DCCs have been incorporated in image reconstruction procedures for positron emission tomography, single photon emission computed tomography, and x-ray computed tomography (CT). Building on published fanbeam DCCs for nontruncated projections along a line, the authors recently announced new DCCs that can be applied to truncated parallel projections in classical (two-dimensional) image reconstruction. These DCCs take the form of polynomial expressions for a weighted backprojection of the projections. The purpose of this work was to present the new DCCs for truncated parallel projections, to extend these conditions to truncated fanbeam projections on a circular trajectory, to verify the conditions with numerical examples, and to present a model of how DCCs could be applied with a toy problem in patient motion estimation with truncated projections.

METHODS

A mathematical derivation of the new parallel DCCs was performed by substituting the underlying imaging equation into the mathematical expression for the weighted backprojection and demonstrating the resulting polynomial form. This DCC result was extended to fanbeam projections by a substitution of parallel to fanbeam variables. Ideal fanbeam projections of a simple mathematical phantom were simulated and the DCCs for these projections were evaluated by fitting polynomials to the weighted backprojection. For the motion estimation problem, a parametrized motion was simulated using a dynamic version of the mathematical phantom, and both noiseless and noisy fanbeam projections were simulated for a full circular trajectory. The fanbeam DCCs were applied to extract the motion parameters, which allowed the motion contamination to be removed from the projections. A reconstruction was performed from the corrected projections.

RESULTS

The mathematical derivation revealed the anticipated polynomial behavior. The conversion to fanbeam variables led to a straight-forward weighted fanbeam backprojection which yielded the same function and therefore the same polynomial behavior as occurred in the parallel case. Plots of the numerically calculated DCCs showed polynomial behavior visually indistinguishable from the fitted polynomials. For the motion estimation problem, the motion parameters were satisfactorily recovered and ten times more accurately for the noise-free case. The reconstructed images showed that only a faint trace of the motion blur was still visible after correction from the noisy motion-contaminated projections.

CONCLUSIONS

New DCCs have been established for fanbeam and parallel projections, and these conditions have been validated using numerical experiments with truncated projections. It has been shown how these DCCs could be applied to extract parameters of unwanted physical effects in tomographic imaging, even with truncated projections.