Statistical inversion for medical x-ray tomography with few radiographs: II. Application to dental radiology

Iterative tomographic reconstruction with TV prior for low-dose CBCT dental imaging

Objective.Cone-beam computed tomography is becoming more and more popular in applications such as 3D dental imaging. Iterative methods compared to the standard Feldkamp algorithm have shown improvements in image quality of reconstruction of low-dose acquired data despite their long computing time. An interesting aspect of iterative methods is their ability to include prior information such as sparsity-constraint. While a large panel of optimization algorithms along with their adaptation to tomographic problems are available, they are mainly studied on 2D parallel or fan-beam data. The issues raised by 3D CBCT and moreover by truncated projections are still poorly understood.Approach.We compare different carefully designed optimization schemes in the context of realistic 3D dental imaging. Besides some known algorithms, SIRT-TV and MLEM, we investigate the primal-dual hybrid gradient (PDHG) approach and a newly proposed MLEM-TV optimizer. The last one is alternating EM steps and TV-denoising, combination not yet investigated for CBCT. Experiments are performed on both simulated data from a 3D jaw phantom and data acquired with a dental clinical scanner.Main results.With some adaptations to the specificities of CBCT operators, PDHG and MLEM-TV algorithms provide the best reconstruction quality. These results were obtained by comparing the full-dose image with a low-dose image and an ultra low-dose image.Significance.The convergence speed of the original iterative methods is hampered by the conical geometry and significantly reduced compared to parallel geometries. We promote the pre-conditioned version of PDHG and we propose a pre-conditioned version of the MLEM-TV algorithm. To the best of our knowledge, this is the first time PDHG and convergent MLEM-TV algorithms are evaluated on experimental dental CBCT data, where constraints such as projection truncation and presence of metal have to be jointly overcome.

Multi-Scale Learned Iterative Reconstruction

Model-based learned iterative reconstruction methods have recently been shown to outperform classical reconstruction algorithms. Applicability of these methods to large scale inverse problems is however limited by the available memory for training and extensive training times, the latter due to computationally expensive forward models. As a possible solution to these restrictions we propose a multi-scale learned iterative reconstruction scheme that computes iterates on discretisations of increasing resolution. This procedure does not only reduce memory requirements, it also considerably speeds up reconstruction and training times, but most importantly is scalable to large scale inverse problems with non-trivial forward operators, such as those that arise in many 3D tomographic applications. In particular, we propose a hybrid network that combines the multiscale iterative approach with a particularly expressive network architecture which in combination exhibits excellent scalability in 3D. Applicability of the algorithm is demonstrated for 3D cone beam computed tomography from real measurement data of an organic phantom. Additionally, we examine scalability and reconstruction quality in comparison to established learned reconstruction methods in two dimensions for low dose computed tomography on human phantoms.

Impact of statistical reconstruction and compressed sensing algorithms on projection data elimination during X-ray CT image reconstruction

OBJECTIVES

To examine the effect of incomplete, or total elimination of, projection data on computed tomography (CT) images subjected to statistical reconstruction and/or compressed sensing algorithms.

METHODS

Multidetector row CT images were used. The algebraic reconstruction technique (ART) and the maximum likelihood-expectation maximization (ML-EM) method were compared with filtered back-projection (FBP). Effects on reconstructed images were studied when the projection data of 360° (360 projections) were decreased to 180 or 90 projections by reducing the collection angle or thinning the image data. The total variation (TV) regularization method using compressed sensing was applied to images processed by the ART. Image noise was subjectively evaluated using the root-mean-square error and signal-to-noise ratio.

RESULTS

When projection data were reduced by one-half or three-quarters, ART and ML-EM produced better image quality than FBP. Both ART and ML-EM resulted in high quality at a spread of 90 projections over 180° rotation. Computational loading was high for statistical reconstruction, but not for ML-EM, compared with the ART. TV regularization made it possible to use only 36 projections while still achieving acceptable image quality.

CONCLUSIONS

Incomplete projection data-accomplished by reducing the angle to collect image data or thinning the projection data without reducing the angle of rotation over which it is collected-made it possible to reduce the radiation dose while retaining image quality with statistical reconstruction algorithms and/or compressed sensing. Despite heavier computational calculation loading, these methods should be considered for reducing radiation doses.

ℓ0 Gradient Minimization Based Image Reconstruction for Limited-Angle Computed Tomography

In medical and industrial applications of computed tomography (CT) imaging, limited by the scanning environment and the risk of excessive X-ray radiation exposure imposed to the patients, reconstructing high quality CT images from limited projection data has become a hot topic. X-ray imaging in limited scanning angular range is an effective imaging modality to reduce the radiation dose to the patients. As the projection data available in this modality are incomplete, limited-angle CT image reconstruction is actually an ill-posed inverse problem. To solve the problem, image reconstructed by conventional filtered back projection (FBP) algorithm frequently results in conspicuous streak artifacts and gradual changed artifacts nearby edges. Image reconstruction based on total variation minimization (TVM) can significantly reduce streak artifacts in few-view CT, but it suffers from the gradual changed artifacts nearby edges in limited-angle CT. To suppress this kind of artifacts, we develop an image reconstruction algorithm based on ℓ₀ gradient minimization for limited-angle CT in this paper. The ℓ₀-norm of the image gradient is taken as the regularization function in the framework of developed reconstruction model. We transformed the optimization problem into a few optimization sub-problems and then, solved these sub-problems in the manner of alternating iteration. Numerical experiments are performed to validate the efficiency and the feasibility of the developed algorithm. From the statistical analysis results of the performance evaluations peak signal-to-noise ratio (PSNR) and normalized root mean square distance (NRMSD), it shows that there are significant statistical differences between different algorithms from different scanning angular ranges (p<0.0001). From the experimental results, it also indicates that the developed algorithm outperforms classical reconstruction algorithms in suppressing the streak artifacts and the gradual changed artifacts nearby edges simultaneously.

Statistical image reconstruction from limited projection data with intensity priors

The radiation dose generated from x-ray computed tomography (CT) scans and its responsibility for increasing the risk of malignancy became a major concern in the medical imaging community. Accordingly, investigating possible approaches for image reconstruction from low-dose imaging protocols, which minimize the patient radiation exposure without affecting image quality, has become an issue of interest. Statistical reconstruction (SR) methods are known to achieve a superior image quality compared with conventional analytical methods. Effective physical noise modeling and possibilities of incorporating priors in the image reconstruction problem are the main advantages of the SR methods. Nevertheless, the high computation cost limits its wide use in clinical scanners. This paper presents a framework for SR in x-ray CT when the angular sampling rate of the projection data is low. The proposed framework is based on the fact that, in many CT imaging applications, some physical and anatomical structures and the corresponding attenuation information of the scanned object can be a priori known. Therefore, the x-ray attenuation distribution in some regions of the object can be expected prior to the reconstruction. Under this assumption, the proposed method is developed by incorporating this prior information into the image reconstruction objective function to suppress streak artifacts. We limit the prior information to only a set of intensity values that represent the average intensity of the normal and expected homogeneous regions within the scanned object. This prior information can be easily computed in several x-ray CT applications. Considering the theory of compressed sensing, the objective function is formulated using the ℓ(1) norm distance between the reconstructed image and the available intensity priors. Experimental comparative studies applied to simulated data and real data are used to evaluate the proposed method. The comparison indicates a significant improvement in image quality when the proposed method is used.

A patient set-up protocol based on partially blocked cone-beam CT

Three-dimensional x-ray cone-beam CT (CBCT) is being increasingly used in radiation therapy. Since the whole treatment course typically lasts several weeks, the repetitive x-ray imaging results in large radiation dose delivered on the patient. In the current radiation therapy treatment, CBCT is mainly used for patient set-up, and a rigid transformation of the CBCT data from the planning CT data is also assumed. For an accurate rigid registration, it is not necessary to acquire a full 3D image. In this paper, we propose a patient set-up protocol based on partially blocked CBCT. A sheet of lead strips is inserted between the x-ray source and the scanned patient. From the incomplete projection data, only several axial slices are reconstructed and used in the image registration for patient set-up. Since the radiation is partially blocked, the dose delivered onto the patient is significantly reduced, with an additional benefit of reduced scatter signals. The proposed approach is validated using experiments on two anthropomorphic phantoms. As x-ray beam blocking ratio increases, more dose reduction is achieved, while the patient set-up error also increases. To investigate this tradeoff, two lead sheets with different strip widths are implemented, which correspond to radiation dose reduction of approximately 6 and approximately 11, respectively. We compare the registration results using the partially blocked CBCT with those using the regular CBCT. Both lead sheets achieve high patient set-up accuracies. It is seen that, using the lead sheet with radiation dose reduction by a factor of approximately 11, the patient set-up error is still less than 1mm in translation and less than 0.2 degrees in rotation. The comparison of the reconstructed images also shows that the image quality of the illuminated slices in the partially blocked CBCT is much improved over that in the regular CBCT.

Performance comparison between total variation (TV)-based compressed sensing and statistical iterative reconstruction algorithms

Of all available reconstruction methods, statistical iterative reconstruction algorithms appear particularly promising since they enable accurate physical noise modeling. The newly developed compressive sampling/compressed sensing (CS) algorithm has shown the potential to accurately reconstruct images from highly undersampled data. The CS algorithm can be implemented in the statistical reconstruction framework as well. In this study, we compared the performance of two standard statistical reconstruction algorithms (penalized weighted least squares and q-GGMRF) to the CS algorithm. In assessing the image quality using these iterative reconstructions, it is critical to utilize realistic background anatomy as the reconstruction results are object dependent. A cadaver head was scanned on a Varian Trilogy system at different dose levels. Several figures of merit including the relative root mean square error and a quality factor which accounts for the noise performance and the spatial resolution were introduced to objectively evaluate reconstruction performance. A comparison is presented between the three algorithms for a constant undersampling factor comparing different algorithms at several dose levels. To facilitate this comparison, the original CS method was formulated in the framework of the statistical image reconstruction algorithms. Important conclusions of the measurements from our studies are that (1) for realistic neuro-anatomy, over 100 projections are required to avoid streak artifacts in the reconstructed images even with CS reconstruction, (2) regardless of the algorithm employed, it is beneficial to distribute the total dose to more views as long as each view remains quantum noise limited and (3) the total variation-based CS method is not appropriate for very low dose levels because while it can mitigate streaking artifacts, the images exhibit patchy behavior, which is potentially harmful for medical diagnosis.

Digital tomosynthesis aided by low-resolution exact computed tomography

Tomosynthesis reconstructs 3-dimensional images of an object from a significantly fewer number of projections as compared with that required by computed tomography (CT). A major problem with tomosynthesis is image artifacts associated with the data incompleteness. In this article, we propose a hybrid tomosynthesis approach to achieve higher image quality as compared with competing methods. In this approach, a low-resolution CT scan is followed by a high-resolution tomosynthesis scan. Then, both scans are combined to reconstruct images. To evaluate the image quality of the proposed method, we design a new breast phantom for numerical simulation and physical experiments. The results show that images obtained by our approach are clearly better than those obtained without such a CT scan.

A Bayesian reconstruction method for micro-rotation imaging in light microscopy

The authors present a three-dimensional (3D) reconstruction algorithm and reconstruction-based deblurring method for light microscopy using a micro-rotation device. In contrast to conventional 3D optical imaging where the focal plane is shifted along the optical axis, micro-rotation imaging employs dielectric fields to rotate the object inside a fixed optical set-up. To address this entirely new 3D-imaging modality, the authors present a reconstruction algorithm based on Bayesian inversion theory and use the total variation function as a structure prior. The spectral properties of the reconstruction by simulations that illustrate the strengths and the weaknesses of the micro-rotation approach, compared with conventional 3D optical imaging, were studied. The reconstruction from real data sets shows that this method is promising for 3D reconstruction and offers itself as a deblurring method using a reconstruction-based procedure for removing out-of-focus light from the micro-rotation image series.

Structure-from-motion without correspondence from tomographic projections by Bayesian inversion theory

In conventional tomography, the interior of an object is reconstructed from tomographic projections such as X-ray or transmission electron microscope images. All the current reconstruction methods assume that projection geometry of the imaging device is either known or solved in advance by using e.g., fiducial or nonfiducial feature points in the images. In this paper, we propose a novel approach where the imaging geometry is solved simultaneously with the volume reconstruction problem while no correspondence information is needed. Our approach is a direct application of Bayesian inversion theory and produces the maximum likelihood or maximum a posteriori estimates for the motion parameters under the selected noise and prior distributions. In this paper, the method is implemented for a two-dimensional model problem with one-dimensional affine projection data. The performance of the method is tested with simulated and measured X-ray projection data.

Multiresolution local tomography in dental radiology using wavelets

A Bayesian multiresolution model for local tomography in dental radiology is proposed. In this model a wavelet basis is used to present dental structures and the prior information is modeled in terms of Besov norm penalty. The proposed wavelet-based multiresolution method is used to reduce the number of unknowns in the reconstruction problem by abandoning fine-scale wavelets outside the region of interest (ROI). This multiresolution model allows significant reduction in the number of unknowns without the loss of reconstruction accuracy inside the ROI. The feasibility of the proposed method is tested with two-dimensional (2D) examples using simulated and experimental projection data from dental specimens.

Wavelet-based reconstruction for limited-angle X-ray tomography

The aim of X-ray tomography is to reconstruct an unknown physical body from a collection of projection images. When the projection images are only available from a limited angle of view, the reconstruction problem is a severely ill-posed inverse problem. Statistical inversion allows stable solution of the limited-angle tomography problem by complementing the measurement data by a priori information. In this work, the unknown attenuation distribution inside the body is represented as a wavelet expansion, and a Besov space prior distribution together with positivity constraint is used. The wavelet expansion is thresholded before reconstruction to reduce the dimension of the computational problem. Feasibility of the method is demonstrated by numerical examples using in vitro data from mammography and dental radiology.

Parallelized Bayesian inversion for three-dimensional dental X-ray imaging

Diagnostic and operational tasks based on dental radiology often require three-dimensional (3-D) information that is not available in a single X-ray projection image. Comprehensive 3-D information about tissues can be obtained by computerized tomography (CT) imaging. However, in dental imaging a conventional CT scan may not be available or practical because of high radiation dose, low-resolution or the cost of the CT scanner equipment. In this paper, we consider a novel type of 3-D imaging modality for dental radiology. We consider situations in which projection images of the teeth are taken from a few sparsely distributed projection directions using the dentist's regular (digital) X-ray equipment and the 3-D X-ray attenuation function is reconstructed. A complication in these experiments is that the reconstruction of the 3-D structure based on a few projection images becomes an ill-posed inverse problem. Bayesian inversion is a well suited framework for reconstruction from such incomplete data. In Bayesian inversion, the ill-posed reconstruction problem is formulated in a well-posed probabilistic form in which a priori information is used to compensate for the incomplete information of the projection data. In this paper we propose a Bayesian method for 3-D reconstruction in dental radiology. The method is partially based on Kolehmainen et al. 2003. The prior model for dental structures consist of a weighted l1 and total variation (TV)-prior together with the positivity prior. The inverse problem is stated as finding the maximum a posteriori (MAP) estimate. To make the 3-D reconstruction computationally feasible, a parallelized version of an optimization algorithm is implemented for a Beowulf cluster computer. The method is tested with projection data from dental specimens and patient data. Tomosynthetic reconstructions are given as reference for the proposed method.

Software for automated application of a reference-based method for a posteriori determination of the effective radiographic imaging geometry

OBJECTIVES

Presentation and validation of software developed for automated and accurate application of a reference-based algorithm (reference sphere method: RSM) inferring the effective imaging geometry from quantitative radiographic image analysis.

METHODS

The software uses modern pattern recognition and computer vision algorithms adapted for the particular application of automated detection of the reference sphere shadows (ellipses) with subpixel accuracy. It applies the RSM algorithm to the shadows detected, thereby providing three-dimensional Cartesian coordinates of the spheres. If the three sphere centres do not lie on one line, they uniquely determine the imaging geometry. Accuracy of the computed coordinates is investigated in a set of 28 charge-coupled device (CCD)-based radiographs of two human mandible segments produced on an optical bench. Each specimen contained three reference spheres (two different radii r1=1.5 mm, r2=2.5 mm). True sphere coordinates were assessed with a manually operated calliper. Software accuracy was investigated for a weighted and unweighted algebraic ellipse-fitting algorithm.

RESULTS

The critical depth- (z-) coordinates revealed mean absolute errors ranging between 1.1+/-0.7 mm (unweighted version; r=2.5 mm) and 1.4+/-1.4 mm (weighted version, r=2.5 mm), corresponding to mean relative errors between 5% and 6%. Outliers resulted from complete circular dense structure superimposition and one obviously deformed reference sphere.

CONCLUSIONS

The software provides information fundamentally important for the image formation and geometric image registration, which is a crucial step for three-dimensional reconstruction from > or =2 two-dimensional views.