Exact interior reconstruction with cone-beam CT

Deep-Interior: A new pathway to interior tomographic image reconstruction via a weighted backprojection and deep learning

BACKGROUND

In recent years, deep learning strategies have been combined with either the filtered backprojection or iterative methods or the direct projection-to-image by deep learning only to reconstruct images. Some of these methods can be applied to address the interior reconstruction problems for centered regions of interest (ROIs) with fixed sizes. Developing a method to enable interior tomography with arbitrarily located ROIs with nearly arbitrary ROI sizes inside a scanning field of view (FOV) remains an open question.

PURPOSE

To develop a new pathway to enable interior tomographic reconstruction for arbitrarily located ROIs with arbitrary sizes using a single trained deep neural network model.

METHODS

The method consists of two steps. First, an analytical weighted backprojection reconstruction algorithm was developed to perform domain transform from divergent fan-beam projection data to an intermediate image feature space,B ( x ⃗ ) $B(\vec{x})$ , for an arbitrary size ROI at an arbitrary location inside the FOV. Second, a supervised learning technique was developed to train a deep neural network architecture to perform deconvolution to obtain the true imagef ( x ⃗ ) $f(\vec{x})$ from the new feature spaceB ( x ⃗ ) $B(\vec{x})$ . This two-step method is referred to as Deep-Interior for convenience. Both numerical simulations and experimental studies were performed to validate the proposed Deep-Interior method.

RESULTS

The results showed that ROIs as small as a diameter of 5 cm could be accurately reconstructed (similarity index 0.985 ± 0.018 on internal testing data and 0.940 ± 0.025 on external testing data) at arbitrary locations within an imaging object covering a wide variety of anatomical structures of different body parts. Besides, ROIs of arbitrary size can be reconstructed by stitching small ROIs without additional training.

CONCLUSION

The developed Deep-Interior framework can enable interior tomographic reconstruction from divergent fan-beam projections for short-scan and super-short-scan acquisitions for small ROIs (with a diameter larger than 5 cm) at an arbitrary location inside the scanning FOV with high quantitative reconstruction accuracy.

Suppressing multi-material and streak artifacts with an accelerated 3D iterative image reconstruction algorithm for in-line X-ray phase-contrast computed tomography

In-line X-ray phase-contrast computed tomography typically contains two independent procedures: phase retrieval and computed tomography reconstruction, in which multi-material and streak artifacts are two important problems. To address these problems simultaneously, an accelerated 3D iterative image reconstruction algorithm is proposed. It merges the above-mentioned two procedures into one step, and establishes the data fidelity term in raw projection domain while introducing 3D total variation regularization term in image domain. Specifically, a transport-of-intensity equation (TIE)-based phase retrieval method is updated alternately for different areas of the multi-material sample. Simulation and experimental results validate the effectiveness and efficiency of the proposed algorithm.

Diagonalization of the finite Hilbert transform on two adjacent intervals: the Riemann-Hilbert approach

In this paper we study the spectra of bounded self-adjoint linear operators that are related to finite Hilbert transformsH L : L 2 ( [ b L , 0 ] ) → L 2 ( [ 0 , b R ] ) andH R : L 2 ( [ 0 , b R ] ) → L 2 ( [ b L , 0 ] ) . These operators arise when one studies the interior problem of tomography. The diagonalization ofH R , H L has been previously obtained, but only asymptotically whenb L ≠ - b R . We implement a novel approach based on the method of matrix Riemann-Hilbert problems (RHP) which diagonalizesH R , H L explicitly. We also find the asymptotics of the solution to a related RHP and obtain error estimates.

An adaptive reconstruction algorithm for spectral CT regularized by a reference image

The photon counting detector based spectral CT system is attracting increasing attention in the CT field. However, the spectral CT is still premature in terms of both hardware and software. To reconstruct high quality spectral images from low-dose projections, an adaptive image reconstruction algorithm is proposed that assumes a known reference image (RI). The idea is motivated by the fact that the reconstructed images from different spectral channels are highly correlated. If a high quality image of the same object is known, it can be used to improve the low-dose reconstruction of each individual channel. This is implemented by maximizing the patch-wise correlation between the object image and the RI. Extensive numerical simulations and preclinical mouse study demonstrate the feasibility and merits of the proposed algorithm. It also performs well for truncated local projections, and the surrounding area of the region- of-interest (ROI) can be more accurately reconstructed. Furthermore, a method is introduced to adaptively choose the step length, making the algorithm more feasible and easier for applications.

Interior Tomography from Differential Phase Contrast Data via Hilbert Transform Based on Spline Functions

X-ray phase contrast imaging is an important mode due to its sensitivity to subtle features of soft biological tissues. Grating-based differential phase contrast (DPC) imaging is one of the most promising phase imaging techniques because it works with a normal x-ray tube of a large focal spot at a high flux rate. However, a main obstacle before this paradigm shift is the fabrication of large-area gratings of a small period and a high aspect ratio. Imaging large objects with a size-limited grating results in data truncation which is a new type of the interior problem. While the interior problem was solved for conventional x-ray CT through analytic extension, compressed sensing and iterative reconstruction, the difficulty for interior reconstruction from DPC data lies in that the implementation of the system matrix requires the differential operation on the detector array, which is often inaccurate and unstable in the case of noisy data. Here, we propose an iterative method based on spline functions. The differential data are first back-projected to the image space. Then, a system matrix is calculated whose components are the Hilbert transforms of the spline bases. The system matrix takes the whole image as an input and outputs the back-projected interior data. Prior information normally assumed for compressed sensing is enforced to iteratively solve this inverse problem. Our results demonstrate that the proposed algorithm can successfully reconstruct an interior region of interest (ROI) from the differential phase data through the ROI.

Towards clinical application of a Laplace operator-based region of interest reconstruction algorithm in C-arm CT

It is known that a reduction of the field-of-view in 3-D X-ray imaging is proportional to a reduction in radiation dose. The resulting truncation, however, is incompatible with conventional reconstruction algorithms. Recently, a novel method for region of interest reconstruction that uses neither prior knowledge nor extrapolation has been published, named approximated truncation robust algorithm for computed tomography (ATRACT). It is based on a decomposition of the standard ramp filter into a 2-D Laplace filtering and a 2-D Radon-based residual filtering step. In this paper, we present two variants of the original ATRACT. One is based on expressing the residual filter as an efficient 2-D convolution with an analytically derived kernel. The second variant is to apply ATRACT in 1-D to further reduce computational complexity. The proposed algorithms were evaluated by using a reconstruction benchmark, as well as two clinical data sets. The results are encouraging since the proposed algorithms achieve a speed-up factor of up to 245 compared to the 2-D Radon-based ATRACT. Reconstructions of high accuracy are obtained, e.g., even real-data reconstruction in the presence of severe truncation achieve a relative root mean square error of as little as 0.92% with respect to nontruncated data.

Gel'fand-Graev's reconstruction formula in the 3D real space

PURPOSE

Gel'fand and Graev performed classical work on the inversion of integral transforms in different spaces [Gel'fand and Graev, Funct. Anal. Appl. 25(1) 1-5 (1991)]. This paper discusses their key results for further research and development.

METHODS

The Gel'fand-Graev inversion formula reveals a fundamental relationship between projection data and the Hilbert transform of an image to be reconstructed. This differential backprojection (DBP)∕backprojection filtration (BPF) approach was rediscovered in the CT field, and applied in important applications such as reconstruction from truncated projections, interior tomography, and limited-angle tomography. Here the authors present the Gel'fand-Graev inversion formula in a 3D setting assuming the 1D x-ray transform.

RESULTS

The pseudodifferential operator is a powerful theoretical tool. There is a fundamental mathematical link between the Gel'fand-Graev formula and the DBP (or BPF) approach in the case of the 1D x-ray transform in a 3D real space.

CONCLUSIONS

This paper shows the power of mathematics for tomographic imaging and the value of a pure theoretical finding, which may appear quite irrelevant to daily healthcare at the first glance.

The meaning of interior tomography

The classic imaging geometry for computed tomography is for the collection of un-truncated projections and the reconstruction of a global image, with the Fourier transform as the theoretical foundation that is intrinsically non-local. Recently, interior tomography research has led to theoretically exact relationships between localities in the projection and image spaces and practically promising reconstruction algorithms. Initially, interior tomography was developed for x-ray computed tomography. Then, it was elevated to have the status of a general imaging principle. Finally, a novel framework known as 'omni-tomography' is being developed for a grand fusion of multiple imaging modalities, allowing tomographic synchrony of diversified features.

Interior tomography with continuous singular value decomposition

The long-standing interior problem has important mathematical and practical implications. The recently developed interior tomography methods have produced encouraging results. A particular scenario for theoretically exact interior reconstruction from truncated projections is that there is a known sub-region in the ROI. In this paper, we improve a novel continuous singular value decomposition (SVD) method for interior reconstruction assuming a known sub-region. First, two sets of orthogonal eigen-functions are calculated for the Hilbert and image spaces respectively. Then, after the interior Hilbert data are calculated from projection data through the ROI, they are projected onto the eigen-functions in the Hilbert space, and an interior image is recovered by a linear combination of the eigen-functions with the resulting coefficients. Finally, the interior image is compensated for the ambiguity due to the null space utilizing the prior sub-region knowledge. Experiments with simulated and real data demonstrate the advantages of our approach relative to the POCS type interior reconstructions.

Time-resolved cardiac interventional cone-beam CT reconstruction from fully truncated projections using the prior image constrained compressed sensing (PICCS) algorithm

C-arm cone-beam CT could replace preoperative multi-detector CT scans in the cardiac interventional setting. However, cardiac gating results in view angle undersampling and the small size of the detector results in projection data truncation. These problems are incompatible with conventional tomographic reconstruction algorithms. In this paper, the prior image constrained compressed sensing (PICCS) reconstruction method was adapted to solve these issues. The performance of the proposed method was compared to that of FDK, FDK with extrapolated projection data (E-FDK), and total variation-based compressed sensing. A canine projection dataset acquired using a clinical C-arm imaging system supplied realistic cardiac motion and anatomy for this evaluation. Three different levels of truncation were simulated. The relative root mean squared error and the universal image quality index were used to quantify the reconstruction accuracy. Three main conclusions were reached. (1) The adapted version of the PICCS algorithm offered the highest image quality and reconstruction accuracy. (2) No meaningful variation in performance was observed when the amount of truncation was changed. (3) This study showed evidence that accurate interior tomography with an undersampled acquisition is possible for realistic objects if a prior image with minimal artifacts is available.

Completeness map evaluation demonstrated with candidate next-generation cardiac CT architectures

PURPOSE

In this report, the authors introduce the general concept of the completeness map, as a means to evaluate the completeness of data acquired by a given CT system design (architecture and scan mode). They illustrate the utility of completeness map by applying the completeness map concept to a number of candidate CT system designs, as part of a study to advance the state-of-the-art in cardiac CT.

METHODS

In order to optimally reconstruct a point within a volume of interest (VOI), the Radon transform on all possible planes through that point should be measured. The authors quantified the extent to which this ideal condition is satisfied for the entire image volume. They first determined a Radon completeness number for each point in the VOI, as the percentage of possible planes that is actually measured. A completeness map is then defined as a 3D matrix of the completeness numbers for the entire VOI. The authors proposed algorithms to analyze the projection datasets in Radon space and compute the completeness number for a fixed point and apply these algorithms to various architectures and scan modes that they are evaluating. In this report, the authors consider four selected candidate architectures, operating with different scan modes, for a total of five system design alternatives. Each of these alternatives is evaluated using completeness map.

RESULTS

If the detector size and cone angle are large enough to cover the entire cardiac VOI, a single-source circular scan can have ≥99% completeness over the entire VOI. However, only the central z-slice can be exactly reconstructed, which corresponds to 100% completeness. For a typical single-source architecture, if the detector is limited to an axial dimension of 40 mm, a helical scan needs about five rotations to form an exact reconstruction region covering the cardiac VOI, while a triple-source helical scan only requires two rotations, leading to a 2.5x improvement in temporal resolution. If the source and detector of an inverse-geometry (IGCT) system have the same axial extent, and the spacing of source points in the axial and transaxial directions is sufficiently small, the IGCT can also form an exact reconstruction region for the cardiac VOI. If the VOI can be covered by the x-ray beam in any view, a composite-circling scan can generate an exact reconstruction region covering the VOI.

CONCLUSIONS

The completeness map evaluation provides useful information for selecting the next-generation cardiac CT system design. The proposed completeness map method provides a practical tool for analyzing complex scanning trajectories, where the theoretical image quality for some complex system designs is impossible to predict, without yet-undeveloped reconstruction algorithms.

Image reconstruction for hybrid true-color micro-CT

X-ray micro-CT is an important imaging tool for biomedical researchers. Our group has recently proposed a hybrid "true-color" micro-CT system to improve contrast resolution with lower system cost and radiation dose. The system incorporates an energy-resolved photon-counting true-color detector into a conventional micro-CT configuration, and can be used for material decomposition. In this paper, we demonstrate an interior color-CT image reconstruction algorithm developed for this hybrid true-color micro-CT system. A compressive sensing-based statistical interior tomography method is employed to reconstruct each channel in the local spectral imaging chain, where the reconstructed global gray-scale image from the conventional imaging chain served as the initial guess. Principal component analysis was used to map the spectral reconstructions into the color space. The proposed algorithm was evaluated by numerical simulations, physical phantom experiments, and animal studies. The results confirm the merits of the proposed algorithm, and demonstrate the feasibility of the hybrid true-color micro-CT system. Additionally, a "color diffusion" phenomenon was observed whereby high-quality true-color images are produced not only inside the region of interest, but also in neighboring regions. It appears harnessing that this phenomenon could potentially reduce the color detector size for a given ROI, further reducing system cost and radiation dose.

High-order total variation minimization for interior SPECT

Recently, we developed an approach for solving the computed tomography (CT) interior problem based on the high-order TV (HOT) minimization, assuming that a region-of-interest (ROI) is piecewise polynomial. In this paper, we generalize this finding from the CT field to the single-photon emission computed tomography (SPECT) field, and prove that if an ROI is piecewise polynomial, then the ROI can be uniquely reconstructed from the SPECT projection data associated with the ROI through the HOT minimization. Also, we propose a new formulation of HOT, which has an explicit formula for any n-order piecewise polynomial function, while the original formulation has no explicit formula for n ≥ 2. Finally, we verify our theoretical results in numerical simulation, and discuss relevant issues.

STABILITY OF THE INTERIOR PROBLEM FOR POLYNOMIAL REGION OF INTEREST

In many practical applications, it is desirable to solve the interior problem of tomography without requiring knowledge of the attenuation function f_a on an open set within the region of interest (ROI). It was proved recently that the interior problem has a unique solution if f_a is assumed to be piecewise polynomial on the ROI. In this paper, we tackle the related question of stability. It is well-known that lambda tomography allows one to stably recover the locations and values of the jumps of f_a inside the ROI from only the local data. Hence, we consider here only the case of a polynomial, rather than piecewise polynomial, f_a on the ROI. Assuming that the degree of the polynomial is known, along with some other fairly mild assumptions on f_a , we prove a stability estimate for the interior problem. Additionally, we prove the following general uniqueness result. If there is an open set U on which f_a is the restriction of a real-analytic function, then f_a is uniquely determined by only the line integrals through U. It turns out that two known uniqueness theorems are corollaries of this result.

Four-dimensional volume-of-interest reconstruction for cone-beam computed tomography-guided radiation therapy

PURPOSE

Data sufficiency are a major problem in four-dimensional cone-beam computed tomography (4D-CBCT) on linear accelerator-integrated scanners for image-guided radiotherapy. Scan times must be in the range of 4-6 min to avoid undersampling artifacts. Various image reconstruction algorithms have been proposed to accommodate undersampled data acquisitions, but these algorithms are computationally expensive, may require long reconstruction times, and may require algorithm parameters to be optimized. The authors present a novel reconstruction method, 4D volume-of-interest (4D-VOI) reconstruction which suppresses undersampling artifacts and resolves lung tumor motion for undersampled 1-min scans. The 4D-VOI reconstruction is much less computationally expensive than other 4D-CBCT algorithms.

METHODS

The 4D-VOI method uses respiration-correlated projection data to reconstruct a four-dimensional (4D) image inside a VOI containing the moving tumor, and uncorrelated projection data to reconstruct a three-dimensional (3D) image outside the VOI. Anatomical motion is resolved inside the VOI and blurred outside the VOI. The authors acquired a 1-min. scan of an anthropomorphic chest phantom containing a moving water-filled sphere. The authors also used previously acquired 1-min scans for two lung cancer patients who had received CBCT-guided radiation therapy. The same raw data were used to test and compare the 4D-VOI reconstruction with the standard 4D reconstruction and the McKinnon-Bates (MB) reconstruction algorithms.

RESULTS

Both the 4D-VOI and the MB reconstructions suppress nearly all the streak artifacts compared with the standard 4D reconstruction, but the 4D-VOI has 3-8 times greater contrast-to-noise ratio than the MB reconstruction. In the dynamic chest phantom study, the 4D-VOI and the standard 4D reconstructions both resolved a moving sphere with an 18 mm displacement. The 4D-VOI reconstruction shows a motion blur of only 3 mm, whereas the MB reconstruction shows a motion blur of 13 mm. With graphics processing unit hardware used to accelerate computations, the 4D-VOI reconstruction required a 40-s reconstruction time.

CONCLUSIONS

4D-VOI reconstruction effectively reduces undersampling artifacts and resolves lung tumor motion in 4D-CBCT. The 4D-VOI reconstruction is computationally inexpensive compared with more sophisticated iterative algorithms. Compared with these algorithms, our 4D-VOI reconstruction is an attractive alternative in 4D-CBCT for reconstructing target motion without generating numerous streak artifacts.

Compressive sensing-based interior tomography: preliminary clinical application

Compressive sensing (CS)-based interior tomography is a state-of-the-art method for accurate image reconstruction from only locally truncated projections. Here, we report our preliminary interior tomography results reconstructed from raw projections of a patient acquired on a GE Discovery CT750 HD scanner. This is the first clinical application of the CS-based interior reconstruction techniques, and the results show an excellent match with those reconstructed from global projections.

Image reconstruction from limited angle projections collected by multisource interior x-ray imaging systems

A multisource x-ray interior imaging system with limited angle scanning is investigated to study the possibility of building an ultrafast micro-CT for dynamic small animal imaging, and two methods are employed to perform interior reconstruction from a limited number of projections collected by the multisource interior x-ray system. The first is total variation minimization with the steepest descent search (TVM-SD) and the second is total difference minimization with soft-threshold filtering (TDM-STF). Comprehensive numerical simulations and animal studies are performed to validate the associated reconstructed methods and demonstrate the feasibility and application of the proposed system configuration. The image reconstruction results show that both of the two reconstruction methods can significantly improve the image quality and the TDM-SFT is slightly superior to the TVM-SD. Finally, quantitative image analysis shows that it is possible to make an ultrafast micro-CT using a multisource interior x-ray system scheme combined with the state-of-the-art interior tomography.

Short echo-time 3D radial gradient-echo MRI using concurrent dephasing and excitation

Ultrashort echo-time imaging and sweep imaging with Fourier transformation are powerful techniques developed for imaging ultrashort T(2) species. However, it can be challenging to implement them on standard clinical MRI systems due to demanding hardware requirements. In this article, the limits of what is possible in terms of the minimum echo-time and repetition time with 3D radial gradient-echo sequences, which can be readily implemented on a standard clinical scanner, are investigated. Additionally, a new 3D radial gradient-echo sequence is introduced, called COncurrent Dephasing and Excitation (CODE). The unique feature of CODE is that the initial dephasing of the readout gradient is performed during RF excitation, which allows CODE to effectively achieve echo-times on the order of ∼0.2 ms and larger in a clinical setting. The minimum echo-time achievable with CODE is analytically described and compared with a standard 3D radial gradient-echo sequence. CODE was implemented on a clinical 3 T scanner (Siemens 3 T MAGNETOM Trio), and both phantom and in vivo human knee images are shown for demonstration.

Statistical interior tomography

This paper presents a statistical interior tomography (SIT) approach making use of compressed sensing (CS) theory. With the projection data modeled by the Poisson distribution, an objective function with a total variation (TV) regularization term is formulated in the maximization of a posteriori (MAP) framework to solve the interior problem. An alternating minimization method is used to optimize the objective function with an initial image from the direct inversion of the truncated Hilbert transform. The proposed SIT approach is extensively evaluated with both numerical and real datasets. The results demonstrate that SIT is robust with respect to data noise and down-sampling, and has better resolution and less bias than its deterministic counterpart in the case of low count data.

Exact interior reconstruction from truncated limited-angle projection data

Using filtered backprojection (FBP) and an analytic continuation approach, we prove that exact interior reconstruction is possible and unique from truncated limited-angle projection data, if we assume a prior knowledge on a subregion or subvolume within an object to be reconstructed. Our results show that (i) the interior region-of-interest (ROI) problem and interior volume-of-interest (VOI) problem can be exactly reconstructed from a limited-angle scan of the ROI/VOI and a 180 degree PI-scan of the subregion or subvolume and (ii) the whole object function can be exactly reconstructed from nontruncated projections from a limited-angle scan. These results improve the classical theory of Hamaker et al. (1980).

Interior SPECT- Exact and Stable ROI Reconstruction from Uniformly Attenuated Local Projections

Single photon emission computed tomography (SPECT) is an important biomedical imaging modality. However, since gamma cameras are expensive and bulky, truncated projection data are either preferred or unavoidable. Inspired by the recent results on interior tomography in the x-ray CT field, here we present the interior SPECT approach for exact and stable reconstruction of a region of interest (ROI) from uniformly attenuated local projection data, aided by prior knowledge of a sub-region in the ROI. First, by analytic continuation we prove that interior SPECT is both exact and stable, and by singular value decomposition (SVD) we establish the stability of interior SPECT. Then, given the constant attenuation coefficient and object boundary, our interior SPECT reconstruction is achieved by inverting a generalized truncated Hilbert transform using the SVD technique. Preliminary numerical simulation data demonstrate that our work has practical utilities. The theoretical generalization of our work to the variable attenuation case is underway, and the same numerical approach can be applied.

Compressed sensing based interior tomography

While conventional wisdom is that the interior problem does not have a unique solution, by analytic continuation we recently showed that the interior problem can be uniquely and stably solved if we have a known sub-region inside a region of interest (ROI). However, such a known sub-region is not always readily available, and it is even impossible to find in some cases. Based on compressed sensing theory, here we prove that if an object under reconstruction is essentially piecewise constant, a local ROI can be exactly and stably reconstructed via the total variation minimization. Because many objects in computed tomography (CT) applications can be approximately modeled as piecewise constant, our approach is practically useful and suggests a new research direction for interior tomography. To illustrate the merits of our finding, we develop an iterative interior reconstruction algorithm that minimizes the total variation of a reconstructed image and evaluate the performance in numerical simulation.

A filtered backprojection algorithm for triple-source helical cone-beam CT

Multisource cone-beam computed tomography (CT) is an attractive approach of choice for superior temporal resolution, which is critically important for cardiac imaging and contrast enhanced studies. In this paper, we present a filtered-backprojection (FBP) algorithm for triple-source helical cone-beam CT. The algorithm is both exact and efficient. It utilizes data from three inter-helix PI-arcs associated with the inter-helix PI-lines and the minimum detection windows defined for the triple-source configuration. The proof of the formula is based on the geometric relations specific to triple-source helical cone-beam scanning. Simulation results demonstrate the validity of the reconstruction algorithm. This algorithm is also extended to a multisource version for (2N + 1)-source helical cone-beam CT. With parallel computing, the proposed FBP algorithms can be significantly faster than our previously published multisource backprojection-filtration algorithms. Thus, the FBP algorithms are promising in applications of triple-source helical cone-beam CT.

Compressive sampling based interior reconstruction for dynamic carbon nanotube micro-CT

In the computed tomography (CT) field, one recent invention is the so-called carbon nanotube (CNT) based field emission x-ray technology. On the other hand, compressive sampling (CS) based interior tomography is a new innovation. Combining the strengths of these two novel subjects, we apply the interior tomography technique to local mouse cardiac imaging using respiration and cardiac gating with a CNT based micro-CT scanner. The major features of our method are: (1) it does not need exact prior knowledge inside an ROI; and (2) two orthogonal scout projections are employed to regularize the reconstruction. Both numerical simulations and in vivo mouse studies are performed to demonstrate the feasibility of our methodology.

SOLVING THE INTERIOR PROBLEM OF COMPUTED TOMOGRAPHY USING A PRIORI KNOWLEDGE

The case of incomplete tomographic data for a compactly supported attenuation function is studied. When the attenuation function is a priori known in a subregion, we show that a reduced set of measurements is enough to uniquely determine the attenuation function over all the space. Furthermore, we found stability estimates showing that reconstruction can be stable near the region where the attenuation is known. These estimates also suggest that reconstruction stability collapses quickly when approaching the set of points that are viewed under less than 180 degrees. This paper may be seen as a continuation of the work "Truncated Hilbert transform and Image reconstruction from limited tomographic data" that was published in Inverse Problems in 2006. This continuation tackles new cases of incomplete data that could be of interest in applications of computed tomography.

An outlook on x-ray CT research and development

Over the past decade, computed tomography (CT) theory, techniques and applications have undergone a rapid development. Since CT is so practical and useful, undoubtedly CT technology will continue advancing biomedical and non-biomedical applications. In this outlook article, we share our opinions on the research and development in this field, emphasizing 12 topics we expect to be critical in the next decade: analytic reconstruction, iterative reconstruction, local/interior reconstruction, flat-panel based CT, dual-source CT, multi-source CT, novel scanning modes, energy-sensitive CT, nano-CT, artifact reduction, modality fusion, and phase-contrast CT. We also sketch several representative biomedical applications.