Low-dose x-ray phase-contrast and absorption CT using equally sloped tomography

Accurate real space iterative reconstruction (RESIRE) algorithm for tomography

Tomography has made a revolutionary impact on the physical, biological and medical sciences. The mathematical foundation of tomography is to reconstruct a three-dimensional (3D) object from a set of two-dimensional (2D) projections. As the number of projections that can be measured from a sample is usually limited by the tolerable radiation dose and/or the geometric constraint on the tilt range, a main challenge in tomography is to achieve the best possible 3D reconstruction from a limited number of projections with noise. Over the years, a number of tomographic reconstruction methods have been developed including direct inversion, real-space, and Fourier-based iterative algorithms. Here, we report the development of a real-space iterative reconstruction (RESIRE) algorithm for accurate tomographic reconstruction. RESIRE iterates between the update of a reconstructed 3D object and the measured projections using a forward and back projection step. The forward projection step is implemented by the Fourier slice theorem or the Radon transform, and the back projection step by a linear transformation. Our numerical and experimental results demonstrate that RESIRE performs more accurate 3D reconstructions than other existing tomographic algorithms, when there are a limited number of projections with noise. Furthermore, RESIRE can be used to reconstruct the 3D structure of extended objects as demonstrated by the determination of the 3D atomic structure of an amorphous Ta thin film. We expect that RESIRE can be widely employed in the tomography applications in different fields. Finally, to make the method accessible to the general user community, the MATLAB source code of RESIRE and all the simulated and experimental data are available at  https://zenodo.org/record/7273314 .

NUDIM: A non-uniform fast Fourier transform based dual-space constraint iterative reconstruction method in biological electron tomography

Electron tomography, a powerful imaging tool for studying 3D structures of macromolecular assemblies, always suffers from imperfect reconstruction with limited resolution due to the intrinsic low signal-to-noise ratio (SNR) and inaccessibility to certain tilt angles induced by radiation damage or mechanical limitation. In order to compensate for such insufficient data with low SNR and further improve imaging resolution, prior knowledge constraints about the objects in both real space and reciprocal space are thus exploited during tomographic reconstruction. However, direct Fast Fourier transform (FFT) between real space and reciprocal space remains extraordinarily challenging owing to their inconsistent grid sampling modes, e.g. regular and uniform grid sampling in real space whereas radial or polar grid sampling in reciprocal space. In order to solve such problem, a technique of non-uniform fast Fourier transform (NFFT) has been developed to transform efficiently between non-uniformly sampled grids in real and reciprocal space with sufficient accuracy. In this work, a Non-Uniform fast Fourier transform based Dual-space constraint Iterative reconstruction Method (NUDIM) applicable to biological electron tomography is proposed with a combination of basic concepts from equally sloped tomography (EST) and NFFT based reconstruction. In NUDIM, the use of NFFT can circumvent such grid sampling inconsistency and thus alleviate the stringent equally-sloped sampling requirement in EST reconstruction, while the dual-space constraint iterative procedure can dramatically enhance reconstruction quality. In comparison with conventional reconstruction methods, NUDIM is numerically and experimentally demonstrated to produce superior reconstruction quality with higher contrast, less noise and reduced missing wedge artifacts. More importantly, it is also capable of retrieving part of missing information from a limited number of projections.

GENFIRE: A generalized Fourier iterative reconstruction algorithm for high-resolution 3D imaging

Tomography has made a radical impact on diverse fields ranging from the study of 3D atomic arrangements in matter to the study of human health in medicine. Despite its very diverse applications, the core of tomography remains the same, that is, a mathematical method must be implemented to reconstruct the 3D structure of an object from a number of 2D projections. Here, we present the mathematical implementation of a tomographic algorithm, termed GENeralized Fourier Iterative REconstruction (GENFIRE), for high-resolution 3D reconstruction from a limited number of 2D projections. GENFIRE first assembles a 3D Fourier grid with oversampling and then iterates between real and reciprocal space to search for a global solution that is concurrently consistent with the measured data and general physical constraints. The algorithm requires minimal human intervention and also incorporates angular refinement to reduce the tilt angle error. We demonstrate that GENFIRE can produce superior results relative to several other popular tomographic reconstruction techniques through numerical simulations and by experimentally reconstructing the 3D structure of a porous material and a frozen-hydrated marine cyanobacterium. Equipped with a graphical user interface, GENFIRE is freely available from our website and is expected to find broad applications across different disciplines.

A modified equally sloped algorithm based on the total variation algorithm in computed tomography for insufficient data

Computed tomography (CT) has become an important technique for analyzing the inner structures of material, biological and energy fields. However, there are often challenges in the practical application of CT due to insufficient data. For example, the maximum rotation angle of the sample stage is limited by sample space or image reconstruction from the limited number of views required to reduce the X-ray dose delivered to the sample. Therefore, it is difficult to acquire CT images with complete data. In this work, an iterative reconstruction algorithm based on the minimization of the image total variation (TV) has been utilized to develop equally sloped tomography (EST), and the reconstruction was carried out from limited-angle, few-view and noisy data. A synchrotron CT experiment on hydroxyapatite was also carried out to demonstrate the ability of the TV-EST algorithm. The results indicated that the new TV-EST algorithm was capable of achieving high-quality reconstructions from projections with insufficient data.

Monochromatic-beam-based dynamic X-ray microtomography based on OSEM-TV algorithm

Monochromatic-beam-based dynamic X-ray computed microtomography (CT) was developed to observe evolution of microstructure inside samples. However, the low flux density results in low efficiency in data collection. To increase efficiency, reducing the number of projections should be a practical solution. However, it has disadvantages of low image reconstruction quality using the traditional filtered back projection (FBP) algorithm. In this study, an iterative reconstruction method using an ordered subset expectation maximization-total variation (OSEM-TV) algorithm was employed to address and solve this problem. The simulated results demonstrated that normalized mean square error of the image slices reconstructed by the OSEM-TV algorithm was about 1/4 of that by FBP. Experimental results also demonstrated that the density resolution of OSEM-TV was high enough to resolve different materials with the number of projections less than 100. As a result, with the introduction of OSEM-TV, the monochromatic-beam-based dynamic X-ray microtomography is potentially practicable for the quantitative and non-destructive analysis to the evolution of microstructure with acceptable efficiency in data collection and reconstructed image quality.

Frontier methods in coherent X-ray diffraction for high-resolution structure determination

In 1912, Max von Laue and collaborators first observed diffraction spots from a millimeter-sized crystal of copper sulfate using an X-ray tube. Crystallography was born of this experiment, and since then, diffraction by both X-rays and electrons has revealed a myriad of inorganic and organic structures, including structures of complex protein assemblies. Advancements in X-ray sources have spurred a revolution in structure determination, facilitated by the development of new methods. This review explores some of the frontier methods that are shaping the future of X-ray diffraction, including coherent diffractive imaging, serial femtosecond X-ray crystallography and small-angle X-ray scattering. Collectively, these methods expand the current limits of structure determination in biological systems across multiple length and time scales.

In-line phase-contrast breast tomosynthesis: a phantom feasibility study at a synchrotron radiation facility

The major objective is to adopt, apply and test developed in-house algorithms for volumetric breast reconstructions from projection images, obtained in in-line phase-contrast mode. Four angular sets, each consisting of 17 projection images obtained from four physical phantoms, were acquired at beamline ID17, European Synchroton Radiation Facility, Grenoble, France. The tomosynthesis arc was  ±32°. The physical phantoms differed in complexity of texture and introduced features of interest. Three of the used phantoms were in-house developed, and made of epoxy resin, polymethyl-methacrylate and paraffin wax, while the fourth phantom was the CIRS BR3D. The projection images had a pixel size of 47 µm  ×  47 µm. Tomosynthesis images were reconstructed with standard shift-and-add (SAA) and filtered backprojection (FBP) algorithms. It was found that the edge enhancement observed in planar x-ray images is preserved in tomosynthesis images from both phantoms with homogeneous and highly heterogeneous backgrounds. In case of BR3D, it was found that features not visible in the planar case were well outlined in the tomosynthesis slices. In addition, the edge enhancement index calculated for features of interest was found to be much higher in tomosynthesis images reconstructed with FBP than in planar images and tomosynthesis images reconstructed with SAA. The comparison between images reconstructed by the two reconstruction algorithms shows an advantage for the FBP method in terms of better edge enhancement. Phase-contrast breast tomosynthesis realized in in-line mode benefits the detection of suspicious areas in mammography images by adding the edge enhancement effect to the reconstructed slices.

Three-dimensional visualisation of soft biological structures by X-ray computed micro-tomography

Whereas the two-dimensional (2D) visualisation of biological samples is routine, three-dimensional (3D) imaging remains a time-consuming and relatively specialised pursuit. Current commonly adopted techniques for characterising the 3D structure of non-calcified tissues and biomaterials include optical and electron microscopy of serial sections and sectioned block faces, and the visualisation of intact samples by confocal microscopy or electron tomography. As an alternative to these approaches, X-ray computed micro-tomography (microCT) can both rapidly image the internal 3D structure of macroscopic volumes at sub-micron resolutions and visualise dynamic changes in living tissues at a microsecond scale. In this Commentary, we discuss the history and current capabilities of microCT. To that end, we present four case studies to illustrate the ability of microCT to visualise and quantify: (1) pressure-induced changes in the internal structure of unstained rat arteries, (2) the differential morphology of stained collagen fascicles in tendon and ligament, (3) the development of Vanessa cardui chrysalises, and (4) the distribution of cells within a tissue-engineering construct. Future developments in detector design and the use of synchrotron X-ray sources might enable real-time 3D imaging of dynamically remodelling biological samples.

Accelerating an Ordered-Subset Low-Dose X-Ray Cone Beam Computed Tomography Image Reconstruction with a Power Factor and Total Variation Minimization

In recent years, there has been increased interest in low-dose X-ray cone beam computed tomography (CBCT) in many fields, including dentistry, guided radiotherapy and small animal imaging. Despite reducing the radiation dose, low-dose CBCT has not gained widespread acceptance in routine clinical practice. In addition to performing more evaluation studies, developing a fast and high-quality reconstruction algorithm is required. In this work, we propose an iterative reconstruction method that accelerates ordered-subsets (OS) reconstruction using a power factor. Furthermore, we combine it with the total-variation (TV) minimization method. Both simulation and phantom studies were conducted to evaluate the performance of the proposed method. Results show that the proposed method can accelerate conventional OS methods, greatly increase the convergence speed in early iterations. Moreover, applying the TV minimization to the power acceleration scheme can further improve the image quality while preserving the fast convergence rate.

Sparse sampling in helical cone-beam CT perfect reconstruction algorithms

In the current paper we consider the Helical Cone Beam CT. This scanning method exposes the patient to large quantities of radiation and results in very large amounts of data being collected and stored. Both these facts are prime motivators for the development of an efficient, reduced rate, sampling pattern. We calculate bounds on the support in the frequency domain of the collected data and use these to suggest an efficient sampling pattern. A reduction of up to a factor of 2 in sampling rate is suggested. Indeed, we show that reconstruction quality is not affected by this reduction of sampling rates.

X-ray propagation-based equally sloped tomography for mouse brain

BACKGROUND

The outstanding functional importance of the brain implies a strong need for brain imaging modalities. However, the current imaging approaches that target the brain in rodents remain suboptimal.

OBJECTIVE AND METHODS

In this paper, X-ray propagation-based phase contrast imaging combined with equally sloped tomography (PPCI-EST) was employed to nondestructively investigate the mouse brain.

RESULTS

The grey and white matters, which have extremely small differences in electron density, were clearly discriminated. The fine structures, including the corpus callosum (cc), the optic chiasma (ox) and the caudate putamen (CPu), were revealed. Compared to the filtered back projection reconstruction, the PPCI-EST significantly reduce projection number while maintaining sufficient image quality.

CONCLUSIONS

It could be a potential tool for fast and low-dose phase-contrast imaging to biomedical specimens.

Analytic 3D imaging of mammalian nucleus at nanoscale using coherent x-rays and optical fluorescence microscopy

Despite the notable progress that has been made with nano-bio imaging probes, quantitative nanoscale imaging of multistructured specimens such as mammalian cells remains challenging due to their inherent structural complexity. Here, we successfully performed three-dimensional (3D) imaging of mammalian nuclei by combining coherent x-ray diffraction microscopy, explicitly visualizing nuclear substructures at several tens of nanometer resolution, and optical fluorescence microscopy, cross confirming the substructures with immunostaining. This demonstrates the successful application of coherent x-rays to obtain the 3D ultrastructure of mammalian nuclei and establishes a solid route to nanoscale imaging of complex specimens.

Sparse-view x-ray CT reconstruction via total generalized variation regularization

Sparse-view CT reconstruction algorithms via total variation (TV) optimize the data iteratively on the basis of a noise- and artifact-reducing model, resulting in significant radiation dose reduction while maintaining image quality. However, the piecewise constant assumption of TV minimization often leads to the appearance of noticeable patchy artifacts in reconstructed images. To obviate this drawback, we present a penalized weighted least-squares (PWLS) scheme to retain the image quality by incorporating the new concept of total generalized variation (TGV) regularization. We refer to the proposed scheme as 'PWLS-TGV' for simplicity. Specifically, TGV regularization utilizes higher order derivatives of the objective image, and the weighted least-squares term considers data-dependent variance estimation, which fully contribute to improving the image quality with sparse-view projection measurement. Subsequently, an alternating optimization algorithm was adopted to minimize the associative objective function. To evaluate the PWLS-TGV method, both qualitative and quantitative studies were conducted by using digital and physical phantoms. Experimental results show that the present PWLS-TGV method can achieve images with several noticeable gains over the original TV-based method in terms of accuracy and resolution properties.

Radiation dose reduction in medical x-ray CT via Fourier-based iterative reconstruction

PURPOSE

A Fourier-based iterative reconstruction technique, termed Equally Sloped Tomography (EST), is developed in conjunction with advanced mathematical regularization to investigate radiation dose reduction in x-ray CT. The method is experimentally implemented on fan-beam CT and evaluated as a function of imaging dose on a series of image quality phantoms and anonymous pediatric patient data sets. Numerical simulation experiments are also performed to explore the extension of EST to helical cone-beam geometry.

METHODS

EST is a Fourier based iterative algorithm, which iterates back and forth between real and Fourier space utilizing the algebraically exact pseudopolar fast Fourier transform (PPFFT). In each iteration, physical constraints and mathematical regularization are applied in real space, while the measured data are enforced in Fourier space. The algorithm is automatically terminated when a proposed termination criterion is met. Experimentally, fan-beam projections were acquired by the Siemens z-flying focal spot technology, and subsequently interleaved and rebinned to a pseudopolar grid. Image quality phantoms were scanned at systematically varied mAs settings, reconstructed by EST and conventional reconstruction methods such as filtered back projection (FBP), and quantified using metrics including resolution, signal-to-noise ratios (SNRs), and contrast-to-noise ratios (CNRs). Pediatric data sets were reconstructed at their original acquisition settings and additionally simulated to lower dose settings for comparison and evaluation of the potential for radiation dose reduction. Numerical experiments were conducted to quantify EST and other iterative methods in terms of image quality and computation time. The extension of EST to helical cone-beam CT was implemented by using the advanced single-slice rebinning (ASSR) method.

RESULTS

Based on the phantom and pediatric patient fan-beam CT data, it is demonstrated that EST reconstructions with the lowest scanner flux setting of 39 mAs produce comparable image quality, resolution, and contrast relative to FBP with the 140 mAs flux setting. Compared to the algebraic reconstruction technique and the expectation maximization statistical reconstruction algorithm, a significant reduction in computation time is achieved with EST. Finally, numerical experiments on helical cone-beam CT data suggest that the combination of EST and ASSR produces reconstructions with higher image quality and lower noise than the Feldkamp Davis and Kress (FDK) method and the conventional ASSR approach.

CONCLUSIONS

A Fourier-based iterative method has been applied to the reconstruction of fan-bean CT data with reduced x-ray fluence. This method incorporates advantageous features in both real and Fourier space iterative schemes: using a fast and algebraically exact method to calculate forward projection, enforcing the measured data in Fourier space, and applying physical constraints and flexible regularization in real space. Our results suggest that EST can be utilized for radiation dose reduction in x-ray CT via the readily implementable technique of lowering mAs settings. Numerical experiments further indicate that EST requires less computation time than several other iterative algorithms and can, in principle, be extended to helical cone-beam geometry in combination with the ASSR method.

Correlative nanoscale 3D imaging of structure and composition in extended objects

Structure and composition at the nanoscale determine the behavior of biological systems and engineered materials. The drive to understand and control this behavior has placed strong demands on developing methods for high resolution imaging. In general, the improvement of three-dimensional (3D) resolution is accomplished by tightening constraints: reduced manageable specimen sizes, decreasing analyzable volumes, degrading contrasts, and increasing sample preparation efforts. Aiming to overcome these limitations, we present a non-destructive and multiple-contrast imaging technique, using principles of X-ray laminography, thus generalizing tomography towards laterally extended objects. We retain advantages that are usually restricted to 2D microscopic imaging, such as scanning of large areas and subsequent zooming-in towards a region of interest at the highest possible resolution. Our technique permits correlating the 3D structure and the elemental distribution yielding a high sensitivity to variations of the electron density via coherent imaging and to local trace element quantification through X-ray fluorescence. We demonstrate the method by imaging a lithographic nanostructure and an aluminum alloy. Analyzing a biological system, we visualize in lung tissue the subcellular response to toxic stress after exposure to nanotubes. We show that most of the nanotubes are trapped inside alveolar macrophages, while a small portion of the nanotubes has crossed the barrier to the cellular space of the alveolar wall. In general, our method is non-destructive and can be combined with different sample environmental or loading conditions. We therefore anticipate that correlative X-ray nano-laminography will enable a variety of in situ and in operando 3D studies.

High-resolution, low-dose phase contrast X-ray tomography for 3D diagnosis of human breast cancers

Mammography is the primary imaging tool for screening and diagnosis of human breast cancers, but ~10-20% of palpable tumors are not detectable on mammograms and only about 40% of biopsied lesions are malignant. Here we report a high-resolution, low-dose phase contrast X-ray tomographic method for 3D diagnosis of human breast cancers. By combining phase contrast X-ray imaging with an image reconstruction method known as equally sloped tomography, we imaged a human breast in three dimensions and identified a malignant cancer with a pixel size of 92 μm and a radiation dose less than that of dual-view mammography. According to a blind evaluation by five independent radiologists, our method can reduce the radiation dose and acquisition time by ~74% relative to conventional phase contrast X-ray tomography, while maintaining high image resolution and image contrast. These results demonstrate that high-resolution 3D diagnostic imaging of human breast cancers can, in principle, be performed at clinical compatible doses.

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.

CT reconstruction from parallel and fan-beam projections by a 2-D discrete Radon transform

The discrete Radon transform (DRT) was defined by Abervuch as an analog of the continuous Radon transform for discrete data. Both the DRT and its inverse are computable in O(n(2) log n) operations for images of size n × n. In this paper, we demonstrate the applicability of the inverse DRT for the reconstruction of a 2-D object from its continuous projections. The DRT and its inverse are shown to model accurately the continuum as the number of samples increases. Numerical results for the reconstruction from parallel projections are presented. We also show that the inverse DRT can be used for reconstruction from fan-beam projections with equispaced detectors.

Robust principal component analysis-based four-dimensional computed tomography

The purpose of this paper for four-dimensional (4D) computed tomography (CT) is threefold. (1) A new spatiotemporal model is presented from the matrix perspective with the row dimension in space and the column dimension in time, namely the robust PCA (principal component analysis)-based 4D CT model. That is, instead of viewing the 4D object as a temporal collection of three-dimensional (3D) images and looking for local coherence in time or space independently, we perceive it as a mixture of low-rank matrix and sparse matrix to explore the maximum temporal coherence of the spatial structure among phases. Here the low-rank matrix corresponds to the 'background' or reference state, which is stationary over time or similar in structure; the sparse matrix stands for the 'motion' or time-varying component, e.g., heart motion in cardiac imaging, which is often either approximately sparse itself or can be sparsified in the proper basis. Besides 4D CT, this robust PCA-based 4D CT model should be applicable in other imaging problems for motion reduction or/and change detection with the least amount of data, such as multi-energy CT, cardiac MRI, and hyperspectral imaging. (2) A dynamic strategy for data acquisition, i.e. a temporally spiral scheme, is proposed that can potentially maintain similar reconstruction accuracy with far fewer projections of the data. The key point of this dynamic scheme is to reduce the total number of measurements, and hence the radiation dose, by acquiring complementary data in different phases while reducing redundant measurements of the common background structure. (3) An accurate, efficient, yet simple-to-implement algorithm based on the split Bregman method is developed for solving the model problem with sparse representation in tight frames.