Three-dimensional reconstruction from projections: a review of algorithms

Characterization of MXene-Based Materials by X-Ray Computed Tomography

MXenes are a class of 2D materials that have gained significant attention for their potential applications in energy storage, electromagnetic interference shielding, biomedicine, and (opto)electronics. Despite their broad range of applications, a detailed understanding of the internal architecture of MXene-based materials remains limited due to the lack of effective 3D imaging techniques. This work demonstrates the application of X-ray micro-computed tomography (micro-CT) to investigate various MXene systems, including nanocomposites, coated textiles, and aerogels. Micro-CT enables high-resolution, 3D visualization of the internal microstructure, MXene distribution, infiltration patterns, and defect formations, which significantly influence the material's performance. Moreover, the typical technical challenges and limitations encountered during sample preparation, scanning, and post-processing of micro-CT data are discussed. The information obtained from optical and electron microscopy is also compared with micro-CT, highlighting the unique advantages of micro-CT in providing comprehensive 3D imaging and quantitative data. This study highlights micro-CT as a powerful and nondestructive imaging tool for characterizing MXene-based materials, providing insights into material optimization and guidelines for developing future advanced applications.

Algorithm for Modern Electron Microscopic Examination of the Golgi Complex

The Golgi complex (GC) is an essential organelle of the eukaryotic exocytic pathway. It has a very complexed structure and thus localization of its resident proteins is not trivial. Fast development of microscopic methods generates a huge difficulty for Golgi researchers to select the best protocol to use. Modern methods of light microscopy, such as super-resolution light microscopy (SRLM) and electron microscopy (EM), open new possibilities in analysis of various biological structures at organelle, cell, and organ levels. Nowadays, new generation of EM methods became available for the study of the GC; these include three-dimensional EM (3DEM), correlative light-EM (CLEM), immune EM, and new estimators within stereology that allow realization of maximal goal of any morphological study, namely, to achieve a three-dimensional model of the sample with optimal level of resolution and quantitative determination of its chemical composition. Methods of 3DEM have partially overlapping capabilities. This requires a careful comparison of these methods, identification of their strengths and weaknesses, and formulation of recommendations for their application to cell or tissue samples. Here, we present an overview of 3DEM methods for the study of the GC and some basics for how the images are formed and how the image quality can be improved.

Source-detector trajectory optimization in cone-beam computed tomography: a comprehensive review on today’s state-of-the-art

Cone-beam computed tomography (CBCT) imaging is becoming increasingly important for a wide range of applications such as image-guided surgery, image-guided radiation therapy as well as diagnostic imaging such as breast and orthopaedic imaging. The potential benefits of non-circular source-detector trajectories was recognized in early work to improve the completeness of CBCT sampling and extend the field of view (FOV). Another important feature of interventional imaging is that prior knowledge of patient anatomy such as a preoperative CBCT or prior CT is commonly available. This provides the opportunity to integrate such prior information into the image acquisition process by customized CBCT source-detector trajectories. Such customized trajectories can be designed in order to optimize task-specific imaging performance, providing intervention or patient-specific imaging settings. The recently developed robotic CBCT C-arms as well as novel multi-source CBCT imaging systems with additional degrees of freedom provide the possibility to largely expand the scanning geometries beyond the conventional circular source-detector trajectory. This recent development has inspired the research community to innovate enhanced image quality by modifying image geometry, as opposed to hardware or algorithms. The recently proposed techniques in this field facilitate image quality improvement, FOV extension, radiation dose reduction, metal artifact reduction as well as 3D imaging under kinematic constraints. Because of the great practical value and the increasing importance of CBCT imaging in image-guided therapy for clinical and preclinical applications as well as in industry, this paper focuses on the review and discussion of the available literature in the CBCT trajectory optimization field. To the best of our knowledge, this paper is the first study that provides an exhaustive literature review regarding customized CBCT algorithms and tries to update the community with the clarification of in-depth information on the current progress and future trends.

An Image Reconstruction System Applied to High Voltage Microscopy

Increased specimen penetration; the principle advantage of high voltage microscopy, is accompanied by an increased need to utilize information on three dimensional specimen structure available in the form of two dimensional projections (i.e. micrographs). We are engaged in a program to develop methods which allow the maximum use of information contained in a through tilt series of micrographs to determine three dimensional speciman structure.

In general, we are dealing with structures lacking in symmetry and with projections available from only a limited span of angles (±60°). For these reasons, we must make maximum use of any prior information available about the specimen. To do this in the most efficient manner, we have concentrated on iterative, real space methods rather than Fourier methods of reconstruction. The particular iterative algorithm we have developed is given in detail in ref. 3. A block diagram of the complete reconstruction system is shown in fig. 1.

Micro-CT - a digital 3D microstructural voyage into scaffolds: a systematic review of the reported methods and results

BACKGROUND

Cell behavior is the key to tissue regeneration. Given the fact that most of the cells used in tissue engineering are anchorage-dependent, their behavior including adhesion, growth, migration, matrix synthesis, and differentiation is related to the design of the scaffolds. Thus, characterization of the scaffolds is highly required. Micro-computed tomography (micro-CT) provides a powerful platform to analyze, visualize, and explore any portion of interest in the scaffold in a 3D fashion without cutting or destroying it with the benefit of almost no sample preparation need.

MAIN BODY

This review highlights the relationship between the scaffold microstructure and cell behavior, and provides the basics of the micro-CT method. In this work, we also analyzed the original papers that were published in 2016 through a systematic search to address the need for specific improvements in the methods section of the papers including the amount of provided information from the obtained results.

CONCLUSION

Micro-CT offers a unique microstructural analysis of biomaterials, notwithstanding the associated challenges and limitations. Future studies that will include micro-CT characterization of scaffolds should report the important details of the method, and the derived quantitative and qualitative information can be maximized.

A Survey of the Use of Iterative Reconstruction Algorithms in Electron Microscopy

One of the key steps in Electron Microscopy is the tomographic reconstruction of a three-dimensional (3D) map of the specimen being studied from a set of two-dimensional (2D) projections acquired at the microscope. This tomographic reconstruction may be performed with different reconstruction algorithms that can be grouped into several large families: direct Fourier inversion methods, back-projection methods, Radon methods, or iterative algorithms. In this review, we focus on the latter family of algorithms, explaining the mathematical rationale behind the different algorithms in this family as they have been introduced in the field of Electron Microscopy. We cover their use in Single Particle Analysis (SPA) as well as in Electron Tomography (ET).

CT brush and CancerZap!: two video games for computed tomography dose minimization

BACKGROUND

X-ray dose from computed tomography (CT) scanners has become a significant public health concern. All CT scanners spray x-ray photons across a patient, including those using compressive sensing algorithms. New technologies make it possible to aim x-ray beams where they are most needed to form a diagnostic or screening image. We have designed a computer game, CT Brush, that takes advantage of this new flexibility. It uses a standard MART algorithm (Multiplicative Algebraic Reconstruction Technique), but with a user defined dynamically selected subset of the rays. The image appears as the player moves the CT brush over an initially blank scene, with dose accumulating with every "mouse down" move. The goal is to find the "tumor" with as few moves (least dose) as possible.

RESULTS

We have successfully implemented CT Brush in Java and made it available publicly, requesting crowdsourced feedback on improving the open source code. With this experience, we also outline a "shoot 'em up game" CancerZap! for photon limited CT.

CONCLUSIONS

We anticipate that human computing games like these, analyzed by methods similar to those used to understand eye tracking, will lead to new object dependent CT algorithms that will require significantly less dose than object independent nonlinear and compressive sensing algorithms that depend on sprayed photons. Preliminary results suggest substantial dose reduction is achievable.

Analysis of electrohydrodynamic jetting behaviors using three-dimensional shadowgraphic tomography

Electrohydrodynamic jetting behaviors of liquid menisci were analyzed experimentally by three-dimensional optical shadowgraphic tomography. The tomographic algorithm was developed after a series of multiplicative algebraic reconstruction techniques updated the object's intensities by using a cubic cosine basis function to determine the weighting coefficients of the projection matrix. The algorithm was evaluated initially by using a synthesized three-dimensional droplet phantom. Three-dimensional reconstructions of several jetting modes were built based on three images of projection data captured by three high-speed cameras, which were positioned at an offset angle of 45° relative to one another.

Multifunctional in vivo vascular imaging using near-infrared II fluorescence

In vivo real-time epifluorescence imaging of mouse hind limb vasculatures in the second near-infrared region (NIR-II) is performed using single-walled carbon nanotubes as fluorophores. Both high spatial (∼30 μm) and temporal (<200 ms per frame) resolution for small-vessel imaging are achieved at 1-3 mm deep in the hind limb owing to the beneficial NIR-II optical window that affords deep anatomical penetration and low scattering. This spatial resolution is unattainable by traditional NIR imaging (NIR-I) or microscopic computed tomography, and the temporal resolution far exceeds scanning microscopic imaging techniques. Arterial and venous vessels are unambiguously differentiated using a dynamic contrast-enhanced NIR-II imaging technique on the basis of their distinct hemodynamics. Further, the deep tissue penetration and high spatial and temporal resolution of NIR-II imaging allow for precise quantifications of blood velocity in both normal and ischemic femoral arteries, which are beyond the capabilities of ultrasonography at lower blood velocities.

Comparative study on basis functions for projection matrix of three-dimensional tomographic reconstruction for analysis of dropletbehavior from electrohydrodynamic jet

Three-dimensional optical tomography techniques were developed to reconstruct three-dimensional objects using a set of two-dimensional projection images. Five basis functions, such as cubic B-spline, o-Moms, keys, and cosine functions and Gaussian basis functions, were used to calculate the weighting coefficients for a projection matrix. Two different forms of a multiplicative algebraic reconstruction technique were also used to solve inverse problems. The reconstruction algorithm was examined by using several phantoms, which included droplet behaviors and random distributions of particles in a volume. The three-dimensional volume comprised of particles was reconstructed from four projection angles, which were positioned at an offset angle of 45° between each other. Then, three-dimensional velocity fields were obtained from the reconstructed particle volume by three-dimensional cross correlation. The velocity field of the synthetic vortex flow was reconstructed to analyze the three-dimensional tomography algorithm.

In vivo fluorescence imaging in the second near-infrared window with long circulating carbon nanotubes capable of ultrahigh tumor uptake

Cancer imaging requires selective high accumulation of contrast agents in the tumor region and correspondingly low uptake in healthy tissues. Here, by making use of a novel synthetic polymer to solubilize single-walled carbon nanotubes (SWNTs), we prepared a well-functionalized SWNT formulation with long blood circulation (half-life of ∼30 h) in vivo to achieve ultrahigh accumulation of ∼30% injected dose (ID)/g in 4T1 murine breast tumors in Balb/c mice. Functionalization dependent blood circulation and tumor uptake were investigated through comparisons with phospholipid-PEG solubilized SWNTs. For the first time, we performed video-rate imaging of tumors based on the intrinsic fluorescence of SWNTs in the second near-infrared (NIR-II, 1.1-1.4 μm) window. We carried out dynamic contrast imaging through principal component analysis (PCA) to immediately pinpoint the tumor within ∼20 s after injection. Imaging over time revealed increasing tumor contrast up to 72 h after injection, allowing for its unambiguous identification. The 3D reconstruction of the SWNTs distribution based on their stable photoluminescence inside the tumor revealed a high degree of colocalization of SWNTs and blood vessels, suggesting enhanced permeability and retention (EPR) effect as the main cause of high passive tumor uptake of the nanotubes.

High-resolution three-dimensional probes of biomaterials and their interfaces

Interfacial relationships between biomaterials and tissues strongly influence the success of implant materials and their long-term functionality. Owing to the inhomogeneity of biological tissues at an interface, in particular bone tissue, two-dimensional images often lack detail on the interfacial morphological complexity. Furthermore, the increasing use of nanotechnology in the design and production of biomaterials demands characterization techniques on a similar length scale. Electron tomography (ET) can meet these challenges by enabling high-resolution three-dimensional imaging of biomaterial interfaces. In this article, we review the fundamentals of ET and highlight its recent applications in probing the three-dimensional structure of bioceramics and their interfaces, with particular focus on the hydroxyapatite-bone interface, titanium dioxide-bone interface and a mesoporous titania coating for controlled drug release.

On the implementation of computed laminography using synchrotron radiation

Hard x rays from a synchrotron source are used in this implementation of computed laminography for three-dimensional (3D) imaging of flat, laterally extended objects. Due to outstanding properties of synchrotron light, high spatial resolution down to the micrometer scale can be attained, even for specimens having lateral dimensions of several decimeters. Operating either with a monochromatic or with a white synchrotron beam, the method can be optimized to attain high sensitivity or considerable inspection throughput in synchrotron user and small-batch industrial experiments. The article describes the details of experimental setups, alignment procedures, and the underlying reconstruction principles. Imaging of interconnections in flip-chip and wire-bonded devices illustrates the peculiarities of the method compared to its alternatives and demonstrates the wide application potential for the 3D inspection and quality assessment in microsystem technology.

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.

Cone-beam tomography with discrete data sets

Sufficiency conditions for cone-beam data are well known for the case of continuous data collection along a cone-vertex curve with continuous detectors. These continuous conditions are inadequate for real-world data where discrete vertex geometries and discrete detector arrays are used. In this paper we present a theoretical formulation of cone-beam tomography with arbitrary discrete arrays of detectors and vertices. The theory models the imaging system as a linear continuous-to-discrete mapping and represents the continuous object exactly as a Fourier series. The reconstruction problem is posed as the estimation of some subset of the Fourier coefficients. The main goal of the theory is to determine which Fourier coefficients can be reliably determined from the data delivered by a specific discrete design. A Fourier component will be well determined by the data if it satisfies two conditions: it makes a strong contribution to the data, and this contribution is relatively independent of the contribution of other Fourier components. To make these considerations precise, we introduce a concept called the cross-talk matrix. A diagonal element of this matrix measures the strength of a Fourier component in the data, while an off-diagonal element quantifies the dependence or aliasing of two different components. One reasonable approach to system design is to attempt to make the diagonal elements of this matrix large and the off-diagonal elements small for some set of Fourier components. If this goal can be achieved, simple linear reconstruction algorithms are available for estimating the Fourier coefficients. To illustrate the usefulness of this approach, numerical results on the cross-talk matrix are presented for different discrete geometries derived from a continuous helical vertex orbit, and simulated images reconstructed with two linear algorithms are presented.

Experiments with the nonlinear and chaotic behaviour of the multiplicative algebraic reconstruction technique (MART) algorithm for computed tomography

Among the iterative reconstruction algorithms for tomography, the multiplicative algebraic reconstruction technique (MART) has two advantages that make it stand out from other algorithms: it confines the image (and therefore the projection data) to the convex hull of the patient, and it maximizes entropy. In this paper, we have undertaken a series of experiments to determine the importance of MART nonlinearity to image quality. Variants of MART were implemented aiming to exploit and exaggerate the nonlinear properties of the algorithm. We introduce the Power MART, Boxcar Averaging MART and Bouncing MART algorithms. Power MART is linked to the relaxation concept. Its behaviour is similar to that of the chaos of a logistic equation. There appears to be an antagonism between increasing nonlinearity and noise in the projection data. The experiments confirm our general observation that regularization as a means of solving simultaneous linear equations that are underdetermined is suboptimal: it does not necessarily select the correct image from the hyperplane of solutions, and so does not maximize the image quality:x-ray dose ratio. Our investigations prove that there is scope to optimize CT algorithms and thereby achieve greater dose reduction.

Performance evaluation of iterative tomography algorithms for incomplete projection data

Projection data obtained through optical techniques for tomographic measurements, such as interferometry for refractive-index-based measurements, are often incomplete. This is due to limitations in the optical system, data storage, and alignment and vignette issues. Algebraic iterative reconstruction techniques are usually favored for such incomplete projections. A number of iterative algorithms, based on additive and multiplicative corrections, are used with a known simulated phantom and noise source to assess the reconstruction performance of incomplete data sets. In addition, we present reconstructions using experimental data obtained from a coherent gradient sensing interferometer for a steady temperature field in a fluid medium. We tested the algorithms using the simulated data set for incompleteness conditions similar to those found in the experimental data, and the best-performing algorithm is identified.

Orientation Refinement of Virus Structures with Unknown Symmetry

Structural biology, in particular the structure determination of viruses and other large macromolecular complexes leads to data- and compute-intensive problems that require resources well beyond those available on a single system. Thus, there is an imperative need to develop parallel algorithms and programs for clusters and computational grids. We present one of the most challenging computational problems posed by the three-dimensional structure determination of viruses, the orientation refinement.

Development of separable electron spin resonance-computed tomography imaging for multiple radical species: an application to .OH and .NO

A method of separable ESR-CT (electron spin resonance-computed tomography) imaging for multiple radical species was developed and applied to imaging of .OH and .NO. The algorithm was improved by combining filtered back-projection with a modified algebraic reconstruction technique to enhance accuracy and shorten calculation time. With this algorithm, spectral-spatial images of the phantom consisting of 3-carbamoyl-2,2,5,5,-tetramethylpyrrolidine-N-oxyl and 2-phenyl-4,4,5,5,-tetramethylimidazoline-3-oxide-1-oxyl could be obtained in different directions by rotating the spatial axis. The spatial function of individual radicals was extracted by each of the two methods from each spectral-spatial image. The separative 2D images of each radical were individually constructed using the spatial function obtained with the two methods. By comparing the separative images with the phantom sample, the algorithm for separable ESR-CT imaging was established. This ESR-CT technique was combined with L-band ESR spectroscopy and applied to the separative imaging of .OH and .NO, which were spin trapped with 5, 5-dimethyl-1-pyrroline-N-oxide (DMPO) and Fe(2+)-N-methyl-D-glucamine dithiocarbamate complex, respectively. The ESR signal of DMPO-OH decreased gradually during data acquisition, and the decrease was calibrated by extrapolating the signal intensity to the beginning of data sampling. Both the position and size of the individual images for .OH and .NO were in very good agreement with the findings for the sample.

High resolution light diffraction tomography: nearfield measurements of 10 MHz continuous wave ultrasound

Light diffraction tomography is an ultrasound measurement method that offers possibilities to make high spatial resolution measurements. The aim of this study was to investigate the spatial resolution of light diffraction tomography by characterizing the complex pressure and phase patterns 0.5-1 mm from the surface of a 5 mm diameter 10 MHz ultrasound transducer. Three different transducers were measured with a detection width of 5 microm, sample distances of 20-40 microm and 50-100 projections/180 degrees. They were excited with a 10 Vpp CW-signal. To verify the results the optical measurements were compared with 75 microm diameter hydrophone measurements and with theoretical calculations. The light diffraction tomography results show very good agreement with the hydrophone measurements and pressure peaks separated 160 microm were resolved. Comparison with the theoretical calculation shows that small distortions, caused by defects in the matching layer or in the piezo-electric disc, disturb the symmetric ring-pattern characteristic of an ideal piston source.

Evolution escapes rugged fitness landscapes by gene or genome doubling: the blessing of higher dimensionality

Evolution on rugged landscapes ceases when a local maximum is attained. This poses the problem of how evolution could approach or attain a global maximum, especially for large genomes for which quasispecies are ineffective. I show how increasing the dimensionality of the landscape, which occurs every time there is a gene or higher order duplication (up to polyploidy), may solve this problem. Epistasis or complementarity between the duplicated genes provides an all uphill pathway towards the global maximum. The evolution of hemoglobin and other dimeric and tetrameric proteins provides a testable case, since fitness is readily defined.

Limited-data computed tomography algorithms for the physical sciences

Five limited-data computed tomography algorithms are compared. The algorithms used are adapted versions of the algebraic reconstruction technique, the multiplicative algebraic reconstruction technique, the Gerchberg-Papoulis algorithm, a spectral extrapolation algorithm descended from that of Harris [J. Opt. Soc. Am. 54, 931-936 (1964)], and an algorithm based on the singular value decomposition technique. These algorithms were used to reconstruct phantom data with realistic levels of noise from a number of different imaging geometries. The phantoms, the imaging geometries, and the noise were chosen to simulate the conditions encountered in typical computed tomography applications in the physical sciences, and the implementations of the algorithms were optimized for these applications. The multiplicative algebraic reconstruction technique algorithm gave the best results overall; the algebraic reconstruction technique gave the best results for very smooth objects or very noisy (20-dB signal-to-noise ratio) data. My implementations of both of these algorithms incorporate apriori knowledge of the sign of the object, its extent, and its smoothness. The smoothness of the reconstruction is enforced through the use of an appropriate object model (by use of cubic B-spline basis functions and a number of object coefficients appropriate to the object being reconstructed). The average reconstruction error was 1.7% of the maximum phantom value with the multiplicative algebraic reconstruction technique of a phantom with moderate-to-steep gradients by use of data from five viewing angles with a 30-dB signal-to-noise ratio.

Algebraic reconstruction in CT from limited views

The author presents an algebraic reconstruction technique (ART) as a viable alternative in computerized tomography (CT) from limited views. Recently, algorithms of iterative reconstruction-reprojection (IRR) based on the method of convolution-backprojection have been proposed for application in limited-view CT. Reprojection was used in an iterative fashion alternating with backprojection as a means of estimating projection values within the sector of missing views. In algebraic methods of reconstruction for CT, only those projections corresponding to known data are required. Reprojection along missing views would merely serve to introduce redundant equations. Computer simulation studies are presented which demonstrate significantly improved reconstructed images achieved by an ART algorithm as compared to IRR methods.

A three-dimensional approach to mitotic chromosome structure: evidence for a complex hierarchical organization

We describe findings on the architecture of Drosophila melanogaster mitotic chromosomes, made using a three-dimensional-oriented structural approach. Using high-voltage and conventional transmission electron microscopy combined with axial tomography and digital contrast-enhancement techniques, we have for the first time visualized significant structural detail within minimally perturbed mitotic chromosomes. Chromosomes prepared by several different preparative procedures showed a consistent size hierarchy of discrete chromatin structural domains with cross-sectional diameters of 120, 240, 400-500, and 800-1,000 A. In fully condensed, metaphase-arrested chromosomes, there is evidence for even larger-scale structural organization in the range of 1,300-3,000-A size. The observed intrachromosomal arrangements of these higher-order structural domains show that both the radial loop and sequential helical coiling models of chromosome structure are over-simplifications of the true situation. Finally, our results suggest that the pathway of chromatin condensation through mitosis consists of concurrent changes occurring at several levels of chromatin organization, rather than a strictly sequential folding process.

Reconstruction from NMR data acquired with imaging gradients having arbitrary time dependence

An algebraic reconstruction method for NMR imaging from data acquired with imaging gradients having essentially arbitrary time dependence is presented. With reasonable discretization, an overdetermined linear system is obtained for which there exist simple factorized 1-inverses of the coefficient matrices; a computationally efficient reconstruction scheme then follows. Some numerical simulations are also described.

Electron microscopy and image processing applied to the study of protein structure and protein-protein interactions

We review the application of electron microscopy and image processing at the molecular level to an ever increasing range of biological specimens. Although recent advances have been due in part to development of more sophisticated instrumentation and/or processing algorithms, widespread application of the well-known techniques of image enhancement and structure reconstruction has depended on new strategies of in vitro crystallization and polymerization, some of which are outlined here. We also discuss the use of stoichiometric labeling and/or "cocrystallization" in identifying the different subunits in multisubunit complexes and in studying protein-protein interactions.