Direct Reconstruction Algorithm Research Articles

A multithreaded real-time solution for 2D EIT reconstruction with the D-bar algorithm

Electrical impedance tomography (EIT) is a non-ionizing real-time imaging tool for bedside pulmonary imaging. Dynamic cross-sectional images of ventilation and perfusion can be produced in real time with fast algorithms from measured voltage data arising from current applied on electrodes placed around the circumference of the patient’s chest. The D-bar method is a direct (non-iterative) reconstruction algorithm for EIT that uses equations of inverse scattering to independently reconstruct the conductivity at each point in the region of interest. However, this nonlinear reconstruction method has a computational complexity that makes real-time imaging a challenge. At the same time, it has the attributes that it does not require successive solutions of the forward problem, as is the case with iterative methods, and it is trivially parallelizable in the spatial variable. Here, we present a novel multithreaded implementation of the D-bar method with a front-end MATLAB/Octave program interfaced with C code that uses the pthreads library. The implementation is analyzed on several different platforms with concurrent threads ranging from 2 to 32 for spatial grids of several sizes. We demonstrate that on a CPU with many cores, very high frame rates (50 frames/s) with high resolution (6000 grid points) are achievable, and on a single AMD CPU real-time reconstructions (faster than 30 frames/s) are achieved with this implementation using 10 or more threads on a grid of 1969 points.

PET Image Reconstruction Incorporating Deep Image Prior and a Forward Projection Model

Convolutional neural networks (CNNs) have recently achieved remarkable performance in positron emission tomography (PET) image reconstruction. In particular, CNN-based direct PET image reconstruction, which directly generates the reconstructed image from the sinogram, has potential applicability to PET image enhancements because it does not require image reconstruction algorithms, which often produce some artifacts. However, these deep learning-based, direct PET image reconstruction algorithms have the disadvantage that they require a large number of high-quality training datasets. In this study, we propose an unsupervised direct PET image reconstruction method that incorporates a deep image prior framework. Our proposed method incorporates a forward projection model with a loss function to achieve unsupervised direct PET image reconstruction from sinograms. To compare our proposed direct reconstruction method with the filtered back projection (FBP) and maximum likelihood expectation maximization (ML-EM) algorithms, we evaluated using Monte Carlo simulation data of brain [$^{18}$F]FDG PET scans. The results demonstrate that our proposed direct reconstruction quantitatively and qualitatively outperforms the FBP and ML-EM algorithms with respect to peak signal-to-noise ratio and structural similarity index.

Projection Deconvolution for Proton CT Using the Spatially Variant Path Uncertainty

Proton computed tomography (pCT) suffers from a lower spatial resolution compared to X-ray CT due to the stochastic nonlinear proton paths. The most likely path (MLP) formalism provides an estimate for the proton path as well as the uncertainty around this estimate. Using the MLP estimate for the image reconstruction instead of straight integration lines has been shown to improve the spatial resolution of pCT images. In this work, the aim is to further increase the spatial resolution by also including the path uncertainty in the reconstruction algorithm. We proposed a projection-based deconvolution method, applied within the framework of a direct reconstruction algorithm based on distance-driven binning. We used an MLP formalism accounting for tracker resolution in addition to multiple Coulomb scattering (MCS). We investigated deconvolution artifacts and proposed a method to mitigate them via spatial regularization. Our method was tested on Monte Carlo simulated data, using a water cylinder with aluminium inserts and two slices of an anthropomorphic phantom. Our results showed an improvement of spatial resolution in all cases (up to 29% or 60% for the cylindrical phantom, depending on whether deconvolution artifacts were corrected for or not). Overshoot artifacts were observed in the case of the cylindrical phantom but were less prominent in the case of the anthropomorphic phantom. In conclusion, we have shown that including the path uncertainty in the reconstruction can notably improve the spatial resolution.

In vivo accurate detection of the liver tumor with pharmacokinetic parametric images from dynamic fluorescence molecular tomography.

.SignificancePharmacokinetic parametric images in dynamic fluorescence molecular tomography (FMT) can describe three-dimensional (3D) physiological and pathological information inside biological tissues, potentially providing quantitative assessment tools for biological research and drug development.AimIn vivo imaging of the liver tumor with pharmacokinetic parametric images from dynamic FMT based on the differences in metabolic properties of indocyanine green (ICG) between normal liver cells and tumor liver cells inside biological tissues.ApproachFirst, an orthotopic liver tumor mouse model was constructed. Then, with the help of the FMT/computer tomography (CT) dual-modality imaging system and the direct reconstruction algorithm, 3D imaging of liver metabolic parameters in nude mice was achieved to distinguish liver tumors from normal tissues. Finally, pharmacokinetic parametric imaging results were validated against in vitro anatomical results.ResultsThis letter demonstrates the ability of dynamic FMT to monitor the pharmacokinetic delivery of the fluorescent dye ICG in vivo, thus, enabling the distinction between normal and tumor tissues based on the pharmacokinetic parametric images derived from dynamic FMT.ConclusionsCompared with CT structural imaging technology, dynamic FMT combined with compartmental modeling as an analytical method can obtain quantitative images of pharmacokinetic parameters, thus providing a more powerful research tool for organ function assessment, disease diagnosis and new drug development.

Improved resolution of D-bar images of ventilation using a Schur complement property and an anatomical atlas.

BackgroundElectrical impedance tomography (EIT) is a nonionizing imaging technique for real‐time imaging of ventilation of patients with respiratory distress. Cross‐sectional dynamic images are formed by reconstructing the conductivity distribution from measured voltage data arising from applied alternating currents on electrodes placed circumferentially around the chest. Since the conductivity of lung tissue depends on air content, blood flow, and the presence of pathology, the dynamic images provide regional information about ventilation, pulsatile perfusion, and abnormalities. However, due to the ill‐posedness of the inverse conductivity problem, EIT images have a coarse spatial resolution. One method of improving the resolution is to include prior information in the reconstruction.PurposeIn this work, we propose a technique in which a statistical prior built from an anatomical atlas is used to postprocess EIT reconstructions of human chest data. The effectiveness of the method is demonstrated on data from two patients with cystic fibrosis.MethodsA direct reconstruction algorithm known as the D‐bar method was used to compute a two‐dimensional reconstruction of the conductivity distribution in the plane of the electrodes. Reconstructions using one step in an iterative (regularized) Newton's method were also computed for comparison. An anatomical atlas consisting of 1589 synthetic EIT images computed from X‐ray computed tomography (CT) scans of 74 adult male subjects was computed for use as a statistical prior. The resolution of the D‐bar images was then improved by maximizing the conditional probability density function of an image that is consistent with the a priori information and the statistical model. A new method to evaluate the accuracy of the EIT images using CT scans of the imaged patient as ground truth is presented. The novel approach is tested on data from two patients with cystic fibrosis.Results and ConclusionsThe D‐bar images resulted in better structural similarity index measures (SSIM) and multiscale (MS) SSIM measures for both subjects using the mask or amplitude evaluation approach than the one‐step (regularized) Newton's method. Further improvement was achieved using the Schur complement (SC) approach, with MS‐SSIM values of 0.718 and 0.682 using SC evaluated with the mask and amplitude approach, respectively, for Patient 1, and MS‐SSIM values of 0.726 and 0.692 using SC evaluated with the mask and amplitude approach, respectively, for Patient 2. The results from applying an anatomical atlas and statistical prior to EIT data from two patients with cystic fibrosis suggest that the spatial resolution of the EIT image can be improved to reveal pathology that may be difficult to discern in the original EIT image. The novel metric of evaluation is consistent with the appearance of improved spatial resolution and provides a new way to evaluate the accuracy of EIT reconstructions when a CT scan is available.

Evaluating the usefulness of the direct density reconstruction algorithm for intensity modulated and passively scattered proton therapy: Validation using an anthropomorphic phantom.

Accurate calculation of the proton beam range inside a patient is an important topic in proton therapy. In recent times, a computed tomography (CT) image reconstruction algorithm was developed for treatment planning to reduce the impact of the variation of the CT number with changes in imaging conditions. In this study, we investigated the usefulness of this new reconstruction algorithm (DirectDensity™: DD) in proton therapy based on its comparison with filtered back projection (FBP). We evaluated the effects of variations in the X-ray tube potential and target size on the FBP- and DD-image values and investigated the usefulness of the DD algorithm based on the range variations and dosimetric quantity variations. For X-ray tube potential variations, the range variation in the case of FBP was up to 12.5mm (20.8%), whereas that of DD was up to 3.3mm (5.6%). Meanwhile, for target size variations, the range variation in the case of FBP was up to 2.2mm (2.5%), whereas that of DD was up to 0.9mm (1.4%). Moreover, the variations observed in the case of DD were smaller than those of FBP for all dosimetric quantities. The dose distributions obtained using DD were more robust against variations in the CT imaging conditions (X-ray tube potential and target size) than those obtained using FBP, and the range variations were often less than the dose calculation grid (2mm). Therefore, the DD algorithm is effective in a robust workflow and reduces uncertainty in range calculations.

Cycling Rate-Induced Spatially-Resolved Heterogeneities in Commercial Cylindrical Li-Ion Batteries.

Synchrotron high-energy X-ray diffraction computed tomography has been employed to investigate, for the first time, commercial cylindrical Li-ion batteries electrochemically cycled over the two cycling rates of C/2and C/20. This technique yields maps of the crystalline components and chemical species as a cross-section of the cell with high spatiotemporal resolution (550 × 550 images with 20 × 20 × 3 µm3 voxel size in ca. 1 h). The recently developed Direct Least-Squares Reconstruction algorithm is used to overcome the well-known parallax problem and led to accurate lattice parameter maps for the device cathode. Chemical heterogeneities are revealed at both electrodes and are attributed to uneven Li and current distributions in the cells. It is shown that this technique has the potential to become an invaluable diagnostic tool for real-world commercial batteries and for their characterization under operating conditions, leading to unique insights into "real" battery degradation mechanisms as they occur.

A direct reconstruction algorithm for the anisotropic inverse conductivity problem based on Calderón’s method in the plane

A direct reconstruction algorithm based on Calderón’s linearization method for the reconstruction of isotropic conductivities is proposed for anisotropic conductivities in two-dimensions. To overcome the non-uniqueness of the anisotropic inverse conductivity problem, the entries of the unperturbed anisotropic tensors are assumed known a priori, and it remains to reconstruct the multiplicative scalar field. The quasi-conformal map in the plane facilitates the Calderón-based approach for anisotropic conductivities. The method is demonstrated on discontinuous radially symmetric conductivities of high and low contrast.

Improved Photoacoustic Imaging of Numerical Bone Model Based on Attention Block U-Net Deep Learning Network

Photoacoustic (PA) imaging can provide both chemical and micro-architectural information for biological tissues. However, photoacoustic imaging for bone tissue remains a challenging topic due to complicated ultrasonic propagations in the porous bone. In this paper, we proposed a post-processing method based on the convolution neural network (CNN) to improve the image quality of PA bone imaging in a numerical model. To be more adaptive for imaging bone samples with complex structure, an attention block U-net (AB-U-Net) network was designed from the standard U-net by integrating the attention blocks in the feature extraction part. The k-wave toolbox was used for the simulation of photoacoustic wave fields, and then the direct reconstruction algorithm—time reversal was adopted for generating a dataset of deep learning. The performance of the proposed AB-U-Net network on the reconstruction of photoacoustic bone imaging was analyzed. The results show that the AB-U-Net based deep learning method can obtain the image presented as a clear bone micro-structure. Compared with the traditional photoacoustic reconstruction method, the AB-U-Net-based reconstruction algorithm can achieve better performance, which greatly improves image quality on test set with peak signal to noise ratio (PSNR) and structural similarity increased (SSIM) by 3.83 dB and 0.17, respectively. The deep learning method holds great potential in enhancing PA imaging technology for bone disease detection.

Partial data inverse problem with 𝐿^{𝑛/2} potentials

We construct an explicit Green’s function for the conjugated Laplacian e − ω ⋅ x / h Δ e − ω ⋅ x / h e^{-\omega \cdot x/h}\Delta e^{-\omega \cdot x/h} , which lets us control our solutions on roughly half of the boundary. We apply the Green’s function to solve a partial data inverse problem for the Schrödinger equation with potential q ∈ L n / 2 q \in L^{n/2} . Separately, we also use this Green’s function to derive L p L^p Carleman estimates similar to the ones in Kenig-Ruiz-Sogge [Duke Math. J. 55 (1987), pp. 329–347], but for functions with support up to part of the boundary. Unlike many previous results, we did not obtain the partial data result from the boundary Carleman estimate—rather, both results stem from the same explicit construction of the Green’s function. This explicit Green’s function has potential future applications in obtaining direct numerical reconstruction algorithms for partial data Calderón problems which is presently only accessible with full data [Inverse Problems 27 (2011)].

Inverse scattering for the one-dimensional Helmholtz equation with piecewise constant wave speed

This paper analyzes inverse scattering for the one-dimensional Helmholtz equation in the case where the wave speed is piecewise constant. Scattering data recorded for an arbitrarily small interval of frequencies is shown to determine the wave speed uniquely, and a direct reconstruction algorithm is presented. The algorithm is exact provided data is recorded for a sufficiently wide range of frequencies and the jump points of the wave speed are equally spaced with respect to travel time. Numerical examples show that the algorithm works also in the general case of arbitrary wave speed (either with jumps or continuously varying etc) giving progressively more accurate approximations as the range of recorded frequencies increases. A key underlying theoretical insight is to associate scattering data to compositions of automorphisms of the unit disk, which are in turn related to orthogonal polynomials on the unit circle. The algorithm exploits the three-term recurrence of orthogonal polynomials to reduce the required computation.

A comparison of direct reconstruction algorithms in proton computed tomography

Several direct algorithms have been proposed to take into account the non-linear path of protons in the reconstruction of a proton CT (pCT) image. This paper presents a comparison between five of them, in terms of spatial resolution and relative stopping power (RSP) accuracy. Our comparison includes (1) a distance-driven algorithm extending the filtered backprojection to non-linear trajectories (DD), (2) an algorithm reconstructing a pCT image from optimized projections (ML), (3) a backproject-then-filter approach using a 2D cone filter (BTF), (4) a differentiated backprojection algorithm based on the inversion of the Hilbert transform (DBP), and (5) an algorithm using a 2D directional ramp filter (DR). We have simulated a single tracking pCT set-up using Geant4 through GATE, with a proton source and two position, direction and energy detectors upstream and downstream from the object. Tracker uncertainties were added on the position and direction measurements. A Catphan 528 phantom and a spiral phantom were simulated to measure the spatial resolution and a Gammex 467 phantom was used for the RSP accuracy. Each proton’s trajectory was estimated using a most likely path (MLP) formalism. The spatial resolution was evaluated using the frequency corresponding to a modulation transfer function of 10% of its peak value and the RSP accuracy using the mean values in the inserts of the Gammex phantom. In terms of spatial resolution, it was shown that, for ideal trackers, the DR and BTF methods offer a slightly better resolution since each proton is directly binned in the image grid according to its MLP. However, all methods but the ML show comparable resolution when using realistic trackers. Regarding the RSP, three algorithms (DR, DD and BTF) show a mean relative error inside the inserts about 0.1%. As the DR and BTF methods are more computationally expensive, the DD—which allows the same spatial resolution in realistic conditions and the same accuracy—and the DBP—which has a fairly good accuracy (<0.2%) and allows reconstruction from truncated data—can be used for a reduced reconstruction time.

Direct List Mode Parametric Reconstruction for Dynamic Cardiac SPECT.

Recently introduced stationary dedicated cardiac SPECT scanners provide new opportunities to quantify myocardial blood flow (MBF) using dynamic SPECT. However, comparing to PET, the low sensitivity of SPECT scanners affects MBF quantification due to the high noise level, especially for 201 Thallium (201Tl) due to its typically low injected dose. The conventional indirect method for generating parametric images typically starts by reconstructing a time series of frame images followed by fitting the time-activity curve (TAC) for each voxel or segment with an appropriate kinetic model. The indirect method is simple and easy to implement; however, it usually suffers from substantial image noise that could also lead to bias. In this paper, we developed a list mode direct parametric image reconstruction algorithm to substantially reduce noise in MBF quantification using dynamic SPECT and allow for patient radiation dose reduction. GPU-based parallel computing was used to achieve more than 2000-fold acceleration. The proposed method was evaluated in both simulation and in vivo canine studies. Compared with the indirect method, the proposed direct method achieved substantially lower image noise and variability, particularly at large number of iterations and at low-count levels.

A GPU acceleration of 3-D Fourier reconstruction in cryo-EM

Cryo-electron microscopy is a popular method for macromolecules structure determination. Reconstruction of a 3-D volume from raw data obtained from a microscope is highly computationally demanding. Thus, acceleration of the reconstruction has a great practical value. In this article, we introduce a novel graphics processing unit (GPU)-friendly algorithm for direct Fourier reconstruction, one of the main computational bottlenecks in the 3-D volume reconstruction pipeline for some experimental cases (particularly those with a large number of images and a high internal symmetry). Contrary to the state of the art, our algorithm uses a gather memory pattern, improving cache locality and removing race conditions in parallel writing into the 3-D volume. We also introduce a finely tuned CUDA implementation of our algorithm, using auto-tuning to search for a combination of optimization parameters maximizing performance on a given GPU architecture. Our CUDA implementation is integrated in widely used software Xmipp, version 3.19, reaching 11.4× speedup compared to the original parallel CPU implementation using GPU with comparable power consumption. Moreover, we have reached 31.7× speedup using four GPUs and 2.14×–5.96× speedup compared to optimized GPU implementation based on a scatter memory pattern.

Direct three-dimensional tomographic reconstruction and phase retrieval of far-field coherent diffraction patterns

We present an alternative numerical reconstruction algorithm for direct tomographic reconstruction of a sample refractive indices from the measured intensities of its far-field coherent diffraction patterns. We formulate the well-known phase-retrieval problem in ptychography in a tomographic framework which allows for simultaneous reconstruction of the illumination function and the sample refractive indices in three dimensions. Our iterative reconstruction algorithm is based on the Levenberg-Marquardt algorithm. We demonstrate the performance of our proposed method with simulation studies.

Complex exponential reconstruction algorithm accelerated by cascadic multigrid method

When laser beam propagates through the turbulent atmosphere, there are branch points in wavefront, which are caused by deep turbulence or long propagation distance. Conventional least-square reconstruction algorithms cannot restore the discontinuous wavefront, which severely limits correction performance of an adaptive optics system. If the incoming wavefront contains a branch cut, there is &lt;inline-formula&gt;&lt;tex-math id="M1"&gt;\begin{document}$ {\rm{2}}n{\text{π}} $\end{document}&lt;/tex-math&gt;&lt;alternatives&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="10-20182137_M1.jpg"/&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="10-20182137_M1.png"/&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt; difference between the measured phase difference and the principle phase difference, which is the reason why conventional least-square reconstruction algorithms cannot reconstruct wavefront with branch points. The complex exponential reconstructor is developed to restore the discontinuous wavefront with phase difference replaced by complex exponents. However, thousands of iterations are required by the complex exponential reconstructor before converging to an acceptable solution. In order to speed up the iterative calculation, the cascadic multigrid method (CMG) is introduced in the process of wavefront reconstruction. The proposed method can be used to restore discontinuous wavefront with lower residual error similar to those reconstructed by the direct iteration. The number of float point multiplications required by the CMG method is nearly 2 orders of magnitude lower than that required by the direct iteration. The acceleration of the CMG method increases with the number of subapertures increasing. The performance of CMG method to recover continuous wavefront is also investigated and compared with conventional wavefront reconstruction algorithm based on successive over-relaxation. It is shown that the CMG method has good capability for wavefront reconstruction with high precision and low computation cost no matter whether it is applied to discontinuous or continuous wavefront. Furthermore, the CMG method is used in the adaptive optics for correcting the turbulence aberration. The direct slope wavefront reconstruction algorithm based on the assumption that the measured slope and the control voltage satisfy the linear relationship cannot restore the wavefront with branch points. As a result, the adaptive optics system with the CMG method doubles the correction quality evaluated by the Strehl ratio compared with that with the direct slope wavefront reconstruction algorithm.

Improving Tomographic Reconstruction from Limited Data Using Mixed-Scale Dense Convolutional Neural Networks

In many applications of tomography, the acquired data are limited in one or more ways due to unavoidable experimental constraints. In such cases, popular direct reconstruction algorithms tend to produce inaccurate images, and more accurate iterative algorithms often have prohibitively high computational costs. Using machine learning to improve the image quality of direct algorithms is a recently proposed alternative, for which promising results have been shown. However, previous attempts have focused on using encoder–decoder networks, which have several disadvantages when applied to large tomographic images, preventing wide application in practice. Here, we propose the use of the Mixed-Scale Dense convolutional neural network architecture, which was specifically designed to avoid these disadvantages, to improve tomographic reconstruction from limited data. Results are shown for various types of data limitations and object types, for both simulated data and large-scale real-world experimental data. The results are compared with popular tomographic reconstruction algorithms and machine learning algorithms, showing that Mixed-Scale Dense networks are able to significantly improve reconstruction quality even with severely limited data, and produce more accurate results than existing algorithms.

Iterative Reconstruction Algorithm for Electrical Capacitance Tomography Based on Calderon’s Method

In this paper, the closed-loop control strategy was facilitated to iteratively reconstruct images of high quality for electrical capacitance tomography (ECT). The classical direct reconstruction algorithm, e.g., Calderon’s method, was first used to roughly estimate the permittivity distributions. The yielded distributions were subsequently utilized to generate the mutual capacitances between different electrodes. A fast capacitance extraction method with constraints of equal potential on all nodes in each electrode was also introduced for the first time to calculate the mutual capacitances in terms of the discretized Dirichlet-to-Neumann map. In the method, only the stiffness matrix in the finite-element model was involved to calculate the capacitances between electrodes regarded as floating potentials, and the consumption time can be dramatically reduced. The calculated capacitances from the reconstructed distributions were then compared with the measured data from the ECT sensor. The deviations were used to correct the reconstructed distributions. A proportional-integral-derivative controller was introduced to perform a closed-loop control strategy for image correction. Similar to the classical feedback control systems, Calderon’s method was treated as the control object to generate an output. The output of the retrieved distributions passes the capacitance calculation and the capacitances are used as the negative feedback. In addition, the invariant property of the conformal transformation was strictly proved for the first time for the ECT sensor with a cross-section of any simply-connected region. As a result, the proposed iterative reconstruction algorithm can be promoted to the cases of any non-circular ECT sensors, e.g., square sensor. Simulated data contaminated with and without noises and real data from the experiments were used to demonstrate the feasibility and effectiveness of the proposed iterative reconstruction algorithm for ECT.

Modulation-nonmodulation pyramid wavefront sensor with direct gradient reconstruction algorithm on the closed-loop adaptive optics system.

A modulation-nonmodulation pyramid wavefront sensor with direct gradient reconstruction algorithm using on the adaptive optics is reported in this paper. This means that the interaction matrix of the system is obtained by using a modulated pyramid and the closed-loop control process is performed with a nonmodulated pyramid. The theoretical basis and simulation analysis of the direct gradient reconstruction algorithm are described in detail, and laboratory results show that the pyramid wavefront sensor based on this algorithm can work as expected in a closed-loop adaptive optics system.

Using a plenoptic sensor to reconstruct vortex phase structures.

A branch point problem and its solution commonly involve recognizing and reconstructing a vortex phase structure around a singular point. In laser beam propagation through random media, the destructive phase contributions from various parts of a vortex phase structure will cause a dark area in the center of the beam's intensity profile. This null of intensity can, in turn, prevent the vortex phase structure from being recognized. In this Letter, we show how to use a plenoptic sensor to transform the light field of a vortex beam so that a simple and direct reconstruction algorithm can be applied to reveal the vortex phase structure. As a result, we show that the plenoptic sensor is effective in detecting branch points and can be used to reconstruct phase distortion in a beam in a wide sense.