Direct Reconstruction Method Research Articles

An advanced multi-camera system for automatic, high-precision and efficient tube profile measurement

In this paper, an advanced multi-camera system based on stereo vision is developed for automatic, highly accurate and efficient tube three-dimensional profile measurement of tubes, which can be applied to tube pre-assembly or in-process tube profile inspection. Multiple cameras are designed to acquire the tube images from different angles to help avoid image understanding disaster in case of tube overlap of complex tubes. A high precision calibration method based on photography is adopted to provide basic accuracy guarantee of the developed system. Then importantly, a new approach to tube model reconstruction is proposed, including a creative and adaptive tube seed growing method for accurate tube axis reconstruction, and a direct tube end reconstruction method. Furthermore, an automatic method of tube features calculation to help improve the efficiency of tube measurement. Finally, several experiments are performed to verify the reliability, efficiency and high-precision of the developed system. According to the experimental results, the tube measurement process can be finished in about 1.5 s/m. The average deviation of the radial deviation and axial deviation is within 0.062 mm respect to its reference value with the maximum standard deviation of 0.025 mm. For the bending angle and rotate angle, the average deviations from the reference values are within 0.025° and 0.031° with the maximum standard deviation of 0.011° and 0.012°, respectively.

Direct Reconstruction of Linear Parametric Images From Dynamic PET Using Nonlocal Deep Image Prior.

Direct reconstruction methods have been developed to estimate parametric images directly from the measured PET sinograms by combining the PET imaging model and tracer kinetics in an integrated framework. Due to limited counts received, signal-to-noise-ratio (SNR) and resolution of parametric images produced by direct reconstruction frameworks are still limited. Recently supervised deep learning methods have been successfully applied to medical imaging denoising/reconstruction when large number of high-quality training labels are available. For static PET imaging, high-quality training labels can be acquired by extending the scanning time. However, this is not feasible for dynamic PET imaging, where the scanning time is already long enough. In this work, we proposed an unsupervised deep learning framework for direct parametric reconstruction from dynamic PET, which was tested on the Patlak model and the relative equilibrium Logan model. The training objective function was based on the PET statistical model. The patient's anatomical prior image, which is readily available from PET/CT or PET/MR scans, was supplied as the network input to provide a manifold constraint, and also utilized to construct a kernel layer to perform non-local feature denoising. The linear kinetic model was embedded in the network structure as a 1 ×1 ×1 convolution layer. Evaluations based on dynamic datasets of 18F-FDG and 11C-PiB tracers show that the proposed framework can outperform the traditional and the kernel method-based direct reconstruction methods.

Direct multi-dimensional Chebyshev polynomial based reconstruction for magnetic particle imaging

Magnetic Particle Imaging is a tomographic imaging technique that measures the voltage induced due to magnetization changes of magnetic nanoparticle distributions. The relationship between the received signal and the distribution of the nanoparticels is described by the system function. A common method for image reconstruction is using a measured system function to create a system matrix and set up a regularized linear system of equations. Since the measurement of the system matrix is time-consuming, different methods for acceleration have been proposed. These include modeling the system matrix or using a direct reconstruction method in time, known as X-space reconstruction. In this work, based on the simplified Langevin model of paramagnetism and certain approximations, a direct reconstruction technique for Magnetic Particle Imaging in the frequency domain with two- and three-dimensional Lissajous trajectory excitation is presented. The approach uses Chebyshev polynomials of second kind. During reconstruction, they are weighted with the frequency components of the voltage signal and additional factors and then summed up. To obtain the final nanoparticle distribution, this result is rescaled and deconvolved. It is shown that the approach works for both simulated data and real measurements. The obtained image quality is comparable to a modeled system matrix approach using the same simplified physical assumptions and no relaxation effects. The reconstruction of a 31 × 31 × 31 volume takes less than a second and is up to 25 times faster than the state-of-the-art Kaczmarz reconstruction. Besides, the derivation of the proposed method shows some new theoretical aspects of the system function and its well-known observed similarity to tensor products of Chebyshev polynomials of second kind.

Direct regularized reconstruction for the three-dimensional Calderón problem

<p style='text-indent:20px;'>Electrical Impedance Tomography gives rise to the severely ill-posed Calderón problem of determining the electrical conductivity distribution in a bounded domain from knowledge of the associated Dirichlet-to-Neumann map for the governing equation. The uniqueness and stability questions for the three-dimensional problem were largely answered in the affirmative in the 1980's using complex geometrical optics solutions, and this led further to a direct reconstruction method relying on a non-physical scattering transform. In this paper, the reconstruction problem is taken one step further towards practical applications by considering data contaminated by noise. Indeed, a regularization strategy for the three-dimensional Calderón problem is presented based on a suitable and explicit truncation of the scattering transform. This gives a certified, stable and direct reconstruction method that is robust to small perturbations of the data. Numerical tests on simulated noisy data illustrate the feasibility and regularizing effect of the method, and suggest that the numerical implementation performs better than predicted by theory.</p>

Automated filtering in the nonlinear Fourier domain of systematic artifacts in 2D electrical impedance tomography

<p style='text-indent:20px;'>For patients undergoing mechanical ventilation due to respiratory failure, 2D electrical impedance tomography (EIT) is emerging as a means to provide functional monitoring of pulmonary processes. In EIT, electrical current is applied to the body, and the internal conductivity distribution is reconstructed based on subsequent voltage measurements. However, EIT images are known to often suffer from large systematic artifacts arising from various limitations and exacerbated by the ill-posedness of the inverse problem. The direct D-bar reconstruction method admits a nonlinear Fourier analysis of the EIT problem, providing the ability to process and filter reconstructions in the nonphysical frequency regime. In this work, a technique is introduced for automated Fourier-domain filtering of known systematic artifacts in 2D D-bar reconstructions. The new method is validated using three numerically simulated static thoracic datasets with induced artifacts, plus two experimental dynamic human ventilation datasets containing systematic artifacts. Application of the method is shown to significantly reduce the appearance of artifacts and improve the shape of the lung regions in all datasets.</p>

High-order numerical solutions to the shallow-water equations on the rotated cubed-sphere grid

A novel numerical approach to solving the shallow-water equations on the sphere using high-order numerical discretizations in both space and time is proposed. A space-time tensor formalism is used to express the equations of motion covariantly and to describe the geometry of the rotated cubed-sphere grid. The spatial discretization is done with the direct flux reconstruction method, which is an alternative formulation to the discontinuous Galerkin approach. The equations of motion are solved in differential form and the resulting discretization is free from quadrature rules. It is well known that the time step of traditional explicit methods is limited by the phase velocity of the fastest waves. Exponential integration is employed to enable integrations with significantly larger time step sizes and improve the efficiency of the overall time integration. New multistep-type exponential propagation iterative methods of orders 4, 5 and 6 are constructed and applied to integrate the shallow-water equations in time. These new schemes enable time integration with high-order accuracy but without significant increases in computational time compared to low-order methods. The exponential matrix functions-vector products used in the exponential schemes are approximated using the complex-step approximation of the Jacobian in the Krylov-based KIOPS (Krylov with incomplete orthogonalization procedure solver) algorithm. Performance of the new numerical methods is evaluated using a set of standard benchmark tests.

Calderón's Method with a Spatial Prior for 2-D EIT Imaging of Ventilation and Perfusion.

Bedside imaging of ventilation and perfusion is a leading application of 2-D medical electrical impedance tomography (EIT), in which dynamic cross-sectional images of the torso are created by numerically solving the inverse problem of computing the conductivity from voltage measurements arising on electrodes due to currents applied on electrodes on the surface. Methods of reconstruction may be direct or iterative. Calderón’s method is a direct reconstruction method based on complex geometrical optics solutions to Laplace’s equation capable of providing real-time reconstructions in a region of interest. In this paper, the importance of accurate modeling of the electrode location on the body is demonstrated on simulated and experimental data, and a method of including a priori spatial information in dynamic human subject data is presented. The results of accurate electrode modeling and a spatial prior are shown to improve detection of inhomogeneities not included in the prior and to improve the resolution of ventilation and perfusion images in a human subject.

Rapid high-quality PET Patlak parametric image generation based on direct reconstruction and temporal nonlocal neural network

Parametric imaging based on dynamic positron emission tomography (PET) has wide applications in neurology. Compared to indirect methods, direct reconstruction methods, which reconstruct parametric images directly from the raw PET data, have superior image quality due to better noise modeling and richer information extracted from the PET raw data. For low-dose scenarios, the advantages of direct methods are more obvious. However, the wide adoption of direct reconstruction is inevitably impeded by the excessive computational demand and deficiency of the accessible raw data. In addition, motion modeling inside dynamic PET image reconstruction raises more computational challenges for direct reconstruction methods. In this work, we focused on the 18F-FDG Patlak model, and proposed a data-driven approach which can estimate the motion corrected full-dose direct Patlak images from the dynamic PET reconstruction series, based on a proposed novel temporal non-local convolutional neural network. During network training, direct reconstruction with motion correction based on full-dose dynamic PET sinograms was performed to obtain the training labels. The reconstructed full-dose /low-dose dynamic PET images were supplied as the network input. In addition, a temporal non-local block based on the dynamic PET images was proposed to better recover the structural information and reduce the image noise. During testing, the proposed network can directly output high-quality Patlak parametric images from the full-dose /low-dose dynamic PET images in seconds. Experiments based on 15 full-dose and 15 low-dose 18F-FDG brain datasets were conducted and analyzed to validate the feasibility of the proposed framework. Results show that the proposed framework can generate better image quality than reference methods.

Direct and indirect parameter imaging methods for dynamic PET

The method of reconstructing parametric images from dynamic positron emission tomography (PET) data with the linear Patlak model has been widely used in scientific research and clinical practice. Whether for direct or indirect image reconstruction, researchers have deeply investigated the associated methods and effects. Among the existing methods, the traditional maximum likelihood expectation maximization (MLEM) reconstruction algorithm is fast but produces a substantial amount of noise. If the parameter images obtained by the MLEM algorithm are postfiltered, a large amount of image edge information is lost. Additionally, although the kernel method has a better noise reduction effect, its calculation costs are very high due to the complexity of the algorithm. Therefore, to obtain parametric images with a high signal-to-noise ratio (SNR) and good retention of detailed information, here, we use guided kernel means (GKM) and dynamic PET image information to conduct guided filtering and perform parametric image reconstruction. We apply this method to direct and indirect reconstruction, and through computer simulations, we show that our proposed method has higher identifiability and a greater SNR than conventional direct and indirect reconstruction methods. We also show that our method produces better images with direct than with indirect reconstruction.

Direct reconstruction of two ultrashort pulses based on non-interferometric frequency-resolved optical gating

We describe a non-interferometric ultrashort-pulse measurement technique based on frequency-resolved optical gating (FROG) with which pulses can be reconstructed directly, i.e. non-iteratively. Two different FROG spectrograms are measured, which represent the only information required to reconstruct the amplitudes and phases of two independent input pulses. The direct reconstruction method is demonstrated with a single-shot FROG setup used to obtain the spectrograms generated from two synchronized input pulses. To demonstrate and determine the reconstruction quality for complex pulses, a programmable pulse shaper is used to modify the pulses sourced from a Kerr-lens mode-locked Ti:sapphire oscillator.

Time-frequency analysis of high harmonic generation using a probe XUV pulse.

Interpretation of strong-field phenomena is mostly based on the analysis of classical electron trajectories in an intense laser field, whose specific properties determine general features of nonlinear laser-matter interaction. Currently, the visualization of closed electron trajectories contributing to high harmonic generation (HHG) of the laser field is the prerogative of a theoretical analysis based on the time-frequency spectrogram of the induced dipole acceleration. Here, we propose a method for direct reconstruction of the HHG time-frequency spectrogram using a time-delayed probe XUV pulse. Our analytical theory and ab initio numerical simulations demonstrate that the XUV-assisted HHG yield as a function of time delay and harmonic energy mimics the short-time Fourier transform of the dipole acceleration induced by the laser field, thereby providing possible in-situ experimental access for tracing electron dynamics in strong-field phenomena.

Direct reconstruction method for discontinuous Galerkin methods on higher-order mixed-curved meshes III. Code optimization via tensor contraction

The present study deals with the code optimization and its implementation of the direct reconstruction method (DRM) using the complete-search tensor contraction (CsTC) framework to extract the best performance of high-order methods on modern computing architectures. DRM was originally proposed to overcome severe computational costs of the physical domain-based discontinuous Galerkin (DG) method on mixed-curved meshes. In this work, the performance of DRM is further enhanced through the code optimization via the CsTC technique. Required kernels for tensor operations in the DRM solution algorithm are analyzed and optimized by completely searching all candidates of GEMM (General Matrix Multiplication) subroutines. The computational performance is thoroughly examined by simulating a turbulent flow over a circular cylinder at ReD=3900 by DG-P3 and -P5 approximations. Compared to a quadrature-based approach with the full integration, the optimized DRM significantly reduces the memory requirements and the number of floating-point operations to compute the DG residual on a linear mesh as well as high-order curved meshes. On a P3-mesh, the optimized DRM provides 13.74× and 23.03× speed-ups in DG-P3 and -P5, respectively, while the amount of memory required is reduced to 1/16.6 and 1/19.9. On a linear mesh, it even yields 1.25× and 1.12× speed-ups in DG-P3 and -P5, respectively. The memory requirement is reduced to 1/1.27 and 1/1.15, respectively. In particular, it is observed that the optimized DRM on a P3-mesh performs better than the optimized quadrature-based method on a P1-mesh.

Modified forward and inverse Born series for the Calderon and diffuse-wave problems

We propose a new direct reconstruction method based on series inversion for electrical impedance tomography (EIT) and the inverse scattering problem for diffuse waves. The standard Born series for the forward problem has the limitation that the series requires that the contrast lies within a certain radius for convergence. Here, we instead propose a modified Born series which converges for the forward problem unconditionally. We then invert this modified Born series and compare reconstructions with the usual inverse Born series. We also show that the modified inverse Born series has a larger radius of convergence.

Linear scheme for the direct reconstruction of noncontact time-domain fluorescence molecular lifetime tomography.

Direct reconstruction of the noncontact time-domain fluorescence molecular lifetime tomography (TD-FMLT) with current nonlinear algorithms has suffered from complexity and heavy computation loads of the physical model for a large imaging area in TD-FMLT. In this work, we discretize the system matrix along time points and apply a linearized reconstruction algorithm using the fused least absolute shrinkage and selection operator method. The reconstructed yield map and object geometry are used as a priori information to mitigate the ill conditions. This approach is implemented on a fully noncontact TD-FMLT system equipped with a femtosecond pulse laser and a high-speed, time-gated camera. We validate the methodology using both numerical simulations and inhomogeneous phantom experiments. The results exhibit good localization accuracy for fluorescent targets and an efficient computation capability for the reconstruction of fluorescence lifetime in noncontact TD-FMLT. We envision that the proposed linear scheme for the direct reconstruction method in noncontact TD-FMLT has a significant potential for in vivo preclinical studies.

Comparison of iterative parametric and indirect deep learning-based reconstruction methods in highly undersampled DCE-MR Imaging of the breast.

To compare the performance of iterative direct and indirect parametric reconstruction methods with indirect deep learning-based reconstruction methods in estimating tracer-kinetic parameters from highly undersampled DCE-MR Imaging breast data and provide a systematic comparison of the same. Estimation of tracer-kinetic parameters using indirect methods from undersampled data requires to reconstruct the anatomical images initially by solving an inverse problem. This reconstructed images gets utilized in turn to estimate the tracer-kinetic parameters. In direct estimation, the parameters are estimated without reconstructing the anatomical images. Both problems are ill-posed and are typically solved using prior-based regularization or using deep learning. In this study, for indirect estimation, two deep learning-based reconstruction frameworks namely, ISTA-Net+ and MODL, were utilized. For direct and indirect parametric estimation, sparsity inducing priors (L1 and Total-Variation) and limited memory Broyden-Fletcher-Goldfarb-Shanno algorithm as solver was deployed. The performance of these techniques were compared systematically in estimation of vascular permeability ( ) from undersampled DCE-MRI breast data using Patlak as pharmaco-kinetic model. The experiments involved retrospective undersampling of the data 20×, 50×, and 100× and compared the results using PSNR, nRMSE, SSIM, and Xydeas metrics. The maps estimated from fully sampled data were utilized as ground truth. The developed code was made available as https://github.com/Medical-Imaging-Group/DCE-MRI-Compare open-source for enthusiastic users. The reconstruction methods performance was evaluated using ten patients breast data (five patients each for training and testing). Consistent with other studies, the results indicate that direct parametric reconstruction methods provide improved performance compared to the indirect parameteric reconstruction methods. The results also indicate that for 20× undersampling, deep learning-based methods performs better or at par with direct estimation in terms of PSNR, SSIM, and nRMSE. However, for higher undersampling rates (50× and 100×) direct estimation performs better in all metrics. For all undersampling rates, direct reconstruction performed better in terms of Xydeas metric, which indicated fidelity in magnitude and orientation of edges. Deep learning-based indirect techniques perform at par with direct estimation techniques for lower undersampling rates in the breast DCE-MR imaging. At higher undersampling rates, they are not able to provide much needed generalization. Direct estimation techniques are able to provide more accurate results than both deep learning- and parametric-based indirect methods in these high undersampling scenarios.

Reconstruction of piecewise constant layered conductivities in electrical impedance tomography

This work presents a new constructive uniqueness proof for Calderón’s inverse problem of electrical impedance tomography, subject to local Cauchy data, for a large class of piecewise constant conductivities that we call piecewise constant layered conductivities (PCLC). The resulting reconstruction method only relies on the physically intuitive monotonicity principles of the local Neumann-to-Dirichlet map, and therefore the method lends itself well to efficient numerical implementation and generalization to electrode models. Several direct reconstruction methods exist for the related problem of inclusion detection, however they share the property that “holes in inclusions” or “inclusions-within-inclusions” cannot be determined. One such method is the monotonicity method of Harrach, Seo, and Ullrich, and in fact the method presented here is a modified variant of the monotonicity method which overcomes this problem. More precisely, the presented method abuses that a PCLC type conductivity can be decomposed into nested layers of positive and/or negative perturbations that, layer-by-layer, can be determined via the monotonicity method. The conductivity values on each layer are found via basic one-dimensional optimization problems constrained by monotonicity relations.

Direct reconstruction method for discontinuous Galerkin methods on higher-order mixed-curved meshes II. Surface integration

As the second extension of the direct reconstruction method (DRM), DRM for the surface integration of the discontinuous Galerkin (DG) weak formulation is presented on multi-dimensional high-order mixed meshes. From the previous study, DRM was successfully extended to the volume integration of the DG method, and produced a significant reduction in both the computational cost and memory overhead, particularly for high-order solution approximations on 3-D high-order meshes. DRM has the potential to reduce the computational cost and memory overhead further by treating the surface integration term consistently. To realize this, we design a linear dependency detector and formulate an adaptive orthonormalization process from the modified Gram-Schmidt (MGS) process. DRM is then applied to the restricted function space of the surface integration in an adaptive and optimal manner. The resulting methods, namely the DRM surface integration with the brute force points (BFP) and shape function points (SFP) methods, perform very well on high-order mixed-curved meshes. In addition, the adaptive MGS process is robust and accurate in extracting linear faces/edges from a given set of mixed-curved meshes. Various benchmark tests from the compressible Euler and Navier-Stokes equations verify the impact of DRM for 3-D large-scale computations on mixed-curved meshes. As an example, the DRM framework applied to the 3-D surface and volume integrations with DG-P3 approximation on P3-mesh may yield O(10)× speed-up, compared to conventional quadrature-based methods.

Discontinuous Galerkin via Interpolation: The Direct Flux Reconstruction Method

The discontinuous Galerkin (DG) method is based on the idea of projection using integration. The recent direct flux reconstruction (DFR) method by Romero et al. (J Sci Comput 67(1):351–374, 2016) is derived via interpolation and results in a scheme identical to DG (on hexahedral meshes). The DFR method is further studied and developed here. Two proofs for its equivalence with the DG scheme considerably simpler than the original proof are presented. The first proof employs the $$ 2K - 1 $$ degree of precision by a $$ K $$-point Gauss quadrature. The second shows the equivalence of DG, FR, and DFR by using the property that the derivative of the degree $$ K + 1 $$ Lobatto polynomial vanishes at the $$ K $$ Gauss points. Fourier analysis for these schemes are presented using an approach more geometric compared with existing analytic approaches. The effects of nonuniform mesh and those of high-order mesh transformation (a precursor for curved meshes in two and three spatial dimensions) on stability and accuracy are examined. These nonstandard analyses are obtained via an in-depth study of the behavior of eigenvalues and eigenvectors.

Direct Ellipsoidal Fitting of Discrete Multi-Dimensional Data

Multi-dimensional distributions of discrete data that resemble ellipsoids arise in numerous areas of science, statistics, and computational geometry. We describe a complete algebraic algorithm to determine the quadratic form specifying the equation of ellipsoid for the boundary of such multi-dimensional discrete distribution. In this approach, the equation of an ellipsoid is reconstructed using a set of matrix equations from low-dimensional projections of the input data. We provide a Mathematica program realizing the full implementation of the ellipsoid reconstruction algorithm in an arbitrary number of dimensions. To demonstrate its many potential uses, the direct reconstruction method is applied to quasi-Gaussian statistical distributions arising in elementary particle production at the Large Hadron Collider.

A Fourier approach to the inverse source problem in an absorbing and anisotropic scattering medium

We revisit the inverse source problem in a two dimensional absorbing and scattering medium and present a direct reconstruction method, which does not require iterative solvability of the forward problem, using measurements of the radiating flux at the boundary. The attenuation and scattering coefficients are known and the unknown source is isotropic. The approach is based on the Cauchy problem for a Beltrami-like equation for the sequence valued maps, and extends the original ideas of Bukhgeim from the non-scattering to the scattering media. We demonstrate the feasibility of the method in a numerical experiment in which the scattering is modeled by the two dimensional Henyey–Greenstein kernel with parameters meaningful in optical tomography.