Conjugate Gradient Least Squares Research Articles

A CGLS-based method for solving magnetic moments of hybrid-model magnetic targets

In order to solve the problem of magnetic moment estimation of magnetic targets, the inverse problem of magnetic moment estimation was constructed based on the hybrid model of ellipsoid and magnetic dipole array. In order to solve the ill-posed problem of the magnetic moment estimation equations, the method was designed to estimate the magnetic moment parameters of the ellipsoid and magnetic dipole array, and the improved discrepancy principle and the maximum chi-square distribution stop criterion were introduced to improve the semi-convergence behavior of the conjugate gradient least squares (CGLS) method. Through simulation examples of magnetic target magnetic moment estimation and ship model measurement data, the performance of two CGLS methods, Tikhonov algorithm and stepwise regression method, were compared and analyzed from four aspects: relative error of magnetic moment estimation, relative error of magnetic field fitting, relative error of magnetic field extrapolation, and computational time complexity. The CGLS method has the advantages of high accuracy of magnetic moment estimation, high immunity of magnetic moment estimation to interference, high accuracy of magnetic field fitting and extrapolation, and low computational time complexity.

A transformer-based dual-domain network for reconstructing FOV extended cone-beam CT images from truncated sinograms in radiation therapy

Background and objectiveCone-beam computed tomography (CBCT) is widely used in clinical radiotherapy, but its small field of view (sFOV) limits its application potential. In this study, a transformer-based dual-domain network (dual_swin), which combined image domain restoration and sinogram domain restoration, was proposed for the reconstruction of complete CBCT images with extended FOV from truncated sinograms. MethodsThe planning CT images with large FOV (LFOV) of 330 patients who received radiation therapy were collected. The synthetic CBCT (sCBCT) images with LFOV were generated from CT images by the trained cycleGAN network, and CBCT images with sFOV were obtained through forward projection, projection truncation, and filtered back projection (FBP), comprising the training and test data. The proposed dual_swin includes sinogram domain restoration, image domain restoration, and FBP layer, and the swin transformer blocks were used as the basic feature extraction module in the network to improve the global feature extraction ability. The proposed dual_swin was compared with the image domain method, the sinogram domain method, the U-Net based dual domain network (dual_Unet), and the traditional iterative reconstruction method based on prior image and conjugate gradient least-squares (CGLS) in the test of sCBCT images and clinical CBCT images. The HU accuracy and body contour accuracy of the predicted images by each method were evaluated. ResultsThe images generated using the CGLS method were fuzzy and obtained the lowest structural similarity (SSIM) among all methods in the test of sCBCT and clinical CBCT images. The predicted images by the image domain methods are quite different from the ground truth and have low accuracy on HU value and body contour. In comparison with image domain methods, sinogram domain methods improved the accuracy of HU value and body contour but introduced secondary artifacts and distorted bone tissue. The proposed dual_swin achieved the highest HU and contour accuracy with mean absolute error (MAE) of 23.0 HU, SSIM of 95.7%, dice similarity coefficient (DSC) of 99.6%, and Hausdorff distance (HD) of 4.1 mm in the test of sCBCT images. In the test of clinical patients, images that were predicted by dual_swin yielded MAE, SSIM, DSC, and HD of 38.2 HU, 91.7%, 99.0%, and 5.4 mm, respectively. The predicted images by the proposed dual_swin has significantly higher accuracy than the other methods (P < 0.05). ConclusionsThe proposed dual_swin can accurately reconstruct FOV extended CBCT images from the truncated sinogram to improve the application potential of CBCT images in radiotherapy.

Visualization of the distortion induced by nonlinear noise reduction in computed tomography.

We developed a method to visualize the image distortion induced by nonlinear noise reduction algorithms in computed tomography (CT) systems. Nonlinear distortion was defined as the induced residual when testing a reconstruction algorithm by the criteria for a linear system. Two types of images were developed: a nonlinear distortion of an object () image and a nonlinear distortion of noise () image to visualize the nonlinear distortion induced by an algorithm. Calculation of the images requires access to the sinogram data, which is seldomly fully provided. Hence, an approximation of the image was estimated. Using simulated CT acquisitions, four noise levels were added onto forward projected sinograms of a typical CT image; these were noise reduced using a median filter with the simultaneous iterative reconstruction technique or a total variation filter with the conjugate gradient least-squares algorithm. The linear reconstruction technique filtered back-projection was also analyzed for comparison. Structures in the image indicated contrast and resolution reduction of the nonlinear denoising. Although the approximated image represented the original image well, it had a higher random uncertainty. The image for the median filter indicated both stochastic variations and structures reminding of the object while for the total variation filter only stochastic variations were indicated. The developed images visualize nonlinear distortions of denoising algorithms. The object may be distorted by the noise and vice versa. Analyzing the distortion correlated to the object is more critical than analyzing a distortion of stochastic variations. The absence of nonlinear distortion may measure the robustness of the denoising algorithm.

Resolution enhancement of 2D controlled-source electromagnetic images by use of point-spread function inversion

The marine controlled-source electromagnetic technique is employed both in large-scale geophysical applications as well as within the exploration of hydrocarbons and gas hydrates. Because of the diffusive character of the EM field, only very low frequencies are used, leading to inversion results with low resolution. In this paper, we calculated the resolution matrix associated with the inversion and derived the corresponding point-spread functions. The PSFs provided information about how much the actual inversion was blurred. Using a space-varying deconvolution can thus further improve the inversion result. The actual deblurring was carried out using the nonnegative flexible conjugate gradient least-squares (NN-FCGLS) algorithm, which is a fast iterative restoration technique. To attain completeness, we also introduced the results obtained using a blind deconvolution algorithm based on the maximum likelihood estimation with unknown PSFs. The potential of the proposed approach has been demonstrated using both complex synthetic data and field data acquired at the Wisting oil field in the Barents Sea. In both cases, the resolution of the final inversion result was improved and showed greater agreement with the known target area.

Comparative study on the applicability of regularization method in ship’s magnetic signature modeling based on monopole array

In this paper, we first study the applicability and stability of the Tikhonov regularization method (TRM) and the truncated singular value decomposition method in ship’s magnetic signature inversion modeling. To further improve the noise immunity and robustness of the model, we apply an iterative regularization algorithm based on the conjugate gradient least squares (CGLS) method. Numerical simulation shows that the proposed algorithm is more applicable and can obtain better results of ship’s magnetic field extrapolation under different noise backgrounds. In the ship model experiment, the errors of the TRM and CGLS method for magnetic signature extrapolation are, respectively, 4.77% and 3.67%, showing that the proposed algorithm can improve the inversion effect of the ship model magnetic field.

Estimation of a Fission Product Concentration Distribution in a Spherical Body from Surface Release Measurements

The diffusion of fission products (FPs) in reactor materials affects the nuclear source term. The diffusion coefficient itself is measured through various techniques. In the release method, it is of interest to know the initial distribution of the FPs in nuclear graphite or other materials from an exterior measurement like mass surface flux or cumulative mass release. In this paper, a Fredholm integral of the first kind is considered, relating the initial distribution to the cumulative release fraction of a diffusant from a spherically symmetric body. The Tikhonov regularization, conjugate gradient least-squares (CGLS) method, and algebraic reconstruction technique (ART) with nonnegativity and conserved mass constraints were compared to fractional release data from a simulated linear profile using data for Cs diffusion in a 0.32-cm sphere NBG-18 at 1090 K. The Tikhonov regularization was shown to provide a better estimation of the initial linear distribution than the CGLS and ART methods. The performance of the Tikhonov regularization was further examined with simulated constant, quartic, and exponential initial distributions. The Tikhonov regularization provided a reasonable recovery of the exponential profile, while the estimation of the linear, constant, and quartic profiles suffers from several issues.

Fast Convolutional Sparse Dictionary Learning Based on LocOMP and Its Application to Bearing Fault Detection

Sparse representations based on convolutional sparse dictionary learning (CSDL) provides an excellent framework for extracting fault impulse response caused by bearing faults. In order to achieve fast dictionary learning, most CSDL-based fault diagnosis techniques recommend truncating the original data. However, the choice of truncation length is very difficult. An improper truncation length will lead to the problems of fault pattern rupture and uneven sparsity distribution. By contrast, if the data are not truncated, these problems will not occur. However, this will result in significant memory and computation consumption of CSDL. In order to overcome these problems, a novel CSDL method, fast CSDL algorithm (FCSDL), is proposed by combining local orthogonal matching pursuit (OMP) algorithm with conjugate gradient least squares (CGLS) algorithm. The new method can achieve fast dictionary learning without truncating data, and occupies very little memory. On this basis, an adaptive bearing fault diagnosis method based on envelope spectrum Kurtosis optimization is further proposed. When the sparsity is unknown, the new method can accurately search the optimal sparsity and quickly recover the fault impact submerged in noise. The performance of the proposed fault diagnosis method is verified by using simulated signals, open bearing data and wheelset bearing experimental data. It is compared with the union of convolutional dictionary learning algorithm (UC-DLA) to highlight the advantages of the proposed method. The test code of reproducible research can be downloaded at https://github.com/aresmiki/ FastCSDL.

Reducing axial truncation artifacts in iterative cone-beam CT for radiation therapy using a priori preconditioned information.

Cone-beam computed tomography (CBCT) is increasingly utilized in radiation therapy for image guidance and adaptive applications. While iterative reconstruction algorithms have been shown to outperform traditional filtered back-projection methods in improving image quality and reducing imaging dose, they cannot handle data truncation in the axial view, which frequently occurs in the full-fan partial-trajectory acquisition mode. This proof-of-concept study presents a novel approach on truncation artifact reduction by utilizing a priori preconditioned information as the initial input for the iterative algorithm. Projections containing axial truncation were used for image reconstruction in extended axial field-of-view (AFOV) using the conjugate gradient least-squares (CGLS) algorithm. A priori information in the form of a planning fan-beam CT (FBCT) was repositioned in the expected CBCT imaging geometry, then further processed to dampen high-density features and convolved with a cubic Gaussian kernel to ensure differentiability for the gradient descent method. Anatomical and positional differences between the estimated and the actual imaging object were introduced to verify the efficacy of the proposed method. Extending the reconstruction AFOV alone could partially reduce truncation artifact. Using a priori information directly resulted in ghosting artifact when there were anatomical and positional differences between the estimated and the actual imaging object. Using a priori preconditioned information was shown to effectively reduce truncation artifact and recover peripheral information. Using a priori preconditioned information can effectively alleviate truncation artifact and assist recovery of peripheral information in iterative CBCT reconstruction.

Ant Colony-Based Hyperparameter Optimisation in Total Variation Reconstruction in X-ray Computed Tomography.

In this paper, a computer-aided training method for hyperparameter selection of limited data X-ray computed tomography (XCT) reconstruction was proposed. The proposed method employed the ant colony optimisation (ACO) approach to assist in hyperparameter selection for the adaptive-weighted projection-controlled steepest descent (AwPCSD) algorithm, which is a total-variation (TV) based regularisation algorithm. During the implementation, there was a colony of artificial ants that swarm through the AwPCSD algorithm. Each ant chose a set of hyperparameters required for its iterative CT reconstruction and the correlation coefficient (CC) score was given for reconstructed images compared to the reference image. A colony of ants in one generation left a pheromone through its chosen path representing a choice of hyperparameters. Higher score means stronger pheromones/probabilities to attract more ants in the next generations. At the end of the implementation, the hyperparameter configuration with the highest score was chosen as an optimal set of hyperparameters. In the experimental results section, the reconstruction using hyperparameters from the proposed method was compared with results from three other cases: the conjugate gradient least square (CGLS), the AwPCSD algorithm using the set of arbitrary hyperparameters and the cross-validation method.The experiments showed that the results from the proposed method were superior to those of the CGLS algorithm and the AwPCSD algorithm using the set of arbitrary hyperparameters. Although the results of the ACO algorithm were slightly inferior to those of the cross-validation method as measured by the quantitative metrics, the ACO algorithm was over 10 times faster than cross—Validation. The optimal set of hyperparameters from the proposed method was also robust against an increase of noise in the data and can be applicable to different imaging samples with similar context. The ACO approach in the proposed method was able to identify optimal values of hyperparameters for a dataset and, as a result, produced a good quality reconstructed image from limited number of projection data. The proposed method in this work successfully solves a problem of hyperparameters selection, which is a major challenge in an implementation of TV based reconstruction algorithms.

The Conjugate Gradient Least Square Algorithm in Terahertz Tomography

Terahertz tomography allows for non-contact tomographic inspection of dielectric materials without the need for radiation protection measures. Terahertz tomography offers the opportunity to inspect such objects from multiple angles not only by measuring the absorption but also by acquiring the time-of-flight of the radiation. Hence, this technique facilitates the reconstruction of the complete complex refractive index of a sample under test. Even complicated surface structures can be imaged, provided the feature size is above the diffraction limit roughly given by the wavelength of the terahertz radiation in use. For industrial applications, computational efficiency and imaging performance are crucial. Therefore, we apply the iterative conjugate gradient least square (CGLS) algorithm to reconstruct images from terahertz tomography data. To ensure reliable convergence of this semi-convergent CGLS algorithm a stopping mechanism based on the L-curve criterion is implemented. The result is a fast-converging, parallelizable method, which offers the flexibility to adapt to the specifics of terahertz tomography. As an example of this adaptability, we implement a non-negativity constraint, suppressing noise in the image and significantly enhancing reconstruction quality.

A comparison of algebraic reconstruction techniques for a single-detector muon computed tomography system

An attenuation-based muon computed tomography algorithm has been developed to enable imaging of large infrastructure using measurements from a single muon tracking module. The performance of the algorithm depends on the choice of reconstruction technique that is incorporated, and is evaluated for seven different reconstruction techniques from the computed tomography literature. The L1-Norm reconstruction technique proposed by Kudo et al. is found to provide the best overall performance. The least-squares conjugate gradient (LSCG) technique and the Simultaneous Iterative Reconstruction Technique (SIRT) are also capable of identifying special nuclear materials, but typically display higher levels of reconstruction artifacts in comparison with L1-Norm. A muon-tomography reconstruction of a simplified simulated reactor core using the L1-Norm algorithm is able to accurately reproduce the geometry, including the identification of missing fuel channels.

Time-domain elastic Gauss–Newton full-waveform inversion: a matrix-free approach

SUMMARYWe present a time-domain matrix-free elastic Gauss–Newton full-waveform inversion (FWI) algorithm. Our algorithm consists of a Gauss–Newton update with a search direction calculated via elastic least-squares reverse time migration (LSRTM). The conjugate gradient least-squares (CGLS) method solves the LSRTM problem with forward and adjoint operators derived via the elastic Born approximation. The Hessian of the Gauss–Newton method is never explicitly formed or saved in memory. In other words, the CGLS algorithm solves for the Gauss–Newton direction via the application of implicit-form forward and adjoint operators which are equivalent to elastic Born modelling and elastic reverse time migration, respectively. We provide numerical examples to test the proposed algorithm where we invert for P- and S-wave velocities simultaneously. The proposed algorithm performs positively on mid-size problems where we report solutions of slight improvement than those computed using the conventional non-linear conjugate gradient method. In spite of the aforementioned limited gain, the theory developed in this paper contributes to a better understanding of time-domain elastic Gauss–Newton FWI.

Mapping the dipole orientation distribution within a super-resolution scale via fluorescence polarization modulation.

Conventional fluorescence polarization microscopy has been largely used to monitor the orientation and the structural information of biomolecules labeled with fluorescence dipoles but suffers from the optical diffraction limit. Here, we put forward a novel algorithm to simultaneously acquire the super-resolution image and the effective orientation distribution information of dipole clusters at corresponding super-resolution. In this paper, the orientation distribution of dipole clusters is statistically modeled by its mean orientation and orientation deviation, which are, respectively, represented by the middle direction and the opening angle of a sector shape. According to this model and microscopy imaging theory, the joint reconstruction algorithm is deduced mathematically in detail based on the conjugate gradient least-squares method. By applying this algorithm to different samples, the reconstructed results prove more than twice the resolution of wide-field images and the orientation distribution information at corresponding spatial resolution. Furthermore, the high accuracy of this algorithm in reconstructing super-resolution orientation distribution information is verified by Monte Carlo simulations.

A Symmetric Successive Overrelaxation (SSOR) based Gauss-Seidel Massive MIMO Detection Algorithm

With an increasing number of antennas and users, the complexity increases dramatically, in order to simplify the process. In this paper, We propose existed algorithms discussing their principles and their advantages as well as the comparison between each method and then we provide the possible improvement in theory changing steps in iteration process to avoid the matrix inversion, by introducing a pre-conditioned gradient (PCG) method and further improved Jacobi method and using Neumann-series terms to estimate an approximate matrix inversion and the Gauss-Seidel (GS) method with a diagonal-approximate initial solution to the method and biconjugate gradient stabilized (BICGSTAB) method and above all of the methods are based on the minimum mean square error (MMSE) detector including the application in multi-user MIMO system. Another approach is from a signal detector based on symmetric successive over relaxation (SSOR) method without matrix inversion and Neumann series based on the zero forcing signal detector. The third algorithm uses a modified version of the conjugate gradient least square (CGLS). These methods are mainly replacing the inversion process into iteration process to reduce the complexity of the algorithm not only replacing the stages but optimizing the speed from setting the initial solution as well as the improvement using the Neumann-series to replace the iteration part. Finally, we proposed a symmetric successive overrelaxation (SSOR) based Gauss-Seidel for massive MIMO detection.

Image Restoration Using a Fixed-Point Method for a TVL2 Regularization Problem

In this paper, we first propose a new TVL2 regularization model for image restoration, and then we propose two iterative methods, which are fixed-point and fixed-point-like methods, using CGLS (Conjugate Gradient Least Squares method) for solving the new proposed TVL2 problem. We also provide convergence analysis for the fixed-point method. Lastly, numerical experiments for several test problems are provided to evaluate the effectiveness of the proposed two iterative methods. Numerical results show that the new proposed TVL2 model is preferred over an existing TVL2 model and the proposed fixed-point-like method is well suited for the new TVL2 model.

Dynamic Force Identification in Peripheral Milling Based on CGLS Using Filtered Acceleration Signals and Averaged Transfer Functions

Dynamic milling forces have been widely used to monitor the condition of the milling process. However, it is very difficult to measure milling forces directly in operation, particularly in the industrial scene. In this paper, a dynamic force identification method in time domain, conjugate gradient least square (CGLS), is employed for reconstructing the time history of milling forces using acceleration signals in the peripheral milling process. CGLS is adopted for force identification because of its high accuracy and efficiency, which handles the ill-conditioned matrix well. In the milling process, the tool with high-speed rotation has different transfer functions between tool nose and accelerometers at different angular positions. Based on this fact, the averaged transfer functions are employed to reduce the error amplification of regularization processing for milling force identification. Moreover, in order to eliminate the effect of idling and high-frequency components on identification accuracy, the Butterworth band-pass filter is adopted for acceleration signals preprocessing. Finally, the proposed method is validated by milling tests under different cutting parameters. Experimental results demonstrate that the identified and measured milling forces are in good agreement on the whole time domain, which verifies the effectiveness and generalization of the indirect method for milling force measuring. In addition, the Tikhonov regularization method is also implemented for comparison, which shows that CGLS has higher accuracy and efficiency.

Laminography in the lab: imaging planar objects using a conventional x-ray CT scanner

Whilst x-ray micro-computed tomography (CT) machines have developed into a popular laboratory tool for non-destructive 3D imaging of materials, they are not well-suited for scanning flat objects, for which there is an increasing demand. Computed laminography (CL) techniques have been developed for imaging planar samples such as fossils, paintings, printed circuit boards and composite panels. This paper is the first work demonstrating how CL may be implemented on a conventional industrial laboratory micro-CT scanner, without the need for specialist equipment. A rotary CL scan configuration (also known as tilted or oblique CT) was successfully implemented, and the projection images reconstructed using the conjugate gradient least squares (CGLS) method. Computer simulations were used to analyse the causes of image artefacts seen in the cone-beam CL reconstructions, and to devise corrections for them. The resulting laboratory CL method produces reconstructions with fewer artefacts and a more isotropic resolution compared to limited-angle scan configurations otherwise used for planar samples. This demonstrates that CL can be performed in the laboratory without specialist equipment beyond that required for standard CT.

Inverse Analysis of Radiative Flux Maps for the Characterization of High Flux Sources

The reconstruction of the angular and spatial intensity distribution from radiative flux maps measured in high flux solar simulators (HFSS) or optical concentrators is an ill-posed inverse problem requiring special solution strategies. We aimed at providing a solution strategy for the determination of intensity distributions of arbitrarily complicated concentrating facilities. The approach consists of the inverse reconstruction of the intensities from multiple radiative flux maps recorded at various positions around the focal plane. The approach was validated by three test cases including uniform spatial, Gaussian spatial, and uniform angular distributions for which we successfully predicted the intensity for a square-shaped target with edge length of 0.5 m and for a displacement range spanning ±1.5 m at a resolution of 3.2 × 106 elements, yielding relative errors between 19.8–26.4% and 15.7–25.6% when using Tikhonov regularization and the conjugate gradient least square (CGLS) method, respectively. The latter method showed superior performance and was used at a resolution of 2.35 × 107 elements to analyze EPFL's HFSS comprising 18 lamps. The inverse solution for a single lamp retrieved from experimentally measured and simulated radiative flux maps showed peak intensities of 13.7 MW/m2/sr and 16.0 MW/m2/sr, respectively, with a relative error of 81.1%. The inverse reconstruction of the entire simulator by superimposing the single lamp intensities retrieved from simulated flux maps resulted in a maximum intensity of 18.8 MW/m2/sr with a relative error of 80%. The inverse method proved to provide reasonable intensity predictions with limited resolution of details imposed by the high gradients in the radiative flux maps.

Mxpfit: A library for finding optimal multi-exponential approximations

mxpfit is a library implemented in C++ to find optimal approximations of functions by multi-exponential sums with complex-valued parameters. The library provides an interface for evaluating exponents and coefficients from sampling data on a uniform grid using the fast Estimation of Signal Parameters via Rotational Invariance Techniques (ESPRIT) algorithm originally proposed by Potts and Tasche (2015). The parameters can be estimated efficiently from a sampling data even including noise. A modified balanced truncation algorithm to find the multi-exponential sum with a smaller order is also provided. These features are useful for finding optimal exponential sum approximations of analytic functions or large-scale numerically sampled data set. Program summaryProgram Title: mxpfitProgram Files doi:http://dx.doi.org/10.17632/vjygkwpss4.1Licensing provisions: MITProgramming language: C++Nature of problem: Nonlinear curve fitting by multi-exponential function, frequency estimation of signals, and reduction of multi-exponential sumSolution method: The modified fast Estimation of Signal Parameters via Rotational Invariance Techniques (ESPRIT) algorithm originally proposed in [1] is utilized for the parameter estimation of multi-exponential sums from sampled data on a uniform grid. In this method, partial Lanczos bidiagonalization is used to find a low-rank approximation of a Hankel matrix constructed from sampled data. An overdetermined Vandermonde system is solved by the conjugate gradient least square (CGLS) method with preconditioner to obtain the coefficients of the multi-exponential sum. The truncation of the multi-exponential sum is performed by a modified balanced truncation method in [2]. The projection matrix to obtain the truncated system is determined by computing con-eigenvalues of a Gramian matrix, which can be computed efficiently with highly relative accuracy using the algorithm in [3].Additional comments: A C++ compiler supporting the C++11 standard is required to compile programs using the mxpfit library. mxpfit depends on the Eigen (http://eigen.tuxfamily.org/) and FFTW3 (http://www.fftw.org/) libraries. CMake (https://cmake.org/) is used for building example programs.

Nonparametric blind SAR image super resolution based on combination of the compressive sensing and sparse priors

This paper by proposing a novel approach, is one the first works that addresses the highly ill-posed problem of nonparametric blind single image super resolution (SISR) of the synthetic aperture radar (SAR) images. Combination of an adaptive compressive sensing (CS) technique and some effective sparse priors, as a powerful regularizer in the both high resolution (HR) image reconstruction and the point spread function (PSF) estimation domains is the fundamental idea of the proposed method. This task is formulated as a new cost function to be minimized with respect to an intermediate reconstructed HR image patch and a nonparametric PSF kernel, according to the alternative minimization (AM) algorithm. To solve the optimization of cost function, a numerical scheme based on the conjugate gradient least squares (CGLS) method is proposed. Experimental results for the both synthetic and realistic low resolution (LR) SAR images demonstrate that the proposed method achieves the state-of-the-art performance.