Landweber Iteration Algorithm Research Articles

Numerical Reconstruction of Heat Source on a Semi-Infinite Rod

Inverse heat conduction on an unbounded domain has always been the focus and difficulty in engineering and mechanical fields. In this paper, we consider the inverse problem of identifying the heat source by using the terminal measured temperature data. Being different from other ordinary heat conduction problems, our mathematical model is defined in a semi-infinite region, which may lead to many numerical methods that cannot be used directly. Based on the artificial boundary method, we first introduce the truncated boundary and derive the exact boundary conditions on it. Then we use the finite difference method to establish an approximate difference scheme for the forward problem. For the inverse problem, we design the Landweber and conjugate gradient iterative algorithm and do numerical experiments. Numerical results verify the correctness and effectiveness of our algorithm.

A 12-Electrode ECT Sensor with Radial Screen for Fluid Diagnosis

In this paper, the use of radial screens in an Electrical Capacitance Tomography (ECT) sensor is investigated with the aim of resulting in higher-resolution images. The effect of the screens on the quality of the reconstructed images was simulated. 12-electrode ECT sensors with and without screens were designed and tested. The capacitance between different electrode pairs was measured for two permittivity distributions. The obtained capacitance data were used to reconstruct images using the Projected Landweber Iteration algorithm based on linear and semi-linear ECT models. The sensitivity of the measurements and the accuracy of the obtained images for the ECT sensors with and without screens were analyzed. The main conclusion is that a little improvement in the quality of images can be achieved with the use of 12-Electrode ECT sensors with radial screen.

Solution of the backward problem for the space-time fractional diffusion equation related to the release history of a groundwater contaminant

Abstract Finding the history of a groundwater contaminant plume from final measurements is an ill-posed problem and, consequently, its solution is extremely sensitive to errors in the input data. In this paper, we study this problem mathematically. So, firstly, existence and uniqueness theorems of a quasi-solution in an appropriate class of admissible initial data are given. Secondly, in order to overcome the ill-posedness of the problem and also approximate the quasi-solution, two approaches (computational and iterative algorithms) are provided. In the computational algorithm, the finite element method and TSVD regularization are applied. This method is tested by two numerical examples. The results reveal the efficiency and applicability of the proposed method. Also, in order to construct the iterative methods, an explicit formula for the gradient of the cost functional J is given. This result helps us to construct two iterative methods, i.e., the conjugate gradient algorithm and Landweber iteration algorithm. We prove the Lipschitz continuity of the gradient of the cost functional, monotonicity and convergence of the iterative methods. At the end of the paper, a numerical example is given to show the validation of the iterative algorithms.

Super-resolution reconstruction algorithm for optical-resolution photoacoustic microscopy images based on sparsity and deconvolution.

The lateral resolution of the optical-resolution photoacoustic microscopy (OR-PAM) system depends on the focusing diameter of the probe beam. By increasing the numerical aperture (NA) of optical focusing, the lateral resolution of OR-PAM can be improved. However, the increase in NA results in smaller working distances, and the entire imaging system becomes very sensitive to small optical imperfections. The existing deconvolution-based algorithms are limited by the image signal-to-noise ratio when improving the resolution of OR-PAM images. In this paper, a super-resolution reconstruction algorithm for OR-PAM images based on sparsity and deconvolution is proposed. The OR-PAM image is sparsely reconstructed according to the constructed loss function, which utilizes the sparsity of the image to combat the decrease in the resolution. The gradient accelerated Landweber iterative algorithm is used to deconvolve to obtain high-resolution OR-PAM images. Experimental results show that the proposed algorithm can improve the resolution of mouse retinal images by approximately 1.7 times without increasing the NA of the imaging system. In addition, compared to the Richardson-Lucy algorithm, the proposed algorithm can further improve the image resolution and maintain better imaging quality, which provides a foundation for the development of OR-PAM in clinical research.

Experimental imaging and algorithm optimization based on deep neural network for electrical capacitance tomography for LN2-VN2 flow

The deep neural network (DNN) method is applied to better map the phase distribution compared with the conventical reconstruction algorithm for nonlinear problems of electrical capacitance tomography (ECT). Three DNNs are trained and evaluated in the numerical experiment for cryogenic two-phase flow. A capacitance acquisition unit is designed. And a cryogenic experiment with liquid nitrogen and nitrogen vapor (LN2-VN2) as the working fluids is conducted to verify the imaging solution for practical application. The DNN with the input of the measured capacitance data (DNN-C) has shown its generalization ability for the practical imaging application. Further, the DNN optimal method is coupled to the conventional LBP and Landweber iteration algorithm by evaluating the nonlinearity between the error capacitance vector and the real phase distribution vector (DNN-EC). The comparison shows that the DNN modification method can significantly improve the phase distribution closed to the pattern in the training input samples with an elimination of the artifacts and sharpening interface, but it fails to recognize the flow pattern which is not included in the samples. The result shows that compared with the DNN-EC, DNN-C is a better choice for the image reconstruction for the cryogenic application.

Coupling RBF-based meshless method and Landweber iteration algorithm for approximating a space-dependent source term in a time fractional diffusion equation

The problem of determining a space-dependent source term in a time fractional diffusion equation is considered from the measured data at the final time. To find an approximation of the source term, a methodology involving minimization of the cost functional is applied. Also, in order to construct the Landweber iteration algorithm, an explicit formula for the gradient of the cost functional J is given via the solution of an adjoint problem. The resulting adjoint problem is treated by a radial basis function method for spatial dimension and a finite difference scheme for the time fractional derivative followed by an iterative domain decomposition method to achieve a desired accuracy. In addition, Lipschitz continuity of the gradient of the cost functional, monotonicity and convergence of Landweber iteration algorithm are proved. At the end, a numerical example is given to show the validation of the iterative algorithm.

Image Stripe Noise Removal Based on Compressed Sensing

The sensors or electronic components are vulnerable to interference in the camera’s imaging process, usually leading to random directional stripes. Therefore, a method of stripe noise removal based on compressed sensing is proposed. First, the measurement matrix of the image with stripe noise is established, which makes the stripe images equivalent to the observation of the original image. Second, the relationships between the corresponding coefficients of adjacent scales are defined. On this basis, the bivariate threshold function is set in the curvelet sparse domain to represent the features of images. Finally, the Landweber iteration algorithm of alternating convex projection and filtering operation is achieved. Furthermore, to accelerate the noise removal at the initial stage of iteration and preserve the image details later, the exponential threshold function is utilized. This method does not need many samples, which is different from the current deep learning method. The experimental results show that the proposed algorithm represents excellent performance in removing the stripes and preserving the texture details. In addition, the PSNR of the denoised image has been dramatically improved compared with similar algorithms.

Robust Automated Stopping Criterion for Semi-Convergent Image and Velocity Reconstruction in Electrical Capacitance Volume Tomography

Abstract—Electrical Capacitance Volume Tomography (ECVT) is a low-cost and high-speed sensing modality with great potential for industrial multiphase flow monitoring. In this work, we examine the use of the slope of the capacitance vector residual curve, as directly provided by the measurement data, to formulate a robust stopping criterion for the iterative image reconstruction in ECVT. The methodology is illustrated based on experimental data from a gas-solid fluidized bed. We show that the proposed stopping criterion can improve the performance of both the image reconstruction and the flow velocity profiling in ECVT applications. For concreteness, we focus on the popular Landweber iterative reconstruction algorithm although other reconstruction algorithms exhibiting semi-convergent behavior might benefit from the present analysis as well.

Image reconstruction method for electrical capacitance tomography using adaptive simulated annealing algorithm.

Image reconstruction of electrical capacitance tomography (ECT) is an ill-posed inverse problem. The simulated annealing (SA) algorithm is suitable for the solution of the ECT inverse problem. However, selection of related parameters of the SA algorithm will influence the reconstruction performance of ECT. We present an ECT image reconstruction method based on the adaptive simulated annealing (ASA) algorithm. We adopt the bat algorithm in the new solution generation strategy of the ASA algorithm. Moreover, the definition of the energy function introduces the sparsity of the reconstructed image. As a result, an adaptive annealing strategy is proposed to select the appropriate annealing rate. We made reconstructed image comparisons among linear back-projection, Landweber iteration, SA, and ASA algorithms via simulation and static experiment. Results show that improved reconstructed images can be obtained using the ASA algorithm.

Image Reconstruction of Electrical Capacitance Tomography Based on Adaptive Support Driven Bayesian Reweighted Algorithm

Image reconstruction of electrical capacitance tomography (ECT) is a nonlinear and ill-posed inverse problem. Therefore, how to introduce an effective algorithm to reduce the ill conditioned degree of ECT imaging, thereby improving the imaging accuracy is an important subject of ECT algorithm research. In order to further study the subject, a novel ECT image reconstruction algorithm based on an adaptive support driven Bayesian reweighted (ASDBR) algorithm was proposed in this paper. The great advantage of this algorithm is that it can accurately extract the main features of the flow pattern and remove redundant information. This algorithm transforms the original problem into a series of subproblems with iteratively reweighted weights, and solves these subproblems by the iterative shrinkage-thresholding algorithm (ISTA). Comparisons are made among the ASDBR algorithm, the Landweber iterative algorithm, the sparse Bayesian learning (SBL) algorithm, and Lasso. Both simulation and experiment results show that the proposed new method considerably enhances the quality of the reconstructed image.

Optimized Landweber iterative fast image reconstruction algorithm for electromagnetic tomography

Optimized Landweber iterative fast image reconstruction algorithm for electromagnetic tomography

Calibration method of instrument line shape for infrared radiometer

Interferometric infrared spectral radiometer has high luminous flux and large passing aperture, so the spectral data collected by the instrument are the convolution of the target spectral data and the instrument line shape (ILS). The main factors affecting the ILS include truncation effect, finite field of view effect, off-axis effect, defocus effect and relative position of detector and so on. In this paper the truncation effect, finite field of view effect, off-axis effect and defocus effect on ILS are expounded. These ILS errors all cause the ideal spectrum to drift towards the low wave number and widen. In this paper, the ideal absorption spectrum of water vapor is simulated by the line-by-line integration, through using MATLAB software and combining with the data of water vapor spectrum in HITRAN database, and the quantitative relationship between the error of instrument linear function and spectral distortion is established. According to the simulation results, a factor weight correction algorithm is proposed. It is believed that with this method, the error spectrum caused by ILS has the same optical properties as the ideal spectrum, but the optical path difference is different. Therefore, the coefficient matrix <i> <b>H</b> </i> can be constructed to establish the quantitative relationship between the error spectrum and the ideal spectrum, and the error spectrum can be corrected by using Landweber iterative algorithm. In this paper, the simulation data of water vapor absorption are used to verify the factor weight correction algorithm, and the spectral drift decreases from 0.51 cm<sup>–1</sup> to less than 0.01 cm<sup>–1</sup>. The accuracy of the factor weight correction algorithm is verified by the experimental observation of the standard black body with self-developed interferometric infrared spectral radiometer. The spectral drift of the measured data decreases from 0.226 cm<sup>–1</sup> to 0.012 cm<sup>–1</sup>, and the corrected spectral data are more accurate.

Investigation of Spatial Resolution of Electrical Capacitance Tomography Based on Coupling Simulation

Electrical capacitance tomography (ECT) has been developed for imaging permittivity distribution of two-phase flows in a pipe or vessel with high temporal resolution. The spatial resolution of ECT is relatively low due to its soft-field, ill-posed, and ill-conditioned nature. So far, there is no comprehensive analysis of the spatial resolution of ECT for complex two-phase flows. To assess the merits of ECT systematically, 2-D modeling is carried out based on fluid-field and electrostatic-field coupling simulation for liquid–solids two-phase flows. The real permittivity distribution and the corresponding capacitance measurements can be obtained simultaneously. The effect of the number of electrodes, the mesh used for reconstruction, and the initialization method for the Landweber iteration algorithm on image quality are investigated. In terms of the correlation coefficient between the reconstructed permittivity distribution and the real permittivity distribution obtained from the coupling simulation model, the accuracy and the stability of reconstructed images are quantitatively evaluated. Furthermore, the sensitivity matrices and point spread functions calculated without capacitance measurements are used to compare the performance of ECT sensors and validate the results with capacitance measurements from the coupling simulation. The two methods, with and without capacitance measurements, are used to evaluate the change in spatial resolution of ECT with a different number of electrodes.

Electrical capacitance tomography image reconstruction by improved orthogonal matching pursuit algorithm

In order to improve the quality of reconstructed images in electrical capacitance tomography (ECT), the image reconstruction method based on compressed sensing for ECT is studied. First, the traditional discrete Fourier transform and discrete cosine transform are used as a sparsity basis to make the grey vectors of the typical two-phase flow distributions sparse. The energy loss of the sparse signals under different sparsity degrees is calculated, and the effect of energy loss on the quality of reconstructed images is studied. Then, using the natural sparsity of the original signal, an improved orthogonal matching pursuit algorithm for ECT image reconstruction is proposed. There are two main improvements in the proposed algorithm. First, multiple columns instead of one column in each iteration are selected for improving the reconstruction speed. Second, a regularisation solution instead of the least-squares solution is used for improving the adaptability to ill-posed inverse problems. Simulation and experimental tests are carried out and the results show that the proposed method can effectively improve the reconstructed images quality, and on the whole, obtain better reconstruction results than the Landweber iteration algorithm, the Tikhonov regularisation algorithm, and the gradient projection for sparse reconstruction algorithm.

Research on Direct 3D Electromagnetic Tomography Technique

Electromagnetic tomography (EMT) is an emerging imaging modality which attempts to visualize the distribution of electrically conductive or magnetically permeable materials using measurements obtained from coils distributed around the vessels or pipelines. The traditional EMT research concerns about the two-dimensional (2D) cross-sectional imaging. However, hardly any three-dimensional (3D) information of the target objects can be obtained. Effective 3D EMT imaging technique can provide further axial information and intuitional images of the target objects, which is especially suitable for processes with significant change of conductivity distribution in the axial direction. The direct 3D EMT imaging technique, which uses measurements obtained from both the same layers and the different layers and can directly acquire the 3D reconstructed images, is studied in this paper including issues such as 3D sensor layout, measurement strategy, simulation research, 3D image reconstruction algorithm. The 3D EMT finite element simulation model is established to conduct the forward problem calculations and further verify the feasibility of the sensitivity matrix. In the simulation process, a three-plane cylindrical EMT sensor with 12 coils is designed to verify the feasibility and effectiveness of the direct 3D EMT imaging technique. The images are reconstructed by three image reconstruction algorithms including Tikhonov regularization method, Projected Landweber iterative algorithm and total variation regularization method. Compared with the other two algorithms, total variation regularization method is superior in terms of the eight typical patterns due to the fact that it possesses good edge-preserving property.

Analytic time-of-flight positron emission tomography reconstruction: two-dimensional case

In a positron emission tomography (PET) scanner, the time-of-flight (TOF) information gives us rough event position along the line-of-response (LOR). Using the TOF information for PET image reconstruction is able to reduce image noise. The state-of-the-art TOF PET image reconstruction uses iterative algorithms. Analytical image reconstruction algorithm exits for TOF PET which emulates the iterative Landweber algorithm. This paper introduces such an algorithm, focusing on two-dimensional (2D) reconstruction. The proposed algorithm is in the form of backprojection filtering, in which the backprojection is performed first, and then a 2D filter is applied to the backprojected image. For the list-mode data, the backprojection is carried out in the event-by-event fashion, and a profile function may be used along the projection LOR. The 2D filter depends on the TOF timing resolution as well as the backprojection profile function. In order to emulate the iterative algorithm effects, a Fourier-domain window function is suggested. This window function has a parameter, k, which corresponds to the iteration number in an iterative algorithm.

Image Noise Covariance Can be Adjusted by a Noise Weighted Filtered Backprojection Algorithm.

It was believed that the noise covariance of an image reconstructed with the filtered backprojection (FBP) algorithm is anisotropic. This paper shows that the noise-weighted FBP algorithm is able to alter the noise covariance and make it approximately isotropic. For the noise-weighted FBP algorithm, this paper develops a closed-form expression for the noise variance image and a closed-form expression for the noise covariance image. Computer simulations are carried out to evaluate the noise covariance with and without the noise weighting in the FBP algorithm. Transmission and emission noise models are used in computer simulations. The noise weighted FBP algorithm has a parameter that emulates the iteration number in the iterative Landweber algorithm. It is observed that the noise covariance can be altered by this emulated iteration number when noise weighting is used. The noise weighting in the noise-weighted FBP algorithm is able to change the noise covariance and a proper selection of the emulated iteration number may be able to give an approximately isotropic image noise covariance function.

A novel algorithm based on L1-Lp norm for inverse problem of electromagnetic tomography

Electromagnetic tomography (EMT) is a novel imaging modality of electrical tomography, which appears to be very promising. The precision and imaging speed of image reconstruction algorithms of EMT are the keys to its application in industrial and biomedical fields. Image reconstruction in EMT is a typical ill-posed and ill-conditioned inverse problem. Following analyzing the advantages and disadvantages of traditional Tikhonov regularization algorithm and total variation algorithm, a new objective functional is introduced with L1 norm on the data term and Lp norm on the regularization term in this paper, which transforms the inverse problem of EMT into an optimization problem. Besides, the L1-Lp optimization framework is solved by the approximation Gauss-Newton algorithm. Both numerical simulation and experimental results demonstrate that the proposed algorithm is capable of enhancing spatial resolution. It also proves that the proposed algorithm has better performance in terms of the typical patterns, compared with the traditional image reconstruction algorithms such as linear back projection (LBP), standard Tikhonov regularization algorithm and the projected Landweber iterative algorithm.

Reconstruction of two-dimensional velocity distribution in scramjet by laser absorption spectroscopy tomography.

We propose a method to reconstruct the two-dimensional (2D) velocity distribution of flow fields by laser absorption spectroscopy tomography. A mathematic model is established to reveal the dependence of spectral absorbance on line-of-sight velocity distribution. Then, with multiple laser beams from different angular views covering the region of interest, a nonlinear equation set of 2D velocity distribution is established according to the model. The integrated absorbance coefficient distribution is reconstructed using the Landweber iteration algorithm and substituted into the nonlinear equation set for further simplification. Finally, the velocity distribution is reconstructed by solving the simplified equation group via the interior point algorithm. The proposed method is validated numerically by reconstructing the velocity distribution of water molecules, as calculated by computational fluid dynamics, over a cross section of a double-mode scramjet combustor. The method does not require adding extra tracer particles and avoids issues arising from the short lifetime of molecular tags. It is suitable for diagnosis of high-speed flow fields.

Estimation of the Optimal Iteration Number for Minimal Image Discrepancy.

Due to noise, the iterative image reconstruction algorithms must stop early before reaching the convergence. There is an optimal stopping point, at which the discrepancy of the reconstruction to the true image reaches minimum. It is still an open problem to find this optimal stopping point. This paper establishes two approximate relationships towards solving this open problem. The first approximate relationship is between the iterative Landweber algorithm and an iteration-number-emulated filtered backprojection (FBP) algorithm. The second approximate relationship is between the optimal iteration-number-emulated FBP reconstruction and the optimal projection-domain filtered data. These two relationships can help us to estimate the optimal stopping point.