<title>Reconstruction from limited projection data</title>

Hybrid Taylor and block-pulse functions operational matrix algorithm and its application to obtain the approximate solution of stochastic evolution equation driven by fractional Brownian motion

Joint reconstruction of multiecho MR images using correlated sparsity

Super-pixel algorithm and group sparsity regularization method for compressed sensing MR image reconstruction

The article aims to propose the Lerch operational matrix method to solve a stochastic fractional differential equation. In this approach, the Lerch polynomials have been used as a basis function. Then, the product operational matrix, integral operational matrix, stochastic operational matrix, and operational matrix of fractional integral based on the Lerch polynomials have been constructed. The main characteristic of this method is to reduce the stochastic fractional differential equation into a system of algebraic equations by using derived operational matrices and suitable collocation points. Moreover, the convergence and error analysis of the presented method is also discussed in detail. Additionally, the applicability of the proposed technique is also demonstrated by solving some examples. To confirm the accuracy and effectiveness of the suggested technique, a comparison between the results produced by the proposed method and those obtained by other methods has been provided.

In matrix recovery, an unknown matrix can be reconstructed by a small number of limited and noisy measurements. Deep learning-based methods, such as deep generative models, pro-vide stronger priors that can serve to mitigate the pressure of sampling during image recovery. But such methods require that the recovered data be limited to the scope of the generator, otherwise it will lead to large recovery error. To circumvent this problem, in this paper, a framework for matrix recovery from limited measurements is proposed, which employs low rank approximation to characterize the deviation of generator, referred to as Low-Rank-Gen. Theoretically, we propose Matrix Set-Restricted Eigenvalue Condition (M-S-REC), and further prove the existence of decoders and upper bound of reconstruction error using certain number of measurements corresponding to such decoder. Empirically, we observe consistent improvements in reconstruction accuracy, PSNR index over competing approaches.

In this study, a simplified spherical approximated higher-order diffusion model is employed for three-dimensional diffuse optical tomography of osteoarthritis in the finger joints. We found that the use of higher-order diffusion model in a stand-alone framework provides significant improvement in reconstruction accuracy over the diffusion approximation model. However, we also found that this is not the case in the image-guided setting when spatial prior knowledge from x-rays is incorporated.

A novel algebraic iterative algorithm based on deflection tomography is presented. This algorithm is derived from the essentials of deflection tomography with a linear expansion of the local basis functions. By use of this algorithm the tomographic problem is finally reduced to the solution of a set of linear equations. The algorithm is demonstrated by mapping a three-peak Gaussian simulative temperature field. Compared with reconstruction results obtained by other traditional deflection algorithms, its reconstruction results provide a significant improvement in reconstruction accuracy, especially in cases with noisy data added. In the density diagnosis of a hypersonic wind tunnel, this algorithm is adopted to reconstruct density distributions of an axial symmetry flow field. One cross section of the reconstruction results is selected to be compared with the inverse Abel transform algorithm. Results show that the novel algorithm can achieve an accuracy equivalent to the inverse Abel transform algorithm. However, the novel algorithm is more versatile because it is applicable to arbitrary kinds of distribution.

Electrical Impedance Tomography (EIT) is an emerging non-invasive imaging technique with significant potential for detecting tissue anomalies; however, its performance is highly sensitive to variations in the frequency and amplitude of the injected electrical signals, which can lead to challenges in accurately differentiating between tissue types and detecting subtle pathological changes. This study aims to optimize EIT performance by systematically investigating the impact of signal frequency and amplitude on image reconstruction quality, thereby enhancing diagnostic accuracy. A portable multi-frequency EIT system was developed using Analog Discovery 2 and MATLAB, featuring a 16-electrode configuration arranged evenly around a tissue phantom, with beef tissue serving as an analog for human tissue due to its comparable conductivity properties. The experimental protocol varied signal amplitudes from 0.4 mA to 1.0 mA and frequencies from 50 kHz to 120 kHz, while two reconstruction algorithms the Gauss-Newton method and the GREIT algorithm were employed to evaluate image quality. Results demonstrated that the Gauss-Newton method achieved superior image clarity, with an approximate 18% improvement in reconstruction accuracy and a 20% reduction in noise at an optimal setting of 100 kHz frequency and 0.8 mA amplitude. Although the GREIT method provided faster reconstruction times, its lower sensitivity to amplitude variations resulted in less detailed anomaly detection. Overall, these findings underscore the critical importance of optimizing electrical parameters in EIT systems to enhance diagnostic capabilities. Future research should focus on integrating machine learning algorithms for real-time image processing and expanding the evaluation to include diverse tissue models to further improve the clinical applicability and robustness of EIT-based diagnostics.

The modes in the electrocardiogram (ECG) signal can be divided into stable modes and unstable modes. The unstable modes are of great significance for signal analysis and classification. When the traditional dynamic mode decomposition (DMD) method is directly applied to these signals, it is difficult to effectively extract these key modes due to the rank mismatch problem of the data matrix. In order to better capture and analyze unstable modes, this study introduced the Hankel matrix to expand ECG data and used it as the input of DMD to propose the Hankel dynamic mode decomposition (HDMD) method. Although HDMD has been applied in other physiological signal processing, this study is the first to successfully apply it to multi-lead ECG signal analysis. By optimizing the delay parameters of the Hankel matrix and retaining the modes that account for 90% of the singular value decomposition energy, we significantly improve the effectiveness of feature extraction. Each type of abnormal ECG signal data is classified on the PTB Diagnostic ECG database. The experimental results of the proposed model are compared with the direct use of DMD for modal extraction. The highest classification accuracy of multi-modal ECG signal data extracted by HDMD is more than 10% higher than that using DMD. At the same time, the mean squared error (MSE) of the reconstruction using HDMD is 0.282 lower than that of DMD, indicating a significant improvement in reconstruction accuracy. Further illustrating the proposed method, the HDMD method can better capture and analyze the unstable modes in the ECG signal, thereby significantly improving the accuracy and robustness of signal classification. Future work will focus on further optimizing the parameter selection of the HDMD model and exploring its application potential in real-time heart disease monitoring and early warning systems.

Risk-based group decision making regarding renewable energy schemes using a stochastic graphical matrix model

Tomographic absorption spectroscopy (TAS) typically employs multiple projections to reconstruct the spatial distribution of temperature and species concentration, making it a promising method for combustion diagnostics. However, the geometric arrangement of the laser beams significantly affects its accuracy, especially under limited beam conditions. In this work, we propose a beam arrangement optimization method to maximize the utilization efficiency of limited beams, thereby improving reconstruction accuracy in a simpler and more intuitive manner. We designed a cost function that combines the beam number matrix (BNM) and the total weight matrix (TWM). This approach ensures that the maximum number of rays passes through each grid while also maximizing the optical path length in the regions covered by the beam distribution. The optimization is performed using a simulated annealing algorithm to obtain the optimal beam arrangement. For a TAS sensor comprising 12 laser beams, numerical simulations demonstrate that the BNM-TWM optimized beam arrangement achieves lower reconstruction errors across various synthetic temperature fields when compared to traditional orthogonal beam arrangements and those optimized for orthogonality (OD). This improvement is particularly significant when the performance of the orthogonal and OD-optimized arrangements is suboptimal. Finally, we demonstrate the application of the BNM-TWM optimized beam arrangement for reconstructing temperature distributions of asymmetric butane flames, where parallel beam distributions were ineffective. The results effectively pinpoint the location of the flame, showing a temperature difference of less than 3% compared to thermocouple measurements for the center of the flame. These findings indicate that the developed beam arrangement optimization method has the potential to enhance the accuracy of TAS reconstruction under limited beam conditions and could be extended to other tomographic imaging systems.

Application of orthonormal Bernstein polynomials to construct a efficient scheme for solving fractional stochastic integro-differential equation

A computer-assisted characterization of 296 Micrococcaceae isolates obtained from aerobically chill-stored lamb carcasses was carried out using a probability matrix and Bayesian identification theorems, complemented with cluster analysis. Preliminary identification was done with an original probability matrix comprising 37 previously described taxa and 32 tests. Although its statistical quality was adequate, the percentage of identification of field strains to species level was only 70% (96.6% identified with genera). To achieve an improved characterization, cluster analysis was subsequently performed on this group and an additional 26% could be associated with defined species, with five more taxa defined. The combined use of both approaches was judged positive as new identifications and better discrimination could be achieved. The majority of our isolates belonged to the Staphylococcus species group. Many species and groups of staphylococci increased as the spoilage progressed.

In this paper, a new computational method based on the second kind Chebyshev wavelets (SKCWs) together with the Galerkin method is proposed for solving a class of stochastic heat equation. For this purpose, a new stochastic operational matrix for the SKCWs is derived. A collocation method based on block pulse functions is employed to derive a general procedure for forming this matrix. The SKCWs and their operational matrices of integration and stochastic Itô-integration are used to transform the under consideration problem into the corresponding linear system of algebraic equations which can be simply solved to achieve the solution of the problem. The proposed method is very convenient for solving such problems, since the initial and boundary conditions are taken into account automatically. Moreover, the efficiency of the proposed method is shown for some concrete examples. The results reveal that the proposed method is very accurate and efficient.

In this paper, an efficient and accurate computational method based on the Legendre wavelets (LWs) together with the Galerkin method is proposed for solving a class of nonlinear stochastic Ito–Volterra integral equations. For this purpose, a new stochastic operational matrix (SOM) for LWs is derived. A collocation method based on hat functions (HFs) is employed to derive a general procedure for forming this matrix. The LWs and their operational matrices of integration and stochastic Ito-integration and also some useful properties of these basis functions are used to transform such problems into corresponding nonlinear systems of algebraic equations, which can be simply solved to achieve the solution of such problems. Moreover, the efficiency of the proposed method is shown for some concrete examples. The results reveal that the proposed method is very accurate and efficient. Furthermore as some useful applications, the proposed method is applied to obtain approximate solutions for some stochastic problems in the mathematics finance, biology, physics and chemistry.