Truncated Singular Value Decomposition Research Articles

Segmentation-based truncated-SVD for effective feature extraction in hyperspectral image classification

ABSTRACT Remote sensing hyperspectral images (HSIs) are rich sources of information about land cover captured across hundreds of narrow, contiguous spectral wavelength bands. However, using the entire original HSI for practical applications can lead to suboptimal classification accuracy. To address this, band reduction techniques, categorized as feature extraction and feature selection methods, are employed to enhance classification results. One commonly used feature extraction approach for HSIs is Principal Component Analysis (PCA). However, PCA may fall short of capturing the local and specific characteristics present in the HSI data. In this paper, we introduce two novel feature extraction methods: Segmented Truncated Singular Value Decomposition (STSVD) and Spectrally Segmented Truncated Singular Value Decomposition (SSTSVD) to improve classification performance. Segmentation is carried out based on highly correlated bands’ segments and spectral bands’ segments within the HSI data. Our study evaluates and compares these newly proposed methods against classical feature extraction methods, including PCA, Incremental PCA, Sparse-PCA, Kernel PCA, Segmented-PCA (SPCA), and Truncated Singular Value Decomposition (TSVD). We perform this analysis on three distinct HSI datasets, namely the Indian Pines HSI, the Pavia University HSI, and the Kennedy Space Center HSI, using per-pixel Support Vector Machine (SVM) and Random Forest (RF) classification. The experimental results demonstrate the superiority of our proposed methods for all three datasets. The best-performing feature extraction methods when classification is performed using an SVM classifier are STSVD3 (89.03%), SSTSVD2 (95.55%), and STSVD3 (97.74%) for the Indian Pines, Pavia University, and Kennedy Space Center datasets, respectively. Similarly, for the RF classifier, the best-performing feature extraction methods are SSTSVD4 (88.98%), SSTSVD3 (96.04%), and SSTSVD4 (96.09%) for Indian Pines, Pavia University, and Kennedy Space Center datasets, respectively.

The relationship between singular value decomposition orthogonal space vector and sound field reconstruction in equivalent source method

In acoustic holography, the equivalent source method represents a typical ill-posed inverse problem, characterized by the significant impact of measurement errors on the sought solution. To minimize the interference of noise with the sought solution, we analyze the relationship between orthogonal space vectors obtained through singular value decomposition and acoustic field reconstruction. The research findings indicate that the primary information of the acoustic field exhibits strong correlation with a small number of space vectors, while noise information demonstrates random correlation with all space vectors. Therefore, we retain the space vectors with higher correlation to the acoustic field while filtering out redundant ones, thereby stabilizing and accurately reconstructing the acoustic field. Numerical analysis of the acoustic field reconstruction demonstrates that compared to regularization methods such as Tikhonov and truncated singular value decomposition (TSVD), this approach achieves higher computational accuracy. Furthermore, the text combines practical examples to further explain the reasons behind the superior accuracy of this method.

Accurate and efficient daily carbon emission forecasting based on improved ARIMA

Major nations across the globe are increasingly concerned about the rising trends in carbon dioxide (CO2) emissions, particularly in societies of varying scales. Against this backdrop, precise prediction of carbon emissions becomes critically important, especially for the formulation and adjustment of near-term carbon reduction policies. However, the non-linear, non-stationary, and complex nature of daily carbon emission data poses a great challenge for daily-level forecasting especially in the big data context. To address this issue, we propose a novel composite forecasting approach named DCEF dedicated to the estimation of daily carbon emissions. In concrete, our approach employs the Empirical Mode Decomposition (EMD) for data stabilization and the Auto-regressive Integrated Moving Average (ARIMA) model for forecasting, while integrating the Truncated singular value decomposition(TSVD) technique for data compression and mitigating noise. Finally, DCEF is empirically validated with real daily carbon emission datasets collected from 6 sectors in 13 countries of varying sizes. Experimental results demonstrate the advantages of our approach compared to other baseline models in terms of prediction accuracy and efficiency.

Synthesis of higher-B0 CEST Z-spectra from lower-B0 data via deep learning and singular value decomposition.

Chemical exchange saturation transfer (CEST) MRI at 3 T suffers from low specificity due to overlapping CEST effects from multiple metabolites, while higher field strengths (B0) allow for better separation of Z-spectral "peaks," aiding signal interpretation and quantification. However, data acquisition at higher B0 is restricted by equipment access, field inhomogeneity and safety issues. Herein, we aim to synthesize higher-B0 Z-spectra from readily available data acquired with 3 T clinical scanners using a deep learning framework. Trained with simulation data using models based on Bloch-McConnell equations, this framework comprised two deep neural networks (DNNs) and a singular value decomposition (SVD) module. The first DNN identified B0 shifts in Z-spectra and aligned them to correct frequencies. After B0 correction, the lower-B0 Z-spectra were streamlined to the second DNN, casting into the key feature representations of higher-B0 Z-spectra, obtained through SVD truncation. Finally, the complete higher-B0 Z-spectra were recovered from inverse SVD, given the low-rank property of Z-spectra. This study constructed and validated two models, a phosphocreatine (PCr) model and a pseudo-in-vivo one. Each experimental dataset, including PCr phantoms, egg white phantoms, and in vivo rat brains, was sequentially acquired on a 3 T human and a 9.4 T animal scanner. Results demonstrated that the synthetic 9.4 T Z-spectra were almost identical to the experimental ground truth, showing low RMSE (0.11% ± 0.0013% for seven PCr tubes, 1.8% ± 0.2% for three egg white tubes, and 0.79% ± 0.54% for three rat slices) and high R2 (>0.99). The synthesized amide and NOE contrast maps, calculated using the Lorentzian difference, were also well matched with the experiments. Additionally, the synthesis model exhibited robustness to B0 inhomogeneities, noise, and other acquisition imperfections. In conclusion, the proposed framework enables synthesis of higher-B0 Z-spectra from lower-B0 ones, which may facilitate CEST MRI quantification and applications.

Adaptive Truncation Threshold Determination for Multimode Fiber Single-Pixel Imaging

Truncated singular value decomposition (TSVD) is a popular recovery algorithm for multimode fiber single-pixel imaging (MMF-SPI), and it uses truncation thresholds to suppress noise influences. However, due to the sensitivity of MMF relative to stochastic disturbances, the threshold requires frequent re-determination as noise levels dynamically fluctuate. In response, we design an adaptive truncation threshold determination (ATTD) method for TSVD-based MMF-SPI in disturbed environments. Simulations and experiments reveal that ATTD approaches the performance of ideal clairvoyant benchmarks, and it corresponds to the best possible image recovery under certain noise levels and surpasses both traditional truncation threshold determination methods with less computation—fixed threshold and Stein’s unbiased risk estimator (SURE)—specifically under high noise levels. Moreover, target insensitivity is demonstrated via numerical simulations, and the robustness of the self-contained parameters is explored. Finally, we also compare and discuss the performance of TSVD-based MMF-SPI, which uses ATTD, and machine learning-based MMF-SPI, which uses diffusion models, to provide a comprehensive understanding of ATTD.

Regularization methods for the inverse initial value problem for the time-fractional diffusion equation with robin boundary condition ★ ★ The project is supported by Postgraduate Research & Practice Innovation Program of Jiangsu Province (KYCX23_2549) and the Natural Science Foundation of Jiangsu Province, China (Grant No. BK20190578).

In the article, we focus on the inverse initial value problem for the time-fractional diffusion equation with robin boundary condition on a cylindrical symmetric field. The ill-posedness of this problem is proved. We introduce the modified Landweber iteration method(MLIM), the Truncated singular value decomposition(TSVD) method for solving it and propose a new regularization method, named as the TSVD-modified Landweber iteration method(TMLIM). The error estimates between the exact solution and the regularized approximate solution are presented by using two regularization parameter selection rules. Finally, numerical examples are provided to demonstrate the effectiveness and feasibility of the regularization methods.

Research of electrical impedance tomography based on multilayer artificial neural network optimized by Hadamard product for human-chest models

Electrical impedance tomography (EIT) is a non-radiation, non-invasive visual diagnostic technique. In order to improve the imaging resolution and the removing artifacts capability of the reconstruction algorithms for electrical impedance imaging in human-chest models, the HMANN algorithm was proposed using the Hadamard product to optimize multilayer artificial neural networks (MANN). The reconstructed images of the HMANN algorithm were compared with those of the generalized vector sampled pattern matching (GVSPM) algorithm, truncated singular value decomposition (TSVD) algorithm, backpropagation (BP) neural network algorithm, and traditional MANN algorithm. The simulation results showed that the correlation coefficient of the reconstructed images obtained by the HMANN algorithm was increased by 17.30% in the circular cross-section models compared with the MANN algorithm. It was increased by 13.98% in the lung cross-section models. In the lung cross-section models, some of the correlation coefficients obtained by the HMANN algorithm would decrease. Nevertheless, the HMANN algorithm retained the image information of the MANN algorithm in all models, and the HMANN algorithm had fewer artifacts in the reconstructed images. The distinguishability between the objects and the background was better compared with the traditional MANN algorithm. The algorithm could improve the correlation coefficient of the reconstructed images, and effectively remove the artifacts, which provides a new direction to effectively improve the quality of the reconstructed images for EIT.

A New CCERT System With Shielding for Gas-Liquid Two-Phase Flow

This work presents a new capacitively coupled electrical resistance tomography (CCERT) system with shielding for gas-liquid two-phase flow. The new system is developed from the aspects of hardware, image reconstruction algorithm and experiments. The hardware includes a 12-electrode CCERT sensor with shielding, a measurement circuit and an image reconstruction computer. The sensor with shielding is designed and the measurement circuit is developed to obtain the resistance information (real part of the impedance). Based on truncated singular value decomposition (TSVD) and density peaks clustering (DPC), a new image reconstruction algorithm is proposed. TSVD is introduced to obtain the initial image. DPC is improved and used to obtain the final image. Image reconstruction experiments were carried out with the new 12-electrode CCERT system with shielding. Experimental results show that the new CCERT system has good imaging performance. The design of the hardware is successful, and the new proposed image reconstruction algorithm is effective. Compared with the conventional image reconstruction algorithms, the proposed algorithm has higher imaging quality. The maximum relative image errors of the new system are 0.2887 under bubble flow and 0.4177 under stratified flow. Research results also indicate that the shielding design contributes to better imaging stability in interference environment.

A dynamic light scattering inversion method based on regularization matrix reconstruction for flowing aerosol measurement.

Tikhonov regularization, or truncated singular value decomposition (TSVD), is usually used for dynamic light scattering (DLS) inversion of particles in suspension. The Tikhonov regularization method uses a regularization matrix to modify all singular values in the kernel matrix. The modification of large singular values cannot effectively reduce the variance of the estimated values but may introduce bias in the solution, resulting in poor disturbance resistance in the inversion results. The TSVD method, on the other hand, truncates all small singular values, which leads to the loss of particle size information during the inversion process. The shortcomings of the two methods mentioned above do not have a significant impact on the inversion of high signal-to-noise ratio data. However, compared to the classical DLS particle size inversion for non-flowing suspended particles, the DLS inversion of flowing aerosols is more significantly affected by noise, and the extraction of particle size information is more difficult due to the effect of flow velocity, resulting in worse inversion results with increasing aerosol flow velocity for both methods. To improve the accuracy of the particle size distribution (PSD) of flowing aerosols, we introduced a kernel matrix into the regularization matrix, and based on the principles of the two methods, the spectral information of the kernel matrix was utilized to make the modification of small singular values by the regularization matrix. Correspondingly, weak or no modification was made according to the values of large singular values to reduce the introduction of bias. The inversion results of simulated and measured data indicate that the reconstruction of the regularization matrix improves the anti-disturbance performance and avoids the loss of particle size information during the regularization inversion process, thereby significantly improving the PSD accuracy, which is affected by the dual effects of flow velocity and noise in the DLS measurement of flowing particles. The peak error and distribution error of the inversion results by reconstructing the regularization matrix are lower than those of Tikhonov regularization.

Robust state of charge estimation for lithium-ion batteries in variable temperatures using truncated singular value decomposition Cubature Kalman Filter algorithm

Advanced battery management systems play a significant role in the safe and efficient use of battery energy storage systems. While the accurate and reliable estimation of state of charge (SOC) is essential for improving the utilization efficiency of batteries and extending their service life. However, the working environment of battery energy storage systems is highly complex, and temperature changes can often affect the accuracy of modeling and estimation. The traditional methods of SOC estimation often ignore the changes in temperature of the whole system, thus resulting in the cumulative errors in SOC estimation. To address this problem, an improved method is proposed in this paper to estimate the SOC of lithium-ion batteries. It requires the use of Cubature Kalman Filter (CKF) algorithm that is based on Truncated Singular Value Decomposition (TSVD). Additionally, the Forgetting Factor Recursive Least Squares (FFRLS) method is used to determine the parameters of the Thevenin equivalent model for lithium batteries. This algorithm is then validated through simulation analysis and comparison with the experimental data obtained under cyclic operating conditions at a temperature of 5 ℃, 25 ℃, and 40 ℃, respectively. The results demonstrate that the proposed TSVD-CKF algorithm is accurate in reflecting the impact of ambient temperature on the model parameters and is applicable to SOC estimation in a wide range of temperatures, showing a strong robustness.

Application of Regularization Methods in the Sky Map Reconstruction of the Tianlai Cylinder Pathfinder Array

The Tianlai cylinder pathfinder is a radio interferometer array to test 21 cm intensity mapping techniques in the post-reionization era. It works in passive drift scan mode to survey the sky visible in the northern hemisphere. To deal with the large instantaneous field of view and the spherical sky, we decompose the drift scan data into m-modes, which are linearly related to the sky intensity. The sky map is reconstructed by solving the linear interferometer equations. Due to incomplete uv coverage of the interferometer baselines, this inverse problem is usually ill-posed, and regularization method is needed for its solution. In this paper, we use simulation to investigate two frequently used regularization methods, the Truncated Singular Value Decomposition (TSVD), and the Tikhonov regularization techniques. Choosing the regularization parameter is very important for its application. We employ the generalized cross validation method and the L-curve method to determine the optimal value. We compare the resulting maps obtained with the different regularization methods, and for the different parameters derived using the different criteria. While both methods can yield good maps for a range of regularization parameters, in the Tikhonov method the suppression of noisy modes are more gradually applied, produce more smooth maps which avoids some visual artefacts in the maps generated with the TSVD method.

Developing analytical models for the post‐detection SINR of TSVD‐based techniques with imperfect CSI

SummaryIn massive multiple input multiple output (m‐MIMO) uplink, the performance gap between zero forcing (ZF) or direct pseudo‐inverse and minimum mean square error (MMSE) detection schemes increases with an increase in the load factor. To report this issue, truncated singular value decomposition (TSVD) is used in both ZF and direct pseudo‐inverse detection techniques using dynamic threshold. Moreover, the analytical models for the post‐detection signal‐to‐interference‐plus‐noise ratio (PDSINR) of TSVD‐based detection schemes are derived with imperfect channel state information (CSI), and subsequently, bit error rate (BER) is evaluated. Later, an optimum hard threshold for the TSVD‐based detection schemes is deduced from the empirical analysis. Further, the tightness of the analysis is verified through Monte Carlo trails. The simulation result shows that at lower values of average signal‐to‐noise ratio (SNR), TSVD‐based detection schemes with imperfect CSI offer comparable performance and low complexity when compared with the MMSE technique. However, the BER of TSVD‐based detection schemes decreases and gets saturated with an increase in the average SNR, whereas the BER of the MMSE detection scheme increases and approaches the BER of ZF or direct pseudo‐inverse detection schemes.

A method to calculate inverse solutions for steady open channel free-surface flow

The inverse problem of steady two-dimensional open channel free-surface flow is considered, with the focus on determining two types of disturbances: a surface pressure distribution and solid channel bottom topography. A closed-form expression for the inverse surface pressure is derived, and a linear Fredholm equation of the first kind is shown to describe the inverse topography problem, which then needs to be descretised and solved numerically. However, the equation for the channel bottom is prone to instability, so the truncated singular value decomposition (TSVD) method is proposed as a way to stabilise the associated discrete solution. The effectiveness of the TSVD method is demonstrated through several numerical examples, and its performance in the presence of error-contaminated input data is also examined. The results show that the TSVD method can recover the topography accurately from the forward free-surface problem, and provide good approximations even with noisy input data.

Interval strategy-based regularization approach for force reconstruction with multi-source uncertainties

A force reconstruction method for uncertain structures instead of direct measurement is proposed in this paper. Considering the multi-source uncertainties as unknown-but-bounded (UBB) parameters including unknown structural dynamics and measurement errors, this study proposed a novel force reconstruction method using an interval strategy-based regularization approach with truncated singular value decomposition (TSVD). To overcome the drawbacks of incompleted models and limited information, the overall modelling and inversion of the interval force are reconstructed based on the non-probabilistic uncertainty theory. First, The interval uncertainty-based kernel function for force inversion is deduced using the second-order interval perturbation method, in which the uncertainties of the reconstructed force originate from the uncertain singular values and measurement errors. Based on the regularization of the TSVD, the interval bounds of the singular values can be accurately obtained. Combining the two typical regularization approaches, a novel interval strategy-based regularization approach is proposed to optimize the best regularization parameter based on the uncertain interval distribution of singular values. This novel strategy can not only balance the two typical regularization approaches but also reduce the burden of force inversion. Two numerical examples including a truss and space capsule are applied to access the proposed method, and the estimated accurate force can prove the effectiveness.

Active control of electromagnetic fields in layered media

This article presents a numerical strategy for actively manipulating electromagnetic (EM) fields in layered media. In particular, we develop a scheme to characterize an EM source that will generate some predetermined field patterns in prescribed disjoint exterior regions in layered media. The proposed question of specifying such an EM source is not an inverse source problem (ISP) since the existence of a solution is not guaranteed. Moreover, our problem allows for the possibility of prescribing different EM fields in mutually disjoint exterior regions. This question involves a linear inverse problem that requires solving a severely ill-posed optimization problem (i.e. suffering from possible non-existence or non-uniqueness of a solution). The forward operator is defined by expressing the EM fields as a function of the current at the source using the layered media Green's function (LMGF), accounting for the physical parameters of the layered media. This results to integral equations that are then discretized using the method of moments (MoM), yielding an ill-posed system of linear equations. Unlike in ISPs, stability with respect to data is not an issue here since no data is measured. Rather, stability with respect to input current approximation is important. To get such stable solutions, we applied two regularization methods, namely, the truncated singular value decomposition (TSVD) method and the Tikhonov regularization method with the Morozov Discrepancy Principle. We performed several numerical simulations to support the theoretical framework and analyzes, and to demonstrate the accuracy and feasibility of the proposed numerical algorithms.

Analysis of underwater acoustic propagation induced by structural vibration in arctic ocean environment based on hybrid FEM-WSM solver

This paper investigates the underwater acoustic propagation characteristics induced by structural vibrations in the arctic ocean environment. For this purpose, a hybrid solver based on the finite element method (FEM) and the wave superposition method (WSM) is proposed. Leveraging the unique strengths of each method, the FEM is employed to perform structural-acoustic coupling analysis, while the WSM is used to analyze underwater acoustic propagation in the arctic ocean environment. In addition, the truncated singular value decomposition (TSVD) technique is introduced to mitigate the ill-conditioned effects of the Green's function matrix provided by the WSM. Numerical results demonstrate the accuracy and efficiency of the proposed solver in comparison to COMSOL Multiphysics finite element analysis software. Furthermore, the influence of sound speed stratification and ice layer thickness on the underwater acoustic propagation characteristics is numerically investigated in detail. These findings may provide valuable insights for the application of underwater acoustic communication and ocean exploration in the Arctic region.

Color ghost imaging based on optimized random speckles and truncated singular value decomposition

The ambient noise always harms the image quality in ghost imaging. In this letter, we propose a scheme of color ghost imaging to reduce the noise impact. In addition to the method of truncated singular value decomposition (TSVD), the measurement matrix is optimized with low-pass filters. In experiment, four (Gaussian, averaging, circular averaging, and median) filters are applied to the patterns of random speckles, which reforms the measurement matrix. By using the TSVD method, the pseudo-inverse matrix of the optimized measurement matrix is obtained and used to reconstruct the images. With proper truncation rates, the mean square errors of the reconstructed images are greatly decreased. Therefore, the image quality in our scheme is greatly improved. Further simulation analysis shows that the point spread functions are greatly optimized after filtering the random speckles. Our technique strengthens the anti-noise ability of ghost imaging, especially in complex environment.

Evaluation of the Resolution in Inverse Scattering of Dielectric Cylinders for Medical Applications.

The inverse scattering problem has numerous significant applications, including in geophysical explorations, medical imaging, and radar imaging. To achieve better performance of the imaging system, theoretical knowledge of the resolution of the algorithm is required for most of these applications. However, analytical investigations about the resolution presently feel inadequate. In order to estimate the achievable resolution, we address the point spread function (PSF) evaluation of the scattered field for a single frequency and the multi-view case both for the near and the far fields and the scalar case when the angular domain of the incident field and observation ranges is a round angle. Instead of the common free space condition, an inhomogeneous background medium, consisting of a homogeneous dielectric cylinder with a circular cross-section in free space, is assumed. In addition, since the exact evaluation of the PSF can only be accomplished numerically, an analytical approximation of the resolution is also considered. For the sake of its comparison, the truncated singular value decomposition (TSVD) algorithm can be used to implement the exact PSF. We show how the behavior of the singular values and the resolution change by varying the permittivity of the background medium. The usefulness of the theoretical discussion is demonstrated in localizing point-like scatterers within a dielectric cylinder, so mimicking a scenario that may occur in breast cancer imaging. Numerical results are provided to validate the analytical investigations.

Enhancing Kinematic Calibration Accuracy for Parallel Manipulators Based on Truncated Total Least-Square Regularization

Abstract This article proposes a novel kinematic calibration approach to address the ill-conditioned identification matrix problem and enhance the accuracy of parallel manipulators (PMs). The kinematic calibration approach applies the truncated total least-square (T-TLS) regularization method to the inverse kinematic calibration of PMs and does not have the unit inconsistency issue and configuration selection issue. The performance of the T-TLS method in the inverse kinematic calibration of PMs that have the ill-conditioned identification matrix problem is compared to that of classical regularization methods, including the truncated singular value decomposition (TSVD) method and Tikhonov method, in simulations. Then, the kinematic calibration approach is applied to a 3-PSS/S spherical PM prototype in the real world. For the 3-PSS/S spherical PM prototype, the kinematic calibration approach can reduce the absolute angular errors of the end-effector from over 1.0 deg before calibration to less than 0.1 deg after calibration.

System Matrix Reconstruction Algorithm for Thermoacoustic Imaging With Magnetic Nanoparticles Based on Acoustic Reciprocity Theorem.

According to the acoustic reciprocity theorem (ART), we propose a system matrix reconstruction algorithm of thermoacoustic imaging for magnetic nanoparticles (MNPs) by a single-pulse magnetic field. In both cases of inhomogeneous and homogeneous acoustic velocity, we respectively derive the linear equation between the sound pressure detection value and the distribution of MNPs. The image reconstruction problem is converted to an inverse matrix solution by using the truncated singular value decomposition (TSVD) method. In forward problem, the calculated forward results are consistent with the simulated thermoacoustic signal signals. In inverse problem, we build the two-dimensional breast cancer model. The TSVD method based on the ART faithfully reflects the distribution of abnormal tissue labeled by the MNPs. In the experiment, the biological sample injected with the MNPs is used as the imaging target. The reconstructed image well reflects the cross-sectional images of the MNPs area. The TSVD method based on the ART takes into account energy attenuation and inhomogeneous acoustic velocity, and use a non-focused broadband ultrasonic transducer as the receiver to obtain a larger imaging field-of-view (FOV). By comparing the image metrics, we prove that the algorithm is superior to the traditional time reversal method. The TSVD method based on the ART can better suppress noise, which is expected to reduce the cost by reducing the number of detectors. It is of great significance for future clinical applications.