Gram-Schmidt Orthogonalization Method Research Articles

Semi-analytical modeling and vibration characteristics analysis of the orthogonally stiffened cylindrical shell with variable cross-sections of stiffeners

The stiffened shell with variable cross-sections of stiffeners (SS-VC) has important applications in aerospace and other fields. Its excellent mechanical properties provide new possibilities for the design and performance improvement of stiffened structural parts. However, its dynamic modeling problems have been urgent to be solved. In this study, the dynamic model of the orthogonally stiffened cylindrical shell with variable cross-sections of stiffeners (OSCS-VC) is established by the semi-analytical method (SAM) and it can be described as follows. The displacement allowable functions of the structure are constructed by using the Gram-Schmidt orthogonalization method. Based on Sanders shell theory, the stress-strain relationships of the longitudinal and ring stiffeners with variable cross-sections are derived under the variable limit integration. The boundary spring stiffness is obtained by the inverse identification technique. The dynamic equation of OSCS-VC is established and solved by using the Lagrange equation. Then, a case study is carried out, the rationality of the semi-analytical dynamic model of OSCS-VC is verified by ANSYS engineering software, literature and the experiment system. Finally, based on the semi-analytical model of OSCS-VC, the influence of the characteristic parameters of cross-sectional functions (CSF) for the longitudinal and ring stiffeners on the natural frequencies is analyzed.

Study of GSO-GL based secondary source configuration optimization method for broadband sound field reproduction

Sound field reproduction is a method that uses multiple secondary sound sources to reproduce a predefined desired sound field in a listening area. The configuration of the secondary sound sources is a key factor affecting the reproduction performances. To optimize the secondary source configuration, this study proposes a new method that combines the Gram-Schmidt orthogonalization method with the group Lasso method, which is called GSO-GL method. Firstly, it selects a certain number of the secondary sources from all the alternative secondary sources using the Gram-Schmidt orthogonalization method. The excitation sources are then further selected from the secondary sources selected in the previous step using the group Lasso method. The sound field simulation from the 2D room model show that the GSO-GL method outperforms the Gram-Schmidt orthogonalization method or the group Lasso method alone, combining the high system stability of the Gram-Schmidt orthogonalization method with the high reproduction accuracy of the group Lasso method.

Conterminous United States Landsat-8 top of atmosphere and surface reflectance tasseled cap transformation coefficients

The tasseled cap transformation (TCT) has been widely used to decompose satellite multi-spectral information into “brightness”, “greenness”, and “wetness” components. Published TCT coefficients for the Landsat sensor series have mainly been derived using top of atmosphere (TOA) reflectance and sparse data sets. Studies to derive TCT coefficients for Landsat surface reflectance (SR) are lacking. In this study, the TCT coefficients were derived independently for Landsat-8 Operational Land Imager (OLI) SR and TOA reflectance using the Gram-Schmidt orthogonalization (GSO) method. To ensure that the derived TCT coefficients are robust and broadly applicable, representative samples of soil, vegetation, and water were selected from summer and autumn Landsat-8 OLI Analysis Ready Data (ARD) sampled from 40.4 million 30 m pixel locations across the conterminous United States (CONUS). Given that the blue band is susceptible to atmospheric contamination due to its shorter wavelength, two groups of TCT coefficients were derived: one from 6 bands (Blue, Green, Red, NIR, SWIR1, SWIR2) and one from 5 bands without the blue band. As TCT results cannot be validated in a formal way, the TCT components for CONUS summer TOA and SR composites were generated and compared with National Land Cover Database (NLCD) land cover classes to provide a synoptic assessment and provide confidence in the results. In addition, three ARD tiles selected to encompass a mix of land cover types, predominantly, desert in Nevada, wetland and urban in Florida, and agriculture in North Dakota, were used to analyze the seasonal variation of the TCT components. The results demonstrate that the derived Landsat-8 TCT coefficients can effectively characterize brightness, greenness, and wetness components across the CONUS, and show good consistency for discrimination of land cover types and track seasonal surface variations. There was no significant difference between each TCT component derived using the 6-band and 5-band TCT coefficients considering a large sample of CONUS pixels. Therefore, the 5-band TCT coefficients provided in this study are recommended for use, as the blue band is atmospherically sensitive and difficult to atmospherically correct reliably.

Comparative study of loudspeaker position optimization techniques for multizone sound field reproduction

Mutlizone sound field reproduction aims to generate personal sound zones in a shared space with multiple loudspeakers. Conventionally, loudspeakers are placed to form a regular pattern such as circular, arc or linear array, which are empirical rather than optimal mainly for the convenience of physical placement. Recently, several algorithms have been proposed to select a fixed number of loudspeaker locations from a large set of candidate positions, such as the sparse regularization (i.e. Lasso and Elastic Net) methods, the Constrained Match Pursuit (CMP) method, the Gram-Schmidt Orthogonalization (GSO) method etc. Most of these methods were investigated for single-zone rather than mulit-zone sound field reproduction based on the pressure matching techniques. This paper compares the performance of the state-of-the-art techniques for loudspeaker position optimization in a multizone sound field reproduction system in terms of reproduction error, acoustic contrast and array effort. Simulation results demonstrate that the CMP-LS method shows the best performance in terms of lower MSE and higher AC while the Lasso method needs the lowest AE.

Channel Contributions of EEG in Emotion Modelling Based on Multivariate Adaptive Orthogonal Signal Decomposition

Empirical Mode Decomposition (EMD) provides an adaptive signal processing tool, and its multivariate extension is useful to model multichannel signals. Recently, EMD and multivariate EMD have successfully been applied to solve different signal processing problems. Electroencephalogram signals are often employed to explore the emotional concepts for human-machine interaction. In this paper, an emotion recognition model is presented via EEG signal decomposition by utilizing multivariate EMD. Intrinsic Mode Functions extracted by the multivariate EMD algorithm are quasi-orthogonal. Hence the Gram-Schmidt Orthogonalization method is applied to the extracted IMFs. The number of orthogonal components reveals the number of modes used in the second step of the proposed method, where the Empirical Wavelet Transform is used to explore different features of the IMFs. By applying Ensemble and Decision Tree classifiers on the calculated features, the emotional states are classified as high-low arousal, valence, and dominance with 72.7%, 62.0%, and 64.7% highest classification performances using the selected channels, respectively.

A Comparative Analysis of Index-Based Methods for Impervious Surface Mapping Using Multiseasonal Sentinel-2 Satellite Data

Studies have shown that Sentinel-2 images have advantages over Landsat images in impervious surface area (ISA) extraction. The performance of index-based methods can be affected by different binary methods and subject to seasonal variation. This study marks the first attempt to assess the performance of different spectral indices for ISA extraction using multiseasonal Sentinel-2 images. Specifically, five indices (i.e., the biophysical composition index calculated using the Gram-Schmidt orthogonalization method, biophysical composition index calculated using a principal component-based Procrustes analysis, Normalized Built-up Area Index (NBAI), combinational build-up index, and perpendicular impervious surface index (PISI)) and three impervious surface binary methods (i.e., Otsu's method, manual method, and Iterative Self-Organizing Data Analysis Technique Algorithm (ISODATA) classification method) were tested on multi-seasonal Sentinel-2 images in the main urban area of Wuhan, China. Results showed that PISI combined with the ISODATA classification method achieved the highest accuracy with 92.64% OA, and 0.8410 Kappa coefficient, and NBAI combined with Otsu's method achieved the lowest accuracy with 35.37% OA, and 0.013 Kappa coefficient. Regarding the seasonal sensitivity, PISI is relatively more stable than the other methods. The superior performance of PISI is largely due to its capability in separating ISA from soil and vegetation. In addition, summer is the best season to map ISA from Sentinel-2 images when the impervious surface is generally less confused with bare soil. This study serves as a reference for the selection of spectral indices for ISA extraction from Sentinel-2 images in relation to binary methods and seasonal effects.

Feature Selection Optimization for Mahalanobis-Taguchi System Using Chaos Quantum-Behavior Particle Swarm

The computational speed in the feature selection of Mahalanobis-Taguchi system (MTS) using standard binary particle swarm optimization (BPSO) is slow and it is easy to fall into the locally optimal solution. This paper proposes an MTS variable optimization method based on chaos quantum-behavior particle swarm. In order to avoid the influence of complex collinearity on the distance measurement results, the Gram-Schmidt orthogonalization method is first used to calculate the Mahalanobis distance (MD) value. Then, the optimal threshold point of the system classification is determined through the receiver operating characteristic (ROC) curve; the misclassification rate and the selected variables are defined; the multi-objective mixed programming model is built. The chaos quantum-behavior particle swarm optimization (CQPSO) algorithm is proposed to solve the optimization combination, and the algorithm performs binary coding on the particle based on probability. Using the optimized combination of variables, a new Mahalanobis-Taguchi metric based prediction system is established to complete the task of precise discrimination. Finally, a fault diagnosis for the steel plate is taken as an example. The experimental results show that the proposed method can effectively enhance the iterative speed and optimization precision of the particles, and the prediction accuracy of the optimized MTS is significantly improved.

Global and Adaptive Thresholding Technique for White Blood Cell Image Segmentation

Global and local thresholding are two thresholding approaches for white blood cell (WBC) image segmentation. Global thresholding determines the threshold value based on the histogram of the overall pixel intensity distribution of the image. In contrast, adaptive thresholding computes the threshold value for each fractional region of the image, so that each fractional region has a different threshold value. In this work, we are assessing both of these approaches for two threshold values. We extended the Otsu’s equation to calculate more than one threshold as it originally designed to find only a single threshold value. Adaptive thresholding first divides an image into fractional-image by considering an imaginary bounding box that surrounds the location of WBC, which involves the Gram-Schmidt orthogonalization method. For segmentation performance evaluation, we compare 35 blood smear test images which segmented by our proposed method, with their corresponding ground truth image to representing them in Zijdenbos Similarity Index (ZSI), precision, and recall measurement. Experimental results show that adaptive thresholding achieves average ZSI, precision and recall, 92.5%, 91.79%, 94.03%, while global thresholding achieves 30.72%, 23.38%, and 99.39% respectively.

Application of Variable Projection Method based on Gram-Schmidt Orthogonalization in Spatial Cartesian Coordinate Transformation Model

For the linear and nonlinear parameters that can be separated in the spatial Cartesian coordinate transformation model, we use the variable projection algorithm in this paper to represent the linear parameters with nonlinear parameters, which are transformed into least squares problems with only nonlinear parameters. We simplify the matrix of the nonlinear function by the Gram-Schmidt orthogonalization method, and combine the nonlinear least squares iterative method with the Levenberg-Marquardt (LM) algorithm to solve for the coordinate transformation parameters. Experiments are carried out by solving for the coordinate transformation parameters of the independent spatial Cartesian coordinate system and the CGCS2000 coordinate system. We compare the solution results of the four methods (parameter non-separation method, traditional variable projection method, variable projection method based on QR decomposition, and variable projection method based on Gram-Schmidt orthogonal decomposition) with respect to the calculated results, the number of iterations and the computation time. The experimental results show that the proposed method in this paper requires a lower computation time and achieves higher computational efficiency when obtaining the same solution results and with the same number of iterations.

Derivation of Tasseled Cap Transformation Coefficients for Sentinel-2 MSI At-Sensor Reflectance Data

Tasseled cap transformation (TCT) is a commonly used remote-sensing technique and has been successfully used in various remote sensing-related applications. However, the TCT coefficient set is sensor-specific, and therefore, in this article, we developed the TCT coefficients specifically for Sentinel-2 multispectral instrument at-sensor reflectance data. A total of ten synchronous image pairs of Sentinel-2 and Landsat-8, collected from the different parts of the world, were used for this approach. Instead of using the traditional Gram-Schmidt orthogonalization (GSO) method, we derived the coefficients using a principal component-based Procrustes analysis (PCP) method. This was done by rotating principal component axes of Sentinel-2 data to align to Landsat-8 Operational Land Imager tasseled cap axes via the Procrustes analysis. The results show that the TCT coefficients derived from our PCP method can effectively enhance brightness, greenness, and wetness characteristics of the Sentinel-2 imagery. The results have also been compared with those of a previous study that derived the Sentinel-2 TCT data using the GSO method instead. Landsat-8 and moderate resolution imaging spectroradiometer (MODIS)-derived TCT data have been used as references for comparison. The comparison shows that the PCP method outperforms the GSO method, as the PCP's results generally have lower root mean square errors and higher correlation coefficients (R) when compared with the corresponding TC components of Landsat-8 and MODIS data, respectively. The greatest difference between the PCP and GSO methods lies in the wetness component. The PCP method can correctly highlight vegetation and soil moisture, as well as water features in the component.

Fault Diagnosis of Tennessee-Eastman Process Using Orthogonal Incremental Extreme Learning Machine Based on Driving Amount.

Fault diagnosis is important to the industrial process. This paper proposes an orthogonal incremental extreme learning machine based on driving amount (DAOI-ELM) for recognizing the faults of the Tennessee-Eastman process (TEP). The basic idea of DAOI-ELM is to incorporate the Gram-Schmidt orthogonalization method and driving amount into an incremental extreme learning machine (I-ELM). The case study for the 2-D nonlinear function and regression problems from the UCI dataset results show that DAOI-ELM can obtain better generalization ability and a more compact structure of ELM than I-ELM, convex I-ELM (CI-ELM), orthogonal I-ELM (OI-ELM), and bidirectional ELM. The experimental training and testing data are derived from the simulations of TEP. The performance of DAOI-ELM is evaluated and compared with that of the back propagation neural network, support vector machine, I-ELM, CI-ELM, and OI-ELM. The simulation results show that DAOI-ELM diagnoses the TEP faults better than other methods.

Improving the Efficiency of Beyond-RPA Methods within the Dielectric Matrix Formulation: Algorithms and Applications to the A24 and S22 Test Sets.

Within a formalism based on dielectric matrices, the electron-hole time-dependent Hartree-Fock (eh-TDHF) and the adiabatic connection second-order screened exchange (AC-SOSEX) are promising approximations to improve ground-state correlation energies by including exchange effects beyond the random phase approximation (RPA). We introduce here an algorithm based on a Gram-Schmidt orthogonalization (GSO) procedure that significantly reduce the number of matrix elements to be computed to evaluate the response functions that enter in the formulation of these two methods. By considering the A24 test set, we show that this approach does not lead to a significant loss of accuracy and can be effectively applied to compute the small interaction energies involved in weakly bound dimers. Importantly, the GSO method significantly extends the applicability of the eh-TDHF and AC-SOSEX to large systems. This is shown by considering the S22 test set, which includes dimers with up to one hundred valence electrons requiring hundreds of thousands of plane-waves in the basis set. By comparing our results to coupled-cluster benchmark values, we show that the inclusion of exchange effects beyond the RPA significantly improves the accuracy, with mean absolute errors that decrease by almost 40% for the A24 test set and by almost 50% for the S22 test set. This approach based on dielectric matrices is particularly suited for plane-wave implementations and might be used in the future to improve the description of the correlation energy in solid state applications.

Modal wavefront reconstruction in radial shearing interferometry with general aperture shapes.

Wavefront reconstruction in radial shearing interferometry with general aperture shapes is challenging because the problem may be ill-conditioned. Here we propose a Gram-Schmidt orthogonalization method to cope with off-axis wavefront reconstruction with any aperture type. The proposed method constructs a set of orthogonal basis functions and computes the corresponding expansion coefficients, which are converted into another set of expansion coefficients to reproduce the original wavefront. The method can effectively alleviate the ill-conditioning of the problem, and is numerically stable compared with the classic least-squares method, especially for non-circular apertures and in the presence of noise. Computer simulation and experimental results are presented to demonstrate the performance of the algorithm.

Deep Network Based on Stacked Orthogonal Convex Incremental ELM Autoencoders

Extreme learning machine (ELM) as an emerging technology has recently attracted many researchers’ interest due to its fast learning speed and state-of-the-art generalization ability in the implementation. Meanwhile, the incremental extreme learning machine (I-ELM) based on incremental learning algorithm was proposed which outperforms many popular learning algorithms. However, the incremental algorithms with ELM do not recalculate the output weights of all the existing nodes when a new node is added and cannot obtain the least-squares solution of output weight vectors. In this paper, we propose orthogonal convex incremental learning machine (OCI-ELM) with Gram-Schmidt orthogonalization method and Barron’s convex optimization learning method to solve the nonconvex optimization problem and least-squares solution problem, and then we give the rigorous proofs in theory. Moreover, in this paper, we propose a deep architecture based on stacked OCI-ELM autoencoders according to stacked generalization philosophy for solving large and complex data problems. The experimental results verified with both UCI datasets and large datasets demonstrate that the deep network based on stacked OCI-ELM autoencoders (DOC-IELM-AEs) outperforms the other methods mentioned in the paper with better performance on regression and classification problems.

Comparison of Weibull parameters computation methods and analytical estimation of wind turbine capacity factor using polynomial power curve model: case study of a wind farm

Wind speed probability at a site has to be modeled for evaluating the energy generation potential of a wind farm. Analytical computation of wind turbine capacity factor at the planning stage of a wind farm is very crucial. Thus, the comparison of Weibull parameters estimation methods and computation of wind turbine capacity factor are the focus of this paper. Soda wind farm used in this case study is located in the Jaisalmer district of western Rajasthan in India. Modeling of wind speed probability and power curve of wind turbines installed at Soda site were done for analytically estimating the capacity factor of wind turbine. Estimated capacity factors were then compared with the measured values of wind farm for validation of results. Four numerical methods namely graphical, empirical, modified maximum likelihood, and energy pattern factor were used for month-wise Weibull parameters estimation at hub height of 65 m. Power curve of the wind turbine installed at site was modeled using eighth-degree polynomial. Coefficients of polynomial were calculated from the combined use of linear least square method and QR decomposition using Gram-Schmidt orthogonalization method. Results show that the percentage error in annual capacity factor estimation using Weibull parameters estimated from graphical, empirical, modified maximum likelihood, and energy pattern factor methods were +9.98%, −1.59%, −1.22%, and −1.29%, respectively. Annual capacity factor that was estimated using the Weibull parameters calculated from modified maximum likelihood method matched best with the measured values. Graphical method gave the most erroneous results.

Optimization and Construction of Single-side Nuclear Magnetic Resonance Magnet

Single-sided NMR devices can operate under conditions inaccessible to conventional NMR while featuring portability and the ability to analyze arbitrary-sized objects. In this paper, a semi-elliptic Halbach magnet array was designed and built for single-side Nuclear Magnetic Resonance (NMR). We present an easy-to-implement target field algorithm for single-side NMR magnet design based on Gram-Schmidt Orthogonal method. The creating magnetic field of designed magnet structure could achieve best flatness in the region of interesting for NMR applications. The optimizing result shows that the best magnet structure can generate magnetic fields which flatly distributed in the horizontal direction and the gradient was distributed in the vertical direction with gradient of 2mT/mm. The field strength and gradient were measured by a three dimensions Hall probe and agreed well with the simulations. DOI: http://dx.doi.org/10.11591/telkomnika.v11i10.3460

GPU Implementation of an Automatic Target Detection and Classification Algorithm for Hyperspectral Image Analysis

The detection of (moving or static) targets in remotely sensed hyperspectral images often requires real-time responses for swift decisions that depend upon high computing performance of algorithm analysis. The automatic target detection and classification algorithm (ATDCA) has been widely used for this purpose. In this letter, we develop several optimizations for accelerating the computational performance of ATDCA. The first one focuses on the use of the Gram-Schmidt orthogonalization method instead of the orthogonal projection process adopted by the classic algorithm. The second one is focused on the development of a new implementation of the algorithm on commodity graphics processing units (GPUs). The proposed GPU implementation properly exploits the GPU architecture at low level, including shared memory, and provides coalesced accesses to memory that lead to very significant speedup factors, thus taking full advantage of the computational power of GPUs. The GPU implementation is specifically tailored to hyperspectral imagery and the special characteristics of this kind of data, achieving real-time performance of ATDCA for the first time in the literature. The proposed optimizations are evaluated not only in terms of target detection accuracy but also in terms of computational performance using two different GPU architectures by NVIDIA: Tesla C1060 and GeForce GTX 580, taking advantage of the performance of operations in single-precision floating point. Experiments are conducted using hyperspectral data sets collected by three different hyperspectral imaging instruments. These results reveal considerable acceleration factors while retaining the same target detection accuracy for the algorithm.

A New Spectral Representation of Earthquake Recordings Using an Improved Hilbert-Huang Transform

A new spectral representation method of earthquake recordings using an improved Hilbert-Huang transform (HHT) is proposed in the paper. Firstly, the problem that the intrinsic mode functions (IMFs) decomposed by the empirical mode decomposition (EMD) in HHT is not exactly orthogonal is pointed out and improved through the Gram-Schmidt orthogonalization method which is referred as the orthogonal empirical mode decomposition (OEMD). Combined the OEMD and the Hilbert transform (HT) which is referred as the improved Hilbert-Huang transform (IHHT), the orthogonal intrinsic mode functions (OIMFs) and the orthogonal Hilbert spectrum (OHS) and the orthogonal Hilbert marginal spectrum (OHMS) are obtained. Then, the IHHT has been applied for the analysis of the El Centro earthquake recording. The obtained spectral representation result shows that the OHS gives more detailed and accurate information in a time–frequency–energy presentation than the Hilbert spectrum (HS) and the OHMS gives more faithful low-frequency energy presentation than the Fourier spectrum (FS) and the Hilbert marginal spectrum (HMS).

Dynamic analysis of a micro-resonator driven by electrostatic combs

The dynamic response of a micro-resonator driven by electrostatic combs is investigated in this work. The micro-resonator is assumed to consist of eight flexible beams and three rigid bodies. The nonlinear partial differential equations that govern the motions of the flexible beams are obtained, as well as their boundary and matching conditions. The natural matching conditions for the flexible beams are the governing equations for the rigid bodies. The undamped natural frequencies and mode shapes of the linearized model of the micro-resonator are determined, and the orthogonality relation of the undamped global mode shapes is established. The modified Newton iterative method is used to simultaneously solve for the frequency equation and identify repeated natural frequencies that can occur in the micro-resonator and their multiplicities. The Gram-Schmidt orthogonalization method is extended to orthogonalize the mode shapes of the continuous system corresponding to the repeated natural frequencies. The undamped global mode shapes are used to spatially discretize the nonlinear partial differential equations of the micro-resonator. The simulation results show that the geometric nonlinearities of the flexible beams can have a significant effect on the dynamic response of the micro-resonator.

Synthesizing Pattern with Broad Nulls through Orthogonal Method

Based on the Gram-Schmidt orthogonalization method utilized for pattern synthesis of linear array without constraints, null derivative constraints are proposed to implement pattern synthesis with broad nulls. The advantage of the simplified computation resulted from steering vector orthogonalization in unconstrained orthogonal approach is maintained, and the uniform or non-uniform linear array pattern synthesis can be realized by this method. The experimental results show that the null constrained orthogonal approach can fulfill linear array pattern synthesis with constrains such as null constrains, first-order and second-order derivative null constraints.