Journal articles: 'Gram Schmidt method'

1

Matsuo, Yoichi, and Takashi Nodera. "Block symplectic Gram–Schmidt method." ANZIAM Journal 56 (February 22, 2016): 416. http://dx.doi.org/10.21914/anziamj.v56i0.9380.

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Tobocman, W. "Iterative inverse scattering method employing Gram‐Schmidt orthogonalization." Journal of the Acoustical Society of America 79, S1 (May 1986): S78. http://dx.doi.org/10.1121/1.2023394.

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Kohaupt, L. "Introduction to a Gram-Schmidt-type biorthogonalization method." Rocky Mountain Journal of Mathematics 44, no. 4 (August 2014): 1265–79. http://dx.doi.org/10.1216/rmj-2014-44-4-1265.

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Tobocman, W. "Iterative inverse scattering method employing Gram-Schmidt orthogonalization." Journal of Computational Physics 64, no. 1 (May 1986): 230–45. http://dx.doi.org/10.1016/0021-9991(86)90027-6.

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Piza, D. M., and S. N. Romanenko. "ADVANCED GRAM-SCHMIDT METHOD FOR RADAR SIGNAL PROCESSING." Radio Electronics, Computer Science, Control, no. 4 (January 5, 2022): 26–33. http://dx.doi.org/10.15588/1607-3274-2021-4-3.

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Context. When protecting radar stations from active noise interference acting along the side lobes of the antenna directional pattern, spatial filtering of signals is used, which is realized by using antennas that are spaced apart in space. In this case, the difference in the directions of reception of the useful signal and the interference makes it possible to form the optimal value of the weighting coefficients of the adaptive spatial filters to suppress the interference. However, if the interfering source moves into the main beam region, then the spatial differences between the wanted signal and the interference are reduced. This leads to significant distortion of the main antenna radiation pattern. As a result, the accuracy of measuring the angular coordinates deteriorates, as well as the sensitivity of the radar receiver. The article proposes a structural-parametric method for adapting a spatial filter, which ensures effective operation of the radar when exposed to the active noise interference both from the direction of the side lobes and from the direction of the main beam. Goal. Improving the efficiency of the radar when the active noise interference source is shifted from the direction to the side lobes to the direction of the main beam. Method. The proposed method makes it possible, due to the structural adaptation of the multichannel spatial filter, to exclude the distortion of the main beam of the radiation pattern of the radar antenna and to ensure its operation under conditions of possible interference from the main beam. Structural adaptation of the spatial filter is realized by the current analysis of the weighting coefficients of the compensation blocks. Results. The structural diagram of the multichannel spatial filter by the Gram-Schmidt method with structural-parametric adaptation, as well as the structural diagram of the compensation block, has been improved. As a result of the simulation, the possibility of eliminating distortions of the radiation pattern of the main antenna of the radar in conditions of the possible impact of the active noise interference along the main beam of the radiation pattern of the radar has been confirmed. Conclusions. The scientific novelty of the work consists in the improvement of the signal-processing algorithm at spatial filtering both when exposed to the active noise interference from the direction of the side lobes, and when the interference source is shifted to the direction of the main beam of radar. The practical novelty of the work lies in the development of a structural diagram and a mathematical model of an improved spatial filter with structural-parametric adaptation.

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Wardani, Irma Budi, and Hartanto Sunardi. "ORTONORMALISASI VEKTOR BASIS DENGAN PROSES GRAM SCHMIDT." Buana Matematika : Jurnal Ilmiah Matematika dan Pendidikan Matematika 5, no. 2: (December 28, 2016): 1–8. http://dx.doi.org/10.36456/buanamatematika.v5i2:.391.

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Gram schmidt process is one of linear algebra roles that associated by basis vector. This thesis aims to determine theoretically step by step in the process of gram schmidt. Gram schmidt process is a method that used to convert an arbitrary basis vector into an orthogonal basis vector. After orthogonal basis vector had been obtained, the orthogonal basis vector was compiled into an orthonormal basis through step by step. A vector on will be expressed as a basis vector if the vector if the vectors in are linear independently and spinning against . And a basis vector can be expressed as a set of orthonormal vectors, then the vector is an orthogonalvector and has norm = 1. If the basis vector has norm 1, to normalize the basis vector by using gram schmidt process. Keywords: vector base, ortonormalisasi, gram schmidt proses.

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Matsuo, Yoichi, and Takashi Nodera. "An efficient implementation of the block Gram--Schmidt method." ANZIAM Journal 54 (August 27, 2013): 476. http://dx.doi.org/10.21914/anziamj.v54i0.6327.

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Shakya, Piyush, Makarand S. Kulkarni, and Ashish K. Darpe. "Bearing diagnosis based on Mahalanobis–Taguchi–Gram–Schmidt method." Journal of Sound and Vibration 337 (February 2015): 342–62. http://dx.doi.org/10.1016/j.jsv.2014.10.034.

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Zeng, Guoqiang, Baihai Zhang, Fenxi Yao, and Senchun Chai. "Modified bidirectional extreme learning machine with Gram–Schmidt orthogonalization method." Neurocomputing 316 (November 2018): 405–14. http://dx.doi.org/10.1016/j.neucom.2018.08.029.

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Vargas, J., J. Antonio Quiroga, C. O. S. Sorzano, J. C. Estrada, and J. M. Carazo. "Two-step demodulation based on the Gram–Schmidt orthonormalization method." Optics Letters 37, no. 3 (February 1, 2012): 443. http://dx.doi.org/10.1364/ol.37.000443.

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Quan, Yinghui, Yingping Tong, Wei Feng, Gabriel Dauphin, Wenjiang Huang, and Mengdao Xing. "A Novel Image Fusion Method of Multi-Spectral and SAR Images for Land Cover Classification." Remote Sensing 12, no. 22 (November 19, 2020): 3801. http://dx.doi.org/10.3390/rs12223801.

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The fusion of multi-spectral and synthetic aperture radar (SAR) images could retain the advantages of each data, hence benefiting accurate land cover classification. However, some current image fusion methods face the challenge of producing unexpected noise. To overcome the aforementioned problem, this paper proposes a novel fusion method based on weighted median filter and Gram–Schmidt transform. In the proposed method, Sentinel-2A images and GF-3 images are respectively subjected to different preprocessing processes. Since weighted median filter does not strongly blur edges while filtering, it is applied to Sentinel-2A images for reducing noise. The processed Sentinel images are then transformed by Gram–Schmidt with GF-3 images. Two popular methods, principal component analysis method and traditional Gram–Schmidt transform, are used as the comparison methods in the experiment. In addition, random forest, a powerful ensemble model, is adopted as the land cover classifier due to its fast training speed and excellent classification performance. The overall accuracy, Kappa coefficient and classification map of the random forest are used as the evaluation criteria of the fusion method. Experiments conducted on five datasets demonstrate the superiority of the proposed method in both objective metrics and visual impressions. The experimental results indicate that the proposed method can improve the overall accuracy by up to 5% compared to using the original Sentinel-2A and has the potential to improve the satellite-based land cover classification accuracy.

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Jamai, Rédouane, and Noureddine Damil. "Influence of iterated Gram–Schmidt orthonormalization in the asymptotic numerical method." Comptes Rendus Mécanique 331, no. 5 (May 2003): 351–56. http://dx.doi.org/10.1016/s1631-0721(03)00072-x.

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Drygas, Hilmar. "On the relationship between the method of least squares and Gram–Schmidt orthogonalization." Acta et Commentationes Universitatis Tartuensis de Mathematica 15, no. 1 (December 11, 2020): 3–13. http://dx.doi.org/10.12697/acutm.2011.15.01.

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A method for solving Least Squares Problems is developed which automatically results in the appearance of the Gram–Schmidt orthogonalizers. Given these orthogonalizers an induction-proof is available for solving Least Squares Problems.

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KAISER, ALEXANDER, WOLFRAM SCHENCK, and RALF MÖLLER. "COUPLED SINGULAR VALUE DECOMPOSITION OF A CROSS-COVARIANCE MATRIX." International Journal of Neural Systems 20, no. 04 (August 2010): 293–318. http://dx.doi.org/10.1142/s0129065710002437.

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We derive coupled on-line learning rules for the singular value decomposition (SVD) of a cross-covariance matrix. In coupled SVD rules, the singular value is estimated alongside the singular vectors, and the effective learning rates for the singular vector rules are influenced by the singular value estimates. In addition, we use a first-order approximation of Gram-Schmidt orthonormalization as decorrelation method for the estimation of multiple singular vectors and singular values. Experiments on synthetic data show that coupled learning rules converge faster than Hebbian learning rules and that the first-order approximation of Gram-Schmidt orthonormalization produces more precise estimates and better orthonormality than the standard deflation method.

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Chen, Guang-Yong, Min Gan, Feng Ding, and C. L. Philip Chen. "Modified Gram–Schmidt Method-Based Variable Projection Algorithm for Separable Nonlinear Models." IEEE Transactions on Neural Networks and Learning Systems 30, no. 8 (August 2019): 2410–18. http://dx.doi.org/10.1109/tnnls.2018.2884909.

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Ajiz, Manzur Ali. "Implementation of gram-schmidt triangularization method using sequential column pointer data structure." Computers & Structures 38, no. 2 (January 1991): 227–37. http://dx.doi.org/10.1016/0045-7949(91)90099-8.

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Liang, C. G., and George M. Lance. "A Differentiable Null Space Method for Constrained Dynamic Analysis." Journal of Mechanisms, Transmissions, and Automation in Design 109, no. 3 (September 1, 1987): 405–11. http://dx.doi.org/10.1115/1.3258810.

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A geometric approach to the solution of the dynamic response of constrained mechanical systems is proposed. A continuous and differentiable basis of the constraint null space is automatically generated using the Gram-Schmidt process. The independent coordinates are obtained by transforming the physical velocity coordinates to the tangent hyperplane of the constraint surface. As a result the independent coordinates lie on the constraint surface and no constraint violation control is necessary.

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Mizukami, Yuichi, Toshiro Nishimori, Junko Okamoto, and Kazuyuki Aihara. "Forecasting daily peak load by a deterministic prediction method employing Gram-Schmidt orthonormalization." Electrical Engineering in Japan 116, no. 1 (1996): 70–79. http://dx.doi.org/10.1002/eej.4391160107.

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Huang, Zehua, Qi Chen, Qihao Chen, and Xiuguo Liu. "Variational Pansharpening for Hyperspectral Imagery Constrained by Spectral Shape and Gram–Schmidt Transformation." Sensors 18, no. 12 (December 7, 2018): 4330. http://dx.doi.org/10.3390/s18124330.

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Image pansharpening can generate a high-resolution hyperspectral (HS) image by combining a high-resolution panchromatic image and a HS image. In this paper, we propose a variational pansharpening method for HS imagery constrained by spectral shape and Gram–Schmidt (GS) transformation. The main novelties of the proposed method are the additional spectral and correlation fidelity terms. First, we design the spectral fidelity term, which utilizes the spectral shape feature of the neighboring pixels with a new weight distribution strategy to reduce spectral distortion caused by the change in spatial resolution. Second, we consider that the correlation fidelity term uses the result of GS adaptive (GSA) to constrain the correlation, thereby preventing the low correlation between the pansharpened image and the reference image. Then, the pansharpening is formulized as the minimization of a new energy function, whose solution is the pansharpened image. In comparative trials, the proposed method outperforms GSA, guided filter principal component analysis, modulation transfer function, smoothing filter-based intensity modulation, the classic and the band-decoupled variational methods. Compared with the classic variation pansharpening, our method decreases the spectral angle from 3.9795 to 3.2789, decreases the root-mean-square error from 309.6987 to 228.6753, and also increases the correlation coefficient from 0.9040 to 0.9367.

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Hidayati, Iswari Nur, Eni Susanti, and Westi Utami. "Analisis Pan-Sharpening untuk Meningkatkan Kualitas Spasial Citra Penginderaan Jauh dalam Klasifikasi Tata Guna Tanah." BHUMI: Jurnal Agraria dan Pertanahan 3, no. 1 (May 29, 2017): 122. http://dx.doi.org/10.31292/jb.v3i1.95.

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Pan-sharpened transformation methods improve the quality of spatial resolution remote sensing imagery. This study used pan-sharpened analysis to improve the quality of image. Pan-sharpening method was used to increase spatial quality of each research object. The aims of reserach were to study image sharpening using Quickbird imagery (multispectral band and pancromatic band) and to calculate overall accuracy of land use classification base on pan-sharpened imagery classification. This study used Brovey transformation and Gram-Schmidt transformation for pan-sharpened process. The classification system used Suharyadi Classification scheme (2001) for urban areas. The results showed that Brovey transformation better than gram-schmidt transformation for the elements of texture, shape, pattern, height, and shading. Gram-Schmidt method was more suitable for the analysis concerned to its original color combination associated with the color or hue of the elements of visual interpretation. The accuracy of the study is 90.70%.Sebagian besar proses citra pan-sharpened yang diperoleh dengan formulasi berbagai algoritma yang sudah ditentukan merupakan representasi antara hubungan karakteristik dari resolusi spektral untuk meningkatkan kualitas secara visual dari citra itu sendiri. Hasil dari beberapa citra pan-sharpened tersebut menjadikan salah satu alternative untuk analisis visual citra penginderaan jauh. Penelitian ini mencoba melakukan analisis pan-sharpened untuk mendapatkan hasil yang lebih maksimal dalam berbagai kenampakan untuk setiap tata guna tanah perkotaan. Pemanfaatan pan-sharpening untuk meningkatkan kualitas spasial dari tiap objek penelitian akan dikaji agar mendapatkan masukan dalam pengembangan metode pan-sharpening untuk klasifikasi tata guna tanah di perkotaan. Penelitian ini menggunakan metode transformasi Brovey dan Gram-Schimdt untuk proses pan-sharpened. Sistem klasifikasi yang digunakan adalah sistem Klasifikasi Suharyadi (2001) untuk daerah perkotaan. Hasil penelitian ini menunjukkan bahwa metode Brovey lebih baik dalam penyajian untuk tekstur, bentuk, pola, tinggi, dan bayangan. Metode Gram-Schimdt lebih cocok untuk analisis yang lebih mementingkan perpaduan warna (komposit) aslinya terkait dengan warna ataupun rona dalam unsur interpretasi visual. Hasil akurasi penelitian penggunaan tanah yang digunakan dalam penelitian ini sebesar 90,70%.

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Mizukami, Yuichi, Toshiro Nishimori, Junko Okamoto, and Kazuyuki Aihara. "Forecasting Daily Peak Load by a Deterministic Prediction Method with the Gram-Schmidt Orthonormalization." IEEJ Transactions on Electronics, Information and Systems 115, no. 6 (1995): 792–97. http://dx.doi.org/10.1541/ieejeiss1987.115.6_792.

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Pu, Yang, Laura A. Sordillo, Yuanlong Yang, and R. R. Alfano. "Key native fluorophores analysis of human breast cancer tissues using Gram–Schmidt subspace method." Optics Letters 39, no. 24 (December 4, 2014): 6787. http://dx.doi.org/10.1364/ol.39.006787.

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Zhang, Hangying, Hong Zhao, Zixin Zhao, Yiying Zhuang, and Chen Fan. "Two-frame fringe pattern phase demodulation using Gram-Schmidt orthonormalization with least squares method." Optics Express 27, no. 8 (April 1, 2019): 10495. http://dx.doi.org/10.1364/oe.27.010495.

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Mikhailenko, V. I., and V. V. Mikhal'chuk. "Spectrum resolution into elementary symmetric bands by means of the gram-schmidt orthogonalization method." Journal of Applied Spectroscopy 45, no. 3 (September 1986): 995–99. http://dx.doi.org/10.1007/bf00663529.

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Huang, Long Yang, and Wei Jun Pan. "Synthesizing Pattern with Broad Nulls through Orthogonal Method." Advanced Materials Research 121-122 (June 2010): 992–95. http://dx.doi.org/10.4028/www.scientific.net/amr.121-122.992.

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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.

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Yasuda, Masaya, Kazuhiro Yokoyama, Takeshi Shimoyama, Jun Kogure, and Takeshi Koshiba. "Analysis of decreasing squared-sum of Gram–Schmidt lengths for short lattice vectors." Journal of Mathematical Cryptology 11, no. 1 (March 1, 2017): 1–24. http://dx.doi.org/10.1515/jmc-2016-0008.

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AbstractIn 2015, Fukase and Kashiwabara proposed an efficient method to find a very short lattice vector. Their method has been applied to solve Darmstadt shortest vector problems of dimensions 134 to 150. Their method is based on Schnorr’s random sampling, but their preprocessing is different from others. It aims to decrease the sum of the squared lengths of the Gram–Schmidt vectors of a lattice basis, before executing random sampling of short lattice vectors. The effect is substantiated from their statistical analysis, and it implies that the smaller the sum becomes, the shorter sampled vectors can be. However, no guarantee is known to strictly decrease the sum. In this paper, we study Fukase–Kashiwabara’s method in both theory and practice, and give a heuristic but practical condition that the sum is strictly decreased. We believe that our condition would enable one to monotonically decrease the sum and to find a very short lattice vector in fewer steps.

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Ghomanjani, Fateme, and Stanford Shateyi. "Alternative methods for solving nonlinear two-point boundary value problems." Open Physics 16, no. 1 (June 26, 2018): 371–74. http://dx.doi.org/10.1515/phys-2018-0050.

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Abstract In this sequel, the numerical solution of nonlinear two-point boundary value problems (NTBVPs) for ordinary differential equations (ODEs) is found by Bezier curve method (BCM) and orthonormal Bernstein polynomials (OBPs). OBPs will be constructed by Gram-Schmidt technique. Stated methods are more easier and applicable for linear and nonlinear problems. Some numerical examples are solved and they are stated the accurate findings.

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Hirani, Jayanti B., Mayank K. Pandya, and Suresh B. Koradiya. "Synthesis of Diverse Fused Tetracyclic Thiazepine-Chalcone Derivatives by Claisen-Schmidt Condensation Reaction and their Antimicrobial Activity." Asian Journal of Chemistry 32, no. 11 (2020): 2860–64. http://dx.doi.org/10.14233/ajchem.2020.22867.

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To develop antimicrobial agent, a series of thiazepine-chalcones was synthesized by Claisen-Schmidt condensation between the couplings of aryl ketone in three steps protocol and different aromatic aldehydes under strong base catalyst at room temperature. The characterization of final products were carried out by IR, 1H & 13C NMR and elemental analysis. The synthesized compounds were also evaluated for their antibacterial and antifungal activities using specific Gram positive and Gram-negative bacterial strains using cup plate method.

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Akbari, P. M., and V. R. Shah. "Synthesis, Characterization and Biological Evaluation of Ethyl-4'-(cyclopropane carboxamido-N-yl)-5-ary-3-oxo-3,4,5,6-tetrahydro-biphenyl-4-carboxylate." Asian Journal of Organic & Medicinal Chemistry 4, no. 4 (2019): 240–43. http://dx.doi.org/10.14233/ajomc.2019.ajomc-p196.

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A series of new substituted cyclohexenone derivatives have been synthesized by the reaction of various substituted chalcones with ethylacetoacetate. Some new N-(4-(3-aryl-acryloyl)phenyl)cyclopropane carboxamide were prepared by Claisen-Schmidt condensation method in presence of sodium hydroxide in ethanol solvent under stirring. The synthesized compounds were characterized by their spectral (IR, NMR, Mass) data and screened for their antimicrobial activities against Gram-positive and Gram-negative bacteria by using standard antimicrobial drugs.

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Labecca, William, Osvaldo Guimarães, and José Roberto C. Piqueira. "Coadjoint Formalism: Nonorthogonal Basis Problems." Mathematical Problems in Engineering 2016 (2016): 1–9. http://dx.doi.org/10.1155/2016/9718962.

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Using nonorthogonal bases in spectral methods demands considerable effort, because applying the Gram-Schmidt process is a fundamental condition for calculations. However, operational matrices numerical methods are being used in an increasing way and extensions for nonorthogonal bases appear, requiring simplified procedures. Here, extending previous work, an efficient tensorial method is presented, in order to simplify the calculations related to the use of nonorthogonal bases in spectral numerical problems. The method is calledcoadjoint formalismand is based onbracketDirac’s formulation of quantum mechanics. Some examples are presented, showing how simple it is to use the method.

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Maurer, T. "HOW TO PAN-SHARPEN IMAGES USING THE GRAM-SCHMIDT PAN-SHARPEN METHOD – A RECIPE." ISPRS - International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences XL-1/W1 (May 2, 2013): 239–44. http://dx.doi.org/10.5194/isprsarchives-xl-1-w1-239-2013.

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Lei, Hebing, Yong Yao, Haopeng Liu, Yiting Tian, Yanfu Yang, and Yinglong Gu. "Accurate phase extraction algorithm based on Gram–Schmidt orthonormalization and least square ellipse fitting method." Journal of Modern Optics 65, no. 10 (January 30, 2018): 1199–209. http://dx.doi.org/10.1080/09500340.2018.1429681.

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Kiani, M. "Extensions to the modified Gram–Schmidt strategy and its application in the steepest ascent method." Journal of Statistical Computation and Simulation 80, no. 4 (April 2010): 389–400. http://dx.doi.org/10.1080/00949650802662813.

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KAH, WONG SOU, KOHBALAN MOORTHY, MOHD SABERI MOHAMAD, SHAHREEN KASIM, SAFAAI DERIS, SIGERU OMATU, and MICHIFUMI YOSHIOKA. "BIOLOGICAL ANALYSIS OF MICROARRAY DATA USING ORTHOGONAL FORWARD SELECTION WITH A CLUSTERING APPROACH." Journal of Biological Systems 23, no. 02 (May 28, 2015): 275–88. http://dx.doi.org/10.1142/s021833901550014x.

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DNA microarray technology allows researchers to monitor the expression level of thousands of genes under various conditions in microarray experiments. However, high-dimensional data in microarray is a major challenge as the irrelevant genes often reduce the detection capability and increase the computation time. Many learning algorithms are not specifically developed to deal with the noisy genes, thus, incorporating them with gene selection techniques has become a necessity. In this paper, we propose a combined method of Gram–Schmidt orthogonal forward selection (OFS) and FunCluster to search for putatively co-regulated biological processes that share the co-expressed genes. There were two datasets used in this research: human white adipose tissue and human skeletal muscle. This study aimed to find a small subset of strongly correlated genes from the raw datasets to maximize the detection capability of cluster analysis. This method was found able to detect the clusters of biological categories that were overlooked in the previous research. Some clusters represented minor functions of the datasets and indicated more specific biological processes. Further, the computation time for both datasets was reduced using this proposed method, as the Gram–Schmidt OFS significantly reduced the dimensionality of the datasets.

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Ren, Jintong, Wunian Yang, Xin Yang, Xiaoyu Deng, He Zhao, Fang Wang, and Lei Wang. "Optimization of Fusion Method for GF-2 Satellite Remote Sensing Images based on the Classification Effect." Earth Sciences Research Journal 23, no. 2 (April 1, 2019): 163–69. http://dx.doi.org/10.15446/esrj.v23n2.80281.

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With the successful launch of China’s GF series satellites, it is more important to study the image data quality, the adaptability of processing method and information extraction method. The panchromatic and multi-spectral data which is based on the GF-2 images data of Chinese sub-meter high-resolution remote sensing satellite is fused by PCA, Pansharp, Gram-Schmidt and NNDiffuse fusion. Then, the quality of the fusion images were evaluated subjectively and objectively. In order to evaluate the applicability of different classification algorithms to the classification, the object-oriented classification algorithm which is based on machine learning algorithm, such as KNN, SVM and Random Trees were used to classify the different GF-2 fusion images. The results showed that: (1) The best visual effect of GF-2 fusion image was the Pansharp fusion image; The quantitative evaluation results showed that the brightness and information retention of Gram-Schmidt fusion image was the best，while the Pansharp fusion image had the highest correlation with the original multi-spectral image; the NNDiffuse fusion image had the highest clarity, and the PCA fusion image quantitative evaluation effect was the worst; (2) According to the applicability analysis of the fusion images based on different classification algorithms with features information extraction, it could be seen that the NNDiffuse fusion method was used for the fusion of GF-2 image data, and the classification of the fusion images was more suitable by using KNN or Random Trees classification algorithm.

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Farzaneh Javan, Sedigheh, Saeid Abbasbandy, and M. Ali Fariborzi Araghi. "Application of Reproducing Kernel Hilbert Space Method for Solving a Class of Nonlinear Integral Equations." Mathematical Problems in Engineering 2017 (2017): 1–10. http://dx.doi.org/10.1155/2017/7498136.

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A new approach based on the Reproducing Kernel Hilbert Space Method is proposed to approximate the solution of the second-kind nonlinear integral equations. In this case, the Gram-Schmidt process is substituted by another process so that a satisfactory result is obtained. In this method, the solution is expressed in the form of a series. Furthermore, the convergence of the proposed technique is proved. In order to illustrate the effectiveness and efficiency of the method, four sample integral equations arising in electromagnetics are solved via the given algorithm.

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Peng, Xiaohong, Rui Zheng, and Jiufu Liu. "Variable Screening Optimization Algorithm for Mahalanobis-Taguchi System." Journal of Physics: Conference Series 2179, no. 1 (January 1, 2022): 012036. http://dx.doi.org/10.1088/1742-6596/2179/1/012036.

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Abstract This paper proposes a Mahalanobis-Taguchi system variable screening optimization method based on binary quantum behavior particle swarm.The main procedures and methods are as follows, Firstly, the Mahalanobis distance value is calculated by the Gram-Schmidt orthogonalization method.We build the multi-objective mixed planning model. The binary quantum behavior particle swarm optimization algorithm is used to solve the optimal combination. A new prediction system based on Mahalanobis-Taguchi metric is established, and the task of accurate discrimination is accomplished.

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Zhang, Yu, Xiaobo Tian, and Rongguang Liang. "Two-step random phase retrieval approach based on Gram-Schmidt orthonormalization and Lissajous ellipse fitting method." Optics Express 27, no. 3 (January 28, 2019): 2575. http://dx.doi.org/10.1364/oe.27.002575.

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Ghodsi, Seyed Roholah, and Mohammad Taeibi-Rahni. "A Novel Parallel Algorithm Based on the Gram-Schmidt Method for Tridiagonal Linear Systems of Equations." Mathematical Problems in Engineering 2010 (2010): 1–17. http://dx.doi.org/10.1155/2010/268093.

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This paper introduces a new parallel algorithm based on the Gram-Schmidt orthogonalization method. This parallel algorithm can find almost exact solutions of tridiagonal linear systems of equations in an efficient way. The system of equations is partitioned proportional to number of processors, and each partition is solved by a processor with a minimum request from the other partitions' data. The considerable reduction in data communication between processors causes interesting speedup. The relationships between partitions approximately disappear if some columns are switched. Hence, the speed of computation increases, and the computational cost decreases. Consequently, obtained results show that the suggested algorithm is considerably scalable. In addition, this method of partitioning can significantly decrease the computational cost on a single processor and make it possible to solve greater systems of equations. To evaluate the performance of the parallel algorithm, speedup and efficiency are presented. The results reveal that the proposed algorithm is practical and efficient.

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Wang, Luyao, Guolin Liu, Qiuxiang Tao, and Ke Wang. "Application of Variable Projection Method based on Gram-Schmidt Orthogonalization in Spatial Cartesian Coordinate Transformation Model." KSCE Journal of Civil Engineering 23, no. 12 (October 24, 2019): 5194–200. http://dx.doi.org/10.1007/s12205-019-1255-z.

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Acharya, Kuldip, Dibyendu Ghoshal, and Bidyut K. Bhattacharyya. "Segmentation of images through curve fitting analysis by modified Vandermonde matrix and modified Gram‐Schmidt method." IET Image Processing 14, no. 17 (December 2020): 4588–98. http://dx.doi.org/10.1049/iet-ipr.2020.0520.

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null, Arisara Chaikittiratana, and Nuttawit Wattanasakulpong. "Dynamic Loadings Induced Vibration of Third Order Shear Deformable FG-CNTRC Beams: Gram-Schmidt-Ritz Method." Advances in Applied Mathematics and Mechanics 14, no. 4 (June 2022): 816–41. http://dx.doi.org/10.4208/aamm.oa-2020-0177.

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Siwi, Sukentyas Estuti. "EVALUATION OF SPOT-5 IMAGE FUSION USING MODIFIED PAN-SHARPENING METHODS." International Journal of Remote Sensing and Earth Sciences (IJReSES) 11, no. 2 (April 12, 2017): 117. http://dx.doi.org/10.30536/j.ijreses.2014.v11.a2609.

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Image fusion, commonly known as pan-sharpening, is a method that combines two data: a panchromatic image that has geometric detail information with the highest spatial resolution and multi-spectral image that has the highest color information but with the lowest resolution. Pan-sharpeningis very important for various remote sensing applications, such as to improve the image classification, to change the detection using temporal data, to increase the geometric, image segmentation, and to improve the visibility of certain object that does not appear on certain data. This study aims to compare the existing pan-sharpening methods such as Brovey, Brovey modification using green and red band, Gram-Schmidt, HPF, Multiplicative, and SFIM.The quality of the pan-sharpening result should be evaluated, this study used Universal Image Quality Index (UIQI/Q index); this evaluation methodgives the opportunity to choose which method is best to provide the most similar spectral information with the original multispectral image. A pan-sharpening qualitative analysis shows that there has been a sharpening process on all pan-sharpening images. Based on spectral visualization (color display), several pan-sharpening methods such as HPF multiplicative method provides brighter colorsand Brovey transformation method displays dark colors. Gram-Schmidt method also provides a different color from the original multispectral image. A pan-sharpening quantitative analysis shows that the best pan-sharpening method with UIQI value> 0.9 is Brovey modification using green and red band. This is due to the green band (500-590 nm) and the red band(610-680 nm) wavelength are in the panchromatic band (480-710 nm) of the SPOT-5 Data.

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Back, Andrew D., and Ah Chung Tsoi. "An Adaptive Lattice Architecture for Dynamic Multilayer Perceptrons." Neural Computation 4, no. 6 (November 1992): 922–31. http://dx.doi.org/10.1162/neco.1992.4.6.922.

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Time-series modeling is a topic of growing interest in neural network research. Various methods have been proposed for extending the nonlinear approximation capabilities to time-series modeling problems. A multilayer perceptron (MLP) with a global-feedforward local-recurrent structure was recently introduced as a new approach to modeling dynamic systems. The network uses adaptive infinite impulse response (IIR) synapses (it is thus termed an IIR MLP), and was shown to have good modeling performance. One problem with linear IIR filters is that the rate of convergence depends on the covariance matrix of the input data. This extends to the IIR MLP: it learns well for white input signals, but converges more slowly with nonwhite inputs. To solve this problem, the adaptive lattice multilayer perceptron (AL MLP), is introduced. The network structure performs Gram-Schmidt orthogonalization on the input data to each synapse. The method is based on the same principles as the Gram-Schmidt neural net proposed by Orfanidis (1990b), but instead of using a network layer for the orthogonalization, each synapse comprises an adaptive lattice filter. A learning algorithm is derived for the network that minimizes a mean square error criterion. Simulations are presented to show that the network architecture significantly improves the learning rate when correlated input signals are present.

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Kong, Yingying, Fang Hong, Henry Leung, and Xiangyang Peng. "A Fusion Method of Optical Image and SAR Image Based on Dense-UGAN and Gram–Schmidt Transformation." Remote Sensing 13, no. 21 (October 24, 2021): 4274. http://dx.doi.org/10.3390/rs13214274.

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To solve the problems such as obvious speckle noise and serious spectral distortion when existing fusion methods are applied to the fusion of optical and SAR images, this paper proposes a fusion method for optical and SAR images based on Dense-UGAN and Gram–Schmidt transformation. Firstly, dense connection with U-shaped network (Dense-UGAN) are used in GAN generator to deepen the network structure and obtain deeper source image information. Secondly, according to the particularity of SAR imaging mechanism, SGLCM loss for preserving SAR texture features and PSNR loss for reducing SAR speckle noise are introduced into the generator loss function. Meanwhile in order to keep more SAR image structure, SSIM loss is introduced to discriminator loss function to make the generated image retain more spatial features. In this way, the generated high-resolution image has both optical contour characteristics and SAR texture characteristics. Finally, the GS transformation of optical and generated image retains the necessary spectral properties. Experimental results show that the proposed method can well preserve the spectral information of optical images and texture information of SAR images, and also reduce the generation of speckle noise at the same time. The metrics are superior to other algorithms that currently perform well.

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Liang, Xing Zhu, Yu E. Lin, and Jing Zhao Li. "Two-Dimensional Orthogonal Unsupervised Discriminant Projection." Advanced Materials Research 542-543 (June 2012): 1343–46. http://dx.doi.org/10.4028/www.scientific.net/amr.542-543.1343.

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Unsupervised Discriminant Projection (UDP) is one of the most promising feature extraction methods. However, UDP suffers from the small sample size problem and the optimal basis vectors obtained by the UDP are nonorthogonal. In this paper, we present a new method called Two-dimensional Orthogonal Unsupervised Discriminant Projection (2DOUDP), which is not necessary to convert the image matrix into high-dimensional image vector and does not suffer the small sample size problem. To further improve the recognition performance, the orthogonal projection matrix obtained based on Gram–Schmidt orthogonalization is given. Experimental results on ORL database indicate that the proposed 2DOUDP method achieves better recognition rate than the UDP method and some other orthogonal feature extraction algorithms.

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Liew, K. M., and K. Y. Lam. "A Set of Orthogonal Plate Functions for Flexural Vibration of Regular Polygonal Plates." Journal of Vibration and Acoustics 113, no. 2 (April 1, 1991): 182–86. http://dx.doi.org/10.1115/1.2930167.

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A computationally efficient and very accurate numerical method is proposed for vibration analysis of regular polygonal plates with any combinations of clamped, simply-supported and free boundary conditions. The method involves the use of two-dimensional orthogonal polynomials generated by the Gram-Schmidt recurrence procedure. For the cases of simply supported and fully clamped hexagonal and octagonal plates, the results obtained agreed very well with those existing in the literature. The frequencies and mode shapes for several hexagonal and octagonal plates subjected to mixed boundary conditions are also presented.

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Kopal, Jiří, Miroslav Rozložník, and Miroslav Tůma. "A note on adaptivity in factorized approximate inverse preconditioning." Analele Universitatii "Ovidius" Constanta - Seria Matematica 28, no. 2 (July 1, 2020): 149–59. http://dx.doi.org/10.2478/auom-2020-0024.

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AbstractThe problem of solving large-scale systems of linear algebraic equations arises in a wide range of applications. In many cases the preconditioned iterative method is a method of choice. This paper deals with the approximate inverse preconditioning AINV/SAINV based on the incomplete generalized Gram–Schmidt process. This type of the approximate inverse preconditioning has been repeatedly used for matrix diagonalization in computation of electronic structures but approximating inverses is of an interest in parallel computations in general. Our approach uses adaptive dropping of the matrix entries with the control based on the computed intermediate quantities. Strategy has been introduced as a way to solve di cult application problems and it is motivated by recent theoretical results on the loss of orthogonality in the generalized Gram– Schmidt process. Nevertheless, there are more aspects of the approach that need to be better understood. The diagonal pivoting based on a rough estimation of condition numbers of leading principal submatrices can sometimes provide inefficient preconditioners. This short study proposes another type of pivoting, namely the pivoting that exploits incremental condition estimation based on monitoring both direct and inverse factors of the approximate factorization. Such pivoting remains rather cheap and it can provide in many cases more reliable preconditioner. Numerical examples from real-world problems, small enough to enable a full analysis, are used to illustrate the potential gains of the new approach.

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Yuan, L., and G. Zhu. "RESEARCH ON REMOTE SENSING IMAGE CLASSIFICATION BASED ON FEATURE LEVEL FUSION." ISPRS - International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences XLII-3 (April 30, 2018): 2185–89. http://dx.doi.org/10.5194/isprs-archives-xlii-3-2185-2018.

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Remote sensing image classification, as an important direction of remote sensing image processing and application, has been widely studied. However, in the process of existing classification algorithms, there still exists the phenomenon of misclassification and missing points, which leads to the final classification accuracy is not high. In this paper, we selected Sentinel-1A and Landsat8 OLI images as data sources, and propose a classification method based on feature level fusion. Compare three kind of feature level fusion algorithms (i.e., Gram-Schmidt spectral sharpening, Principal Component Analysis transform and Brovey transform), and then select the best fused image for the classification experimental. In the classification process, we choose four kinds of image classification algorithms (i.e. Minimum distance, Mahalanobis distance, Support Vector Machine and ISODATA) to do contrast experiment. We use overall classification precision and Kappa coefficient as the classification accuracy evaluation criteria, and the four classification results of fused image are analysed. The experimental results show that the fusion effect of Gram-Schmidt spectral sharpening is better than other methods. In four kinds of classification algorithms, the fused image has the best applicability to Support Vector Machine classification, the overall classification precision is 94.01&thinsp;% and the Kappa coefficients is 0.91. The fused image with Sentinel-1A and Landsat8 OLI is not only have more spatial information and spectral texture characteristics, but also enhances the distinguishing features of the images. The proposed method is beneficial to improve the accuracy and stability of remote sensing image classification.

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Zhang, Yu, Xiaobo Tian, and Rongguang Liang. "Random three-step phase retrieval approach based on difference map Gram–Schmidt orthonormalization and Lissajous ellipse fitting method." Optics and Lasers in Engineering 121 (October 2019): 11–17. http://dx.doi.org/10.1016/j.optlaseng.2019.03.012.

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Journal articles on the topic 'Gram Schmidt method'

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