Wavelet-based iterative data enhancement for implementation in purification of modal frequency for extremely noisy ambient vibration tests in Shiraz-Iran

Abstract

The main purpose of the present study is to enhance high-level noisy data by a wavelet-based iterative filtering algorithm for identification of natural frequencies during ambient wind vibrational tests on a petrochemical process tower. Most of denoising methods fail to filter such noise properly. Both the signal-to-noise ratio and the peak signal-to-noise ratio are small. Multiresolution-based one-step and variational-based filtering methods fail to denoise properly with thresholds obtained by theoretical or empirical method. Due to the fact that it is impossible to completely denoise such high-level noisy data, the enhancing approach is used to improve the data quality, which is the main novelty from the application point of view here. For this iterative method, a simple computational approach is proposed to estimate the dynamic threshold values. Hence, different thresholds can be obtained for different recorded signals in one ambient test. This is in contrast to commonly used approaches recommending one global threshold estimated mainly by an empirical method. After the enhancements, modal frequencies are directly detected by the cross wavelet transform (XWT), the spectral power density and autocorrelation of wavelet coefficients. Estimated frequencies are then compared with those of an undamaged-model, simulated by the finite element method.

Keywords

ambient vibration test / high level noise / iterative signal enhancement / wavelet / cross and autocorrelation of wavelets

References

[1] Doebling S W, Farrar C R, Prime M B. A summary review of vibration-based damage identification methods. Journal of Shock and Vibration, 1998, 30(2): 91–105

[2] Doebling S W, Farrar C R, Prime M B, Shevitz D W. Damage Identification and Health Monitoring of Structural and Mechanical Systems from Changes in Their Vibration Characteristics: A Literature Review. OSTI.GOV Technical Report LA-13070-MS ON: DE96012168; TRN: 96:003834. 1996

[3] Le T H, Tamura Y. Modal identification of ambient vibration structure using frequency domain decomposition and wavelet transform. In: Proceedings of the 7th Asia-Pacific Conference on Wind Engineering. Taipei, China: APCWE, 2009

[4] Wijesundara K K, Negulescu C, Foerster E, Monfort Climent D. Estimation of modal properties of structures through ambient excitation measurements using continuous wavelet transform. In: Proceedings of 15WCEE. Lisbon: SPES, 2012, 26: 15–18

[5] Abdel-Ghaffar A M, Scanlan R H. Ambient vibration studies of golden gate bridge: I. Suspended structure. Journal of Engineering Mechanics, 1985, 111(4): 463–482

[6] Harik I, Allen D, Street R, Guo M, Graves R, Harison J, Gawry M. Free and ambient vibration of Brent-Spence Bridge. Journal of Structural Engineering, 1997, 123(9): 1262–1268

[7] Farrar C, James G. System identification from ambient vibration measurements on a bridge. Journal of Sound and Vibration, 1997, 205(1): 1–18

[8] Siringoringo D M, Fujino Y. System identification of suspension bridge from ambient vibration response. Engineering Structures, 2008, 30(2): 462–477

[9] Sohn H. A Review of Structural Health Monitoring Literature: 1996–2001. Los Alamos National Laboratory Report. 2004

[10] Kijewski T, Kareem A. Wavelet transforms for system identification in civil engineering. Computer-Aided Civil and Infrastructure Engineering, 2003, 18(5): 339–355

[11] Lin J, Qu L. Feature extraction based on Morlet wavelet and its application for mechanical fault diagnosis. Journal of Sound and Vibration, 2000, 234(1): 135–148

[12] Al-Raheem K F, Roy A, Ramachandran K P, Harrison D K, Grainger S. Rolling element bearing faults diagnosis based on autocorrelation of optimized: Wavelet de-noising technique. International Journal of Advanced Manufacturing Technology, 2009, 40(3–4): 393–402

[13] Yan B, Miyamoto A, Brühwiler E. Wavelet transform-based modal parameter identification considering uncertainty. Journal of Sound and Vibration, 2006, 291(1–2): 285–301

[14] Jiang X, Adeli H. Pseudospectra, MUSIC, and dynamic wavelet neural network for damage detection of highrise buildings. International Journal for Numerical Methods in Engineering, 2007, 71(5): 606–629

[15] Osornio-Rios R A, Amezquita-Sanchez J P, Romero-Troncoso R J, Garcia-Perez A. MUSIC-ANN analysis for locating structural damages in a truss-type structure by means of vibrations. Computer-Aided Civil and Infrastructure Engineering, 2012, 27(9): 687–698

[16] Carassale L, Percivale F. POD-based modal identification of wind-excited structures. In: Proceedings of the 12th International Conference on Wind Engineering. Cairns, 2007, 1239–1246

[17] Tamura Y. Advanced Structural Wind Engineering. Tokyo: Springer, 2013, 347–376

[18] Meo M, Zumpano G, Meng X, Cosser E, Roberts G, Dodson A. Measurements of dynamic properties of a medium span suspension bridge by using the wavelet transforms. Mechanical Systems and Signal Processing, 2006, 20(5): 1112–1133

[19] Lardies J, Gouttebroze S. Identification of modal parameters using the wavelet transform. International Journal of Mechanical Sciences, 2002, 44(11): 2263–2283

[20] He X, Moaveni B, Conte J P, Elgamal A, Masri S F. Modal identification study of Vincent Thomas bridge using simulated wind-induced ambient vibration data. Computer-Aided Civil and Infrastructure Engineering, 2008, 23(5): 373–388

[21] Ni Y C, Lu X L, Lu W S. Field dynamic test and Bayesian modal identification of a special structure—The Palms Together Dagoba. Structural Control and Health Monitoring, 2016, 23(5): 838–856

[22] Zhang F L, Ventura C E, Xiong H B, Lu W S, Pan Y X, Cao J X. Evaluation of the dynamic characteristics of a super tall building using data from ambient vibration and shake table tests by a Bayesian approach. Structural Control and Health Monitoring, 2017, 25(2): 1– 18

[23] Kang N, Kim H, Sunyoung Choi & Seongwoo Jo, Hwang J S, Yu E. Performance evaluation of TMD under typhoon using system identification and inverse wind load estimation. Computer-Aided Civil and Infrastructure Engineering, 2012, 27(6): 455–473

[24] Wenzel H, Pichler D. Ambient Vibration Monitoring. Vienna: John Wiley & Sons, 2005

[25] Brownjohn J M W. Structural health monitoring of civil infrastructure. Philosophical Transactions of the Royal Society A, 2007, 365(1851): 589–622

[26] He X H, Hua X G, Chen Z Q, Huang F L. EMD-based random decrement technique for modal parameter identification of an existing railway bridge. Engineering Structures, 2011, 33(4): 1348–1356

[27] Huang C S, Hung S L, Lin C I, Su W C. A wavelet-based approach to identifying structural modal parameters from seismic response and free vibration data. Computer-Aided Civil and Infrastructure Engineering, 2005, 20(6): 408–423

[28] Ivanovic S, Trifunac M D, Novikova E I, Gladkov A A, Todorovska M I. Instrumented 7-Storey Reinforced Concrete Building in Van Nuys, California: Ambient Vibration Survey Following the Damage from the 1994 Northridge Earthquake. Report No. CE 9903. 1999

[29] Ivanovic S S, Trifunac M D, Todorovska M I. Ambient vibration tests of structures—A review. ISET Journal of Earthquake Technology, 2000, 37: 165–197

[30] Brownjohn J M W, De Stefano A, Xu Y L, Wenzel H, Aktan A E. Vibration-based monitoring of civil infrastructure: Challenges and successes. Journal of Civil Structural Health Monitoring, 2011, 1(3–4): 79–95

[31] Roeck G D. The state-of-the-art of damage detection by vibration monitoring: The SIMCES experience. Structural Control and Health Monitoring, 2003, 10(2): 127–134

[32] He D, Wang X, Friswell M I, Lin J. Identification of modal parameters from noisy transient response signals. Structural Control and Health Monitoring, 2017, 24(11): 1–10

[33] Juang J N, Pappa R S. Effects of noise on modal parameters identified by the eigensystem realization algorithm. Journal of Guidance, Control, and Dynamics, 1986, 9(3): 294–303

[34] Dorvash S, Pakzad S N. Effects of measurement noise on modal parameter identification. Smart Materials and Structures, 2012, 21(6): 065008

[35] Li P, Hu S L J, Li H J. Noise issues of modal identification using eigensystem realization algorithm. Procedia Engineering, 2011, 14: 1681–1689

[36] Yoshitomi S, Takewaki I. Noise-effect compensation method for physical-parameter system identification under stationary random input. Structural Control and Health Monitoring, 2009, 16(3): 350–373

[37] Huang C S, Su W C. Identification of modal parameters of a time invariant linear system by continuous wavelet transformation. Mechanical Systems and Signal Processing, 2007, 21(4): 1642–1664

[38] Yan B, Miyamoto A. A comparative study of modal parameter identification based on wavelet and Hilbert-Huang transforms. Computer-Aided Civil and Infrastructure Engineering, 2006, 21(1): 9–23

[39] Su W C, Huang C S, Chen C H, Liu C Y, Huang H C, Le Q T. Identifying the modal parameters of a structure from ambient vibration data via the stationary wavelet packet. Computer-Aided Civil and Infrastructure Engineering, 2014, 29(10): 738–757

[40] Su W C, Liu C Y, Huang C S. Identification of instantaneous modal parameter of time-varying systems via a wavelet-based approach and its application. Computer-Aided Civil and Infrastructure Engineering, 2014, 29(4): 279–298

[41] Chen S L, Liu J J, Lai H C. Wavelet analysis for identification of damping ratios and natural frequencies. Journal of Sound and Vibration, 2009, 323(1–2): 130–147

[42] Yi T H, Li H N, Zhao X Y. Noise smoothing for structural vibration test signals using an improved wavelet thresholding technique. Sensors (Basel), 2012, 12(8): 11205–11220

[43] Huang N E. Hilbert-Huang Transform and its Applications. Singapore: World Scientific, 2011, 1–26

[44] Teolis A. Computational Signal Processing with Wavelets. Basel: Springer Science & Business Media, 2012

[45] Misiti M, Misiti Y, Oppenheim G, Poggi J M. Wavelets and their Applications. Wiltshire: John Wiley & Sons, 2013

[46] Mallat S. Wavelet Analysis & Its Applications. London: Academic Press, 1999

[47] Van Fleet P. Discrete Wavelet Transformations: An Elementary Approach with Applications. New Jersey: John Wiley & Sons, 2011

[48] Jansen M. Noise Reduction by Wavelet Thresholding. New York: Springer Science & Business Media, 2012

[49] Soman K P. Insight into Wavelets: From Theory to Practice. New Delhi: PHI Learning Pvt. Ltd., 2010

[50] Jiang X, Mahadevan S, Adeli H. Bayesian wavelet packet denoising for structural system identification. Structural Control and Health Monitoring, 2007, 14(2): 333–356

[51] Coifman R R, Wickerhauser M V. Adapted waveform “de-Noising” for medical signals and images. IEEE Engineering in Medicine and Biology Magazine, 1995, 14(5): 578–586

[52] Coifman R R, Wickerhauser M V. Experiments with adapted wavelet de-noising for medical signals and images. In: Time Frequency and Wavelets in Biomedical Signal Processing, IEEE press series in Biomedical Engineering. New York: Wiley-IEEE Press, 1998

[53] Hadjileontiadis L J, Panas S M. Separation of discontinuous adventitious sounds from vesicular sounds using a wavelet-based filter. IEEE Transactions on Biomedical Engineering, 1997, 44(12): 1269–1281

[54] Hadjileontiadis L J, Liatsos C N, Mavrogiannis C C, Rokkas T A, Panas S M. Enhancement of bowel sounds by wavelet-based filtering. IEEE Transactions on Biomedical Engineering, 2000, 47(7): 876–886

[55] Ranta R, Heinrich C, Louis-Dorr V, Wolf D. Interpretation and improvement of an iterative wavelet-based denoising method. IEEE Signal Processing Letters, 2003, 10(8): 239–241

[56] Ranta R, Louis-Dorr V, Heinrich C, Wolf D. Iterative wavelet-based denoising methods and robust outlier detection. IEEE Signal Processing Letters, 2005, 12(8): 557–560

[57] Starck J L, Bijaoui A. Filtering and deconvolution by the wavelet transform. Signal Processing, 1994, 35(3): 195–211

[58] Peyré G. Advanced Signal, Image and Surface Processing-Numerical Tours. Université Paris-Dauphine, 2010

[59] Grinsted A, Moore J C, Jevrejeva S. Application of the cross wavelet transform and wavelet coherence to geophysical time series. Nonlinear Processes in Geophysics, 2004, 11(5/6): 561–566

[60] Rafiee J, Tse P W. Use of autocorrelation of wavelet coefficients for fault diagnosis. Mechanical Systems and Signal Processing, 2009, 23(5): 1554–1572

[61] Jiang X, Adeli H. Wavelet packet-autocorrelation function method for traffic flow pattern analysis. Computer-Aided Civil and Infrastructure Engineering, 2004, 19(5): 324–337

[62] Bruns A. Fourier-, Hilbert-and wavelet-based signal analysis: Are they really different approaches? Journal of Neuroscience Methods, 2004, 137(2): 321–332

[63] Le Van Quyen M, Foucher J, Lachaux J P, Rodriguez E, Lutz A, Martinerie J, Varela F J. Comparison of Hilbert transform and wavelet methods for the analysis of neuronal synchrony. Journal of Neuroscience Methods, 2001, 111(2): 83–98

[64] Rainieri C, Fabbrocino G. Operational Modal Analysis of Civil Engineering Structures. New York: Springer, 2014

[65] Brownjohn J M W. Ambient vibration studies for system identification of tall buildings. Earthquake Engineering & Structural Dynamics, 2003, 32(1): 71–95

[66] Mahato S, Teja M V, Chakraborty A. Adaptive HHT (AHHT) based modal parameter estimation from limited measurements of an RC—framed building under multi—component earthquake excitations. Structural Control and Health Monitoring, 2015, 22(7): 984–1001

[67] Peng Z K, Tse P W, Chu F L. An improved Hilbert–Huang transform and its application in vibration signal analysis. Journal of Sound and Vibration, 2005, 286(1–2): 187–205

[68] Yang W X. Interpretation of mechanical signals using an improved Hilbert-Huang transform. Mechanical Systems and Signal Processing, 2008, 22(5): 1061–1071

[69] Bao C, Hao H, Li Z X, Zhu X. Time-varying system identification using a newly improved HHT algorithm. Computers & Structures, 2009, 87(23–24): 1611–1623

[70] Wu Z, Huang N E. Ensemble empirical mode decomposition: A noise-assisted data analysis method. Advances in Data Science and Adaptive Analysis, 2009, 1(1): 1–41

[71] Daubechies I, Lu J, Wu H T. Synchrosqueezed wavelet transforms: An empirical mode decomposition-like tool. Applied and Computational Harmonic Analysis, 2011, 30(2): 243–261

[72] Brevdo E, Wu H T, Thakur G, Fuckar N S. Synchrosqueezing and its applications in the analysis of signals with time-varying spectrum. Proceedings of the National Academy of Sciences of the United States of America, 2011, 93: 1079–1094

[73] Perez-Ramirez C A, Amezquita-Sanchez J P, Adeli H, Valtierra-Rodriguez M, Camarena-Martinez D, Romero-Troncoso R J. New methodology for modal parameters identification of smart civil structures using ambient vibrations and synchrosqueezed wavelet transform. Engineering Applications of Artificial Intelligence, 2016, 48: 1–12

[74] Li C, Liang M. Time-frequency signal analysis for gearbox fault diagnosis using a generalized synchrosqueezing transform. Mechanical Systems and Signal Processing, 2012, 26: 205–217

[75] Feng Z, Chen X, Liang M. Iterative generalized synchrosqueezing transform for fault diagnosis of wind turbine planetary gearbox under nonstationary conditions. Mechanical Systems and Signal Processing, 2015, 52–53: 360–375

[76] Staszewski W J. Identification of damping in MDOF systems using time-scale decomposition. Journal of Sound and Vibration, 1997, 203(2): 283–305

[77] Areias P, Rabczuk T, Camanho P. Finite strain fracture of 2D problems with injected anisotropic softening elements. Theoretical and Applied Fracture Mechanics, 2014, 72: 50–63

[78] Nanthakumar S, Lahmer T, Zhuang X, Zi G, Rabczuk T. Detection of material interfaces using a regularized level set method in piezoelectric structures. Inverse Problems in Science and Engineering, 2016, 24(1): 153–176

[79] Hamdia K M, Silani M, Zhuang X, He P, Rabczuk T. Stochastic analysis of the fracture toughness of polymeric nanoparticle composites using polynomial chaos expansions. International Journal of Fracture, 2017, 206(2): 215–227

[80] Hamdia K M, Ghasemi H, Zhuang X, Alajlan N, Rabczuk T. Sensitivity and uncertainty analysis for flexoelectric nanostructures. Computer Methods in Applied Mechanics and Engineering, 2018, 337: 95–109

[81] Vu-Bac N, Lahmer T, Zhuang X, Nguyen-Thoi T, Rabczuk T. A software framework for probabilistic sensitivity analysis for computationally expensive models. Advances in Engineering Software, 2016, 100: 19–31

[82] Areias P, Rabczuk T, Dias-da-Costa D. Element-wise fracture algorithm based on rotation of edges. Engineering Fracture Mechanics, 2013, 110: 113–137

[83] Areias P, Rabczuk T. Finite strain fracture of plates and shells with configurational forces and edge rotations. International Journal for Numerical Methods in Engineering, 2013, 94(12): 1099–1122

[84] Areias P, Msekh M, Rabczuk T. Damage and fracture algorithm using the screened Poisson equation and local remeshing. Engineering Fracture Mechanics, 2016, 158: 116–143

[85] Areias P, Rabczuk T. Steiner-point free edge cutting of tetrahedral meshes with applications in fracture. Finite Elements in Analysis and Design, 2017, 132: 27–41

[86] Areias P, Reinoso J, Camanho P P, César de Sá J, Rabczuk T. Effective 2D and 3D crack propagation with local mesh refinement and the screened Poisson equation. Engineering Fracture Mechanics, 2018, 189: 339–360

[87] Anitescu C, Hossain M N, Rabczuk T. Recovery-based error estimation and adaptivity using high-order splines over hierarchical T-meshes. Computer Methods in Applied Mechanics and Engineering, 2018, 328: 638–662

[88] Chau-Dinh T, Zi G, Lee P S, Rabczuk T, Song J H. Phantom-node method for shell models with arbitrary cracks. Computers & Structures, 2012, 92–93: 242–256

[89] Budarapu P R, Gracie R, Bordas S P, Rabczuk T. An adaptive multiscale method for quasi-static crack growth. Computational Mechanics, 2014, 53(6): 1129–1148

[90] Talebi H, Silani M, Bordas S P, Kerfriden P, Rabczuk T. A computational library for multiscale modeling of material failure. Computational Mechanics, 2014, 53(5): 1047–1071

[91] Budarapu P R, Gracie R, Yang S W, Zhuang X, Rabczuk T. Efficient coarse graining in multiscale modeling of fracture. Theoretical and Applied Fracture Mechanics, 2014, 69: 126–143

[92] Talebi H, Silani M, Rabczuk T. Concurrent multiscale modeling of three dimensional crack and dislocation propagation. Advances in Engineering Software, 2015, 80: 82–92

[93] Amiri F, Millán D, Shen Y, Rabczuk T, Arroyo M. Phase-field modeling of fracture in linear thin shells. Theoretical and Applied Fracture Mechanics, 2014, 69: 102–109

[94] Areias P, Rabczuk T, Msekh M. Phase-field analysis of finite-strain plates and shells including element subdivision. Computer Methods in Applied Mechanics and Engineering, 2016, 312: 322–350

[95] Ren H, Zhuang X, Cai Y, Rabczuk T. Dual-horizon peridynamics. International Journal for Numerical Methods in Engineering, 2016, 108(12): 1451–1476

[96] Ren H, Zhuang X, Rabczuk T. Dual-horizon peridynamics: A stable solution to varying horizons. Computer Methods in Applied Mechanics and Engineering, 2017, 318: 762–782

[97] Rabczuk T, Areias P, Belytschko T. A meshfree thin shell method for non-linear dynamic fracture. International Journal for Numerical Methods in Engineering, 2007, 72(5): 524–548

[98] Rabczuk T, Belytschko T. A three-dimensional large deformation meshfree method for arbitrary evolving cracks. Computer Methods in Applied Mechanics and Engineering, 2007, 196(29–30): 2777–2799

[99] Rabczuk T, Gracie R, Song J H, Belytschko T. Immersed particle method for fluid-structure interaction. International Journal for Numerical Methods in Engineering, 2010, 81: 48–71

[100] Rabczuk T, Bordas S, Zi G. On three-dimensional modelling of crack growth using partition of unity methods. Computers & Structures, 2010, 88(23–24): 1391–1411

[101] Rabczuk T, Zi G, Bordas S, Nguyen-Xuan H. A simple and robust three-dimensional cracking-particle method without enrichment. Computer Methods in Applied Mechanics and Engineering, 2010, 199(37–40): 2437–2455

[102] Rabczuk T, Belytschko T. Cracking particles: A simplified meshfree method for arbitrary evolving cracks. International Journal for Numerical Methods in Engineering, 2004, 61(13): 2316–2343

[103] Rabczuk T, Belytschko T, Xiao S. Stable particle methods based on Lagrangian kernels. Computer Methods in Applied Mechanics and Engineering, 2004, 193(12–14): 1035–1063

[104] Amiri F, Anitescu C, Arroyo M, Bordas S P A, Rabczuk T. XLME interpolants, a seamless bridge between XFEM and enriched meshless methods. Computational Mechanics, 2014, 53(1): 45–57

[105] Hughes T J, Cottrell J A, Bazilevs Y. Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement. Computer Methods in Applied Mechanics and Engineering, 2005, 194(39–41): 4135–4195

[106] Nguyen-Thanh N, Nguyen-Xuan H, Bordas S P A, Rabczuk T. Isogeometric analysis using polynomial splines over hierarchical T-meshes for two-dimensional elastic solids. Computer Methods in Applied Mechanics and Engineering, 2011, 200(21–22): 1892–1908

[107] Nguyen V P, Anitescu C, Bordas S P, Rabczuk T. Isogeometric analysis: An overview and computer implementation aspects. Mathematics and Computers in Simulation, 2015, 117: 89–116

[108] Ghasemi H, Park H S, Rabczuk T. A level-set based IGA formulation for topology optimization of flexoelectric materials. Computer Methods in Applied Mechanics and Engineering, 2017, 313: 239–258

[109] Nguyen-Thanh N, Kiendl J, Nguyen-Xuan H, Wüchner R, Bletzinger K U, Bazilevs Y, Rabczuk T. Rotation free isogeometric thin shell analysis using PHT-splines. Computer Methods in Applied Mechanics and Engineering, 2011, 200(47–48): 3410–3424

[110] Nguyen-Thanh N, Valizadeh N, Nguyen M, Nguyen-Xuan H, Zhuang X, Areias P, Zi G, Bazilevs Y, De Lorenzis L, Rabczuk T. An extended isogeometric thin shell analysis based on Kirchhoff-Love theory. Computer Methods in Applied Mechanics and Engineering, 2015, 284: 265–291

[111] Nguyen-Thanh N, Zhou K, Zhuang X, Areias P, Nguyen-Xuan H, Bazilevs Y, Rabczuk T. Isogeometric analysis of large-deformation thin shells using RHT-splines for multiple-patch coupling. Computer Methods in Applied Mechanics and Engineering, 2017, 316: 1157–1178

[112] Vu-Bac N, Duong T, Lahmer T, Zhuang X, Sauer R, Park H, Rabczuk T. A NURBS-based inverse analysis for reconstruction of nonlinear deformations of thin shell structures. Computer Methods in Applied Mechanics and Engineering, 2018, 331: 427–455

[113] Ghasemi H, Park H S, Rabczuk T. A multi-material level set-based topology optimization of flexoelectric composites. Computer Methods in Applied Mechanics and Engineering, 2018, 332: 47–62

[114] Ghorashi S S, Valizadeh N, Mohammadi S, Rabczuk T. T-spline based XIGA for fracture analysis of orthotropic media. Computers & Structures, 2015, 147: 138–146

[115] Kumari S, Vijay R. Effect of symlet filter order on denoising of still images. Advances in Computers, 2012, 3(1): 137–143

[116] Rousseeuw P J, Leroy A M. Robust Regression & Outlier Detection. Hoboken: John Wiley & Sons, 1987

[117] Samadi J. Seismic Behavior of Structure-equipment in a petrochemical complex to evaluate vulnerability assessment: A case study. Thesis for the Master’s Degree. Tehran: Civil Engineering, University of Tehran, 2010

[118] Jensen A, la Cour-Harbo A. Ripples in Mathematics: The Discrete Wavelet Transform. Heidelberg: Springer Science & Business Media. 2001

[119] Soman K. Insight into Wavelets: From Theory to Practice. New Delhi: PHI Learning Pvt. Ltd., 2010

[120] Wickerhauser M V. Adapted Wavelet Analysis: From Theory to Software. Natick: AK Peters/CRC Press, 1996

[121] Donoho D L, Johnstone J M. Ideal spatial adaptation by wavelet shrinkage. Biometrika, 1994, 81(3): 425–455

[122] Stein C M. Estimation of the mean of a multivariate normal distribution. Annals of Statistics, 1981, 9(6): 1135–1151

[123] Yousefi H, Ghorashi S S, Rabczuk T. Directly simulation of second order hyperbolic systems in second order form via the regularization concept. Communications in Computational Physics, 2016, 20(1): 86–135

[124] Yousefi H, Noorzad A, Farjoodi J. Multiresolution based adaptive schemes for second order hyperbolic PDEs in elastodynamic problems. Applied Mathematical Modelling, 2013, 37(12–13): 7095–7127

[125] Selesnick I W, Bayram I. Total Variation Filtering, White paper, Connexions Web site. 2010

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