Abstract
The determination of threshold and the construction of thresholding function would directly affect the signal denoising quality in wavelet transform denoising techniques. However, some deficiencies exist in the conventional methods, such as fixed threshold value and the inflexible thresholding functions. To overcome the defects of the traditional wavelet thresholding techniques, a modified particle swarm optimization (MPSO) algorithm-based parametric wavelet thresholding approach is proposed for signal denoising. Firstly, a kind of parametric wavelet thresholding function construction method is proposed on the basis of conventional thresholding functions. With mathematical derivation, the properties of the constructed function are proved. Three dynamic adjustment strategies are then employed to modify the PSO algorithm. The mean square error (MSE) between the original signal and the reconstructed signal is minimized by the MPSO algorithm. Finally, the performances of the proposed approach and the existing methods are simulated by denoising four benchmark signals with different noise levels. The simulation results show that the proposed MPSO-based parametric wavelet thresholding can obtain lower MSE, higher signal-to-noise ratio, and noise suppression ratio compared to the other algorithms. Besides, the denoising visual results also indicate the superiority of the proposed approach in terms of the signal denoising capability.
Similar content being viewed by others
References
O.T. Altinoz, A.E. Yilmaz, G.W. Weber, Application of chaos embedded PSO for PID parameter tuning. Int. J. Comput. Commun. Control 7(2), 204–217 (2014)
G.G. Bhutada, R.S. Anand, S.C. Saxena, PSO-based learning of sub-band adaptive thresholding function for image denoising. Signal Image Video Process. 6, 1–7 (2012)
R.R. Coifman, D.L. Donoho, Translation-invariant de-noising, in Wavelets and Statistics, ed. by A. Antoniadis, G. Oppenheim, Springer Lecture Notes in Statistics, vol. 103 (Springer, New York, 1995), pp. 125–150
G.Y. Chen, T.D. Bui, Multiwavelets denoising using neighboring coefficients. Signal Process. Lett. 10(7), 211–214 (2003)
S.G. Chang, B. Yu, M. Vetterli, Adaptive wavelet thresholding for image denoising and compression. IEEE Trans. Image Process. 9(9), 1532–1546 (2000)
A. Chatterjee, P. Siarry, Nonlinear inertia weight variation for dynamic adaptation in particle swarm optimization. Comput. Oper. Res. 33, 859–871 (2006)
Y. Dong, S. Xu, A new directional weighted median filter for removal of random-valued impulse noise. IEEE Signal Process. Lett. 14(3), 193–196 (2007)
D.L. Donoho, I.M. Johnstone, Ideal spatial adaptation by wavelet shrinkage. Biometrika 81(3), 425–455 (1994)
D.L. Donoho, I.M. Johnstone, Adapting to unknown smoothness via wavelet shrinkage. J. Am. Stat. Assoc. 90(432), 1200–1224 (1995)
D.L. Donoho, De-noising by soft thresholding. IEEE Trans. Inf. Theory 42(3), 613–627 (1995)
R.C. Eberhart , Y.H. Shi, Particle swarm optimization: developments, applications and resources, in Proceedings of the IEEE Congress on Evolutionary Computation, Seoul, Korea (2001), pp. 81–86
R.C. Gonzalez, R.E. Woods, Digital Image Processing, 2nd edn. (Pearson Prentice-Hall, Singapore, 2002)
V. Giurgiutiu, L. Yu, Comparison of short-time Fourier transform and wavelet transform of transient and tone burst wave propagation signals for structural health monitoring, in Proceedings of the 4th International Workshop on Structural Health Monitoring, Stanford, CA, USA (2003), pp. 1267–1274
H.Y. Gao, Wavelet shrinkage denoising using the non-negative garrote. J. Comput. Graph. Stat. 7(4), 469–488 (1998)
H.Y. Gao, A.G. Bruce, WaveShrink with firm shrinkage. Stat. Sin. 7(4), 855–874 (1997)
S. Gupta, S. Devi, Modified PSO algorithm with high exploration. Int. J. Softw. Eng. Res. Pract. 1(1), 15–19 (2011)
C. He, J.C. Xing, J.L. Li, Q.L. Yang, R.H. Wang, X. Zhang, A new wavelet thresholding function based on hyperbolic tangent function. Math. Probl. Eng. Article ID 528656 (2014)
D.J. Jwo, T.S. Cho, Critical remarks on the linearised and extended Kalman filters with geodetic navigation examples. Measurement 43(9), 1077–1089 (2010)
H. Jia, X. Zhang, J. Bai, A continuous differentiable wavelet threshold function for speech enhancement. J. Cent. South Univ. 20, 2219–2225 (2013)
J. Kennedy, R.C. Eberhart, Particle swarm optimization, in Proceedings of the IEEE International Conference on Neural Networks, Perth, Australia (1995), pp. 1942–1948
H.W. Lee, J.W. Lee, W.G. Jung, G.K. Lee, The periodic moving average filter for removing motion artifacts from PPG signals. Int. J. Control Autom. Syst. 5(6), 701–706 (2007)
I.R. Legarreta, P.S. Addison, N. Grubb, G.R. Clegg, C.E. Robertson, K.A.A. Fox, J.N. Watson, R-wave detection using continuous wavelet modulus maxima, in Proceedings of the IEEE Computers in Cardiology, Thessaloniki Chalkidiki, Greece (2003), pp. 565–568
S. Mallat, W.L. Hwang, Singularity detection and processing with wavelets. IEEE Trans. Inf. Theory 38(2), 617–643 (1992)
S. Mallat, Theory for multi-resolution signal decomposition: the wavelet representation. IEEE Trans. Pattern Anal. Mach. Intell. 11(7), 674–693 (1989)
M. Nasri, H. Nezamabadi-pour, Image denoising in the wavelet domain using a new adaptive thresholding function. Neurocomputing 72(4), 1012–1025 (2009)
G.P. Nason, B.W. Silverman, The stationary wavelet transform and some statistical applications, in Wavelets and Statistics, ed. by A. Antoniadis, G. Oppenheim, Springer Lecture Notes in Statistics, vol. 103 (Springer, New York, 1995), pp. 281–299
J.C. Pesquet, H. Krim, H. Carfantan, Time-invariant orthonormal wavelet representations. IEEE Trans. Signal Process. 44(8), 1964–1970 (1996)
A. Ratnaweera, S.K. Halgamuge, H.C. Watson, Self-organizing hierarchical particle swarm optimizer with time-varying acceleration coefficients. IEEE Trans. Evol. Comput. 8(3), 240–255 (2004)
Y. Soon, S.N. Koh, Speech enhancement using 2-D Fourier transform. IEEE Trans. Speech Audio Process. 11(6), 717–724 (2003)
M. Souden, J. Benesty, S. Affes, On the global output SNR of the parameterized frequency-domain multichannel noise reduction Wiener filter. Signal Process. Lett. 17(5), 425–428 (2010)
A. Sumithra, B. Thanushkodi, Performance evaluation of different thresholding methods in time adaptive wavelet based speech enhancement. IACSIT Int. J. Eng. Technol. 1(5), 42–51 (2009)
F. Sattar, L. Floreby, G. Salomonsson, B. Lövström, Image enhancement based on a nonlinear multiscale method. IEEE Trans. Image Process. 6(6), 888–895 (1997)
Y. Yang, Y. Wei, Neighboring coefficients preservation for signal denoising. Circuits Syst. Signal Process. 31(2), 827–832 (2012)
Y. Yang, Y. Wei, Signal denoising based on the adaptive shrinkage function and neighborhood characteristics. Circuits Syst. Signal Process. 33(12), 3921–3930 (2014)
T.H. Yi, H.N. Li, X.Y. Zhao, Noise smoothing for structural vibration test signals using an improved wavelet thresholding technique. Sensors 12(8), 11205–11220 (2012)
X.P. Zhang, M.D. Desai, Adaptive denoising based on SURE risk. Signal Process. Lett. 5(10), 265–267 (1998)
S. Zhong, S.O. Oyadiji, Crack detection in simply supported beams using stationary wavelet transform of modal data. Struct. Control Health Monit. 18(2), 169–190 (2011)
X. Zhang, J.L. Li, J.C. Xing, P. Wang, Q.L. Yang, R.H. Wang, C. He, Optimal sensor placement for latticed shell structure based on an improved particle swarm optimization algorithm. Math. Probl. Eng. Article ID 743904 (2014)
Acknowledgments
This project was supported by the National Natural Science Foundation of China (Grant No. 61321491) and Natural Science Foundation of Jiangsu Province (Grant No. BK20151451), China. We would like to express our appreciation to the anonymous referees and the associate editor for their valuable comments and suggestions.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that there is no conflict of interests regarding the publication of this paper.
Rights and permissions
About this article
Cite this article
Zhang, X., Li, J., Xing, J. et al. A Particle Swarm Optimization Technique-Based Parametric Wavelet Thresholding Function for Signal Denoising. Circuits Syst Signal Process 36, 247–269 (2017). https://doi.org/10.1007/s00034-016-0303-x
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00034-016-0303-x