A Review of Wavelet Networks, Wavenets, Fuzzy Wavenets and their Applications

The combination of wavelet theory and neural networks has lead to the development of wavelet networks. Wavelet networks are feed-forward neural networks using wavelets as activation function. They have been used in classification and identification problems with some success. Their strength lies on catching essential features in “frequency-rich” signals. In wavelet networks, both the position and the dilation of the wavelets are optimized besides the weights. Wavenet is another term to describe wavelet networks. Originally, wavenets did refer to neural networks using dyadic wavelets. In wavenets, the position and dilation of the wavelets are fixed and the weights are optimized by the network. We propose to adopt this terminology. The theory of wavenets has been generalized by the author to biorthogonal wavelets. This extension to biorthogonal wavelets has lead to the development of fuzzy wavenets. A serious difficulty with most neurofuzzy methods is that they do often furnish rules without a transparent interpretation. A solution to this problem is furnished by multiresolution techniques. The most appropriate membership functions are chosen from a dictionary of membership functions forming a multiresolution. The dictionary contains a number of membership functions that have the property to be symmetric, everywhere positive and with a single maxima. This family includes among others splines and some radial functions. The main advantage of using a dictionary of membership functions is that each term, such as “small”, “large” is well defined beforehand and is not modified during learning. The multiresolution properties of the membership functions in the dictionary function permit to fuse or split membership functions quite easily so as to express the rules under a linguistically understandable and intuitive form for the human expert. Different techniques, generally referred by the term “fuzzy-wavelet”, have been developed for data on a regular grid. Fuzzy wavenets extend these techniques to online learning. A major advantage of fuzzy wavenets techniques in comparison to most neurofuzzy methods is that the rules are validated online during learning by using a simple algorithm based on the fast wavelet decomposition algorithm. Significant applications of wavelet networks and fuzzy wavenets are discussed to illustrate the potential of thcse methods.

[1]  Bernard Dubuisson,et al.  Engine Knock Detection from Vibration Signals Using Pattern Recognition , 1997 .

[2]  Harold H. Szu,et al.  Neural network adaptive wavelets for signal representation and classification , 1992 .

[3]  Charles A. Micchelli,et al.  Using the Refinement Equations for the Construction of Pre-Wavelets II: Powers of Two , 1991, Curves and Surfaces.

[4]  A. Prochazka,et al.  Time series prediction using genetically trained wavelet networks , 1994, Proceedings of IEEE Workshop on Neural Networks for Signal Processing.

[5]  S. S. Rao,et al.  Recurrent wavelet networks , 1994, Proceedings of 1994 IEEE International Conference on Neural Networks (ICNN'94).

[6]  Sagar V. Kamarthi,et al.  Wavelet networks for sensor signal classification in flank wear assessment , 1998, J. Intell. Manuf..

[7]  R. L. Peskin,et al.  Wavelet neural networks and receptive field partitioning , 1993, IEEE International Conference on Neural Networks.

[8]  Yagyensh C. Pati,et al.  Analysis and synthesis of feedforward neural networks using discrete affine wavelet transformations , 1993, IEEE Trans. Neural Networks.

[9]  Hartmut Dickhaus,et al.  Identification of high risk patients in cardiology by wavelet networks , 1996, Proceedings of 18th Annual International Conference of the IEEE Engineering in Medicine and Biology Society.

[10]  Kurt Hornik,et al.  Multilayer feedforward networks are universal approximators , 1989, Neural Networks.

[11]  P. Chang,et al.  Nonlinear communication channel equalization using wavelet neural networks , 1994, Proceedings of 1994 IEEE International Conference on Neural Networks (ICNN'94).

[12]  Y. C. Pati,et al.  Discrete Affine Wavelet Transforms For Anaylsis And Synthesis Of Feedfoward Neural Networks , 1990, NIPS 1990.

[13]  Wenbo Wang,et al.  A new wavelet network architecture for echo cancellation , 1996, Proceedings of Digital Processing Applications (TENCON '96).

[14]  Bhavik R. Bakshi,et al.  Wave-Nets: novel learning techniques, and the induction of physically interpretable models , 1994, Defense, Security, and Sensing.

[15]  Michio Sugeno,et al.  Fuzzy identification of systems and its applications to modeling and control , 1985, IEEE Transactions on Systems, Man, and Cybernetics.

[16]  Marc Thuillard,et al.  FUZZY LOGIC IN THE WAVELET FRAMEWORK , 2000 .

[17]  M. Vukobratovic,et al.  Wavelet neural network approach for control of non-contact and contact robotic tasks , 1997, Proceedings of 12th IEEE International Symposium on Intelligent Control.

[18]  Liangyue Cao,et al.  Predicting chaotic time series with wavelet networks , 1995 .

[19]  Marc Thuillard.  Wavelets in Soft Computing , 2001, World Scientific Series in Robotics and Intelligent Systems.

[20]  Qinghua Zhang,et al.  Multidimensional wavelet frames , 1995, IEEE Trans. Neural Networks.

[21]  B. E. Miller,et al.  Using a wavelet network to characterize real environments for haptic display , 1998 .

[22]  Shaohua Tan,et al.  Near-optimal construction of wavelet networks for nonlinear system modeling , 1996, 1996 IEEE International Symposium on Circuits and Systems. Circuits and Systems Connecting the World. ISCAS 96.

[23]  Liangyue Cao,et al.  Predicting economic time series using a nonlinear deterministic technique , 1996 .

[24]  L. F. Yeung,et al.  Multi-input system identification and its applications using wavelet constructive method , 1996, Proceedings of 35th IEEE Conference on Decision and Control.

[25]  Brian A. Telfer,et al.  Wavelet transforms and neural networks for compression and recognition , 1996, Neural Networks.

[26]  S. Mallat.  A wavelet tour of signal processing , 1998 .

[27]  Jelena Kovacevic,et al.  Wavelet families of increasing order in arbitrary dimensions , 2000, IEEE Trans. Image Process..

[28]  G. Sommer,et al.  Affine real-time face tracking using a wavelet network , 1999, Proceedings International Workshop on Recognition, Analysis, and Tracking of Faces and Gestures in Real-Time Systems. In Conjunction with ICCV'99 (Cat. No.PR00378).

[29]  John G. Daugman,et al.  Complete discrete 2-D Gabor transforms by neural networks for image analysis and compression , 1988, IEEE Trans. Acoust. Speech Signal Process..

[30]  Vladik Kreinovich,et al.  Wavelet neural networks are asymptotically optimal approximators for functions of one variable , 1994, Proceedings of 1994 IEEE International Conference on Neural Networks (ICNN'94).

[31]  R. Cavicchi,et al.  Optimization of temperature programmed sensing for gas identification using micro-hotplate sensors , 1998 .

[32]  Germano C. Vasconcelos,et al.  Function approximation by polynomial wavelets generated from powers of sigmoids , 1996, Defense + Commercial Sensing.

[33]  Martin D. Buhmann.  Pre-Wavelets on Scattered Knots and from Radial Function Spaces: A Review , 1994, IMA Conference on the Mathematics of Surfaces.

[34]  Marc Pierre Thuillard,et al.  Applications of fuzzy wavelets and wavenets in soft computing illustrated with the example of fire detectors , 2000, SPIE Defense + Commercial Sensing.

[35]  Kurt Hornik,et al.  FEED FORWARD NETWORKS ARE UNIVERSAL APPROXIMATORS , 1989 .

[36]  Harold H. Szu,et al.  Continuous speech segmentation determined by blind source separation , 1998, Defense, Security, and Sensing.