Q. Zhang, A. Benveniste: "Wavelet Networks" IEEE Transactions on Neural Networks, Vol 3, No. 6, November 1992  Home/Search   Document Not in Database   Summary   Related Articles   Check

....is not very robust with respect to background variations when the segmentation is not done properly. 1.2 Contribution This dissertation presents GWNs as an object representation approach that is both feature based and template based. Even though Wavelet Networks were already introduced by [Zhang and Benveniste, 1992], they have hardly been used or even mentioned in active research. We will argue that the potential of GWNs has been underestimated. This thesis contributes a thor ough evaluation of their properties as well as their advantages and disadvantages for real appli cations. In detail, GWNs supply a ....

....a and = cosO sinO h Ro sinO cosO ] the resolution of identity (2.3) then becomes O)Oa,b,O dO da db (2.11) f C (Lvf) a,b, 1 Note that the dilation parameter a is the same for both dimensions. This, however, can be relaxed for functions f L a (iRa) that are separable in every coordinate [Zhang and Benveniste, 1992]: f(x) fl(Xl) X f2(x2) For such functions each component is handled separately in the integral, so that for any such function f L 2 (11 2) the continuous 2 D wavelet transform is given by (Lf) c,s,O) j 2f(x) SR(x c) dx = f, 0,s) f, q ) 2.12) with the rotation matrix R, the ....

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Q. Zhang and A. Benveniste. Wavelet networks. IEEE Trans. Neural Networks', 3:889-898, 1992.

....Les simulations montrent que l approximation de f(x) avec un reseau ayant 4 unites cachees est un probleme di#cile pour un algorithme de descente de gradient en version o# line. Pour comparer notre approche avec d autres, nous avons choisi comme fonction a apprendre la fonction utilisee dans [13]. Une description de cette fonction g(x) et de l ensemble d apprentissage est fournie par la figure 2. g(x) est definie par: g(x) 2.186x 12.864 si # 2 = 4.246x si = 10e 0.05x 0.5 sin( 0.03x 0.7)x) si 0 3 2 1 0 1 2 3 4 5 3 2 1 0 1 2 3 4 Entr ee Ensemble ....

....1: Sortie d un perceptron avec 4 unites cachees. La sortie d un perceptron ayant deux couches avec un neurone lineaire dans la derniere couche est donnee par: # i #(## i . #x # i ) # 0 , 1) ou #x est l entree du reseau et # est la fonction de transfert des neurones caches. Comme dans [13] et [10] nous proposons d utiliser une methode de parametrisation geometrique. Nous reecrivons l equation 1 en: # i # # d i . #x # t i ) # 0 (2) La methode de descente de gradient par gradient conjugue de Polak Ribiere ( PRCG ) initialisee aleatoirement ou avec l AGDR a ete ....

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Q. Zhang and A. Benveniste. Wavelet networks. IEEE Trans. On Neural Networks, 3(6), November 1992.

....la representation. Les possibilites d extension de ces simulateurs sont en general assez limitees par cette absence d unification : la notion de retro propagation est ainsi etroitement liee a celle de MLP, ce qui veut dire que la mise en place d un modele comme celui des reseaux d ondelettes (cf. [17]) impose de programmer a nouveau la retro propagation pour celuici. En fait, l extensibilite est assuree au niveau des fonctions annexes (comme la gestion des simulations) mais pas au niveau des fondements mathematiques. D autre part, cette extensibilite reste tres couteuse en terme de temps de ....

Qinghua Zhang and Albert Benveniste. Wavelet networks. IEEE Trans. On Neural Networks, 3(6):889--898, November 1992.

....that the wavelet layer exhibits superior performance than the conventional competitive neural layers when patterns exhibit a low signal to noise ratio. 1. Introduction Recently, important bridges have been established between the field of artificial neural networks (ANN s) and wavelets [1] [2]. Wavelet Theory comprises a set of techniques aimed at developing efficient representations of signals through the use of elementary functions that are localized both in frequency and in time [3] A remarkable feature of wavelet based signal processing is that it mimics several natural phenomena ....

....groups of K wavelets, each group with its own set of scales A g ,translations B g , and weights W g . Henceforth, to help clarify the relation between such architecture and conventional neural layers, the linear combination of the wavelets in a group will be called a wavelet neuron, or wavelon [2]. The g wavelon in the layer will be denoted by y g : Figure 6: Architecture of an unsupervised wavelet layer with G wavelons . When a training pattern is presented to the CWL, its distance (norm of the difference) to each wavelon is computed. The wavelon with the smallest distance is ....

Q. Zhang and A. Benveniste. Wavelet networks. IEEE Trans. Neural Networks, 3(6):889--898, Nov. 1992.

....(for which GVS performs better) 2. 3 Feed forward neural network case We have demonstrated in a previous paper [4] that an extended back propagation algorithm can be defined for arbitrary feed forward neural networks (including in the same framework MLP, RBF networks and Wavelet Networks [10] for instance) This algorithm allows to compute e#ciently the di#erential of F (x, w) with respect to its input x, x, w) if F is the output of a neural network (with w as generalized weight vector) Therefore, our algorithm can be applied to any feed forward neural network. 3 Experiment on ....

Qinghua Zhang and Albert Benveniste. Wavelet networks. IEEE Trans. On Neural Networks, 3(6):889--898, November 1992.

....the pose of a face. In the last section we will conclude with some nal remarks. 2 Introduction to Gabor Wavelet Networks In this section we want to propose, as a major contribution of this work, the Gabor Wavelet Network for image representation. The idea of the wavelet network is inspired by [30], and the use of Gabor functions is inspired by the fact that they provide the best possible trade o between spatial resolution and frequency resolution. Furthermore, the use of Gabor lters in image analysis is biologically motivated as they model the response of the receptive elds of the ....

.... (y c y ) cos ) sin : 1) Here, c x , c y denote the translation of the Gabor wavelet, s x , s y denote the dilation and denotes the orientation. The parameters vector n (translation, orientation and dilation) of the wavelets may be chosen arbitrarily at this point. According to [30], any function f 2 L ) can be represented by a wavelet network. We are therefore going to interpret the image f to be a function of the space L ) and assume further, without loss of generality that f is DC free. In order to nd the GWN for image f we minimize the energy functional n i ....

Q. Zhang and A. Benviste. Wavelet networks. IEEE Trans. Neural Networks, 3(6):889-898, Nov. 1992. 27

....suited to the attribute suppression goal. 2. 5 Feed forward neural network case We have demonstrated in a previous paper [3] that an extended back propagation algorithm can be defined for arbitrary feed forward neural networks (including in the same framework MLP, RBFN and Wavelet Networks [17] for instance) This algorithm allows to compute e#ciently the di#erential of F (x, w) with respect to its input x, if F is the output of a neural network (with w as generalized weight vector) It allows also to compute (x, w) where E(x, w) is the error made by the network on example x (this ....

Qinghua Zhang and Albert Benveniste. Wavelet networks. IEEE Trans. On Neural Networks, 3(6):889--898, November 1992.

....In order to fulfill these requirements, we propose to switch from the standard MLP parametrization In Proceedings of the IEEE International Conference on Neural Networks, Orlando, June 94. Available at http: apiacoa.org publications 1994 icnn94.pdf to a geometrical one, as proposed in [15]. The initialization of the network is carried out by a simplified version of the methods proposed in [10] and [14] This new training process is faster than the classic one provided a good starting point is chosen among the possible ones. With the bad starting points, the network can get very ....

....the hidden neurons. To speed up the training process, we propose to switch from the classic point of view to a more geometrical and intuitive one. We rewrite equation 1 in : # i # # 0 (2) We call d i the dilatation parameter and # t i the translation parameter. As pointed out in [15], the partial derivative ### i = # i # # (## i . #x # i )#x, 3) depends on the value #x, whereas the partial derivative # # = # i # # (#x # t i ) 4) depends on the value (#x # t i ) and is thus less dependent from translation of all the input space and therefore more stable. ....

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Qinghua Zhang and Albert Benveniste. Wavelet networks. IEEE Trans. On Neural Networks, 3(6):889--898, November 1992.

....with affine deformations. In section 3, we will introduce our subspace tracking approach, discuss the details and conclude the paper with the experiments in section 4 and concluding remarks. 2. Introduction to Gabor Wavelet Networks The basic idea of the wavelet networks was first stated in [11], and the use of Gabor functions is inspired by the fact that they are recognized to be good feature detectors [8, 9] Number of Wavelets 116 216 original 16 52 Figure 1. Face images reconstructed with different number of wavelets To define a GWN, we start out, generally speaking, by taking a ....

Q. Zhang and A. Benveniste. Wavelet networks. IEEE Trans. Neural Networks, 3:889--898, 1992.

....face recognition. In the last section we will conclude with some final remarks. 2 Introduction to Gabor Wavelet Networks In this section we want to propose, as a major contribution of this work, the Gabor Wavelet Network for image representation. The idea of the wavelet network is inspired by [15], and the use of Gabor functions is inspired by the fact that they provide the best possible trade off between spatial resolution and frequency resolution. Furthermore, the Gabor filters are recognized to be good feature detectors [9] An image representation with Gabor Wavelet Networks has the ....

.... (y cy)sin) 1) with n = cx,cy,9, sx,sy) T. Here, cx, cy denote the translation of the Gabor wavelet, sx, sy denote the dilation and 9 denotes the orientation. The parameters ni (translation, orientation and dilation) of the wavelets may be chosen arbitrarily at this point. According to [15], any function f 2(2) can be represented by a wavelet network. We are therefore going to interpret the image f to be a function of the space 2(2) and assume further, without loss of generality that f is DC free. In order to find the GWN for image f we minimize the energy function E = min [If ....

Q. Zhang and A. Benviste. Wavelet networks. IEEE Trans. Neural Networks, 3(6):889-898, Nov. 1992.

....weights w h are frozen (i.e. they do not depend on the function f(x) Let us call such neural networks, for which s(x) is a wavelet, w h = 2 j , and b h = Gammak, wavelet neural networks. Remark. The relationship between wavelets and neural networks has been explored in [18] 24] 19] [25], 20] 21] C. Wavelet Neural Network as an Approximation Scheme We have already mentioned that an arbitrary smooth function f(x) on (0,1) can be represented by a series f(x) P j;k C jk e jk (x) with C jk = R f(x)e jk (x) dx. We can use the coefficients C jk as a record from which we ....

Q. Zhang and A. Benveniste, "Wavelet networks ", IEEE Trans. on Neural Networks, Vol. 3, pp. 889--898, 1992.

....; y i ) where x i is the pixel position and y i is the pixel intensity. Thus, our objective is to determine a continuous function f : R 2 R that approximates f , i.e. a continuous representation for the face template. The method used to obtain the face representation is a wavelet network [10], which is an alternative to feedforward neural networks for approximating continuous functions. In the following subsections, we first discuss neural networks and after we introduce the wavelet networks for function approximation. 3.1 Neural Networks Feedforward neural networks have been ....

Q. Zhang and A. Benviste, "Wavelet networks", IEEE Trans. on Neural Networks, 3(6):889-898, November 1992.

....with affine deformations. In section 3, we will introduce our subspace tracking approach, discuss the details and conclude the paper with the experiments in section 4 and concluding remarks. 2. Introduction to Gabor Wavelet Networks The basic idea of the wavelet networks was first stated in [11], and the use of Gabor functions is inspired by the fact that they are recognized to be good feature detectors [8, 9] Number of Wavelets 116 216 original 16 52 Figure 1. Face images reconstructed with di erent number of wavelets To define a GWN, we start out, generally speaking, by taking a ....

Q. Zhang and A. Benveniste. Wavelet networks. IEEE Trans. Neural Networks, 3:889--898, 1992.

....and the progressive attention property to speed up the response time of the system and to optimize the training of the neural network. In the last section we will conclude with some nal remarks. 2 Introduction to Gabor Wavelet Networks The basic idea of the wavelet networks is rst stated by [27], and the use of Gabor functions is inspired by the fact that they are recognized to be good feature detectors [13] To de ne a GWN, we start out, generally speaking, by taking a family of N odd Gabor wavelet functions = f n1 ; nN g of the form n x; y = exp 1 2 h sx ( x ....

Q. Zhang and A. Benveniste. Wavelet networks. IEEE Trans. Neural Networks, 3:889-898, 1992.

....of the human brain, a major part of neural network research has turned towards a more statistical point of view, where neural networks are considered as graphs of non linear regressors. In that way, their main justi cation is based on what is often called their universal approximation capability ([12, 18, 17, 19, 23]) This property is often established (explicitly or implicitly) thanks to the possibility to use a neural network to perform discrete convolutions in a frequency domain. The convolution terms are a direct consequence of the architecture of onehidden layer neural networks. Therefore the only ....

....tools, since [12, 18, 17] This regression is performed by means of discrete frequential convolutions based on neuron transfer functions. Such convolutions clearly appear in some works about models that use localized neural functions: for example [19] for RBF networks (radial basis function) or [23] for wavelet networks 1 . Convolution is more hidden when it is dealt with the most standard feedforward neural model, called multilayer perceptrons (MLP) Each neuron in a layer of a MLP applies a sigmoidal transfer function to a weighted sum of the outputs of all neurons in the previous ....

Q. Zhang and A. Benveniste. Wavelet networks. IEEE Trans. on Neural Networks, 3(6):889-898, Nov. 1992.

....the parameters of the neurons and for choosing the network structure. Recently, due to the similarity between discrete inverse wavelet transform and one hidden layer neural network, the idea of combining both wavelets and neural networks has been proposed by Q. Zhang and A. Benveniste in [9], and further developed in subsequent works [10, 11] In [9] the wavelet network was introduced that is a class of feedforward networks composed of wavelets. There are several advantages by combining wavelets and neural networks, for instance, results from This work has been partially supported by ....

....network structure. Recently, due to the similarity between discrete inverse wavelet transform and one hidden layer neural network, the idea of combining both wavelets and neural networks has been proposed by Q. Zhang and A. Benveniste in [9] and further developed in subsequent works [10, 11] In [9] the wavelet network was introduced that is a class of feedforward networks composed of wavelets. There are several advantages by combining wavelets and neural networks, for instance, results from This work has been partially supported by Alcatel Alsthom Recherche and European Gas Turbine SA. 1 ....

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Q. Zhang and A. Benveniste, "Wavelet networks," IEEE Trans. on Neural Networks, vol. 3, pp. 889--898, Nov. 1992.

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Q. Zhang, A. Benveniste: "Wavelet Networks" IEEE Transactions on Neural Networks, Vol 3, No. 6, November 1992

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Q. Zhang and A. Benveniste. Wavelet networks. IEEE Trans. Neural Networks, 3:889--898, 1992. 6

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Q. Zhang and A. Benveniste. Wavelet networks. IEEE Trans. Neural Networks, 3(6):889--898, Nov. 1992.

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Q. Zhang and A. Benveniste, "Wavelet Networks", IEEE Trans. Neural Networks, vol. 3, no. 6, Nov 1992, pp. 889-898.

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Q. Zhang and A. Benveniste, "Wavelet Networks", IEEE Trans. Neural Networks, vol. 3, no. 6, Nov 1992, pp. 889-898.

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Zhang Q. and Benveniste,A. (1992) Wavelet Network, IEEE Trans. Neural Networks 3, 889.

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Q. Zhang and A. Benveniste. Wavelet networks. IEEE Trans. Neural Networks, 3:889-898, 1992.

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Zhang,Q., Benveniste,A., (1992). Wavelet networks, IEEE Trans. on Neural Networks, 3, 889--898.

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Zhang,Q., Benveniste,A., (1992). Wavelet networks, IEEE Trans. on Neural Networks, 3, 889-898.

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