Abstract—An application of Beta wavelet networks to synthesize pass-high and pass-low wavelet filters is investigated in this work. A Beta wavelet network is constructed using a parametric function called Beta function in order to resolve some nonlinear approximation problem. We combine the filter design theory with wavelet network approximation to synthesize perfect filter reconstruction. The order filter is given by the number of neurons in the hidden layer of the neural network. In this paper we use only the first derivative of Beta function to illustrate the proposed design procedures and exhibit its performance. Keywords—Beta wavelets, Wavenet, multiresolution analysis, perfect filter reconstruction, salient point detect, repeatability. I. INTRODUCTION ECENTLY, the subject of wavelet analysis has attracted much attention from both mathematicians and engineers alike. Wavelets have been applied successfully to multiscale analysis and synthesis, time-frequency signal analysis in signal processing, function approximation, approximation in solving partial differential equations. Wavelets are well suited to depicting functions with local nonlinearities and fast variations because of their intrinsic properties of finite support and self-similarity. The relationship between the scaling function and the wavelet function is now clear. The scaling function provides a set of basis function to approximate a signal at a certain resolution and the wavelet provides a set of basis functions for the detail signal. When the detail signal is added to the signal approximation, the result is the signal approximation at the next higher level of resolution. For a general continuous time signal f(t), these successive additions of detail signals to create the next higher resolution approximation must continue forever to get an accurate representation of f(t). This problem is neatly fixed when dealing with discrete time signals as they are already defined with finite time resolution and can be accurately represented in some subspace V k where k < +∝. In this paper, we propose a novel method to generate wavelet filters using Beta Wavelet Neural Network (BWNN). Wajdi Bellil is with University of Sfax, National Engineering School of Sfax, B.P. W, 3038, Sfax, Tunisia (phone: 216-98212934; e-mail: Wajdi.bellil@ieee.org). Chokri Ben Amar is with University of Sfax, National Engineering School of Sfax, B.P. W, 3038, Sfax, Tunisia (phone: 216-98638417; e-mail: chokri.benamar@ieee.org). Adel M. Alimi is with University of Sfax, National Engineering School of Sfax, B.P. W, 3038, Sfax, Tunisia (phone: 216-98667682; e-mail: adel.alimi@ieee.org). The advantage of the proposed method is demonstrated by computer simulations. This paper is organized as follows. Section 2 presents the theory of Beta wavelet. Section 3 shows the discrete wavelet transform, MRA and filter implementation. Section 4 illustrates the reason why a WNN is needed to synthesis wavelet filters. Section 5 demonstrates the simulation results on Beta wavelet filters and some others. Section 6 concludes this paper. II. A NOVOL BETA WAVELET FAMILY The Beta function [1, 2] is defined as: q p qx px x and x x q p with else x x x if x x x x x x x x x x q p x c q c p c + + = ℜ ∈ < ⎪ ⎩ ⎪ ⎨ ⎧ ∈ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − − ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − − = 0 1 1 0 1 0 1 1 0 0 1 0 , , 0 ] , [ ) , , , ; ( β (1) We have proved in [1, 2] that all derivatives of Beta function ∈ L²(IR) and are of class C∞, so they have the property of universal approximation. The general form of the nth derivative of Beta function is: () () x x P dx x d x n n n n β β 1 1 1 ) ( ) ( + + + = Ψ (2) ) ( ) ( ) ( )! ( ) ( )! ( ) 1 ( ) ( ) ( ) ( ) ( ! ) ( ! ) 1 ( 1 1 1 1 1 0 1 1 1 1 0 x x P x x q i n x x p i n C x P x P x x x q n x x p n i n i i n i n i n i n n n n n β β β ∑ − = − + − + − + + ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − − − − − − + + ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ − − − − = x x q x x p x P Where − − − = 1 0 1 ) ( 1 ) ( ) ( ) ( ) ( ' 1 1 1 > ∀ + = − − n x P x P x P x P and n n n (3) III. DISCRETE WAVELET TRANSFORM, MRA AND FILTER IMPLEMENTATION The DWT will transform a discrete time signal to a discrete wavelet representation [3]. The first step is to discretize the wavelet parameters. This is commonly done with the dyadic sampling grid, defined by: ( ) Z n m n t t m m n m ∈ − Ψ = Ψ , , 2 2 ) ( 2 / , (4) This reduces the previously continuous set to a now discrete, orthogonal set. The analysis formula becomes Z n m t t f W n m n m ∈ Ψ = , , ) ( ), ( , , (5) Synthesis of Wavelet Filters using Wavelet Neural Networks Wajdi Bellil, Chokri Ben Amar, and Adel M. Alimi R World Academy of Science, Engineering and Technology 13 2006 108