In this paper, we develop the theory of the wavelet transform over Galois elds. To avoid the limitations inherent in the number theoretic Fourier transform over nite elds, our wavelet transform relies on a basis decomposition in the time domain rather than in the frequency domain. First, we characterize the in nite dimensional vector spaces for which an orthonormal basis expansion of any sequen...

Generalizing the result of Bownik and Speegle [Approximation Theory X: Wavelets, Splines and Applications, Vanderbilt University Press, pp. 63–85, 2002], we provide plenty of non-MSF A-wavelets with the help of a given A-wavelet set. Further, by showing that the dimension function of the non-MSF A-wavelet constructed through an A-wavelet set W coincides with the dimension function of W , we con...

Wavelet theory has become popular in modeling echo state network. One of the most promising directions is usage of the wavelets as membership activation functions of its reservoir. However, only Symlets wavelet seems to be suitable for hybrid wavelet-sigmoid activation functions. To enhance a systematic study of the field, we concentrate on developing more wavelets towards the typical ESN repre...

We propose a wavelet-based approach to measure the Shannon entropy in the context of spatial point patterns. The method uses the fully anisotropic Morlet wavelet to estimate the energy distribution at different directions and scales. The spatial heterogeneity and complexity of spatial point patterns is then analyzed using the multiscale anisotropic wavelet entropy. The efficacy of the approach ...

It is well known that for irreducible, square-integrable representations of a locally compact group, there exist so-called admissible vectors which allow the construction of generalized continuous wavelet transforms. In this paper we discuss when the irreducibility requirement can be dropped, using a connection between generalized wavelet transforms and Plancherel theory. For unimodular groups ...

Most simple nonlinear thresholding rules for wavelet-based denoising assume that the wavelet coefficients are independent. However, wavelet coefficients of natural images have significant dependencies. In this paper, we will only consider the dependencies between the coefficients and their parents in detail. For this purpose, new non-Gaussian bivariate distributions are proposed, and correspond...

The wavelet network 22, 23] has been introduced as a special feedforward neural network supported by the wavelet theory. Such network can be directly used in function approximation problems, and consequently can be applied to nonlinear system modeling by means of nonlinear black-box identiication. In this paper the construction of feedforward neural networks is discussed from both identiication...

This paper is to construct Riesz wavelets with short support. Riesz wavelets with short support are of interests in both theory and application. In theory, it is known that a B-spline of order m has the shortest support among all compactly supported refinable functions with the same regularity. However, it remained open whether a Riesz wavelet with the shortest support and m vanishing moments c...