A theory of constructing locally supported radial wavelet frame and its application

A new method for constructing locally supported radial wavelet frame or basis, which is different from the multiresolution analysis, is proposed. A continuously differentiable radial function with a local support is chosen at first. Then a radial wavelet is obtained by the first and second derivatives of the radial function. If the radial function is both locally supported and infinitely differentiable, so is the radial wavelet. It is shown that the radial wavelet is a multidimensional dyadic one. I. Daubechies' wavelet frame Theorem is extended from one dimension to n dimensions. It is proven that the family generated by dilations and translations from a single radial wavelet can constitute a frame in L²(ℝⁿ). Consequently, it is concluded that the radial wavelet family generated by dilations and translations combined with their linear combination can constitute an orthonormal basis in L²(ℝⁿ). Finally, an example of the radial wavelet, which is inseparable and with a local support and infinitely high regularity, is given based on the framework given here. As an application, a class of wavelet network for image denoising is designed, and an underrelaxation iterative fast-learning algorithm with varied learning rate is given as well.