Locally supported wavelet with infinitely high regularity and fast algorithm for edge detection

According to I. Daubechies' theory, the regularity of orthonormal wavelet bases with compact support increases linearly with the support width. By relaxing the orthogonality, much more freedom on the choice of wavelet function is gained. An infinitely differentiable wavelet with a local support is proposed. The wavelet is derived from the first- or second-order derivatives of a radial function. It is then proven that the wavelet is a two-dimensional dyadic one. An arbitrary square integrable function can be reconstructed from its dyadic wavelet transformation, and the reconstruction is stable. Different from tensor product multidimensional wavelets, this wavelet is inseparable. As an example of application, an algorithm is developed for detecting edges in image by finding the local maxima of its wavelet transform modulus. The computation of the wavelet transform in image processing depends only on the wavelet functional values at a few integer points. For a N×N image, the computational complexity of the detection scheme is significantly better than that of Mallat's fast wavelet algorithm.