(Anti)symmetric compactly supported tight wavelet frames constructed from quasi-interpolatory subdivision masks and their application to denoising

Abstract

Since the Unitary Extension Principle (UEP) [11] emerged into the real world, studies about the construction of tight wavelet frames have been prevailed. Among these studies, (anti)symmetric compactly supported tight wavelet frames are paid a good attention in both theory and practice. Petukhov proved in [10] that an interpolatory symbol h0 admits (anti)symmetric solutions to UEP if and only if $h_0(z) = (z^(1-2N) + 2 + z^(2N-1))=4$. These solutions generate a refinable function only when N = 1, i.e., the refinable function is the piecewise linear B-spline. Motivated by this result, we considered the quasi-interpolatory filters of [2], which generate the refinable functions that reproduce polynomials up to cer- tain degrees. These filters (or masks) are generated with a free tension parameter w by the quasi-interpolatory subdivision scheme [2]. A general formula for even degrees was presented in [9]. This thesis deals only with the case of degree L = 1, 2, 3, 4, and 5 because of the computation complexity. For the practical advantages, three generators of (anti)symmetric compactly supported tight wavelet frames were employed with a time-frequency overcom- plete filter bank [12]. We determined all the possible tension parameter $\omega$``s which provide the (anti)symmetric compactly supported tight wavelet frames in each degree. The smoothness, approximation order, and vanishing moments of each case were computed. In addition to the construction of the tight wavelet frames, some application to signal and image denoising are presented.