Semi-regular Dubuc-Deslauriers wavelet tight frames

In this paper, we construct wavelet tight frames with n vanishing moments for Dubuc-Deslauriers 2npoint semi-regular interpolatory subdivision schemes. Our motivation for this construction is its practical use for further regularity analysis of wide classes of semi-regular subdivision. Our constructive tools are local eigenvalue convergence analysis for semi-regular Dubuc-Deslauriers subdivision, the Unitary Extension Principle and the generalization of the Oblique Extension Principle to the irregular setting by Chui, He and St\"ockler. This group of authors derives suitable approximation of the inverse Gramian for irregular Bspline subdivision. Our main contribution is the derivation of the appropriate approximation of the inverse Gramian for the semi-regular Dubuc-Deslauriers scaling functions ensuring n vanishing moments of the corresponding framelets

Wavelet frame bijectivity on Lebesgue and Hardy spaces

We prove a sufficient condition for frame-type wavelet series in $L^p$, the Hardy space $H^1$, and BMO. For example, functions in these spaces are shown to have expansions in terms of the Mexican hat wavelet, thus giving a strong answer to an old question of Meyer. Bijectivity of the wavelet frame operator acting on Hardy space is established with the help of new frequency-domain estimates on the Calder\'on-Zygmund constants of the frame kernel.Comment: 23 pages, 7 figure

Gröbner bases and wavelet design

AbstractIn this paper, we detail the use of symbolic methods in order to solve some advanced design problems arising in signal processing. Our interest lies especially in the construction of wavelet filters for which the usual spectral factorization approach (used for example to construct the well-known Daubechies filters) is not applicable. In these problems, we show how the design equations can be written as multivariate polynomial systems of equations and accordingly how Gröbner algorithms offer an effective way to obtain solutions in some of these cases

Highly Symmetric Multiple Bi-Frames for Curve and Surface Multiresolution Processing

Wavelets and wavelet frames are important and useful mathematical tools in numerous applications, such as signal and image processing, and numerical analysis. Recently, the theory of wavelet frames plays an essential role in signal processing, image processing, sampling theory, and harmonic analysis. However, multiwavelets and multiple frames are more flexible and have more freedom in their construction which can provide more desired properties than the scalar case, such as short compact support, orthogonality, high approximation order, and symmetry. These properties are useful in several applications, such as curve and surface noise-removing as studied in this dissertation. Thus, the study of multiwavelets and multiple frames construction has more advantages for many applications. Recently, the construction of highly symmetric bi-frames for curve and surface multiresolution processing has been investigated. The 6-fold symmetric bi-frames, which lead to highly symmetric analysis and synthesis bi-frame algorithms, have been introduced. Moreover, these multiple bi-frame algorithms play an important role on curve and surface multiresolution processing. This dissertation is an extension of the study of construction of univariate biorthogonal wavelet frames (bi-frames for short) or dual wavelet frames with each framelet being symmetric in the scalar case. We will expand the study of biorthogonal wavelets and bi-frames construction from the scalar case to the vector case to construct biorthogonal multiwavelets and multiple bi-frames in one-dimension. In addition, we will extend the study of highly symmetric bi-frames for triangle surface multiresolution processing from the scalar case to the vector case. More precisely, the objective of this research is to construct highly symmetric biorthogonal multiwavelets and multiple bi-frames in one and two dimensions for curve and surface multiresolution processing. It runs in parallel with the scalar case. We mainly present the methods of constructing biorthogonal multiwavelets and multiple bi-frames in both dimensions by using the idea of lifting scheme. On the whole, we discuss several topics include a brief introduction and discussion of multiwavelets theory, multiresolution analysis, scalar wavelet frames, multiple frames, and the lifting scheme. Then, we present and discuss some results of one-dimensional biorthogonal multiwavelets and multiple bi-frames for curve multiresolution processing with uniform symmetry: type I and type II along with biorthogonality, sum rule orders, vanishing moments, and uniform symmetry for both types. In addition, we present and discuss some results of two-dimensional biorthogonal multiwavelets and multiple bi-frames and the multiresolution algorithms for surface multiresolution processing. Finally, we show experimental results on curve and surface noise-removing by applying our multiple bi-frame algorithms