Dual Gramian analysis: Duality principle and unitary extension principle

Tight wavelet frames in low dimensions with canonical filters

One of the major driven forces in the area of applied and computational harmonic analysis over the last decade or longer is the development of redundant systems that have sparse approximations of various classes of functions. Such redundant systems include framelet (tight wavelet frame), ridgelet, curvelet, shearlet and so on. This paper mainly focuses on a special class of such redundant systems: tight wavelet frames, especially, those tight wavelet frames generated via a multiresolution analysis. In particular, we will survey the development of the unitary extension principle and its generalizations. A few examples of tight wavelet frame systems generated by the unitary extension principle are given. The unitary extension principle makes constructions of tight wavelet frame systems straightforward and painless which, in turn, makes a wide usage of the tight wavelet frames possible. Applications of wavelet frame, especially frame based image restorations, are also discussed in details.

Digital Gabor filters do generate MRA-based wavelet tight frames

Tight wavelet frames for irregular multiresolution analysis

The material presented in this book naturally splits in two parts: a functional analytic treatment of frames in general Hilbert spaces, and a more direct approach to structured frames like Gabor frames and wavelet frames. For the second part the most general results were presented in Chapter 21, in the setting of generalized shift-invariant systems on an LCA group.The current chapter is in a certain sense a natural continuation of both tracks. We consider connections between frame theory and abstract harmonic analysis and show how we can construct frames in Hilbert spaces via the theory for group representations. In special cases the general approach will bring us back to the Gabor systems and wavelet systems. The abstract framework adds another new aspect to the theory: we will not only obtain expansions in Hilbert spaces but also in a class of Banach spaces.

Recent advances in real algebraic geometry and in the theory of polynomial optimization are applied to answer some open questions in the theory of multivariate tight wavelet frames whose generators have at least one vanishing moment. Namely, several equivalent formulations of the so-called Unitary Extension Principle (UEP) are given in terms of Hermitian sums of squares of certain nonnegative Laurent polynomials and in terms of semidefinite programming. These formulations merge recent advances in real algebraic geometry and wavelet frame theory and lead to an affirmative answer to the long-standing open question of the existence of tight wavelet frames in dimension d=2. They also provide, for every d, efficient numerical methods for checking the existence of tight wavelet frames and for their construction. A class of counterexamples in dimension d=3 show that, in general, the so-called sub-QMF condition is not sufficient for the existence of tight wavelet frames. Stronger sufficient conditions for determining the existence of tight wavelet frames in dimension d≥3 are derived. The results are illustrated on several examples.

A finite frame is said to be scalable if its vectors can be rescaled so that the resulting set of vectors is a tight frame. The theory of scalable frame has been extended to the setting of Laplacian pyramids which are based on (rectangular) paraunitary matrices whose column vectors are Laurent polynomial vectors. This is equivalent to scaling the polyphase matrices of the associated filter banks. Consequently, tight wavelet frames can be constructed by appropriately scaling the columns of these paraunitary matrices by diagonal matrices whose diagonal entries are square magnitude of Laurent polynomials. In this paper we present examples of tight wavelet frames constructed in this manner and discuss some of their properties in comparison to the (non tight) wavelet frames they arise from.

Information science focuses on understanding problems from the perspective of the stake holders involved and then applying information and other technologies as needed. A necessary and sufficient condition is identified in term of refinement masks for applying the unitary extension principle for periodic functions to construct tight wavelet frames. Then a theory on the approximation order of truncated tight frame series is established, which facilitates construction of tight periodic wavelet frames with desirable approximation order. The pyramid decomposition scheme is derived based on the generalized multiresolution structure. Introduction and Concepts The setup of tight wavelet frames provides great flexibility in approximating and representing periodic functions. Fundamentals issues involved include the construction of tight periodic wavelet frames, approximation powers of such wavelet frames, and whether wavelet frames lead to sparse representat of locally smooth periodic functions. The frame theory plays an important role in the modern time-frequency analysis. It has been developed very fast over the last twenty years, espe cially in the context of wavelets and Gabor systems. This scientific field investigates the relationship betwenrse represeen the structure of materials at atomic or molecular scales and their macroscopic properties. Wavelet theory has been applied to signal processing, image compression, and so on. Frames for a separable Hilbert space were formally defined by Duffin and Schaeffer [1] in 1952 to study some deep problems in nonharmonic Fourier series. Basically, Duffin and Schaeffer abstracted the fundamental notion of Gabor for studying signal processing [2]. These ideas did not seem to generate much general interest outside of nonharmonic Fourier series however (see Young's [3]) until the landmark paper of Daubechies, Grossmann, and Meyer [4] in 1986. After this groundbreaking work, the theory of frames began to be more widely studied both in theory and in applications [5,6], such as signal processing, image processing, data compression, sampling theory. We begin with the unitary extension pinciple and formulate a general procedure for constructing wavelet frames. The emphasis is on having refinement masks as the starting point. The condition for this can be easily verified and also provide insight to the refinement masks that enable the construction process. Let [0, 2 ] L a be the space of all 2a -periodic square-integrable complex-valued func tions over the real line R with inner product , given by 2 0 , 1/(2 ) ( ) ( ) a h v a h x v x dx = ∫ where 2 ( ), ( ) [0, 2 ] h x v x L a ∈ , and norm 1/ 2 2 || || , u u u = . For a function 2 ( ) [0, 2 ] h x L a ∈ , we denote its Fourier series as ( ) in n Z n e h ∈ ∑ , where ( ) , in n h e h = , n Z ∈ , are its Fourier coefficient. For any 0 , u Z ≤ ∈ we define the shift operator 2 2 [0, 2 ] [0, 2 ] : u S L a L a  S by ( ) : ( 2 / 2 ) u u S h x h x a = −   . For 2 ( ) [0, 2 ] h x L a ∈ , since ( ) h x is a periodic function, it suffices to consider the shifts ( ) u k S h x ∈Λ   , , where k Λ is given by International Conference on Materials Engineering and Information Technology Applications (MEITA 2015) © 2015. The authors Published by Atlantis Press 240 1 1 1 1 { 2 1, 2 2, , 2 1, 2 } k k k k k − − − − Λ = − + − + −  : . Let (2 ) k ∆ be the 2 − periodic complex sequences k b , that is k k k b u b   ( +2 )= ( ) for all ,u Z ∈  . We denote the discrete Fourier transform of (2 ) k k b ∈∆ by 2 /2 ( ) : ( ) k k ai n k k b n b e − ∈Ω =∑ 

Generalized frames (in short, $g$-frames) are a natural generalization of standard frames in separable Hilbert spaces. Motivated by the concept of weaving frames in separable Hilbert spaces by Bemrose, Casazza, Grochenig, Lammers and Lynch in the context of distributed signal processing, we study weaving properties of $g$-frames. Firstly, we present necessary and sufficient con\-ditions for weaving $g$-frames in Hilbert spaces. We extend some results of \cite Bemrose, Casazza, Grochenig, Lammers and Lynch, and Casazza and Lynch regarding conversion of standard weaving frames to $g$-weaving frames. Some Paley-Wiener type perturbation results for weaving $g$-frames are obtained. Finally, we give necessary and sufficient conditions for weaving $g$-Riesz bases.

The concept of R-duals of a frame was introduced by Casazza, Kutyniok and Lammers in 2004, with the motivation to obtain a general version of the duality principle in Gabor analysis. For tight Gabor frames and Gabor Riesz bases the three authors were actually able to show that the duality principle is a special case of general results for R-duals. In this paper we introduce various alternative R-duals, with focus on what we call R-duals of type II and III. We show how they are related and provide characterizations of the R-duals of type II and III. In particular, we prove that for tight frames these classes coincide with the R-duals by Casazza et el., which is desirable in the sense that the motivating case of tight Gabor frames already is well covered by these R-duals. On the other hand, all the introduced types of R-duals generalize the duality principle for larger classes of Gabor frames than just the tight frames and the Riesz bases; in particular, the R-duals of type III cover the duality principle for all Gabor frames.

The constructions of pseudo-spline tight frames were first introduced for the first time by I. Daubechies et al to construct tight framelets with desired approximation orders via the unitary extension principle (UEP). Pseudo-splines provide a rich family of refutable functions and the tight frame system derived from them normally gives better approximation order and other good properties than that derived from B-splines which are one of the special classes of pseudo-splines. This paper discusses the construction of tight wavelet frame with some essential properties, such as regularity and high approximation, and presents the construction of tight wavelet frames derived from the pseudo-splines via UEP and OEP (oblique extension principle)

Information science focuses on understanding problems from the perspective of the stake holders involved and then applying information and other technologies as needed. A necessary and sufficient condition is identified in term of refinement masks for applying the unitary extension principle for periodic functions to construct tight wavelet frames. Then a theory on the approxi-mation order of truncated tight frame series is established, which facilitates construction of tight periodic wavelet frames with desirable approximation order. The pyramid decomposition scheme is derived based on the generalized multiresolution structure.

A super wavelet of length n is an n-tuple (? 1,? 2,?,? n ) in the product space $\prod_{j=1}^{n} L^{2}(\mathbb{R})$ , such that the coordinated dilates of all its coordinated translates form an orthonormal basis for $\prod_{j=1}^{n} L^{2} (\mathbb{R})$ . This concept is generalized to the so-called super frame wavelets, super tight frame wavelets and super normalized tight frame wavelets (or super Parseval frame wavelets), namely an n-tuple (? 1,? 2,?,? n ) in $\prod_{j=1}^{n}L^{2} (\mathbb{R})$ such that the coordinated dilates of all its coordinated translates form a frame, a tight frame, or a normalized tight frame for $\prod_{j=1}^{n} L^{2}(\mathbb{R})$ . In this paper, we study the super frame wavelets and the super tight frame wavelets whose Fourier transforms are defined by set theoretical functions (called s-elementary frame wavelets). An m-tuple of sets (E 1,E 2,?,E m ) is said to be ?-disjoint if the E j 's are pair-wise disjoint under the 2?-translations. We prove that a ?-disjoint m-tuple (E 1,E 2,?,E m ) of frame sets (i.e., ? j defined by $\widehat{\eta_{j}}=\frac{1}{\sqrt{2\pi}}\chi_{E_{j}}$ is a frame wavelet for L 2(?) for each j) lead to a super frame wavelet (? 1,? 2,?,? m ) for $\prod_{j=1}^{m} L^{2} (\mathbb{R})$ where $\widehat{\eta_{j}}=\frac{1}{\sqrt{2\pi}}\chi_{E_{j}}$ . In the case of super tight frame wavelets, we prove that (? 1,? 2,?,? m ), defined by $\widehat{\eta_{j}}=\frac{1}{\sqrt{2\pi}}\chi_{E_{j}}$ , is a super tight frame wavelet for ?1?j?m L 2(?) with frame bound k 0 if and only if each ? j is a tight frame wavelet for L 2(?) with frame bound k 0 and that (E 1,E 2,?,E m ) is ?-disjoint. Denote the set of all ?-disjoint s-elementary super frame wavelets for ?1?j?m L 2(?) by $\mathfrak{S}(m)$ and the set of all s-elementary super tight frame wavelets (with the same frame bound k 0) for ?1?j?m L 2(?) by $\mathfrak{S}^{k_{0}}(m)$ . We further prove that $\mathfrak{S}(m)$ and $\mathfrak{S}^{k_{0}}(m)$ are both path-connected under the ?1?j?m L 2(?) norm, for any given positive integers m and k 0.

Nonstationary tight wavelet frames, I: Bounded intervals

In this paper, we construct a length-four parameterized tight wavelet frame with three passes based on the design of low pass filter using Sub-QMF. Comparing to the usual methods, it not only keeps some advantages of parameterized tight wavelet frame such as more freedom, good smoothness and symmetries, but also has the least passes. By applying it to the image fusion, we further discovery that it is more effective than Laplacian pyramid, Haar wavelet, and tight frame with four passes.Keywordsparameterized designtight wavelet frameSub-QMFimage fusion