Highly symmetric bi-frames for triangle surface multiresolution processing

In this paper we investigate the construction of dyadic affine (wavelet) bi-frames for triangular-mesh surface multiresolution processing. We introduce 6-fold symmetric bi-frames with 4 framelets (frame generators). 6-fold symmetric bi-frames yield frame decomposition and reconstruction algorithms (for regular vertices) with high symmetry, which is required for the design of the corresponding frame multiresolution algorithms for extraordinary vertices on the triangular mesh. Compared with biorthogonal wavelets, the constructed bi-frames have better smoothness and smaller supports. In addition, we also provide frame multiresolution algorithms for extraordinary vertices. All the frame algorithms considered in this paper are given by templates (stencils) so that they are implementable. Furthermore, we present some preliminary experimental results on surface processing with frame algorithms constructed in this paper.     

The template‐based procedure of biorthogonal ternary wavelets for curve multiresolution processing

Curve multiresolution processing techniques have been widely discussed in the study of subdivision schemes and many applications, such as surface progressive transmission and compression. The ternary subdivision scheme is the more appealing one because it can possess the symmetry, smaller topological support, and certain smoothness, simultaneously. So biorthogonal ternary wavelets are discussed in this paper, in which refinable functions are designed for cure and surface multiresolution processing of ternary subdivision schemes. Moreover, by the help of lifting techniques, the template‐based procedure is established for constructing ternary refinable systems with certain symmetry, and it also gives a clear geometric templates of corresponding multiresolution algorithms by several iterative steps. Some examples with certain smoothness are constructed.     

Constructing pairs of dual bandlimited framelets with desired time localization

For sufficiently small translation parameters, we prove that any bandlimited function ψ, for which the dilations of its Fourier transform form a partition of unity, generates a wavelet frame with a dual frame also having the wavelet structure. This dual frame is generated by a finite linear combination of dilations of ψ with explicitly given coefficients. The result allows a simple construction procedure for pairs of dual wavelet frames whose generators have compact support in the Fourier domain and desired time localization. The construction is based on characterizing equations for dual wavelet frames and relies on a technical condition. We exhibit a general class of function satisfying this condition; in particular, we construct piecewise polynomial functions satisfying the condition.      

Frames of p-adic wavelets and orbits of the affine group

The general construction of frames of p-adic wavelets is described. We consider the orbit of a generic mean zero locally constant function with compact support (mean zero test function) with respect to the action of the p-adic affine group and show that this orbit is a uniform tight frame. We discuss the relations of this result with the multiresolution wavelet analysis. The text was submitted by the authors in English.     

Entourage uniformities for frames

The purpose of this paper is to define a uniformity on a frame using entourages. A definition of uniformities in terms of covers has been given byA. Pultr ([8], [10]). We introduce two new symmetry conditions for a quasi-uniform space, open-set symmetry and small-set symmetry. We prove that a quasi-uniformityU is a uniformity if and only if it is both open-set symmetric and small-set symmetric. The category of (covering) uniform frames is isomorphic with the category of entourage uniform frames.     

Multiwavelet Frames from Refinable Function Vectors

Starting from any two compactly supported d-refinable function vectors in (L ₂(R)) ^r with multiplicity r and dilation factor d, we show that it is always possible to construct 2rd wavelet functions with compact support such that they generate a pair of dual d-wavelet frames in L ₂(R) and they achieve the best possible orders of vanishing moments. When all the components of the two real-valued d-refinable function vectors are either symmetric or antisymmetric with their symmetry centers differing by half integers, such 2rd wavelet functions, which generate a pair of dual d-wavelet frames, can be real-valued and be either symmetric or antisymmetric with the same symmetry center. Wavelet frames from any d-refinable function vector are also considered. This paper generalizes the work in [5,12,13] on constructing dual wavelet frames from scalar refinable functions to the multiwavelet case. Examples are provided to illustrate the construction in this paper.     

Tight Frames and Their Symmetries

The aim of this paper is to investigate the symmetry properties of tight frames, with a view to constructing tight frames of orthogonal polynomials in several variables which share the symmetries of the weight function, and other similar applications. This is achieved by using representation theory to give methods for constructing tight frames as orbits of groups of unitary transformations acting on a given finite-dimensional Hilbert space. Along the way, we show that a tight frame is determined by its Gram matrix and discuss how the symmetries of a tight frame are related to its Gram matrix. We also give a complete classification of those tight frames which arise as orbits of an abelian group of symmetries.     

Quasi-Biorthogonal Frame Multiresolution Analyses and Wavelets

We introduce the concepts of quasi-biorthogonal frame multiresolution analyses and quasi-biorthogonal frame wavelets which are natural generalizations of biorthogonal multiresolution analyses and biorthogonal wavelets, respectively. Necessary and sufficient conditions for quasi-biorthogonal frame multiresolution analyses to admit quasi-biorthogonal wavelet frames are given, and a non-trivial example of quasi-biorthogonal frame multiresolution analyses admitting quasi-biorthogonal frame wavelets is constructed. Finally, we characterize the pair of quasi-biorthogonal frame wavelets that is associated with quasi-biorthogonal frame multiresolution analyses.     

On Dual Gabor Frame Pairs Generated by Polynomials

We provide explicit constructions of particularly convenient dual pairs of Gabor frames. We prove that arbitrary polynomials restricted to sufficiently large intervals will generate Gabor frames, at least for small modulation parameters. Unfortunately, no similar function can generate a dual Gabor frame, but we prove that almost any such frame has a dual generated by a B-spline. Finally, for frames generated by any compactly supported function φ whose integer-translates form a partition of unity, e.g., a B-spline, we construct a class of dual frame generators, formed by linear combinations of translates of φ. This allows us to chose a dual generator with special properties, for example, the one with shortest support, or a symmetric one in case the frame itself is generated by a symmetric function. One of these dual generators has the property of being constant on the support of the frame generator.     

Irregular wavelet frames

For the non-band-limited functionΨ, a sufficient condition is presented under which is a frame for L²(R). The stability of these frames is studied. For the wavelets frequently used in signal processing, some concrete results are given.     

Some New uniform frames with block size four and index one or three

In this paper, some new uniform frames with block size four and index one or three are obtained. The known existence results for (4,1)‐frames and (4,3)‐frames are both improved to the extent that only a finite number of possible exceptions remain. © 2003 Wiley Periodicals, Inc.     

Tight wavelet frames for irregular multiresolution analysis

An important tool for the construction of tight wavelet frames is the Unitary Extension Principle first formulated in the Fourier-domain by Ron and Shen. We show that the time-domain analogue of this principle provides a unified approach to the construction of tight frames based on many variations of multiresolution analyses, e.g., regular refinements of bounded L-shaped domains, refinements of subdivision surfaces around irregular vertices, and nonstationary subdivision. We consider the case of nonnegative refinement coefficients and develop a fully local construction method for tight frames. Especially, in the shift-invariant setting, our construction produces the same tight frame generators as the Unitary Extension Principle.     

Minimally Supported Frequency Composite Dilation Parseval Frame Wavelets

A composite dilation Parseval frame wavelet is a collection of functions generating a Parseval frame for L ²(ℝ ⁿ ) under the actions of translations from a full rank lattice and dilations by products of elements of groups A and B. A minimally supported frequency composite dilation Parseval frame wavelet has generating functions whose Fourier transforms are characteristic functions of sets contained in a lattice tiling set. Constructive proofs are used to establish the existence of minimally supported frequency composite dilation Parseval frame wavelets in arbitrary dimension using any finite group B, any full rank lattice, and an expanding matrix generating the group A and normalizing the group B. Moreover, every such system is derived from a Parseval frame multiresolution analysis. Multiple examples are provided including examples that capture directional information.      

ON THE STABILITY OF FUSION FRAMES （FRAMES OF SUBSPACES）

A frame is an orthonormal basis-like collection of vectors in a Hilbert space, but need not be a basis or orthonormal. A fusion frame (frame of subspaces) is a frame-like collection of subspaces in a Hilbert space, thereby constructing a frame for the whole space by joining sequences of frames for subspaces. Moreover the notion of fusion frames provide a framework for applications and providing efficient and robust information processing algorithms.In this paper we study the conditions under which removing an element from a fusion frame, again we obtain another fusion frame. We give another proof of [5, Corollary 3.3(iii)] with extra information about the bounds.     

Pairs of Dual Wavelet Frames from Any Two Refinable Functions

Starting from any two compactly supported refinable functions in L₂(R) with dilation factor d,we show that it is always possible to construct 2d wavelet functions with compact support such that they generate a pair of dual d-wavelet frames in L₂(R). Moreover, the number of vanishing moments of each of these wavelet frames is equal to the approximation order of the dual MRA; this is the highest possible. In particular, when we consider symmetric refinable functions, the constructed dual wavelets are also symmetric or antisymmetric. As a consequence, for any compactly supported refinable function in L₂(R), it is possible to construct, explicitly and easily, wavelets that are finite linear combinations of translates (d · – k), and that generate a wavelet frame with an arbitrarily preassigned number of vanishing moments.We illustrate the general theory by examples of such pairs of dual wavelet frames derived from B-spline functions.     

The symmetry group of a finite frame

We define the symmetry group of a finite frame as a group of permutations on its index set. This group is closely related to the symmetry group of Vale and Waldron (2005) [12] for tight frames: they are isomorphic when the frame is tight and has distinct vectors. The symmetry group is the same for all similar frames, in particular for a frame, its dual and canonical tight frames. It can easily be calculated from the Gramian matrix of the canonical tight frame. Further, a frame and its complementary frame have the same symmetry group. We exploit this last property to construct and classify some classes of highly symmetric tight frames.     

Parseval frame wavelets with -dilations

We study Parseval frame wavelets in with matrix dilations of the form , where A is an arbitrary expanding n×n matrix with integer coefficients, such that |detA|=2. We show that each A-MRA admits either Parseval frame wavelets, or Parseval frame bi-wavelets. The minimal number of generators for a Parseval frame associated with an A-MRA (i.e. 1 or 2) is determined in terms of a scaling function. All Parseval frame (bi)wavelets associated with A-MRA's are described. We then introduce new classes of filter induced wavelets and bi-wavelets. It is proved that these new classes strictly contain the classes of all A-MRA Parseval frame wavelets and bi-wavelets, respectively. Finally, we demonstrate a method of constructing all filter induced Parseval frame (bi)wavelets from generalized low-pass filters.