Refinable Functions Research Articles

D-Refinable (dual) pseudo-splines and their regularities

In this paper, we introduce the concept of [Formula: see text]-dilation (dual) pseudo-splines in [Formula: see text] with dilation [Formula: see text], characterize the linear independence of the integer shifts, the subdivision schemes of polynomial reproduction and Hölder regularity for (dual) pseudo-splines with general dilation. We present a new method to determine the Hölder regularity of refinable functions with general dilation [Formula: see text]. Furthermore, we compare the regularities between pseudo-splines and dual ones.

Subdivision and multiresolution for PUPs

Partition of Unity Parametrics (PUPs) is a generalization of NURBS that permits the use of arbitrary basis functions to model parametric curves and surfaces. An interesting problem for PUPs is the identification of subdivision, reverse subdivision, and multiresolution schemes for this recently developed and flexible class of parametric curves and surfaces.In this paper, we introduce a systematic approach to derive uniform subdivision schemes for PUPs curves and tensor-product surfaces. Our approach formulates PUPs subdivision as a least squares problem. This allows us to find exact subdivision filters for refinable basis functions and optimal approximate schemes for irrefinable ones. Additionally, we derive PUPs multiresolution masks based on their subdivision filters. We formulate the problem as a constrained least squares optimization, such that the resulting multiresolution schemes are banded and optimal in terms of minimizing multiresolution reconstruction error.Finally, to illustrate our methods, we provide sample subdivision and multiresolution schemes with different properties. These include specific examples targeted towards applications of PUPs multiresolution schemes for compression, feature transfer, and macroscopic editing.

Weak Nonhomogeneous Wavelet Bi-Frames for Reducing Subspaces of Sobolev Spaces

ABSTRACTAs homogeneous wavelet bi-frames derived from a pair of refinable functions, nonhomogeneous wavelet bi-frames admit fast wavelet transform, which is one of main concerns in wavelet analysis. Simultaneously, for a nonhomogeneous wavelet bi-frame in a pair of dual spaces with s≠0, smoothness and vanishing moment requirements are separated from each other, that is, one system is for smoothness and the other for vanishing moments. This gives us more flexibility to construct nonhomogeneous wavelet bi-frames than in . However, the requirement of the Bessel property of affine systems is always needed. For refinable-function based wavelet bi-frames, such a requirement is imposed on refinable functions in different ways. This is not what we expect. So a natural question involves what we are expecting from a pair of general refinable functions. For this purpose, we introduce the notion of the reducing subspace of a Sobolev space and the notion of the weak nonhomogeneous wavelet bi-frame in a general pair of dual reducing subspaces of . We characterize reducing subspaces of a Sobolev space. Then, under the setting of reducing subspaces, we study the properties of weak nonhomogeneous wavelet bi-frames, obtain a construction of weak nonhomogeneous wavelet bi-frames from a pair of refinable functions, and derive a fast wavelet algorithm associated with such weak nonhomogeneous wavelet bi-frames.

Totally positive refinable functions with general dilation M

We construct a new class of approximating functions that are M-refinable and provide shape preserving approximations. The refinable functions in the class are smooth, compactly supported, centrally symmetric and totally positive. Moreover, their refinable masks are associated with convergent subdivision schemes. The presence of one or more shape parameters gives a great flexibility in the applications. Some examples for dilation M=4 and M=5 are also given.

Analysis and construction of a family of refinable functions based on generalized Bernstein polynomials

In this paper, we construct a new family of refinable functions from generalized Bernstein polynomials, which include pseudo-splines of Type II. A comprehensive analysis of the refinable functions is carried out. We then prove the convergence of cascade algorithms associated with the new masks and construct Riesz wavelets whose dilation and translation form a Riesz basis for $L_{2}(\mathbb{R})$ . Stability of the subdivision schemes, regularity and approximation orders are obtained. We also illustrate the symmetry of the corresponding refinable functions.

Wavelet bi‐frames with uniform symmetry

Wavelet bi‐frames with uniform symmetry are discussed in this paper. Every refinable function in the bi‐frame system is symmetric, which is very useful in the image processing and curve and surface multiresolution processing. By the aid of the lifting scheme, bi‐frame multiresolution algorithms can be divided into several iterative steps, and each step can be shown by a symmetric template. The template‐based procedure is established for constructing bi‐frames with uniform symmetry and N > 2 generators. In particular, we take the bi‐frame with three generators as an example to provide a clearer picture of the template‐based procedure for constructing bi‐frames. Three types of bi‐frames with three generators are studied, and some examples with certain smoothness are constructed. These examples include some bi‐frames with interpolating property. Copyright © 2016 John Wiley & Sons, Ltd.

Compactly Supported Tight and Sibling Frames Based on Generalized Bernstein Polynomials

We obtain a family of refinable functions based on generalized Bernstein polynomials to provide derived properties. The convergence of cascade algorithms associated with the new masks is proved, which guarantees the existence of refinable functions. Then, we analyze the symmetry, regularity, and approximation order of the refinable functions, which are of importance. Tight and sibling frames are constructed and interorthogonality of sibling frames is demonstrated. Finally, we give numerical examples to explicitly illustrate the construction of the proposed approach.

Some smooth compactly supported tight framelets associated to the quincunx matrix

We construct several families of tight wavelet frames in L2(R2) associated to the quincunx matrix. A couple of those families has five generators. Moreover, we construct a family of tight wavelet frames with three generators. Finally, we show families with only two generators. The generators have compact support, any given degree of regularity, and any fixed number of vanishing moments. Our construction is made in Fourier space and involves some refinable functions, the Oblique Extension Principle and a slight generalization of a theorem of Lai and Stöckler. In addition, we will use well known results on construction of tight wavelet frames with two generators on R with the dyadic dilation. The refinable functions we use are constructed from the Daubechies low pass filters and are compactly supported. The main difference between these families is that while the refinable functions associated to the five generators have many symmetries, the refinable functions used in the construction of the others families are merely even.

MULTIRESOLUTION ANALYSES ON QUASILATTICES

We derive relations between geometric means of the Fourier moduli of a refinable distribution and of a related polynomial. We use Pisot-Vijayaraghavan numbers to construct families of one dimension quasilattices and multiresolution analyses spanned by distributions that are refinable with respect to dilation by the PV numbers and translation by quasilattice points. We conjecture that scalar valued refinable distributions are never integrable, construct piecewise constant vector valued refinable functions, and discuss multidimensional extensions.

Extension Principles for Dual Multiwavelet Frames of $$L_2(\mathbb {R}^s)$$ L 2 ( R s ) constructed from Multirefinable Generators

In this work we prove that any pair of homogeneous dual multiwavelet frames of $$L_2(\mathbb {R}^s)$$ constructed from a pair of refinable function vectors gives rise to a pair of nonhomogeneous dual multiwavelet frames and vice versa. We also prove that the Mixed Oblique Extension Principle characterizes dual multiwavelet frames. Our results extend recent characterizations of affine dual frames derived from scalar refinable functions obtained in [3].

Some Smooth Compactly Supported Tight Wavelet Frames with Vanishing Moments

Let $$A \in \mathbb {R}^{d \times d}$$ , $$d \ge 1$$ be a dilation matrix with integer entries and $$| \det A|=2$$ . We construct several families of compactly supported Parseval framelets associated to A having any desired number of vanishing moments. The first family has a single generator and its construction is based on refinable functions associated to Daubechies low pass filters and a theorem of Bownik. For the construction of the second family we adapt methods employed by Chui and He and Petukhov for dyadic dilations to any dilation matrix A. The third family of Parseval framelets has the additional property that we can find members of that family having any desired degree of regularity. The number of generators is $$2^d+d$$ and its construction involves some compactly supported refinable functions, the Oblique Extension Principle and a slight generalization of a theorem of Lai and Stockler. For the particular case $$d=2$$ and based on the previous construction, we present two families of compactly supported Parseval framelets with any desired number of vanishing moments and degree of regularity. None of these framelet families have been obtained by means of tensor products of lower-dimensional functions. One of the families has only two generators, whereas the other family has only three generators. Some of the generators associated with these constructions are even and therefore symmetric. All have even absolute values.

Sparse representation on graphs by tight wavelet frames and applications

In this paper, we introduce a new (constructive) characterization of tight wavelet frames on non-flat domains in both continuum setting, i.e. on manifolds, and discrete setting, i.e. on graphs; discuss how fast tight wavelet frame transforms can be computed and how they can be effectively used to process graph data. We start with defining the quasi-affine systems on a given manifold $\cM$ that is formed by generalized dilations and shifts of a finite collection of wavelet functions $\Psi:=\{\psi_j: 1\le j\le r\}\subset L_2(\R)$. We further require that $\psi_j$ is generated by some refinable function $\phi$ with mask $a_j$. We present the condition needed for the masks $\{a_j: 0\le j\le r\}$ so that the associated quasi-affine system generated by $\Psi$ is a tight frame for $L_2(\cM)$. Then, we discuss how the transition from the continuum (manifolds) to the discrete setting (graphs) can be naturally done. In order for the proposed discrete tight wavelet frame transforms to be useful in applications, we show how the transforms can be computed efficiently and accurately by proposing the fast tight wavelet frame transforms for graph data (WFTG). Finally, we consider two specific applications of the proposed WFTG: graph data denoising and semi-supervised clustering. Utilizing the sparse representation provided by the WFTG, we propose $\ell_1$-norm based optimization models on graphs for denoising and semi-supervised clustering. On one hand, our numerical results show significant advantage of the WFTG over the spectral graph wavelet transform (SGWT) by [1] for both applications. On the other hand, numerical experiments on two real data sets show that the proposed semi-supervised clustering model using the WFTG is overall competitive with the state-of-the-art methods developed in the literature of high-dimensional data classification, and is superior to some of these methods.

Interpolating refinable function vectors with reflection properties

In this paper, we investigate interpolating refinable function vectors whose component functions constitute reflection pairs. A necessary condition for an interpolating refinable function vector to satisfy the reflection property is provided for any dilation factor and multiplicity. Although, the converse is not true in general, for some special cases it becomes a sufficient condition as well. We concentrate on the relation between a dilation factor and multiplicity, and provide the necessary and sufficient condition on the coefficients of its refinement mask in order for such refinable function vector to form a reflection pair. Finally, trivial solutions consisting of translated versions of one component function which is symmetric by itself are discussed. To illustrate the results, various numerical examples are provided.

On the volume of sets bounded by refinable functions

We present a method for the precise determination of the volume of subsets of Rd which are bounded by a hypersurface parametrized by a set of refinable functions. The derivation is based on the linear refinement equations rather than on closed form expressions of these functions, which may not be available. In particular, our approach makes it possible to compute the area of planar domains bounded by subdivision curves or the volume of spatial domains bounded by subdivision surfaces.

Refinable functions, functionals, and iterated function systems

Invariant measures of iterated function systems and refinable functions are mathematical tools of vast usage, that, although usually considered in different contexts, are deeply linked. We outline these links, in particular with respect to existence and regularity of these objects.

Biorthogonal Multiwavelets with Sampling Property and Application in Image Compression

This paper discusses biorthogonal multiwavelets with sampling property. In such systems, vector-valued refinable functions act as the sinc function in the Shannon sampling theorem, and their corresponding matrix-valued masks possess a special structure. In particular, for the multiplicity $$r=2$$r=2, a biorthogonal multifilter bank can be reduced to two scalar-valued filters. Moreover, if the vector-valued scaling functions are interpolating, three different concepts: balancing order, approximation order and analysis-ready order, will be shown to be equivalent. Based on this result, we introduce the transferring armlet order for constructing biorthogonal balanced multiwavelets with sampling property. Also, some balanced biorthogonal multiwavelets will be obtained. Finally, application of biorthogonal interpolating multiwavelets in image compression is discussed. Experiments show that for the same length, the biorthogonal multifilter bank is superior to the orthogonal case. Moreover, certain biorthogonal interpolating multiwavelets are also better than the classical Daubechies wavelets.

Duality for Frames

The subject of this article is the duality principle, which, well beyond its stand at the heart of Gabor analysis, is a universal principle in frame theory that gives insight into many phenomena. Its fiber matrix formulation for Gabor systems is the driving principle behind seemingly different results. We show how the classical duality identities, operator representations and constructions for dual Gabor frames are in fact aspects of the dual Gramian matrix fiberization and its sole duality principle, giving a unified view to all of them. We show that the same duality principle, via dual Gramian matrix analysis, holds for dual (or bi-) systems in abstract Hilbert spaces. The essence of the duality principle is the unitary equivalence of the frame operator and the Gramian of certain adjoint systems. An immediate consequence is, for example, that, even on this level of generality, dual frames are characterized in terms of biorthogonality relations of adjoint systems. We formulate the duality principle for irregular Gabor systems which have no structure whatsoever to the sampling of the shifts and modulations of the generating window. In case the shifts and modulations are sampled from lattices we show how the abstract matrices can be reduced to the simple structured fiber matrices of shift-invariant systems, thus arriving back in the well understood territory. Moreover, in the arena of multiresolution analysis (MRA)-wavelet frames, the mixed unitary extension principle can be viewed as the duality principle in a sequence space. This perspective leads to a construction scheme for dual wavelet frames which is strikingly simple in the sense that it only needs the completion of an invertible constant matrix. Under minimal conditions on the MRA, our construction guarantees the existence and easy constructability of non-separable multivariate dual MRA-wavelet frames. The wavelets have compact support and we show examples for multivariate interpolatory refinable functions. Finally, we generalize the duality principle to the case of transforms that are no longer defined by discrete systems, but may have discrete adjoint systems.

Riesz multiresolution analysis on zero-dimensional groups

We construct wavelet Riesz bases with non- compact support on zero- dimensional groups. For Vilenkin groups, we obtain an algorithm for constructing refinable functions in terms of trees.

Compactly Supported Tensor Product Complex Tight Framelets with Directionality

Although tensor product real-valued wavelets have been successfully applied to many high-dimensional problems, they can capture well only edge singularities along the coordinate axis directions. As an alternative and improvement of tensor product real-valued wavelets and dual tree complex wavelet transform, recently tensor product complex tight framelets with increasing directionality have been introduced in [B. Han, Math. Model. Nat. Phenom., 8 (2013), pp. 18--47] and applied to image denoising in [B. Han and Z. Zhao, SIAM J. Imaging Sci., 7 (2014), pp. 997--1034]. Despite several desirable properties, those directional tensor product complex tight framelets are bandlimited and do not have compact support in the space/time domain. Since compactly supported wavelets and framelets are of great interest and importance in both theory and application, it remains as an unsolved problem whether there exist compactly supported tensor product complex tight framelets with directionality. In this paper, we shall answer this question by proving a theoretical result on directionality of tight framelets and by introducing an algorithm to construct compactly supported complex tight framelets with directionality. Our examples show that compactly supported complex tight framelets with directionality can be easily derived from any given eligible low-pass filters and refinable functions. Several examples of compactly supported tensor product complex tight framelets with directionality have been presented.

Computation of quadrature rules for integration with respect to refinable functions on assigned nodes

Integrals involving refinable functions are of interest in several applications ranging from discretization of PDEs to wavelet analysis. We present a procedure to construct quadrature rules with assigned nodes for these integrals. The process requires in input the refinement mask coefficients and the sequence of nodes only. The corresponding weights are computed by an iterative procedure that does not involve the solution of linear systems. The proposed approach is deeply based on the strong connection between balanced measures and integrals of refinable functions.