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Standard interpolatory subdivision schemes and their underlying interpolating refinable functions are of interest in CAGD, numerical PDEs, and approximation theory. Generalizing these notions, we introduce and study n s -step interpolatory M -subdivision schemes and their interpolating M -refinable functions with n s ∈ N ∪ { ∞ } and a dilation factor M ∈ N \ { 1 } . We completely characterize C m -convergence and smoothness of n s -step interpolatory subdivision schemes and their interpolating M -refinable functions in terms of their masks. Inspired by n s -step interpolatory stationary subdivision schemes, we further introduce the notion of r-mask quasi-stationary subdivision schemes, and then we characterize their C m -convergence and smoothness properties using only their masks. Moreover, combining n s -step interpolatory subdivision schemes with r-mask quasi-stationary subdivision schemes, we can obtain r n s -step interpolatory subdivision schemes. Examples and construction procedures of convergent n s -step interpolatory M -subdivision schemes are provided to illustrate our results with dilation factors M = 2 , 3 , 4 . In addition, for the dyadic dilation M = 2 and r = 2 , 3 , using r masks with only two-ring stencils, we provide examples of C r -convergent r-step interpolatory r-mask quasi-stationary dyadic subdivision schemes. [ABSTRACT FROM AUTHOR]

When constructing multivariate framelets, it is often unavoidable to work with matrices of multivariate trigonometric polynomials and complicated matrix decomposition problems. These problems become even harder when good properties such as high-order vanishing moments are required on the framelets. In this paper, we establish a new method for constructing multivariate dual framelets with high-order vanishing moments. The underlying scheme of our algorithm is the famous Mixed Extension Principle that allows us to derive the high-pass filters (or framelet generators) from a given pair of refinable filters with high-order linear-phase moments. Our method only involves two steps: (1) directly constructing the first few pairs of high-pass filters by using the linear-phase moment conditions of the refinement filters; (2) solving a system of linear equations to obtain the rest of the high-pass filters. Both are easy to implement for scientific computation, regardless of what dimension or dilation matrix we work with. Apart from high-order vanishing moments, we will see that if the refinement filters take coefficients from some subfield of ℂ that is closed under complex conjugation, so do the high-pass filters. Furthermore, our algorithm gives the upper bounds for the number of high-pass filters in arbitrary dimensions. At the end of the paper, we will give several illustrative examples, from which we can also see that the support sizes of the high-pass filters are comparable with those of the refinement filters. [ABSTRACT FROM AUTHOR]

In this article, we obtain methods for constructing step tight frames on an arbitrary locally compact zero-dimensional group. To do this, we use the principle of unitary extension. First, we indicate a method for constructing a step scaling function on an arbitrary zero-dimensional group. To construct the scaling function, we use an oriented tree and specify the conditions on the tree under which the tree generates the mask $m_0$ of a scaling function. Then we find conditions on the masks $m_0, m_1,\ldots , m_q$ under which the corresponding wavelet functions $\psi_1, \psi_2,\ldots ,\psi_q$ generate a tight frame. Using these conditions, we indicate a way of constructing such masks. In conclusion, we give examples of the construction of tight frames.

The construction of wavelet bases and frames on the Vilenkin group Gp is studied. Wavelet systems consisting of functions that are compactly supported and band-limited at the same time are of our interest. A complete description of all refinable functions providing such systems is given. [ABSTRACT FROM AUTHOR]

In recent years, we found that some multiscale methods applied to fractional differential problems, are easy and efficient to implement, when we use some fractional refinable functions introduced in the literature. In fact, these functions not only generate a multiresolution on R , but also have fractional (non-integer) derivative satisfying a very convenient recursive relation. For this reason, in this paper, we describe this class of refinable functions and focus our attention on their approximating properties. [ABSTRACT FROM AUTHOR]

In recent years, we found that some multiscale methods applied to fractional differential problems, are easy and efficient to implement, when we use some fractional refinable functions introduced in the literature. In fact, these functions not only generate a multiresolution on R, but also have fractional (non-integer) derivative satisfying a very convenient recursive relation. For this reason, in this paper, we describe this class of refinable functions and focus our attention on their approximating properties.

In this paper we characterize all subspaces of analytic functions in finitely generated shift-invariant spaces with compactly supported generators and provide explicit descriptions of their elements. We illustrate the differences between our characterizations and Strang-Fix or zero conditions on several examples. Consequently, we depict the analytic functions generated by scalar or vector subdivision with masks of bounded and unbounded support. In particular, we prove that exponential polynomials are indeed the only analytic limits of level dependent scalar subdivision schemes with finitely supported masks. [ABSTRACT FROM AUTHOR]

A sequence of continuous real functions (f m) m ∈ N 0 , with exponential decay at infinity that implies their Fourier-Laplace transforms F m (i z) are analytic on | ℜ (z) | < B , z ∈ C , generates a sequence of polynomials (Q k) k ∈ N 0 , that are biorthogonal to the distributional derivatives μ m : = (− 1) m f m (m) , m ∈ N 0. The generating function is a generalized Taylor series expansion of the Fourier-Laplace kernel e x z in terms of μ ˆ m (i z) = z m F m (i z) , m ∈ N 0. If μ m = ω P m , where P m are polynomials of degree m and ω is a weight function, Q m is a constant multiple of P m and they are orthogonal polynomials. In the case where f m are uniform or quasi-uniform B -splines, the corresponding biorthogonal polynomials exhibit properties reminiscent of Appell sequences, while in the case where f m are refinable functions they are eigenfunctions of the adjoint of the corresponding refinement operators. For nonuniform B -splines, a suitable choice of the knot sequence defines (f m) m ∈ N 0 in which the corresponding polynomials Q k reduce to the Jacobi polynomials and the biorthogonal spline functions μ m reduce to the corresponding weighted Jacobi polynomials, if there are no interior knots. [ABSTRACT FROM AUTHOR]

In this paper, we propose a new method for the construction of multi-dimensional, wavelet-like families of affine frames, commonly referred to as framelets, with specific directional characteristics, small and compact support in space, directional vanishing moments (DVM), and axial symmetries or anti-symmetries. The framelets we construct arise from readily available refinable functions. The filters defining these framelets have few non-zero coefficients, custom-selected orientations and can act as finite-difference operators. [ABSTRACT FROM AUTHOR]

A rigorous convergence analysis is presented for arbitrary order cardinal spline subdivision with general integer arity, for which the binary case, with arity two, is a well-studied subject. Explicit geometric convergence rates are derived, and particular attention is devoted to the rendering of cardinal spline graphs and parametric curves.

This paper presents a detailed regularity analysis of multivariate refinable functions with general dilation matrices, with emphasis on the anisotropic case. In the univariate setting, the smoothness of refinable functions is well understood by means of the matrix approach. In the multivariate setting, this approach has been extended only to the special case of isotropic refinement with the dilation matrix all of whose eigenvalues are equal in the absolute value. The general anisotropic case has resisted to be fully understood: the matrix approach can determine whether a refinable function belongs to C (R s) or L p (R s) , 1 ≤ p < ∞ , but its Hölder regularity remained mysteriously unattainable. We show how to compute the Hölder regularity in C (R s) or L p (R s) , 1 ≤ p < ∞. In the anisotropic case, our expression for the exact Hölder exponent of a refinable function reflects the impact of the variable moduli of the eigenvalues of the corresponding dilation matrix. In the isotropic case, our results reduce to the well-known facts from the literature. We also analyze the higher regularity of anisotropic refinable functions. We illustrate our results with several examples. [ABSTRACT FROM AUTHOR]

Abstract A directional compactly supported d -dimensional Haar tight framelet is constructed such that all its high-pass filters in its underlying tight framelet filter bank have only two nonzero coefficients with opposite signs and they exhibit totally (3 d − 1) ∕ 2 directions in dimension d. Furthermore, applying the projection method to such a tight framelet, a directional compactly supported box spline tight framelet with simple geometric structure is built such that all the high-pass filters in its underlying tight framelet filter bank have only two nonzero coefficients with opposite signs as well. Moreover, such compactly supported box spline tight framelets can achieve arbitrarily high numbers of directions by using refinable box splines with increasing supports. Their application to pMRI with good performance is presented. [ABSTRACT FROM AUTHOR]

Abstract This paper is concerned with general symmetric 2 n -point interpolatory subdivision scheme (ISS) with polynomial reproduction of arbitrary order m ≤ n. An explicit formulation is derived for the corresponding refinement symbol, and convergence intervals are obtained for the one-parameter case, the left hand endpoints of which improve on previous such lower parameter convergence bounds. [ABSTRACT FROM AUTHOR]

Abstract Mixed Oblique Extension Principles (MOEP) provide an important method to construct affine dual frames from refinable functions. This paper addresses MOEP under the setting of reducing subspaces of L 2 (R d). We obtain an MOEP for (non)homogeneous affine dual frames and (non)homogeneous affine Parseval frames. [ABSTRACT FROM AUTHOR]

Abstract 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 ( R ) $L_{2}(\mathbb{R})$ . Stability of the subdivision schemes, regularity and approximation orders are obtained. We also illustrate the symmetry of the corresponding refinable functions.

We consider piecewise Chebyshevian splines, in the sense of splines with pieces taken from any different five-dimensional Extended Chebyshev spaces, and with connection matrices at the knots. In this large context we establish necessary and sufficient conditions for the existence of totally positive refinable B-spline bases. These conditions are applied in many important special cases, e.g. symmetric cardinal geometrically continuous quartic B-spline, parametrically continuous mixed L-splines. The great variety of illustrations provided proves the richness of this class of splines for design. This richness can be exploited in various other fields as well. [ABSTRACT FROM AUTHOR]

We construct a simple volume-preserving map from the cube [0,b]3 to the tetrahedron {(x,y,z)∈R3, x≥0, y≥0, z≥0, x+y+z≤a}, with a=b63. This map will allow us to construct equal-volume subdivisions of arbitrary tetrahedrons and arbitrary convex polyhedrons into polyhedral cells. Moreover, mapping the regular octahedron onto the ball using a volume-preserving map previously constructed by the authors, one can obtain uniform and refinable grids on the 3D ball by a simple procedure, starting from appropriate grids on the cube. [ABSTRACT FROM AUTHOR]

To be directly useful both for shape design and a thin shell analysis, a surface representation has to satisfy three properties: (1) be compatible with CAD surface representations, (2) yield generically a good highlight distribution, and (3) offer a refinable space of functions on the surface. Here we propose a new construction, based on a number of recently-developed techniques, that satisfies all three criteria. The construction converts quad meshes with irregularities, where more or fewer than four quads meet, to C 1 (or, at the cost of more pieces, C 2 ) bi-4 surface consisting of subdivision rings for the main body completed by a tiny G 1 cap. [ABSTRACT FROM AUTHOR]

For refinable function-based affine bi-frames, nonhomogeneous ones admit fast algorithms and have extension principles as homogeneous ones. But all extension principles are based on some restrictions on refinable functions. So it is natural to ask what are expected from general refinable functions. In this paper, we introduce the notion of weak nonhomogeneous affine bi-frame (WNABF). Under the setting of reducing subspaces of L (R), we characterize WNABFs and obtain a mixed oblique extension principle for WNABFs based on general refinable functions. [ABSTRACT FROM AUTHOR]

In this paper, we generalize the family of Deslauriers-Dubuc's interpolatory masks from dimension one to arbitrary dimensions with respect to the quincunx dilation matrices, thereby providing a family of quincunx fundamental refinable functions in arbitrary dimensions. We show that a family of unique quincunx interpolatory masks exists and such a family of masks is of real value and has the full-axis symmetry property. In dimension d = 2, we give the explicit form of such unique quincunx interpolatory masks, which implies the nonnegativity property of such a family of masks. [ABSTRACT FROM AUTHOR]

Let be a dilation matrix, an expansive matrix that maps into itself. Let be a finite subset of and for let be complex matrices. The refinement equation corresponding to and is A solution if one exists, is called a refinable vector function or a vector scaling function of multiplicity This paper characterizes the higher-order smoothness of compactly supported solutions of the refinement equation, in terms of the -norm joint spectral radius of a finite set of finite matrices determined by the coefficients [ABSTRACT FROM AUTHOR]

To date, singularly-parameterized surface constructions suffer from poor highlight line distributions, ruling them out as a surface representation of choice for primary design surfaces. This paper explores graded, many-piece, everywhere C 1 singularly-parameterized surface caps that mimic the shape of a high-quality guide surface. The approach illustrates the trade-off between polynomial degree and surface quality. For bi-degree 5, minor flaws in the highlight line distribution are still visible when zooming in on the singularity, but the distribution is good at the macroscopic level. Constructions of degree bi-4 or bi-3 may require one or more steps of guided subdivision to reach the same macroscopic quality. Akin to subdivision surfaces, singularly-parameterized functions on the surfaces are straightforward to refine. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

As 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 spaceswiths≠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. [ABSTRACT FROM AUTHOR]

We study the approximation properties of the class of nonstationary refinable ripplets introduced in Gori and Pitolli (2008). These functions are the solution of an infinite set of nonstationary refinable equations and are defined through sequences of scaling masks that have an explicit expression. Moreover, they are variation-diminishing and highly localized in the scale-time plane, properties that make them particularly attractive in applications. Here, we prove that they enjoy Strang-Fix conditions, convolution and differentiation rules and that they are bell-shaped. Then, we construct the corresponding minimally supported nonstationary prewavelets and give an iterative algorithm to evaluate the prewavelet masks. Finally, we give a procedure to construct the associated nonstationary biorthogonal bases and filters to be used in efficient decomposition and reconstruction algorithms. As an example, we calculate the prewavelet masks and the nonstationary biorthogonal filter pairs corresponding to the C nonstationary scaling functions in the class and construct the corresponding prewavelets and biorthogonal bases. A simple test showing their good performances in the analysis of a spike-like signal is also presented. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new family of nonstationary refinable functions from Generalized Bernstein Polynomials, which include a class of nonstationary refinable functions generated from the family of masks for the pseudo splines of type II (see [17]). Furthermore, a proof of the convergence of nonstationary cascade algorithms associated with the new masks is completed. We then construct symmetric compacted supported nonstationary C∞ tight wavelet frames in L2(R) with the spectral frame approximation order. [ABSTRACT FROM AUTHOR]

For any symmetry group , any appropriate matrix dilation (compatible with ) and any appropriate symmetry center we give an explicit method for the construction of -symmetric with respect to the center refinable masks which have sum rule of an arbitrary order . Moreover, we give a description of all these masks. For any symmetry group , any appropriate matrix dilation (compatible with ) and any appropriate row of symmetry centers we give two explicit methods for the construction of -symmetric with respect to the row of centers refinable matrix masks which have sum rule of an arbitrary order . A description of all such matrix masks is also presented. [ABSTRACT FROM AUTHOR]

Stability is an expected property for refinable vectors, which is widely considered in the study of refinement equations. There are two types of stability for refinable vectors. One isthe ordinary-stability, another isthe vector-stability.The ordinary-stabilityconsiders the stability of entries of refinable vectors, butthe vector-stabilityconsiders the stability of refinable vectors when they are considered as elements of super-Hilbert spaces. In this article, we give a necessary and sufficient condition for refinable vectors to be vector-stable. Our results improve on some known ones. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

A constructive existence result for a class of polynomial identities is established and applied in two related contexts. First, an algorithm is developed for the explicit construction of a sequence of local interpolation operators mapping the space of continuous functions on the real line into the nested sequence of refinement spaces generated by the shifts of a given refinable function, and where optimal polynomial exactness, as governed by the order of the sum-rule condition satisfied by the corresponding refinement sequence, is achieved. The above algorithm requires as input only the values at the integers of the refinable function, and we proceed, secondly, to derive sufficient conditions for the existence of a refinable function with prescribed values at the integers. As our main examples, we consider the cardinal B-spline case, as well as refinable functions with normalized binomial coefficient values at the integers. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We present a method for the precise determination of the volume of subsets of R d 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. [ABSTRACT FROM AUTHOR]

In this paper we study the problem of constructing refinable orthonormal cardinal functions from Blaschke products. The digital filter perspective of our construction corresponds to what is called an infinite impulse response (IIR) filter. We show how to construct, at least numerically, stable filters of this type in contrast to the Butterworth filter which is the maximally flat filter in our class.

Refinable functions play a main role in the construction of wavelets; here we present a large class of refinable functions which can be identified through the explicit expression of their masks. These refinable functions can be considered as a generalization of cardinal B-splines, from which they borrow a few nice properties, like symmetry and total positivity, that are of relevant interest in most filtering applications. Moreover, each of the refinable functions here considered generates a multiresolution analysis. The associated wavelet bases and the corresponding biorthogonal bases are constructed, too.

A multi-level spline technique known as partial shape preserving splines (PSPS) (Li and Tian, 2011) has recently been developed for the design of piecewise polynomial freeform geometric surfaces, where the basis functions of the PSPS can be directly built from an arbitrary set of polygons that partitions a giving parametric domain. This paper addresses a special type of PSPS, the triangular PSPS (Tri-PSPS), where all spline basis functions are constructed from a set of triangles. Compared with other triangular spline techniques, Tri-PSPS have several distinctive features. Firstly, for each given triangle, the corresponding spline basis function for any required degree of smoothness can be expressed in closed-form and directly written out in full explicitly as piecewise bivariate polynomials. Secondly, Tri-PSPS are an additive triangular spline technique, where the spline function built from a given triangle can be replaced with a set of refined spline functions built on a set of smaller triangles that partition the initial given triangle. In addition, Tri-PSPS are a multilevel spline technique, Tri-PSPS surfaces can be designed to have a continuously varying levels of detail, achieved simply by specifying a proper value for the smoothing parameter introduced in the spline functions. In terms of practical implementation, Tri-PSPS are a parallel computing friendly spline scheme, which can be easily implemented on modern programmable GPUs or on high performance computer clusters, since each of the basis functions of Tri-PSPS can be directly computed independent of each other in parallel. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

This paper concerns the construction of quadrature rules based on the use of suitable refinable quasi-interpolatory operators introduced here. Convergence analysis of the obtained quadrature rules is developed and numerical examples are included. [ABSTRACT FROM AUTHOR]

We analyse the properties of a particular class of symmetric, compactly supported, totally positive refinable functions. We show that these refinable functions can be used in the generalized cardinal interpolation problem for which there exists a unique exceptional value which can be evaluated exactly. Some numerical examples concerning interpolation and construction of semi-orthogonal wavelets by means of these refinable functions are displayed.

In this paper, a class of refinable functions is given by smoothening pseudo-splines in order to get divergence free and curl free wavelets. The regularity and stability of them are discussed. Based on that, the corresponding Riesz wavelets are constructed. [ABSTRACT FROM AUTHOR]

The research about two-direction refinable functions and two-direction wavelets with dilation factor 2 and their properties are extended to dilation factor m. The condition that the two-direction refinable functions and its corresponding two-direction wavelet are symmetric and antisymmetric is given. Finally, an algorithm for constructing the orthonormal two-direction wavelets with orthonormal scaling functions and the corresponding orthonrmal wavelets is presented and an example is given. [ABSTRACT FROM AUTHOR]