Algebraic conditions on non-stationary subdivision symbols for exponential polynomial reproduction

We present an accurate investigation of the algebraic conditions that the symbols of a non-singular, univariate, binary, non-stationary subdivision scheme should fulfill in order to reproduce spaces of exponential polynomials. A subdivision scheme is said to possess the property of reproducing exponential polynomials if, for any initial data uniformly sampled from some exponential polynomial function, the scheme yields the same function in the limit. The importance of this property is due to the fact that several curves obtained by combinations of exponential polynomials (such as conic sections, spirals or special trigonometric and hyperbolic functions) are considered of interest in geometric modeling. Since the space of exponential polynomials trivially includes standard polynomials, this work extends the theory on polynomial reproduction to the non-stationary context. A significant application of the derived algebraic conditions on the subdivision symbols is the construction of new non-stationary subdivision schemes with specific reproduction properties.     

Noninterpolatory Hermite subdivision schemes

Bivariate interpolatory Hermite subdivision schemes have recently been applied to build free-form subdivision surfaces. It is well known to geometric modelling practitioners that interpolatory schemes typically lead to ``unfair" surfaces--surfaces with unwanted wiggles or undulations--and noninterpolatory (a.k.a. approximating in the CAGD community) schemes are much preferred in geometric modelling applications. In this article, we introduce, analyze and construct noninterpolatory Hermite subdivision schemes, a class of vector subdivision schemes which can be applied to iteratively refine Hermite data in a not necessarily interpolatory fashion. We also study symmetry properties of such subdivision schemes which are crucial for application in free-form subdivision surfaces.

A key step in our mathematical analysis of Hermite type subdivision schemes is that we make use of the strong convergence theory of refinement equations to convert a prescribed geometric condition on the subdivision scheme--namely, the subdivision scheme is of Hermite type--to an algebraic condition on the subdivision mask. The latter algebraic condition can then be used in a computational framework to construct specific schemes.

     

CONVERGENCE RATE OF VECTOR SUBDIVISION SCHEME

In this paper,some characterizations on the convergence rate of both the homoge- neous and nonhomogeneous subdivision schemes in Sobolev space are studied and given.     

On multivariate subdivision schemes with nonnegative finite masks

We study the convergence of multivariate subdivision schemes with nonnegative finite masks. Consequently, the convergence problem for the multivariate subdivision schemes with nonnegative finite masks supported on centered zonotopes is solved. Roughly speaking, the subdivision schemes defined by these masks are always convergent, which gives an answer to a question raised by Cavaretta, Dahmen and Micchelli in 1991.

     

Stationary binary subdivision schemes using radial basis function interpolation

A new family of interpolatory stationary subdivision schemes is introduced by using radial basis function interpolation. This work extends earlier studies on interpolatory stationary subdivision schemes in two aspects. First, it provides a wider class of interpolatory schemes; each 2L-point interpolatory scheme has the freedom of choosing a degree (say, m) of polynomial reproducing. Depending on the combination (2L,m), the proposed scheme suggests different subdivision rules. Second, the scheme turns out to be a 2L-point interpolatory scheme with a tension parameter. The conditions for convergence and smoothness are also studied. Dedicated to Prof. Charles A. Micchelli on the occasion of his 60th birthday Mathematics subject classifications (2000) 41A05, 41A25, 41A30, 65D10, 65D17. Byung-Gook Lee: This work was done as a part of Information & Communication fundamental Technology Research Program supported by Ministry of the Information & Communication in Republic of Korea. Jungho Yoon: Corresponding author. Supported by the Korea Science and Engineering Foundation grant (KOSEF R06-2002-012-01001).     

Modified form of binary and ternary 3-point subdivision schemes

Binary 3-point scheme, developed by Hormann and Sabin [Hormann, K. and Sabin, Malcolm A., 2008, A family of subdivision schemes with cubic precision, Computer Aided Geometric Design, 25, 41-52], has been modified by introducing a tension parameter which generates a family of C¹ limiting curves for certain range of tension parameter. Ternary 3-point scheme, introduced by Siddiqi and Rehan [Siddiqi, Shahid S. and Rehan, K., 2009, A ternary three point scheme for curve designing, International Journal of Computer Mathematics, In Press, DOI: 10.1080/00207160802428220], has also been modified by introducing a tension parameter which generates family of C¹ and C² limiting curves for certain range of tension parameter. Laurent polynomial method is used to investigate the continuity of the subdivision schemes. The performance of modified schemes has been demonstrated by considering different examples along with its comparison with the established subdivision schemes.     

On refinable functions and subdivision with positive masks

We consider aspects of the analysis of refinement equations with positive mask coefficients. First we derive, explicitly in terms of the mask, estimates for the geometric convergence rate of both the cascade algorithm and the corresponding subdivision scheme, as well as the Hölder continuity exponent of the resulting refinable function. Moreover, we show that the subdivision scheme converges for a class of unbounded initial sequences. Finally, we present a regularity result containing sufficient conditions on the mask for the refinable function to possess continuous derivatives up to a given order.     

Vector subdivision schemes and multiple wavelets

We consider solutions of a system of refinement equations written in the form

where the vector of functions is in and is a finitely supported sequence of matrices called the refinement mask. Associated with the mask is a linear operator defined on by . This paper is concerned with the convergence of the subdivision scheme associated with , i.e., the convergence of the sequence in the -norm.

Our main result characterizes the convergence of a subdivision scheme associated with the mask in terms of the joint spectral radius of two finite matrices derived from the mask. Along the way, properties of the joint spectral radius and its relation to the subdivision scheme are discussed. In particular, the -convergence of the subdivision scheme is characterized in terms of the spectral radius of the transition operator restricted to a certain invariant subspace. We analyze convergence of the subdivision scheme explicitly for several interesting classes of vector refinement equations.

Finally, the theory of vector subdivision schemes is used to characterize orthonormality of multiple refinable functions. This leads us to construct a class of continuous orthogonal double wavelets with symmetry.

     

On the matching polynomial of subdivision graphs

Let G be a simple graph and let S(G) be the subdivision graph of G, which is obtained from G by replacing each edge of G by a path of length two. In this paper, by the Principle of Inclusion and Exclusion we express the matching polynomial and Hosoya index of S(G) in terms of the matchings of G. Particularly, if G is a regular graph or a semi-regular bipartite graph, then the closed formulae of the matching polynomial and Hosoya index of S(G) are obtained. As an application, we prove a combinatorial identity.     

On spherical designs obtained from Q‐polynomial association schemes

We characterize that the image of the embedding of the Q ‐polynomial association scheme into the first eigenspace by primitive idempotent E ₁ is a spherical t‐design in terms of the Krein numbers. Furthermore, we show that the strengths of P‐ and Q‐polynomial schemes as spherical designs are bounded by a constant. Copyright © 2011 John Wiley & Sons, Ltd. 19:167‐177, 2011     

The independence of initial vectors in the subdivision schemes

Starting with an initial vector λ = (λ(κ))κ∈z ∈ ep(Z), the subdivision scheme generates asequence (Snaλ)∞n=1 of vectors by the subdivision operator Saλ(κ) = ∑λ(j)a(k - 2j), k ∈ Z. j∈zSubdivision schemes play an important role in computer graphics and wavelet analysis. It is very interesting tounderstand under what conditions the sequence (Snaλ)∞n=1 converges to an Lp-function in an appropriate sense.This problem has been studied extensively. In this paper we show that the subdivision scheme converges forany initial vector in ep(Z) provided that it does for one nonzero vector in that space. Moreover, if the integertranslates of the refinable function are stable, the smoothness of the limit function corresponding to the vectorλ is also independent of λ.     

Characterization of some convergent bivariate subdivision schemes with nonnegative masks

Knowing that the convergence of a multivariate subdivision scheme with a nonnegative mask can be characterized by whether or not some finite products of row-stochastic matrices induced by this mask have a positive column. However, the number of those products is exponential with respect to the size of matrices. For nonnegative univariate subdivision, this problem is completely solved. Thus, the convergence in this case can be checked in linear time with respect to the size of a square matrix. This paper will demonstrate the necessary and sufficient conditions for the convergence of some nonnegative bivariate subdivision schemes by means of the so-called connectivity of a square matrix, which is derived by a given mask. Moreover, the connectivity can be examined in linear time with respect to the size of this matrix.     

A unified three point approximating subdivision scheme

In this paper,we propose a three point approximating subdivision scheme,with three shape parameters,that unifies three different existing three point approximating schemes.Some sufficient conditions for subdivision curve C0 to C3 continuity and convergence of the scheme for generating tensor product surfaces for certain ranges of parameters by using Laurent polynomial method are discussed.The systems of curve and surface design based on our scheme have been developed successfully in garment CAD especially for clothes modelling.     

Some sufficient conditions for convergent multivariate subdivision schemes with nonnegative finite masks

It is well known that the convergence of multivariate subdivision schemes with finite masks can be characterized via joint spectral radius. For nonnegative masks, we will present in this paper some computable simply sufficient conditions for the convergence, which will cover a substantially large class of schemes.     

On the regularity analysis of interpolatory Hermite subdivision schemes

It is well known that the critical Hölder regularity of a subdivision schemes can typically be expressed in terms of the joint-spectral radius (JSR) of two operators restricted to a common finite-dimensional invariant subspace. In this article, we investigate interpolatory Hermite subdivision schemes in dimension one and specifically those with optimal accuracy orders. The latter include as special cases the well-known Lagrange interpolatory subdivision schemes by Deslauriers and Dubuc. We first show how to express the critical Hölder regularity of such a scheme in terms of the joint-spectral radius of a matrix pair {F₀,F₁} given in a very explicit form. While the so-called finiteness conjecture for JSR is known to be not true in general, we conjecture that for such matrix pairs arising from Hermite interpolatory schemes of optimal accuracy orders a “strong finiteness conjecture” holds: ρ(F₀,F₁)=ρ(F₀)=ρ(F₁). We prove that this conjecture is a consequence of another conjectured property of Hermite interpolatory schemes which, in turn, is connected to a kind of positivity property of matrix polynomials. We also prove these conjectures in certain new cases using both time and frequency domain arguments; our study here strongly suggests the existence of a notion of “positive definiteness” for non-Hermitian matrices.     

Approximating exponential and logarithmic functions using polynomial interpolation

This article takes a closer look at the problem of approximating the exponential and logarithmic functions using polynomials. Either as an alternative to or a precursor to Taylor polynomial approximations at the precalculus level, interpolating polynomials are considered. A measure of error is given and the behaviour of the error function is analysed. The results of interpolating polynomials are compared with those of Taylor polynomials.     

Full rank positive matrix symbols: interpolation and orthogonality

We investigate full rank interpolatory vector subdivision schemes whose masks are positive definite on the unit circle except the point z=1. Such masks are known to give rise to convergent schemes with a cardinal limit function in the scalar case. In the full rank vector case, we show that there also exists a cardinal refinable function based on this mask, however, with respect to a different notion of refinability which nevertheless also leads to an iterative scheme for the computation of vector fields. Moreover, we show the existence of orthogonal scaling functions for multichannel wavelets and give a constructive method to obtain these scaling functions. AMS subject classification (2000) 42C40, 65T60, 65D05     

Algebraic conditions on non-stationary subdivision symbols for exponential polynomial reproduction

We present an accurate investigation of the algebraic conditions that the symbols of a non-singular, univariate, binary, non-stationary subdivision scheme should fulfill in order to reproduce spaces of exponential polynomials. A subdivision scheme is said to possess the property of reproducing exponential polynomials if, for any initial data uniformly sampled from some exponential polynomial function, the scheme yields the same function in the limit. The importance of this property is due to the fact that several curves obtained by combinations of exponential polynomials (such as conic sections, spirals or special trigonometric and hyperbolic functions) are considered of interest in geometric modeling. Since the space of exponential polynomials trivially includes standard polynomials, this work extends the theory on polynomial reproduction to the non-stationary context. A significant application of the derived algebraic conditions on the subdivision symbols is the construction of new non-stationary subdivision schemes with specific reproduction properties.     

A piecewise polynomial approach to analyzing interpolatory subdivision

The four-point interpolatory subdivision scheme of Dubuc and its generalizations to irregularly spaced data studied by Warren and by Daubechies, Guskov, and Sweldens are based on fitting cubic polynomials locally. In this paper, we analyze the convergence of the scheme by viewing the limit function as the limit of piecewise cubic functions arising from the scheme. This allows us to recover the regularity results of Daubechies et al. in a simpler way and to obtain the approximation order of the scheme and its first derivative.     

Analysis of subdivision schemes for nets of functions by proximity and controllability

In this paper we develop tools for the analysis of net subdivision schemes, schemes which recursively refine nets of bivariate continuous functions defined on grids of lines, and generate denser and denser nets. Sufficient conditions for the convergence of such a sequence of refined nets, and for the smoothness of the limit function, are derived in terms of proximity to a bivariate linear subdivision scheme refining points, under conditions controlling some aspects of the univariate functions of the generated nets. Approximation orders of net subdivision schemes, which are in proximity with positive schemes refining points are also derived. The paper concludes with the construction of a family of blending spline-type net subdivision schemes, and with their analysis by the tools presented in the paper. This family is a new example of net subdivision schemes generating C¹ limits with approximation order 2.