Convergence Of Subdivision Schemes Research Articles

The Family of Multiparameter Quaternary Subdivision Schemes

In the field of subdivision, the smoothness increases as the arity of schemes increases. The family of high arity schemes gives high smoothness comparative to low arity schemes. In this paper, we propose a simple and generalized formula for a family of multiparameter quaternary subdivision schemes. The conditions for convergence of subdivision schemes are also presented. Moreover, we derive subdivision schemes after substituting the different values of parameters. We also analyzed the important properties of the proposed family of subdivision schemes. After comparison with existing schemes, we analyze that the proposed family of subdivision schemes gives better smoothness and approximation compared with the existing subdivision schemes.

Convergence of subdivision schemes on Riemannian manifolds with nonpositive sectional curvature

This paper studies well-definedness and convergence of subdivision schemes which operate on Riemannian manifolds with nonpositive sectional curvature. These schemes are constructed from linear ones by replacing affine averages by the Riemannian centre of mass. In contrast to previous work, we consider schemes without any sign restriction on the mask, and our results apply to all input data. We also analyse the Hölder continuity of the resulting limit curves. Our main result states that if the norm of the derived scheme (resp. iterated derived scheme) is smaller than the corresponding dilation factor then the adapted scheme converges. In this way, we establish that convergence of a linear subdivision scheme is almost equivalent to convergence of its nonlinear manifold counterpart.

Convergence and smoothness of tensor-product of two non-uniform linear subdivision schemes

The aim of this short note is to provide a rigorous proof that the tensor product of two non-uniform linear convergent subdivision schemes converges and has the same regularity as the minimal regularity of the univariate schemes. It extends results that are known for the uniform linear case and are based on symbols, a notion which is no longer available in the non-uniform setting.

3n-Point Quaternary Shape Preserving Subdivision Schemes

In this paper, an algorithm is defined to construct 3n-point quaternary approximating subdivision schemes which are useful to design different geometric objects in the field of geometric modeling. We are going to establish a family of approximating schemes because approximating scheme provide maximum smoothness as compare to the interpolating schemes. It is to be observed that the proposed schemes satisfying the basic sum rules with bell-shaped mask go up to the convergent subdivision schemes which preserve monotonicity. We analyze the shape-preserving properties such that convexity and concavity of proposed schemes. We also show that quaternary schemes associated to the certain refinable functions with dilation 4 have higher order shape preserving properties. We also calculated the polynomial reproduction of proposed quaternary approximating subdivision schemes. The proposed schemes have tension parameter, so by choosing different values of the tension parameter we can get different limit curves of initial control polygon. We show in the table form that the proposed schemes are better than the existing schemes by comparing them on the behalf of their support and continuity. The visual quality of proposed schemes is demonstrated by different snapshots.

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.

Quasi-interpolation operators on hexagonal grids with high approximation orders in spaces of polyharmonic splines

In this paper we discuss the problem of approximating data given on hexagonal lattices. The construction of continuous representation from sampled data is an essential element of many applications as, for instance, image resampling, numerical solution of PDE boundary problems, etc. A useful tool is quasi-interpolation that doesn’t need the solution of a linear system. It is then important to have quasi-interpolation operators with high approximation orders, and capable to provide an efficient computation of the quasi interpolant function. To this end, we show that the idea proposed by Bozzini et al. [1] for the construction of quasi-interpolation operators in spaces of m-harmonic splines with knots in Z2 which reproduce polynomials of high degree, can be generalized to any spaces of m-harmonic splines with knots on a lattice Γ of R2 and in particular on hexagonal grids. Then by a simple procedure which starts from a generator ϕ0Γ with corresponding quasi-interpolation operator reproducing only linear polynomials, it is possible to define recursively generators ϕ1Γ,⋯,ϕm−1Γ with corresponding quasi-interpolation operators reproducing polynomials up to degree 3,5,⋯,2m−1. We show that this new generators of quasi-interpolation operators on a general lattice are positive definite functions, and are scaling functions whenever ϕ0Γ has those properties. Moreover we are able to associate with ϕjΓ a dyadic convergent subdivision scheme that allows a fast computation of the quasi-interpolant.

Properties of generators of quasi-interpolation operators of high approximation orders in spaces of polyharmonic splines

We have presented in Bozzini et al. (2011) a procedure in spaces of m-harmonic splines in Rd that starts from a simple generator ϕ0 and recursively defines generators ϕ1,ϕ2,…,ϕm−1 with corresponding quasi-interpolation operators reproducing polynomials of degrees 3, 5,…,2m−1 respectively. In this paper we study the properties of generators ϕj, and we prove that these new generators are positive definite functions, and are scaling functions whenever ϕ0 has those properties. Moreover ϕ0 and ϕj generate the same multiresolution analysis. We show that it is possible to define a convergent subdivision scheme, and to provide in this way a fast computation of the quasi-interpolant.

Polynomial reproduction for univariate subdivision schemes of any arity

In this paper, we study the ability of convergent subdivision schemes to reproduce polynomials in the sense that for initial data, which is sampled from some polynomial function, the scheme yields the same polynomial in the limit. This property is desirable because the reproduction of polynomials up to some degree d implies that a scheme has approximation order d + 1 . We first show that any convergent, linear, uniform, and stationary subdivision scheme reproduces linear functions with respect to an appropriately chosen parameterization. We then present a simple algebraic condition for polynomial reproduction of higher order. All results are given for subdivision schemes of any arity m ≥ 2 and we use them to derive a unified definition of general m -ary pseudo-splines. Our framework also covers non-symmetric schemes and we give an example where the smoothness of the limit functions can be increased by giving up symmetry.

Convergence analysis of [formula omitted] Hermite interpolatory subdivision schemes by explicit joint spectral radius formulas

Subdivision schemes are popular iterative processes to build graphs of functions, curves and surfaces. We analyze the 2-point Hermite C 2 subdivision scheme introduced by Merrien in [26]. For the analysis of its convergence and its smoothness properties we are concerned with the computation of the joint spectral radius of a family of 2 matrices associated with the scheme. In this paper, by an explicit computation of the joint spectral radius of such pairs of matrices, we determine necessary and sufficient conditions for the scheme to be C 2 convergent, whenever it reproduces cubic polynomials. In addition, we present two one-parameter families of convergent subdivision schemes belonging to the class in [26] possessing interesting properties from the shape control point of view.

Analysis of a class of nonlinear subdivision schemes and associated multiresolution transforms

This paper is devoted to the convergence and stability analysis of a class of nonlinear subdivision schemes and associated multiresolution transforms. As soon as a nonlinear scheme can be written as a specific perturbation of a linear and convergent subdivision scheme, we show that if some contractivity properties are satisfied, then stability and convergence can be achieved. This approach is applied to various schemes, which give different new results. More precisely, we study uncentered Lagrange interpolatory linear schemes, WENO scheme (Liu et al., J Comput Phys 115:200---212, 1994), PPH and Power-P schemes (Amat and Liandrat, Appl Comput Harmon Anal 18(2):198---206, 2005; Serna and Marquina, J Comput Phys 194:632---658, 2004) and a nonlinear scheme using local spherical coordinates (Aspert et al., Comput Aided Geom Des 20:165---187, 2003). Finally, a stability proof is given for the multiresolution transform associated to a nonlinear scheme of Marinov et al. (2005).

Convergence rates of vector cascade algorithms in [formula omitted

We investigate the solutions of vector refinement equations of the form ϕ = ∑ α ∈ Z s a ( α ) ϕ ( M · - α ) , where the vector of functions ϕ = ( ϕ 1 , … , ϕ r ) T is in ( L p ( R s ) ) r , 1 ⩽ p ⩽ ∞ , a ≕ ( a ( α ) ) α ∈ Z s is a finitely supported sequence of r × r matrices called the refinement mask, and M is an s × s integer matrix such that lim n → ∞ M - n = 0 . Associated with the mask a and M is a linear operator Q a defined on ( L p ( R s ) ) r by Q a ψ ≔ ∑ β ∈ Z s a ( β ) ψ ( M · - β ) . The iteration scheme ( Q a n ψ ) n = 1 , 2 , … is called a cascade algorithm (see [D.R. Chen, R.Q. Jia, S.D. Riemenschneider, Convergence of vector subdivision schemes in Sobolev spaces, Appl. Comput. Harmon. Anal. 12 (2002) 128–149; B. Han, The initial functions in a cascade algorithm, in: D.X. Zhou (Ed.), Proceeding of International Conference of Computational Harmonic Analysis in Hong Kong, 2002; B. Han, R.Q. Jia, Multivariate refinement equations and convergence of subdivision schemes, SIAM J. Math. Anal. 29 (1998) 1177–1199; R.Q. Jia, Subdivision schemes in L p spaces, Adv. Comput. Math. 3 (1995) 309–341; R.Q. Jia, S.D. Riemenschneider, D.X. Zhou, Vector subdivision schemes and multiple wavelets, Math. Comp. 67 (1998) 1533–1363; S. Li, Characterization of smoothness of multivariate refinable functions and convergence of cascade algorithms associated with nonhomogeneous refinement equations, Adv. Comput. Math. 20 (2004) 311–331; Q. Sun, Convergence and boundedness of cascade algorithm in Besov space and Triebel–Lizorkin space I, Adv. Math. (China) 29 (2000) 507–526]). Cascade algorithm is an important issue to wavelets analysis and computer graphics. Main results of this paper are related to the convergence and convergence rates of vector cascade algorithm in ( L p ( R s ) ) r ( 1 ⩽ p ⩽ ∞ ) . We give some characterizations on convergence of cascade algorithm and also give estimates on convergence rates of this cascade algorithm with M being isotropic dilation matrix. It is well known that smoothness is a very important property of a multiple refinable function. A characterization of L p ( 1 ⩽ p ⩽ ∞ ) smoothness of multiple refinable functions is also presented when M = qI s × s , where I s × s is the s × s identity matrix, and q ⩾ 2 is an integer. In particular, the smoothness results given in [R.Q. Jia, S.D. Riemenschneider, D.X. Zhou, Smoothness of multiple refinable functions and multiple wavelets, SIAM J. Matrix Anal. Appl. 21 (1999) 1–28] is a special case of this paper.

Convergence of subdivision schemes and smoothness of limit functions

Starting with vector λ = ( λ ( k ) ) k ∈ Z ∈ ℓ p ( Z ) , the subdivision scheme generates a sequence { S a n λ } n = 1 ∞ of vectors by the subdivision operator S a λ ( k ) = ∑ j ∈ Z λ ( j ) a ( k − 2 j ) , k ∈ Z . Subdivision schemes play an important role in computer graphics and wavelet analysis. It is very interesting to understand under what conditions the sequence { S a n λ } n = 1 ∞ converges to a L p -function in an appropriate sense. This problem has been studied extensively. In this paper, we consider the convergence of subdivision scheme in Sobolev spaces with the tool of joint spectral radius. Firstly, the conditions under which the sequence { S a n λ } n = 1 ∞ converges to a W p k -function in an appropriate sense are given. Then, we show that the subdivision scheme converges for any initial vector in W p k ( R ) provided that it does for one nonzero vector in that space. Moreover, if the shifts of the refinable function are stable, the smoothness of the limit function corresponding to the vector λ is also independent of λ, where the smoothness of a given function is measured by the generalized Lipschitz space.

On the convergence of convolved vector subdivision schemes

The aim of this paper is to discuss the use of convolution to build new families of refinable functions and thus new families of subdivision algorithms. In particular, we first generalise a result given in [C. Conti, K. Jetter, in: A. Cohen, R. Rabut, L.L. Schumaker (Eds.), Curves and Surfaces Fitting, Saint Malo, 1999, Vanderbilt University Press, Nashville, TN, 2000, pp. 135–142] where, following a too restrictive definition of convergence, it is proved that convolution of two refinable function vectors, which give rise to convergent subdivision schemes, generates a convolved scheme which is again convergent. Then, we discuss the use of convolution to construct refinable bases suitable to interpolate directions given by difference of control points. This turns out to be useful, for example, in algebraic numerical grid generation methods such as the one discussed in [C. Conti, R. Morandi, in: Proceedings of MASCOT/02, IMACS Ser. Comput. Appl. Math., IMACS, New Brunswick, NJ, 2003, pp. 75–82].

Poly-scale refinability and subdivision

A stationary subdivision scheme is a two-scale process, where values at the next level of refinement are computed from the values of the current level using a single given mask P={p k} k∈ Z d . Under a certain restriction on the mask it can be shown that there exists a distributional solution for the functional equation φ=∑ k∈ Z d p kφ(2·−k) . It is well known that the limit of a convergent subdivision scheme initialized by data f 0={f 0 k} k∈ Z d can be represented as ∑ k∈ Z d f k 0φ(x−k) , where φ is a continuous solution of the functional equation. In this work we generalize this framework in the following sense. The (poly) M-scale subdivision scheme computes the next level of refinement from the M−1 scales of the previous level, using M−1 given masks, P m={p m,k} k∈ Z d , m=1,…, M−1. With a certain restriction on the masks there exists a distributional solution for the poly-scale functional equation φ=∑ m=1 M−1 ∑ k∈ Z d p m,kφ(2 m·−k) . We show that a convergent poly-scale subdivision process initialized by data f 0={f 0 k} k∈ Z d converges to ∑ k∈ Z d f 0 kφ(x−k) , where φ is a continuous solution of the poly-scale functional equation. In applications, the poly-scale framework allows the design of subdivision schemes with features that are not possible in the standard two-scale case.

Convergence of subdivision schemes in L p spaces

In this paper it is proved that Lp solutions of a refinement equation exist if and only if the corresponding subdivision scheme with suitable initial function converges in Lp without any assumption on the stability of the solutions of the refinement equation. A characterization for convergence of subdivision scheme is also given in terms of the refinement mask. Thus a complete answer to the relation between the existence of Lp solutions of the refinement equation and the convergence of the corresponding subdivision schemes is given.

Convergence of Subdivision Schemes Associated with Nonnegative Masks

This paper is concerned with refinement equations of the type $$ f = \sum_{\alpha\in\bbbz^s} a(\alpha) f({M\kern .12em\cdot}-\alpha), $$ where f is the unknown function defined on the s-dimensional Euclidean space $\bbbr^s$, a is a finitely supported sequence on $\bbbz^s$, and M is an s x s dilation matrix with m := |det M|. The solution of a refinement equation can be obtained by using the subdivision scheme associated with the mask. In this paper we give a characterization for the convergence of the subdivision scheme when the mask is nonnegative. Our method is to relate the problem of convergence to m column-stochastic matrices induced by the mask. In this way, the convergence of the subdivision scheme can be determined in a finite number of steps by checking whether each finite product of those column-stochastic matrices has a positive row. As a consequence of our characterization, we show that the convergence of the subdivision scheme with a nonnegative mask depends only on the location of its positive coefficients. Several examples are provided to demonstrate the power and applicability of our approach.

Multivariate Refinement Equations and Convergence of Subdivision Schemes

Refinement equations play an important role in computer graphics and wavelet analysis. In this paper we investigate multivariate refinement equations associated with a dilation matrix and a finitely supported refinement mask. We characterize the Lp-convergence of a subdivision scheme in terms of the p-norm joint spectral radius of a collection of matrices associated with the refinement mask. In particular, the 2-norm joint spectral radius can be easily computed by calculating the eigenvalues of a certain linear operator on a finite dimensional linear space. Examples are provided to illustrate the general theory.

Characterization of covergent subdivision schemes

Subdivision schemes provide important techniques for the fast generation of curves and surfaces. A recusive refinement of a given control polygon will lead in the limit to a desired visually smooth object. These methods play also an important role in wavelet analysis. In this paper, we use a rather simple way to characterize the convergence of subdivision schemes for multivariate cases. The results will be used to investigate the regularity of the solutions for dilation equations.

Simple Regularity Criteria for Subdivision Schemes

Convergent subdivision schemes arise in several fields of applied mathematics (computer-aided geometric design, fractals, compactly supported wavelets) and signal processing (multiresolution decomposition, filter banks). In this paper, a polynomial description is used to study the existence and Hölder regularity of limit functions of binary subdivision schemes. Sharp regularity estimates are derived; they are optimal in most cases. They can easily be implemented on a computer, and simulations show that the exact regularity order is accurately determined after a few iterations. Connection is made to regularity estimates of solutions to two-scale difference equations as derived by Daubechies and Lagarias, and other known Fourier-based estimates. The former are often optimal, while the latter are optimal only for a subclass of symmetric limit functions.

Using parameters to increase smoothness of curves and surfaces generated by subdivision

Some recent subdivision schemes for curves and surfaces are viewed as perturbations of classical schemes for B-splines, and their convergence analysis is presented. It is shown that, under certain mild conditions, a small perturbation of a convergent subdivision scheme does not impair the smoothness of the limit function. Moreover, a proper perturbation can even increase the smoothness and thus provide a one-parameter class of schemes with better continuity properties.