Vector Refinement Equations Research Articles

Vector refinement equation and subdivision schemes in L p spaces

In this paper we will first prove that the nontrivial L p solutions of the vector refinement e-quation exist if and only if the corresponding subdivision scheme with a suitable initial function converges in L p without assumption of the stability of the solutions. Then we obtain a characterization of the convergence of the subdivision scheme in terms of the mask. This gives a complete answer to the existence of L p solutions of the refinement equation and the convergence of the corresponding subdivision schemes.

Image Denoising in Industry Based on Multiwavelet Riesz Bases

Damage testing of components is a key point in many industry fields. In some cases, endoscope is used to inspect the damage part, while the images are often noised. In this paper, we focus on industrial image denoising based on multiwavelet Riesz bases. Starting from compactly supported vector refinement equation, we provide a characterization to form two Riesz bases and an example is given. Based on example Riesz bases, we research industrial endoscope image denoising and get satisfying result.

Vector subdivision schemes with polynomially decaying masks

We investigate the solutions of vector refinement equations with polynomially decaying masks, which play an important role for characterizations of Riesz wavelet bases and biorthogonal wavelets. A vector refinement equation is of the form: where φ = ( φ 1, . . . , φ r )T is the unknown vector of functions on R s , a := ( a ( α )) α ∈Z s is a polynomially decaying sequence of r × r matrices called refinement mask. Associated with the mask a is a linear operator Qa defined on ( Lp (R s )) r by: Qaf := Σ α ∈Z s a ( α ) f (2 ·- α ). The iterative scheme ( Q a n f ) n =1,2,... is called a vector subdivi- sion scheme or a vector cascade algorithm. In this paper, we shall prove that if the vector subdivision scheme converges in ( L2 (R s )) r , then its resulting limit function automatically has polynomial decay. Moreover, under some appropriate conditions on the initial function, the convergence rates of the vector subdivision scheme will be given.

Solutions of multiple vector refinement equations with infinite mask

This paper is devoted to investigating the solutions of refinement equations of the form $$ \varphi (x) = \sum\limits_{\alpha \in Z^s } {a(a)\varphi (Mx - \alpha ),x \in R^s ,} $$ where the vector of functions ϕ = (ϕ1, ..., ϕr)T is in (L1(Rs))r, \( a = (a(\alpha ))_{\alpha \in Z^s } \) is an infinitely supported sequence of r× r matrices called the refinement mask, and M is an s× s integer matrix such that \( \mathop {\lim }\limits_{n \to \infty } \)M−n = 0, with m = detM. Some properties about the solutions of refinement equations are obtained.

Riesz multiwavelet bases generated by vector refinement equation

In this paper, we investigate compactly supported Riesz multiwavelet sequences and Riesz multiwavelet bases for L2(ℝs). Suppose ψ = (ψ1,..., ψr)T and \( \tilde \psi = (\tilde \psi ^1 ,...,\tilde \psi ^r )^T \) are two compactly supported vectors of functions in the Sobolev space (Hμ(ℝs))r for some μ > 0. We provide a characterization for the sequences {ψjkl: l = 1,...,r, j e ℤ, k e ℤs} and \( \tilde \psi _{jk}^\ell :\ell = 1,...,r,j \in \mathbb{Z},k \in \mathbb{Z}^s \) to form two Riesz sequences for L2(ℝs), where ψjkl = mj/2ψl(Mj·−k) and \( \tilde \psi _{jk}^\ell = m^{{j \mathord{\left/ {\vphantom {j 2}} \right. \kern-\nulldelimiterspace} 2}} \tilde \psi ^\ell (M^j \cdot - k) \), M is an s × s integer matrix such that limn→∞M−n = 0 and m = |detM|. Furthermore, let ϕ = (ϕ1,...,ϕr)T and \( \tilde \phi = (\tilde \phi ^1 ,...,\tilde \phi ^r )^T \) be a pair of compactly supported biorthogonal refinable vectors of functions associated with the refinement masks a, \( \tilde a \) and M, where a and \( \tilde a \) are finitely supported sequences of r × r matrices. We obtain a general principle for characterizing vectors of functions ψν = (ψν1,...,ψνr)T and \( \tilde \psi ^\nu = (\tilde \psi ^{\nu 1} ,...,\tilde \psi ^{\nu r} )^T \), ν = 1,..., m − 1 such that two sequences {ψjkνl: ν = 1,..., m − 1, l = 1,...,r, j e ℤ, k e ℤs} and {\( \tilde \psi _{jk}^\nu \) : ν=1,...,m−1,l=1,...,r, j ∈ ℤ, k ∈ ℤs} form two Riesz multiwavelet bases for L2(ℝs). The bracket product [f, g] of two vectors of functions f, g in (L2(ℝs))r is an indispensable tool for our characterization.

Subdivision schemes with polynomially decaying masks

In this paper we investigate the L2-solutions of vector refinement equations with polynomially decaying masks and a general dilation matrix, which plays a vital role for characterizations of wavelets and biorthogonal wavelets with infinite support. A vector refinement equation with polynomially decaying masks and a general dilation matrix is the form: $$ \phi(x)=\sum_{\alpha\in\Bbb Z^s}a(\alpha)\medspace\phi(Mx-\alpha),\quad x\in\Bbb R^s, $$ where the vector of functions \(\phi=(\phi_{1},\cdots,\phi_{r})^{T}\) is in \((L_{2}(\Bbb R^s))^{r},\)\(a:=(a(\alpha))_{\alpha\in\Bbb Z^s}\) is a polynomially decaying sequence of r×r matrices called refinement mask and M is an s×s integer matrix such that \(\lim_{n\to\infty}M^{-n}=0.\) The corresponding cascade operator on \((L_2(\Bbb R^s))^r\) is given by: $$ Q_{a}f(x):=\sum_{\alpha\in\Bbb Z^s}a(\alpha)f(Mx-\alpha),\quad x\in\Bbb R^s, \quad f=(f_1,...,f_r)^T\in (L_2(\Bbb R^s))^r. $$ The iterative scheme \((Q_a^nf)_{n=1,2,\cdots,}\) is called vector cascade algorithm. In this paper we give a complete characterization of convergence of the sequence \((Q_a^nf)_{n=1,2\cdots}\) in L2-norm. Some properties of the transition operator restricted to a certain linear space are discussed. As an application of convergence, we also obtain a characterization of smoothness of solutions of refinement equation mentioned above for the case r = 1.

Mean size formula of wavelet subdivision tree on Heisenberg group

The purpose of this paper is to investigate the mean size formula of wavelet packets (wavelet subdivision tree) on Heisenberg group. The formula is given in terms of the p-norm joint spectral radius. The vector refinement equations on Heisenberg group and the subdivision tree on the Heisenberg group are discussed. The mean size formula of wavelet packets can be used to describe the asymptotic behavior of norm of the subdivision tree.

Biorthogonal multiple wavelets generated by vector refinement equation

Biorthogonal multiple wavelets are generated from refinable function vectors by using the multiresolution analysis. In this paper we provide a general method for the construction of compactly supported biorthogonal multiple wavelets by refinable function vectors which are the solutions of vector refinement equations of the form $$\varphi (x) = \sum\limits_{\alpha \in \mathbb{Z}^s } {a(\alpha )\varphi (Mx - \alpha ), x \in \mathbb{R}^s } ,$$ where the vector of functions ϕ = (ϕ 1, …, ϕ r)T is in \((L_2 (\mathbb{R}^s ))^r ,a = :(a(\alpha ))_{\alpha \in \mathbb{Z}^s } \) 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. Our characterizations are in the general setting and the main results of this paper are the real extensions of some known results.

Vector refinement equations with infinitely supported masks

In this paper we investigate the L 2 -solutions of vector refinement equations with exponentially decaying masks and a general dilation matrix. A vector refinement equation with a general dilation matrix and exponentially decaying masks is of the form φ ( x ) = ∑ α ∈ Z s a ( α ) φ ( Mx - α ) , x ∈ R s , where the vector of functions φ = ( φ 1 , … , φ r ) T is in ( L 2 ( R s ) ) r , a ≕ ( a ( α ) ) α ∈ Z s is an exponentially decaying sequence of r × r matrices called refinement mask and M is an s × s integer matrix such that lim n → ∞ M - n = 0 . Associated with the mask a and dilation matrix M is a linear operator Q a on ( L 2 ( R s ) ) r given by Q a f ( x ) ≔ ∑ α ∈ Z s a ( α ) f ( Mx - α ) , x ∈ R s , f = ( f 1 , … , f r ) T ∈ ( L 2 ( R s ) ) r . The iterative scheme ( Q a n f ) n = 1 , 2 , … , is called vector subdivision scheme or vector cascade algorithm. The purpose of this paper is to provide a necessary and sufficient condition to guarantee the sequence ( Q a n f ) n = 1 , 2 , … to converge in L 2 -norm. As an application, we also characterize biorthogonal multiple refinable functions, which extends some main results in [B. Han, R.Q. Jia, Characterization of Riesz bases of wavelets generated from multiresolution analysis, Appl. Comput. Harmon. Anal., to appear] and [R.Q. Jia, Convergence of vector subdivision schemes and construction of biorthogonal multiple wavelets, Advances in Wavelet (Hong Kong, 1997), Springer, Singapore, 1998, pp. 199–227] to the general setting.

Subdivisions with infinitely supported mask

In this paper we investigate the convergence of subdivision schemes associated with masks being polynomially decay sequences. Two-scale vector refinement equations are the form φ ( x ) = ∑ α ∈ Z a ( α ) φ ( 2 x - α ) , x ∈ R , where the vector of functions φ = ( φ 1 , … , φ r ) T is in ( L 2 ( R ) ) r and a ≕ ( a ( α ) ) α ∈ Z is polynomially decay sequence of r × r matrices called refinement mask. Associated with the mask a is a linear operator on ( L 2 ( R ) ) r given by Q a f ( x ) ≔ ∑ α ∈ Z a ( α ) f ( 2 x - α ) , x ∈ R , f = ( f 1 , … , f r ) T ∈ ( L 2 ( R ) ) r . By using same methods in [B. Han, R. Q. Jia, Characterization of Riesz bases of wavelets generated from multiresolution analysis, manuscript]; [B. Han, Refinable functions and cascade algorithms in weighted spaces with infinitely supported masks, manuscript]; [R.Q. Jia, Q.T. Jiang, Z.W. Shen, Convergence of cascade algorithms associated with nonhomogeneous refinement equations, Proc. Amer. Math. Soc. 129 (2001) 415–427]; [R.Q. Jia, Convergence of vector subdivision schemes and construction of biorthogonal multiple wavelets, in: Advances in Wavelet, Hong Kong,1997, Springer, Singapore, 1998, pp. 199–227], a characterization of convergence of the sequences ( Q a n f ) n = 1 , 2 , … in the L 2 -norm is given, which extends the main results in [R.Q. Jia, S.D. Riemenschneider, D.X. Zhou, Vector subdivision schemes and multiple wavelets, Math. Comp. 67 (1998) 1533–1563] on convergence of the subdivision schemes associated with a finitely supported mask to the case in which mask a is polynomially decay sequence. As an application, we also obtain a characterization of smoothness of solutions of the refinement equation mentioned above for the case r = 1 .

Solutions in Sobolev spaces of vector refinement equations with a general dilation matrix

In this paper, we present a necessary and sufficient condition for the existence of solutions in a Sobolev space Wpk(ℝs) (1≤p≤∞) to a vector refinement equation with a general dilation matrix. The criterion is constructive and can be implemented. Rate of convergence of vector cascade algorithms in a Sobolev space Wpk(ℝs) will be investigated. When the dilation matrix is isotropic, a characterization will be given for the Lp (1≤p≤∞) critical smoothness exponent of a refinable function vector without the assumption of stability on the refinable function vector. As a consequence, we show that if a compactly supported function vector φ∈Lp(ℝs) (φ∈C(ℝs) when p=∞) satisfies a refinement equation with a finitely supported matrix mask, then all the components of φ must belong to a Lipschitz space Lip(ν,Lp(ℝs)) for some ν>0. This paper generalizes the results in R.Q. Jia, K.S. Lau and D.X. Zhou (J. Fourier Anal. Appl. 7 (2001) 143–167) in the univariate setting to the multivariate setting.

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.

Matrix-valued symmetric templates for interpolatory surface subdivisions: I. Regular vertices

The objective of this paper is to introduce a general procedure for deriving interpolatory surface subdivision schemes with “symmetric subdivision templates” (SSTs) for regular vertices. While the precise definition of “symmetry” will be clarified in the paper, the property of SSTs is instrumental to facilitate application of the standard procedure for finding symmetric weights for taking weighted averages to accommodate extraordinary (or irregular) vertices in surface subdivisions, a topic to be studied in a continuation paper. By allowing the use of matrices as weights, the SSTs introduced in this paper may be constructed to overcome the size barrier limited to scalar-valued interpolatory subdivision templates, and thus avoiding the unnecessary surface oscillation artifacts. On the other hand, while the old vertices in a (scalar) interpolatory subdivision scheme do not require a subdivision template, we will see that this is not the case for the matrix-valued setting. Here, we employ the same definition of interpolation subdivisions as in the usual scalar consideration, simply by requiring the old vertices to be stationary in the definition of matrix-valued interpolatory subdivisions. Hence, there would be another complication when the templates are extended to accommodate extraordinary vertices if the template sizes are not small. In this paper, we show that even for C 2 interpolatory subdivisions, only one “ring” is sufficient in general, for both old and new vertices. For example, for 1-to-4 split C 2 interpolatory surface subdivisions, we obtain matrix-valued symmetric interpolatory subdivision templates (SISTs) for both triangular and quadrilateral meshes with sizes that agree with those of the Loop and Catmull–Clark schemes, respectively. Matrix-valued SISTs of similar sizes are also constructed for C 2 interpolatory 3 and 2 subdivision schemes in this paper. In addition to small template sizes, an obvious feature of matrix-valued weights is the flexibility for introducing shape-control parameters. Another significance is that, in contrast to the usual scalar setting, matrix-valued SISTs can be formulated in terms of the coefficient sequence of some vector refinement equation of interpolating bivariate C 2 splines with small support. For example, by modifying the spline function vectors introduced in our previous work [C.K. Chui, Q.T. Jiang, Surface subdivision schemes generated by refinable bivariate spline function vectors, Appl. Comput. Harmon. Anal. 15 (2003) 147–162; C.K. Chui, Q.T. Jiang, Refinable bivariate quartic and quintic C 2 -splines for quadrilateral subdivisions, Preprint, 2004], C 2 symmetric interpolatory subdivision schemes associated with refinement equations of C 2 cubic and quartic splines on the 6-directional and 4-directional meshes, respectively, are also constructed in this paper.

L p –Solutions of Vector Refinement Equations with General Dilation Matrix

L p –Solutions of Vector Refinement Equations with General Dilation Matrix

Spectral Analysis of the Transition Operator and Its Applications to Smoothness Analysis of Wavelets

This paper investigates spectral properties of the transition operator associated to a multivariate vector refinement equation and their applications to the study of smoothness of the corresponding refinable vector of functions. Let $\Phi=(\phi_1,\ldots,\phi_r)^T$ be an $r \times 1$ vector of compactly supported functions in $L_2(\Rs)$ satisfying $\,\Phi = \sum_{\ga\in\Zs} a(\ga) \Phi({M\kern .1em \cdot}-\ga)$, where M is an expansive integer matrix. The smoothness of $\Phi$ is measured by the Sobolev critical exponent $\gl (\Phi) := \sup \bigl\{ \gl: \int_{\Rs} |\hat \phi_j(\xi)|^2(1+|\xi|^\gl)^2\,d\xi<\infty, \, 1 \le j\le r \bigr\}$. Suppose %the mask $a$ is finitely supported, i.e., %the set $M$ is similar to %a diagonal matrix $\hbox{diag}(\gs_1,\ldots,\gs_s)$ with $|\gs_1|=\cdots=|\gs_s|$ and $\supp a := \{\ga\in\Zs: a(\ga) \ne 0\}$ is finite. For $\mu=(\mu_1,\ldots,\mu_s)\in\NN_0^s$, define $\gs^{-\mu} := \gs_1^{-\mu_1} \cdots \gs_s^{-\mu_s}$. Let $A := \sum_{\ga\in\Zs} a(\ga)/|\det M|$ and $b(\ga) := \sum_{\gb\in\Zs} \overline{a(\gb)} \otimes a(\ga+\gb)/|\det M|$, $\ga\in\Zs$, where $\,\otimes\,$ denotes the (right) Kronecker product. Suppose that the highest total degree of polynomials reproduced by $\Phi$ is $k-1$ and $\spec(A)$ (the spectrum of $A$) is $\{\eta_1,\eta_2, \ldots,\eta_r\}$ with $\eta_1=1$ and $\eta_j \ne 1, 2\le j\le r$. Set $$ E_k := \{ \eta_j \overline{\gs^{-\mu}}, \overline{\eta_j} \gs^{-\mu}: |\mu|

Interpolatory orthogonal multiwavelets and refinable functions

Multiwavelet bases of L/sub 2/ consist of families of functions {2/sup j/2//spl psi//sub 2/(2/sup j/x-k)}. By allowing more than one function {/spl psi//sub 1/,/spl psi//sub 2/}, multiwavelets provide some useful applications in signal processing and nice features such as symmetry and orthogonality. The elementary structure for multiwavelets is the multiresolution analysis of multiplicity two {V/sub j/} generated by dilating the basic subspace V/sub 0/. This subspace V/sub 0/ is generated by a multiple refinable function /spl phi/=(/spl phi//sub 1/,/spl phi//sub 2/)/sup T/ (refinable vector of functions) satisfying a vector refinement equation /spl phi/(x)=/spl Sigma/a(k)/spl phi/(2x-k). Here, each a(k) is a 2/spl times/2 matrix. In this paper, we investigate interpolatory orthogonal multiple refinable functions and multiwavelets. The interpolatory property here means that /spl phi//sub 1/ and /spl phi//sub 2/ vanish at all integers and half integers, except that /spl phi//sub 1/(0)=1 and /spl phi//sub 2/(1/2)=1. When /spl phi/ is both interpolatory and orthogonal (which is impossible for scalar refinable functions), the coefficients in the multiresolution representation can be realized by sampling instead of inner products. If f(x)=/spl Sigma/{c/sub 1/(k)/spl phi//sub 1/(2/sup N/x-k)+c/sub 2/(k)/spl phi//sub 2/(2/sup N/x-k)}, then c/sub 1/(k)=f(k/2/sup N/) and c/sub 2/(k)=f(k/2/sup N/+1/2/sup N+1/) for k/spl isin/Z. What is more, the orthogonal multiwavelets we construct here are also interpolatory. We show that the refinement mask for an interpolatory orthogonal multiple refinable function and multiwavelets (filterbank) is reduced to a scalar CQF. The approximation order of interpolatory multiple refinable functions is described. A complete characterization of interpolatory orthogonal multiple refinable functions is given in this paper. However, interpolatory orthogonal multiple refinable functions cannot be symmetric. Examples are presented to illustrate the general theory.

L p solutions of refinement equations

In the recent characterizations of the Lp solution of the refinement equation in terms of the “p-norm joint spectral radius,” there are problems in choosing the initial function for iteration [3, 23], or in addition, requiring stability of the refinable function [13, 17]. In this article we overcome these difficulties and give a more complete characterization of this nature. The criterion is constructive and can be implemented. It can be used to describe the regularity of the solution without assuming stability. This has significant advantages over the previous work. The corresponding results for vector refinement equations are also discussed.

Vector subdivision schemes and multiple wavelets

We consider solutions of a system of refinement equations written in the form ϕ = ∑ α ∈ Z a ( α ) ϕ ( 2 ⋅ − α ) , \begin{equation*}\phi = \sum _{\alpha \in \mathbb {Z}} a(\alpha )\phi (2\cdot -\alpha ),\end{equation*} where the vector of functions ϕ = ( ϕ 1 , … , ϕ r ) T \phi =(\phi ^{1},\ldots ,\phi ^{r})^{T} is in ( L p ( R ) ) r (L_{p}(\mathbb {R}))^{r} and a a is a finitely supported sequence of r × r r\times r matrices called the refinement mask. Associated with the mask a a is a linear operator Q a Q_{a} defined on ( L p ( R ) ) r (L_{p}(\mathbb {R}))^{r} by Q a f := ∑ α ∈ Z a ( α ) f ( 2 ⋅ − α ) Q_{a} f := \sum _{\alpha \in \mathbb {Z}} a(\alpha )f(2\cdot -\alpha ) . This paper is concerned with the convergence of the subdivision scheme associated with a a , i.e., the convergence of the sequence ( Q a n f ) n = 1 , 2 , … (Q_{a}^{n}f)_{n=1,2,\ldots } in the L p L_{p} -norm. Our main result characterizes the convergence of a subdivision scheme associated with the mask a a 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 L 2 L_{2} -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.

Regularity of refinable function vectors

We study the existence and regularity of compactly supported solutions φ = (φv) v= /r− of vector refinement equations. The space spanned by the translates of φv can only provide approximation order if the refinement maskP has certain particular factorization properties. We show, how the factorization ofP can lead to decay of |v(u)| as |u| → ∞. The results on decay are used to prove uniqueness of solutions and convergence of the cascade algorithm.