Orthogonal Refinable Functions Research Articles

Bivariate two-band wavelets demystified

There are several well known constructions of bivariate, compactly supported wavelets based on orthogonal refinable functions with dilation matrices of determinant 2 or -2. The corresponding filterbank consists of only two subbands: low-pass and high-pass. We unify these constructions and express their intrinsic structure via normal forms of the corresponding bivariate polyphase representations. A normal form is sparse and is obtained from the polyphase representation via a suitable unitary transformation. We characterize certain normal forms of bi-degree (1,1) and (2,2) and show that their non-zero elements correspond to the solutions of the univariate Quadrature Mirror Filter conditions. The unitary transformations are chosen to ensure sum rule conditions of certain order. We illustrate our results on some examples and address the quest for characterizing bivariate wavelet constructions of higher bi-degree.

High Balanced Biorthogonal Multiwavelets with Symmetry

Balanced multiwavelet transform can process the vector-valued data sparsely while preserving a polynomial signal. Yang et al. (2006) constructed balanced multiwavelets from the existing nonbalanced ones. It will be proved, however, in this paper that if the nonbalanced multiwavelets have antisymmetric component, it is impossible for the balanced multiwavelets by the method mentioned above to have symmetry. In this paper, we give an algorithm for constructing a pair of biorthogonal symmetric refinable function vectors from any orthogonal refinable function vector, which has symmetric and antisymmetric components. Then, a general scheme is given for high balanced biorthogonal multiwavelets with symmetry from the constructed pair of biorthogonal refinable function vectors. Moreover, we discuss the approximation orders of the biorthogonal symmetric refinable function vectors. An example is given to illustrate our results.

Construction of biorthogonal two-direction refinable function and two-direction wavelet with dilation factor m

An algorithm for constructing a pair of biorthogonal two-direction refinable function and the corresponding biorthogonal two-direction wavelet is presented. In particular, the algorithm can also be used to construct two-direction orthogonal refinable function. Finally, we give an example illustrating how to use our method to construct biorthogonal two-direction scaling function with dilation factor 3.

Pseudo-splines, wavelets and framelets

The first type of pseudo-splines were introduced in [I. Daubechies, B. Han, A. Ron, Z. Shen, Framelets: MRA-based constructions of wavelet frames, Appl. Comput. Harmon. Anal. 14 (1) (2003) 1–46; I. Selesnick, Smooth wavelet tight frames with zero moments, Appl. Comput. Harmon. Anal. 10 (2) (2001) 163–181] to construct tight framelets with desired approximation orders via the unitary extension principle of [A. Ron, Z. Shen, Affine systems in L 2 ( R d ) : The analysis of the analysis operator, J. Funct. Anal. 148 (2) (1997) 408–447]. In the spirit of the first type of pseudo-splines, we introduce here a new type (the second type) of pseudo-splines to construct symmetric or antisymmetric tight framelets with desired approximation orders. Pseudo-splines provide a rich family of refinable functions. B-splines are one of the special classes of pseudo-splines; orthogonal refinable functions (whose shifts form an orthonormal system given in [I. Daubechies, Orthonormal bases of compactly supported wavelets, Comm. Pure Appl. Math. 41 (1988) 909–996]) are another class of pseudo-splines; and so are the interpolatory refinable functions (which are the Lagrange interpolatory functions at Z and were first discussed in [S. Dubuc, Interpolation through an iterative scheme, J. Math. Anal. Appl. 114 (1986) 185–204]). The other pseudo-splines with various orders fill in the gaps between the B-splines and orthogonal refinable functions for the first type and between B-splines and interpolatory refinable functions for the second type. This gives a wide range of choices of refinable functions that meets various demands for balancing the approximation power, the length of the support, and the regularity in applications. This paper will give a regularity analysis of pseudo-splines of the both types and provide various constructions of wavelets and framelets. It is easy to see that the regularity of the first type of pseudo-splines is between B-spline and orthogonal refinable function of the same order. However, there is no precise regularity estimate for pseudo-splines in general. In this paper, an optimal estimate of the decay of the Fourier transform of the pseudo-splines is given. The regularity of pseudo-splines can then be deduced and hence, the regularity of the corresponding wavelets and framelets. The asymptotical regularity analysis, as the order of the pseudo-splines goes to infinity, is also provided. Furthermore, we show that in all tight frame systems constructed from pseudo-splines by methods provided both in [I. Daubechies, B. Han, A. Ron, Z. Shen, Framelets: MRA-based constructions of wavelet frames, Appl. Comput. Harmon. Anal. 14 (1) (2003) 1–46] and this paper, there is one tight framelet from the generating set of the tight frame system whose dilations and shifts already form a Riesz basis for L 2 ( R ) .

On a conjecture about MRA Riesz wavelet bases

Let ϕ \phi be a compactly supported refinable function in L 2 ( R ) L_2(\mathbb {R}) such that the shifts of ϕ \phi are stable and ϕ ^ ( 2 ξ ) = a ^ ( ξ ) ϕ ^ ( ξ ) \hat \phi (2\xi )=\hat a(\xi )\hat \phi (\xi ) for a 2 π 2\pi -periodic trigonometric polynomial a ^ \hat a . A wavelet function ψ \psi can be derived from ϕ \phi by ψ ^ ( 2 ξ ) := e − i ξ a ^ ( ξ + π ) ¯ ϕ ^ ( ξ ) \hat \psi (2\xi ):=e^{-i\xi }\overline {\hat a(\xi +\pi )} \hat \phi (\xi ) . If ϕ \phi is an orthogonal refinable function, then it is well known that ψ \psi generates an orthonormal wavelet basis in L 2 ( R ) L_2(\mathbb {R}) . Recently, it has been shown in the literature that if ϕ \phi is a B B -spline or pseudo-spline refinable function, then ψ \psi always generates a Riesz wavelet basis in L 2 ( R ) L_2(\mathbb {R}) . It was an open problem whether ψ \psi can always generate a Riesz wavelet basis in L 2 ( R ) L_2(\mathbb {R}) for any compactly supported refinable function in L 2 ( R ) L_2(\mathbb {R}) with stable shifts. In this paper, we settle this problem by proving that for a family of arbitrarily smooth refinable functions with stable shifts, the derived wavelet function ψ \psi does not generate a Riesz wavelet basis in L 2 ( R ) L_2(\mathbb {R}) . Our proof is based on some necessary and sufficient conditions on the 2 π 2\pi -periodic functions a ^ \hat a and b ^ \hat b in C ∞ ( R ) C^{\infty }(\mathbb {R}) such that the wavelet function ψ \psi , defined by ψ ^ ( 2 ξ ) := b ^ ( ξ ) ϕ ^ ( ξ ) \hat \psi (2\xi ):=\hat b(\xi )\hat \phi (\xi ) , generates a Riesz wavelet basis in L 2 ( R ) L_2(\mathbb {R}) .

A Class of Orthogonal Refinable Functions and Wavelets

We give a construction, for any n ≥ 2, of a space S of spline functions of degree n − 1 with simple knots in 1 Z which is generated by a triple of refinable, orthogonal functions with compact support. Indeed, the result holds more generally by replacing the B-spline of degree n − 1 with simple knots at the integers by any continuous refinable function whose mask is a Hurwitz polynomial of degree n .A simple construction is also given for the corresponding wavelets.

Orthogonality criteria for compactly supported refinable functions and refinable function vectors

A refinable function φ(x):ℝn→ℝ or, more generally, a refinable function vector Φ(x)=[φ1(x),...,φr(x)]T is an L1 solution of a system of (vector-valued) refinement equations involving expansion by a dilation matrix A, which is an expanding integer matrix. A refinable function vector is called orthogonal if {φj(x−α):α∈ℤn, 1≤j≤r form an orthogonal set of functions in L2(ℝn). Compactly supported orthogonal refinable functions and function vectors can be used to construct orthonormal wavelet and multiwavelet bases of L2(ℝn). In this paper we give a comprehensive set of necessary and sufficient conditions for the orthogonality of compactly supported refinable functions and refinable function vectors.