Bin Han Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta, Canada

Bivariate (Two-dimensional) Wavelets Bin Han Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta, Canada Article Outline Glossary I. Definitions II. Introduction III. Bivariate Refinable Functions and Their Properties IV. The Projection Method V. Bivariate Orthonormal and Biorthogonal Wavelets VI. Bivariate Riesz Wavelets VII. Pairs of Dual Wavelet Frames VIII. Future Directions. IX. Bibliography Glossary Dilation matrix A 2 2 matrix M is called a dilation matrix if all the entries of M are integers and all the eigenvalues of M are greater than one in modulus. Isotropic dilation matrix A dilation matrix M is said to be isotropic if M is similar to a diagonal matrix and all its eigenvalues have the same modulus. Wavelet system A wavelet system is a collection of square integrable functions that are generated from a finite set of functions (which are called wavelets) by using integer shifts and dilations. I. Definitions Throughout this article, R, C, Z denote the real line, the complex plane, and the set of all integers, respectively. For 1 p, L p (R 2 ) denotes the set of all Lebesgue measurable bivariate functions f such that f p L p (R 2 ) := R 2 f(x) p dx <. In particular, the space L 2 (R 2 ) of square integrable functions is a Hilbert space under the inner product f, g := f(x)g(x) dx, f, g L 2 (R 2 ), R 2 where g(x) denotes the complex conjugate of the complex number g(x). In applications such as image processing and computer graphics, the following are commonly used isotropic dilation matrices: M 2 = [ 1 1 1 1 ], Q 2 = [ 1 1 1 1 ] 2 1, M 3 =, di 1 2 2 = d 0, (1) 0 d where d is an integer with d > 1. Using a dilation matrix M, a bivariate M-wavelet system is generated by integer shifts and dilates from a finite set {ψ 1,..., ψ L } of functions in L 2 (R 2 ). More precisely, the set of 1

2 all the basic wavelet building blocks of an M-wavelet system generated by {ψ 1,..., ψ L } is given by X M ({ψ 1,..., ψ L }) := {ψ l,m : j Z, k Z 2, l = 1,..., L}, (2) where ψ l,m (x) := detm j/2 ψ l (M j x k), x R 2. (3) One of the main goals in wavelet analysis is to find wavelet systems X M ({ψ 1,..., ψ L }) with some desirable properties such that any two-dimensional function or signal f L 2 (R 2 ) can be sparsely and efficiently represented under the M-wavelet system X M ({ψ 1,..., ψ L }): L f = h l (f)ψ l,m, (4) l=1 j Z k Z 2 where h l : L 2(R 2 ) C are linear functionals. There are many types of wavelet systems studied in the literature. In the following, let us outline some of the most important types of wavelet systems. Orthonormal wavelets. We say that {ψ 1,..., ψ L } generates an orthonormal M-wavelet basis in L 2 (R 2 ) if the system X M ({ψ 1,..., ψ L }) is an orthonormal basis of the Hilbert space L 2 (R 2 ). That is, the linear span of elements in X M ({ψ 1,..., ψ L }) is dense in L 2 (R 2 ) and ψ l,m,,m ψl j,k = δ l l δ j j δ k k, j, j Z, k, k Z 2, l, l = 1,..., L, (5) where δ denotes the Dirac sequence such that δ 0 = 1 and δ k = 0 for all k 0. For an orthonormal wavelet basis X M ({ψ 1,..., ψ L }), the linear functional h l in (4) is given by h l (f) = f, ψl,m and the representation in (4) becomes f = L l=1 j Z k Z 2 f, ψ l,m ψl,m, f L 2(R 2 ) (6) with the series converging in L 2 (R 2 ). Riesz wavelets. We say that {ψ 1,..., ψ L } generates a Riesz M-wavelet basis in L 2 (R 2 ) if the system X M ({ψ 1,..., ψ L }) is a Riesz basis of L 2 (R 2 ). That is, the linear span of elements in X M ({ψ 1,..., ψ L }) is dense in L 2 (R 2 ) and there exist two positive constants C 1 and C 2 such that L 2 C 1 c l 2 L c l ψ l,m L C 2 c l 2 l=1 j Z k Z 2 l=1 j Z k Z 2 l=1 j Z k Z 2 L 2 (R 2 ) for all finitely supported sequences {c l } j Z,k Z 2,l=1,...,L. Clearly, a Riesz M-wavelet generalizes an orthonormal M-wavelet by relaxing the orthogonality requirement in (5). For a Riesz basis X M ({ψ 1,..., ψ L }), it is well-known that there exists a dual Riesz basis { ψ l, : j Z, k Z 2, l = 1,..., L} of elements in L 2 (R 2 ) (this set is not necessarily generated by integer shifts and dilates from some finite set of functions) such that (5) still holds after replacing ψ l,m j,k by ψ l,j,k. For a Riesz wavelet basis, the linear functional in (4) becomes h l (f) = f, ψ l,. In fact, ψl, := F 1 (ψ l,m ), where F : L 2(R 2 ) L 2 (R 2 ) is defined to be F(f) := L l=1 j Z k Z 2 f, ψ l,m ψl,m, f L 2(R 2 ). (7)

3 Wavelet frames. A further generalization of a Riesz wavelet is a wavelet frame. We say that {ψ 1,..., ψ L } generates an M-wavelet frame in L 2 (R 2 ) if the system X M ({ψ 1,..., ψ L }) is a frame of L 2 (R 2 ). That is, there exist two positive constants C 1 and C 2 such that C 1 f 2 L 2 (R 2 ) L l=1 j Z k Z 2 f, ψ l,m 2 C 2 f 2 L 2 (R 2 ), f L 2(R 2 ). (8) It is not difficult to check that a Riesz M-wavelet is an M-wavelet frame. Then (8) guarantees that the frame operator F in (7) is a bounded and invertible linear operator. For an M-wavelet frame, the linear functional in (4) can be chosen to be h l (f) = f, F 1 (ψ l,m ) ; however, such functionals may not be unique. The fundamental difference between a Riesz wavelet and a wavelet frame lies in that for any given function f L 2 (R 2 ), the representation in (4) is unique under a Riesz wavelet while it may not be unique under a wavelet frame. The representation in (4) with the choice h l (f) = f, F 1 (ψ l,m ) is called the canonical representation of a wavelet frame. In other words, {F 1 (ψ l,m ) : j Z, k Z2, l = 1,..., L} is called the canonical dual frame of the given wavelet frame X M ({ψ 1,..., ψ L }). Biorthogonal wavelets. We say that ({ψ 1,..., ψ L }, { ψ 1,..., ψ L }) generates a pair of biorthogonal M-wavelet bases in L 2 (R 2 ) if each of X M ({ψ 1,..., ψ L }) and X M ({ ψ 1,..., ψ L }) is a Riesz basis of L 2 (R 2 ) and (5) still holds after replacing ψ l,m j,k by ψ l,m j,k. In other words, the dual Riesz basis of X M ({ψ 1,..., ψ L }) has the wavelet structure and is given by X M ({ ψ 1,..., ψ L }). For a biorthogonal wavelet, the wavelet representation in (4) becomes f = L l=1 f, j Z k Z 2 ψ l,m ψl,m, f L 2(R 2 ). (9) Obviously, {ψ 1,..., ψ L } generates an orthonormal M-wavelet basis in L 2 (R 2 ) if and only if ({ψ 1,..., ψ L }, {ψ 1,..., ψ L }) generates a pair of biorthogonal M-wavelet bases in L 2 (R 2 ). Dual wavelet frames. Similarly, we have the notion of a pair of dual wavelet frames. We say that ({ψ 1,..., ψ L }, { ψ 1,..., ψ L }) generates a pair of dual M-wavelet frames in L 2 (R 2 ) if each of X M ({ψ 1,..., ψ L }) and X M ({ ψ 1,..., ψ L }) is an M-wavelet frame in L 2 (R 2 ) and f, g = L l=1 f, j Z k Z 2 ψ l,m ψl,m, g, f, g L 2(R 2 ). It follows from the above identity that (9) still holds for a pair of dual M-wavelet frames. We say that {ψ 1,..., ψ L } generates a tight M-wavelet frame if ({ψ 1,..., ψ L }, {ψ 1,..., ψ L }) generates a pair of dual M-wavelet frames in L 2 (R 2 ). For a tight wavelet frame {ψ 1,..., ψ L }, the wavelet representation in (6) still holds. So, a tight wavelet frame is a generalization of an orthonormal wavelet basis. II. Introduction Bivariate (two-dimensional) wavelets are of interest in representing and processing two dimensional data such as images and surfaces. In this article we shall discuss some basic background and results on bivariate wavelets. We denote Π J the set of all bivariate polynomials of total degree at most J. For compactly supported functions ψ 1,..., ψ L, we say that {ψ 1,..., ψ L } has J vanishing moments if P, ψ l = 0 for all l = 1,..., L and all polynomials P Π J 1.

4 The advantages of wavelet representations largely lie in the following aspects: (1) There is a fast wavelet transform (FWT) for computing the wavelet coefficients h l (f) in the wavelet representation (4). (2) The wavelet representation has good time and frequency localization. Roughly speaking, the basic building blocks ψ l,m have good time localization and smoothness. (3) The wavelet representation is sparse. For a smooth function f, most wavelet coefficients are negligible. Generally, the wavelet coefficient h l (f) only depends on the information of f in a small neighborhood of the support of ψ l,m (more precisely, the support of ψl,m if h l l,m (f) = f, ψ ). If f in a small neighborhood of the support of ψ l,m is smooth or behaves like a polynomial to certain degree, then by the vanishing moments of the dual wavelet functions, h l (f) is often negligible. (4) For a variety of function spaces B such as Sobolev and Besov spaces, its norm f B, f B, is equivalent to certain weighted sequence norm of its wavelet coefficients {h l (f)} j Z,k Z 2,l=1,...,L. For more details on advantages and applications of wavelets, see [1, 4, 5, 6, 7, 10]. For a dilation matrix M = di 2, the easiest way for obtaining a bivariate wavelet is to use the tensor product method. In other words, all the functions in the generator set {ψ 1,..., ψ L } L 2 (R 2 ) take the form ψ l (x, y) = f l (x)g l (y), x, y R, where f l and g l are some univariate functions in L 2 (R). However, tensor product (also called separable) bivariate wavelets give preference to the horizontal and vertical directions which may not be desirable in applications such as image processing ([7, 10, 17, 52]). Also, for a nondiagonal dilation matrix, it is difficult or impossible to use the tensor product method to obtain bivariate wavelets. Therefore, nonseparable bivariate wavelets themselves are of importance and interest in both theory and application. In this article, we shall address several aspects of bivariate wavelets with a general two-dimensional dilation matrix. III. Bivariate Refinable Functions and Their Properties In order to have a fast wavelet transform to compute the wavelet coefficients in (4), the generators ψ 1,..., ψ L in a wavelet system are generally obtained from a refinable function φ via a multiresolution analysis ([4, 7, 10, 21]). Let M be a 2 2 dilation matrix. For a function φ in L 2 (R 2 ), we say that φ is M-refinable if it satisfies the following refinement equation φ(x) = detm k Z 2 a k φ(mx k), a.e. x R 2, (10) where a : Z 2 C is a finitely supported sequence on Z 2 satisfying k Z a 2 k = 1. Such a sequence a is often called a mask in wavelet analysis and computer graphics, or a low-pass filter in signal processing. Using the Fourier transform ˆφ of φ L 2 (R 2 ) L 1 (R 2 ) and the Fourier series â of a, which are defined to be ˆφ(ξ) := φ(x)e ix ξ dx and â(ξ) := a k e ik ξ, ξ R 2, (11) R 2 k Z 2 the refinement equation (10) can be equivalently rewritten as ˆφ(M T ξ) = â(ξ) ˆφ(ξ), ξ R 2, (12)

5 where M T denotes the transpose of the dilation matrix M. Since â(0) = 1 and â is a trigonometric polynomial, one can define a function ˆφ by ˆφ(ξ) := â((m T ) j ξ), ξ R 2, (13) j=1 with the series converging uniformly on any compact set of R 2. It is known ([3, 7]) that φ is a compactly supported tempered distribution and clearly φ satisfies the refinement equation in (12) with mask a. We call φ the standard refinable function associated with mask a and dilation M. Now the generators ψ 1,..., ψ L of a wavelet system are often obtained from the refinable function φ via ψ l (M T ξ) := âl (ξ) ˆφ(ξ), ξ R 2, l = 1,..., L, for some 2π-periodic trigonometric polynomials âl. Such sequences a l are called wavelet masks or high-pass filters. Except for very few special masks a such as masks for box spline refinable functions (see [8] for box splines), most refinable functions φ, obtained in (13) from mask a and dilation M, do not have explicit analytic expressions and a so-called cascade algorithm or subdivision scheme is used to approximate the refinable function φ. Let B be a Banach space of bivariate functions. Starting with a suitable initial function f B, we iteratively compute a sequence {Q n a,m f} n=0 of functions, where Q a,m f := detm k Z 2 a k f(m k). (14) If the sequence {Q n a,m f} n=0 of functions converges to some f B in the Banach space B, then we have Q a,m f = f. That is, as a fixed point of the cascade operator Q a,m, f is a solution to the refinement equation (10). A cascade algorithm plays an important role in the study of refinable functions in wavelet analysis and of subdivision surfaces in computer graphics. For more details on cascade algorithms and subdivision schemes, see [1, 3, 9, 24, 25, 33, 38, 55] and numerous references therein. One of the most important properties of a refinable function φ is its smoothness, which is measured by its L p smoothness exponent ν p (φ) and is defined to be ν p (φ) := sup{ν : φ W ν p (R 2 )}, 1 p, (15) where Wp ν (R 2 ) denotes the fractional Sobolev space of order ν > 0 in L p (R 2 ). The notion of sum rules of a mask is closely related to the vanishing moments of a wavelet system ([7, 46]). For a bivariate mask a, we say that a satisfies the sum rules of order J with the dilation matrix M if a j+mk P (j + Mk) = a Mk P (Mk), P Π J 1. (16) k Z 2 k Z 2 Or equivalently, µ 1 1 µ 2 2 â(2πγ) = 0 for all µ 1 + µ 2 < J and γ [(M T ) 1 Z 2 ]\Z 2, where 1 and 2 denote the partial derivatives in the first and second coordinate, respectively. Throughout the article, we denote sr(a, M) the highest order of sum rules satisfied by the mask a with the dilation matrix M. To investigate various properties of refinable functions, next we introduce a quantity ν p (a, M) in wavelet analysis (see [33]). For a bivariate mask a and a dilation matrix M,

6 we denote ν p (a, M) := log ρ(m) [ detm 1 1/p ρ(a, M, p)], 1 p, (17) where ρ(m) denotes the spectral radius of M and { } ρ(a, M, p) := max lim sup a n,(µ 1,µ 2 ) 1/n l p (Z n 2 ) : µ 1 + µ 2 = sr(a, M), µ 1, µ 2 N {0}, where the sequence a n,(µ 1,µ 2 ) is defined by its Fourier series as follows: for ξ = (ξ 1, ξ 2 ) R 2, a n,(µ 1,µ 2) (ξ) := (1 e iξ 1 ) µ 1 (1 e iξ 2 ) µ2â((m T ) n 1 ξ)â((m T ) n 2 ξ) â(m T ξ)â(ξ). It is known that ([33,38]) ν p (a, M) ν q (a, M) ν p (a, M) + (1/q 1/p) log ρ(m) detm for all 1 p q. Parts of the following result essentially appeared in various forms in many papers in the literature. For the study of refinable functions and cascade algorithms, see [3, 7, 9, 16, 19, 21, 25, 27, 31, 33, 47, 50, 56] and references therein. The following is from [33]. Theorem 1. Let M be a 2 2 isotropic dilation matrix and a be a finitely supported bivariate mask. Let φ denote the standard refinable function associated with mask a and dilation M. Then ν p (φ) ν p (a, M) for all 1 p. If the shifts of φ are stable, that is, {φ( k) : k Z 2 } is a Riesz system in L p (R 2 ), then ν p (φ) = ν p (a, M). Moreover, for every nonnegative integer J, the following statements are equivalent (1) For every compactly supported function f Wp J (R 2 ) (if p =, we require f C J (R 2 )) such that ˆf(0) = 1 and µ 1 1 µ 2 2 ˆf(2πk) = 0 for all µ 1 + µ 2 < J and k Z 2 \{0}, the cascade sequence {Q n a,m f} n=0 converges in the Sobolev space Wp J (R 2 ) (in fact, the limit function is φ). (2) ν p (a, M) > J. Symmetry of a refinable function is also another important property of a wavelet system in applications. For example, symmetry is desirable in order to handle the boundary of an image or to improve the visual quality of a surface ([1, 7, 9, 10, 34, 52]). Let M be a 2 2 dilation matrix and G be a finite set of 2 2 integer matrices. We say that G is a symmetry group with respect to M ([29, 31]) if G is a group under matrix multiplication and MEM 1 G for all E G. In dimension two, two commonly used symmetry groups are D 4 and D 6 : and D 6 := { ± D 4 := 1 0, ± 0 1 { ± [ 1 0 0 1 ] 0 1, ± 1 1 1 0, ±, ± 0 1 1 1, ± 1 0 0 1, ± 1 0 0 1, ± 1 0 [ 0 ]} 1 1 0 1 1, ± 0 1 } 1 0. 1 1 For symmetric refinable functions, we have the following result ([34, Proposition 2.1]): Proposition 2. Let M be a 2 2 dilation matrix and G be a symmetry group with respect to M. Let a be a bivariate mask and φ be the standard refinable function associated with mask a and dilation M. Then a is G-symmetric with center c a : a E(k ca )+c a = a k for all k Z 2 and E G, if and only if, φ is G-symmetric with center c: φ(e( c) + c) = φ, E G with c := (M I 2 ) 1 c a. In the following, let us present some examples of bivariate refinable functions.

7 Example 3. Let M = 2I 2 and â(ξ 1, ξ 2 ) := cos 4 (ξ 1 /2) cos 4 (ξ 2 /2). Then a is a tensor product mask with sr(a, 2I 2 ) = 4 and a is D 4 -symmetric with center 0. The associated standard refinable function φ is the tensor product spline of order 4. The Catmull-Clark subdivision scheme for quadrilateral meshes in computer graphics is based on this mask a ([1]). Example 4. Let M = 2I 2 and â(ξ 1, ξ 2 ) := cos 2 (ξ 1 /2) cos 2 (ξ 2 /2) cos 2 (ξ 1 /2 + ξ 2 /2). Then sr(a, 2I 2 ) = 4 and a is D 6 -symmetric with center 0. The associated refinable function φ is the convolution of the three direction box spline with itself ([8]). The Loop subdivision scheme for triangular meshes in computer graphics is based on this mask a ([1]). Another important property of a refinable function is interpolation. We say that a bivariate function φ is interpolating if φ is continuous and φ(k) = δ k for all k Z 2. Theorem 5. Let M be a 2 2 dilation matrix and a be a finitely supported bivariate mask. Let φ denote the standard refinable function associated with mask a and dilation M. Then φ is an interpolating function if and only if ν (a, M) > 0 and a is an interpolatory mask with dilation M, that is, a 0 = detm 1 and a Mk = 0 for all k Z 2 \{0}. Example 6. Let M = 2I 2 and â(ξ 1, ξ 2 ) := cos(ξ 1 /2) cos(ξ 2 /2) cos(ξ 1 /2 + ξ 2 /2) [ 1 + 2 cos(ξ 1 ) + 2 cos(ξ 2 ) + 2 cos(ξ 1 + ξ 2 ) cos(2ξ 1 + ξ 2 ) cos(ξ 1 + 2ξ 2 ) cos(ξ 1 ξ 2 )]/4. Then sr(a, 2I 2 ) = 4, a is D 6 -symmetric with center 0 and a is an interpolatory mask with dilation 2I 2. Since ν 2 (a, 2I 2 ) 2.44077, so, ν (a, 2I 2 ) ν 2 (a, 2I 2 ) 1 1.44077 > 0. By Theorem 5, the associated standard refinable function φ is interpolating. The butterfly interpolatory subdivision scheme for triangular meshes is based on this mask a (see [25]). For more details on bivariate interpolatory masks and interpolating refinable functions, see [3, 9, 24, 25, 27, 28, 29, 33, 39, 40, 41, 60]. IV. The Projection Method In applications, one is interested in analyzing some optimal properties of multivariate wavelets. The projection method is useful for this purpose. Let r and s be two positive integers with r s. Let P be an r s real-valued matrix. For a compactly supported function φ and a finitely supported mask a in dimension s, we define the projected function P φ and projected mask P a in dimension r by (18) P φ(ξ) := ˆφ(P t T ξ) and P a(ξ) := â(p T ξ + 2πε j ), ξ R r, (19) where ˆφ and â are understood to be continuous and {ε 1,..., ε t } is a complete set of representatives of the distinct cosets of [P T R r ]/Z s. If P is an integer matrix, then P a(ξ) = â(p T ξ). Now we have the following result on projected refinable functions ([29, 30, 36, 37]). Theorem 7. Let N be an s s dilation matrix and M be an r r dilation matrix. Let P be an r s integer matrix such that P N = MP and P Z s = Z r. Let â be a 2πperiodic trigonometric polynomials in s-variables with â(0) = 1 and φ the standard N- refinable function associated with mask a. Then sr(a, N) sr(p a, M) and P φ(m T ξ) = P a(ξ) P φ(ξ). That is, P φ is M-refinable with mask P a. Moreover, for all 1 p, ν p (φ) ν p (P φ) and det M 1 1/p ρ(p a, M, p) det N 1 1/p ρ(a, N, p). j=1

8 If we further assume that ρ(m) = ρ(n), then ν p (a, N) ν p (P a, M). As pointed out in [36, 37], the projection method is closely related to box splines. For a given r s (direction) integer matrix Ξ of rank r with r s, the Fourier transform of its associated box spline M Ξ and its mask a Ξ are given by (see [8]) M Ξ (ξ) := k Ξ 1 e ik ξ ik ξ and â Ξ (ξ) := 1 + e ik ξ, ξ R r, (20) 2 k Ξ where k Ξ means that k is a column vector of Ξ and k goes through all the columns of Ξ once and only once. Let χ [0,1] s denote the characteristic function of the unit cube [0, 1] s. From (20), it is evident that the box spline M Ξ is just the projected function Ξχ [0,1] s, since Ξχ [0,1] s = M Ξ by Ξχ [0,1] s(ξ) = χ [0,1] s(ξ T ξ), ξ R r. Note that M Ξ is 2I r -refinable with the mask a Ξ, since M Ξ (2ξ) = â Ξ (ξ) M Ξ (ξ). As an application of the projection method in Theorem 7, we have ([27, Theorem 3.5]): Corollary 8. Let M = 2I s and a be an interpolatory mask with dilation M such that a is supported inside [ 3, 3] s. Then ν (a, 2I s ) 2 and therefore, φ C 2 (R s ), where φ is the standard refinable function associated with mask a and dilation 2I s. We use proof by contradiction. Suppose ν (a, 2I s ) > 2. Let P = [1, 0,..., 0] be a 1 s matrix. Then we must have sr(a, 2I s ) 3, which, combining with other assumptions on a, will force P a(ξ) = 1/2+9/16 cos(ξ) 1/16 cos(3ξ). Since ν (P a, 2) = 2, by Theorem 7, we must have ν (a, 2I s ) ν (P a, 2) = 2. A contradiction. So, ν (a, 2I s ) 2. In particular, the refinable function in the butterfly subdivision scheme in Example 6 is not C 2. The projection method can be used to construct interpolatory masks painlessly ([29]). Theorem 9. Let M be an r r dilation matrix. Then there is an r r integer matrix H such that MZ r = HZ r and H r = detm I r. Let P := detm 1 H. Then for any (tensor product) interpolatory mask a with dilation detm I r, the projected mask P a is an interpolatory mask with dilation M and sr(p a, M) sr(a, detm I r ). Example 10. For M = M 2 or M = Q 2, we can take H := M 2 in Theorem 9. For more details on the projection method, see [27, 29, 30, 34, 36, 37]. V. Bivariate Orthonormal and Biorthogonal Wavelets In this section, we shall discuss the analysis and construction of bivariate orthonormal and biorthogonal wavelets. For analysis of biorthogonal wavelets, the following result is well-known (see [7, 17, 18, 26, 28, 33, 48, 53, 56] and references therein): Theorem 11. Let M be a 2 2 dilation matrix and a, ã be two finitely supported bivariate masks. Let φ and φ be the standard M-refinable functions associated with masks a and ã, respectively. Then φ, φ L 2 (R 2 ) and satisfy the biorthogonality relation φ, φ( k) = δ k, k Z 2, (21) if and only if, ν 2 (a, M) > 0, ν 2 (ã, M) > 0 and (a, ã) is a pair of dual masks: â(ξ + 2πγ)ˆã(ξ + 2πγ) = 1, ξ R 2, (22) γ Γ M T

where Γ M T is a complete set of representatives of distinct cosets of [(M T ) 1 Z 2 ]/Z 2 with 0 Γ M T. Moreover, if (21) holds and there exist 2π-periodic trigonometric polynomials â 1,..., âm 1, ã 1,..., ã m 1 with m := detm such that where for ξ R 2, M [â, (ξ) := â 1,...,âm 1 ] with {γ 0,..., γ m 1 } := Γ M T ψ l (M T ξ) := âl (ξ) ˆφ(ξ) M [â, â 1,...,âm 1 ] (ξ)m [ˆã, ã 1,..., ã m 1 ] (ξ)t = I m, ξ R 2, (23) â(ξ + 2πγ 0 ) â 1 (ξ + 2πγ 0 ) â m 1 (ξ + 2πγ 0 ) â(ξ + 2πγ 1 ) â 1 (ξ + 2πγ 1 ) â m 1 (ξ + 2πγ 1 )...... â(ξ + 2πγ m 1 ) â 1 (ξ + 2πγ m 1 ) â m 1 (ξ + 2πγ m 1 ) and γ 0 := 0. Define ψ 1,..., ψ m 1, ψ 1,..., ψ m 1 by and 9 (24) ψ l (M T ξ) := ã l (ξ) ˆ φ(ξ), l = 1,..., m 1. (25) Then ({ψ 1,..., ψ m 1 }, { ψ 1,..., ψ m 1 }) generates a pair of biorthogonal M-wavelet bases in L 2 (R 2 ). As a direct consequence of Theorem 11, for orthonormal wavelets, we have Corollary 12. Let M be a 2 2 dilation matrix and a be a finitely supported bivariate mask. Let φ be the standard refinable function associated with mask a and dilation M. Then φ has orthonormal shifts, that is, φ, φ( k) = δ k for all k Z 2, if and only if, ν 2 (a, M) > 0 and a is an orthogonal mask (that is, (22) is satisfied with ˆã being replaced by â). If in addition there exist 2π-periodic trigonometric polynomials â1,..., âm 1 with m := detm such that M [â, â 1,...,âm 1 ] (ξ)m [â,â1,...,âm 1 ](ξ) T = I m. Then {ψ 1,..., ψ m 1 }, defined in (25), generates an orthonormal M-wavelet basis in L 2 (R 2 ). The masks a and ã are called low-pass filters and the masks a 1,..., a m 1, ã 1,..., ã m 1 are called high-pass filters in the engineering literature. The equation in (23) is called the matrix extension problem in the wavelet literature (see [4, 7, 10, 48]). Given a pair of dual masks a and ã, though the existence of â1,..., âm 1, ã 1,..., ã m 1 (without symmetry) in the matrix extension problem in (23) is guaranteed by the Quillen-Suslin theorem, it is far from a trivial task to construct them (in particular, with symmetry) algorithmically. For the orthogonal case, the general theory of the matrix extension problem in (23) with ã = a and ã l = a l, l = 1,..., m 1, remains unanswered. However, when detm = 2, the matrix extension problem is trivial. In fact, letting γ Γ M T \{0} (that is, Γ M T = {0, γ}, one defines â1 (ξ) := e iη ξˆã(ξ + 2πγ) and ã 1 (ξ) := e iη ξ â(ξ + 2πγ), where η Z 2 satisfies η γ = 1/2. It is not an easy task to construct nonseparable orthogonal masks with desirable properties for a general dilation matrix. For example, for the dilation matrix Q 2 in (1), it is not known so far in the literature ([17]) whether there is a compactly supported C 1 Q 2-refinable function with orthonormal shifts. See [11, 12, 17, 27, 29, 32, 34, 52, 53, 56, 59] for more details on construction of orthogonal masks and orthonormal refinable functions. However, for any dilation matrix, separable orthogonal masks with arbitrarily high orders of sum rules can be easily obtained via the projection method. Let M be a 2 2 dilation matrix. Then M = Ediag(d 1, d 2 )F for some integer matrices E and F such that dete = detf = 1 and d 1, d 2 N. Let a be a tensor product orthogonal mask with the

10 diagonal dilation matrix diag(d 1, d 2 ) satisfying the sum rules of any preassigned order J. Then the projected mask Ea (that is, (Ea) k := a E 1 k, k Z 2 ) is an orthogonal mask with dilation M and sr(ea, M) J. See [29, Corollary 3.4] for more detail. Using the high-pass filters in the matrix extension problem, for a given mask a, dual masks ã (without any guaranteed order of sum rules of the dual masks) of a can be obtained by the lifting scheme in [64] and the stable completion method in [13]. Special families of dual masks with sum rules can be obtained by the convolution method in [26, Proposition 3.7] and [45], but the constructed dual masks generally have longer supports with respect to their orders of sum rules. Another method, which we shall cite here, for constructing all finitely supported dual masks with any preassigned orders of sum rules is the CBC (coset by coset) algorithm proposed in [14, 27, 28]. For µ = (µ 1, µ 2 ) and ν = (ν 1, ν 2 ), we say ν µ if ν 1 µ 1 and ν 2 µ 2. Also we denote µ = µ 1 + µ 2, µ! := µ 1!µ 2! and x µ := x µ 1 1 x µ 2 2 for x = (x 1, x 2 ). We denote Ω M a complete set of representatives of distinct cosets of Z 2 /[MZ 2 ] with 0 Ω M. The following result is from [27]. Theorem 13. (CBC Algorithm) Let M be a 2 2 dilation matrix and a be an interpolatory mask with dilation M. Let J be any preassigned positive integer. (1) Compute the quantities h a µ, µ < J, from the mask a by the recursive formula: h a µ := δ µ ( 1) µ ν µ! ν!(µ ν)! ha ν a k k µ ν. 0 ν<µ k Z 2 (2) For every nonzero coset γ Ω M \{0} and pre-selected subsets E γ Z 2, construct ã on the coset γ + MZ 2, with ã γ+mk = 0 for k Z 2 \E γ, satisfying the equation: k E γ ã γ+mk (γ + Mk) µ = detm 1 h a µ, µ < J. (3) Construct ã at the zero coset MZ 2 by ã Mj := detm 1 δ k a Mk Mj+γ ã γ+mk, j Z 2. k E γ γ Ω M \{0} Then mask ã is a dual mask of a and satisfies the sum rules of order J with dilation M. For a given interpolatory mask a and an arbitrary integer J, in fact any finitely supported dual mask ã of a, with ã satisfying the sum rules of order J, must come from the above CBC algorithm. The CBC algorithm also works well with symmetry. When a is symmetric, a dual mask ã with symmetry and sum rules can be easily obtained via the CBC algorithm by appropriately selecting the sets E γ. A general CBC algorithm is given in [28, Page 33 and Theorem 3.4] for constructing finitely supported dual masks ã with arbitrarily high orders of sum rules as long as the primal mask a possesses at least one finitely supported dual mask which may not have any order of sum rules. See [14, 27, 28] for more details on the CBC algorithm. The projection method can be used to obtain some optimal properties of bivariate orthonormal and biorthogonal wavelets ([27, 30, 36, 37]). Theorem 14. Let M = di 2 and (a, ã) be a pair of dual masks with dilation M. Let P = [1, 0]. Suppose that the mask a is projectable, that is, â(ξ, γπ/d) = 0 for all ξ R and γ = 1,..., d 1. Then (P a, P ã) is also a pair of dual masks with dilation d. If (21) holds

11 for φ and φ, where φ and φ are standard M-refinable functions with masks a and ã, then (21) still holds with φ and φ being replaced by P φ and P φ. Moreover, ν p (φ) ν p (P φ), ν p ( φ) ν p (P φ), ν p (a, di 2 ) ν p (P a, d) and ν p (ã, di 2 ) ν p (P ã, d) for all 1 p. Let us present a simple example to illustrate the CBC algorithm and the projection method on construction and analysis of bivariate biorthogonal wavelets. Example 15. Let M = 2I 2 and a be the interpolatory mask given in (18). Note that the mask a is projectable. Let P = [1, 0]. Then P a(ξ) = (2 cos(ξ)) cos 4 (ξ/2). By the CBC algorithm, we see that the shortest symmetric dual mask b of P a such that b satisfies the sum rules of order at least one is supported on [ 4, 4] and is uniquely given by ˆb(ξ) := cos 2 (ξ/2)[14 5 cos(ξ) 2 cos(2ξ) + cos(3ξ)]/8. By calculation, we have ν 2 (b, 2) 0.59322. Take J = 2 since sr(b, 2) = 2. By the CBC algorithm, any D 6 - symmetric dual mask ã of a, such that ã is supported inside [ 4, 4] 2 and sr(ã, 2I 2 ) = 2, must take the form 0 0 0 0 t 2 0 1 2t 2 0 t 2 0 0 0 0 t 3 t 3 t 3 t 3 0 1 64 0 0 1 2t 2 t 3 t 1 t 4 t 1 t 3 1 2t 2 0 0 t 3 t 4 28 + 4t 1 28 + 4t 1 t 4 t 3 0 t 2 t 3 t 1 28 + 4t 1 10 6t 1 + 6t 2 28 + 4t 1 t 1 t 3 t 2 0 t 3 t 4 28 + 4t 1 28 + 4t 1 t 4 t 3 0 0 1 2t 2 t 3 t 1 t 4 t 1 t 3 1 2t 2 0 0 0 t 3 t 3 t 3 t 3 0 0 0 0 t 2 0 1 2t 2 0 t 2 0 0 0 0 with t 3 := 10 + 2t 1 + 4t 2 and t 4 := 30 6t 1 4t 2, where ã (0,0) = 10 6t 1 + 6t 2. By Theorem 14 and P ã = b, we have ν 2 (ã, 2I 2 ) ν 2 (b, 2) 0.59322. Take t 1 = 17/4 and t 2 = 1/2, by calculation, we have ν 2 (ã, 2I 2 ) 0.56715. So, the conditions in Theorem 11 are satisfied and ã is near best in terms of the L 2 smoothness with respect to the support of the dual mask. VI. Bivariate Riesz Wavelets Wavelet-based numerical algorithms have been successfully used in numerical solutions to partial differential equations and integral equations ([5, 6]). The wavelets used in such wavelet-based methods are Riesz wavelets. For analysis on Riesz wavelets, see [16, 35, 42, 43, 44, 49, 58]. For applications based on Riesz wavelets, see [5, 6, 44, 51]. Theorem 16. Let M be a 2 2 dilation matrix with m := detm and a be a finitely supported bivariate mask. Let â1,..., âm 1 be some 2π-periodic trigonometric polynomials such that det M [â, â 1,...,âm 1 ] (ξ) 0 for all ξ R2 and â1 (0) = = âm 1 (0) = 0. Let φ be the standard refinable function associated with mask a and dilation M. If ν 2 (a, M) > 0 and ν 2 (ã, M) > 0, where ˆã is the (1, 1)-entry of the matrix [M [â, â 1,...,âm 1 ] (ξ)t ] 1, then {ψ 1,..., ψ m 1 }, which are defined in (25), generates a Riesz M-wavelet basis in L 2 (R 2 ). Example 17. Let M = 2I 2 and â(ξ 1, ξ 2 ) := cos 2 (ξ 1 /2) cos 2 (ξ 2 /2) cos 2 (ξ 1 /2 + ξ 2 /2) in Example 4. Then sr(a, 2I 2 ) = 4 and a is D 6 -symmetric with center 0. Define ([44, 61]) â 1 (ξ 1, ξ 2 ) := e i(ξ 1+ξ 2)â(ξ 1 + π, ξ 2 ), â 3 (ξ 1, ξ 2 ) := e iξ1â(ξ 1 + π, ξ 2 + π). â 2 (ξ 1, ξ 2 ) := e iξ2â(ξ 1, ξ 2 + π),

12 Then all the conditions in Theorem 16 are satisfied and {ψ 1, ψ 2, ψ 3 } generates a Riesz 2I 2 - wavelet basis in L 2 (R 2 ) ([44]). This Riesz wavelet derived from the Loop scheme has been used in [51] for mesh compression in computer graphics with impressive performance. VII. Pairs of Dual Wavelet Frames In this section, we mention a method for constructing bivariate dual wavelet frames. The following Oblique Extension Principle (OEP) has been proposed in [23] (and independently in [15]. Also see [22]) for constructing pairs of dual wavelet frames. Theorem 18. Let M be a 2 2 dilation matrix and a, ã be two finitely supported masks. Let φ, φ be the standard M-refinable functions with masks a and ã, respectively, such that φ, φ L 2 (R 2 ). If there exist 2π-periodic trigonometric polynomials Θ, â1,..., âl, ã 1,..., ã L such that Θ(0) = 1, â1 (0) = = âl (0) = ã 1 (0) = = ã L (0) = 0, and M [Θ(M T )â,â1,...,âl ] (ξ)m [ˆã, ã 1,..., ã L ] (ξ)t = diag(θ(ξ + γ 0 ),..., Θ(ξ + γ m 1 )), (26) where {γ 0,..., γ m 1 } = Γ M T with γ 0 = 0 in Theorem 11. Then ({ψ 1,..., ψ L }, { ψ 1,..., ψ L }), defined in (25), generates a pair of dual M-wavelet frames in L 2 (R 2 ). For dimension one, many interesting tight wavelet frames and pairs of dual wavelet frames in L 2 (R) have been constructed via the OEP method in the literature, for more details, see [15, 22, 23, 26, 32, 54, 62, 63] and references therein. The application of the OEP in high dimensions is much more difficult, mainly due to the matrix extension problem in (26). The projection method can also be used to obtain pairs of dual wavelet frames ([36, 37]). Theorem 19. Let M be an r r dilation matrix and N be an s s dilation matrix with r s. Let P be an r s integer matrix of rank r such that MP = P N and P T (Z r \[M T Z r ]) Z s \[N T Z s ]. Let ψ 1,..., ψ L, ψ 1,..., ψ L be compactly supported functions in L 2 (R s ) such that ν 2 (ψ l ) > 0 and ν 2 ( ψ l ) > 0 l = 1,..., L. If ({ψ 1,..., ψ L }, { ψ 1,..., ψ L }) generates a pair of dual N-wavelet frames in L 2 (R s ), then ({P ψ 1,..., P ψ L }, {P ψ 1,..., P ψ L }) generates a pair of dual M-wavelet frames in L 2 (R r ). Example 20. Let Ξ be an r s (direction) integer matrix such that Ξ T (Z r \[2Z r ]) Z s \[2Z 2 ]. Let M = 2I r and N = 2I s. Let {ψ 1,..., ψ 2s 1 } be the generators of the tensor product Haar orthonormal wavelet in dimension s, derived from the Haar orthonormal refinable function φ := χ [0,1] s. Then by Theorem 19, {Ξψ 1,..., Ξψ 2s 1 } generates a tight 2I r -wavelet frame in L 2 (R r ) and all the projected wavelet functions are derived from the refinable box spline function Ξφ = M Ξ. VIII. Future Directions. There are still many challenging problems on bivariate wavelets. In the following, we only mention a few here. (1) For any 2 2 dilation matrix M, e.g., M = Q 2 in (1), can one always construct a family of MRA compactly supported orthonormal M-wavelet bases with arbitrarily high smoothness (and with symmetry if detm > 2)? (2) The matrix extension problem in (23) for orthogonal masks. That is, for a given orthogonal mask a with dilation M, find finitely supported high-pass filters a 1,..., a m 1 (with symmetry if possible) such that (23) holds with ã = a and ã l = a l.

13 (3) The matrix extension problem in (23) for a given pair of dual masks with symmetry. That is, for a given pair of symmetric dual masks a and ã with dilation M, find finitely supported symmetric high-pass filters a 1,..., a m 1, ã 1,..., ã m 1 such that (23) is satisfied. (4) Directional bivariate wavelets. In order to handle edges of different orientations in images, directional wavelets are of interest in applications. See [20, 57] and many references on this topic. IX. Bibliography Books and Reviews 1. Subdivision for Modeling and Animation, course notes by T. DeRose, D. R. Forsey, L. Kobbelt, M. Lounsbery, J. Peters, P. Schröder, and D. Zorin, (1998) 2. C. Cabrelli, C. Heil and U. Molter, Self-similarity and multiwavelets in higher dimensions, Mem. Amer. Math. Soc. 170 (2004), no. 807. 3. A. S. Cavaretta, W. Dahmen and C. A. Micchelli, Stationary subdivision, Mem. Amer. Math. Soc. 93 (1991), no. 453. 4. C. K. Chui, An introduction to wavelets. Academic Press, Inc., Boston, MA, (1992). 5. A. Cohen, Numerical analysis of wavelet methods, North-Holland Publishing Co., Amsterdam, (2003). 6. W. Dahmen, Wavelet and multiscale methods for operator equations, Acta Numer. 6 (1997), 55 228. 7. I. Daubechies, Ten Lectures on Wavelets, CBMS-NSF Series, SIAM, Philadelphia, 1992. 8. C. de Boor, K. H ollig and S. D. Riemenschneider, Box splines. Springer-Verlag, New York, (1993). 9. N. Dyn and D. Levin, Subdivision schemes in geometric modeling, Acta Numerica, 11 (2002), 73 144. 10. S. Mallat, A wavelet tour of signal processing, Academic Press, Inc., San Diego, CA, (1998). Primary Literature 11. A. Ayache, Some methods for constructing nonseparable, orthonormal, compactly supported wavelet bases, Appl. Comput. Harmon. Anal. 10 (2001), 99 111. 12. E. Belogay and Y. Wang, Arbitrarily smooth orthogonal nonseparable wavelets in R 2, SIAM J. Math. Anal. 30 (1999), 678 697. 13. J. M. Carnicer, W. Dahmen and J. M. Peña, Local decomposition of refinable spaces and wavelets, Appl. Comput. Harmon. Anal. 3 (1996), 127 153. 14. D. R. Chen, B. Han and S. D. Riemenschneider, Construction of multivariate biorthogonal wavelets with arbitrary vanishing moments, Adv. Comput. Math. 13 (2000), 131 165. 15. C. K. Chui, W. He and J. Stöckler, Compactly supported tight and sibling frames with maximum vanishing moments, Appl. Comput. Harmon. Anal. 13 (2002), 224 262. 16. A. Cohen and I. Daubechies, A new technique to estimate the regularity of refinable functions, Rev. Mat. Iberoamericana 12 (1996), 527 591. 17. A. Cohen and I. Daubechies, Nonseparable bidimensional wavelet bases, Rev. Mat. Iberoamericana 9 (1993), 51 137. 18. A. Cohen, I. Daubechies and J.C. Feauveau, Biorthogonal bases of compactly supported wavelets, Comm. Pure Appl. Math. 45 (1992), 485 560. 19. A. Cohen, K. Gröchenig and L. F. Villemoes, Regularity of multivariate refinable functions, Constr. Approx. 15 (1999), 241 255.

14 20. A. Cohen and J. M. Schlenker, Compactly supported bidimensional wavelet bases with hexagonal symmetry, Constr. Approx. 9 (1993), 209 236. 21. I. Daubechies, Orthonormal bases of compactly supported wavelets, Comm. Pure Appl. Math. 41 (1988), 909 996. 22. I. Daubechies and B. Han, Pairs of dual wavelet frames from any two refinable functions, Constr. Approx. 20 (2004), 325 352. 23. I. Daubechies, B. Han, A. Ron, and Z. Shen, Framelets: MRA-based constructions of wavelet frames, Appl. Comput. Harmon. Anal. 14 (2003), 1 46. 24. S. Dahlke, K. Gröchenig and P. Maass, A new approach to interpolating scaling functions, Appl. Anal. 72 (1999), 485 500. 25. N. Dyn, J. A. Gregory and D. Levin, A butterfly subdivision scheme for surface interpolation with tension control, ACM Trans. Graphics, 9 (1990), 160 169. 26. B. Han, On dual wavelet tight frames, Appl. Comput. Harmon. Anal. 4 (1997), 380 413. 27. B. Han, Analysis and construction of optimal multivariate biorthogonal wavelets with compact support, SIAM Math. Anal. 31 (2000), 274 304. 28. B. Han, Approximation properties and construction of Hermite interpolants and biorthogonal multiwavelets, J. Approx. Theory 110 (2001), 18 53. 29. B. Han, Symmetry property and construction of wavelets with a general dilation matrix, Linear Algebra Appl. 353, (2002), 207 225. 30. B. Han, Projectable multivariate refinable functions and biorthogonal wavelets, Appl. Comput. Harmon. Anal. 13 (2002), 89 102. 31. B. Han, Computing the smoothness exponent of a symmetric multivariate refinable function, SIAM J. Matrix Anal. Appl. 24 (2003), 693 714. 32. B. Han, Compactly supported tight wavelet frames and orthonormal wavelets of exponential decay with a general dilation matrix, J. Comput. Appl. Math. 155 (2003), 43 67. 33. B. Han, Vector cascade algorithms and refinable function vectors in Sobolev spaces, J. Approx. Theory 124 (2003), 44 88. 34. B. Han, Symmetric multivariate orthogonal refinable functions, Appl. Comput. Harmon. Anal., 17 (2004), 277 292. 35. B. Han, On a conjecture about MRA Riesz wavelet bases, em Proc. Amer. Math. Soc., 134 (2006), 1973-1983. 36. B. Han, The projection method in wavelet analysis, Mod. Methods Math., G. Chen and M.J. Lai eds, Nashboro Press, Brentwood, TN, (2006), 202 225. 37. B. Han, Construction of wavelets and framelets by the projection method, International J. Math. Sci., 1 (2007), to appear. 38. B. Han and R. Q. Jia, Multivariate refinement equations and convergence of subdivision schemes, SIAM J. Math. Anal. 29 (1998), 1177 1199. 39. B. Han and R. Q. Jia, Optimal interpolatory subdivision schemes in multidimensional spaces, SIAM J. Numer. Anal. 36 (1999), 105 124. 40. B. Han and R. Q. Jia, Quincunx fundamental refinable functions and quincunx biorthogonal wavelets, Math. Comp. 71 (2002), 165 196. 41. B. Han and R. Q. Jia, Optimal C 2 two-dimensional interpolatory ternary subdivision schemes with two-ring stencils, Math. Comp. 75 (2006), 1287 1308. 42. B. Han and R. Q. Jia, Characterization of Riesz bases of wavelets generated from multiresolution analysis, Appl. Comput. Harmon. Anal., to appear.

15 43. B. Han and Z. Shen, Wavelets with short support, SIAM J. Math. Anal. 38 (2006), 530 556. 44. B. Han and Z. Shen, Wavelets from the Loop scheme, J. Fourier Anal. Appl. 11 (2005), 615 637. 45. H. Ji, S. D. Riemenschneider and Z. Shen, Multivariate compactly supported fundamental refinable functions, duals, and biorthogonal wavelets, Stud. Appl. Math. 102 (1999), 173 204. 46. R. Q. Jia, Approximation properties of multivariate wavelets, Comp. Math. 67 (1998), 647-665. 47. R. Q. Jia, Characterization of smoothness of multivariate refinable functions in Sobolev spaces, Trans. Amer. Math. Soc. 351 (1999), 4089 4112. 48. R. Q. Jia and C. A. Micchelli, Using the refinement equation for the construction of pre-wavelets II: Power of two, Curves and Surfaces (P.J. Laurent, A. Le Méhauté and L. L. Schumaker eds.), Academic Press, New York, (1991), 209 246. 49. R. Q. Jia, J. Z. Wang, and D. X. Zhou, Compactly supported wavelet bases for Sobolev spaces, Appl. Comput. Harmon. Anal. 15 (2003), 224 241. 50. Q. T. Jiang, On the regularity of matrix refinable functions, SIAM J. Math. Anal. 29 (1998), 1157 1176 51. A. Khodakovsky, P. Schröder, and W. Sweldens, Progressive geometry compression, Proceedings of SIGGRAPH 2000. 52. J. Kovačević and M. Vetterli, Nonseparable multidimensional perfect reconstruction filter banks and wavelet bases for R n, IEEE Trans. Inform. Theory 38 (1992), 533 555. 53. M. J. Lai, Construction of multivariate compactly supported orthonormal wavelets, Adv. Comput. Math. 25 (2006), 41 56. 54. M. J. Lai and J. Stöckler, Construction of multivariate compactly supported tight wavelet frames, Appl. Comput. Harmon. Anal. 21 (2006), 324 348. 55. W. Lawton, S. L. Lee and Z. Shen, Convergence of multidimensional cascade algorithm, Numer. Math. 78 (1998), 427 438. 56. W. Lawton, S. L. Lee and Z. Shen, Stability and orthonormality of multivariate refinable functions, SIAM J. Math. Anal. 28 (1997), 999 1014. 57. E. Le Pennec and S. Mallat, Sparse geometric image representations with bandelets, IEEE Trans. Image Process. 14 (2005), 423 438. 58. R. Lorentz and P. Oswald, Criteria for hierarchical bases in Sobolev spaces, Appl. Comput. Harmon. Anal. 8 (2000), 32 85. 59. P. Maass, Families of orthogonal two-dimensional wavelets, SIAM J. Math. Anal. 27 (1996), 1454 1481. 60. S. D. Riemenschneider and Z. Shen, Multidimensional interpolatory subdivision schemes, SIAM J. Numer. Anal. 34 (1997), 2357 2381. 61. S. D. Riemenschneider and Z. W. Shen, Wavelets and pre-wavelets in low dimensions, J. Approx. Theory, 71 (1992), 18 38. 62. A. Ron and Z. Shen, Affine systems in L 2 (R d ): the analysis of the analysis operator, J. Funct. Anal. 148 (1997), 408 447. 63. A. Ron and Z. Shen, Affine systems in L 2 (R d ) II: dual systems, J. Fourier Anal. Appl. 3 (1997), 617 637. 64. W. Sweldens, The lifting scheme: a custom-design construction of biorthogonal wavelets, Appl. Comput. Harmon. Anal. 3 (1996), 186 200.