A computer algebra approach to orthonormal wavelets Fr´ ed´ eric Chyzak
[email protected] INRIA Rocquencourt, F–78153 Le Chesnay, France Peter Paule
[email protected] Research Institute for Symbolic Computation Johannes Kepler University, A–4040 Linz, Austria Otmar Scherzer
[email protected] Institut f¨ ur Angewandte Mathematik LMU–M¨ unchen, D–80333 M¨ unchen, Germany On leave from Industrial Mathematics Institute, Linz, Austria Armin Schoisswohl
[email protected] Industrial Mathematics Institute Johannes Kepler University, A–4040 Linz, Austria Burkhard Zimmermann
[email protected] Research Institute for Symbolic Computation Johannes Kepler University, A–4040 Linz, Austria February 16, 2000
1
Introduction
This paper is based on [CPSSZ00] which is to appear in Experimental Mathematics. The name wavelet was made up by French researchers [MAFG82, Mor83, GM84] for a particular class of functions. The existence of wavelet-like functions has been known since the beginning of the century (a notable example is what is known as the Haar wavelet today [Haa10]). However, only recently the unifying concepts necessary for a general understanding of wavelets were provided
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[Mey85, Mal86, Dau88, Bat87, Lem88]. Since then there have been numerous contributions to wavelets, both in theory and applications. Wavelets are widely used in many practical applications such as data compression [BBH93, DJL92] and the solution of partial differential equations [Jaf92], to give a few references. We attempt to provide the basics of wavelets with the emphasize on the needs of computer algebra researchers who may wish to apply their area to wavelet research. The biggest barrier to enter wavelet research are the constitutive equations. A vector solving the algebraic system of constitutive equations uniquely determines a wavelet function. The solution vector is called the refinement mask, the components of the vector are called filter coefficients. In practical applications, any algorithm relying on wavelets uses the filter coefficients only and not the wavelet function itself. In this paper we review the constitutive equations and outline how this equations can be solved by using computer algebra, in particular using Gr¨ obner bases. Gr¨obner bases also showed up in a different setting in the design of two-dimensional wavelets [FMR00]. The most popular wavelets due to Daubechies [Dau88] form an orthonormal basis for the space of square integrable functions on the real line. Following a construction of Meyer [Mey91] we outline the construction of filter coefficients of wavelets on (compact) intervals using a matrix analytical approach. Again, by this approach the filter coefficients can be calculated using computer algebra.
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Orthonormal wavelets on R
The construction of compactly supported orthonormal wavelet bases for L2 (R) is well understood (see e.g. [Dau88, Dau92, CD93, Dau93]). It is related to the construction of a scaling function φ, which satisfies that for fixed m ∈ Z the functions φm,k (x) := 2−m/2 φ(2−m x − k), k ∈ Z are orthonormal with respect to L2 (R). Moreover, the spaces Vm := span{φm,k , k ∈ Z} constitute a multiresolution analysis for L2 (R), i.e., Vm ⊂ Vm−1 ,
for m ∈ Z,
with \
Vm = {0} and
[
Vm = L2 (R) .
Since Vm is contained in Vm−1 the scaling function φ must satisfy the so-called dilation equation X (1) φ(x) = hk φ(2x − k) . The wavelet spaces Wm are spanned by the functions ψm,k (x) 2−m/2 ψ(2−m x − k), k ∈ Z, where the mother wavelet ψ satisfies X (2) ψ(x) = (−1)k hk φ(2x − k) .
:=
This in particular guarantees that Wm is the orthogonal complement of Vm in Vm−1 . 2
The existence of a compactly supported smooth scaling function satisfying that its integer translates are orthonormal can be expressed in terms of the filter coefficients {hk }. In the case of the famous Daubechies wavelets [Dau88, Dau92, CD93, Dau93] this leads to the following system of algebraic equations: X hk = 2 X (3) hk hk−2l = 2 δ0,l X (−1)k h1−k k l = 0, (l = 0, . . . , N − 1) where hk = 0 for k < 1 − N or k > N .
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Closed form representation of filter coefficients
Gr¨ obner bases are the obvious tools to use for solving systems of polynomial equations symbolically. After their original introduction by B. Buchberger [Buc65] to answer ideal-theoretic questions, the solving of algebraic systems was soon realized to be one of their natural domains of application [Buc70]. We illustrate the Gr¨ obner bases method for N = 2. In this case we are interested in all common roots of the five polynomials in the four variables h−1 , h0 , h1 , h2 : (4)
−2 + h−1 + h0 + h1 + h2 , −2 + h2−1 + h20 + h21 + h22 , h−1 h1 + h0 h2 , h−1 − h0 + h1 − h2 , 2 h−1 − h0 + h2 .
Let I be the ideal in the polynomial ring C[h−1 , h0 , h1 , h2 ] generated by the polynomials from (4). Applying Buchberger’s algorithm with respect to a certain order imposed on the monomials of C[h−1 , h0 , h1 , h2 ] (here: “lexicographic with h2 > h1 > h0 > h−1 ”) delivers an alternative description of the ideal I, namely by the generators: (5)
−1 − 4 h−1 + 8 h2−1 , −1 − 2 h−1 + 2 h0 , −1 + h−1 + h1 , −1 + 2 h−1 + 2 h2 .
The polynomials (5) again generate the ideal I and in particular share the same variety of common roots as the generators from (4). But additionally they form a Gr¨ obner basis of I. Due to the choice of a lexicographic monomial order, they furthermore possess the following “elimination property”: the first polynomial in the Gr¨ obner basis is a univariate polynomial (here in h−1 ), the second one a bivariate polynomial that involves only one further variable (here h−1 and h0 ), and so on. In other words, the role of the Gr¨obner basis algorithm in solving systems of algebraic equations is the same as that of Gaussian elimination in solving systems of linear equations, namely to triangularize the system or to carry out the elimination, respectively. Remarkably, in our situation of solving filter coefficient equations an even nicer pattern emerges. Namely, given the first univariate Gr¨ obner basis polynomial p1 (h−1 ) in h−1 only, the second Gr¨ obner basis polynomial is the sum of a univariate polynomial in h−1 and a linear polynomial in h0 ; the third Gr¨ obner basis 3
Figure 1: Daubechies scaling function (left) and wavelet (right) for N = 2 polynomial is the sum of a univariate polynomial in h−1 and a linear polynomial in h1 , and so on. So far we have strong computational evidence that this observation holds also for arbitrary N . We conclude our informal discussion on Gr¨ obner bases by stating the solution for N = 2 explicitly. In this case the first polynomial in the lexicographic √ Gr¨ obner basis is given by −1 − 4h−1 + 8h2−1 . Its real roots are (1 + 3)/4 and √ (1 − 3)/4. Therefore we obtain two solutions for the filter coefficients, namely: √ √ √ ! √ 1+ 3 3+ 3 3− 3 1− 3 , , , (h−1 , h0 , h1 , h2 ) = 4 4 4 4 and (h−1 , h0 , h1 , h2 ) =
√ √ √ √ ! 1− 3 3− 3 3+ 3 1+ 3 , , , . 4 4 4 4
Figure 1 shows plots of the corresponding scaling function and wavelet which are recursively defined by (1) and (2). Furthermore we remark that it is possible to transform the system (3) of 2N + 1 equations in 2N unknowns into an equivalent system of N equations in N unknowns, thus being able to reduce the computational effort dramatically. For further details on this topic we refer to [CPSSZ00].
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Wavelets on the interval
To our knowledge the first construction of orthogonal wavelets on subintervals is due to Meyer [Mey91]. His construction restricts compactly supported scaling functions and wavelets on R (as considered in section 2) to the subinterval of interest and orthonormalizes them. We consider wavelets on R+ : Let ( 0 if x < 0, half φm,k (x) := φm,k (x) if x ≥ 0. half The orthonormalization of φhalf m := {φm,k , k ∈ Z} is realized by a linear basis transform
φedge = A φhalf m m . 4
The basis transform A can be obtained from a Cholesky factorization Λ = A−1
A−1
t
,
D E half where the matrix Λ is given by the inner products Λk,l := φhalf m,k , φm,l . Note that due to the truncation and orthonormalization the scaling functions φedge m,k are no longer given by the translates and dilates of a single function. However, each scaling function fulfills a dilation equation X edge edge φedge Hk,l φm−1,l . m,k = l
edge In analogy to (2) the wavelets ψm,k on R+ satisfy edge ψm,k =
X
edge Gedge k,l φm−1,l .
l
The matrix Gedge can be computed from H edge using elementary linear algebra. For more details on the calculation of the matrices H edge and Gedge we refer to [CPSSZ00]. Our derivation of the refinement matrices H edge and Gedge for wavelets on the interval shows that each single step can be performed with computer algebra methods. This in turn shows that there exist closed form representations for the refinement matrices. For an application of wavelets on the interval to data compression we refer to [SSK98]. Figure 2 shows plots of the scaling functions and wavelets on R+ for N = 2.
Acknowledgement The work of the authors is partially supported by the Austrian Fonds zur F¨ orderung der Wissenschaftlichen Forschung, SFB 013, Grants F1305 and F1310.
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Figure 2: Daubechies scaling functions (left) and wavelets (right) on R+ for N = 2. In this case we have adapted scaling functions and wavelets at the boundary (k = 0, 1). The scaling functions and wavelets for k ≥ 2 are still given by the translates and dilates of the original functions φ and ψ, respectively.
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