Quadrature Prefilters for the Discrete Wavelet Transform. Bruce R. Johnson. James L. Kinsey. Abstract

Transcription

1 Quadrature Prefilters for the Discrete Wavelet Transform Bruce R. Johnson James L. Kinsey Abstract Discrepancies between the Discrete Wavelet Transform and the coefficients of the Wavelet Series are known to be reducible by initialization of input data. Prefilters based on Lagrange interpolants are derived here for biorthogonal compact support wavelet systems, providing exact subspace proection in cases of local polynomial smoothness. The resulting convergence acceleration in a non-polynomial test case is examined. Irregular sampling rates are also accommodated. * This work was supported by the Robert A. Welch Foundation and by Grant CHE from the National Science Foundation. * The authors are with the Department of Chemistry and the Rice Quantum Institute, MS #600, Rice University, Houston, TX EDICS Classification: SP I. Introduction The use of FIR filters based on compact support wavelet families allows for efficient digital multiresolution analysis of time series and other signals [1-4]. A general signal f ( t) may be expanded approximately in a finite Wavelet Series (WS) ( ) f t 0 φ k, f 0 φk ( t ) + ψ k, f ψ k ( t ), (1) k J 1 =0 where the synthesis scaling functions obey k

2 φ k ( t ) = 2 /2 φ( 2 /2 t k ), (2) and similar relations hold for the wavelets ψ k and the biorthogonal analysis functions φ k and ψ k. All of the expansion coefficients in subspaces J 1,J 2,...,0 can be generated from the proection integrals φ k J, f using the two-scale relations [5] φ k 1 t ( ) = ( t), ψ 1 k t k h k φ 2k+ k ( ) = ( t). (3) k g k ψ 2k+ k Thus it is the evaluation of the integrals φ k J, f from discrete input data which is addressed here. Their accurate calculation is important in order to exploit the computational efficiency of the Discrete Wavelet Transform (DWT) for signal analysis. For nonsingular functions f ( t), one may approximate the integrals on the finest octave by simple function samples, φ J k, f f k / 2 J ( ) = [ k]. (4) This underlies the standard Mallat Algorithm [2] [6], fast pyramidal filtering of an input data sequence. Shensa [7] and Rioul and Duhamel [8] clarified the point that, for general compact support wavelets, this procedure does not precisely produce the coefficients of the WS. The essential difficulty is that of ensuring accurate proection to V J, the subspace spanned by the φ k J, when limited to knowledge only of discrete data that is not strongly oversampled. The Shensa Algorithm [7] [8] focuses instead on a function ˆ ( t) = [ k] χ J ( t k ), (5) k which is defined in terms of the J level samples and a sampling function χ J designed to make ˆ in some manner close to f. The V J proections of this function are then given by φ J k, f ˆ J = [ l] dt φ J k t l ( ) χj t l ( ). (6) 2

3 J One approach takes χ J ( t l )=φ l ( t ), for which the right hand side of Eq. (6) reduces to [ k]; that is, ˆ ( t) is entirely in V J and its expansion coefficients are exactly given by the samples of f ( t) [7]. A different approach requires the samples of f ( t) and ˆ ( t) to coincide, using interpolating functions with the property χ J ( n) =δ n,0. Thus, for instance, the use of sinc functions has been explored [9] [10]. For this case, the integral in Eq. (6), which constitutes a prefiltering of the data, is more complicated. Other choices of prefiltering have also been examined, in some cases optimized to the specific data [11] [Chan, 1996 #29]. II. Lagrange Interpolation It is natural to adopt polynomial interpolation here since the scaling functions span loworder polynomials by construction. Such a procedure is directly connected to numerical quadrature estimation of the proection integrals [12-16]. The interpolants in this case are Lagrange polynomials, already familiar in wavelet-related contexts [17] [7] [Donoho, 1992 #30]. A unique level of generality may be obtained here since it is not necessary to assume uniform sample spacing for the derivation [16]. Thus, for some ordered set of distinct points t q in the neighborhood of a particular φ k J, ˆ r ( ) ( t) = f t q L r,q ( t ), (7) q=1 where the Lagrange polynomials are L r,q ( t ) = r q q t t q t q t q (8) If f ( t) is a polynomial of order less than r, then ˆ ( t) exactly equals f ( t); otherwise they are only equal at the sample points. A different set of points is generally to be used for the next value of k, though the sets may overlap; this reduces to a simple unit translation [cf., Eq. (5)] in the uniform-spacing case. The general-spacing analog of Eq. (6) is now 3

4 r ( ) dt φ J k, f ˆ J J = f t q φ k ( t ) L r,q t q=1 ( ) r ( ) = f t q q=1 r 1 p= 0 1 p! ( p) L r,q ( k /2 J ) dt t k /2 J ( ) p ( ) φ k J t r ( ) = ω q f t q, (9) q=1 where r 1 ω q = 2 J/2 Jp m p p! p= 0 ( p) L r,q ( k /2 J ), (10) m p = dt t p φ ( t ) = 1 2 p p 1 ( ) 1 p p m p µ p p, (11) p =0 µ p = 1 2 L 1 h k k p. (12) k=0 Equation (9) reflects the well-known connection between Lagrange interpolation of a function and numerical quadrature of its integrals. The quadrature weights ω q (related to those derived by Sweldens and Piessens [14]) are obtained by Taylor expansion of the L r,q around the position of φ k J. The scaling function moments m p can be iteratively calculated [Eq. (11)] using the discrete moments µ p as shown in [12]. Thus, the weights are broken into parts m p (calculated once) ( p) specific to the wavelet basis and L r,q specific to k and the associated points tq. If the latter have ( p) the same relative configuration for each k, as in the case of uniform sample spacing, the L r,q also only need to be calculated once. We will concentrate on the uniform case below. III. Simulation Results For testing purposes, the symmetric biorthogonal wavelet system with recursion lengths ( L, L) = (9, 7) described by Cohen, et al. [5] and Antonini, et al. [18] is used. Symmetry forces 4

5 the odd moments to be exactly zero here in contrast to the situation for orthogonal wavelet families [1], [12]. A damped and chirped sine wave, ( ) = e t 2 /8 sin t ( 1 + t ) f t [ ], (13) is chosen as a test function and illustrated in Fig. 1. There are several oscillations within the support, rendering inaccurate any few-point quadrature at the = 0 level. To avoid undue complexity in the quadrature computations, samples were chosen symmetrically distributed around the center of the support (2 J k) and with spacing 2 J. For odd r, these sets were of the form 2 J k, 2 J (k±1), 2 J (k±2), etc. For even r, they were of the form 2 J (k±1/2), 2 J (k±3/2), etc. Table 1 shows the values of the = k = 0 integral φ, f calculated by quadrature on different octaves J and iterating with the low-pass filter h k back to = 0. Natural sampling [cf. Eq. (4)] is equivalent to quadrature of order r = 1; a 7-point series of calculations is also shown. Both series ultimately lead to the same final result, but natural sampling proceeds at a very slow rate by comparison. The rate of convergence has been examined as a function of both quadrature order r and starting octave J. Figure 2 shows the relative error obtained for odd r with respect to the converged value For J = 0 (no recursion), the error is always large, as expected, except for the fortuitously accurate case r = 3; higher order r accomplishes nothing except to place new sample points outside of the range of importance of f(x) since the spacing is large on this scale. The picture changes dramatically with higher J, where strong dependence of the accuracy on r is found. One finds from detailed examination the following two patterns: for a given even value of r (not shown here), the asymptotic error decreases as 2 Jr with increasing J while, for a given odd value of r, the error decreases as 2 J(r +1). The former case is the expected asymptotic behavior [19], [14]. The accelerated convergence in the odd case can be understood from consideration of Eq. (9) in the case of monomials centered around k /2 J r ( ) p ω q t q k / 2 J = 2 J/2 Jp m p, 0 p r 1. (14) q =1 5

6 The symmetrical distribution of sample points and the vanishing of m r for odd r leads to automatic satisfaction of one higher equation (p = r ). (This situation is similar to an accelerated convergence found with a specific one-point quadrature for Daubechies scaling functions [12].) For r = 1(no prefiltering), the asymptotic convergence rate is 4 J rather than 2 J since the sample is taken at the center of the support. For higher r the convergence is very strong, drastically reducing the number of scales that must be traversed to meet a given convergence threshold. IV. Conclusion Where are these considerations important in practical applications? They are certainly critical in numerical analysis, solution of partial differential equations and some cases of spectral analysis where natural sampling would lead to the need for excessive refinement in pursuit of convergence. On the other hand, many signal and image processing applications do not seek such accuracy as a result of noise, inherent limits of precision, and/or lack of any known functions underlying discrete experimental data. In such cases one may use the standard DWT with natural sampling and regard the resulting multiresolution analysis as a sampling of the exact wavelet transform of the function ˆ J ( t)= [ k]φ k ( t ), as discussed above. This has the appearance of k reducing the matter to one of interpretation. In actuality, however, the behavior of ˆ output of its DWT are sensitive to the value o until the scale is so fine that all relevant frequencies are resolved (as in the first column of Table 1). The occasionally-encountered ( t) and the statement that single wavelets do not require use of prefilters is really a statement about the case for sufficiently large J. In many cases where the sampling rate has a hard upper limit, the proectionbased prefilter described above can be regarded as extrapolating the ordinary DWT algorithm to higher resolution. The order r 1of polynomials for which the Lagrange-polynomial-based prefilter gives exact subspace proection is not limited by the approximating power of the wavelets chosen. In this sense, r is a design parameter. To some extent, the same may be said of the sample positions used. Numerical experiments verify a certain level of robustness of the calculated WS coefficients 6

7 with respect to irregular sampling. This can be useful for specific applications, e.g., unevenly spaced antenna arrays, or for generic situations such as drop-outs in data sequences. As expected for any polynomial interpolation, there is reasonable stability as long as sample points do not approach each other too closely and the interpolated point is roughly within their range. On the other hand, numerical experiments including strong noise contamination show no greater stability than for natural sampling, indicating an avenue needing further exploration. Acknowledgment Many thanks are due to R. O. Wells, Jr., C. S. Burrus, J. E. Odegard, and other members of the Computational Mathematics Laboratory and the Digital Signal Processing group at Rice University for extended interactions. Useful conversations with W. Sweldens and G. Strang are also gratefully acknowledged. References [1] I. Daubechies, Ten Lectures on Wavelets, vol. 61. Philadelphia: SIAM Publications, [2] S. G. Mallat, A theory for multiresolution signal decomposition: The wavelet representation, IEEE Trans. Pattern Anal. Machine Intell., vol. 11, pp , [3] G. Strang and T. Nguyen, Wavelets and Filter Banks. Wellesley: Wellesley-Cambridge Press, [4] C. S. Burrus, R. A. Gopinath, and H. Guo, Introduction to Wavelets and Wavelet Transforms: Prentice-Hall, [5] A. Cohen, I. Daubechies, and J.-C. Feauveau, Biorthogonal Bases of Compactly Supported Wavelets, Comm. Pure Appl. Math., vol. 45, pp , [6] S. Mallat, Multifrequency channel decomposition of images and wavelet methods, IEEE Trans. Acoust. Speech, Signal Processing, vol. 37, pp , [7] M. J. Shensa, The Discrete Wavelet Transform: Wedding the À Trous and Mallat Algorithms, IEEE Trans. Signal Proc., vol. 40, pp , [8] O. Rioul and P. Duhamel, Fast Algorithms for Discrete and Continuous Wavelet Transforms, IEEE Trans. Inform. Theory, vol. 38, pp ,

8 [9] P. Abry and P. Flandrin, On the Initialization of the Discrete Wavelet Transform Algorithm, IEEE Sig. Proc. Lett., vol. 1, pp , [10] X.-P. Zhang, L.-S. Tian, and Y.-N. Peng, From the Wavelet Series to the Discrete Wavelet Transform - The Initialization, IEEE Trans. Signal Proc., vol. 44, pp , [11] X.-G. Xia, C.-C. J. Kuo, and Z. Zhang, Wavelet Coefficient Computation with Optimal Prefiltering, IEEE Trans. Signal Proc., vol. 42, pp , [12] R. A. Gopinath and C. S. Burris, On the moments of the scaling function φ, presented at Proceedings of the ISCAS-92, San Diego, [13] G. Beylkin, R. Coifman, and V. Rokhlin, Fast Wavelet Transforms and Numerical Algorithms I, Comm. Pure and Appl. Math, vol. 44, pp , [14] W. Sweldens and R. Piessens, Quadrature Formulae and Asymptotic Error Expansions for Wavelet Approximations of Smooth Functions, SIAM J. Numer. Anal., vol. 31, pp , [15] W. Sweldens and R. Piessens, Asymptotic error expansion of wavelet approximations of smooth functions II, Numerische Mathematik, vol. 68, pp. 377, [16] B. R. Johnson, J. P. Modisette, P. J. Nordlander, and J. L. Kinsey, Quadrature Integration for Orthogonal Wavelet Systems, J. Chem. Phys., vol. 110, pp , [17] G. Deslauriers and S. Dubuc, Symmetric Iterative Interpolation Processes, Constr. Approx., vol. 5, pp , [18] M. Antonini, M. Barlaud, P. Mathieu, and I. Daubechies, Image Coding Using Wavelet Transform, IEEE Trans. Image Proc., vol. 1, pp , [19] G. Strang, Wavelets and dilation equations: A brief introduction, SIAM Review, vol. 31, pp ,

9 Table Table 1. Integral φ 0 0, f for the symmetric (9,7) biorthogonal Wavelet System from Refs. [5] and [18] and the signal of Eq. (13), as calculated starting from various octaves J. J Natural Sampling Wavelet quadrature (r = 7) Figures t Figure 1. Damped and chirped sine wave. 9

10 J = 0 J = 1 J = 2 J = 3 J = Quadrature Order r Figure 2. Relative error φ [, f φ, f calc exact ] / φ, f exact for different quadrature orders and different finest scales used with recursion. The function f ( t) is as given in Fig