Littlewood Paley Spline Wavelets

Transcription

1 Proceedings of the 6th WSEAS International Conference on Wavelet Analysis & Multirate Systems, Bucharest, Romania, October 6-8, 26 5 Littlewood Paley Spline Wavelets E. SERRANO and C.E. D ATTELLIS Escuela de Ciencia y Tecnología Universidad Nacional de General San Martín M. Irigoyen 3 - (65) San Martín - Pcia de Buenos Aires ARGENTINE eserrano@unsam.edu.ar ; ceda@favaloro.edu.ar Abstract: In this work we propose a simple and efficient method to compute the Semi-discrete Wavelet Transform using appropriate spline wavelets in the Littlewood Paley decomposition scheme. These wavelets are compactly supported and oscillating functions. Moreover, the scheme can be refined in the frequency domain and a fine approximate on the Continuous Transform is achieved. Key-Words:- Littlewood Paley Analysis - Wavelet Transform - Spline functions - Spline wavelets Introduction Nowadays, wavelets play a significant role in pure and applied mathematics. In physics or signal processing applications wavelet analysis can be consider as a time-scale technique. Therefore,for many problems, it performs better than Fourier methods. However, it is a multifaceted tool. For instance, the Discrete Wavelet Transform (DWT) is correlated with the atomic decomposition of the given signal, using an orthonormal wavelet basis in the context of a multiresolution scheme. On the other hand, when the shiftinvariance is needed or large scales phenomena are as important as small scales ones, the correct tool for analyzing is the Continuous (CWT) or the Semi-Discrete Wavelet Transform (SDWT). Recently, Littlewood - Paley analysis seems to have the better performance for analizing turbulent phenomena, fractional Brownian processes or frequency modulated signals (chirps). We refer to [4,5] for details. Littlewood-Paley analysis can be considered as a particular semi-discrete wavelet transform. Following the above mentioned reference [5], let us give a quick survey. Let a function ϕ S, the Schwartz class in L 2 (R), and assume that its Fourier transform ϕ vanishes outside the interval ω 2π and ϕ(ω) =on ω π. Given a signal s, we define, in the frecuency domain, the convolution operator L j, for each j Z, as: [L j f](ω) = ϕ(2 j ω)ŝ(ω) and the difference operator: Observe that D j s = (L j+ L j )s D j s = (2 (j+) ϕ(2 j+.) 2 j ϕ(2 j.)) s and defining the Littlewood-Paley wavelet: we have: ψ(x) = 2ϕ(2x) ϕ(x) D j s =2 j ψ(2 j.) s In summary, we have the remarkable decomposition: s = L s + j= D j s where each component is a convolution and its Fourier transform is concentrated in the two side band 2 j π ω 2 j+ π.

2 Proceedings of the 6th WSEAS International Conference on Wavelet Analysis & Multirate Systems, Bucharest, Romania, October 6-8, 26 6 The function ψ is and admissible wavelet. On the other hand, we remark that ϕ is not compactly supported because its Fourier transform has finite support. Furthemore, in general, the function ϕ doesn t verify a finite double-scale equation. These circunstances may be a drawback for the numerical applications. For these reasons we explore alternatives. Here we propose to replace the function ϕ in the Schwartz class for an appropriate spline function φ with compact support and oscillating. As we expect, using splines we can exploit self- similarity properties, particularly two scale relations. This give us numerical adventages. Indeed, beyond the semidiscrete transform and the diadic scales, those properties led us to design a refinated scheme of analysis in the frecuency domain, in a tight approximate of the Continuous Wavelet Transform, [3]. 2. Basic Functions and Wavelets For integer odd numbers m, the Q-spline functions are defined from [,7]: ( ) senω/2 m+ Q m (ω) = ω/2 or, in the time domain: Q m (x) = Q... Q }{{ (x) } m+ times being Q the characteristic function of [ /2, /2]. Let us enphatize that Q m can be avaluated using closed formulas, [7]. The functions Q m are symmetrical, supported on the intervall [ (m+)/2, (m+)/2] and their Fourier transforms, Q m (ω) are well concentrated on [ π, π] and they decay as ω (m+). Moreover, we reacall that each family {Q m (x k); k Z}is a Riesz Basis for the subspace of spline functions of order m with integer knots and finite energy. We refer to [,3] for details. Next, for each m and any positive integer n we define φ = φ m,n, the basic spline function as: φ(ω) = Q m (ω)f m,n (ω) where: F m,n (ω) = n f m,n [] + f m,n [k] cos(kω) k= and the coefficients f m,n give us the unique solution for the linear equations: φ() =, φ (2i) () = ; i =,...,n The basic function φ V m is a nice alternative for the function ϕ S. Depending on m and n, it is supported on the interval [ n (m +)/2, n +(m +)/2] and φ is an almost ideal low pass-filter on [ π, π]. Following, we define ψ = ψ m,n, the associate Littlewood-Paley wavelet, as: ψ(x) = 2φ(2x) φ(x) This is a spline function of order m with knots in Z/2. It is symmetric, centered on x = and it is also supported on [ n (m+)/2,n+ (m +)/2]. The Fourier transform ψ(ω) is well localized on the two side band π/2 ω π and it decays as / ω m+. It has a pic in ω pic.5π. Since ψ (l) () = ; l =,...,2n + the wavelet has 2n + null moments. In summary, the proposed spline wavelets seem similar to the original ones in the Schwartz class. We expect that they will constitute an efficient tool for the applications. 3. Semidiscrete Transform We recall that, for odd integers m, thetwo scale relations for the Q-spline are: Q m (2ω) = Q m (ω) cos m+ (ω/2) Q m (x/2) = 2 k (m+)/2 h m [k]q m (x k)

3 Proceedings of the 6th WSEAS International Conference on Wavelet Analysis & Multirate Systems, Bucharest, Romania, October 6-8, 26 7 where: h m [k] = ( 2 m+ m + k +(m +)/2 Then, for each ψ = ψ m,n we can write: where ψ(ω) = Q m (ω/2)g(ω) G(ω) = F m,n (ω/2) F m,n (ω) cos m+ (ω/4) This is a 4π-periodic trigonometrical polynomial of the form: G(ω) = g m,n [p] exp( ipω/2) where P =2n +(m +)/2 and its coefficients can be easily computed from the above formulas. Finally, we simply obtain: ψ(x) = 2 g m,n [p]q m (2x p) Given m and n and a locally integrable signal s, the Semidiscrete Transform, Ds = D m,n s, is defined as: D j s =2 j ψ(2 j.) s for j Z. As usual, we will compute the transform for j =, 2,...,J min. Assume that m and n are chosen and denote: Then, recalling that: 2 j ψ(x) = 2 j+ we have: D j s(x) =2 j+ S j = Q m (2 j.) s ) g m,n [p]q m (2 j+ x p) g m,n [p]s j+ (x p2 (j+) ) Particullarly, for integer values q: D j s(q) = 2 j+ g m,n [p]s j+ (q p2 (j+) ) Now using the two-scale relation for the Q m spline we deduce the recursive scheme: S j (q) = 2 h m [k]s j+ (q 2 ( j+) k) k (m+)/2 for any q Z. At this point we recall that these discrete convolutions are computed intercaling 2 ( j+) zeros in the filter g m,n or h m. In analogous form we deduce the recursive scheme to compute the convolutions L j s: L j s(q) = 2 j b m,n [p]s j (q 2 (j) p) where b m,n [] = f m,n [] and b m,n [p] = /2f m,n [ p ]. Therefore, we can decompose the signal in the frequency components D j s. Also we obtain the filtered signal L j s. In the first step we must compute the initial values S [q]. Many alternatives are opened for this purpose. One simple choice, assuming that s[k] are the sampled valued of the signal, is to compute the discrete convolution: S [q] = k Q m [k]s[q k] Observe that, in this case, S j, L j s and D j s are spline functions of order m. Particullarly, L s is a representation of the signal in V m. In the opposite, we can reconstruct this representation as: L [q] = j=j min D j [q]+l Jmin [q] As it is well known the semidiscrete transform gives us a shift-invariant time-scale decomposition of the signal. It is a recommended tool for some applications, as turbulence analysis or pattern recognition, for instance. Followig we seek for a refined scale providing a more precise transform in the frequency domain. 4. Extended Scheme Assume that the sampled rate is R and we start with the sampled values s[k/r].

4 Proceedings of the 6th WSEAS International Conference on Wavelet Analysis & Multirate Systems, Bucharest, Romania, October 6-8, 26 8 Now, for j and r =,...R, define: j r = j + log 2 (R/(R + r)) Note that j j r <j and f(2 j rx) =f(2 j (R/(R + r)x) Also we remak that (j +) r = j r +. In accordance with these definitions, we have: φ jr (x) =2 jr φ(2 jr x) ψ jr (x) =2 jr ψ(2 jr x) and the associated transforms L jr s and D jr s. Now the filters ψ jr (ω) are well localized on the two side bands 2 jr π ω 2 jr π and their pics are ω jr,pic 3.2 jr π. Denoting: S jr = Q m (2 jr.) s in analogous way we deduce that: S jr (x) = 2 h m [k]. and for integer values q: k (m+)/2. S jr+(x 2 (jr+) k) D jr s(q/r) = 2 jr+ g m,n [p].. S jr+(q/r p2 (jr+) ) L jr s(q/r) = 2 jr b m,n [p].. S jr (q/r 2 (jr) p) At this point let us summarize: Alghoritm: Given the signal s decide the parameters m, n and R decide J min <, q min q q max. Then: For j =and r<r, Compute S r [q/r] For j =, 2,...,J min, and r< R, Compute L jr+s[q/r] Compute D jr s[q/r] Compute S jr [q/r] To compute the initial values S r [q/r], we propose the model: s(x) = R s[k/r]δ(x/r k) k Then: S r (x) = R Q m (.) s(x) R + r = R R s[k/r](δ(./r k) Q m ( R + r.) k and finally: S r (x) = R k s[k/r]q m ( q k R + r ) The values Q m ((q k)/(r+r)) can be easely computed usig the closed formulas for the Q m splines. We also remark that, according with our proposal, S j r, L jr s and D jr s are spline functions of order m with knots in Z/R. we can reconstruct the averaged representation of the signal: L r [q/r] = R R r= j r=j min,r + L Jmin,r [q/r] D jr [q/r]+ 5. Example For instance, let us consider the case m =3 and n = 2. First, the explicit formulas for the cubic Q 3 -spline, are: 2 x 3 x x 3 Q 3 (x) = 6 x 3 + x 2 2 x + 4 x 2 3 otherwise It is supported on [ 2, 2]. The two scale coefficientes are: h 3 [] = 6/8

5 Proceedings of the 6th WSEAS International Conference on Wavelet Analysis & Multirate Systems, Bucharest, Romania, October 6-8, h 3 [ ] = h 3 [] = /2 h 3 [ 2] = h 3 [2] =/ In accordance we have the basic function: φ 3,2 (x) = b 3,2 [k]q 3 (x k) k Figure : Function φ 3,2 (x) where b 3,2 [] = 2/72 b 3,2 [ ] = b 3,2 [] = 7/6 b 3,2 [ 2] = h 3,2 [2] = 8/ and ψ 3,2 (x) = k 6 g 3,2 [k]q 3 (2x k) Figure 2: Wavelet ψ 3,2 (x) where g 3,2 [] = 33/6 g 3,2 [ ] = g 3,2 [] = 283/24 g 3,2 [ 2] = g 3,2 [2] = 5/92 g 3,2 [ 3] = g 3,2 [3] = 6/48 g 3,2 [ 4] = g 3,2 [4] = 3/96 g 3,2 [ 5] = g 3,2 [5] = 7/48 g 3,2 [ 6] = g 3,2 [6] = 7/ Figure 3: Fourier Transform φ 3,2 (ω) We illustrate these functions in the figures. and 2. The respective Fourier transforms are showed in the figures 3. and 4. Finally, we give a simple application. Figure 5. shows the chirp: s(x) = cos( x ) Figure 4: Fourier Transform ψ 3,2 (ω) We choice R = 7 and compute the transform D jr for j =,..., 5 and r =,...,6on the interval [, 5]. See figure 5. As expected, the maxima modulus map give us a shape conciding with the instantaneous frequency og the chirp, [3].

6 Proceedings of the 6th WSEAS International Conference on Wavelet Analysis & Multirate Systems, Bucharest, Romania, October 6-8, Figure 5: Function chirp cos( x ) Schonberg I., Cardinal Spline Interpolation, SIAM, Philadelphia, M. Unser, A. Aldroubi and M. Eden, B- Spline Signal Proccesing: Part I- Theory, IEEE Trans. Acoust., Speech and Signal Process. 4, N o 2, pp , M. Unser, A. Aldroubi and M. Eden, B- Spline Signal Proccesing: Part II- Efficient Design and Applications, IEEE Trans. Acoust., Speech and Signal Process. 4, N o 2, pp , M. Unser, A. Aldroubi, Polynomial Splines and Wavelets-A Signal Processing Perpective in Wavelets: A Tutorial in Theory and Applications, San Diego: Ac. Press. Inc., pp 72-9, Figure 6: Wavelet transform and instantaneous frequencies (solid line) 6. Acknowledgments The authors are thankful to Fundación Antorchas ( ) for the support. ( ) This work was supported by Fundación Antorchas (Argentine). References:. Blu T., Splines: a perfect fit for signal and image processing, IEEE Signal and Image Processing Magazine, Vol. 6, No 6, Daubechies I.,, Ten Lectures on Wavelets,SIAM, Philadelphia, Mallat M., A Wavelet Tour of Signal Processing, Academic Press, Philadelphia, Meyer Y., Wavelets, Algorithms and Applications,SIAM, Philadelphia, Meyer Y., Oscillating Patterns in Image Processing and Nonlinear Evolutions Equations, University Lecture Series, AMS, Serrano E. and Fabio F., Undecimated Wavelet Transform from Orthogonal Spline Wavelets in Wavelets Theory and Harmonic Analysis in Applied Sciences, Birkhauser Ed., 996.