410 CHAPTER 6, WAVELET TRANSFORM Equations (6.38) and (6.34) allow us to write Subtracting Eq. (6.39) from Eq. (6.37) and rearranging gives where Thus, the projection of / onto W m +i is representable as a linear combination of translates and dilates of the mother function ip(t). Another important observation is the relationships between the wavelet and scaling coefficients at scale ra + 1 and the scaling coefficient at the finer scale m. We have seen that and Prom Eqs. (6.38) and (6.40) we conclude that c m+ i jn and d m+ i, n can be obtained by convolving c m>n with -\/2ho(n) and T/2hi(n), respectively, followed by a 2- fold down-sampling as shown in Fig. 6.9. Hence the interscale coefficients can be represented by a decimated two-band filter bank. The output of the upper decimator represents the coefficients in the approximation of the signal at scale m + 1, while the lower decimator output represents the detail coefficients at that scale. In the next section, we show that any orthonormal wavelet of compact support can be representable in the form of the two-band unitary filter bank developed here. More interesting wavelets with smoother time-frequency representation are also developed in the sequel. 6.2. MULTIRESOL UTION SIGNAL DECOMPOSITION 411 Figure 6.9: Interscale coefficients as a two-band filter bank. 6.2.3 Two-Band Unitary PR-QMF and Wavelet Bases Here we resume the discussion of the interscale basis coefficients in Eq. (6.26). But first, we must account for the time normalization implicit in translation. Hence, with <j>(t) 4 —» $($!) as a Fourier Transform pair, we then have and Taking the Fourier transform of both sides of Eq. (6.26) gives Now with uj = OTo as a normalized frequency and H.Q(e^} as the transform of the sequence {ho(n)} J we obtain 4 The variables O and u in this equation run from — oo to oo. In addition, H^(eP^} is periodic with period 27r. Similarly, for the next two adjacent resolutions, 4 We will use fi as the frequency variable in a continuous-time signal, and u for discrete-time signals, even though Jl — u for TO = 1. 412 CHAPTER 6. WAVELET TRANSFORM Therefore, $(O) of Eq. (6.43) becomes Note that Ho(e^) has a period of 8?r. If we repeat this procedure infinitely many times, and using lim n _*oo O/2 n = 0, we get $(fi) as the iterated product We can show that the completeness property of a rmiltiresolution approxima- tion implies that any scaling function satisfies a nonzero mean constraint (Prob. 6.1) If (j)(t) is real, it is determined uniquely, up to a sign, by the requirement that 4>0n(t) be orthonormal. Therefore, and which is equivalent to Hence the Fourier transform of the continuous-time scaling function is obtained by the infinite resolution product of the discrete-time Fourier transform of the interscale coefficients {ho(n}}. If the duration of the interscale coefficients {ho(n)} is finite, the scaling function cf)(i) is said to be compactly supported. Furthermore, if ho(n) has a duration 0 < n < N — 1, then <f>(t] is also supported within 0 < t < (N — l)Tb- (Prob. 6.2) For convenience, we take Xb = 1 in the sequel (Daubechies, 1988). In the Haar example, we had N — 2, and the duration of ho(n) was 0 < n < 1; accordingly, the support for <p(t) is 0 < t < 1. Next we want to find the constraints Ho(e^ U} ) must satisfy so that (p(t] is a scaling function, and for any given scale ra, the set {4>mn(k)} is orthonormal. In 6.2. MULTIRESOLUTION SIGNAL DECOMPOSITION 413 particular, if {<p(t — n)} spans VQ, then we show in Appendix B that the corre- sponding <J»(Q) must satisfy the unitary condition in frequency Next, after substituting into the preceding orthonormality condition arid after some manipulations (Prob. 6.4), we obtain This can be rewritten as an even and odd indexed sum, This last equation yields the magnitude square condition of the interscale coeffi- cient sequence {ho(n)}, This is recognized as the low-pass filter requirement in a maximally decimated unitary PR-QMF of Eq. (3.129). We proceed in a similar manner to obtain filter requirements for the orthonormal wavelet bases. First, it is observed that if the scaling function (j)(i) is compactly supported on [0, N — 1], the corresponding wavelet ijj(t) generated by Eq. (6.27) is compactly supported on [1 — y, y], Again, for the Haar wavelet, we had N = 2. In that case the duration of h\(ri) is 0 < n < 1, as is the support for ^(t}. Letting h\(n) ^—^ H\(e^} and transforming Eq. (6.27) gives the transform of the wavelet as r^ If we replace the second term on the right-hand side of this equation with the infinite product derived earlier in Eq. (6.44), The orthonormal wavelet bases are complementary to the scaling bases. These satisfy the intra- and interscale orthonormalities 4.14 CHAPTER 6. WAVELET TRANSFORM where m and k are the scale and n and / the translation parameters. Notice that the orthoriormality conditions of wavelets hold for different scales, in addition to the same scale, which is the case for scaling functions. Since {tp(t — n)} forms an orthonormal basis for WQ, their Fourier transforms must satisfy the unitary As before, using in this last equation leads to the expected result or which corresponds to the high-pass requirement in the two-channel unitary PR- QMF, Eq. (3.129). Finally, these scaling and wavelet functions also satisfy the orthonormality condition between themselves, Note that the orthonormality of wavelet and scaling functions is satisfied at dif- ferent scales, as well as at the same scale. This time-domain condition implies its counterpart in the frequency-domain as Now, if we use Eqs. (6.44), (6.50), and (6.54) we can obtain the frequency- domain condition for alias cancellation [see Eq. (3.130)]: or The three conditions required of the transforms of the interscale coefficients, {ho(n}} and {/ii(n)| in Eqs. (6.49), (6.53), and (6.55) in the design of compactly supported orthonormal wavelet and scaling functions are then equivalent to the 6.2. MULTIRESOLUTION SIGNAL DECOMPOSITION 415 requirement that the alias component (AC) matrix HAC(^ U ) of Chapter 3 for the two-band filter bank case, be paraunitary for all a;. In particular, the cross-filter orthonormality, Eq. (6.55), is satisfied by the choice or in the time-domain, In addition, since and we have already argued that then Hi(e ju; ) must be a high-pass filter with Thus the wavelet must be a band-pass function, satisfying the admissibility con- dition j Therefore, HQ(Z) and H\(z) must each have at least one zero at z ~ — I and z = 1, respectively. It is also clear from Eq. (6.57) that if ho(n) is FIR, then so is hi(n). Hence the wavelet function is of compact support if the scaling function is. In summary, compactly supported orthonormal wavelet bases imply a parau- nitary, 2-band FIR PR-QMF bank; conversely, a paraunitary FIR PR-QMF filter pair with the constraint that HQ(Z) have at least one zero at z = — 1 imply a compactly supported orthonormal wavelet basis (summarized in Table 6.1). This is needed to ensure that ^(0) — 0. Orthonormal wavelet bases can be constructed by multiresolution analysis, as described next. 416 CHAPTER 6. WAVELET TRANSFORM Table 6.1: Summary of relationships between paraunitary 2-band FIR PR-QMF's and compactly supported orthonormal wavelets. 6.2.4 Multiresolution Pyramid Decomposition The multiresolution analysis presented in the previous section is now used to de- compose the signal into successive layers at coarser resolutions plus detail signals, also at coarser resolution. The structure of this multiscale decomposition is the same as the pyramid decomposition of a signal, described in Chapter 3. Suppose we have a function / 6 VQ. Then, since {(f>(t ~n)} spans VQ, / can be represented as a superposition of translated scaling functions: Next, since VQ — V\ ® W\, we can express / as the sum of two functions, one lying entirely in V\ and the other in the orthogonal complement W\: 6.2, MULTIRESOL UTION SIGNAL DECOMPOSITION 417 Here, the scaling coefficients CI >H and the wavelet coefficients d\^ n are given by In the example using Haar functions, we saw that for a given starting sequence {co,n}> the coefficients in the next resolution {ci >n } and {d\^ n } can be represented, respectively, as the convolution of co, n with HQ = /IQ(—n) and of co, n with h\(n) — /ii(—ri), followed by down-sampling by 2. Our contention is that this is generally true. To appreciate this, multiply both sides of Eq. (6.62) by (j>i n (t) and integrate But fw(t) is a linear combination of {V ; ifc(^)} 5 each component of which is orthog- onal to (f>i n (t). Therefore, the second inner product in Eq. (6.64) is zero, leaving us with This last integral is zero by orthogonality.) Therefore, Therefore, 418 CHAPTER 6. WAVELET TRANSFORM Figure 6.11: First stage of multiresolution signal decomposition. In a similar way, we can arrive at Figure 6.10 shows twofold decimation and interpolation operators. So our last two equations define convolution followed by subsampling as shown in Fig. 6.11. This is recognized as the first stage of a subband tree where {ho(n), h\(n}} consti- tute a paraunitary FIR pair of filters. The discrete signal d\^ n is just the discrete wavelet transform coefficient at resolution 1/2. It represents the detail or differ- ence information between the original signal co, n and its smoothed down-sampled approximation ci >n . These signals c\^ n and di >n are said to have a resolution of 1/2, if co, n has unity resolution. Every down-sampling by 2 reduces the resolution by that factor. The next stage of decomposition is now easily obtained. We take f£ € V\ — V<2 © W-2 and represent it by a component in ¥2 and another in W%: 6.2. MULTIRESOLUTION SIGNAL DECOMPOSITION 419 Following the procedure outlined, we can obtain the coefficients of the smooth- ed signal (approximation) and of the detail signal (approximation error) at reso- lution 1/4: These relations are shown in the two-stage multiresolution pyramid displayed in Fig. 6.12. The decomposition into coarser, smoothed approximation and detail can be continued as far as we please. Figure 6.12: Multiresolution pyramid decomposition. To close the circle we can now reassemble the signal from its pyramid decom- position. This reconstruction of C0 )n , from its decomposition CI )TI , and d\^ n can be achieved by up-sampling and convolution with the filters /IQ(W), and h\(n) as in Fig. 6.13. This is as expected, since the front end of the one-stage pyramid is simply the analysis section of a two-band, PR-QMF bank. The reconstruction therefore must correspond to the synthesis bank. To prove this, we need to rep- resent /io(?0 and h\(n) in terms of the scaling and wavelet functions. Note that —• J\ analysis filters /i^(n) = hi(—ri) as shown are anticausal when synthesis filters are causal. Recall that {/io(n)|, {/ii(n)}, are the interscale coefficients [...]... 00 892 06 0043042 24 493 1 0012206 1 294 92 -0.0063 696 01011 -0.00182045 891 6 0.000 790 205101 0.0003 296 65174 -0.000050 192 775 -0.000024465734 Uitt f 433 ¥*) 0018563 -.5 492 41 15 797 9 0012787 -.230 390 0282787 0 092 178 399 241 -0.0476 395 90310 0 6 2 5 4 6 4 0 895 92 0232063 0224535 701455 710408 0548335 -.5 497 87 76 298 7 001 292 4 02461 19 •.1007 59 98 796 6 0017212 -.505002 00486 79 -.406841 0022406 0.016744410163 0036831 097 863... pp 2 091 -2110, Dec 198 9 S Mallat, ' 'Multiresolution Approximations and Wavelet Orthonormal Bases of L2(#):' Trans Amer Math Soc., Vol 315, pp 69- 87, Sept 198 9 S G Mallat Ed., A Wavelet Tour of Signal Processing Academic Press, 199 8 A Mertins, Signal Analysis Wiley 199 9 Y Meyer, Ondelettes et Operateurs Tome I Herrmann Ed., 199 0 Y Meyer, Wavelets: Algorithms and Applications SIAM, 199 3 S H Nawab and T... and New Results," Proc IEEE ICASSP, pp 1723-1726, 199 0 M Vetterli and C Herley, "Wavelets arid Filter Banks: Theory and Design," IEEE Trans Signal Processing, Vol 40, No 9, pp 2207 2232, Sept 199 2 M Vetterli and J Kovacevic, Wavelets and Subband Coding Prentice-Hall, 199 5 H Volkmer, "Asymptotic Regularity of Compactly Supported Wavelets, " SIAM J Math Anal, Vol 26, pp 1075-1087, 199 5 J, E Younberg and. .. Press, 199 8 M V Tazebay and A N Akansu, "Progressive Optimality in Hierarchical Filter Banks," Proc IEEE ICIP, pp 825-8 29, 199 4 A H Tewfik and P E Jorgensen, "The Choice of a Wavelet for Signal Coding and Processing," Proc IEEE ICASSP, 199 1 A H Tewfik and M Y Kim, "Multiscale Statistical Signal Processing Algorithms," Proc IEEE ICASSP, pp IV 373-376, 199 2 M Vetterli and C Herley, "Wavelets and Filter... Meyer, and L Raphael, Eds., Wavelets and Their Applications Jones and Bartlett, Boston MA 199 2 442 CHAPTER 6 WAVELET TRANSFORM M J Shensa, "The Discrete Wavelet Transform: Wedding the a trous and Mallat Algorithms," IEEE Trans Signal Processing, Vol 40, No 10, pp 2464 2482, Oct 199 2 G Strarig and T Nguyen, Wavelets and Filter Banks Wellesley-Cambridge Press, 199 6 B Suter, Multirate and Wavelet Signal. .. Grossman, and P Tchamitchian, Eds Wavelets, TimeFrequency Methods and Phase Space Springer-Verlag, 198 9 R E Crochiere and L R Rabiner, Multirate Digital Signal Processing Prentice-Hall, 198 3 I Daubechies, "Orthonormal Bases of Compactly Supported Wavelets, " Communications in Pure and Applied Math., Vol 41, pp 90 9 -99 6, 198 8 I Daubechies, "The Wavelet Transform, Time-Frequency Localization and Signal Analysis,"... Akarisu and M J Medley, Eds., Wavelet, Subband and Block Transforms in Communications and Multimedia Kluwer, 199 9 A N Akansu and M J T Smith, Eds., Subband arid Wavelet Transforms: Design and Applications Kluwer, 199 6 A N Akansu, R A Haddad, and H Caglar, "Perfect Reconstruction Binomial QMF-Wavelet Transform," Proc SPIE Visual Communication and Image Processing, Vol 1360, pp 6 09- 618, Oct 199 0 A N Akansu,... Operators," Proc IEEE ICASSP 199 1 R A Gopinath, J E Odegard, and C S Burrus, "On the Correlation Structure of Multiplicity M Scaling Functions and Wavelets, " Proc IEEE ISCAS, pp 95 9 -96 2, 199 2 P Goupillaud, A Grossmann, and J Morlet, "Cycle-Octave and Related Transforms in Seismic Signal Analysis," Geo€;xploration, Vol 23, pp 85 102 Elsevier Science Pub., 198 4 A Grossmann and J Morlet, "Decomposition of Hardy... in Signal Processing, J S Lim and A V Oppenheirn, Eds Prentice-Hall, 198 8 A Papoulis, Signal Analysis McGraw-Hill, 197 7 M R Portnoff, "Time-Frequency Representation of Digital Signals and Systems Based on the Short-Time Fourier Analysis," IEEE Trans, on ASSP, Vol 28, pp 55- 69 Feb 198 0 O Rioul arid M Vetterli, "Wavelets and Signal Processing," IEEE Signal Processing Magazine, Vol 8, pp 14-38, Oct 199 1... Heil and D F Walnut, "Continuous and Discrete Wavelet Transforms," SIAM Review, Vol 31, pp 628 666, Dec 198 9 IEEE Trans, on Information Theory, Special Issue on Wavelet Transforms and Multiresolution Signal Analysis, March 199 2 IEEE Trans, on Signal Processing, Special Issue on Theory and Application of Filter Banks and Wavelet Transforms April 199 8 6.5, DISCUSSIONS AND CONCLUSION 441 A J E M Janssen, . stage of decomposition is now easily obtained. We take f£ € V — V<2 © W-2 and represent it by a component in ¥2 and another in W%: 6.2. MULTIRESOLUTION SIGNAL DECOMPOSITION . supported orthonormal wavelets. 6.2.4 Multiresolution Pyramid Decomposition The multiresolution analysis presented in the previous section is now used to de- compose the signal into successive. coefficients, {ho(n}} and {/ii(n)| in Eqs. (6. 49) , (6.53), and (6.55) in the design of compactly supported orthonormal wavelet and scaling functions are then equivalent to the 6.2. MULTIRESOLUTION