2 DWT DWT (Complex Discrete Wavelet Transform CDWT) [ ] DWT Hilbert ( ) DWT DWT [8] CDWT Hilbert 1/2 2 Hilbert [9] CDWT [10] Meyer (Perfect Tran

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1 Translation-Invariance Complex Discrete Wavelet Transform (Zhong Zhang) * (Hiroshi Toda) * * (Toyohashi University of Technology) 1 (Discrete Wavelet Transform DWT) DWT Mallat[1] (Multi Resolution Analysis MRA) DWT [2] ( ) MRA DWT DWT Mallat[2] (Matching Pursuit) Femandes [3]

$Invariance PTI) Hilbert [11] ( ) ( ) 2 3 Hilbert [9] 4 CDWT 5 Meyer [11] CDWT 6 CDWT 2 $L^{2}(R)$ ( $R$ : ) $f(t)$ $\int_{-\infty}^{\infty} f(t) ^{2}dt<\infty$$

2 2 DWT DWT (Complex Discrete Wavelet Transform CDWT) [ ] DWT Hilbert ( ) DWT DWT [8] CDWT Hilbert 1/2 2 Hilbert [9] CDWT [10] Meyer (Perfect Translation Invariance PTI) Hilbert [11] ( ) ( ) 2 3 Hilbert [9] 4 CDWT 5 Meyer [11] CDWT 6 CDWT 2 $L^{2}(R)$ ( $R$ : ) $f(t)$ $\int_{-\infty}^{\infty} f(t) ^{2}dt<\infty$ (1)

3 3 21 $f(t)$ $g(t)$ $(f$ $g\}$ $\{f$ $g \rangle=\int_{-\infty}^{\infty}f(t)\overline{g(t)}dt$ (2) $\{f g\rangle$ $ \langle f(t) g(t)\}$ $\{f(u)$ $g(u)\rangle$ $\overline{g(t)}$ $g(t)$ $f(t)$ (Norm) $\Vert f\vert$ $\Vert f\vert=\sqrt{\langle ff\rangle}$ (3) $)$ $\psi_{k}(t)(k\in Z Z:$ $f(t)$ $g(t)= \sum_{k}\langle f$ $\psi_{k}\rangle\psi_{k}(t)$ (4) 1/2 $g(t)= \frac{1}{2}\sum_{k}\{f$ $\psi_{k}\rangle\psi_{k}(t)$ (5) $\psi_{k}(t)(k\in Z)$ $f(t)$ 22 $z$ $z$ $\{h_{n}\}$ $z$ $H_{z}(z)$ $H_{z}(z)$ $=$ $\sum_{n}h_{n}z^{-n}$ (6) $z=e^{i\omega}$ (6) $H_{\omega}(\omega)$ $H_{\omega}(\omega)$ $=$ $H_{z}(e^{i\omega})$ (7) $z$ $h_{n}$ $=$ $\frac{1}{2\pi i}\oint_{c}h_{z}(z)z^{n-1}dz$ (8)

4 4 $c$ 1 $H_{\omega}(\omega)$ $\{h_{n}\}$ $h_{n}$ $=$ $\frac{1}{2\pi}\int_{-\pi}^{\pi}h_{\omega}(\omega)e^{in\omega}d\omega$ (9) $\hat{f}(\omega)$ $f(t)$ $\hat{f}(\omega)$ $=$ $\int_{-\infty}^{\infty}f(t)e^{-i\omega t}dt$ (10) $f(t)$ $=$ $\frac{1}{2\pi}\int_{-\infty}^{\infty}\hat{f}(\omega)e^{i\omega t}dw$ (11) 23 (Kronecker delta) $\delta_{kl}(k l\in Z)$ $\delta_{kl}$ $=$ $\{\begin{array}{ll}1 k=l0 k\neq l\end{array}$ (12) (Dirac delta function) $\delta(t)$ $\delta(t)=\{\begin{array}{ll}\infty t=00 t\neq 0\end{array}$ (13) $\int_{-\infty}^{\infty}\delta(t)dt=1$ (14) $f(t)$ $\int_{-\infty}^{\infty}\delta(t)f(t)dt=f(0)$ (15) 24 Hilbert $\psi^{r}(t)$ $\psi^{i}(t)$ Hilbert [6 9] $\hat{\psi}^{i}(\omega)$ $=$ $\{\begin{array}{ll}i\hat{\psi}^{r}(\omega) \omega<00 \omega=0-i\hat{\psi}^{r}(\omega) \omega>0\end{array}$ (16) $\hat{\psi}^{r}(0)=\hat{\psi}^{i}(0)=0$ (17)

5 5 (16) (17) $\hat{\psi}^{r}(\omega)+i\hat{\psi}^{i}(\omega)=\{\begin{array}{l}0 \omega\leq 02 \hat{\psi}^{r}(\omega) \omega>0\end{array}$ (18) (18) $\psi^{r}(t)$ $\psi^{i}(t)$ Hilbert [6 9] Hilbert 3 Hilbert [9] $]$ 31 $[$ $\phi(t)=\sum_{n}p_{n}\phi(2t-n)$ (19) $\psi(t)=\sum_{n}q_{n}\phi(2t-n)$ (20) $\phi(t)$ $\{q_{n}\}$ $\psi(t)$ 2 $\{p_{n}\}$ $q_{n}=(-1)^{1-n}p_{1-n}$ (21) $\{\phi(t-k)$ $\phi(t-l)\rangle=\delta_{kl}$ (22) $\{\psi(t-k)$ $\psi(t-l)\}=\delta_{kl}$ (23) $\{\psi(t-k)$ $\phi(t-l)\}=0$ (24) 1 $\Vert\phi(t)\Vert=\Vert\psi(t)\Vert=1$ (25)

6 6 CDWT $n\in Z$ $\phi(n)$ $\{p_{n}\}$ $n_{0}\leq n\leq n_{1}\in Z$ (19) $0$ $\phi(t)$ [no $n_{1}$ ] $\phi(n)(n_{0}+1\leq n\leq n_{1}-1\in Z)$ (19) $t=n_{0}+1$ $n_{0}+2$ $\ldots$ $n_{1}-1$ $\phi(n_{0}+1)$ $\phi(n_{0}+2)$ $\ldots$ $\phi(n_{1}-1)$ $P(\phi(n_{1}-1)_{/}\phi(n_{0}+2)\phi(n_{0}+1)^{\backslash }=(\begin{array}{l}\phi(n_{0}+1)\phi(n_{0}+2)\vdots\phi(n_{1}-1)\end{array})$ (26) $P=[^{p}p_{n_{0}}^{no+1}0+3p_{n_{0}+5}p_{no+4}p_{no+2}pn_{0}p_{n_{0}+3}p_{no+1}0\cdot\cdot\cdotp_{n_{1}-1}0::)$ (27) (26) (27) $n\in Z$ $\phi(n)$ 1 $n-1 \sum_{n=no+1}^{1}\phi(n)$ $=$ $1$ (28) $\phi(n)$ $n\in Z$ 32 [9] $\psi^{r}(t)$ $\{p_{n}^{r}\}$ $\psi^{r}(t)$ Hilbert $\psi^{i}(t)$ $p_{n}^{i}$ $\psi^{r}(t)$ $\psi^{i}(t)$ Selesnick[6] $\psi^{r}(t)$ $\{p_{n}^{r}\}$ $\psi^{i}(t)$ $\{p_{n}^{i}\}$ $P_{\omega}^{I}(\omega)$ $=$ $P_{(d}^{R}(\omega)H_{\omega}(\omega)$ (29) $H_{d}(\omega)$ $=$ $e^{-i\omega/2}$ (30) 2 $\psi^{r}(t)$ $\psi^{i}(t)$ Hilbert $P_{\omega}^{R}(\omega)$ $P_{\omega}^{I}(\omega)$ $H_{\omega}(\omega)$ $\{p_{n}^{r}\}$ $\{p_{n}^{i}\}$ $\{h_{n}\}$

7 7 (6) (7) (30) (9) $H_{\omega}(\omega)$ $\omega$ $\{h_{n}\}$ $h_{n}$ $=$ $\frac{1}{2\pi}\int_{-\pi}^{\pi}h_{\omega}(\omega)e^{im_{4})}$ $= \frac{1}{2\pi}\int_{-\pi}^{\pi}e^{i(n-1/2)\omega}d\omega$ $=$ $\frac{1}{2\pi}[\frac{1}{i(n-1/2)}e^{i(n-1/2)\omega}]_{-\pi}^{\pi}$ $=$ $\frac{\sin\{(n-1/2)\pi\}}{(n-1/2)\pi}$ (31) (29) $p_{n}^{i}$ $=$ $\sum_{k}p_{k}^{r}h_{n-k}$ (32) $\{p_{n}^{i}\}$ $\{p_{n}^{r}\}$ (31) $\{h_{n}\}$ $\{p_{n}^{r}\}$ $\{p_{n}^{i}\}$ $\phi^{r}(t)$ $\phi^{i}(t)$ $\psi^{r}(t)$ $\psi^{i}(t)$ CDWT [ ] 1 Daubechies6 [12] $\{p_{n}^{r}\}(0\leq n\leq$ $11\in Z$ $\{p_{n}\}$ ) RI-Daubechies $0$ 6 1 (a) $\phi^{r}(t)$ $\phi^{i}(t)$ (b) $\psi^{r}(t)$ $\psi^{i}(t)$ (c) $ \hat{\psi}^{r}(\omega)+i\hat{\psi}^{i}(\omega) $ $\{p_{n}^{i}\}$ $0$ $n$ $-4\leq n\leq 15\in Z$ ( [9] ) $n$ $\psi^{r}(t)$ Hilbert (18) $\psi^{i}(t)$ $\omega<0$ $ \hat{\psi}^{r}(\omega)+i\hat{\psi}^{i}(\omega) =0$ 1 (c) $\omega<0$ $-660dB$ $0$ $\psi^{r}(t)$ $\psi^{i}(t)$ Hilbert 1 (a) $\phi^{r}(t)$ $\phi^{i}(t)$ 1/2 $-26dB$ ( $-1/2$ 2 $\phi^{i}(t)$ $\phi^{r}(t)$ $-26dB$ ) 1 [8] [9 10] Daubechies6 Daubechies 2 $\sim 10$ Symlet $4\sim 10$ Coiflet2 10[12] $\sim$ [9]

8 8 (a) Scaling function (b) Mother wavelet 2 $ \hat{\psi}^{r}(w)+i\hat{\psi}^{1}(a) $ 1 $0_{-0}$ - $0$ $w/\pi$ (c) Verification of Hilbert transform pair 1: RI-Daubechies 6 4 CDWT (CDWT) [10] CDWT CDWT 41 CDWT [10] $i$ $k$ $\phi_{jk}^{r}(t)$ $\phi_{jk}^{i}$ $\psi_{jk }^{R}$ $\psi_{jk}^{i}$ $\phi_{jk}^{r}(t)=$ $\phi_{jk}^{i}(t)=$ $2^{j}\phi^{R}(2^{j}t-k)$ (33) $2^{j}\phi^{I}(2^{j}t-k)$ (34) $\psi_{jk}^{r}(t)$ $=$ $\sqrt{2}j\psi^{r}(2^{j}t-k)$ (35) $\psi_{jk}^{i}(t)$ $=$ $\sqrt{2}j_{\psi^{i}(2^{j}t-k)}$ (36)

9 9 $g(t)$ CDWT $g(t)= \frac{1}{2}\sum_{jk}\{\{g$ $\psi_{jk}^{r}\}\psi_{jk}^{r}(t)+\langle g\psi_{jk}^{i}\rangle\psi_{jk}^{i}(t)\}$ (37) (37) $\infty$ (13) $\sim(15)$ $\delta(t)$ $f^{\phi}(t)$ $f^{\phi}(t)$ $=$ $\sum_{k}f_{k}^{\phi}\delta(t-k)$ (38) $\{f_{n}^{\phi}\}$ (38) $\{f_{n}\}$ $f^{\phi}(t)$ 1 1 $J(J<0\in Z)$ $J$ $\{d_{jk}^{r}\}$ $\{d_{jk}^{i}\}$ $\{c_{jk}^{r}\}$ $\{c_{jk}^{i}\}$ $f(t)= \sum_{j=j}^{-1}\sum_{k}\{d_{jk}^{r}\psi_{jk}^{r}(t)+d_{jk}^{i}\psi_{jk}^{i}(t)\}$ $+ \sum_{k}\{c_{jk}^{r}\phi_{jk}^{r}(t)+c_{jk}^{i}\phi_{jk}^{i}(t)\}$ (39) $d_{jk}^{r}= \frac{1}{2}\langle f^{\phi}$ $\psi_{jk}^{r}\}$ $d_{jk}^{i}= \frac{1}{2}\{f^{\phi}$ $\psi_{jk}^{i}\}\}$ (40) $c_{jk}^{r}= \frac{1}{2}\langle f^{\phi}$ $\phi_{jk}^{r}\rangle$ $c_{jk}^{i}= \frac{1}{2}\langle f^{\phi}$ $\phi_{jk}^{i}\}$ (41) (39) $f(t)$ $\{f_{n}\}$ $f_{n}=f(n)$ $n\in Z$ $\{f_{n}^{\phi}\}$ (38) $]$ 42 $[$10 (39) $\{f_{n}\}$ $\{f_{n}^{\phi}\}$ (38) [10] $s(t)$ $s(t)=s^{r}(t)+s^{i}(t)$ (42) $s^{r}(t)= \frac{1}{2}\sum_{m}\langle\delta(u)$ $\phi^{r}(u-m)\}\phi^{r}(t-m)$ (43) $s^{i}(t)= \frac{1}{2}\sum_{m}\langle\delta(u)$ $\phi^{i}(u-m)\}\phi^{i}(t-m)$ (44)

10 10 (15) (43) $s^{r}(t)$ $s^{r}(t)$ $=$ $\frac{1}{2}\sum_{m}\int_{-\infty}^{\infty}\delta(u)\overline{\phi^{r}(u-m)}du\phi^{r}(t-m)$ $=$ $\frac{1}{2}\sum_{m}\overline{\phi^{r}(-m)}\phi^{r}(t-m)$ (45) $s^{i}(t)$ $s^{i}(t)$ $=$ $\frac{1}{2}\sum_{m}\overline{\phi^{i}(-m)}\phi^{i}(t-m)$ (46) $s(t)$ $\{f_{n}\}$ $\delta_{n0}$ $\delta_{n0}$ $=$ $\{\begin{array}{ll}1 n=00 n\neq 0\end{array}$ (47) $(n\in Z)$ $\delta_{n0}$ $=$ $\sum_{k}\beta_{k}s(n-k)$ (48) (48) $n\in Z$ $\{\beta_{k}\}(k\in Z)$ $(k$ $n$ $\infty$ ) $\{f_{n}\}$ $s(t)$ ( $f_{n}=f(n)$ $n\in Z$ ) $f(t)$ $=$ $f_{n}^{\phi}$ $=$ $f_{k}^{\phi}s(t-k)$ (49) $\sum_{k}\beta_{k}f_{n-k}$ (50) $\phi^{r}(t-n)$ (49) (42) (45) (46) $\phi^{i}(t-n)$ $f(t)= \sum_{k}\{c_{0k}^{r}\phi^{r}(t-k)+c_{0k}^{i}\phi^{i}(t-k)\}$ (51) $c_{0k}^{r}= \frac{1}{2}\sum_{m}\overline{\phi^{r}(-m)}f_{k-m}^{\phi}$ (52) $c_{0k}^{i}= \frac{1}{2}\sum_{m}\overline{\phi^{i}(-m)}f_{k-m}^{\phi}$ (53)

11 $[$ $]$ 43 $[$10 (42) $\sim(53)$ $\{c_{0k}^{r}\}$ $\{c_{0k}^{i}\}$ $a_{n}^{r}$ $=$ $\frac{1}{\sqrt{2}}\overline{p_{-n}^{r}}$ $b_{n}^{r}= \frac{1}{\sqrt{2}}\overline{q_{n}^{\underline{r}}}$ (54) $a_{n}^{i}$ $=$ $\frac{1}{\sqrt{2}}\overline{p_{-n}^{i}}$ $b_{n}^{i}= \frac{1}{\sqrt{2}}\overline{q_{-n}^{i}}$ (55) $\{a_{n}^{i^{-}}\}$ $\{a_{n}^{r}\}$ $\{b_{n}^{r}\}$ (52) (53) $\{c_{0n}^{r}\}$ $\{c_{0n}^{i}\}$ CDWT [ ] $c_{j-1n}^{r}= \sum_{k}a_{2n-k}^{r}c_{jk}^{r}$ $d_{j-1n}^{r}= \sum_{k}b_{2n-k}^{r}c_{jk}^{r}$ (56) $c_{j-1n}^{i}= \sum_{k}a_{2n-k}^{i}c_{jk}^{i}$ $d_{j-1n}^{i}= \sum_{k}b_{2n-k}^{i}c_{jk}^{i}$ (57) $J$ (39) $\sim(41)$ $\{d_{jk}^{r}\}$ $\{d_{j}^{i_{k}}\}$ $\{c_{jk}^{r}\}$ $\{c_{j\text{ }}^{I}\}$ $]$ 44 $[$10 $g_{n}^{r}$ $=$ $h_{n}^{r}=$ (58) $g_{n}^{i}$ $=$ $\frac{1}{\sqrt{2}}p_{n}^{i}$ $h_{n}^{i}= \frac{1}{\sqrt{2}}q_{n}^{i}$ (59) $\{g_{n}^{i}\}$ $\{g_{n}^{r}\}$ $\{h_{n}^{r}\}$ $\{h_{n}^{i}\}$ $]$ $c_{jn}^{r}= \sum_{k}\{g_{n-2k}^{r}c_{j-1k}^{r}+h_{n-2k}^{r}d_{j-1k}^{r}\}$ (60) $c_{jn}^{i}= \sum_{k}\{g_{n-2k}^{i}c_{j-1k}^{i}+h_{n-2k}^{i}d_{j-1k}^{i}\}$ (61) 43 $\{d_{jk}^{r}\}$ $\{d_{j}^{i_{k}}\}$ $\{c_{jk}^{r}\}$ $\{c_{j_{1}k}^{i}\}$ $0$ $\{c_{0n}^{r}\}$ $f(t)$ (51) $\{f_{n}\}$ $f_{n}=f(n)$ $n\in Z$

12 Meyer 2 Meyer 1/2 (Perfect Translation Invariance PTI) CDWT 4 CDWT 51 Meyer Meyer 1 $\phi^{m}(t)$ $\hat{\phi}^{m}(\omega)=\{\begin{array}{ll}1 \omega \leqq 2\pi/30 \omega \geqq 4\pi/3\end{array}$ (62) $ \hat{\phi}^{m}(\omega-2\pi) ^{2}+ \hat{\phi}^{m}(\omega) ^{2}=1$ $\frac{2\pi}{3}<\omega<\frac{4\pi}{3}$ (63) $\hat{\phi}^{m}(\omega)$ (63) $2\pi/3< \omega <4\pi/3$ Meyer Daubechies[12] $\hat{\phi}^{m}(\omega)$ $=$ $\cos\{\frac{\pi}{2}\nu(\frac{3}{2\pi} \omega -1)\}$ (64) $\nu(x)$ $=$ $x^{4}(35-84x+70x^{2}-20x^{3})$ (65) $\hat{\phi}^{m}(\omega)$ $\phi^{m}(t)$ Meyer Meyer $\{p_{n}^{m}\}$ $\phi^{m}(t)$ [11] $p_{n}^{m}$ $=$ $\phi^{m}(\frac{n}{2})$ (66) Meyer $\phi^{m}(t)$ $b\in R$ $\phi^{b}(t)$ $\phi^{b}(t)$ $=$ $\phi^{m}(t-b)$ $b\in R$ (67)

13 13 $b\in R$ $\phi^{b}(t)$ $\psi^{b}(t)$ $\{p_{n}^{b}\}$ [11] $p_{n}^{b}$ $=$ $\phi^{m}(\frac{n-b}{2})$ (68) $\{p_{n}^{b}\}$ $\{p_{n}^{m}\}$ $P_{\omega}^{b}(\omega)$ [11] $P_{\omega}^{M}(\omega)$ $P_{\omega}^{b}(\omega)$ $=$ $P_{\omega}^{M}(\omega)e^{-ib\omega}$ (69) 52 [11] $b$ (69) 2 2 $b_{1}$ $b_{2}$ $P_{\omega}^{b_{1}}(\omega)$ $P_{\omega}^{b_{2}}(\omega)$ Hilbert $b_{1}$ $b_{2}$ Selesnick[6] Hilbert $b_{1}$ $b_{2}$ (29) (30) 1/2 $b_{1}=b$ $b_{2}=b+1/2$ (70) (69) $b$ $P_{\omega}^{R}(\omega)$ $=$ $P_{\omega}^{b}(\omega)$ $=P_{\omega}^{M}(\omega)e^{-ib\omega}$ (71) $P_{\omega}^{I}(\omega)$ $=$ $P_{\omega}^{b+1/2}(\omega)=P_{\omega}^{M}(\omega)e^{-i(b+1/2)\omega}$ (72) (71) (72) $P_{\omega}^{M}(\omega)$ $P_{\omega}^{I}(\omega)$ $=$ $P_{\omega}^{R}(\omega)e^{-i\omega/2}$ (73) (29) (30) Selesnick Hilbert (71) (72) (68) $p_{n}^{r}$ $=p_{n}^{b}$ $= \phi^{m}(\frac{n-b}{2})$ (74) $p_{n}^{i}$ $=p_{n}^{b+1/2}= \phi^{m}(\frac{n-b-1/2}{2})$ (75) (67) Meyer $\phi^{m}(t)$ $\phi^{r}(t)$ $=$ $\phi^{b}(t)$ $=\phi^{m}(t-b)$ (76) $\phi^{i}(t)$ $=$ $\phi^{b+1/2}(t)=\phi^{m}(t-b-\frac{1}{2})$ (77)

14 14 (a) $b=0$ (b) $b=1/4$ (c) $b=1/2$ 2: PTI $\phi^{r}(t)$ 1/2 $\phi^{i}(t)$ [8] Hilbert $\psi^{r}(t)$ $=$ $\psi^{b}(t)$ (78) $\psi^{i}(t)$ $=$ $\psi^{b+1/2}(t)$ (79) (74) $\psi^{i}(t)$ $\sim(77)$ $\{p_{n}^{i}\}$ $\phi^{r}(t)$ $\phi^{i}(t)$ (20) $b$ 2 $0\leqq b\leqq 1/2$ $b=0$ $\psi^{r}(t)$ $\psi^{i}(t)$ $b=1/2$ $b=1/4$ Kingsbury [5] 1/4 53 CDWT CDWT 4 CDWT (42) $s(t)$ [11] $s(n)=\delta_{n0}$ $n\in Z$ (80) (80) CDWT 42 (38) $\delta(t)$

15 $[$ $f^{\phi}(t)$ $\{f_{n}\}$ $f^{\phi}(t)$ $=$ $\sum_{m}f_{m}\delta(t-m)$ (81) $f^{\phi}(t)$ $\phi^{i}$ $\phi^{r}(t-n)$ ( $n$ ) $f(t)= \sum_{k}\{c_{0k}^{r}\phi^{r}(t-k)+c_{0k}^{i}\phi^{i}(t-k)\}$ (82) $c_{0k}^{r}= \frac{1}{2}\langle f^{\phi}(t)$ $\phi^{r}(t-k)\rangle$ (83) $c_{0k}^{i}= \frac{1}{2}\langle f^{\phi}(t)$ $\phi^{i}(t-k)\}$ (84) (82) $f(t)$ $f_{n}=$ $]$ $f(n)$ $n\in Z$ (83) (84) (81) (15) (82) $\sim(84)$ n $\}$ $f(t)= \sum_{k}\{c_{0k}^{r}\phi^{r}(t-k)+c_{0k}^{i}\phi^{i}(t-k)\}$ (85) $c_{0k}^{r}= \frac{1}{2}\sum_{m}\overline{\phi^{r}(-m)}f_{k-m}$ (86) $c_{0k}^{i}= \frac{1}{2}\sum_{m}\overline{\phi^{i}(-m)}f_{k-m}$ (87) (86) (87) $\{c_{0k}^{r}\}$ $\{c_{0k}^{i}\}$ $0$ $\{c_{0n}^{r}\}$ $\{c_{0n}^{i}\}$ $f(t)$ (85) $\{f_{n}\}$ $f_{n}=f(n)$ $n\in Z$ 6CDWT 1998 Kingsbury [4 5] CDWT CDWT DWT CDWT Kingsbury [5] 1/4 Hilbert Selesnick [6] CDWT Hilbert CDWT CDWT

16 16 Hilbert Hilbert CDWT CDWT CDWT [8] [9] CDWT [10] CDWT Meyer [11] CDWT DWT DWT 3 CDWT 6 [5] [1] S Mallat: A Theory for Multiresolution Signal Decomposition: The Wavelet Representation IEEE Trans on Patten Analysis and Machine Intelligence Vol11 No7 $ (1989)$ [2] S G Mallat and Z Zhang: Matching pursuits with time-frequency dictionaries; IEEE Transactions on Signal Processing Vol41 No12 $ (1993)$ [3] F C A Femandes I W Selesnick R L C Spaendonck and C S Burrus: Complex wavelet transforms with allpass filters; Signal Processing Vol33 No8 $ (2003)$ [4] JFA Magarey and NG Kingsbury: Motion estimation using a complexvalued wavelet transform; IEEE Trans on Signal Processing Vol46 No4 $ (1998)$ [5] N Kingsbury: Complex wavelets for shift invariant analysis and filtering of signals; Joumal of Applied and Computational Harmonic Analysis Vol10 No3 $ (2001)$

17 17 [6] I W Selesnick: The design of approximate Hilbert transform pairs of wavelet bases; IEEE Trans on Signal Processing Vol50 No5 $ (2002)$ [7] : RI-Spline 2 :RI-Spline ; Vo139 No 7 $ (2003)$ [8] : 1 : ;Joumal of Signal Processing Vol11 No5 $ (2007)$ [9] : 2 : :Joumal of Signal Processing Vol11 No5 $ (2007)$ [10] : 3 : ;Journal of Signal Processing Vol11 No 5 $ (2007)$ [11] : ; Joumal of Signal Processing Vol12 No2 $ (2008)$ $[$12 $]$ I Daubechies: Ten lectures on wavelets SIAM Philadelphia 1992 [13] C K Chui : (1993) [14] : ; 2005

1 -- 9 -- 6 6--1 (DFT) 009 DFT: Discrete Fourier Transform 6--1--1 N x[n] DFT N 1 X[k] = x[n]wn kn, k = 0, 1,, N 1 (6 ) n=0 1) W N = e j π N W N twidd

1 -- 9 6 009 (DFT) 6-1 DFT 6- DFT FFT 6-3 DFT 6-4 6-5 c 011 1/(0) 1 -- 9 -- 6 6--1 (DFT) 009 DFT: Discrete Fourier Transform 6--1--1 N x[n] DFT N 1 X[k] = x[n]wn kn, k = 0, 1,, N 1 (6 ) n=0 1) W N = e