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wavelet.tex
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\section{$B%&%'!<%V%l%C%H2r@O$K$D$$$F(B}
\label{sec:wavelet}
$B$3$N@a$r=q$/$?$a$K!"J88%(B\cite{Daubsci_2003},\cite{Toda_2005}$B$r;29M$K$7$?!#(B
\subsection{$B%U!<%j%(JQ49$H%&%'!<%V%l%C%HJQ49(B}
$B:#2s$O!"%9%Q%$%/GH7A$+$iFCD'$rCj=P$9$k$?$a$K%&%'!<%V%l%C%HJQ49$H$$$&(B
$B<jK!$rMQ$$$?!#(B
$B%&%'!<%V%l%C%HJQ49$H$O!"?.9f$N;~4V<~GH?t2r@O$N<jK!$N0l$D$G$"$k!#(B
$B%&%'!<%V%l%C%HJQ49$rMQ$$$k$H!"0l<!85$N?.9f$r<~GH?tJ,2r$9$k;v$,$G$-$k!#(B
$B?.9f$N<~GH?t2r@O$N<jK!$H$7$FM-L>$J$b$N$K$O!"%U!<%j%(JQ49$,$"$k!#(B
$B?.9f$KBP$7%U!<%j%(JQ49$rMQ$$$k;v$K$h$j!"?.9f$r;X?t4X?t$NOB$KJ,2r$7!"(B
$B$=$N?.9f$K$I$l$@$1$N<~GH?t$NGH$,4^$^$l$k$N$+$rD4$Y$k;v$,$G$-$k!#(B
$B%U!<%j%(JQ49$,;X?t4X?t$NOB$H$7$F?.9f$rJ,2r$9$k$N$KBP$7!"%&%'!<%V%l%C%HJQ49$O!"(B
$B%&%'!<%V%l%C%H!J>.$5$$GH!K$H8F$P$l$kFC<l$J4X?t$NOB$G?.9f$rJ,2r$9$k!#(B
$B%U!<%j%(JQ49$H%&%'!<%V%l%C%HJQ49$r?t<0$K$7$F8+Hf$Y$k$H0J2<$N$h$&$K$J$k!#(B
$B!JO"B3!K%U!<%j%(JQ49(B
\begin{equation}
f(\omega)=\int_{\infty}^{- \infty}f(t)e^{-i\omega t} d\omega
\end{equation}
$B!JO"B3!K%&%'!<%V%l%C%HJQ49(B
\begin{eqnarray}
W(\omega) &=& \int_{\infty}^{- \infty}f(t) \psi_{a,b} (t) d\omega \\
\psi_{a,b}(t) &=& \psi (\frac{t-b}{a})
\end{eqnarray}
$B%U!<%j%(JQ49$N4pDl4X?t$O<~4|$,O"B3$7$?;X?t4X?t$G$"$k!#$3$l$KBP$7!"%&%'!<%V%l%C%HJQ49(B
$B$N4pDl4X?t(B$\psi(x)$$B$O!"6u4V$K6I=jE*$J4X?t$G$"$k!#(B
$B%&%'!<%V%l%C%H4X?t$O%U!<%j%(JQ49$N4pDl4X?t$H$O0[$J$j!"MM!9$J<oN`$N$b$N$,$"$k!#(B
$B:#2s$O!"(BDaubechies$B$N4pDl4X?t$H8F$P$l$k%&%'!<%V%l%C%H4X?t$r%9%Q%$%/$NFCD'Cj=P$K:NMQ$7$?!#(B
Daubechies$B$N4pDl4X?t(B(\fgref{fig:Daubechies})$B$O0J2<$N@-<A$r;}$D!"%&%'!<%V%l%C%H4X?t$G$"$k!#(B
(1)$B%3%s%Q%/%H%5%]!<%H$r;}$D(B($BJ?$?$/8@$&$H!"0l$D$N6h4V$G$N$_CM$r$b$A!"B>$G(B
$B$NCM$O(B0),(2)$B@55,D>8r7O!#(B
$B@-<A(B(2)$B$K$h$C$F!"%&%'!<%V%l%C%H4X?t$rMQ$k$H!"B?=E2rA|EY2r@O!J(B\fgref{MRA}$B!K(B
$B$H8F$P$l$kN%;6%&%'!<%V%l%C%HJQ49$N%"%k%4%j%:%`$r;HMQ$G$-$k!#(B
\subsection{$BB?=E2rA|EY2r@O(B}
$B0J2<$O!"N%;6%&%'!<%V%l%C%HJQ49$N%"%k%4%j%:%`$N0l$D$G$"$kB?=E2rA|EY2r@O!J(B\fgref{MRA}$B!K$K$D$$$F@bL@(B
$B$9$k!#(B
$BB?=E2rA|EY2r@O$H$O!"?.9f=hM}E*$K8@$&$H!"2r@O$9$k?.9f$K%m!<%Q%9%U%#%k%?$H%P%s%I%Q%9%U%#%k%?!J%O%$%Q%9%U%#%k%?!K$r$+$1!"%m!<%Q%9%U%#%k%?$G<h$j$@[email protected],$K$5$i$K%m!<%Q%9%U%#%k%?$H%P%s%I%Q%9%U%#%k%?$r$+$1$k!"$H$$$&$3$H$r7+$jJV$9$3$H$K$h$C$F!"N%;6%&%'!<%V%l%C%HJQ49$r9T$&%"%k%4%j%:%`$G$"$k!#(B
$B85$N?.9f$r(Blevel 0$B$H$7(B,$B%U%#%k%?$r$+$1$k$4$H$K!"?.9f$r(Blevel -1, level
-2,$\dots$,$B$H8F$V!#%G!<%?$NMWAG?t$,(B$2^{x}$$B$G$"$k?.9f$O:GBg$G(Blevel -x$B$^$GB?=E2rA|EY2r@O$r9T$&;v$,$G$-$k!#(B
$BB?=E2rA|EY2r@O$G$O!"%&%'!<%V%l%C%HJQ49$N4pDl4X?t$G$"$k%&%'!<%V%l%C%H4X?t(B $\psi(x)$$B$r!"<!$N$h$&$K%9%1!<%j%s%04X?t(B $\phi(x)$$B$+$i9=@.$9$k!#(B
$B%9%1!<%j%s%04X?t(B
\begin{equation}
\phi_{j,n}(t) = \sqrt{2}^{j} \phi(2^{j}t - n)
\end{equation}
$B%&%'!<%V%l%C%H4X?t(B
\begin{equation}
\psi_{j,n}(t) = \sqrt{2}^{j} \psi(2^{j}t - n)
\end{equation}
$BB?=E2rA|EY2r@O$r?.9f=hM}E*$K8@$&$H!"%9%1!<%j%s%04X?t$,%m!<%Q%9%U%#%k%?!"%&%'!<%V%l%C%H4X(B
$B?t$,%P%s%I%Q%9%U%#%k%?!J%O%$%Q%9%U%#%k%?!K$G$"$k!#(B
$B$3$3$G?.9f(Bf$B$O<!$N$h$&$K(B$\phi(t)$$B$GJd40$9$k;v$,$G$-$k!#(B
\begin{eqnarray}
f(t) = \sum_{m} c_{0,m} \phi (t-m)
\end{eqnarray}
level 0$B$N?.9f$NB?=E2rA|EY2r@O$NCf?H$O$3$N(B$c_{0,m}$$B$NNs!"(B
$\{ c_{0,0},c_{0,1},\dots c_{0,2^{x}} \}$
$B%9%1!<%j%s%04X?t$H%&%'!<%V%l%C%H4X?t$O<!$N%D!<%9%1!<%k?tNs$H8F$P$l$k(B
$p_{n},q_{n}$$B$K4X$7$F!"%D!<%9%1!<%k4X78$H8F$P$l$k4X78(B
$B$rK~$?$9!#(B
$B%D!<%9%1!<%k?tNs(B$p_{n},q_{n}$$B$O<!$N$h$&$J?tNs$G$"$k!#(B
\begin{eqnarray}
q_{n} &=& \left( -1\right)^{1-n} p_{1-n} \\
\sum_{n} p_{n} &=& 2
\end{eqnarray}
$B$3$N%D!<%9%1!<%k?tNs6qBNE*$JCM$O!"%&%'!<%V%l%C%H4X?t$N:n<T$K$h$C$F8x3+$5$l$F$$$k!#(B
$B%I%Y%7!<$O%+%9%1!<%I%"%k%4%j%:%`$H$$$&J}K!$G!"%D!<%9%1!<%k?tNs$r5a$a$?!#(B
$B%D!<%9%1!<%k?tNs$O(B$\phi(t),\psi(t)$$B$H!"<!$N%D!<%9%1!<%k4X78$r$_$?$9!#(B
\begin{eqnarray}
\phi(t) &=& \sum_{n} p_{n} \phi(2t - n) \\
\psi(t) &=& \sum_{n} q_{n} \phi(2t - n)
\end{eqnarray}
$B$3$N%D!<%9%1!<%k?tNs$+$i!"<!$NJ,2r?tNs(B$h_{0}[n],h_{1}[n]$$B$,(B
\begin{eqnarray}
h_{0}[n] &=& \frac{1}{\sqrt{2}} \Bar p_{-n} \\
h_{1}[n] &=& \frac{1}{\sqrt{2}} \Bar p_{-n}
\end{eqnarray}
$B$HDj5A$5$l!"J,2r?tNs(B$h_{0}[n],h_{1}[n]$$B$rMQ$$$k;v$K$h$C$F<!$N(B
$BJ,2r%"%k%4%j%:%`(B
\begin{eqnarray}
c_{j-1,n} &=& \sum_{m} h_{0}[2n-m] c_{j,m} \\
d_{j-1,n} &=& \sum_{m} h_{1}[2n-m] c_{j,m}
\end{eqnarray}
$B$,7W;;$G$-$k!#J,2r%"%k%4%j%:%`$+$i7W;;$7$?!"%&%'!<%V%l%C%H78?t(B$d_{-1,n},d_{-2,n} \dots ,d_{-j,n}$$B$H%9%1!<%j%s%04X?t78?t(B$c_{-j,n}$$B$,(BLevel-j$B$GB?=E2rA|EY2r@O$7$?8e$N?.9f$N%G!<%?$NCf?H$G$"$k!#(B
\section{k-means$BK!(B}
\label{sec:k-means}
$BJ,N`$9$k%G!<%?$r(Bn$B8D$H$7!"(B$\mathbf{x}_{i}(i=0 \sim n)$$B$H$9$k!#(B
$B3F%G!<%?$O(Bj$B<!85$H$7!"(B$\mathbf{x}_{i} = (x_{0},x_{1},\dots,x_{j-1})$$B$H$9$k!#(B
$B%/%i%9%?!<$N?t$r(Bk$B8D$H$9$k!#(B
$B%G!<%?4V$N5wN%$rDj5A$9$k!#%G!<%?$NA4$F$NE@$r!"M-8B8D$NBeI=E@$H5wN%$K4p$E$$$F!":G$b6a$$E@$r=8$a(B
$B$F:n$C$?NN0h$r(BVoronoi$BNN0h$H$$$&!#(B
\par
k-menas$BK!$N<j=g$O!"<!$N$h$&$K$J$k!#(B
\begin{quote}
\begin{enumerate}
\item $B3F%G!<%?$NCf$+$i!"(Bk$B8D$NBeI=%Y%/%H%k$r%i%s%@%`$KA*$s$G=i4|CM$H$9$k!#(B
\item $BBeI=%Y%/%H%k$N(BVoronoi$BNN0h$r5a$a!"3F(BVoronoi$BNN0h$NJ?6QCM$r7W;;$9$k!#(B
$BJ?6QCM$K0lHV6a$$%G!<%?$r?7$7$$BeI=%Y%/%H%k$H$9$k!#(B
\item $BBeI=%Y%/%H%k$,<}B+$9$k$^$G!"(B(2)$B$r7+$jJV$9!#(B
\end{enumerate}
\end{quote}