MATH 414 Lecture 31

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Multi-Resolution Analysis MRA

Collection of subspaces of $L^{2}$, $\left\{V_{j}\right\}_{j=-\infty }^{\infty }$, and a function $\phi$, called the scaling function.

The $V_{j}$'s and $\phi$ satisfy the following properties:

1. Nested. $\dots \subset V_{j-1}\subset V_{j}\subset V_{j+1}\subset \dots$
2. Density. $\mathrm {closure} \left(\bigcup _{j=-\infty }^{\infty }\right)=L^{2}$
3. Separation. $\bigcap _{j}V_{j}=\left\{0\right\}$
4. Scaling. $f(x)\in V_{j}$ if and only if $f(2^{-j}\,x)\in V_{0}$
5. Orthonormal Property. $\left\{\phi (x-k),k\in \mathbb {Z} \right\}$ is an orthonormal basis for $V_{0}$.

Haar MRA

The Haar MRA that we've been talking about.

${\begin{aligned}V_{j}=\left\{f\in L^{2}~\mid ~f\ {\mbox{is constant on}}\ 2^{-j}\,k\leq x<2^{-j}(k+1)\right\}\\\phi (x)&={\begin{cases}1&0\leq x<1\\0&{\mbox{otherwise}}\end{cases}}\end{aligned}}$

Shannon MRA

Recall that $f(x)$ is said to be band-limited if and only if ${\hat {f}}(\omega )$ is $0$ when $\left|\omega \right|>\Omega >0$

$f(x)={\frac {1}{\sqrt {2\pi }}}\,\int _{-\Omega }^{\Omega }{\hat {f}}(\omega )\,\mathrm {e} ^{+i\,\omega \,x}\,\mathrm {d} \omega$

The band-limited functions with fixed $\Omega$ is a subspace of $L^{2}$. Why?

• $f(x)\equiv 0={\frac {1}{\sqrt {2\pi }}}\int _{-\Omega }^{\Omega }\,0\cdot \mathrm {e} ^{+i\,\omega \,x}\,\mathrm {d} \omega$ is in the space
• closed over addition
  ${\begin{aligned}f(x)+g(x)&={\frac {1}{\sqrt {2\pi }}}\,\int _{-\Omega }^{\Omega }{\hat {f}}\,\mathrm {e} ^{+i\,\omega \,x}\,\mathrm {d} \omega +{\frac {1}{\sqrt {2\pi }}}\,\int _{-\Omega }^{\Omega }{\hat {g}}\,\mathrm {e} ^{+i\,\omega \,x}\,\mathrm {d} \omega \\&={\frac {1}{\sqrt {2\pi }}}\,\int _{-\Omega }^{\Omega }\left({\hat {f}}+{\hat {g}}\right)\,\mathrm {e} ^{+i\,\omega \,x}\,\mathrm {d} \omega \end{aligned}}$
• closed over multiplication

In the Shannon MRA,

${\begin{aligned}V_{j}&=\left\{f\in L^{2}~\mid ~f\ {\mbox{is band-limited}},\ \Omega =2^{j}\,\pi \right\}\\\phi (x)&=\mathrm {sinc} {x}={\begin{cases}1&x=0\\{\frac {\sin {\pi \,x}}{\pi \,x}}&x\neq 0\end{cases}}\end{aligned}}$

1. Nested: $V_{j-1}$, band is $2^{j-1}\,\pi$, and for every $f\in V_{j-1}$, ${\hat {f}}$ has support contained in $\left[-2^{j-1}\,\pi ,2^{j-1}\,\pi \right]$. Functions in $V_{j}$ have support contained in $\left[-2^{j}\,\pi ,2^{j}\,\pi \right]\supset \left[-2^{j-1}\,\pi ,2^{j-1}\,\pi \right]$
2. Density.
3. Separation.
4. Scaling.
5. Orthonormal Property. $\left\{\phi (x-k)\right\}_{k\in \mathbb {Z} }$ is an orthonormal basis: ${\mathcal {F}}\left[{\sqrt {2\pi }}\,\mathrm {sinc} {x}\right]\left(\omega \right)=\gamma (\omega )={\begin{cases}1&\left|\omega \right|<\pi \\0&{\mbox{otherwise}}\end{cases}}$. To show that $\int _{-\infty }^{\infty }\phi (x-\ell )\,\phi (x-k)\,\mathrm {d} x=\delta _{k,\ell }$, we need Parseval's theorem.

${\begin{aligned}\int _{-\infty }^{\infty }\phi (x-\ell )\,\phi (x-k)\,\mathrm {d} x&=\int _{-\infty }^{\infty }{\mathcal {F}}\left[\phi (x-\ell )\right](\omega )\,{\mathcal {F}}\left[\phi (x-k\right](\omega )\,\mathrm {d} \omega \\&={\frac {1}{2\pi }}\,\int _{-\pi }^{\pi }\mathrm {e} ^{-i\,\omega \,\ell }\,\mathrm {e} ^{-i\,\omega \,k}\,\mathrm {d} \omega \\&={\frac {1}{2\pi }}\,\int _{-\pi }^{\pi }\mathrm {e} ^{-i\,\omega \,\left(\ell -k\right)}\,\mathrm {d} \omega \\&={\begin{cases}1&k=\ell \\0&{\mbox{otherwise}}\end{cases}}\end{aligned}}$

Linear Spline MRA

$\left\{f\in L^{2}~\mid ~f\ {\mbox{is continuous and piecewise linear}}\right\}$

${\begin{aligned}V_{j}&=\left\{f\in L^{2}~\mid ~f\ {\mbox{is a linear spline with possible corners at}}\ x=2^{-j}\,k\right\}\\\phi (x)&={\begin{cases}1+x&-1\leq x<0\\1-x&0\leq x<1\\0&{\mbox{otherwise}}\end{cases}}\end{aligned}}$

For example, $V_{0}$ has corners at integers

Properties:

1. $\left\{2^{\frac {j}{2}}\,\phi \left(2^{j}\,x-k\right)\right\}_{k\in \mathbb {Z} }$ is an orthonormal basis for $V_{j}$
2. Scaling. $\phi (x)\in V_{0}\subset V_{1}$, expand $\phi (x)$ in the orthonormal basis $\left\{2^{\frac {1}{2}}\,\phi \left(2x-k\right)\right\}_{k\in \mathbb {Z} }$

In General

The scaling relation for any wavelet system is given by

$\phi (x)=\sum _{k\in \mathbb {Z} }p_{k}\,\phi (2x-k)$

For Haar wavelets, $p_{0}=p_{1}=1$, $p_{k}=0$ for $k>1$

Support of a Function

The support of a function $f$ is the largest closed interval on which $f$ doesn't vanish.