MATH 414 Lecture 31

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Multi-Resolution Analysis MRA

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

The $V_j$'s and $\varphi$ satisfy the following properties:

1. Nested. $\dots \subset V_{j-1}\subset V_j\subset V_{j+1}\subset \dots$
2. Density. $\mathrm{c}\mathrm{l}\mathrm{o}\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{e}\left(\bigcup _{j=-\mathrm{\infty }}^{\mathrm{\infty }}\right)=L^2$
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. $\{\varphi (x-k),k\in \mathbb{\mathbb{Z}}\}$ is an orthonormal basis for $V_0$.

Haar MRA

The Haar MRA that we've been talking about.

$\begin{array}{r}{V}_j=\{f\in L^2\mid f\text{is constant on}{2}^{-j}k\le x<2^{-j}(k+1)\}\\ \varphi (x)& =\{\begin{array}{ll}1& 0\le x<1\\ 0& \text{otherwise}\end{array}\end{array}$

Shannon MRA

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

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

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

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

In the Shannon MRA,

$\begin{array}{rl}{V}_j& =\{f\in L^2\mid f\text{is band-limited},\mathrm{\Omega }=2^j\pi \}\\ \varphi (x)& =\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{c}x=\{\begin{array}{ll}1& x=0\\ \frac{\sin \pi x}{\pi x}& x\ne 0\end{array}\end{array}$

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 $[-2^{j-1}\pi ,2^{j-1}\pi ]$. Functions in $V_j$ have support contained in $[-2^j\pi ,2^j\pi ]\supset [-2^{j-1}\pi ,2^{j-1}\pi ]$
2. Density.
3. Separation.
4. Scaling.
5. Orthonormal Property. ${\{\varphi (x-k)\}}_{k\in \mathbb{\mathbb{Z}}}$ is an orthonormal basis: ${ℱ}\left[\sqrt{2\pi }\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{c}x\right]\left(\omega \right)=\gamma (\omega )=\{\begin{array}{ll}1& \left|\omega \right|<\pi \\ 0& \text{otherwise}\end{array}$. To show that $\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}\varphi (x-\ell )\varphi (x-k)\mathrm{d}x={\delta }_{k,\ell }$, we need Parseval's theorem.

$\begin{array}{rl}\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}\varphi (x-\ell )\varphi (x-k)\mathrm{d}x& =\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}{ℱ}[\varphi (x-\ell )](\omega ){ℱ}[\varphi (x-k](\omega )\mathrm{d}\omega \\ & =\frac{1}{2\pi }\underset{-\pi }{\overset{\pi }{\int }}{\mathrm{e}}^{-i\omega \ell }{\mathrm{e}}^{-i\omega k}\mathrm{d}\omega \\ & =\frac{1}{2\pi }\underset{-\pi }{\overset{\pi }{\int }}{\mathrm{e}}^{-i\omega (\ell -k)}\mathrm{d}\omega \\ & =\{\begin{array}{ll}1& k=\ell \\ 0& \text{otherwise}\end{array}\end{array}$

Linear Spline MRA

$\{f\in L^2\mid f\text{is continuous and piecewise linear}\}$

$\begin{array}{rl}{V}_j& =\{f\in L^2\mid f\text{is a linear spline with possible corners at}x=2^{-j}k\}\\ \varphi (x)& =\{\begin{array}{ll}1+x& -1\le x<0\\ 1-x& 0\le x<1\\ 0& \text{otherwise}\end{array}\end{array}$

For example, $V_0$ has corners at integers

Properties:

1. ${\left\{2^{\frac{j}{2}}\varphi (2^{j}x-k)\right\}}_{k\in \mathbb{\mathbb{Z}}}$ is an orthonormal basis for $V_j$
2. Scaling. $\varphi (x)\in V_0\subset V_1$, expand $\varphi (x)$ in the orthonormal basis ${\left\{2^{\frac{1}{2}}\varphi (2x-k)\right\}}_{k\in \mathbb{\mathbb{Z}}}$

In General

The scaling relation for any wavelet system is given by

$\varphi (x)=\sum _{k\in \mathbb{\mathbb{Z}}}{p}_k\varphi (2x-k)$

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.