MATH 414 Lecture 31

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Multi-Resolution Analysis MRA

Collection of subspaces of , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left\{ V_j \right\}_{j=-\infty}^\infty} , and a function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi} , called the scaling function.

The Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V_j} 's and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi} satisfy the following properties:

  1. Nested. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \dots \subset V_{j-1} \subset V_j \subset V_{j+1} \subset \dots}
  2. Density. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathrm{closure}\left( \bigcup_{j=-\infty}^\infty \right) = L^2}
  3. Separation. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \bigcap_{j} V_j = \left\{ 0 \right\}}
  4. Scaling. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x) \in V_j} if and only if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(2^{-j} \, x) \in V_0}
  5. Orthonormal Property. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left\{ \phi(x-k), k \in \mathbb{Z} \right\}} is an orthonormal basis for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V_0} .

Haar MRA

The Haar MRA that we've been talking about.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} V_j = \left\{ f \in L^2 ~\mid~ f\ \mbox{is constant on}\ 2^{-j} \, k \le x < 2^{-j}(k+1) \right\} \\ \phi(x) &= \begin{cases} 1 & 0 \le x < 1 \\ 0 & \mbox{otherwise} \end{cases} \end{align}}

Shannon MRA

Recall that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)} is said to be band-limited if and only if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{f}(\omega)} is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0} when Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left| \omega \right| > \Omega > 0}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Omega} is a subspace of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L^2} . Why?

  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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
    Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align}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{align}}
  • closed over multiplication

In the Shannon MRA,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} 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 \ne 0 \end{cases} \end{align}}

  1. Nested: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V_{j-1}} , band is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2^{j-1} \, \pi} , and for every Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f \in V_{j-1}} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{f}} has support contained in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left[ -2^{j-1} \, \pi, 2^{j-1}\,\pi \right]} . Functions in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V_j} have support contained in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left\{ \phi(x-k) \right\}_{k\in \mathbb{Z}}} is an orthonormal basis: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int_{-\infty}^{\infty} \phi(x-\ell) \, \phi(x-k) \,\mathrm{d}x = \delta_{k,\ell}} , we need Parseval's theorem.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \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{align}}


Linear Spline MRA

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left\{ f \in L^2 ~\mid~ f \ \mbox{is continuous and piecewise linear} \right\}}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} 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 \le x < 0 \\ 1 - x & 0 \le x < 1 \\ 0 & \mbox{otherwise} \end{cases} \end{align}}

For example, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V_0} has corners at integers

Properties:

  1. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left\{ 2^{ \frac{j}{2} } \, \phi \left( 2^j \, x - k \right) \right\}_{ k \in \mathbb{Z} }} is an orthonormal basis for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V_j}
  2. Scaling. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi(x) \in V_0 \subset V_1} , expand Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi(x)} in the orthonormal basis Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi(x) = \sum_{ k \in \mathbb{Z} } p_k \, \phi(2x-k)}

For Haar wavelets, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p_0 = p_1 = 1} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p_k = 0} for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k > 1}


Support of a Function

The support of a function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} is the largest closed interval on which Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} doesn't vanish.

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