MATH 414 Lecture 30

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Wavelet Decomposition and Reconstruction

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_j(x) &= \sum_{k\in \mathbb{Z}} a_k^j \, \phi(2^j \, x - k) \\ f_{j-1}(x) &= \mathrm{proj}_{V_{j-1}}{(f_j)} = \sum_{k \in \mathbb{Z}} a_k^{j-1} \, \phi(2^{j-1} \, x - k) \\ w_{j-1}(x) &= \mathrm{proj}_{W_{j-1}}{(f_j)} = \sum_{k \in \mathbb{Z}} b_k^{j-1} \, \psi(2^{j-1} \, x - k) \\ a_k^{j-1} &= \frac{1}{2} \, \left( a_{2k}^j + a_{2k+1}^j \right) \\ b_k^{j-1} &= \frac{1}{2} \, \left( a_{2k}^j - a_{2k+1}^j \right) \end{align}}

Decomposition

Low pass filter 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 \ell = \left( \dots, 0, 0, 0, \underbrace{\frac{1}{2}}_{k=-1}, \underbrace{\frac{1}{2}}_{k=0}, 0, 0, 0, \dots \right)}

High pass filter 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 h = \left( \dots, 0, 0, 0, \underbrace{-\frac{1}{2}}_{k=-1}, \underbrace{\frac{1}{2}}_{k=0}, 0, 0, 0, \dots \right)}


Downsampling:

  1. Form 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 \frac{1}{2} \left( x_k + x_{k+1} \right) = y_k}
  2. Downsample by discarding odd 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}
  3. Call the resulting sequence 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 a_k^{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 \begin{align} y &= \ell * x \\ y_k &= \left( \ell * x \right)_k \\ (Dy)_{2k} &= y_{2k} \\ (Dy)_{2k+1} &= 0 \end{align}}


Thus 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 a^{j-1} = D(\ell * a^j)}

Similarly, 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 b^{j-1} = D(h * a^j)}

Downsampling, high pass 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 h} , nor low pass 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 \ell} depend on the level 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 j} !:


Reconstruction

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} a_{2k}^j &= a_k^{j-1} + b_k^{j-1} \\ a_{2k+1}^j &= a_k^{j-1} - b_k^{j-1} \end{align}}

To handle the even and odd cases in filter form, we need a few more operations:

New low pass filter 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 \tilde{\ell} = \left( \ldots, 0, 0, 0, \underbrace{1}_{k=0}, \underbrace{1}_{k=1}, 0, 0, 0, \dots \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 \left( \tilde{\ell} * z \right)_k = z_k + z_{k-1}}

New high pass filter 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 \tilde{h} = \left( \ldots, 0, 0, 0, \underbrace{1}_{k=0}, \underbrace{-1}_{k=1}, 0, 0, 0, \dots \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 \left( \tilde{h} * z \right)_k = z_k - z_{k-1}}

Upsampling:

  1. Form 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 x = \left( \ldots, x_{-1}, x_0, x_1, \dots \right)}
  2. 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 Ux} inserts zeroes at every odd term: 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( \ldots, 0, x_{-1}, 0, x_0, 0, x_1, 0, \dots \right)}


Summary

MATH 414 Filter-Based Wavelet Decomposition.svg
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