# CDFWavelet

represents a CohenDaubechiesFeauveau wavelet of type "9/7".

CDFWavelet["type"]

represents a CohenDaubechiesFeauveau wavelet of type "type".

# Details

• CDFWavelet defines a set of biorthogonal wavelets.
• The following "type" forms can be used:
•  "5/3" used in lossless JPEG2000 compression "9/7" used in lossy JPEG2000 compression
• The scaling function () and wavelet function () have compact support. The functions are symmetric.
• CDFWavelet can be used with such functions as DiscreteWaveletTransform, WaveletPhi, etc.

# Examples

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## Basic Examples(3)

Scaling function:

Wavelet function:

Filter coefficients:

## Scope(16)

### Basic Uses(10)

Compute primal lowpass filter coefficients:

Dual lowpass filter coefficients:

Primal highpass filter coefficients:

Dual highpass filter coefficients:

Lifting filter coefficients:

Generate function to compute lifting wavelet transform:

Primal scaling function:

Dual scaling function:

Plot scaling function using different levels of recursion:

Primal wavelet function:

Dual wavelet function:

Plot scaling function using different levels of recursion:

### Wavelet Transforms(5)

Compute a DiscreteWaveletTransform:

View the tree of wavelet coefficients:

Get the dimensions of wavelet coefficients:

Plot the wavelet coefficients:

Compute a DiscreteWaveletPacketTransform:

View the tree of wavelet coefficients:

Get the dimensions of wavelet coefficients:

Plot the wavelet coefficients:

Compute a StationaryWaveletTransform:

View the tree of wavelet coefficients:

Get the dimensions of wavelet coefficients:

Plot the wavelet coefficients:

Compute a StationaryWaveletPacketTransform:

View the tree of wavelet coefficients:

Get the dimensions of wavelet coefficients:

Plot the wavelet coefficients:

Compute a LiftingWaveletTransform:

View the tree of wavelet coefficients:

Get the dimensions of wavelet coefficients:

Plot the wavelet coefficients:

### Higher Dimensions(1)

Multivariate scaling and wavelet functions are products of univariate ones:

## Properties & Relations(16)

Lowpass filter coefficients sum to unity; :

Highpass filter coefficients sum to zero; :

Dual lowpass filter coefficients sum to unity; :

Dual highpass filter coefficients sum to zero; :

Scaling function integrates to unity; :

Dual scaling function integrates to unity; :

Wavelet function integrates to zero; :

Dual wavelet function integrates to zero; :

satisfies the recursion equation :

Plot the components and the sum of the recursion:

satisfies the recursion equation :

Plot the components and the sum of the recursion:

satisfies the recursion equation :

Plot the components and the sum of the recursion:

satisfies the recursion equation :

Plot the components and the sum of the recursion:

Frequency response for is given by :

The filter is a lowpass filter:

Fourier transform of is given by :

Frequency response for is given by :

The filter is a dual lowpass filter:

Fourier transform of is given by :

Frequency response for is given by :

The filter is a lowpass filter:

Fourier transform of is given by :

Frequency response for is given by :

The filter is a lowpass filter:

Fourier transform of is given by :

## Neat Examples(2)

Plot translates and dilations of scaling function:

Plot translates and dilations of wavelet function:

Wolfram Research (2010), CDFWavelet, Wolfram Language function, https://reference.wolfram.com/language/ref/CDFWavelet.html.

#### Text

Wolfram Research (2010), CDFWavelet, Wolfram Language function, https://reference.wolfram.com/language/ref/CDFWavelet.html.

#### CMS

Wolfram Language. 2010. "CDFWavelet." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/CDFWavelet.html.

#### APA

Wolfram Language. (2010). CDFWavelet. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/CDFWavelet.html

#### BibTeX

@misc{reference.wolfram_2024_cdfwavelet, author="Wolfram Research", title="{CDFWavelet}", year="2010", howpublished="\url{https://reference.wolfram.com/language/ref/CDFWavelet.html}", note=[Accessed: 23-July-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_cdfwavelet, organization={Wolfram Research}, title={CDFWavelet}, year={2010}, url={https://reference.wolfram.com/language/ref/CDFWavelet.html}, note=[Accessed: 23-July-2024 ]}