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Legacy Documentation

Mathematica 8 (2010)

This is documentation for Mathematica 8, which was
based on an earlier version of Wolfram Language.

View current documentation (Version 14.3)

Mathematica > Data Manipulation > Signal Processing > Wavelet Analysis > WaveletMapIndexed >

Mathematica > Visualization and Graphics > Data Visualization > Wavelet Analysis > WaveletMapIndexed >

Mathematica > Data Manipulation > Numerical Data > Data Transforms and Smoothing > Wavelet Analysis > WaveletMapIndexed >

BUILT-IN MATHEMATICA SYMBOL

ContinuousWaveletData DiscreteWaveletData WaveletThreshold DiscreteWaveletTransform ContinuousWaveletTransform InverseWaveletTransform MapIndexed

WaveletMapIndexed

WaveletMapIndexed
applies the function f to the arrays of coefficients and indices of a ContinuousWaveletData or DiscreteWaveletData object.

WaveletMapIndexed
applies f to the DiscreteWaveletData coefficients specified by wind.

WaveletMapIndexed
applies f to the ContinuousWaveletData coefficients specified by octvoc.

WaveletMapIndexed takes a ContinuousWaveletData or DiscreteWaveletData object and returns an object of the same type.

For dwd with coefficients , WaveletMapIndexed effectively corresponds to DiscreteWaveletData .

For cwd with coefficients , WaveletMapIndexed effectively corresponds to ContinuousWaveletData .

Each should return an array or Image of the same dimensions as .

For Sound and SampledSoundList , each is given as a list of vectors, each corresponding to a channel of sound samples.

The wavelet index convention is the same as explained in ContinuousWaveletData and DiscreteWaveletData .

(2)

Rescale all coefficients of a discrete wavelet transform by 20:

Normal gives the array of coefficients:

Compare with the unmodified coefficients:

Amplify the coefficient of the stationary wavelet transform of an image:

The inverse wavelet transform gives an image with vertical edges sharpened:

Rescale all coefficients of a discrete wavelet transform by 20:

Out[2]=

Normal gives the array of coefficients:

Out[3]=

Compare with the unmodified coefficients:

Out[4]=

Amplify the coefficient of the stationary wavelet transform of an image:

Out[1]=

Out[2]=

The inverse wavelet transform gives an image with vertical edges sharpened:

Out[3]=

(11)

Apply an arbitrary function to all coefficients of a discrete wavelet transform:

Apply a symbolic function that also depends on the wavelet index for each coefficient vector:

WaveletMapIndexed operates on ContinuousWaveletData or DiscreteWaveletData :

The result is a wavelet data object of the same type:

The modified data can be used in other wavelet functions such as inverse wavelet transforms:

Transform only specified coefficients in DiscreteWaveletData :

Apply a function to detail coefficients only, using the index pattern :

Apply a function to coarse coefficients only, using the index pattern :

Transform only specified coefficients in a ContinuousWaveletData :

Apply a function to coefficients in the first octave only:

Apply a function to all coefficients except those in the second octave, first voice :

The function f can depend on the wavelet index as its second argument:

Define a function with an arbitrary dependence on the wavelet index:

Apply the function to continuous wavelet transform coefficients:

For list data, the coefficients supplied as the first argument of f are lists:

Apply a function that transforms lists:

For multidimensional data, the coefficients are arrays of the same depth:

Apply a function that transforms array coefficients of that depth:

For image data, the coefficients are supplied to f as Image objects:

The coefficients have the same number of channels as the original image:

Apply a function that transforms image coefficients:

For sound data, the coefficients are two-dimensional arrays:

Dimensions of one coefficient:

The two dimensions specify the channel number and the wavelet coefficients for that channel:

Apply a function that transforms two-channel data:

Reconstructed Sound data:

(7)

Coefficients with short indices correspond to small-scale structure in the data:

Zero all small-scale coefficients from the stationary wavelet transform of random data:

The inverse wavelet transform varies only on larger scales:

Perform a simple thresholding operation by removing low-amplitude wavelet coefficients:

Compare with the original data:

Blur an image by setting small-scale detail coefficients to zero:

Compare with the original image:

Sharpen an image by amplifying small-scale detail coefficients:

Compare with the original image:

Use a mask image to vary between blurring and sharpening across an image:

Compare with the original image:

Apply a nonlinear function to wavelet coefficients for sound data:

Inverse transform to obtain a reconstructed sound object:

Perform a wavelet-based shrinkage based on conditional mean:

Compute a discrete wavelet transform up to refinement level 6:

Compute the standard deviation for the finest detail coefficients:

Compute the standard deviation for all wavelet coefficients:

Assuming a Gaussian mixture model, variance can be estimated in the proportion to :

Shrinkage estimates of the signal coefficients are given by:

Use WaveletMapIndexed to map over detail coefficients:

Reconstruct thresholded signal coefficients:

(3)

MapIndexed applies f to the parts of any expression:

WaveletMapIndexed applies f to the coefficients in the wavelet data object wd:

WaveletMapIndexed applies f to each part of each coefficient:

MapIndexed gives the part specification as the second argument of f:

WaveletMapIndexed gives the wavelet index specification as the second argument of f:

WaveletMapIndexed transforms arrays of coefficients, giving a new DiscreteWaveletData :

Use Map and Normal [dwd] to transform coefficients into normal expressions:

Or use ReplaceAll ():

(2)

The function f is always passed the index specification as its second argument:

Use a function that operates on its first argument only:

The function f should return an array or image of the same dimensions:

Listable functions return an array of the same dimensions:

Arithmetic operations such as multiplication are Listable :

Use Map for functions that are not Listable :

ContinuousWaveletData bullet DiscreteWaveletData bullet WaveletThreshold bullet DiscreteWaveletTransform bullet ContinuousWaveletTransform bullet InverseWaveletTransform bullet MapIndexed

Wavelet Analysis

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