WaveletMapIndexed

WaveletMapIndexed[f,wd]

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

WaveletMapIndexed[f,dwd,wind]

applies f to the DiscreteWaveletData coefficients specified by wind.

WaveletMapIndexed[f,cwd,octvoc]

applies f to the ContinuousWaveletData coefficients specified by octvoc.

Details

Examples

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Basic Examples  (2)

Rescale all coefficients of a discrete wavelet transform by 20:

Normal gives the array of coefficients:

Compare with the unmodified coefficients:

Amplify the {1} coefficient of the stationary wavelet transform of an image:

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

Scope  (11)

Basic Uses  (3)

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:

Coefficient Specification  (4)

Transform only specified coefficients in DiscreteWaveletData:

Apply a function to detail coefficients only, using the index pattern {___, 1}:

Apply a function to coarse coefficients only, using the index pattern {___, 0}:

Transform only specified coefficients in a ContinuousWaveletData:

Apply a function to coefficients in the first octave {1,_} only:

Apply a function to all coefficients except those in the second octave, first voice {2,1}:

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:

Data  (4)

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:

Applications  (7)

Data Processing  (2)

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:

Image Processing  (3)

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:

Sound Processing  (1)

Apply a nonlinear function to wavelet coefficients for sound data:

Inverse transform to obtain a reconstructed sound object:

Wavelet Thresholding  (1)

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:

Properties & Relations  (3)

MapIndexed[f,expr] applies f to the parts of any expression:

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

WaveletMapIndexed[vMap[f,v],wd] 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 (/.):

Possible Issues  (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:

Introduced in 2010
 (8.0)