BUILT-IN MATHEMATICA SYMBOL

WaveletThreshold

WaveletThreshold[dwd]
thresholds the detail wavelet coefficients in the DiscreteWaveletData object dwd.

WaveletThreshold

thresholds the coefficients using the thresholding specification tspec.

WaveletThreshold

thresholds the wavelet coefficients given by the wavelet indices wind.

MORE INFORMATION

WaveletThreshold[dwd] is effectively WaveletThreshold.

WaveletThreshold indicates which wavelet coefficients to threshold, using the same index convention as described for DiscreteWaveletData.

By default, thresholding is performed on detail coefficients at each refinement level.

The threshold specification tspec can be of the form: tfun, .

Possible tfun names and options include:

	{"Hard",}
	{"Soft",}
	{"Firm",,r,p}
	{"PiecewiseGarrote",}
	{"SmoothGarrote",,n}
	{"Hyperbola",}
	{"LargestCoefficients",k}	keep the largest k coefficients

In all cases is assumed to be a positive number or a thresholding function tfunc to compute . Each should return a positive number.

The parameter conditions for are that is a positive real and a positive real number between 0 and 1.

The parameter conditions for are to have be a positive real number.

The threshold can be automatically computed using the following methods:

	Automatic	thresholding value
	{"FDR",}	false discovery rate at significance level
	"GCV"	minimizes generalized cross-validation function
	"GCVLevel"	performed at each level
	"SURE"	Stein's unbiased risk estimate
	"SUREHybrid"	combination of and thresholding
	"SURELevel"	performed at each level
	"Universal"	Donoho and Johnstone's universal threshold
	"UniversalLevel"	performed at each level

The parameter conditions for are that should be a number between 0 and 1. By default is equivalent to .

The following short tspec forms can be used:

	"FDR"
	"GCV"
	"GCVLevel"
	"SURE"
	"SURELevel"
	"SUREShrink"
	"Universal"
	"UniversalLevel"
	"VisuShrink"
	"VisuShrinkLevel"

EXAMPLES

CLOSE ALL

Basic Examples (2)

DiscreteWaveletTransform of noisy data:

Smooth data by thresholding wavelet coefficients:

Compare original and smoothed data:

Remove noise from a color image:

Threshold wavelet coefficients:

Synthesize smoothed image using InverseWaveletTransform and compare with original:

DiscreteWaveletTransform of noisy data:

In[1]:=

In[2]:=

Smooth data by thresholding wavelet coefficients:

In[3]:=

Out[3]=

Compare original and smoothed data:

In[4]:=

Out[4]=

Remove noise from a color image:

In[1]:=

In[2]:=

Threshold wavelet coefficients:

In[3]:=

Out[3]=

Synthesize smoothed image using InverseWaveletTransform and compare with original:

In[4]:=

Out[4]=

Scope (12)

WaveletThreshold operates on a DiscreteWaveletData object:

The result is a new DiscreteWaveletData object representing thresholded coefficients:

Specify thresholding method:

Use specific threshold function:

Also specify how to choose threshold value:

Inverse transform and compare:

Visualize the effect on wavelet coefficients:

Compare using WaveletListPlot:

Perform thresholding on specific wavelet indexes:

By default only the detail coefficients are thresholded:

Use Automatic to threshold coefficients used in the inverse transform:

Use All to threshold all coefficients:

Use

to fully control which coefficients to threshold:

Compare the resulting coefficients:

Smooth noisy data using automatic thresholding methods:

Compare all named automatic thresholding methods tspec:

Choose specific thresholding function tfun to apply:

Reconstruct data from thresholded coefficients with automatically chosen threshold value:

Use a named method to automatically compute the threshold value

:

Reconstruct data after

thresholding with various threshold value selection methods:

Use a specific numerical threshold value

:

The best smoothing occurs for threshold values

that are similar to the scale of the noise:

Use a function to compute a threshold value

:

Threshold all coefficients below the standard deviation:

Plot the effect of thresholding function tfun on coefficient values ranging from -0.5 to 0.5:

Apply thresholding separately at each refinement level to data varying on different scales:

Different methods for selecting separate threshold values at each level:

Compare with methods that choose one threshold value for all levels:

Perform thesholding by specifying a function and wavelet index to compute

:

Threshold all detail coefficients below the standard deviation:

Generalizations & Extensions (1)

Use FindThreshold to compute thresholding values:

Specifying a thresholding function computes level-dependent thresholding values:

Applications (8)

Data processing pipeline for simple automatic smoothing:

Denoise 1D data:

Denoise 2D data:

Add normally distributed noise with standard deviation

:

Compare effectiveness of automatic smoothing for different noise levels:

Threshold specific components of an image:

Strongly threshold horizontal and vertical detail coefficients

only:

Compare coefficients:

The inverse transform image has mainly diagonal features:

Plot the noise subtracted by different thresholding functions tfun with automatic threshold value:

Subtract smoothed data from original data (giving a noise model):

Compare performance of automatic thresholding methods when smoothing noisy data:

Quantify performance using SNR (signal-to-noise ratio) and peak SNR in decibels (dB):

Smooth by applying thresholding method th to detail coefficients

:

Compare all named automatic thresholding methods; higher value is better:

Plot smoothed data together with perfectly noise-free data:

thresholding value:

Perform a LiftingWaveletTransform:

The mean absolute deviation is computed using detail coefficients at first refinement level

:

Compute universal threshold value

as

and perform a

threshold:

Use WaveletThreshold to compute the

threshold value:

Compare threshold values:

thresholding value:

Perform a LiftingWaveletTransform:

Compute

threshold values

:

Use WaveletThreshold to compute

threshold value:

Compare threshold values:

thresholding value:

Perform a LiftingWaveletTransform:

Estimate noise level:

For each wavelet coefficient

, find its two-sided

-value

:

Order the

according to their size

. Find

, where

is the significance level:

Compute threshold as

:

Use WaveletThreshold to compute

threshold value:

Compare threshold values:

Properties & Relations (9)

Thresholding functions approach Identity for small threshold values

:

thresholding is similar to Chop:

Data values with absolute value below threshold

are set to 0:

thresholding performs a shrinking operation:

Data values below a certain threshold

are set to 0; those above

are "shrunk" by

:

thresholding:

Parameter

(

) controls the range over which firm thresholding interpolates between

and Identity. Parameter

(

) controls where

lies between

and

, with

by default.

Different settings for

:

Different settings for

:

Different settings for

:

thresholding is a compromise between

and

thresholding:

thresholding has uniformly smaller variance than

thresholding:

In the limit

,

threshold performs

thresholding:

In the limit

,

threshold performs

thresholding:

This is similar to

thresholding with the advantage of having a single parameter

:

thresholding:

In the limit

,

goes to

thresholding:

Use WaveletMapIndexed to perform

thresholding:

Possible Issues (3)

Using

thresholding function with

thresholding value:

Noise coefficients may pass through

thresholding:

thresholding function shrinks these spurious peaks:

The noise estimate is computed based on first-level detail coefficients:

A warning is generated if the estimated noise level is

:

WaveletThreshold does not operate on non-numeric wavelet coefficients: