Wavelet Toolbox    

Available Methods for De-noising, Estimation and Compression Using GUI Tools

This section presents the predefined strategies available using the de-noising, estimation and compression GUI tools.

One-Dimensional DWT and SWT De-noising

Level-dependent or interval-dependent thresholding methods are available. Predefined thresholding strategies:

One-Dimensional DWT Compression

  1. Level-dependent or interval-dependent hard thresholding methods are available. Predefined thresholding strategies are:
  2. Global hard thresholding methods with GUI-driven choice are available. Predefined thresholding strategies are:

Two-Dimensional DWT and SWT De-noising

Level-dependent and orientation-dependent (horizontal, vertical, and diagonal) thresholding methods are available. Predefined thresholding strategies are:

Two-Dimensional DWT Compression

Level-dependent and orientation-dependent (horizontal, vertical, and diagonal) thresholding methods are available.

  1. Level-dependent or interval-dependent hard thresholding methods are available. Predefined thresholding strategies are:
  2. Global hard thresholding methods with GUI-driven choice are available. Predefined thresholding strategies are:

One-Dimensional Wavelet Packet De-noising

Global thresholding methods with GUI-driven choice are available. Predefined thresholding strategies are:

One-Dimensional Wavelet Packet Compression

Global hard thresholding methods with GUI-driven choice are available. Predefined thresholding strategies are:

Two-Dimensional Wavelet Packet De-noising

Global thresholding methods with GUI-driven choice are available. Predefined thresholding strategies are:

Two-Dimensional Wavelet Packet Compression

Global thresholding methods with GUI-driven choice are available. Predefined thresholding strategies are:

One-Dimensional Regression Estimation

A preliminary histogram estimator (binning) is used, and then the predefined thresholding strategies described in One-Dimensional DWT and SWT De-noising, are available.

Density Estimation

A preliminary histogram estimator (binning) is used, and then the predefined thresholding strategies are:

- Global threshold

- By level threshold 1, By level threshold 2, By level threshold 3.

The last choice includes a sparsity parameter a (a < 1), the default is 0.6.

More About the Thresholding Strategies

A lot of references are available for this topic of de-noising, estimation, and compression.

For example: [Ant94], [AntP98], [HalPKP97], [AntG99], [Ogd97], [HarKPT98], [DonJ94a&b], [DonJKP95], and [DonJKP96] (see References). A short description of the available methods previously mentioned follows.

Scarce High, Medium, and Low.   

These strategies are based on an approximation result from Birgé and Massart (for more information, see [BirM97]) and are well suited for compression.

Three parameters characterize the strategy:

The strategy is such that:

So the strategy leads to select the highest coefficients in absolute value at each level, the numbers of kept coefficients grow scarcely with J-j.

Typically, a = 1.5 for compression and a = 3 for de-noising.

A natural default value for M is the length of the coarsest approximation coefficients, since the previous formula for j = J+1, leads to M = nJ+1.

Let L denote the length of the coarsest approximation coefficients in the 1-D case and S the size of the coarsest approximation coefficients in the 2-D case.

Three different choices for M are proposed:

The related M-files are wdcbm, wdcbm2, and wthrmngr (for more information, see the corresponding reference pages).

Penalized High, Medium, and Low.   

These strategies are based on a recent de-noising result by Birgé and Massart, and can be viewed as a variant of the fixed form strategy (see the section De-Noising) of the wavelet shrinkage.

The threshold T applied to the detail coefficients for the wavelet case or the wavelet packet coefficients for a given fixed WP tree, is defined by:

with

where

Three different intervals of choices for the sparsity parameter a are proposed:

The related M-files are wbmpen, wpbmpen, and wthrmngr (for more information, see the corresponding reference pages).

Remove Near 0.   

Let c denote the detail coefficients at level 1 obtained from the decomposition of the signal or the image to be compressed, using db1. The threshold value is set to median(abs(c)) or to 0.05*max(abs(c)) if median(abs(c)) = 0.

The related M-files are ddencmp, and wthrmngr (for more information, see the corresponding reference pages).

Balance Sparsity-Norm.   

Let c denote all the detail coefficients, two curves are built associating, for each possible threshold value t, two percentages:

A default is provided for the 1-D case taking t such that the two percentages are equal. Another one is obtained for the 2-D case by taking the square root of the previous t.

The related M-file is wthrmngr (for more information, see the corresponding reference page).

Fixed Form.   

This thresholding strategy comes from Donoho-Johnstone (see References and the 'sqtwolog' option of the wden function in De-Noising), the universal threshold is of the following form:

The related M-files are ddencmp, thselect, wden, wdencmp, and wthrmngr (for more information, see the corresponding reference pages).

Heursure, Rigsure, and Minimax.   

These methods are available for 1-D de-noising tools and come from Donoho-Johnstone (see References).

The related M-files are thselect, wden, wdencmp, and wthrmngr (for more information, see the corresponding reference pages).

Global, and By level 1, 2, 3.   

These options are dedicated to the density estimation problem.

See [HalPKP97], [AntG99], [Ogd97], and [HarKPT98] in References for more details.

Note that:

Then, these options are defined as follow.

  1. Global:

Threshold value is set to

  1. By level 1:

Level dependent thresholds T(j) are defined by:

  1. By level 2:

Level dependent thresholds T(j) are defined by:

  1. By level 3:

Level dependent thresholds T(j) are defined by:

where a is a sparsity parameter ( is the default)


 Function Estimation: Density and Regression Wavelet Packets