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:

- Hard or soft (default) thresholding
- Scaled white noise, unscaled white noise (default) or non-white noise
- Thresholds values are:
- Donoho-Johnstone methods: Fixed-form (default), Heursure, Rigsure, Minimax
- Birgé-Massart method: Penalized high, Penalized medium, Penalized low.

**One-Dimensional DWT Compression**

- Level-dependent or interval-dependent hard thresholding methods are available. Predefined thresholding strategies are:
This method includes a sparsity parameter

*a*(1 <*a*< 5). Using this strategy the default is*a*= 1.5. - 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:

- Hard or soft (default) thresholding
- Scaled white noise, unscaled white noise (default) or non-white noise
- Thresholds values are:
- Donoho-Johnstone method: Fixed form (default)
- Birgé-Massart method: Penalized high, Penalized medium, Penalized low.
- Empirical method: Balance sparsity-norm, default = sqrt

The last three choices include a sparsity parameter *a* (*a* > 1), see One-Dimensional DWT and SWT De-noising.

**Two-Dimensional DWT Compression**

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

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

- Hard or soft (default) thresholding
- Thresholds values:
- Donoho-Johnstone methods: Fixed form (unscaled noise) (default); Fixed form (scaled noise)
- Birgé-Massart method: Penalized high, Penalized medium, Penalized low.

This method includes a sparsity parameter *a* (*a* > 1); see One-Dimensional DWT and SWT De-noising.

**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:

- Hard or soft (default) thresholding
- Thresholds values:
- Donoho-Johnstone methods: Fixed form (unscaled noise) (default); Fixed form (scaled noise)
- Birgé-Massart method: Penalized high, Penalized medium, Penalized low.
- Empirical method: Balance sparsity-norm (sqrt).

The last three choices include a sparsity parameter *a* (*a* > 1); see One-Dimensional DWT and SWT De-noising.

**Two-Dimensional Wavelet Packet Compression**

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

- Empirical methods

**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:

- 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.

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:

*J*the level of the decomposition*M*a positive constant*a*a sparsity parameter (*a*> 1)

- At level
*J*the approximation is kept - For level
*j*from 1 to*J*, the*n*_{j}largest coefficients are kept with

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 = n*_{J+}_{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:

- Scarce high:
*M*=*L*in the 1-D case*M*= 4*prod(*S*) in the 2-D case- Scarce medium:
*M*= 1.5**L*in the 1-D case*M*= 4*4*prod(*S*) / 3 in the 2-D case- Scarce low:
*M*= 2**L*in the 1-D case*M*= 4*8*prod(*S*) / 3 in the 2-D case

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:

- The sparsity parameter
*a*> 1 - The coefficients
*c(k)*are sorted in decreasing order of their absolute value *v*is the noise variance

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

- Penalized high, 2.5 a < 10
- Penalized medium, 1.5 <
*a*< 2.5 - Penalized low, 1 <
*a*< 2

The related M-files are `wbmpen`

, `wpbmpen`

, and `wthrmngr`

(for more information, see the corresponding reference pages).

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).

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

- The 2-norm recovery in percentage
- The relative sparsity in percentage, obtained from the compressed signal by setting to 0 the coefficients less than
*t*in absolute value

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).

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:

- DWT or SWT 1-D, where
*n*is the signal length and*s*is the noise standard deviation. - DWT or SWT 2-D, where [
*n*,*m*] is the image size - WP 1-D,

- WP 2-D,

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).

These options are dedicated to the density estimation problem.

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

*c*is all the detail coefficients of the binned data*d(j)*is the detail coefficients at level*j**n*is the number of bins chosen for the preliminary estimator (binning)

Then, these options are defined as follow.

- Global:

- By level 1:

Level dependent thresholds *T(j)* are defined by:

- By level 2:

Level dependent thresholds *T(j) *are defined by:

- 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 |