# wdcbm

Thresholds for wavelet 1-D using Birgé-Massart strategy

## Syntax

``[thr,nkeep] = wdcbm(C,L,alpha,M)``

## Description

````[thr,nkeep] = wdcbm(C,L,alpha,M)` returns level-dependent thresholds `thr` and numbers of coefficients to be kept `nkeep`, for denoising or compressing a signal. `wdcbm` uses a wavelet coefficients selection rule based on the Birgé-Massart strategy to obtain the thresholds.`[C,L]` is the wavelet decomposition structure of the signal to be denoised or compressed, at level ```N = length(L)-2```. `alpha` and `M` are real numbers greater than 1.`wdcbm(C,L,alpha)` is equivalent to `wdcbm(C,L,alpha,L(1))`.```

example

## Examples

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Load the electrical signal. Select a portion of the signal.

```load leleccum indx = 2600:3100; x = leleccum(indx);```

Obtain the wavelet decomposition of the signal at level 5 using the `db3` wavelet.

```wname = "db3"; lev = 5; [c,l] = wavedec(x,lev,wname);```

Use `wdcbm` to select level-dependent thresholds for signal compression. Use the suggested parameters.

```alpha = 1.5; m = l(1); [thr,nkeep] = wdcbm(c,l,alpha,m)```
```thr = 1×5 19.5569 17.1415 20.2599 42.8959 15.0049 ```
```nkeep = 1×5 1 2 3 4 7 ```

Use `wdencmp` for compressing the signal using the thresholds. Use hard thresholding.

`[xd,cxd,lxd,perf0,perfl2] = wdencmp("lvd",c,l,wname,lev,thr,"h");`

Plot the original and compressed signals.

```subplot(2,1,1) plot(indx,x) title("Original Signal") subplot(2,1,2) plot(indx,xd) title("Compressed Signal") xlab1 = ['2-norm rec.: ',num2str(perfl2)]; xlab2 = [' % -- zero cfs: ',num2str(perf0), ' %']; xlabel([xlab1 xlab2])```

## Input Arguments

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Wavelet decomposition of the signal to be denoised or compressed, specified as a vector. The vector contains the wavelet coefficients. The bookkeeping vector `L` contains the number of coefficients by level. See `wavedec`.

Data Types: `double`

Bookkeeping vector, specified as a vector of positive integers. The bookkeeping vector is used to parse the coefficients in the wavelet decomposition `C` by level. See `wavedec`.

Data Types: `double`

Sparsity parameter to use in the Birgé-Massart strategy, specified as a real-valued scalar greater than 1. Typically, ```alpha = 1.5``` for compression and ```alpha = 3``` for denoising. For more information, see Wavelet Coefficients Selection.

Data Types: `double`

Factor to use in the Birgé-Massart strategy, specified as a real-valued scalar greater than 1. The default value is `L(1)`, the number of the coarsest approximation coefficients. Recommended values for `M` are from `L(1)` to `2*L(1)`. For more information, see Wavelet Coefficients Selection.

Data Types: `double`

## Output Arguments

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Level-dependent thresholds, returned as a vector of length `N`, where ```N = length(L)-2```. `thr(i)` contains the threshold for level i. The thresholds are obtained using a wavelet coefficients selection rule based on the Birgé-Massart strategy. For more information, see Wavelet Coefficients Selection.

Number of coefficients to be kept at each level, returned as a vector of length `N`, where ```N = length(L)-2```. `nkeep(i)` contains the number of coefficients to be kept for level i.

Data Types: `double`

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### Wavelet Coefficients Selection

Thresholds are obtained using a wavelet coefficients selection rule based on the Birgé-Massart strategy. The values `N=length(L)-2`, `M` and `alpha` define the strategy.

• At level `N+1` (and coarser levels), everything is kept.

• For level i from 1 to `N`, the ni largest coefficients are kept, where ```ni = M / (N+2-i)alpha```.

The default value of `M = L(1)` corresponds to the formula ```nN+1 = M / (N+2-(N+1))alpha = M```.

## References

[1] Birgé, Lucien, and Pascal Massart. “From Model Selection to Adaptive Estimation.” In Festschrift for Lucien Le Cam: Research Papers in Probability and Statistics, edited by David Pollard, Erik Torgersen, and Grace L. Yang, 55–87. New York, NY: Springer, 1997. https://doi.org/10.1007/978-1-4612-1880-7_4.

## Version History

Introduced before R2006a