# wdcbm

Thresholds for wavelet 1-D using Birgé-Massart strategy

## Syntax

```[THR,NKEEP] = wdcbm(C,L,ALPHA,M) wdcbm(C,L,ALPHA) wdcbm(C,L,ALPHA,L(1)) ```

## Description

`[THR,NKEEP] = wdcbm(C,L,ALPHA,M)` returns level-dependent thresholds `THR` and numbers of coefficients to be kept `NKEEP`, for denoising or compression. `THR` is obtained using a wavelet coefficients selection rule based on the Birgé-Massart strategy.

`[C,L]` is the wavelet decomposition structure of the signal to be denoised or compressed, at level `j = length(L)-2`. `ALPHA` and `M` must be real numbers greater than 1.

`THR` is a vector of length `j`; `THR(i)` contains the threshold for level i.

`NKEEP` is a vector of length `j`; `NKEEP(i)` contains the number of coefficients to be kept at level i.

j, `M` and `ALPHA` define the strategy:

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

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

Typically `ALPHA` = 1.5 for compression and `ALPHA` = 3 for denoising.

A default value for `M` is `M` = `L`(1), the number of the coarsest approximation coefficients, since the previous formula leads for i = j+1, to nj+1 = `M` = `L`(1). Recommended values for `M` are from `L`(1) to 2*`L`(1).

`wdcbm(C,L,ALPHA)` is equivalent to `wdcbm(C,L,ALPHA,L(1))`.

## Examples

```% Load electrical signal and select a part of it. load leleccum; indx = 2600:3100; x = leleccum(indx); % Perform a wavelet decomposition of the signal % at level 5 using db3. wname = 'db3'; lev = 5; [c,l] = wavedec(x,lev,wname); % Use wdcbm for selecting level dependent thresholds % for signal compression using the adviced parameters. alpha = 1.5; m = l(1); [thr,nkeep] = wdcbm(c,l,alpha,m) thr = 19.5569 17.1415 20.2599 42.8959 15.0049 nkeep = 1 2 3 4 7 % Use wdencmp for compressing the signal using the above % thresholds with hard thresholding. [xd,cxd,lxd,perf0,perfl2] = ... wdencmp('lvd',c,l,wname,lev,thr,'h'); % Plot original and compressed signals. subplot(211), plot(indx,x), title('Original signal'); subplot(212), plot(indx,xd), title('Compressed signal'); xlab1 = ['2-norm rec.: ',num2str(perfl2)]; xlab2 = [' % -- zero cfs: ',num2str(perf0), ' %']; xlabel([xlab1 xlab2]); ```

## References

Birgé, L.; P. Massart (1997), “From model selection to adaptive estimation,” in D. Pollard (ed), Festchrift for L. Le Cam, Springer, pp. 55–88.