# wpdencmp

Denoising or compression using wavelet packets

## Syntax

``[xd,treed,perf0,perfl2] = wpdencmp(x,sorh,n,wname,crit,par,keepapp)``
``[___] = wpdencmp(tree,sorh,crit,par,keepapp)``

## Description

`wpdencmp` performs a denoising or compression process of a signal or image using wavelet packets. The ideas and procedures for denoising and compression using either wavelet or wavelet packet decompositions are the same. See `wdenoise` or `wdencmp` for more information.

example

````[xd,treed,perf0,perfl2] = wpdencmp(x,sorh,n,wname,crit,par,keepapp)` returns a denoised or compressed version `xd` of the input data `x` obtained by wavelet packet coefficient thresholding. `wpdencmp` also returns the wavelet packet best tree decomposition `treed` of `xd` (see `besttree` for more information), and the L2 energy recovery and compression scores in percentages as `perfl2` and `perf0`, respectively.```
````[___] = wpdencmp(tree,sorh,crit,par,keepapp)` uses the wavelet packet decomposition `tree` of the data to be denoised or compressed.```

## Examples

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This example shows how to denoise using wavelet packets.

Use `wnoise` to generate the heavy sine signal and a noisy version.

```init = 1000; [xref,x] = wnoise(5,11,7,init); figure subplot(2,1,1) plot(xref) axis tight title('Heavy Sine') subplot(2,1,2) plot(x) axis tight title('Noisy Heavy Sine')```

Denoise the noisy signal using a four-level wavelet packet decomposition. Use the order 4 Daubechies least asymmetric wavelet.

```n = length(x); thr = sqrt(2*log(n*log(n)/log(2))); xwpd = wpdencmp(x,'s',4,'sym4','sure',thr,1);```

Compare with a wavelet-based denoising result. Use `wdenoise` with comparable input arguments. Plot the differences between the two denoised signals and original signal.

```xwd = wdenoise(x,4,'Wavelet','sym4','DenoisingMethod','UniversalThreshold','ThresholdRule','Hard'); figure subplot(2,1,1) plot(x-xwpd) axis tight ylim([-12 12]) title('Difference Between Wavelet Packet Denoised and Original') subplot(2,1,2) plot(x-xwd) axis tight ylim([-12 12]) title('Difference Between Wavelet Denoised and Original')```

This example shows how to denoise an image using wavelet packets.

Load an image and generate a noisy copy. For reproducibility set the random seed.

```rng default load sinsin x = X/18 + randn(size(X)); imagesc(X) colormap(gray) title('Original Image')```

```figure imagesc(x) colormap(gray) title('Noisy Image')```

Denoise the noisy image using wavelet packet decomposition. Use `ddencmp` to determine denoising parameters. Do a three-level decomposition with the order 4 Daubechies least asymmetric wavelet.

```[thr,sorh,keepapp,crit] = ddencmp('den','wp',x); xd = wpdencmp(x,sorh,3,'sym4',crit,thr,keepapp); figure imagesc(xd) colormap(gray) title('Denoised Image')```

This example shows how to compress a 1-D signal using wavelet packets.

Load a signal. Use `ddencmp` to determine compression values for that signal.

```load sumlichr x = sumlichr; [thr,sorh,keepapp,crit] = ddencmp('cmp','wp',x)```
```thr = 0.5193 ```
```sorh = 'h' ```
```keepapp = 1 ```
```crit = 'threshold' ```

Compress the signal using global thresholding with threshold best basis. Use the order 4 Daubechies least asymmetric wavelet and do a three-level wavelet packet decomposition.

`[xc,wpt,perf0,perfl2] = wpdencmp(x,sorh,3,'sym4',crit,thr,keepapp);`

Compare the original signal with the compressed version.

```subplot(2,1,1) plot(x) title('Original Signal') axis tight subplot(2,1,2) plot(xc) xlabel(['L^2 rec.: ',num2str(perfl2),'% zero cfs.: ',num2str(perf0),'%']) title('Compressed Signal Using Wavelet Packets') axis tight```

Compress the signal again, but this do a three-level wavelet decomposition. Keep all the other parameters the same.

```[thr,sorh,keepapp] = ddencmp('cmp','wv',x); [xcwv,~,~,perf0wv,perfl2wv] = wdencmp('gbl',x,'sym4',3,thr,sorh,keepapp); figure subplot(2,1,1) plot(x) title('Original Signal') axis tight subplot(2,1,2) plot(xc) xlabel(['L^2 rec.: ',num2str(perfl2wv),'% zero cfs.: ',num2str(perf0wv),'%']) title('Compressed Signal Using Wavelets') axis tight```

A larger fraction of coefficients are set equal to 0 when compressing using a wavelet packet decomposition.

## Input Arguments

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Input data to denoise or compress, specified by a real-valued vector or matrix.

Data Types: `double`

Wavelet packet decomposition of the data to be denoised or compressed, specified as a wavelet packet tree. See `wpdec` and `wpdec2` for more information.

Type of thresholding to perform:

• `'s'` — Soft thresholding

• `'h'` — Hard thresholding

See `wthresh` for more information.

Wavelet packet decomposition level, specified as a positive integer.

Name of wavelet, specified as a character vector or string scalar, to use for denoising. See `wavemngr` for more information.

Entropy type, specified as one of the following:

Entropy Type (`crit`)

Threshold Parameter (`par`)

`'shannon'`

`par` is not used.

`'log energy'`

`par` is not used.

`'threshold'``0 ≤ par`

`par` is the threshold.

`'sure'``0 ≤ par`

`par` is the threshold.

`'norm'``1 ≤ par`

`par` is the power.

`'user'`Character vector

`par` is a character vector containing the file name of your own entropy function, with a single input `x`.

'`FunName`'No constraints on `par`

`FunName` is any character vector other than the previous entropy types listed.

`FunName` contains the file name of your own entropy function, with `x` as input and `par` as an additional parameter to your entropy function.

`crit` and threshold parameter `par` together define the entropy criterion used to determine the best decomposition. See `wentropy` for more information.

If `crit = 'nobest'`, no optimization is done, and the current decomposition is thresholded.

Threshold parameter, specified by a real number, character vector, or string scalar. `par` and the entropy type `crit` together define the entropy criterion used to determine the best decomposition. See `wentropy` for more information.

Data Types: `double`

Threshold approximation setting, specified as either `0` or `1`. If `keepapp = 1`, the approximation coefficients cannot be thresholded. If ```keepapp = 0```, the approximation coefficients can be thresholded.

Data Types: `double`

## Output Arguments

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Denoised or compressed data, returned as a real-valued vector or matrix. `xd` and `x` have the same dimensions.

Wavelet packet best tree decomposition of `xd`, returned as a wavelet packet tree.

Compression score, returned as a real number. `perf0` is the percentage of thresholded coefficients that are equal to 0.

L2 energy recovery, returned as a real number. `perfl2` is equal to If `x` is a one-dimensional signal and `wname` an orthogonal wavelet, `perfl2` simplifies to $\frac{100{‖xd‖}^{2}}{{‖x‖}^{2}}.$

## References

[1] Antoniadis, A., and G. Oppenheim, eds. Wavelets and Statistics. Lecture Notes in Statistics. New York: Springer Verlag, 1995.

[2] Coifman, R. R., and M. V. Wickerhauser. “Entropy-Based Algorithms for Best Basis Selection.” IEEE Transactions on Information Theory. Vol. 38, Number 2, 1992, pp. 713–718.

[3] DeVore, R. A., B. Jawerth, and B. J. Lucier. “Image Compression Through Wavelet Transform Coding.” IEEE Transactions on Information Theory. Vol. 38, Number 2, 1992, pp. 719–746.

[4] Donoho, D. L. “Progress in Wavelet Analysis and WVD: A Ten Minute Tour.” Progress in Wavelet Analysis and Applications (Y. Meyer, and S. Roques, eds.). Gif-sur-Yvette: Editions Frontières, 1993.

[5] Donoho, D. L., and I. M. Johnstone. “Ideal Spatial Adaptation by Wavelet Shrinkage.” Biometrika. Vol. 81, 1994, pp. 425–455.

[6] Donoho, D. L., I. M. Johnstone, G. Kerkyacharian, and D. Picard. “Wavelet Shrinkage: Asymptopia?” Journal of the Royal Statistical Society, series B. Vol. 57, Number 2, 1995, pp. 301–369.

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