De-noising or compression using wavelet packets

`[XD,TREED,PERF0,PERFL2] = wpdencmp(X,SORH,N,`

* 'wname'*,CRIT,PAR,KEEPAPP)

[XD,TREED,PERF0,PERFL2] = wpdencmp(TREE,SORH,CRIT,PAR,KEEPAPP)

`wpdencmp`

is a one- or
two-dimensional de-noising and compression oriented function.

`wpdencmp`

performs a
de-noising or compression process of a signal or an image, using wavelet
packet. The ideas and the procedures for de-noising and compression
using wavelet packet decomposition are the same as those used in the
wavelets framework (see `wden`

and `wdencmp`

for more information).

`[XD,TREED,PERF0,PERFL2] = wpdencmp(X,SORH,N,`

returns
a de-noised or compressed version * 'wname'*,CRIT,PAR,KEEPAPP)

`XD`

of input signal `X`

(one-
or two-dimensional) obtained by wavelet packets coefficients thresholding. The additional output argument `TREED`

is the
wavelet packet best tree decomposition (see `besttree`

for
more information) of `XD`

. `PERFL2`

and `PERF0`

are *L ^{2}* energy
recovery and compression scores in percentages.

`PERFL2 = 100 *`

(vector-norm of WP-cfs of `XD`

/
vector-norm of WP-cfs of `X`

^{2}.

If `X`

is a one-dimensional signal and * 'wname'* an
orthogonal wavelet,

`PERFL2`

is reduced to$$\frac{100{\Vert XD\Vert}^{2}}{{\Vert X\Vert}^{2}}$$

`SORH `

equal to `'s'`

or `'h'`

is
for soft or hard thresholding (see `wthresh`

for
more information).

Wavelet packet decomposition is performed at level `N`

and * 'wname'* is
a character vector containing the wavelet name. Best decomposition
is performed using entropy criterion defined by character vector

`CRIT`

and
parameter `PAR`

(see `wentropy`

for
more information). Threshold parameter is also `PAR`

.
If `KEEPAPP = 1`

, approximation coefficients cannot
be thresholded; otherwise, they can be. `[XD,TREED,PERF0,PERFL2] = wpdencmp(TREE,SORH,CRIT,PAR,KEEPAPP)`

has
the same output arguments, using the same options as above, but obtained
directly from the input wavelet packet tree decomposition `TREE`

(see `wpdec`

for more information) of the signal
to be de-noised or compressed.

In addition if `CRIT = 'nobest'`

no optimization
is done and the current decomposition is thresholded.

% The current extension mode is zero-padding (see dwtmode). % Load original signal. load sumlichr; x = sumlichr; % Use wpdencmp for signal compression. % Find default values (see ddencmp). [thr,sorh,keepapp,crit] = ddencmp('cmp','wp',x) thr = 0.5193 sorh = h keepapp = 1 crit = threshold % De-noise signal using global thresholding with % threshold best basis. [xc,wpt,perf0,perfl2] = ... wpdencmp(x,sorh,3,'db2',crit,thr,keepapp); % Using some plotting commands, % the following figure is generated.

% Load original image. load sinsin % Generate noisy image. x = X/18 + randn(size(X)); % Use wpdencmp for image de-noising. % Find default values (see ddencmp). [thr,sorh,keepapp,crit] = ddencmp('den','wp',x) thr = 4.9685 sorh = h keepapp = 1 crit = sure % De-noise image using global thresholding with % SURE best basis. xd = wpdencmp(x,sorh,3,'sym4',crit,thr,keepapp); % Using some plotting commands, % the following figure is generated.

% Generate heavy sine and a noisy version of it. init = 1000; [xref,x] = wnoise(5,11,7,init); % Use wpdencmp for signal de-noising. n = length(x); thr = sqrt(2*log(n*log(n)/log(2))); xwpd = wpdencmp(x,'s',4,'sym4','sure',thr,1); % Compare with wavelet-based de-noising result. xwd = wden(x,'rigrsure','s','one',4,'sym4');

Antoniadis, A.; G. Oppenheim, Eds. (1995), *Wavelets
and statistics*, Lecture Notes in Statistics, 103, Springer
Verlag.

Coifman, R.R.; M.V. Wickerhauser (1992), "Entropy-based
algorithms for best basis selection," *IEEE Trans.
on Inf. Theory*, vol. 38, 2, pp. 713–718.

DeVore, R.A.; B. Jawerth, B.J. Lucier (1992), "Image
compression through wavelet transform coding," *IEEE
Trans. on Inf. Theory*, vol. 38, No 2, pp. 719–746.

Donoho, D.L. (1993), "Progress in wavelet analysis and WVD: a ten minute tour," in Progress in wavelet analysis and applications, Y. Meyer, S. Roques, pp. 109–128. Frontières Ed.

Donoho, D.L.; I.M. Johnstone (1994), "Ideal spatial adaptation
by wavelet shrinkage," *Biometrika*, vol.
81, pp. 425–455.

Donoho, D.L.; I.M. Johnstone, G. Kerkyacharian, D. Picard (1995),
"Wavelet shrinkage: asymptopia," *Jour. Roy.
Stat. Soc.*, series B, vol. 57 no. 2, pp. 301–369.

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