Wavelet Data Compression

The compression features of a given wavelet basis are primarily linked to the relative scarceness of the wavelet domain representation for the signal. The notion behind compression is based on the concept that the regular signal component can be accurately approximated using the following elements: a small number of approximation coefficients (at a suitably chosen level) and some of the detail coefficients.

Like denoising, the compression procedure contains three steps:

Decompose
选择小波,选择一个水平N. Compute the wavelet decomposition of the signalsat levelN.
Threshold detail coefficients
For each level from 1 toN, a threshold is selected and hard thresholding is applied to the detail coefficients.
Reconstruct
Compute wavelet reconstruction using the original approximation coefficients of levelNand the modified detail coefficients of levels from 1 toN.

Thedifference of the denoising procedure is found in step2. There are two compression approaches available. The first consists of taking the wavelet expansion of the signal and keeping the largest absolute value coefficients. In this case, you can set a global threshold, a compression performance, or a relative square norm recovery performance.

因此,只有一个参数需要被选中. The second approach consists of applying visually determined level-dependent thresholds.

Let us examine two real-life examples of compression using global thresholding, for a given and unoptimized wavelet choice, to produce a nearly complete square norm recovery for a signal (seeSignal Compression) and for an image (seeImage Compression).

% Load electrical signal and select a part. load leleccum; indx = 2600:3100; x = leleccum(indx); % Perform wavelet decomposition of the signal. n = 3; w = 'db3'; [c,l] = wavedec(x,n,w); % Compress using a fixed threshold. thr = 35; keepapp = 1; [xd,cxd,lxd,perf0,perfl2] = ... wdencmp('gbl',c,l,w,n,thr,'h',keepapp);

Signal Compression

The result is quite satisfactory, not only because of the norm recovery criterion, but also on a visual perception point of view. The reconstruction uses only 15% of the coefficients.

% Load original image. load woman; x = X(100:200,100:200); nbc = size(map,1); % Wavelet decomposition of x. n = 5; w = 'sym2'; [c,l] = wavedec2(x,n,w); % Wavelet coefficients thresholding. thr = 20; keepapp = 1; [xd,cxd,lxd,perf0,perfl2] = ... wdencmp('gbl',c,l,w,n,thr,'h',keepapp);

Image Compression

If the wavelet representation is too dense, similar strategies can be used in the wavelet packet framework to obtain a sparser representation. You can then determine the best decomposition with respect to a suitably selected entropy-like criterion, which corresponds to the selected purpose (denoising or compression).

Compression Scores

When compressing using orthogonal wavelets, theRetained energyin percentage is defined by

$\frac{100 * {(vector-norm(coeffs of the current decomposition, 2))}^{2}}{{(vector-norm (original signal, 2))}^{2}}$

When compressing using biorthogonal wavelets, the previous definition is not convenient. We use instead theEnergy ratioin percentage defined by

$\frac{100 * {(vector-norm(compressed signal, 2))}^{2}}{{(vector-norm (original signal, 2))}^{2}}$

and as a tuning parameter theNorm cfs recoverydefined by

$\frac{100 * {(vector-norm(coeffs of the current decomposition, 2))}^{2}}{{(vector-norm (coeffs of the original decomposition, 2))}^{2}}$

TheNumber of zerosin percentage is defined by

$\frac{100 * (number of zeros of the current decomposition)}{(number of coefficients)}$