Wavelet Toolbox 

The purpose of this example is to show how to compress an image using twodimensional wavelet analysis. Compression is one of the most important applications of wavelets. Like denoising, the compression procedure contains three steps:
Decompose: Choose a wavelet, choose a level N. Compute the wavelet decomposition of the signal at level N.
Threshold detail coefficients: For each level from 1 to N, a threshold is selected and hard thresholding is applied to the detail coefficients.
Reconstruct: Compute wavelet reconstruction using the original approximation coefficients of level N and the modified detail coefficients of levels from 1 to N.
The difference with the denoising procedure is found in step 2. 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. Thus, only a single parameter needs to be selected.
The second approach consists of applying visually determined leveldependent thresholds.
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Let us examine a reallife example of compression for a given and unoptimized wavelet choice, to produce a nearly complete square norm recovery for an image.
load woman; % Load original image. image(X) title('Original Image') colormap(map) x = X(100:200,100:200); % Select ROI
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.
n = 5; % Decomposition Level w = 'sym8'; % Near symmetric wavelet [c l] = wavedec2(x,n,w); % Multilevel 2D wavelet decomposition.
In this first method, the WDENCMP function performs a compression process from the wavelet decomposition structure [c,l] of the image.
opt = 'gbl'; % Global threshold thr = 20; % Threshold sorh = 'h'; % Hard thresholding keepapp = 1; % Approximation coefficients cannot be thresholded [xd,cxd,lxd,perf0,perfl2] = wdencmp(opt,c,l,w,n,thr,sorh,keepapp); image(x) title('Original Image') colormap(map) figure('Color','white'),image(xd) title('Compressed Image  Global Threshold = 20') colormap(map)
L2norm recovery (%)
perf0
perf0 = 74.3067
Compression score (%)
perfl2
perfl2 = 99.9772
The density of the current decomposition sparse matrix is:
cxd = sparse(cxd); cxd_density = nnz(cxd)/prod(size(cxd))
cxd_density = 0.2569
Method 2: LevelDependent Thresholding
The WDENCMP function also allows level and orientationdependent thresholds. In this case the approximation is kept. The leveldependent thresholds in the three orientations horizontal, diagonal and vertical are as follows:
opt = 'lvd'; % Level dependent thresholds thr_h = [17 18]; % Horizontal thresholds. thr_d = [19 20]; % Diagonal thresholds. thr_v = [21 22]; % Vertical thresholds. thr = [thr_h ; thr_d ; thr_v];
In this second example, notice that the WDENCMP function performs a compression process from the image x.
[xd2,cxd2,lxd2,perf02,perfl22] = wdencmp(opt,x,w,2,thr,sorh); image(x) title('Original Image') colormap(map) figure('Color','white'),image(xd2) title('Compressed Image  LevelDependent Thresholding') colormap(map)
L2norm recovery (%)
perf02
perf02 = 77.3435
Compression score (%)
perfl22
perfl22 = 99.6132
The density of the current decomposition sparse matrix is:
cxd2 = sparse(cxd2); cxd2_density = nnz(cxd2)/prod(size(cxd2))
cxd2_density = 0.2266
By using leveldependent thresholding, the density of the wavelet decomposition was reduced by 3% while improving the L2norm recovery by 3%. 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 entropylike criterion, which corresponds to the selected purpose (denoising or compression).