Data Compression using 2-D Wavelet Analysis

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The purpose of this example is to show how to compress an image using two-dimensional 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 level-dependent thresholds.

Load an Image

Let us examine a real-life 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)

Figure contains an axes object. The axes object with title Original Image contains an object of type image.

x = X(100:200,100:200);  % Select ROI

Method 1: Global Thresholding

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 2-D 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 contains an axes object. The axes object with title Original Image contains an object of type image.

figure
image(xd)
title('Compressed Image - Global Threshold = 20')
colormap(map)

Figure contains an axes object. The axes object with title Compressed Image - Global Threshold = 20 contains an object of type image.

Compression score (%)

perf0

perf0 = 
74.3067

L2-norm recovery (%)

perfl2

perfl2 = 
99.9772

The density of the current decomposition sparse matrix is:

cxd = sparse(cxd);
cxd_density = nnz(cxd)/numel(cxd)

cxd_density = 
0.2569

Method 2: Level-Dependent Thresholding

The WDENCMP function also allows level and orientation-dependent thresholds. In this case the approximation is kept. The level-dependent 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 contains an axes object. The axes object with title Original Image contains an object of type image.

figure
image(xd2)
title('Compressed Image - Level-Dependent Thresholding')
colormap(map)

Figure contains an axes object. The axes object with title Compressed Image - Level-Dependent Thresholding contains an object of type image.

Compression score (%)

perf02

perf02 = 
77.3435

L2-norm recovery (%)

perfl22

perfl22 = 
99.6132

The density of the current decomposition sparse matrix is:

cxd2 = sparse(cxd2);
cxd2_density = nnz(cxd2)/numel(cxd2)

cxd2_density = 
0.2266

Summary

By using level-dependent thresholding, the density of the wavelet decomposition was reduced by 3% while improving the L2-norm 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 entropy-like criterion, which corresponds to the selected purpose (denoising or compression).