WAVELET-BASED IMAGE COMPRESSION

Procedure & Implementation

We used a standard matlab wavelet code package, WAVELAB 802, to perform the transforms. The wavelets we chose to use were the Deslauriers wavelets of polynomial size 3.

The compression scheme we used was to set a threshold value that was some fraction of the norm of the entire wavelet transform matrix. If the magnitude of a wavelet in the representation was not larger than this value, it was not included in the compression. We then rebuilt an image which (to some degree that depended on how many bases we included) resembed the original image by running the inverse transform.


ORIGINAL IMAGE

The fabulous Lena


BASIS VECTORS

These are some basis vectors obtained by running the inverse wavelet transform on a matrix with a single nonzero value in it.
Deslaurier(1,1)
Deslaurier(10,10)
Deslaurier(16,16)
Deslaurier(4,2)


COMPRESSED IMAGES
Threshold = .5, Bases included = 19, Compression ratio = 3400 : 1
Threshold = 1, Bases included = 470, Compression ratio = 140 : 1
Threshold = 2, Bases included = 2383, Compression ratio = 27 : 1
Threshold = 4, Bases included = 6160, Compression ratio = 10 : 1
Threshold = 8, Bases included = 12378, Compression ratio = 5 : 1

MATLAB CODE
load /home/nirav/elec301/lena256.mat;
imagesc(lena256);
colormap(gray(256));

[qmf, dqmf] = MakeBSFilter('Deslauriers', 3);

% The MakeBSFilter function creates biorthonormal filter pairs. The filter
% pairs that we're making is an Interpolating (Deslauriers-Dubuc) filter
% of polynomial degree 3

wc = FWT2_PB(lena256, 1, qmf, dqmf);

% wc correspond to the wavelet coefficients of the sample image
% FWT2_PB is a function that takes a 2 dimensional wavelet transform
% We specify the image matrix, the level of coarseness (1), the quadrature
% mirror filter (qmf), and the dual quadrature mirror filter (dqmf)

% we take a tolerance which is some fraction
% of the norm of the sample image

nl = norm(lena256) / (4 * norm(size(lena256)));

% if the value of the wavelet coefficient matrix at a particular
% row and column is less than the tolerance, we 'throw' it out
% and increment the zero count.

zerocount = 0;
for i = 1:256 
  for j = 1:256
    if ( abs(wc(i,j)) < nl) 
         wc(i,j) = 0;
         zerocount = zerocount + 1;
    end
  end
end

x = IWT2_PB(wc, 1, qmf, dqmf);
imagesc(x);

% here is some sample code to view how these deslauriers wavelets look

[qmf, dqmf] = MakeBSFilter('Deslauriers', 3);
for i = 1:256
  for j = 1:256
    wc(i,j) = 0;
  end
end

% this is the Deslauriers(4,2) matrix

wc(4, 2) = 1000;
x = IWT2_PB(wc, 1, qmf, dqmf);
imagesc(x);

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