In this post, I will discuss the Haar wavelet analysis (wavelet transform) in image processing.

Applications of Haar wavelet transform in image processing are Feature extraction, Texture analysis, Image compression etc.

 

 

 

 

• Wavelet is time frequency resolutions analysis.
• As shown in above figure.1  its have the information of different time frequencies of signal.
• Its like applying a comb of filters bands in signal they are called levels. First we apply full frequency band and have no knowledge of time.
• Apply filter which have half bandwidth of signal lose some frequency information but  have some knowledge of time resolution called first level decomposition.
• Further reduce the filter bandwidth will be second level decomposition.

Haar wavelet is made of two filters one is low pass filter and second high pass filter.

 

Coefficients of Low pass filter  = [0.707, 0.707]

Coefficients of High pass filter  = [-0.707, 0.707]

 

 

 

• Figure .3 shows the block diagram of first level decomposition.
• I is image of size m , n. We take the image column vectors (1d) and apply the low pass filter and high pass filter.
•  Down sample the output and reshape the image I1 of size m, n/2 (size is n/2 because we down sampled the columns by 2).
• Apply low and high pass filter in row wise in image I1 again down sample the output and reshape the image (m/2,n/2).

 

• Figure. 5 shows the result of haar wavelet decomposition results at first level.
• LL filter output will proceeds for next level decomposition. As its contains most information of input image and looks almost same at down sampled resolution.
• HH, LH  and HL will have texture information in vertical, horizontal and ramp directions. HH, LH and HL are called haar features.
• We can further  decompose the LL image into LL1, LH1, HL1 and HH1 images.

 

 

MATLAB provide Wavelet Toolbox GUI to perform the wavelet analysis.

Type wevemenu in command window of  MATLAB.

 

%%%%%%Code Start %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

function [LL, LH, HL, HH] = haar_wave(image_in)

%% function haar_wave calculate the haar wave decomposition of given imput image
%% Image in :- input gray level image
%% LL :- Low Low band output image
%% LH :- Low High band output image
%% HL :- High Low band output image
%% HH :- High High band output image

image_in = im2double(image_in);
filter_L = [.707, .707];
filter_H = [-.707, .707];
[row, col] = size(image_in);
image_vector = image_in(:);

%% apply low pass filter in one d image column wise image
image_low_pass_1D = filter(filter_L,1,image_vector);
image_high_pass_1D = filter(filter_H,1,image_vector);

% Down sample the image low and high pass image.
temp = 1;
Dimage_low_pass_1D = 0;
Dimage_high_pass_1D = 0;

for i = 1:2:length(image_low_pass_1D)
Dimage_low_pass_1D(temp) = image_low_pass_1D(i);
Dimage_high_pass_1D(temp) = image_high_pass_1D(i);
temp = temp+1;
end

%% reshape the image in 2 D, size will be row/2 , col
Dimage_low_pass = reshape(Dimage_low_pass_1D,round(row/2), col)’;
Dimage_high_pass = reshape(Dimage_high_pass_1D, round(row/2),col)’;

%% conver the image in 1 D but in column vector, I transpose Dimage_low_paas image above.

Dimage_low_pass_1D = Dimage_low_pass(:);
Dimage_high_pass_1D = Dimage_high_pass(:);

LL_1D = filter(filter_L,1,Dimage_low_pass_1D);
LH_1D = filter(filter_H,1,Dimage_low_pass_1D);

HL_1D = filter(filter_L,1,Dimage_high_pass_1D);
HH_1D = filter(filter_H,1,Dimage_high_pass_1D);

%Down sample the all 1 _d vector images
temp = 1;
DLL_1D = 0;
DLH_1D = 0;
DHL_1D = 0;
DHH_1D = 0;

for i = 1:2:length(LL_1D)
DLL_1D(temp) = LL_1D(i);
DLH_1D(temp) = LH_1D(i);
DHL_1D(temp) = HL_1D(i);
DHH_1D(temp) = HH_1D(i);
temp = temp+1;
end

%% reshape the images in size row/2 and clo/2

LL = reshape(DLL_1D, round(col/2), round(row/2))’;
LH = reshape(DLH_1D, round(col/2), round(row/2))’;
HL = reshape(DHL_1D, round(col/2), round(row/2))’;
HH = reshape(DHH_1D, round(col/2), round(row/2))’;

end

 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

                      %% call the haar wavelet function and check the result

clc % clear the screen

clear all % clear all variables

close all % close all MATLAB windows

%% read Image

% set path to read image from system

ImagePath = ‘D:\DSUsers\uidp6927\image_processingCode\lena.jpg’;

img_RGB = imread(ImagePath); % read image from path

img_RGB = im2double(img_RGB); % Convert image to double precision

img_gray = rgb2gray(img_RGB); % convert image to gray

% first level decomposition
[LL, LH, HL, HH] = haar_wave(img_gray);

% second level decomposition
[LL1, LH1, HL1, HH1] = haar_wave(LL);

%% Draw all image

figure, imshow(img_gray);

figure,
subplot(2,2,1), imshow(LL,[])
title(‘LL haar wave image’)
subplot(2,2,2), imshow(LH,[])
title(‘LH haar wave image’)
subplot(2,2,3), imshow(HL,[])
title(‘HL haar wave image’)
subplot(2,2,4), imshow(HH,[])
title(‘HH haar wave image’)

figure,
subplot(2,2,1), imshow(LL1,[])
title(‘LL1 haar wave image’)
subplot(2,2,2), imshow(LH1,[])
title(‘LH1 haar wave image’)
subplot(2,2,3), imshow(HL1,[])
title(‘HL1 haar wave image’)
subplot(2,2,4), imshow(HH1,[])
title(‘HH1 haar wave image’)

%%%%%%%%%%%%%%%%%Code End %%%%%%%%%%%%%%%%%%%%%%%%%%%%

 

Happy Learning

Cheers