QUANTIZATION OF 512x512 image WITHOUT USING THE USUAL QUANTIZATION MATRIX-IMAGE COMPRESSION

I have been battling with the method or the idea I can use in quantizing 512x512 image in whcih I am not allowed to use the usual quantization matrix table/predefined quantization matrix.

I need to select my frequency of interest during the qunatization process and turn the frequencies I am not intereted in to 0 especially the high frequency DCT coefficients.

Here is my code, I was told my code has no XY symetry and I do not now how to make it have XY symmetry as well.

   tic
    %% Reading the file
    FileName = convertStringsToChars(FileName);
    O_Image = imread(FileName);
%     imshow(O_Image);


% Threshold=10^(-5);  % The stopping condition to discard quantized DCT coeffiecint below  0.000001
Threshold=10^(-6);
    %% Parameters
    nBit = 8;   % Number of bit    8 bits=1 byte  
    Index = strfind(FileName, '.');   % Search for files that contains dot
    FileName_TXT = append(FileName(1:Index(end)-1), "_Quality", num2str(Quality), "_", Transform , ".txt");
    FileName_JPEG = append(FileName(1:Index(end)), "jpg"); 

    O_Size = size(O_Image);
    % Factor encoded on 1 byte to determine the size of the original image
    O_k1 = ceil(O_Size(1)/(2^(nBit)-1));  %This gives 3 bits which is 1 byte  for the row and the column of the image dim
    O_k2 = ceil(O_Size(2)/(2^(nBit)-1));  
    
    %% Transform
    switch Transform
        case 'DCT'
             O_Value = dct2(O_Image);    %DCT matrix of the original Image
             S_Transform = '0';
             
          
               % Make a quantization matrix Z    
             x=linspace(1,2,prod(O_Size));       % prod(O_Size)=512*512
             y=reshape(x,length(O_Image),length(O_Image));  %reshape(x,[512,512]) returns the 512-by-512 matrix 
                                                            % whose elements are taken columnwise from x

            for i=1:length(O_Image)     
                for j=1:length(O_Image)
                    Z(i,j)=[length(O_Image)^y(i,j)];  % This matrix depends on the location of                  
                end                                           % each element , so every element is treated 
            end                                               % differently .

            % % %     Make the Quantization
            O_Value = O_Value./(Z*Fact);            %Fact=x which is enables us to treat DCT coeefients differently
            A=O_Value;
            
            
%             figure; imshow(log(abs(O_Value)),[]); colormap parula; colorbar;
        case 'DFT'
            O_Value = [real((fftshift(fft(O_Image)))) imag((fftshift(fft(O_Image))))]/(2*prod(O_Size));
            O_Size(2) = 2*O_Size(2);
            O_k2 = 2*O_k2;
            S_Transform = '1';
%             figure; imshow(log(abs(fftshift((fft2(O_Image)))/prod(O_Size))),[]); colormap parula; colorbar;
    end
    

Here is my main code

    FileName = "I_lenagray.tif";
Transform = "DCT";
ReductionFactor = 1; % reduction rate  either 1 or 2  for a good image compression and reconstrcution
Quality = 1:0.1:10; %  Quality range 
 

%%% Initialization
Flag = 1;
Q_Rate = zeros(1, size(Quality,2)); % A vector for rate of compression 
Q_PSNR_OR = zeros(1, size(Quality,2)); % A vector for PSNR  of original and reconstructed  image
Q_PSNR_OJ = zeros(1, size(Quality,2));
Q_PSNR_RJ = zeros(1, size(Quality,2));
Q_MSE_OR = zeros(1, size(Quality,2));
Q_MSE_OJ = zeros(1, size(Quality,2));
Q_MSE_RJ = zeros(1, size(Quality,2));

%%% Loop
Pos_q = 0;      %Indix for interation on the position depending on the quality selected 
r=linspace(100000,100,91); % Remake a quality vector for matrix of quantization
for q = Quality
    Pos_q = Pos_q + 1; %
    
    Fact=r(Pos_q);        %Choose the corresponding quality
%     [R_Image, C_Rate, PSNR, MSE, ~,Threshold,A,B,Z] = MainFunction(FileName, Transform, q,Fact, ReductionFactor);
%     Image_Saving(FileName, R_Image, q, 'tif')
    [R_Image, C_Rate, PSNR, MSE, ~,Threshold] = MainFunction(FileName, Transform, q,Fact, ReductionFactor);

    if Flag
        Dim = size(R_Image);                                     %Size of the reconstrcuted image as compared to 512x512of the original image
        Q_ImageR = uint8(zeros(Dim(1), Dim(2), size(Quality,2))); % Conversion of zeros of the recontsructed image with respect to the size of the quality in uint8 format  
        Flag = 0;

I will really appreciate any idea I can use to achieve this image compression.

Is there anything I can do different to have this code achieve the aim ?

Thanks for your help.

[Jan]

The nature of this forum is to share solutions with the community. Then private messages are not useful.

Answer 1

[prabhat kumar sharma]

Hello Olusola

The quantization matrix Z defined by your current code lacks XY symmetry. Your matrix's quantization is not the same along the diagonal as required by XY symmetry because it was created by reshaping a linearly spaced vector into a square matrix.

You can use a matrix where the value grows as you move further from the top-left corner (which represents the DC coefficient and low-frequency components in the DCT domain) to produce an XY symmetric quantization matrix that focuses on low frequencies.

You can accomplish this by changing your code as follows:

1. Replace the current quantization matrix generation code with a symmetric matrix where the quantization factor increases with the distance from the top-left corner.

2. Apply the quantization matrix to the DCT coefficients.

3. Set the high-frequency DCT coefficients to zero based on the threshold you're interested in.

switch Transform
    case 'DCT'
         O_Value = dct2(O_Image);    % DCT matrix of the original Image
         S_Transform = '0';
         
         % Create a symmetric quantization matrix Z with increasing values from the top-left corner
         [X, Y] = meshgrid(1:length(O_Image), 1:length(O_Image));
         Z = sqrt((X-1).^2 + (Y-1).^2) + 1; % Add 1 to avoid division by zero
         
         % Apply the quantization matrix to the DCT coefficients
         O_Value = O_Value ./ (Z * Fact);
         
         % Threshold high frequencies
         O_Value(abs(O_Value) < Threshold) = 0;
         
         % Your Further code...
end

In the modified code: A symmetric quantization matrix Z is created using meshgrid. It's based on Euclidean distance from the top-left corner, with 1 added to avoid division by zero. The DCT coefficients are divided by Z scaled by Fact, applying quantization. DCT coefficients below the Threshold are zeroed out, removing high-frequency details. This method retains low-frequency information and adjusts compression via Fact and Threshold.

I hope it helps you with your problem!