Help on how to achieve lossless image compression using arithmetic encoding and decoding.

Had this error after passing the coefficients of wavelet image tranform, symbols and probability to the package

Arithmetic Encoding & Decoding

Version 1.0.0.0 (2.67 KB) by Vinay Kumar Tadepalli

Error:

[Tag_bits]= Arithmetic_enc(sym,p,c);

Plese enter proper values!!!

Elapsed time is 0.000439 seconds.

Output argument "Tag_bits" (and possibly others) not assigned a value in the execution with "Arithmetic_enc" function.

code:

function Tag_bits =Arithmetic_enc(sym,p,seq)
tic;  % TIC-TOC commands are used to measure the simulation time of the program.
format long g;
%% ARGUMENTS OF THE FUNCTION
% SYM is a string of symbols of information from the source.
% P represents the array of Probabilities of the corresponding symbols in
% the SYM string.
% SEQ is the string of sequence of symbols to be encoded by arithmetic
% coding.
if(length(sym)==length(p) && sum(p)==1)  
    %% ALGORITHM IMPLEMENTATION
    
    % Calculate the Symbol Intervals.
    Fx=zeros(1,length(sym));
    for i=1:length(sym)
        if i==1
            Fx(i)=p(i);
        else
            Fx(i)=Fx(i-1)+p(i);
        end
    end
    
    % Encoding the Sequence of Symbols. 
    L=0;U=1; % Initial Lower and Upper Intervals.
    Tag_bits=zeros(1,0); % Initializing the Tag Bits.
    for i=1:length(seq)
        j=find(seq(i)==sym);   % Finds the Index of the sequence symbol in the symbol string.
        if(j==1)
            L_new=L;
        else
            L_new=L+(U-L)*Fx(j-1);
        end
        U_new=L+(U-L)*Fx(j);
        L=L_new;
        U=U_new;
        while((L<0.5 && U<0.5) ||(L>=0.5 && U>0.5))
            if(L<0.5 && U<0.5)
                Tag_bits=[Tag_bits,'0'];
                L=2*L;
                U=2*U;
            else
                Tag_bits=[Tag_bits,'1'];
                L=2*(L-0.5);
                U=2*(U-0.5);
            end
        end
    end
    tag=(L+U)/2;
    
    % Embedding the Final Tag Value.
    bits=zeros(1,0);
    if(2*tag>1)
        tag=2*tag-1;
        bits=[bits,'1'];
    else
        tag=2*tag;
        bits=[bits,'0'];
    end
    while(bin2dec(bits)/2^length(bits)<L)
        if(2*tag>1)
            tag=2*tag-1;
            bits=[bits,'1'];
        else
            tag=2*tag;
            bits=[bits,'0'];
        end
    end
    Tag_bits=[Tag_bits,bits];
    
    % Padding of zeros is done to keep the TAG BITS size multiple of 16 bits.
    Tag_bits=[Tag_bits,dec2bin(0,16-rem(length(Tag_bits),16))];
    display('Tag Value is:');
    disp(bin2dec(Tag_bits)/2^length(Tag_bits));
    display('Tag Word is:');
    Tag_bits=[Tag_bits,dec2bin(length(seq),16)];
else
    display('Plese enter proper values!!!');
end
toc;

workspace values for

seq = c

c= 1x349184 double

p = 1x349118 double

sym = 1x349118 double

sum(p)

ans = 0.999999999999999

Help on how to achieve lossless image compression with any other package or this package for arithmetic encoding and decoding.

[DGM]

Since we don't know how you tried to call the function, we don't know whether any other suggestion would work either. All we know is that one of the following is true:

• your list of probabilities does not sum to unity
• your list of symbols is not the same length as the corresponding list of probabilites

If it's the first case, there might be room for float rounding errors to cause problems there.

Documentation on this FEX submission could be better.

[enyawson]

have updated the question. I checked the length of symbols and probabilities but realised the sum of probability doesn't add up 1

Answer 1

[Rahul]

Hi enyawson,

I understand that you are attempting to perform lossless image compression using the 'Arithmetic_enc' function from the referenced File Exchange package, but an error occurs because the function’s initial validation check is not being satisfied.

In the provided implementation, execution proceeds only if both of the following conditions hold:

1. 'length(sym)' is equal to 'length(p)'.
2. 'sum(p)' is exactly equal to 1.

From your provided workspace data, 'length(sym)' and 'length(p)' differ (349118 vs. 349184 in the case of 'seq') and 'sum(p)' is reported as 0.999999999999999 rather than exactly 1. The second case is a typical floating-point precision issue that can occur when probabilities are computed from large datasets.

In MATLAB, such numerical discrepancies can be handled by normalizing or tolerating small deviations. For example:

% Normalize probabilities to sum exactly to 1
'p' = 'p' / sum('p');

% OR: Allow tolerance in the condition
if length('sym') == length('p') && abs(sum('p') - 1) < 1e-12
    % Proceed with encoding
else
    disp('Please enter proper values!');
end

It is also important that 'sym' contains all unique symbols from your sequence and 'p' contains their associated probabilities in corresponding order. If the data originates from a wavelet transform of an image, mapping the transform coefficients into discrete symbols is a prerequisite for using this arithmetic encoding function.

If the goal is to achieve lossless image compression without modifying the original File Exchange code, MATLAB provides other options such as:

• Communications Toolbox: functions like 'arithenco' and 'arithdeco' for arithmetic coding.
• Image Processing Toolbox: lossless formats ('imwrite' with 'png' or 'tiff' in compression mode).
• Wavelet Toolbox: functions for quantization and entropy coding after wavelet decomposition.

You can refer to the following documentation links and examples regarding various optimization techniques:

• Arithmetic Encoding and Decoding in MATLAB: https://www.mathworks.com/help/comm/ref/arithenco.html
• Lossless Image Compression Workflow: https://www.mathworks.com/matlabcentral/answers/121768-matlab-source-code-for-image-compression-algorithm

This approach should ensure that the probabilities are consistent, symbols align correctly, and the encoding process can proceed without the error message.