How to Solve Histogram Equalization Numerical Problem in MATLAB?

Histogram Equalization is a mathematical technique to widen the dynamic range of the histogram. Sometimes the histogram is spanned over a short range, by equalization the span of the histogram is widened. In digital image processing, the contrast of an image is enhanced using this very technique. 

Use of Histogram Equalization:

It is used to increase the spread of the histogram. If the histogram represents the digital image, then by spreading the intensity values over a large dynamic range we can improve the contrast of the image. 

Algorithm:

• Find the frequency of each value represented on the horizontal axis of the histogram i.e. intensity in the case of an image.
• Calculate the probability density function for each intensity value.
• After finding the PDF, calculate the cumulative density function for each intensity's frequency.
• The CDF value is in the range 0-1, so we multiply all CDF values by the largest value of intensity i.e. 255.
• Round off the final values to integer values.

Example:

% MATLAB code for Histogram equalisation % function to return resultant % matrix and summary function [f,r]=HistEq(k,max) Freq=zeros(1,max); [x,y]=size(k); % Calculate frequency of each % intensity value. for i=1:x  for j=1:y  Freq(k(i,j)+1)=Freq(k(i,j)+1)+1;  end end % Calculate PDF for each intensity value. PDF=zeros(1,max); Total=x*y; for i=1:max  PDF(i)=Freq(i)/Total; end % Calculate the CDF for each intensity value. CDF=zeros(1,max); CDF(1)=PDF(1); for i=2:max  CDF(i)=CDF(i-1)+PDF(i); end % Multiply by Maximum intensity value % and round off the result. Result=zeros(1,max); for i=1:max  Result(i)=uint8(CDF(i)*(max-1)); end % Compute the Equalized image/matrix. mat=zeros(size(k)); for i=1:x  for j=1:y  mat(i,j)=Result(k(i,j)+1);  end end   f=mat; r=Result; end % Utility code here. k=[0, 1, 1, 3, 4;   7, 2, 5, 5, 7;   6, 3, 2, 1, 1;   1, 4, 4, 2, 1];   [new_matrix, summary]=HistEq(k,8); 

Output:

A 3-bit image of size 4x5 is shown below. Compute the histogram equalized image. 

Steps:

• Find the range of intensity values.
• Find the frequency of each intensity value.
• Calculate the probability density function for each frequency.
• Calculate the cumulative density function for each frequency.
• Multiply CDF with the highest intensity value possible.
• Round off the values obtained in step-5.

Overview of calculation: Range of intensity values = [0, 1, 2, 3, 4, 5, 6, 7] Frequency of values = [1, 6, 3, 2, 3, 2, 1, 2] total = 20 = 4*5 Calculate PDF = frequency of each intensity/Total sum of all frequencies, for each i value of intensity Calculate CDF =cumulative frequency of each intensity value = sum of all PDF value (<=i) Multiply CDF with 7. Round off the final value of intensity.

The tabular form of the calculation is given here:

Interpretation:  The pixel intensity in the image has modified. 0 intensity is replaced by 0. 1 intensity is replaced by 2. 2 intensity is replaced by 4. 3 intensity is replaced by 4. 4 intensity is replaced by 5. 5 intensity is replaced by 6. 6 intensity is replaced by 6. 7 intensity is replaced by 7.

Output: The new image is as follow: