Histogram Equalization in Digital Image Processing

Histogram equalisation is a contrast enhancement technique that improves image quality by redistributing pixel intensity values across the full available range. It is especially useful for images with poor lighting or compressed intensity distribution.

Expands intensity distribution to utilize the full dynamic range
Reveals hidden details in regions with low contrast

A digital image is represented as a matrix of intensity values, where each pixel stores the brightness level at a specific location. These values range from black to white depending on the intensity scale.

0 \leq r \leq L - 1,\ \text{where } L = 2^m

Intensity Transformation

Intensity transformation is a basic image processing technique where pixel values of an image are mapped to new values using a mathematical function. This process is used to enhance image quality by adjusting brightness and contrast.

It is generally defined as:

s = T(r)

where:

r = input pixel intensity
s = output pixel intensity
r ≥ 0

Different intensity transformation techniques are:

1. Image negation: This reverses the grayscales of an image, making dark pixels whiter and white pixels darker. This is completely analogous to the photographic negative, hence the name.

s = L - 1 - r

2. Log Transform: Here c is some constant. It is used for expanding the dark pixel values in an image.

s = c log(1+r)

3. Power-law Transform: Here c and γ are some arbitrary constants. This transform can be used for a variety of purposes by varying the value of γ.

s = c rγ

Histogram Equalization

The histogram of a digital image represents the distribution of pixel intensities across different gray levels. For an image with intensity levels ranging from 0 to (L−1), the histogram is defined as:

h(r_k) = n_k

where:

r_k = kth intensity level
n_k = number of pixels having intensity r_k

The histogram can also be normalized by dividing by the total number of pixels in the image. For an N × N image, the normalized histogram is defined as:

p( rk ) = nk/N2

This function represents the probability of occurrence of intensity level r_k. Clearly,

∑ p( rk ) = 1

The histogram of an image provides important information about its grayscale distribution and contrast characteristics.

If the histogram is concentrated toward the left, it indicates a dark image.
If the histogram is concentrated toward the right, it indicates a bright image.
A narrow histogram around the center indicates a low-contrast image.
A widely spread histogram indicates a high-contrast image.

In image processing, image contrast is often enhanced using histogram equalization.

Histogram equalization redistributes pixel intensities across the full range to produce a more uniform histogram. This improves overall contrast and reveals hidden details in the image.

Mathematical Derivation of Transformation Function

To design histogram equalization mathematically, we model intensity levels as a continuous random variable r \in [0, L-1].

Let the transformation be:

s = T(r)

with constraints:

T(r) is strictly increasing.
0 \le T(r) \le L-1

These conditions ensure T(r) is invertible.

The cumulative distribution function (CDF) of output intensity is:

F_S(x) = P(s \le x) = P(r \le T^{-1}(x))

From this, the transformation function that produces a uniform distribution is:

s = (L-1)\int_{0}^{r} p_r(x)\,dx

For discrete images, this becomes:

s_k = (L-1)\sum_{j=0}^{k} p(r_j)

This transformation redistributes intensity values to achieve a uniform histogram.

Example 1: Displaying the histogram of the original image to visualize the distribution of pixel intensity values.

Matlab

% Histogram Equalization in MATLAB
% reading image
I = imread("GFG.jfif");
figure
subplot(1,3,1)
imshow(I)
subplot(1,3,2:3)
imhist(I)

Output:

Example 2: Plots the histogram of the input image to analyze the frequency distribution of pixel intensity levels.