Let's say you have the following greyscale image:
\(A =
Histogram
Now the histogram is a function \(H: [0,255] \rightarrow \mathbb{N}_0\).
The histogram of \(A\) is
\(H(x) :=
Accumulated histogram
The accumulated histogram \(H_\alpha: [0,255] \rightarrow \mathbb{N}_0\) is defined as
\(H_\alpha(x) := \sum_{i=0}^x H(i)\)
This means, in the given example you get
\(H_\alpha(x) :=
Normalized histogram
The normalized histogram is defined as \(H_n(x) := \mathrm{round}(\frac{255}{w \cdot h} \cdot H_\alpha(x))\) where \(w\) is the width of the image and \(h\) is the height of the image.
In our example it's:
\(H_n(x) :=
So the resulting image is
\(A =