How to equalize histograms

Question

I am learning some image processing stuff and equalizing a histogram comes up as an important topic. I have followed the procedure listed on Wikipedia but the resulting equalized histogram does not look much better to me.

Let's say I have the following histogram where values only range from 0 to 7:

0 | 2
1 | 6
2 | 12
3 | 13
4 | 6
5 | 11
6 | 7
7 | 7

The equation to equalize the histogram is as follows: $h(v) = round(\frac{cdf(v) - cdf_{min}}{(M*N) - cdf_{min}} * (L - 1))$

So, using this I get the following:

$h(0) = 0$

$h(1) = 1$

$h(2) = 2$

$h(3) = 4$

$h(4) = 4$

$h(5) = 5$

$h(6) = 6$

$h(7) = 7$

Transferring this over to the new histogram, I get the following equalized histogram:

0 | 2
1 | 6
2 | 12
3 | 0
4 | (13 + 6) = 19
5 | 11
6 | 7
7 | 7

This doesn't look right at all... Have I done something incorrectly?

Answer 1

You have correctly applied the formula.

It does not help your example mainly because the values are already spread out. In fact you lose a little information.

But histogram equalization is designed to deal with other types of cases where the data is mainly clumped together in a small part of the distribution, and where you want to increase the contrast across the distribution.

If you had started with

0 | 7
1 | 10
2 | 13
3 | 22
4 | 1
5 | 2
6 | 7
7 | 2

you would have ended with something like

0 | 7
1 | 10
2 | 0
3 | 13
4 | 0
5 | 0
6 | 25
7 | 9

which would have increased the visible contrast (though still losing a little information in the data).