HE in image processing

Hello, I reading image processing book , in Histogram Equalization we use continuous PDf and PDF is suitable for continuous variable. But in Digital image processing we have discrete number. for example 0,1,2 ... 255 Why first we use continuous PDf formula and use integration ? Thanks

Answer 1

[Image Analyst]

In theory, if you have a continuous PDF (because you have continuous gray levels) you can get a perfectly flat histogram after equalization. In practice, you can't and it just moves the bins around so that the bins have the same height as originally (unless you do something to reduce that effect such as adding noise or sorting) and produces a crummy image. I attach some demos for you to run that do the standard method, the Mathworks method (which is better), and my method, which is nearly perfect. My method gives a perfectly flat histogram except in the case where the number of pixels is not a multiple of the number of bins, and in that case all bins are flat except one.

[Image Analyst]

Dani's "Answer" moved here:

Thank you for your reply, But in continuous space,The probability in one point(for exmaple: 127 or 196 or ...) is zero. Whereas in digital image processing , we have probability in one point. For example we have 16 intensity from 255. and our probability is 16/255 and in continuous probability we don't have probability in one point.

[Image Analyst]

The probability (area under the PDF curve) is zero, but the probability density is not. And it's the probability density function that is integrated to give the cumulative distribution function which is then used as a look up table/function to give the transformed output value.