Your formula for increasing contrast in an image throws away data, since everything below middle gray in the original image would become black, since negative values aren’t allowed.
I found the formula I’m talking about, in the area of digital images (about contrast stretching) and it is this:
In the field of images it’s used to obtain negative values but not only for that (mine was just an example).
From this formula many of the formulas used to normalize the votes are obtained, including that of the STLR and also of Cardinal Baldwin.
Given a vote like this:
In STLR it normalizes from the range [0,2.5] to [0,5], obtaining [2,5]
In Cardinal Baldwin from the range [1,2.5] to [0,5], obtaining [0,5]