I am reading the etext on Digital Image Processing by Gonzalez and under the section on Histogram Equalization, an equation which is apparently basic is shown. I have no clue what the equation means and where it is from. I hope someone can point me to it in wiki or something. I have been googling for awhile but I have yet to get any meaningful results.

enter image description here

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    $\begingroup$ Please post the definition of $T(r)$ and condition a) $\endgroup$ – LJSilver May 2 '16 at 10:59
  • $\begingroup$ Search the site for Histogram Equalization. There are already similar ones. May be you can find a more understandable explanation. $\endgroup$ – Fat32 May 2 '16 at 11:10

Assuming that in your question $T(r)$ denotes a map $:\mathbb{R}\to[0,1]$ and assuming that it is continuous and strictly increasing, then it is obviously an isomorphism among $r\in[0,1]$ onto its image.
As a consequence you get the following result: let $r$ be any value in $[0,1)$, and let $a=T(r)$, then for any $\rho\in[r,r+dr]$ it is $\alpha = T(\rho) \in [a,a+da]$, having denoted $a+da = T(r+dr)$. Therefore the probability of $r$ and $a$ to belong their infinitesimal intervals (which is actually the definition of PDF) is the same. hence as $dr\to 0$, also $da\to 0$ and you can write $$ p_r(r) dr = p_a(a) da $$ which yields the result

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  • $\begingroup$ Thanks ! That sounded so simple and I apologize for not posting the conditions. But the conditions you have stated are right. $\endgroup$ – RuiQi May 3 '16 at 7:58

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