Determining Parameters of S Function for Image Global Contrast Enhancement
Okay, I am convinced that there is a typo in Eq. (4) of the cited paper.  It should be:

$$ S(x;a,b,c)=\begin{cases}
0 &amp; x\leq a\\
\frac{(x-a)^2}{(b-a)(c-a)} &amp; a&lt;x\leq b\\
1-\frac{(x-c)^2}{(c-b)(c-a)} &amp; b\leq x\leq c\\
1 &amp; x&gt;c
\end{cases} $$

This sigmoid function or smoothstep function is continuous everywhere and the first derivative is continuous everywhere, but the second derivative is discontinuous when $x=a$ or $x=b$ or $x=c$.

You must have $a &lt; b &lt; c$.

$a$ is defined where $S(x)$ begins to transition from 0 upward.  and $c$ is defined for when the transition is over and $S(x)=1$.  $b$, which is in between $a$ and $c$ is the point of inflection for when the curvature changes from a concave curvature to a convex curvature (or maybe it's the other way around).

There are better sigmoid or smoothstep functions than this that the author of the paper uses.  I dunno why he/she is so enamored with this definition.