Determining Parameters of S Function for Image Global Contrast Enhancement I know that S-function (membership function from fuzzy set theory) is a commonly used function for image enhancement. I am reading a paper titled - A Novel Fuzzy Image Enhancement Using S-Shaped Membership Function. The S-shaped membership function is defined as follows:

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

where the parameters are to be found. In the above mentioned paper, it given that the values of a, b, c are entropy(I), median(I) and (max(I)+mean(I))/2 respectively. But there's no explanation for how and why these values were chosen. I tried to look for it in many papers referred by the above paper but they are of no use. Can someone give me some idea please? Thank you in advance.
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.