From a paper I am using for my Bachelor thesis:
The HMT transform by the pair $(A,B)$ associates to a binary image $X$ the set $X\otimes(A,B)$ of positions where the translate of $A$ fits inside $X$ and at the same time the translate of $B$ fits inside the complement $X^C$ of $X$.
And regarding the gray-level extension of the HMT in particular:
We use a definition of HMT operator that assigns to $A$ and $B$ gray-levels $a$ and $b$, respectively. Following this definition, the gray-level (GL) HMT compares the minimum intensity $a_{min}$ in $A$ to the maximum intensity $b_{max}$ in $B$: $$S_s^t = I_s\otimes(A,B)(p)=\begin{cases}(I_s \ominus A)(p)-a & if\ (I_s \ominus A)(p) \geq (I_s \oplus B)(p)+a-b\neq\infty\\-\infty & \text{otherwise}\end{cases}$$ If $a_{min}>(b_{max}+a-b)$, then the point is selected by the GL HMT.
I have difficulties understanding this definition. There is no more information given.
It is obvious from context, that $S_s^T$ is subset of $\mathbb Z^3$. This is also supported by the point of the first paragraph quoted above, saying that $X\otimes(A,B)$ is a set of positions. From this point my first question arises:
(1) What sense does it make that the left part of the equation above is a subset of $\mathbb Z^3$ whereas the right part of this equation is scalar-valued? The equation formally implies $S_s^t=-\infty$ in certain cases.
I believe that this is the core problem to understanding the definition, and the second obscurity will vanish by answering the first question:
(2) Yet it is unclear, what exactly $a$ and $b$ are: where are their values supposed to come from? And what might $p$ stand for?