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I am reading

where I have encountered these three equations

\begin{gather*} e_l=\lvert L-L_l\rvert>T \\ c_\text{max} = \max(c_t, c_r,c_l,c_r,c_{2l}) \\ e_{l'}=e_l \land c_l > 0.5 c_\text{max}. \end{gather*}

Because $e_l$ is a number being either 0 or 1, and $c_l$ is always going to be a fraction number, I am wondering what is the AND result of these two? Would it be either 1 or 0 or it is going to be the value of $c_l$ when $e_l$ is 1?

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  • $\begingroup$ This is the minimum, not the Boolean ‘and’. $\endgroup$
    – LSpice
    Commented Sep 25, 2022 at 22:51
  • $\begingroup$ @LSpice Thanks for your reply! Would you mind pointing to a reference telling the operator \land is minimum instead of Boolean "and"? I can't seem to find it anywhere $\endgroup$
    – mathnerd
    Commented Sep 25, 2022 at 22:54
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    $\begingroup$ The paper you're reading does it … what more would you want for an acceptable reference? It is, for example, quite common in lattice theory. (Note, by the way, that the Boolean ‘and’ becomes a special case of the minimum if we impose the partial order according to which ${\perp} = 0$ is less than $\top = 1$.) $\endgroup$
    – LSpice
    Commented Sep 25, 2022 at 22:55
  • $\begingroup$ maybe a page says the meaning of ”^" can mean minimum or something... $\endgroup$
    – mathnerd
    Commented Sep 25, 2022 at 23:04
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    $\begingroup$ I suspect it could mean "$e_{l'} = e_l$ and $c_l \gt 0.5c_{max}$", i.e. the conjunction of two conditions, not a boolean operation on numbers. But if the context tells you otherwise, then I would guess LSpice's comment is on-target: $\wedge$ also stands for "meet". $\endgroup$
    – David Roberts ♦
    Commented Sep 25, 2022 at 23:54

1 Answer 1

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The paragraph before says

We calculate the maximum contrast $c_{max}$ for all these edges and compare it with the contrast for the left edge [this is $c_l$]. If the latter is above a threshold of $0.5\cdot c_{max}$ the edge is preserved; otherwise, it is ignored.

So this reads to me like a condition $c_l \gt 0.5\cdot c_{max}$.

We also have the definition of $e_l$:

a straightforward algorithm would calculate $e_l=|L−L_l|\gt T$, where $e_l$ is the boolean value that codes whether the edge is active, [emphasis added]

so we should think of $e_l$ not as $0$ or $1$, but a boolean, $T$ or $F$, giving the truth value of the condition "$|L−L_l|\gt T$", which may or may not be true.

Moreover, after he displayed formulas, we have

where $c_t$, $c_r$, $c_b$, $c_l$, $c_{2l}$ are the contrast deltas for the edges shown in Figure 5, and $e'_l$ represents the final boolean value (active or not) for the left edge boundary.

so again, we have that $e'_l$ is a boolean, and so the condition $e_l = e'_l$ is checking if two booleans are equal (i.e. both true or both false).

The paper is using a symbol, $\wedge$, where a word would do, in a false economy, or else insisting on avoiding a word in an equation environment.

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  • $\begingroup$ But perhaps we shouldn't call the Boolean $T$ or $F$ when testing $\lvert L - L_l\rvert > T$. 😄 (I've always been partial to the marvellously symmetric $\top$ and $\perp$, although they have reasonable arguments against them. Agreed on the false economy. I was swept away by this marvellous way of expressing ideas as an undergraduate, and I suspect I'm not the only one … but I got over it, as I think most sufferers do!) $\endgroup$
    – LSpice
    Commented Sep 27, 2022 at 0:43
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    $\begingroup$ @LSpice oh, I originally wrote $\top$ and $\bot$, but decided to be less fancy and obscure. $\endgroup$
    – David Roberts ♦
    Commented Sep 27, 2022 at 3:21

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