How to represent a given equation more clear, profressional and short form?

Question

I have a image $U_{m \times n}:\Omega \to \mathbb R^2$, the output $P$ can be define as $$P=\mu J_{m \times n} - U$$ where $\mu = \max \{ u_{ij} : 1 \leq i \leq m, 1 \leq j \leq n\}$, $J_{m \times n}$ be the $m \times n$ matrix whose $i, j$th component is $1$: that is, the all-ones matrix. (This notation isn't quite standard, but it's as close to standard as I know. $J$ is often the all-ones matrix)

However, it is so many sentences for expression the above equation. Do we have more short and standard way to represent it? As my found, the $J_{m \times n}$ can be expressed by indicator function such as

$$P=\max (U)\times 1_{\Omega}-U$$

where $1_{\Omega}$ is Indicator function

Does it equivalent with original meaning? If not, please give me a standard and common expression in image processing. Thanks

Answer 1

I think the notation $J_{m,n}$ is pretty standard for an all-ones matrix. So the only possible clarification would be to dispense with $\mu$ and directly specify what it actually is:

$$P=||U||_{\max}J_{m,n}-U$$

where $||.||_{\max}$ denotes the max norm of a matrix.

Answer 2

I would propose \mathbb{1} from the \usepackage{bbold} (or \mathds{1}, see details and options here), with outputs a $1$ with a double bar, like for $\mathbb{R}$ (I cannot have it displayed on SE). I would stick to a simple $\max $ or $\sup$, as it is not a norm without the absolute value. Plus, image values are taken in $\mathbb{R}^2$, as in some astronomical, seismic or processed images. So, something like:

\documentclass{article}
\usepackage{bbold}
\begin{document}
$\max_{i,j} \{ A_{i,j}\} . \mathbb{1}-A$
\end{document}