论文中英文翻译素材

数学

1. 一个齐次坐标P,先让它左乘上一个矩阵 M，再让它的每个分量都除以第 4 个分量。
   To project a homogeneous point (x, y, z, 1), we left-multiply by M and then divide by the fourth coordinate:
   $$\left[\begin{array}{cccc}\frac{n}{r}& 0& 0& 0\\ 0& \frac{n}{t}& 0& 0\\ 0& 0& -\frac{f+n}{f-n}& -\frac{2fn}{f-n}\\ 0& 0& -1& 0\end{array}\right]\left[\begin{array}{c}x\\ y\\ z\\ 1\end{array}\right]=\left[\begin{array}{c}\frac{n}{r}x\\ \frac{n}{t}y\\ -\frac{f+n}{f-n}z-\frac{-2fn}{f-n}\\ -z\end{array}\right]$$

$$project\to \left[\begin{array}{c}\frac{n}{r}\frac{x}{-z}\\ \frac{n}{t}\frac{y}{-z}\\ \frac{f+n}{f-n}-\frac{2fn}{f-n}\frac{1}{-z}\end{array}\right]$$

2. margin
   Point-NeRF achieves the best PSNRs, SSIMs and LPIPSs on most of the scenes and outperforms state-of-the-art methods [2,29,35,53] with a big margin.
   Our Point-NeRF outperforms all these previous studies in all metrics by substantial margins.

3. case sensitive 大小写敏感
   Note that regex engines are case sensitive by default. cat does not match Cat, unless you tell the regex engine to ignore differences in case.

4. Names of Mark in Regex
   the backslash \,
   the caret ^,
   the dollar sign $,
   the period or dot .,
   the vertical bar or pipe symbol |,
   the question mark ?,
   the asterisk or star *,
   the plus sign +,
   the opening parenthesis (,
   the closing parenthesis ),
   the opening square bracket [,
   and the opening curly brace {,
   These special characters are often called “metacharacters”. Most of them are errors when used alone.