Find the area, perimeter, and compactness of the nonzero region in the $3 \times 3$ image.
(i)
$$
\left[\begin{array}{lll}
1 & 1 & 1 \\
1 & 1 & 1 \\
1 & 1 & 1
\end{array}\right]
$$
(ii)
$$
\left[\begin{array}{lll}
0 & 0 & 0 \\
0 & 1 & 1 \\
1 & 1 & 1
\end{array}\right]
$$
(iii)
$$
\left[\begin{array}{lll}
1 & 0 & 0 \\
1 & 1 & 0 \\
1 & 1 & 1
\end{array}\right]
$$