For a nonnegative real matrix $A$ of rank $1$, $A$ can be factored as ${old a old b}^t$ for some vectors ${old a}$ and ${old b}$. The {it perimeter} of $A$ is the number of nonzero entries in both $old a$ and $old b$. If $B$ is a matrix of rank $k$, then $B$ is the sum of $k$ matrices of rank $1$. The perimeter of $B$ is the minimum of the sums of perimeters of $k$ matrices of rank $1$, where the minimum is taken over all possible rank-$1$ decompositions of $B$. In this paper, we obtain characterizations of the linear operators which preserve perimeters $2$ and $k$ for some $kgeq 4$. That is, a linear operator $T$ preserves perimeters $2$ and $k(geq 4)$ if and only if it has the form $T(A)=UAV$, or $T(A)=UA^tV$ with some invertible matrices $U$ and $V$.

Keywords: rank, perimeter, linear operator, $(U,V)$-operator