For a nonnegative real matrix A of rank 1, A can be factored as $ab^t$ for some vectors a and b. The perimeter of A is the number of nonzero entries in both a and 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 $kgeq4$. That is, a linear operator T preserves perimeters 2 and $k(geq4)$ if and only if it has the form T(A) = UAV or T(A) = $UA^tV$ with some invertible matrices U and V.