  This paper investigates the computation of proximity operators for scale and
signed permutation invariant functions. A scale-invariant function remains
unchanged under uniform scaling, while a signed permutation invariant function
retains its structure despite permutations and sign changes applied to its
input variables. Noteworthy examples include the $\ell_0$ function and the
ratios of $\ell_1/\ell_2$ and its square, with their proximity operators being
particularly crucial in sparse signal recovery. We delve into the properties of
scale and signed permutation invariant functions, delineating the computation
of their proximity operators into three sequential steps: the
$\mathbf{w}$-step, $r$-step, and $d$-step. These steps collectively form a
procedure termed as WRD, with the $\mathbf{w}$-step being of utmost importance
and requiring careful treatment. Leveraging this procedure, we present a method
for explicitly computing the proximity operator of $(\ell_1/\ell_2)^2$ and
introduce an efficient algorithm for the proximity operator of $\ell_1/\ell_2$.
