In one dimension one of the most simple filters can be described by
$$\frac{\mathrm{d}}{\mathrm{d}x}g = - a g + f$$
where $f:\Re\mapsto\Re$ and $a$ const. Its frequency response is
$$H(\mathrm{i}\omega) = \frac{1}{\mathrm{i}\omega + a}$$
What would a two-dimensional version with a radial symmetry in the frequency domain look like? If you run one pass for $x$, followed by a pass for $y$, the compound response would look like
$$H(\mathrm{i}\xi)H(\mathrm{i}\eta) = \frac{1}{\mathrm{i}\xi + a_x} \frac{1}{\mathrm{i}\eta + a_y}=\frac{{a_x} {a_y}-\eta \xi }{\left( {{{a_x}}^{2}}+{{\xi }^{2}}\right) \left( {{{a_y}}^{2}}+{{\eta }^{2}}\right) }+\frac{\mathrm{i} \left( -\xi {a_y}-\eta {a_x}\right) }{\left( {{{a_x}}^{2}}+{{\xi }^{2}}\right) \left( {{{a_y}}^{2}}+{{\eta }^{2}}\right) } $$
and
$$|H(\mathrm{i}\xi)H(\mathrm{i}\eta)|^2= \left(|H(\mathrm{i}\xi)||H(\mathrm{i}\eta)|\right)^2= |H(\mathrm{i}\xi)|^2|H(\mathrm{i}\eta)|^2= \frac{1}{\xi^2 + a_x^2}\frac{1}{\eta^2 + a_y^2} = \frac{1}{{{{a_x}}^{2}} {{{a_y}}^{2}}+{{\xi }^{2}}\, {{{a_y}}^{2}}+{{\eta }^{2}}\, {{{a_x}}^{2}}+{{\eta }^{2}}\, {{\xi }^{2}}}$$
This has the radial component $x^2/a_x^2 + x^2/a_y^2 $, but there is also a mix term $\xi^2\eta^2$. What I am looking for is something like
$$|H(\mathrm{i}\xi, \mathrm{i}\eta)|^2 = \frac{1}{\frac{\xi^2}{a_x^2} + \frac{\eta^2}{a_y^2} + 1}$$
I guess I am looking for some PDE.