P is a 2D uniform distribution with density $p(x,y)=\chi_{[0,1]^2}$,
Q is a 4-piecewise-uniform distribution with density
$\begin{equation} q(x,y)=\left\{ \begin{aligned} \frac{1}{2} & & (x,y)&\in[0,\frac{1}{2}]^2 \\ 2 & & (x,y)&\in[0,\frac{1}{2}]\times(\frac{1}{2},1] \\ 1& & (x,y)&\in(\frac{1}{2},1]\times[0,\frac{1}{2}] \\ \frac{1}{2} & & (x,y)&\in(\frac{1}{2},1]^2 \\ \end{aligned} \right. \end{equation}$
I am thinking about using a continuos mapping to get Q from P. Is it possible to do that? I know we can use inverse of cdf to do so in 1D, but it seems cannot generalize this method to 2D cases.