Measure-preserving parametrization of one-dimensional transport maps
= Measure-preserving parametrization of one-dimensional transport maps
{title2=$T=G^{-1}\circ S\circ F$}
When source and target have continuous strictly increasing cumulative distributions, all <transport maps> are obtained by choosing a <Lebesgue-measure-preserving map> $S$ on the unit interval. The maps $F$ and $G$ turn the two <measures> into uniform <measure>. The choice $S=\mathrm{id}$ gives the <monotone rearrangement>.