  Recently it was proved that the group of rough paths modulo tree-like
equivalence is isomorphic to the corresponding signature group through the
signature map S (a generalized notion of taking iterated path integrals).
However, the proof of this uniqueness result does not contain any information
on how to "see" the trajectory of a (tree-reduced) rough path from its
signature, and a constructive understanding on the uniqueness result (in
particular on the inverse of S) has become an interesting and important
question. The aim of the present paper is to reconstruct a rough path from its
signature in an explicit and universal way.
