Inversion of signature for paths of bounded variation
  We develop two methods to reconstruct a path of bounded variation from its signature. The first method gives a simple and explicit expression of any axis path in terms of its signature, but it does not apply directlty to more general ones. The second method, based on an approximation scheme, recovers any tree-reduced path from its signature as the limit of a uniformly convergent sequence of lattice paths. 
Terry Lyons, Weijun Xu
math.CA math.PR