  This paper explores the concept of reparametrization invariant norm
(RPI-norm), that is any norm invariant under composition with diffeomorphisms.
We prove the existence of an infinite family of RPI-norms, called standard
RPI-norms, for which we exhibit both an integral and a discrete
characterization. Our main result states that, for every one-time
differentiable piecewise monotone function with compact support, its standard
RPI-norms allow us to compute the value of any other RPI-norm of the same
function. This is proved by using the standard RPI-norms in order to
reconstruct the function up to reparametrization and an arbitrarily small error
with respect to the total variation norm.
