  Harten's Multiresolution framework has been applied in different contexts,
such as in the numerical simulation of PDE with conservation laws or in image
compression, showing its flexibility to describe and manipulate the data in a
multilevel fashion. Two basic operators form the basis of this theory: the
decimation and the prediction. The decimation is chosen first and determines
the type of data that is being manipulated. For instance, the data could be the
point evaluations or the cell-averages of a function, which are the two
classical environments. Next, the prediction is chosen, and it must be
compatible with the decimation.
  Subdivision schemes can be used as prediction operators, but sometimes they
not fit into one of the two typical environments. In this paper we show how to
invert this order so we can choose a prediction first and then define a
compatible decimation from that prediction. Moreover, we also prove that any
possible decimation can be obtained in this way.
