The Mirror Transform

This paper explains how to describe any finite-energy signal through a unique representation consisting of an ordered set of positions and a sparse set of signals. This is obtained by designing an iterative decomposition through a series of mirror operations around those positions. The purpose is to find at any step of the decomposition the location that provides for the maximum decoupling between the even and odd components of the signal with respect to it. By limiting such even and odd components on three separate information bearing support, the algorithm can be iterated at infinity determining a sequence of positions. The per location information determines the optimal energy decoupling strategy at each stage providing remarkable sparsity in the representation. The resulting transformation leads to a 1-to-1 mapping. The approach is easily extended to finite-energy sequences, and in particular for sequences of finite length $N$, at most $N$ iterations of the decomposition are required. Thanks to the sparsity of the resulting representation, experimental simulations demonstrate superior approximation capabilities of this proposed non-linear Mirror Transform with potential application in many domains such as approximation and coding. Its implementation has been made publicly available.