  We carry out an analysis of $\pi\pi$ scattering in the $IJ=00$, $11$ and $20$
channels in configuration space up to a maximal center-of-mass energy
$\sqrt{s}=1.4$ GeV. We separate the interaction into two regions marked by an
elementarity radius of the system; namely, a long distance region above which
pions can be assumed to interact as elementary particles and a short distance
region where many physical effects cannot be disentangled. The long distance
interaction is described by chiral dynamics, where a two-pion-exchange
potential is identified, computed and compared to lattice calculations. The
short distance piece corresponds to a coarse grained description exemplified by
a superposition of delta-shell potentials sampling the interaction with the
minimal wavelength. We show how the so constructed non-perturbative scattering
amplitude complies with the proper analytic structure, allowing for an explicit
N/D type decomposition in terms of the corresponding Jost functions and
fulfilling dispersion relations without subtractions. We also address
renormalization issues in coordinate space and investigate the role of crossing
when fitting the scattering amplitudes above and below threshold to
Roy-equation results. At higher energies, we show how inelasticities can be
described by one single complex and energy dependent parameter. A successful
description of the data can be achieved with a minimal number of fitting
parameters, suggesting that coarse graining is a viable approach to analyze
hadronic processes.
