  The original Calder\'on problem consists in recovering the potential (or the
conductivity) from the knowledge of the related Neumann to Dirichlet map (or
Dirichlet to Neumann map). Here, we first perturb the medium by injecting
small-scaled and highly heterogeneous particles. Such particles can be bubbles
or droplets in acoustics or nanoparticles in electromagnetism. They are
distributed, periodically for instance, in the whole domain where we want to do
reconstruction. Under critical scales between the size and contrast, these
particles resonate at specific frequencies that can be well computed. Using
incident frequencies that are close to such resonances, we show that
  1) the corresponding Neumann to Dirichlet map of the composite converges to
the one of the homogenized medium. In addition, the equivalent coefficient,
which consist in the sum of the original potential and the effective
coefficient, is negative valued with a controlable amplitude.
  2) as the equivalent coefficient is negative valued, then we can linearize
the corresponding Neumann to Dirichlet map using the effective coefficient's
amplitude.
  3) from the linearized Neumann to Dirichlet map, we reconstruct the original
potential using explicit complex geometrical optics solutions (CGOs).
