  In this paper we consider a novel statistical inverse problem on the
Poincar\'{e}, or Lobachevsky, upper (complex) half plane. Here the Riemannian
structure is hyperbolic and a transitive group action comes from the space of
$2\times2$ real matrices of determinant one via M\"{o}bius transformations. Our
approach is based on a deconvolution technique which relies on the
Helgason--Fourier calculus adapted to this hyperbolic space. This gives a
minimax nonparametric density estimator of a hyperbolic density that is
corrupted by a random M\"{o}bius transform. A motivation for this work comes
from the reconstruction of impedances of capacitors where the above scenario on
the Poincar\'{e} plane exactly describes the physical system that is of
statistical interest.
