  We consider the analytic continuation of Euclidean propagator data obtained
from 4D simulations to Minkowski space. In order to perform this continuation,
the common approach is to first extract the K\"all\'en-Lehmann spectral density
of the field. Once this is known, it can be extended to Minkowski space to
yield the Minkowski propagator. However, obtaining the K\"all\'en-Lehmann
spectral density from propagator data is a well known ill-posed numerical
problem. To regularize this problem we implement an appropriate version of
Tikhonov regularization supplemented with the Morozov discrepancy principle. We
will then apply this to various toy model data to demonstrate the conditions of
validity for this method, and finally to zero temperature gluon and ghost
lattice QCD data. We carefully explain how to deal with the IR singularity of
the massless ghost propagator. We also uncover the numerically different
performance when using two ---mathematically equivalent--- versions of the
K\"all\'en-Lehmann spectral integral.
