  We consider the counting rate estimation of an unknown radioactive source,
which emits photons at times modeled by an homogeneous Poisson process. A
spectrometer converts the energy of incoming photons into electrical pulses,
whose number provides a rough estimate of the intensity of the Poisson process.
When the activity of the source is high, a physical phenomenon known as pileup
effect distorts direct measurements, resulting in a significant bias to the
standard estimators of the source activities used so far in the field. We show
in this paper that the problem of counting rate estimation can be interpreted
as a sparse regression problem. We suggest a post-processed, non-negative,
version of the Least Absolute Shrinkage and Selection Operator (LASSO) to
estimate the photon arrival times. The main difficulty in this problem is that
no theoretical conditions can guarantee consistency in sparsity of LASSO,
because the dictionary is not ideal and the signal is sampled. We therefore
derive theoretical conditions and bounds which illustrate that the proposed
method can none the less provide a good, close to the best attainable, estimate
of the counting rate activity. The good performances of the proposed approach
are studied on simulations and real datasets.
