  In this paper, we consider a nonlinear inverse problem occurring in nuclear
science. Gamma rays randomly hit a semiconductor detector which produces an
impulse response of electric current. Because the sampling period of the
measured current is larger than the mean inter arrival time of photons, the
impulse responses associated to different gamma rays can overlap: this
phenomenon is known as pileup. In this work, it is assumed that the impulse
response is an exponentially decaying function. We propose a novel method to
infer the distribution of gamma photon energies from the indirect measurements
obtained from the detector. This technique is based on a formula linking the
characteristic function of the photon density to a function involving the
characteristic function and its derivative of the observations. We establish
that our estimator converges to the mark density in uniform norm at a
logarithmic rate. A limited Monte-Carlo experiment is provided to support our
findings.
