Resolution of Identity Crisis of Events in Pile-up

Mutually uncorrelated random discrete events, manifesting a common basic process, are examined often in terms of their occurrence rate as a function of one or more of their distinguishing attributes, such as measurements of photon spectrum as a function of energy. Such rate distributions obtained from the observed attribute values for an ensemble of events will correspond to the "true" distribution only if the event occurrence were {\it mutually exclusive}. However, due to finite resolution in such measurements, the problem of event {\it pile-up} is not only unavoidable, but also increases with event rate. Although extensive simulations to estimate the distortion due to pile-up in the observed rate distribution are available, no restoration procedure has yet been suggested. Here we present an elegant analytical solution to recover the underlying {\it true} distribution. Our method, based on Poisson statistics and Fourier transforms, is shown to perform as desired even when applied to distributions that are significantly distorted by pile-up. Our recipes for correction, as well as for prediction, of pile-up are expected to find ready applications in a wide variety of fields, ranging from high-energy physics to medical clinical diagnostics, and involving, but not limited to, measurements of count-rates and/or spectra of incident radiation using Charge Coupled Devices (CCDs) or other similar devices.