This paper describes a parametric deconvolution method (PDPS)
appropriate for a particular class of signals which we call spike-convolution
models. These models arise when a sparse spike train —Dirac deltas
according to our mathematical treatment —is convolved with a fixed
point-spread function, and additive noise or measurement error is
superimposed.We view deconvolution as an estimation problem, regarding the
locations and heights of the underlying spikes, as well as the baseline and the
measurement error variance as unknown parameters.Our estimation scheme consists
of two parts: model fitting and model selection.To fit a spike-convolution
model of a specific order, we estimate peak locations by trigonometric moments,
and heights and the baseline by least squares. The model selection procedure
has two stages. Its first stage is so designed that we expect a model of a
somewhat larger order than the truth to be selected. In the second stage, the
final model is obtained using backwards deletion. This results in not only an
estimate of the model order, but also an estimate of peak locations and heights
with much smaller bias and variation than that found in a direct trigonometric
moment estimate. A more efficient maximum likelihood estimate can be calculated
from these estimates using a Gauss–Newton algorithm. We also present
some relevant results concerning the spectral structure of Toeplitz matrices
which play a key role in the estimation. Finally, we illustrate the behavior of
these estimates using simulated and real DNA sequencing data.