Bayes and Empirical Bayes Semi-Blind Deconvolution using
Eigenfunctions of a Prior Covariance
We consider the semi-blind deconvolution problem;
i.e., estimating an unknown input function to a linear dynamical system
using a finite set of linearly related measurements
where the dynamical system is known up to some system parameters.
Without further assumptions,
this problem is often ill-posed and ill-conditioned.
We overcome this difficulty by modeling the
unknown input as a realization of a stochastic process
with a covariance that is known up to some finite set of covariance parameters.
We first present an empirical Bayes method
where the unknown parameters are estimated by maximizing the
and subsequently the input is reconstructed via a Tikhonov estimator
(with the parameters set to their point estimates).
Next, we introduce a Bayesian method that recovers the
posterior probability distribution, and hence
the minimum variance estimates,
for both the unknown parameters and the unknown input function.
Both of these methods
use the eigenfunctions of the random process covariance
to obtain an efficient representation of the unknown input function and
its probability distributions.
Simulated case studies are used to test the
two methods and compare their relative performance.