  We propose a Bayesian expectation-maximization (EM) algorithm for
reconstructing Markov-tree sparse signals via belief propagation. The
measurements follow an underdetermined linear model where the
regression-coefficient vector is the sum of an unknown approximately sparse
signal and a zero-mean white Gaussian noise with an unknown variance. The
signal is composed of large- and small-magnitude components identified by
binary state variables whose probabilistic dependence structure is described by
a Markov tree. Gaussian priors are assigned to the signal coefficients given
their state variables and the Jeffreys' noninformative prior is assigned to the
noise variance. Our signal reconstruction scheme is based on an EM iteration
that aims at maximizing the posterior distribution of the signal and its state
variables given the noise variance. We construct the missing data for the EM
iteration so that the complete-data posterior distribution corresponds to a
hidden Markov tree (HMT) probabilistic graphical model that contains no loops
and implement its maximization (M) step via a max-product algorithm. This EM
algorithm estimates the vector of state variables as well as solves iteratively
a linear system of equations to obtain the corresponding signal estimate. We
select the noise variance so that the corresponding estimated signal and state
variables obtained upon convergence of the EM iteration have the largest
marginal posterior distribution. We compare the proposed and existing
state-of-the-art reconstruction methods via signal and image reconstruction
experiments.
