We propose a Bayesian expectation-maximization (EM) algorithm for reconstructing structured approximately
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 hidden Markov tree (HMT).
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 employ
a max-product algorithm to implement the maximization (M) step of our EM iteration. The noise variance is
a regularization parameter that controls signal sparsity. 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. Our numerical examples show that the proposed algorithm achieves better reconstruction
performance compared with the state-of-the-art methods.
We propose two hard thresholding schemes for image reconstruction from compressive samples. The measurements
follow an underdetermined linear model, where the regression-coefficient vector is a sum of an unknown
deterministic sparse signal component and a zero-mean white Gaussian component with an unknown variance.
We derived an expectation-conditional maximization either (ECME) iteration that converges to a local maximum
of the likelihood function of the unknown parameters for a given image sparsity level. Here, we present
and analyze a double overrelaxation (DORE) algorithm that applies two successive overrelaxation steps after
one ECME iteration step, with the goal to accelerate the ECME iteration. To analyze the reconstruction accuracy,
we introduce minimum sparse subspace quotient (minimum SSQ), a more flexible measure of the sampling
operator than the well-established restricted isometry property (RIP). We prove that, if the minimum SSQ is
sufficiently large, the DORE algorithm achieves perfect or near-optimal recovery of the true image, provided
that its transform coefficients are sparse or nearly sparse, respectively. We then describe a multiple-initialization
DORE algorithm (DOREMI) that can significantly improve DORE's reconstruction performance. We present
numerical examples where we compare our methods with existing compressive sampling image reconstruction
approaches.
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