4,160 research outputs found
Efficient Gaussian Sampling for Solving Large-Scale Inverse Problems using MCMC Methods
The resolution of many large-scale inverse problems using MCMC methods
requires a step of drawing samples from a high dimensional Gaussian
distribution. While direct Gaussian sampling techniques, such as those based on
Cholesky factorization, induce an excessive numerical complexity and memory
requirement, sequential coordinate sampling methods present a low rate of
convergence. Based on the reversible jump Markov chain framework, this paper
proposes an efficient Gaussian sampling algorithm having a reduced computation
cost and memory usage. The main feature of the algorithm is to perform an
approximate resolution of a linear system with a truncation level adjusted
using a self-tuning adaptive scheme allowing to achieve the minimal computation
cost. The connection between this algorithm and some existing strategies is
discussed and its efficiency is illustrated on a linear inverse problem of
image resolution enhancement.Comment: 20 pages, 10 figures, under review for journal publicatio
A Hierarchical Bayesian Model for Frame Representation
In many signal processing problems, it may be fruitful to represent the
signal under study in a frame. If a probabilistic approach is adopted, it
becomes then necessary to estimate the hyper-parameters characterizing the
probability distribution of the frame coefficients. This problem is difficult
since in general the frame synthesis operator is not bijective. Consequently,
the frame coefficients are not directly observable. This paper introduces a
hierarchical Bayesian model for frame representation. The posterior
distribution of the frame coefficients and model hyper-parameters is derived.
Hybrid Markov Chain Monte Carlo algorithms are subsequently proposed to sample
from this posterior distribution. The generated samples are then exploited to
estimate the hyper-parameters and the frame coefficients of the target signal.
Validation experiments show that the proposed algorithms provide an accurate
estimation of the frame coefficients and hyper-parameters. Application to
practical problems of image denoising show the impact of the resulting Bayesian
estimation on the recovered signal quality
Estimating Signals with Finite Rate of Innovation from Noisy Samples: A Stochastic Algorithm
As an example of the recently-introduced concept of rate of innovation,
signals that are linear combinations of a finite number of Diracs per unit time
can be acquired by linear filtering followed by uniform sampling. However, in
reality, samples are rarely noiseless. In this paper, we introduce a novel
stochastic algorithm to reconstruct a signal with finite rate of innovation
from its noisy samples. Even though variants of this problem has been
approached previously, satisfactory solutions are only available for certain
classes of sampling kernels, for example kernels which satisfy the Strang-Fix
condition. In this paper, we consider the infinite-support Gaussian kernel,
which does not satisfy the Strang-Fix condition. Other classes of kernels can
be employed. Our algorithm is based on Gibbs sampling, a Markov chain Monte
Carlo (MCMC) method. Extensive numerical simulations demonstrate the accuracy
and robustness of our algorithm.Comment: Submitted to IEEE Transactions on Signal Processin
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