529 research outputs found
Fast Markov chain Monte Carlo sampling for sparse Bayesian inference in high-dimensional inverse problems using L1-type priors
Sparsity has become a key concept for solving of high-dimensional inverse
problems using variational regularization techniques. Recently, using similar
sparsity-constraints in the Bayesian framework for inverse problems by encoding
them in the prior distribution has attracted attention. Important questions
about the relation between regularization theory and Bayesian inference still
need to be addressed when using sparsity promoting inversion. A practical
obstacle for these examinations is the lack of fast posterior sampling
algorithms for sparse, high-dimensional Bayesian inversion: Accessing the full
range of Bayesian inference methods requires being able to draw samples from
the posterior probability distribution in a fast and efficient way. This is
usually done using Markov chain Monte Carlo (MCMC) sampling algorithms. In this
article, we develop and examine a new implementation of a single component
Gibbs MCMC sampler for sparse priors relying on L1-norms. We demonstrate that
the efficiency of our Gibbs sampler increases when the level of sparsity or the
dimension of the unknowns is increased. This property is contrary to the
properties of the most commonly applied Metropolis-Hastings (MH) sampling
schemes: We demonstrate that the efficiency of MH schemes for L1-type priors
dramatically decreases when the level of sparsity or the dimension of the
unknowns is increased. Practically, Bayesian inversion for L1-type priors using
MH samplers is not feasible at all. As this is commonly believed to be an
intrinsic feature of MCMC sampling, the performance of our Gibbs sampler also
challenges common beliefs about the applicability of sample based Bayesian
inference.Comment: 33 pages, 14 figure
Refraction-corrected ray-based inversion for three-dimensional ultrasound tomography of the breast
Ultrasound Tomography has seen a revival of interest in the past decade,
especially for breast imaging, due to improvements in both ultrasound and
computing hardware. In particular, three-dimensional ultrasound tomography, a
fully tomographic method in which the medium to be imaged is surrounded by
ultrasound transducers, has become feasible. In this paper, a comprehensive
derivation and study of a robust framework for large-scale bent-ray ultrasound
tomography in 3D for a hemispherical detector array is presented. Two
ray-tracing approaches are derived and compared. More significantly, the
problem of linking the rays between emitters and receivers, which is
challenging in 3D due to the high number of degrees of freedom for the
trajectory of rays, is analysed both as a minimisation and as a root-finding
problem. The ray-linking problem is parameterised for a convex detection
surface and three robust, accurate, and efficient ray-linking algorithms are
formulated and demonstrated. To stabilise these methods, novel
adaptive-smoothing approaches are proposed that control the conditioning of the
update matrices to ensure accurate linking. The nonlinear UST problem of
estimating the sound speed was recast as a series of linearised subproblems,
each solved using the above algorithms and within a steepest descent scheme.
The whole imaging algorithm was demonstrated to be robust and accurate on
realistic data simulated using a full-wave acoustic model and an anatomical
breast phantom, and incorporating the errors due to time-of-flight picking that
would be present with measured data. This method can used to provide a
low-artefact, quantitatively accurate, 3D sound speed maps. In addition to
being useful in their own right, such 3D sound speed maps can be used to
initialise full-wave inversion methods, or as an input to photoacoustic
tomography reconstructions
Fast Gibbs sampling for high-dimensional Bayesian inversion
Solving ill-posed inverse problems by Bayesian inference has recently attracted considerable attention. Compared to deterministic approaches, the probabilistic representation of the solution by the posterior distribution can be exploited to explore and quantify its uncertainties. In applications where the inverse solution is subject to further analysis procedures can be a significant advantage. Alongside theoretical progress, various new computational techniques allow us to sample very high dimensional posterior distributions: in (Lucka 2012 Inverse Problems 28 125012), and a Markov chain Monte Carlo posterior sampler was developed for linear inverse problems with -type priors. In this article, we extend this single component (SC) Gibbs-type sampler to a wide range of priors used in Bayesian inversion, such as general priors with additional hard constraints. In addition, a fast computation of the conditional, SC densities in an explicit, parameterized form, a fast, robust and exact sampling from these one-dimensional densities is key to obtain an efficient algorithm. We demonstrate that a generalization of slice sampling can utilize their specific structure for this task and illustrate the performance of the resulting slice-within-Gibbs samplers by different computed examples. These new samplers allow us to perform sample-based Bayesian inference in high-dimensional scenarios with certain priors for the first time, including the inversion of computed tomography data with the popular isotropic total variation prior
Fast Gibbs sampling for high-dimensional Bayesian inversion
Solving ill-posed inverse problems by Bayesian inference has recently
attracted considerable attention. Compared to deterministic approaches, the
probabilistic representation of the solution by the posterior distribution can
be exploited to explore and quantify its uncertainties. In applications where
the inverse solution is subject to further analysis procedures, this can be a
significant advantage. Alongside theoretical progress, various new
computational techniques allow to sample very high dimensional posterior
distributions: In [Lucka2012], a Markov chain Monte Carlo (MCMC) posterior
sampler was developed for linear inverse problems with -type priors. In
this article, we extend this single component Gibbs-type sampler to a wide
range of priors used in Bayesian inversion, such as general priors
with additional hard constraints. Besides a fast computation of the
conditional, single component densities in an explicit, parameterized form, a
fast, robust and exact sampling from these one-dimensional densities is key to
obtain an efficient algorithm. We demonstrate that a generalization of slice
sampling can utilize their specific structure for this task and illustrate the
performance of the resulting slice-within-Gibbs samplers by different computed
examples. These new samplers allow us to perform sample-based Bayesian
inference in high-dimensional scenarios with certain priors for the first time,
including the inversion of computed tomography (CT) data with the popular
isotropic total variation (TV) prior.Comment: submitted to "Inverse Problems
Sparse Bayesian Inference & Uncertainty Quantification for Inverse Imaging Problems
During the last two decades, sparsity has emerged as a key concept to solve linear and non-linear ill-posed inverse problems, in particular for severely ill-posed problems and applications with incomplete, sub-sampled data. At the same time, there is a growing demand to obtain quantitative instead of just qualitative inverse results together with a systematic assessment of their uncertainties (Uncertainty quantification, UQ). Bayesian inference seems like a suitable framework to combine sparsity and UQ but its application to large-scale inverse problems resulting from fine discretizations of PDE models leads to severe computational and conceptional challenges. In this talk, we will focus on two different Bayesian approaches to model sparsity as a-priori information: Via convex, but non-smooth prior energies such as total variation and Besov space priors and via non-convex but smooth priors arising from hierarchical Bayesian modeling. To illustrate our findings, we will rely on experimental data from challenging biomedical imaging applications such as EEG/MEG source localization and limited-angle CT. We want to share the experiences, results we obtained and the open questions we face from our perspective as researchers coming from a background in biomedical imaging rather than in statistics and hope to stimulate a fruitful discussion for both sides
Refining the U.S. Navy Flight Clearance (Airworthiness Certification) Process: Maximizing Acquisition Reform Benefits for Commercial Derivative Aircraft Acquisitions
The purpose of this thesis is to propose several specific process improvements to the U.S. Navy Flight Clearance process. It also makes recommendations for investment in test and evaluation infrastructure to better support commercial derivative aircraft acquisition for military use. Recent changes to Department of Defense acquisition regulations have increased alternatives available to the services to meet military operational requirements. In order to reduce cost and cycle times, the military services are turning more and more to acquiring commercially developed and certified aircraft to meet military requirements
Approximate k-space models and Deep Learning for fast photoacoustic reconstruction
We present a framework for accelerated iterative reconstructions using a fast
and approximate forward model that is based on k-space methods for
photoacoustic tomography. The approximate model introduces aliasing artefacts
in the gradient information for the iterative reconstruction, but these
artefacts are highly structured and we can train a CNN that can use the
approximate information to perform an iterative reconstruction. We show
feasibility of the method for human in-vivo measurements in a limited-view
geometry. The proposed method is able to produce superior results to total
variation reconstructions with a speed-up of 32 times
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