76,609 research outputs found
From variable density sampling to continuous sampling using Markov chains
International audienceSince its discovery over the last decade, Compressed Sensing (CS) has been successfully applied to Magnetic Resonance Imaging (MRI). It has been shown to be a powerful way to reduce scanning time without sacrificing image quality. MR images are actually strongly compressible in a wavelet basis, the latter being largely incoherent with the k-space or spatial Fourier domain where acquisition is performed. Nevertheless, since its first application to MRI [1], the theoretical justification of actual k-space sampling strategies is questionable. Indeed, the vast majority of k-space sampling distributions have been heuris- tically designed (e.g., variable density) or driven by experimental feasibility considerations (e.g., random radial or spiral sampling to achieve smoothness k-space trajectory). In this paper, we try to reconcile very recent CS results with the MRI specificities (magnetic field gradients) by enforcing the measurements, i.e. samples of k-space, to fit continuous trajectories. To this end, we propose random walk continuous sampling based on Markov chains and we compare the reconstruction quality of this scheme to the state-of-the art
Fast MCMC sampling for Markov jump processes and extensions
Markov jump processes (or continuous-time Markov chains) are a simple and
important class of continuous-time dynamical systems. In this paper, we tackle
the problem of simulating from the posterior distribution over paths in these
models, given partial and noisy observations. Our approach is an auxiliary
variable Gibbs sampler, and is based on the idea of uniformization. This sets
up a Markov chain over paths by alternately sampling a finite set of virtual
jump times given the current path and then sampling a new path given the set of
extant and virtual jump times using a standard hidden Markov model forward
filtering-backward sampling algorithm. Our method is exact and does not involve
approximations like time-discretization. We demonstrate how our sampler extends
naturally to MJP-based models like Markov-modulated Poisson processes and
continuous-time Bayesian networks and show significant computational benefits
over state-of-the-art MCMC samplers for these models.Comment: Accepted at the Journal of Machine Learning Research (JMLR
Variable density sampling based on physically plausible gradient waveform. Application to 3D MRI angiography
Performing k-space variable density sampling is a popular way of reducing
scanning time in Magnetic Resonance Imaging (MRI). Unfortunately, given a
sampling trajectory, it is not clear how to traverse it using gradient
waveforms. In this paper, we actually show that existing methods [1, 2] can
yield large traversal time if the trajectory contains high curvature areas.
Therefore, we consider here a new method for gradient waveform design which is
based on the projection of unrealistic initial trajectory onto the set of
hardware constraints. Next, we show on realistic simulations that this
algorithm allows implementing variable density trajectories resulting from the
piecewise linear solution of the Travelling Salesman Problem in a reasonable
time. Finally, we demonstrate the application of this approach to 2D MRI
reconstruction and 3D angiography in the mouse brain.Comment: IEEE International Symposium on Biomedical Imaging (ISBI), Apr 2015,
New-York, United State
Estimation in discretely observed diffusions killed at a threshold
Parameter estimation in diffusion processes from discrete observations up to
a first-hitting time is clearly of practical relevance, but does not seem to
have been studied so far. In neuroscience, many models for the membrane
potential evolution involve the presence of an upper threshold. Data are
modeled as discretely observed diffusions which are killed when the threshold
is reached. Statistical inference is often based on the misspecified likelihood
ignoring the presence of the threshold causing severe bias, e.g. the bias
incurred in the drift parameters of the Ornstein-Uhlenbeck model for biological
relevant parameters can be up to 25-100%. We calculate or approximate the
likelihood function of the killed process. When estimating from a single
trajectory, considerable bias may still be present, and the distribution of the
estimates can be heavily skewed and with a huge variance. Parametric bootstrap
is effective in correcting the bias. Standard asymptotic results do not apply,
but consistency and asymptotic normality may be recovered when multiple
trajectories are observed, if the mean first-passage time through the threshold
is finite. Numerical examples illustrate the results and an experimental data
set of intracellular recordings of the membrane potential of a motoneuron is
analyzed.Comment: 29 pages, 5 figure
Automatic Differentiation Variational Inference
Probabilistic modeling is iterative. A scientist posits a simple model, fits
it to her data, refines it according to her analysis, and repeats. However,
fitting complex models to large data is a bottleneck in this process. Deriving
algorithms for new models can be both mathematically and computationally
challenging, which makes it difficult to efficiently cycle through the steps.
To this end, we develop automatic differentiation variational inference (ADVI).
Using our method, the scientist only provides a probabilistic model and a
dataset, nothing else. ADVI automatically derives an efficient variational
inference algorithm, freeing the scientist to refine and explore many models.
ADVI supports a broad class of models-no conjugacy assumptions are required. We
study ADVI across ten different models and apply it to a dataset with millions
of observations. ADVI is integrated into Stan, a probabilistic programming
system; it is available for immediate use
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