375 research outputs found
Alternating Deep Low Rank Approach for Exponential Function Reconstruction and Its Biomedical Magnetic Resonance Applications
Exponential function is a fundamental signal form in general signal
processing and biomedical applications, such as magnetic resonance spectroscopy
and imaging. How to reduce the sampling time of these signals is an important
problem. Sub-Nyquist sampling can accelerate signal acquisition but bring in
artifacts. Recently, the low rankness of these exponentials has been applied to
implicitly constrain the deep learning network through the unrolling of low
rank Hankel factorization algorithm. However, only depending on the implicit
low rank constraint cannot provide the robust reconstruction, such as sampling
rate mismatches. In this work, by introducing the explicit low rank prior to
constrain the deep learning, we propose an Alternating Deep Low Rank approach
(ADLR) that utilizes deep learning and optimization solvers alternately. The
former solver accelerates the reconstruction while the latter one corrects the
reconstruction error from the mismatch. The experiments on both general
exponential functions and realistic biomedical magnetic resonance data show
that, compared with the state-of-the-art methods, ADLR can achieve much lower
reconstruction error and effectively alleviates the decrease of reconstruction
quality with sampling rate mismatches.Comment: 14 page
Statistical characterization of residual noise in the low-rank approximation filter framework, general theory and application to hyperpolarized tracer spectroscopy
The use of low-rank approximation filters in the field of NMR is increasing
due to their flexibility and effectiveness. Despite their ability to reduce the
Mean Square Error between the processed signal and the true signal is well
known, the statistical distribution of the residual noise is still undescribed.
In this article, we show that low-rank approximation filters are equivalent to
linear filters, and we calculate the mean and the covariance matrix of the
processed data. We also show how to use this knowledge to build a maximum
likelihood estimator, and we test the estimator's performance with a Montecarlo
simulation of a 13C pyruvate metabolic tracer. While the article focuses on NMR
spectroscopy experiment with hyperpolarized tracer, we also show that the
results can be applied to tensorial data (e.g. using HOSVD) or 1D data (e.g.
Cadzow filter).Comment: 26 pages, 7 figure
Beyond the noise : high fidelity MR signal processing
This thesis describes a variety of methods developed to increase the sensitivity and resolution of liquid state nuclear magnetic resonance (NMR) experiments. NMR is known as one of the most versatile non-invasive analytical techniques yet often suffers from low sensitivity. The main contribution to this low sensitivity issue is a presence of noise and level of noise in the spectrum is expressed numerically as “signal-to-noise ratio”. NMR signal processing involves sensitivity and resolution enhancement achieved by noise reduction using mathematical algorithms. A singular value decomposition based reduced rank matrix method, composite property mapping, in particular is studied extensively in this thesis to present its advantages, limitations, and applications. In theory, when the sum of k noiseless sinusoidal decays is formatted into a specific matrix form (i.e., Toeplitz), the matrix is known to possess k linearly independent columns. This information becomes apparent only after a singular value decomposition of the matrix. Singular value decomposition factorises the large matrix into three smaller submatrices: right and left singular vector matrices, and one diagonal matrix containing singular values. Were k noiseless sinusoidal decays involved, there would be only k nonzero singular values appearing in the diagonal matrix in descending order providing the information of the amplitude of each sinusoidal decay. The number of non-zero singular values or the number of linearly independent columns is known as the rank of the matrix. With real NMR data none of the singular values equals zero and the matrix has full rank. The reduction of the rank of the matrix and thus the noise in the reconstructed NMR data can be achieved by replacing all the singular values except the first k values with zeroes. This noise reduction process becomes difficult when biomolecular NMR data is to be processed due to the number of resonances being unknown and the presence of a large solvent peak
A Stochastic Majorize-Minimize Subspace Algorithm for Online Penalized Least Squares Estimation
Stochastic approximation techniques play an important role in solving many
problems encountered in machine learning or adaptive signal processing. In
these contexts, the statistics of the data are often unknown a priori or their
direct computation is too intensive, and they have thus to be estimated online
from the observed signals. For batch optimization of an objective function
being the sum of a data fidelity term and a penalization (e.g. a sparsity
promoting function), Majorize-Minimize (MM) methods have recently attracted
much interest since they are fast, highly flexible, and effective in ensuring
convergence. The goal of this paper is to show how these methods can be
successfully extended to the case when the data fidelity term corresponds to a
least squares criterion and the cost function is replaced by a sequence of
stochastic approximations of it. In this context, we propose an online version
of an MM subspace algorithm and we study its convergence by using suitable
probabilistic tools. Simulation results illustrate the good practical
performance of the proposed algorithm associated with a memory gradient
subspace, when applied to both non-adaptive and adaptive filter identification
problems
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