502 research outputs found
Constrained Overcomplete Analysis Operator Learning for Cosparse Signal Modelling
We consider the problem of learning a low-dimensional signal model from a
collection of training samples. The mainstream approach would be to learn an
overcomplete dictionary to provide good approximations of the training samples
using sparse synthesis coefficients. This famous sparse model has a less well
known counterpart, in analysis form, called the cosparse analysis model. In
this new model, signals are characterised by their parsimony in a transformed
domain using an overcomplete (linear) analysis operator. We propose to learn an
analysis operator from a training corpus using a constrained optimisation
framework based on L1 optimisation. The reason for introducing a constraint in
the optimisation framework is to exclude trivial solutions. Although there is
no final answer here for which constraint is the most relevant constraint, we
investigate some conventional constraints in the model adaptation field and use
the uniformly normalised tight frame (UNTF) for this purpose. We then derive a
practical learning algorithm, based on projected subgradients and
Douglas-Rachford splitting technique, and demonstrate its ability to robustly
recover a ground truth analysis operator, when provided with a clean training
set, of sufficient size. We also find an analysis operator for images, using
some noisy cosparse signals, which is indeed a more realistic experiment. As
the derived optimisation problem is not a convex program, we often find a local
minimum using such variational methods. Some local optimality conditions are
derived for two different settings, providing preliminary theoretical support
for the well-posedness of the learning problem under appropriate conditions.Comment: 29 pages, 13 figures, accepted to be published in TS
Compressive and Noncompressive Power Spectral Density Estimation from Periodic Nonuniform Samples
This paper presents a novel power spectral density estimation technique for
band-limited, wide-sense stationary signals from sub-Nyquist sampled data. The
technique employs multi-coset sampling and incorporates the advantages of
compressed sensing (CS) when the power spectrum is sparse, but applies to
sparse and nonsparse power spectra alike. The estimates are consistent
piecewise constant approximations whose resolutions (width of the piecewise
constant segments) are controlled by the periodicity of the multi-coset
sampling. We show that compressive estimates exhibit better tradeoffs among the
estimator's resolution, system complexity, and average sampling rate compared
to their noncompressive counterparts. For suitable sampling patterns,
noncompressive estimates are obtained as least squares solutions. Because of
the non-negativity of power spectra, compressive estimates can be computed by
seeking non-negative least squares solutions (provided appropriate sampling
patterns exist) instead of using standard CS recovery algorithms. This
flexibility suggests a reduction in computational overhead for systems
estimating both sparse and nonsparse power spectra because one algorithm can be
used to compute both compressive and noncompressive estimates.Comment: 26 pages, single spaced, 9 figure
Inexact Gradient Projection and Fast Data Driven Compressed Sensing
We study convergence of the iterative projected gradient (IPG) algorithm for
arbitrary (possibly nonconvex) sets and when both the gradient and projection
oracles are computed approximately. We consider different notions of
approximation of which we show that the Progressive Fixed Precision (PFP) and
the -optimal oracles can achieve the same accuracy as for the
exact IPG algorithm. We show that the former scheme is also able to maintain
the (linear) rate of convergence of the exact algorithm, under the same
embedding assumption. In contrast, the -approximate oracle
requires a stronger embedding condition, moderate compression ratios and it
typically slows down the convergence. We apply our results to accelerate
solving a class of data driven compressed sensing problems, where we replace
iterative exhaustive searches over large datasets by fast approximate nearest
neighbour search strategies based on the cover tree data structure. For
datasets with low intrinsic dimensions our proposed algorithm achieves a
complexity logarithmic in terms of the dataset population as opposed to the
linear complexity of a brute force search. By running several numerical
experiments we conclude similar observations as predicted by our theoretical
analysis
Deep Unrolling for Magnetic Resonance Fingerprinting
Magnetic Resonance Fingerprinting (MRF) has emerged as a promising
quantitative MR imaging approach. Deep learning methods have been proposed for
MRF and demonstrated improved performance over classical compressed sensing
algorithms. However many of these end-to-end models are physics-free, while
consistency of the predictions with respect to the physical forward model is
crucial for reliably solving inverse problems. To address this, recently [1]
proposed a proximal gradient descent framework that directly incorporates the
forward acquisition and Bloch dynamic models within an unrolled learning
mechanism. However, [1] only evaluated the unrolled model on synthetic data
using Cartesian sampling trajectories. In this paper, as a complementary to
[1], we investigate other choices of encoders to build the proximal neural
network, and evaluate the deep unrolling algorithm on real accelerated MRF
scans with non-Cartesian k-space sampling trajectories.Comment: Tech report. arXiv admin note: substantial text overlap with
arXiv:2006.1527
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