57 research outputs found
Conjugate gradient acceleration of iteratively re-weighted least squares methods
Iteratively Re-weighted Least Squares (IRLS) is a method for solving
minimization problems involving non-quadratic cost functions, perhaps
non-convex and non-smooth, which however can be described as the infimum over a
family of quadratic functions. This transformation suggests an algorithmic
scheme that solves a sequence of quadratic problems to be tackled efficiently
by tools of numerical linear algebra. Its general scope and its usually simple
implementation, transforming the initial non-convex and non-smooth minimization
problem into a more familiar and easily solvable quadratic optimization
problem, make it a versatile algorithm. However, despite its simplicity,
versatility, and elegant analysis, the complexity of IRLS strongly depends on
the way the solution of the successive quadratic optimizations is addressed.
For the important special case of and sparse
recovery problems in signal processing, we investigate theoretically and
numerically how accurately one needs to solve the quadratic problems by means
of the (CG) method in each iteration in order to
guarantee convergence. The use of the CG method may significantly speed-up the
numerical solution of the quadratic subproblems, in particular, when fast
matrix-vector multiplication (exploiting for instance the FFT) is available for
the matrix involved. In addition, we study convergence rates. Our modified IRLS
method outperforms state of the art first order methods such as Iterative Hard
Thresholding (IHT) or Fast Iterative Soft-Thresholding Algorithm (FISTA) in
many situations, especially in large dimensions. Moreover, IRLS is often able
to recover sparse vectors from fewer measurements than required for IHT and
FISTA.Comment: 40 page
Joint Total Variation ESTATICS for Robust Multi-Parameter Mapping
Quantitative magnetic resonance imaging (qMRI) derives tissue-specific
parameters -- such as the apparent transverse relaxation rate R2*, the
longitudinal relaxation rate R1 and the magnetisation transfer saturation --
that can be compared across sites and scanners and carry important information
about the underlying microstructure. The multi-parameter mapping (MPM) protocol
takes advantage of multi-echo acquisitions with variable flip angles to extract
these parameters in a clinically acceptable scan time. In this context,
ESTATICS performs a joint loglinear fit of multiple echo series to extract R2*
and multiple extrapolated intercepts, thereby improving robustness to motion
and decreasing the variance of the estimators. In this paper, we extend this
model in two ways: (1) by introducing a joint total variation (JTV) prior on
the intercepts and decay, and (2) by deriving a nonlinear maximum \emph{a
posteriori} estimate. We evaluated the proposed algorithm by predicting
left-out echoes in a rich single-subject dataset. In this validation, we
outperformed other state-of-the-art methods and additionally showed that the
proposed approach greatly reduces the variance of the estimated maps, without
introducing bias.Comment: 11 pages, 2 figures, 1 table, conference paper, accepted at MICCAI
202
Symmetrization Techniques in Image Deblurring
This paper presents a couple of preconditioning techniques that can be used
to enhance the performance of iterative regularization methods applied to image
deblurring problems with a variety of point spread functions (PSFs) and
boundary conditions. More precisely, we first consider the anti-identity
preconditioner, which symmetrizes the coefficient matrix associated to problems
with zero boundary conditions, allowing the use of MINRES as a regularization
method. When considering more sophisticated boundary conditions and strongly
nonsymmetric PSFs, the anti-identity preconditioner improves the performance of
GMRES. We then consider both stationary and iteration-dependent regularizing
circulant preconditioners that, applied in connection with the anti-identity
matrix and both standard and flexible Krylov subspaces, speed up the
iterations. A theoretical result about the clustering of the eigenvalues of the
preconditioned matrices is proved in a special case. The results of many
numerical experiments are reported to show the effectiveness of the new
preconditioning techniques, including when considering the deblurring of sparse
images
Semi-local scaling exponent estimation with box-penalty constraints and total-variation regularisation
We here establish and exploit the result that 2-D isotropic self-similar fields beget quasi-decorrelated wavelet coefficients and that the resulting localised log sample second moment statistic is asymptotically normal. This leads to the development of a semi-local scaling exponent estimation framework with optimally modified weights. Furthermore, recent interest in penalty methods for least squares problems and generalised Lasso for scaling exponent estimation inspires the simultaneous incorporation of both bounding box constraints and total variation smoothing into an iteratively reweighted least-squares estimator framework. Numerical results on fractional Brownian fields with global and piecewise constant, semi-local Hurst parameters illustrate the benefits of the new estimators
Cygnus A super-resolved via convex optimisation from VLA data
We leverage the Sparsity Averaging Reweighted Analysis (SARA) approach for
interferometric imaging, that is based on convex optimisation, for the
super-resolution of Cyg A from observations at the frequencies 8.422GHz and
6.678GHz with the Karl G. Jansky Very Large Array (VLA). The associated average
sparsity and positivity priors enable image reconstruction beyond instrumental
resolution. An adaptive Preconditioned Primal-Dual algorithmic structure is
developed for imaging in the presence of unknown noise levels and calibration
errors. We demonstrate the superior performance of the algorithm with respect
to the conventional CLEAN-based methods, reflected in super-resolved images
with high fidelity. The high resolution features of the recovered images are
validated by referring to maps of Cyg A at higher frequencies, more precisely
17.324GHz and 14.252GHz. We also confirm the recent discovery of a radio
transient in Cyg A, revealed in the recovered images of the investigated data
sets. Our matlab code is available online on GitHub.Comment: 14 pages, 7 figures (3/7 animated figures), accepted for publication
in MNRA
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