573 research outputs found
Optimal Convergence Rates for Generalized Alternating Projections
Generalized alternating projections is an algorithm that alternates relaxed
projections onto a finite number of sets to find a point in their intersection.
We consider the special case of two linear subspaces, for which the algorithm
reduces to a matrix teration. For convergent matrix iterations, the asymptotic
rate is linear and decided by the magnitude of the subdominant eigenvalue. In
this paper, we show how to select the three algorithm parameters to optimize
this magnitude, and hence the asymptotic convergence rate. The obtained rate
depends on the Friedrichs angle between the subspaces and is considerably
better than known rates for other methods such as alternating projections and
Douglas-Rachford splitting. We also present an adaptive scheme that, online,
estimates the Friedrichs angle and updates the algorithm parameters based on
this estimate. A numerical example is provided that supports our theoretical
claims and shows very good performance for the adaptive method.Comment: 20 pages, extended version of article submitted to CD
Multiplicative Noise Removal Using L1 Fidelity on Frame Coefficients
We address the denoising of images contaminated with multiplicative noise,
e.g. speckle noise. Classical ways to solve such problems are filtering,
statistical (Bayesian) methods, variational methods, and methods that convert
the multiplicative noise into additive noise (using a logarithmic function),
shrinkage of the coefficients of the log-image data in a wavelet basis or in a
frame, and transform back the result using an exponential function. We propose
a method composed of several stages: we use the log-image data and apply a
reasonable under-optimal hard-thresholding on its curvelet transform; then we
apply a variational method where we minimize a specialized criterion composed
of an data-fitting to the thresholded coefficients and a Total
Variation regularization (TV) term in the image domain; the restored image is
an exponential of the obtained minimizer, weighted in a way that the mean of
the original image is preserved. Our restored images combine the advantages of
shrinkage and variational methods and avoid their main drawbacks. For the
minimization stage, we propose a properly adapted fast minimization scheme
based on Douglas-Rachford splitting. The existence of a minimizer of our
specialized criterion being proven, we demonstrate the convergence of the
minimization scheme. The obtained numerical results outperform the main
alternative methods
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