43 research outputs found
Projections Onto Convex Sets (POCS) Based Optimization by Lifting
Two new optimization techniques based on projections onto convex space (POCS)
framework for solving convex and some non-convex optimization problems are
presented. The dimension of the minimization problem is lifted by one and sets
corresponding to the cost function are defined. If the cost function is a
convex function in R^N the corresponding set is a convex set in R^(N+1). The
iterative optimization approach starts with an arbitrary initial estimate in
R^(N+1) and an orthogonal projection is performed onto one of the sets in a
sequential manner at each step of the optimization problem. The method provides
globally optimal solutions in total-variation, filtered variation, l1, and
entropic cost functions. It is also experimentally observed that cost functions
based on lp, p<1 can be handled by using the supporting hyperplane concept
Focal-plane wavefront sensing with high-order adaptive optics systems
We investigate methods to calibrate the non-common path aberrations at an
adaptive optics system having a wavefront-correcting device working at an
extremely high resolution (larger than 150x150). We use focal-plane images
collected successively, the corresponding phase-diversity information and
numerically efficient algorithms to calculate the required wavefront updates.
The wavefront correction is applied iteratively until the algorithms converge.
Different approaches are studied. In addition of the standard Gerchberg-Saxton
algorithm, we test the extension of the Fast & Furious algorithm that uses
three images and creates an estimate of the pupil amplitudes. We also test
recently proposed phase-retrieval methods based on convex optimisation. The
results indicate that in the framework we consider, the calibration task is
easiest with algorithms similar to the Fast & Furious.Comment: 11 pages, 7 figures, published in SPIE proceeding
Phase and TV Based Convex Sets for Blind Deconvolution of Microscopic Images
In this article, two closed and convex sets for blind deconvolution problem
are proposed. Most blurring functions in microscopy are symmetric with respect
to the origin. Therefore, they do not modify the phase of the Fourier transform
(FT) of the original image. As a result blurred image and the original image
have the same FT phase. Therefore, the set of images with a prescribed FT phase
can be used as a constraint set in blind deconvolution problems. Another convex
set that can be used during the image reconstruction process is the epigraph
set of Total Variation (TV) function. This set does not need a prescribed upper
bound on the total variation of the image. The upper bound is automatically
adjusted according to the current image of the restoration process. Both of
these two closed and convex sets can be used as a part of any blind
deconvolution algorithm. Simulation examples are presented.Comment: Submitted to IEEE Selected Topics in Signal Processin
Stable optimizationless recovery from phaseless linear measurements
We address the problem of recovering an n-vector from m linear measurements
lacking sign or phase information. We show that lifting and semidefinite
relaxation suffice by themselves for stable recovery in the setting of m = O(n
log n) random sensing vectors, with high probability. The recovery method is
optimizationless in the sense that trace minimization in the PhaseLift
procedure is unnecessary. That is, PhaseLift reduces to a feasibility problem.
The optimizationless perspective allows for a Douglas-Rachford numerical
algorithm that is unavailable for PhaseLift. This method exhibits linear
convergence with a favorable convergence rate and without any parameter tuning
Denosing Using Wavelets and Projections onto the L1-Ball
Both wavelet denoising and denosing methods using the concept of sparsity are
based on soft-thresholding. In sparsity based denoising methods, it is assumed
that the original signal is sparse in some transform domains such as the
wavelet domain and the wavelet subsignals of the noisy signal are projected
onto L1-balls to reduce noise. In this lecture note, it is shown that the size
of the L1-ball or equivalently the soft threshold value can be determined using
linear algebra. The key step is an orthogonal projection onto the epigraph set
of the L1-norm cost function.Comment: Submitted to Signal Processing Magazin
Denoising using projections onto the epigraph set of convex cost functions
A new denoising algorithm based on orthogonal projections onto the epigraph set of a convex cost function is presented. In this algorithm, the dimension of the minimization problem is lifted by one and feasibility sets corresponding to the cost function using the epigraph concept are defined. As the utilized cost function is a convex function in RN, the corresponding epigraph set is also a convex set in RN+1. The denoising algorithm starts with an arbitrary initial estimate in RN+1. At each step of the iterative denoising, an orthogonal projection is performed onto one of the constraint sets associated with the cost function in a sequential manner. The method provides globally optimal solutions for total-variation, ℓ1, ℓ2, and entropic cost functions.1 © 2014 IEEE