9 research outputs found
Fast Fourier Optimization: Sparsity Matters
Many interesting and fundamentally practical optimization problems, ranging
from optics, to signal processing, to radar and acoustics, involve constraints
on the Fourier transform of a function. It is well-known that the {\em fast
Fourier transform} (fft) is a recursive algorithm that can dramatically improve
the efficiency for computing the discrete Fourier transform. However, because
it is recursive, it is difficult to embed into a linear optimization problem.
In this paper, we explain the main idea behind the fast Fourier transform and
show how to adapt it in such a manner as to make it encodable as constraints in
an optimization problem. We demonstrate a real-world problem from the field of
high-contrast imaging. On this problem, dramatic improvements are translated to
an ability to solve problems with a much finer grid of discretized points. As
we shall show, in general, the "fast Fourier" version of the optimization
constraints produces a larger but sparser constraint matrix and therefore one
can think of the fast Fourier transform as a method of sparsifying the
constraints in an optimization problem, which is usually a good thing.Comment: 16 pages, 8 figure
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Revisiting compressed sensing: exploiting the efficiency of simplex and sparsification methods
We propose two approaches to solve large-scale compressed sensing problems. The first approach uses the parametric simplex method to recover very sparse signals by taking a small number of simplex pivots, while the second approach reformulates the problem using Kronecker products to achieve faster computation via a sparser problem formulation. In particular, we focus on the computational aspects of these methods in compressed sensing. For the first approach, if the true signal is very sparse and we initialize our solution to be the zero vector, then a customized parametric simplex method usually takes a small number of iterations to converge. Our numerical studies show that this approach is 10 times faster than state-of-the-art methods for recovering very sparse signals. The second approach can be used when the sensing matrix is the Kronecker product of two smaller matrices. We show that the best-known sufficient condition for the Kronecker compressed sensing (KCS) strategy to obtain a perfect recovery is more restrictive than the corresponding condition if using the first approach. However, KCS can be formulated as a linear program with a very sparse constraint matrix, whereas the first approach involves a completely dense constraint matrix. Hence, algorithms that benefit from sparse problem representation, such as interior point methods (IPMs), are expected to have computational advantages for the KCS problem. We numerically demonstrate that KCS combined with IPMs is up to 10 times faster than vanilla IPMs and state-of-the-art methods such as â„“[subscript 1]_â„“[subscript s] and Mirror Prox regardless of the sparsity level or problem size.National Science Foundation (U.S.) (Grant DMS-1005539
Shaped Pupil Lyot Coronagraphs: High-Contrast Solutions for Restricted Focal Planes
Coronagraphs of the apodized pupil and shaped pupil varieties use the
Fraunhofer diffraction properties of amplitude masks to create regions of high
contrast in the vicinity of a target star. Here we present a hybrid coronagraph
architecture in which a binary, hard-edged shaped pupil mask replaces the gray,
smooth apodizer of the apodized pupil Lyot coronagraph (APLC). For any contrast
and bandwidth goal in this configuration, as long as the prescribed region of
contrast is restricted to a finite area in the image, a shaped pupil is the
apodizer with the highest transmission. We relate the starlight cancellation
mechanism to that of the conventional APLC. We introduce a new class of
solutions in which the amplitude profile of the Lyot stop, instead of being
fixed as a padded replica of the telescope aperture, is jointly optimized with
the apodizer. Finally, we describe shaped pupil Lyot coronagraph (SPLC) designs
for the baseline architecture of the Wide-Field Infrared Survey
Telescope-Astrophysics Focused Telescope Assets (WFIRST-AFTA) coronagraph.
These SPLCs help to enable two scientific objectives of the WFIRST-AFTA
mission: (1) broadband spectroscopy to characterize exoplanet atmospheres in
reflected starlight and (2) debris disk imaging.Comment: 41 pages, 15 figures; published in the JATIS special section on
WFIRST-AFTA coronagraph
Optimal pupil apodizations for arbitrary apertures
We present here fully optimized two-dimensional pupil apodizations for which
no specific geometric constraints are put on the pupil plane apodization, apart
from the shape of the aperture itself. Masks for circular and segmented
apertures are displayed, with and without central obstruction and spiders.
Examples of optimal masks are shown for Subaru, SPICA and JWST. Several
high-contrast regions are considered with different sizes, positions, shapes
and contrasts. It is interesting to note that all the masks that result from
these optimizations tend to have a binary transmission profile.Comment: 16 pages, 10 figure
Fast linearized coronagraph optimizer (FALCO) I: a software toolbox for rapid coronagraphic design and wavefront correction
The Fast Linearized Coronagraph Optimizer (FALCO) is an open-source toolbox of routines for coronagraphic focal plane wavefront correction. The goal of FALCO is to provide a free, modular framework for the simulation or testbed operation of several common types of coronagraphs. FALCO includes routines for pair-wise probing estimation of the complex electric field and Electric Field Conjugation (EFC) control, and we ask the community to contribute other wavefront correction algorithms. FALCO utilizes and builds upon PROPER, an established optical propagation library. The key innovation in FALCO is the rapid computation of the linearized response matrix for each deformable mirror (DM), which facilitates re-linearization after each control step for faster DM-integrated coronagraph design and wavefront correction experiments. FALCO is freely available as source code in MATLAB at github.com/ajeldorado/falco-matlab and will be available later this year in Python 3 at github.com/ajeldorado/falco-python
Fast linearized coronagraph optimizer (FALCO) I: a software toolbox for rapid coronagraphic design and wavefront correction
The Fast Linearized Coronagraph Optimizer (FALCO) is an open-source toolbox of routines for coronagraphic focal plane wavefront correction. The goal of FALCO is to provide a free, modular framework for the simulation or testbed operation of several common types of coronagraphs. FALCO includes routines for pair-wise probing estimation of the complex electric field and Electric Field Conjugation (EFC) control, and we ask the community to contribute other wavefront correction algorithms. FALCO utilizes and builds upon PROPER, an established optical propagation library. The key innovation in FALCO is the rapid computation of the linearized response matrix for each deformable mirror (DM), which facilitates re-linearization after each control step for faster DM-integrated coronagraph design and wavefront correction experiments. FALCO is freely available as source code in MATLAB at github.com/ajeldorado/falco-matlab and will be available later this year in Python 3 at github.com/ajeldorado/falco-python
Design, pointing control, and on-sky performance of the mid-infrared vortex coronagraph for the VLT/NEAR experiment
Vortex coronagraphs have been shown to be a promising avenue for high- contrast imaging in the close-in environment of stars at thermal infrared (IR) wavelengths. They are included in the baseline design of the mid-infrared extremely large telescope imager and spectrograph. To ensure good performance of these coronagraphs, a precise control of the centering of the star image in real time is needed. We previously developed and validated the quadrant analysis of coronagraphic images for tip-tilt sensing estimator (QACITS) pointing estimator to address this issue. While this approach is not wavelength-dependent in theory, it was never implemented for mid-IR observations, which leads to specific challenges and limitations. Here, we present the design of the mid-IR vortex coronagraph for the "new Earths in the α Cen Region (NEAR) experiment with the Very Large Telescope (VLT)/Very Large Telescope imager and spectrometer for the mid-infrared (VISIR) instrument and assess the performance of the QACITS estimator for the centering control of the star image onto the vortex coronagraph. We use simulated data and on-sky data obtained with VLT/VISIR, which was recently upgraded for observations assisted by adaptive optics in the context of the NEAR experiment. We demonstrate that the QACITS-based correction loop is able to control the centering of the star image onto the NEAR vortex coronagraph with a stability down to 0.015 λ / D rms over 4 h in good conditions. These results show that QACITS is a robust approach for precisely controlling in real time the centering of vortex coronagraphs for mid-IR observations.Peer reviewe