703 research outputs found
Bispectrum Inversion with Application to Multireference Alignment
We consider the problem of estimating a signal from noisy
circularly-translated versions of itself, called multireference alignment
(MRA). One natural approach to MRA could be to estimate the shifts of the
observations first, and infer the signal by aligning and averaging the data. In
contrast, we consider a method based on estimating the signal directly, using
features of the signal that are invariant under translations. Specifically, we
estimate the power spectrum and the bispectrum of the signal from the
observations. Under mild assumptions, these invariant features contain enough
information to infer the signal. In particular, the bispectrum can be used to
estimate the Fourier phases. To this end, we propose and analyze a few
algorithms. Our main methods consist of non-convex optimization over the smooth
manifold of phases. Empirically, in the absence of noise, these non-convex
algorithms appear to converge to the target signal with random initialization.
The algorithms are also robust to noise. We then suggest three additional
methods. These methods are based on frequency marching, semidefinite relaxation
and integer programming. The first two methods provably recover the phases
exactly in the absence of noise. In the high noise level regime, the invariant
features approach for MRA results in stable estimation if the number of
measurements scales like the cube of the noise variance, which is the
information-theoretic rate. Additionally, it requires only one pass over the
data which is important at low signal-to-noise ratio when the number of
observations must be large
Signal-to-noise ratio of the bispectral analysis of speckle interferometry
Monte Carlo simulations of an atmospheric phase screen, based on a Kolmogorov spectrum of phase fluctuations,
were performed. Speckle patterns produced from the phase screens were used to derive statistical properties of
power spectra and bispectra of speckle interferograms. We present the bispectral modulation transfer function and
its signal-to-noise ratio at high light levels. The results confirm the validity of a heuristic treatment based on an
interferometric picture of speckle pattern formation in deriving the attenuation factor and the signal-to-noise ratio
of the bispectral modulation transfer function in the mid-spatial-frequency range. The derived modulation
transfer function is also interpreted in terms of the signal-to-noise ratio at low light levels. A general expression of
the signal-to-noise ratio of the bispectrum is derived as a function of the transfer functions of the telescope, the
number of speckles, and the mean photon counts in the mid-spatial-frequency range
Higher-order Statistics of Weak Lensing Shear and Flexion
Owing to their more extensive sky coverage and tighter control on systematic
errors, future deep weak lensing surveys should provide a better statistical
picture of the dark matter clustering beyond the level of the power spectrum.
In this context, the study of non-Gaussianity induced by gravity can help
tighten constraints on the background cosmology by breaking parameter
degeneracies, as well as throwing light on the nature of dark matter, dark
energy or alternative gravity theories. Analysis of the shear or flexion
properties of such maps is more complicated than the simpler case of the
convergence due to the spinorial nature of the fields involved. Here we develop
analytical tools for the study of higher-order statistics such as the
bispectrum (or trispectrum) directly using such maps at different source
redshift. The statistics we introduce can be constructed from cumulants of the
shear or flexions, involving the cross-correlation of squared and cubic maps at
different redshifts. Typically, the low signal-to-noise ratio prevents recovery
of the bispectrum or trispectrum mode by mode. We define power spectra
associated with each multi- spectra which compresses some of the available
information of higher order multispectra. We show how these can be recovered
from a noisy observational data even in the presence of arbitrary mask, which
introduces mixing between Electric (E-type) and Magnetic (B-type) polarization,
in an unbiased way. We also introduce higher order cross-correlators which can
cross-correlate lensing shear with different tracers of large scale structures.Comment: 16 pages, 2 figure
Improving fast generation of halo catalogs with higher-order Lagrangian perturbation theory
We present the latest version of Pinocchio, a code that generates catalogues
of DM haloes in an approximate but fast way with respect to an N-body
simulation. This code version extends the computation of particle and halo
displacements up to 3rd-order Lagrangian Perturbation Theory (LPT), in contrast
with previous versions that used Zeldovich approximation (ZA).
We run Pinocchio on the same initial configuration of a reference N-body
simulation, so that the comparison extends to the object-by-object level. We
consider haloes at redshifts 0 and 1, using different LPT orders either for
halo construction - where displacements are needed to decide particle accretion
onto a halo or halo merging - or to compute halo final positions.
We compare the clustering properties of Pinocchio haloes with those from the
simulation by computing the power spectrum and 2-point correlation function
(2PCF) in real and redshift space (monopole and quadrupole), the bispectrum and
the phase difference of halo distributions. We find that 2LPT and 3LPT give
noticeable improvement. 3LPT provides the best agreement with N-body when it is
used to displace haloes, while 2LPT gives better results for constructing
haloes. At the highest orders, linear bias is typically recovered at a few per
cent level.
In Fourier space and using 3LPT for halo displacements, the halo power
spectrum is recovered to within 10 per cent up to Mpc. The
results presented in this paper have interesting implications for the
generation of large ensemble of mock surveys aimed at accurately compute
covariance matrices for clustering statistics.Comment: 20 pages, 20 figures, submitted to MNRA
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