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

Near-Surface Interface Detection for Coal Mining Applications Using Bispectral Features and GPR

The use of ground penetrating radar (GPR) for detecting the presence of near-surface interfaces is a scenario of special interest to the underground coal mining industry. The problem is difficult to solve in practice because the radar echo from the near-surface interface is often dominated by unwanted components such as antenna crosstalk and ringing, ground-bounce effects, clutter, and severe attenuation. These nuisance components are also highly sensitive to subtle variations in ground conditions, rendering the application of standard signal pre-processing techniques such as background subtraction largely ineffective in the unsupervised case. As a solution to this detection problem, we develop a novel pattern recognition-based algorithm which utilizes a neural network to classify features derived from the bispectrum of 1D early time radar data. The binary classifier is used to decide between two key cases, namely whether an interface is within, for example, 5 cm of the surface or not. This go/no-go detection capability is highly valuable for underground coal mining operations, such as longwall mining, where the need to leave a remnant coal section is essential for geological stability. The classifier was trained and tested using real GPR data with ground truth measurements. The real data was acquired from a testbed with coal-clay, coal-shale and shale-clay interfaces, which represents a test mine site. We show that, unlike traditional second order correlation based methods such as matched filtering which can fail even in known conditions, the new method reliably allows the detection of interfaces using GPR to be applied in the near-surface region. In this work, we are not addressing the problem of depth estimation, rather confining ourselves to detecting an interface within a particular depth range

Principles of Image Reconstruction in Interferometry

This book is a collection of 19 articles which reflect the courses given at the Collège de France/Summer school “Reconstruction d'images − Applications astrophysiques“ held in Nice and Fréjus, France, from June 18 to 22, 2012. The articles presented in this volume address emerging concepts and methods that are useful in the complex process of improving our knowledge of the celestial objects, including Earth

Shift-invariant image reconstruction of speckle-degraded using bispectrum estimation

Coherent speckle noise is modeled as a multiplicative noise process that has a negative exponential probability density function. Using a homomorphic transfor mation, this speckle noise is converted to a signal-independent, additive process. The speckled images are randomly jittered from frame-to-frame against a uniform background to simulate image motion and/or platform jitter. Multiple images are logarithmically transformed and ensemble averaged in the bispectral domain. The bispectrum ignores this image motion so no blurring results from the ensemble averaging. Object Fourier magnitude and phase information are also retained in the bispectrum so that the resultant image can be uniquely reconstructed. This value is then exponentiated to complete the image reconstruc tion process. Since speckle masks the resolution of details in the noisy image and effectively destroys the object structure within the image, it is seen that image reconstruction using bispectrum estimation results in images that regain their object structure. Both one-dimensional and two-dimensional images were tested using separate bispectral signal reconstruction algorithms for each

Bispectral reconstruction of speckle-degraded images

The bispectrum of a signal has useful properties such as being zero for a Gaussian random process, retaining both phase and magnitude information of the Fourier transform of a signal, and being insensitive to linear motion. It has found applications in a wide variety of fields. The use of these properties for reducing speckle in coherent imaging systems was investigated. It was found that the bispectrum could be used to restore speckle-degraded images. Coherent speckle noise is modeled as a multiplicative noise process. By using a logarithmic transformation, this speckle noise is converted to a signal independent, additive process which is close to Gaussian when an integrating aperture is used. Bispectral reconstruction of speckle-degraded images is performed on such logarithmically transformed images when we have independent multiple snapshots

Quantum origin of the primordial fluctuation spectrum and its statistics

The usual account for the origin of cosmic structure during inflation is not fully satisfactory, as it lacks a physical mechanism capable of generating the inhomogeneity and anisotropy of our Universe, from an exactly homogeneous and isotropic initial state associated with the early inflationary regime. The proposal in [A. Perez, H. Sahlmann, and D. Sudarsky, Classical Quantum Gravity, 23, 2317, (2006)] considers the spontaneous dynamical collapse of the wave function, as a possible answer to that problem. In this work, we review briefly the difficulties facing the standard approach, as well as the answers provided by the above proposal and explore their relevance to the investigations concerning the characterization of the primordial spectrum and other statistical aspects of the cosmic microwave background and large-scale matter distribution. We will see that the new approach leads to novel ways of considering some of the relevant questions, and, in particular, to distinct characterizations of the non-Gaussianities that might have left imprints on the available data.Comment: 27 pages. Revision to match the published versio

Unveiling the Dynamics of the Universe

We explore the dynamics and evolution of the Universe at early and late times, focusing on both dark energy and extended gravity models and their astrophysical and cosmological consequences. Modified theories of gravity not only provide an alternative explanation for the recent expansion history of the universe, but they also offer a paradigm fundamentally distinct from the simplest dark energy models of cosmic acceleration. In this review, we perform a detailed theoretical and phenomenological analysis of different modified gravity models and investigate their consistency. We also consider the cosmological implications of well motivated physical models of the early universe with a particular emphasis on inflation and topological defects. Astrophysical and cosmological tests over a wide range of scales, from the solar system to the observable horizon, severely restrict the allowed models of the Universe. Here, we review several observational probes -- including gravitational lensing, galaxy clusters, cosmic microwave background temperature and polarization, supernova and baryon acoustic oscillations measurements -- and their relevance in constraining our cosmological description of the Universe.Comment: 94 pages, 14 figures. Review paper accepted for publication in a Special Issue of Symmetry. "Symmetry: Feature Papers 2016". V2: Matches published version, now 79 pages (new format