Improved techniques for bispectral reconstruction of signals

Higher order cumulants and spectra have found a variety of uses in many areas of digital signal processing. The third order spectrum, or bispectrum, is of specific interest to researchers because of some of its properties. The Bispectrum is defined as the fourier transform of the third order cumulant se quence for stochastic processes, and as a triple product of fourier transforms for deterministic signals. In the past, bispectral analysis has been used in applications such as identification of linear filters, quadratic phase coupling problems and detection of deviations from normality. This work is aimed at developing techniques for reconstructing deterministic signals in noise us ing the bispectrum. The bispectrum is zero for many noise processes, and is insensitive to linear phase shifts. The main motivation of this work is to exploit these properties of bispectrum that are potentially useful in signal re covery. The existing bispectral recovery techniques are discussed in the signal reconstruction frame work and their main limitation in handling noisy de terministic signals is brought out. New robust reconstruction procedures are provided in order to use bispectrum in such cases. The developed algorithms are tested over a range of simulated applications to bring out their robustness in handling both deterministic and stochastic signals. The new techniques are compared with existing bispectral methods in various problems. This thesis also discusses some of the tradeoffs involved in using bispectrum based reconstruction approaches against non-bispectral methods

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

Least-squares Fourier phase estimation from the modulo 2Pi bispectrum phase

The recovery of Fourier phases from measurements of the bispectrum occupies a vital role in many astronomical speckle imaging schemes. In arecent paper [J. Opt. Soc. Am. A 7, 14 (1990)] it was suggested that a least-squares solution to this problem must fail if the bispectrum phase is known only modulo 2π. Here an alternative nonlinear least-squares algorithm is presented that differs from the linear method discussed in the aforementioned paper and that permits the fitting of Fourier phases directly to modulo 2π measurements of the bispectrum phase, thus eliminating any need for phase unwrapping. Numerical simulations of this method confirm that it is reliable and robust in the presence of noise and verify its enhanced performance when compared with a linear least-squares method that includes the unwrapping of the bispectral phase before Fourier phase retrieval

Image restoration using HOS and the Radon transform

The authors propose the use of higher-order statistics (HOS) to study the problem of image restoration. They consider images degraded by linear or zero phase blurring point spread functions (PSF) and additive Gaussian noise. The complexity associated with the combination of two-dimensional signal processing and higher-order statistics is reduced by means of the Radon transform. The projection at each angle is an one-dimensional signal that can be processed by any existing 1-D higher-order statistics-based method. They apply two methods that have proven to attain good one-dimensional signal reconstruction, especially in the presence of noise. After the ideal projections have been estimated, the inverse Radon transform gives the restored image. Simulation results are provided.Peer ReviewedPostprint (published version

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

Heterogeneous multireference alignment: a single pass approach

Multireference alignment (MRA) is the problem of estimating a signal from many noisy and cyclically shifted copies of itself. In this paper, we consider an extension called heterogeneous MRA, where $K$ signals must be estimated, and each observation comes from one of those signals, unknown to us. This is a simplified model for the heterogeneity problem notably arising in cryo-electron microscopy. We propose an algorithm which estimates the $K$ signals without estimating either the shifts or the classes of the observations. It requires only one pass over the data and is based on low-order moments that are invariant under cyclic shifts. Given sufficiently many measurements, one can estimate these invariant features averaged over the $K$ signals. We then design a smooth, non-convex optimization problem to compute a set of signals which are consistent with the estimated averaged features. We find that, in many cases, the proposed approach estimates the set of signals accurately despite non-convexity, and conjecture the number of signals $K$ that can be resolved as a function of the signal length $L$ is on the order of $\sqrt{L}$.Comment: 6 pages, 3 figure