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

Estimation in the group action channel

We analyze the problem of estimating a signal from multiple measurements on a \mbox{group action channel} that linearly transforms a signal by a random group action followed by a fixed projection and additive Gaussian noise. This channel is motivated by applications such as multi-reference alignment and cryo-electron microscopy. We focus on the large noise regime prevalent in these applications. We give a lower bound on the mean square error (MSE) of any asymptotically unbiased estimator of the signal's orbit in terms of the signal's moment tensors, which implies that the MSE is bounded away from 0 when $N/\sigma^{2d}$ is bounded from above, where $N$ is the number of observations, $\sigma$ is the noise standard deviation, and $d$ is the so-called \mbox{moment order cutoff}. In contrast, the maximum likelihood estimator is shown to be consistent if $N /\sigma^{2d}$ diverges.Comment: 5 pages, conferenc

Multireference Alignment using Semidefinite Programming

The multireference alignment problem consists of estimating a signal from multiple noisy shifted observations. Inspired by existing Unique-Games approximation algorithms, we provide a semidefinite program (SDP) based relaxation which approximates the maximum likelihood estimator (MLE) for the multireference alignment problem. Although we show that the MLE problem is Unique-Games hard to approximate within any constant, we observe that our poly-time approximation algorithm for the MLE appears to perform quite well in typical instances, outperforming existing methods. In an attempt to explain this behavior we provide stability guarantees for our SDP under a random noise model on the observations. This case is more challenging to analyze than traditional semi-random instances of Unique-Games: the noise model is on vertices of a graph and translates into dependent noise on the edges. Interestingly, we show that if certain positivity constraints in the SDP are dropped, its solution becomes equivalent to performing phase correlation, a popular method used for pairwise alignment in imaging applications. Finally, we show how symmetry reduction techniques from matrix representation theory can simplify the analysis and computation of the SDP, greatly decreasing its computational cost

Doppler radar monitoring of lava dome processes at Merapi Volcano, Indonesia

Merapi volcano in Central Java, Indonesia, is considered one of the most dangerous volcanoes worldwide. Due to the high viscosity of its magma, the lava emerging at the top the volcano cannot flow silently down the flanks of the volcano but builds a lava dome. An indicator for the stability of the lava dome are rockfalls and block and ash flows, which are caused by local instabilities at the dome. When the lava dome reaches a critical size, it collapses. This results in dangerous block and ash flows, which can reach several kilometers into the proximity of the volcano. In the past rockfall and block and ash flow activity has been observed visually or by seismic networks. However, visual observations are often impossible due to bad visibility conditions and until now seismic measurements allow only few insights into the dynamic processes that are involved in instability events, i.e. events of material breaks off the lava dome. In order to enhance monitoring of lava dome activity, a first prototype Doppler radar system has been installed at the western of the Merapi in October 2001. This system consists of a frequency modulated continuous wave (FMCW) 24GHz Doppler radar. The Doppler spectra recorded by the system give a relative measure of the amount of material moving through the beam as well as information about its velocities. Because the radar system is insensitive for clouds, the system provides first continuous "quasi-visual" observations of dome instabilities...thesi

Autonomous Wireless Radar Sensor Mote for Target Material Classification

An autonomous wireless sensor network consisting of different types of sensor modalities is a topic of intense research due to its versatility and portability.These types of autonomous sensor networks commonly include passive sensor nodes such as infrared,acoustic,seismic and magnetic.However,fusion of another active sensor such as Doppler radar in the integrated sensor network may offer powerful capabilities for many different sensing and classification tasks.In this work,we demonstrate the design and implementation of an autonomous wireless sensor network integrating a Doppler sensor into wireless sensor node with commercial off the shelf components.We also investigate the effect of different types of target materials on return radar signal as one of the applications of the newly designed radar-mote network.Usually type of materials can affect the amount of energy reflected back to the source of an electromagnetic wave.We obtain mathematical and simulation models for the reflectivity of different homogeneous non-conducting materials and study the effect of such reflectivity on different types of targets.We validate our simulation results on effect of reflectivity on different types of targets using real toy experiment data collected through our autonomous radar-mote sensor network

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

Spectral Methods from Tensor Networks

A tensor network is a diagram that specifies a way to "multiply" a collection of tensors together to produce another tensor (or matrix). Many existing algorithms for tensor problems (such as tensor decomposition and tensor PCA), although they are not presented this way, can be viewed as spectral methods on matrices built from simple tensor networks. In this work we leverage the full power of this abstraction to design new algorithms for certain continuous tensor decomposition problems. An important and challenging family of tensor problems comes from orbit recovery, a class of inference problems involving group actions (inspired by applications such as cryo-electron microscopy). Orbit recovery problems over finite groups can often be solved via standard tensor methods. However, for infinite groups, no general algorithms are known. We give a new spectral algorithm based on tensor networks for one such problem: continuous multi-reference alignment over the infinite group SO(2). Our algorithm extends to the more general heterogeneous case.Comment: 30 pages, 8 figure

Signal processing algorithms for digital hearing aids

Hearing loss is a problem that severely affects the speech communication and disqualify most hearing-impaired people from holding a normal life. Although the vast majority of hearing loss cases could be corrected by using hearing aids, however, only a scarce of hearing-impaired people who could be benefited from hearing aids purchase one. This irregular use of hearing aids arises from the existence of a problem that, to date, has not been solved effectively and comfortably: the automatic adaptation of the hearing aid to the changing acoustic environment that surrounds its user. There are two approaches aiming to comply with it. On the one hand, the "manual" approach, in which the user has to identify the acoustic situation and choose the adequate amplification program has been found to be very uncomfortable. The second approach requires to include an automatic program selection within the hearing aid. This latter approach is deemed very useful by most hearing aid users, even if its performance is not completely perfect. Although the necessity of the aforementioned sound classification system seems to be clear, its implementation is a very difficult matter. The development of an automatic sound classification system in a digital hearing aid is a challenging goal because of the inherent limitations of the Digital Signal Processor (DSP) the hearing aid is based on. The underlying reason is that most digital hearing aids have very strong constraints in terms of computational capacity, memory and battery, which seriously limit the implementation of advanced algorithms in them. With this in mind, this thesis focuses on the design and implementation of a prototype for a digital hearing aid able to automatically classify the acoustic environments hearing aid users daily face on and select the amplification program that is best adapted to such environment aiming at enhancing the speech intelligibility perceived by the user. The most important contribution of this thesis is the implementation of a prototype for a digital hearing aid that automatically classifies the acoustic environment surrounding its user and selects the most appropriate amplification program for such environment, aiming at enhancing the sound quality perceived by the user. The battery life of this hearing aid is 140 hours, which has been found to be very similar to that of hearing aids in the market, and what is of key importance, there is still about 30% of the DSP resources available for implementing other algorithms