Tuncer C. Aysal

Broadcast Gossip Algorithms for Consensus

Motivated by applications to wireless sensor, peer-to-peer, and ad hoc networks, we study distributed broadcasting algorithms for exchanging information and computing in an arbitrarily connected network of nodes. Specifically, we study a broadcasting-based gossiping algorithm to compute the (possibly weighted) average of the initial measurements of the nodes at every node in the network. We show that the broadcast gossip algorithm converges almost surely to a consensus. We prove that the random consensus value is, in expectation, the average of initial node measurements and that it can be made arbitrarily close to this value in mean squared error sense, under a balanced connectivity model and by trading off convergence speed with accuracy of the computation. We provide theoretical and numerical results on the mean square error performance, on the convergence rate and study the effect of the ldquomixing parameterrdquo on the convergence rate of the broadcast gossip algorithm. The results indicate that the mean squared error strictly decreases through iterations until the consensus is achieved. Finally, we assess and compare the communication cost of the broadcast gossip algorithm to achieve a given distance to consensus through theoretical and numerical results.

Distributed Average Consensus With Dithered Quantization

In this paper, we develop algorithms for distributed computation of averages of the node data over networks with bandwidth/power constraints or large volumes of data. Distributed averaging algorithms fail to achieve consensus when deterministic uniform quantization is adopted. We propose a distributed algorithm in which the nodes utilize probabilistically quantized information, i.e., dithered quantization, to communicate with each other. The algorithm we develop is a dynamical system that generates sequences achieving a consensus at one of the quantization values almost surely. In addition, we show that the expected value of the consensus is equal to the average of the original sensor data. We derive an upper bound on the mean-square-error performance of the probabilistically quantized distributed averaging (PQDA). Moreover, we show that the convergence of the PQDA is monotonic by studying the evolution of the minimum-length interval containing the node values. We reveal that the length of this interval is a monotonically nonincreasing function with limit zero. We also demonstrate that all the node values, in the worst case, converge to the final two quantization bins at the same rate as standard unquantized consensus. Finally, we report the results of simulations conducted to evaluate the behavior and the effectiveness of the proposed algorithm in various scenarios.

Distributed Average Consensus using Probabilistic Quantization

In this paper, we develop algorithms for distributed computation of averages of the node data over networks with bandwidth/power constraints or large volumes of data. Distributed averaging algorithms fail to achieve consensus when deterministic uniform quantization is adopted. We propose a distributed algorithm in which the nodes utilize probabilistically quantized information to communicate with each other. The algorithm we develop is a dynamical system that generates sequences achieving a consensus, which is one of the quantization values. In addition, we show that the expected value of the consensus is equal to the average of the original sensor data. We report the results of simulations conducted to evaluate the behavior and the effectiveness of the proposed algorithm in various scenarios.

Constrained Decentralized Estimation Over Noisy Channels for Sensor Networks

Decentralized estimation of a noise-corrupted source parameter by a bandwidth-constrained sensor network feeding, through a noisy channel, a fusion center is considered. The sensors, due to bandwidth constraints, provide binary representatives of a noise-corrupted source parameter. Recently, proposed decentralized, distributed estimation, and power scheduling methods do no consider errors occurring during the transmission of binary observations from the sensors to fusion center. In this paper, we extend the decentralized estimation model to the case where imperfect transmission channels are considered. The proposed estimator, which operates on additive channel noise corrupted versions of quantized noisy sensor observations, is approached from maximum likelihood (ML) perspective. The resulting ML estimate is a root, in the region of interest (ROI), of a derivative polynomial function. We analyze the natural logarithm of the polynomial within the ROI showing that the function is log-concave, thereby indicating that numerical methods, such as Newtons algorithm, can be utilized to obtain the optimal solution. Due to complexity and implementation issues associated with the numerical methods, we derive and analyze simpler suboptimal solutions, i.e., the two-stage and mean estimators. The two-stage estimator first estimates the binary observations from noisy fusion center observations utilizing a threshold operation, followed by an estimate of the source parameter. The optimal threshold is the maximum a posteriori (MAP) detector for binary detection and minimizes the probability of binary observation estimation error. Optimal threshold expressions for commonly utilized light-(Gaussian) and heavy-tailed (Cauchy) channel noise models are derived. The mean estimator simply averages the noisy fusion center observations. The output variances of means of the proposed suboptimal estimators are derived. In addition, a computational complexity analysis is presented comparing the proposed ML optimal and suboptimal two-stage and mean estimators. Numerical examples evaluating and comparing the performance of proposed ML, two-stage and mean estimators are also presented.

Robust Sampling and Reconstruction Methods for Sparse Signals in the Presence of Impulsive Noise

Recent results in compressed sensing show that a sparse or compressible signal can be reconstructed from a few incoherent measurements. Since noise is always present in practical data acquisition systems, sensing, and reconstruction methods are developed assuming a Gaussian (light-tailed) model for the corrupting noise. However, when the underlying signal and/or the measurements are corrupted by impulsive noise, commonly employed linear sampling operators, coupled with current reconstruction algorithms, fail to recover a close approximation of the signal. In this paper, we propose robust methods for sampling and reconstructing sparse signals in the presence of impulsive noise. To solve the problem of impulsive noise embedded in the underlying signal prior the measurement process, we propose a robust nonlinear measurement operator based on the weighed myriad estimator. In addition, we introduce a geometric optimization problem based on L 1 minimization employing a Lorentzian norm constraint on the residual error to recover sparse signals from noisy measurements. Analysis of the proposed methods show that in impulsive environments when the noise posses infinite variance we have a finite reconstruction error and furthermore these methods yield successful reconstruction of the desired signal. Simulations demonstrate that the proposed methods significantly outperform commonly employed compressed sensing sampling and reconstruction techniques in impulsive environments, while providing comparable performance in less demanding, light-tailed environments.

Rayleigh-Maximum-Likelihood Filtering for Speckle Reduction of Ultrasound Images

Speckle is a multiplicative noise that degrades ultrasound images. Recent advancements in ultrasound instrumentation and portable ultrasound devices necessitate the need for more robust despeckling techniques, for both routine clinical practice and teleconsultation. Methods previously proposed for speckle reduction suffer from two major limitations: 1) noise attenuation is not sufficient, especially in the smooth and background areas; 2) existing methods do not sufficiently preserve or enhance edges-they only inhibit smoothing near edges. In this paper, we propose a novel technique that is capable of reducing the speckle more effectively than previous methods and jointly enhancing the edge information, rather than just inhibiting smoothing. The proposed method utilizes the Rayleigh distribution to model the speckle and adopts the robust maximum-likelihood estimation approach. The resulting estimator is statistically analyzed through first and second moment derivations. A tuning parameter that naturally evolves in the estimation equation is analyzed, and an adaptive method utilizing the instantaneous coefficient of variation is proposed to adjust this parameter. To further tailor performance, a weighted version of the proposed estimator is introduced to exploit varying statistics of input samples. Finally, the proposed method is evaluated and compared to well-accepted methods through simulations utilizing synthetic and real ultrasound data

Meridian Filtering for Robust Signal Processing

A broad range of statistical processes is characterized by the generalized Gaussian statistics. For instance, the Gaussian and Laplacian probability density functions are special cases of generalized Gaussian statistics. Moreover, the linear and median filtering structures are statistically related to the maximum likelihood estimates of location under Gaussian and Laplacian statistics, respectively. In this paper, we investigate the well-established statistical relationship between Gaussian and Cauchy distributions, showing that the random variable formed as the ratio of two independent Gaussian distributed random variables is Cauchy distributed. We also note that the Cauchy distribution is a member of the generalized Cauchy distribution family. Recently proposed myriad filtering is based on the maximum likelihood estimate of location under Cauchy statistics. An analogous relationship is formed here for the Laplacian statistics, as the ratio of Laplacian statistics yields the distribution referred here to as the Meridian. Interestingly, the Meridian distribution is also a member of the generalized Cauchy family. The maximum likelihood estimate under the obtained statistics is analyzed. Motivated by the maximum likelihood estimate under meridian statistics, meridian filtering is proposed. The analysis presented here indicates that the proposed filtering structure exhibits characteristics more robust than that of median and myriad filtering structures. The statistical and deterministic properties essential to signal processing applications of the meridian filter are given. The meridian filtering structure is extended to admit real-valued weights utilizing the sign coupling approach. Finally, simulations are performed to evaluate and compare the proposed meridian filtering structure performance to those of linear, median, and myriad filtering.

Convergence of Consensus Models With Stochastic Disturbances

We consider consensus algorithms in their most general setting and provide conditions under which such algorithms are guaranteed to converge, almost surely, to a consensus. Let {A(t), B(t)} ∈ RN×N be (possibly) stochastic, nonstationary matrices and {x(t), m(t)} 6 RN×1 be state and perturbation vectors, respectively. For any consensus algorithm of the form x(t + 1) = A(t)x(t) + B(t)m(t), we provide conditions under which consensus is achieved almost surely, i.e., Pr-{limt →∞ x(t) = c1} -1 for some c ∈ R. Moreover, we show that this general result subsumes recently reported results for specific consensus algorithms classes, including sum-preserving, nonsum-preserving, quantized, and noisy gossip algorithms. Also provided are the e-converging time for any such converging iterative algorithm, i.e., the earliest time at which the vector x(t) is ε close to consensus, and sufficient conditions for convergence in expectation to the average of the initial node measurements. Finally, mean square error bounds of any consensus algorithm of the form discussed above are presented.

Quadratic Weighted Median Filters for Edge Enhancement of Noisy Images

Quadratic Volterra filters are effective in image sharpening applications. The linear combination of polynomial terms, however, yields poor performance in noisy environments. Weighted median (WM) filters, in contrast, are well known for their outlier suppression and detail preservation properties. The WM sample selection methodology is naturally extended to the quadratic sample case, yielding a filter structure referred to as quadratic weighted median (QWM) that exploits the higher order statistics of the observed samples while simultaneously being robust to outliers arising in the higher order statistics of environment noise. Through statistical analysis of higher order samples, it is shown that, although the parent Gaussian distribution is light tailed, the higher order terms exhibit heavy-tailed distributions. The optimal combination of terms contributing to a quadratic system, i.e., cross and square, is approached from a maximum likelihood perspective which yields the WM processing of these terms. The proposed QWM filter structure is analyzed through determination of the output variance and breakdown probability. The studies show that the QWM exhibits lower variance and breakdown probability indicating the robustness of the proposed structure. The performance of the QWM filter is tested on constant regions, edges and real images, and compared to its weighted-sum dual, the quadratic Volterra filter. The simulation results show that the proposed method simultaneously suppresses the noise and enhances image details. Compared with the quadratic Volterra sharpener, the QWM filter exhibits superior qualitative and quantitative performance in noisy image sharpening

Blind decentralized estimation for bandwidth constrained wireless sensor networks

Recently proposed decentralized, distributed estimation and power scheduling methods for wireless sensor networks (WSNs) do not consider errors occurring during the transmission of binary observations from the sensors to fusion center. In this letter, we extend the decentralized estimation model to the case in which imperfect transmission channels are considered. The proposed estimators, which operate on additive channel noise corrupted versions of quantized noisy sensor observations, are approached from a maximum likelihood (ML) perspective. Complicating this approach is the fact that the noise distribution is rarely fully known to the fusion center. Here we assume the distribution is known but not the defining parameters, e.g., variance. The resulting incomplete data estimation problem is approached from a expectation-maximization (EM) perspective. The critical initialization and convergence aspects of the EM algorithm are investigated. Furthermore, the estimation of the source parameter is extended to the blind case where both the channel and sensor noise parameters are unknown. Finally, numerical experiments are provided to show the effectiveness of the proposed estimators.