Linear Fusion Rule Research Articles

The Multi-Focus-Image-Fusion Method Based on Convolutional Neural Network and Sparse Representation.

Multi-focus-image-fusion is a crucial embranchment of image processing. Many methods have been developed from different perspectives to solve this problem. Among them, the sparse representation (SR)-based and convolutional neural network (CNN)-based fusion methods have been widely used. Fusing the source image patches, the SR-based model is essentially a local method with a nonlinear fusion rule. On the other hand, the direct mapping between the source images follows the decision map which is learned via CNN. The fusion is a global one with a linear fusion rule. Combining the advantages of the above two methods, a novel fusion method that applies CNN to assist SR is proposed for the purpose of gaining a fused image with more precise and abundant information. In the proposed method, source image patches were fused based on SR and the new weight obtained by CNN. Experimental results demonstrate that the proposed method clearly outperforms existing state-of-the-art methods in addition to SR and CNN in terms of both visual perception and objective evaluation metrics, and the computational complexity is greatly reduced. Experimental results demonstrate that the proposed method not only clearly outperforms the SR and CNN methods in terms of visual perception and objective evaluation indicators, but is also significantly better than other state-of-the-art methods since our computational complexity is greatly reduced.

Cramer-Rao Bounds of Localization Estimation for Integrated Radar and Communication System

Instead of only considering the radar estimation error in the traditional radar system (TRS), for the integrated radar and communication system (IRCS), we investigate the Cramer-Rao bound (CRB) of the localization estimation, which is influenced by both radar estimation error and communication transmission error. The functions of radar and communication are operated simultaneously by embedding the communication symbols into the multicarrier radar waveforms. Firstly, we derive the CRB of time/direction of arrival (TOA/DOA) estimation. To minimize the estimation error, we maximize the signal-to-interference-plus-noise ratio (SINR) of radar by iteratively optimizing the radar transmit and receive beamformers (with the constraint of available transmit power). Then, the CRB of localization estimation is derived using hybrid TOA/DOA measurement. The local CRBs from different IRCSs are fused according to the linear fusion rule at the fusion center (FC). Finally, numerical results demonstrate that the additional estimation errors for the IRCS are mainly determined by the channel conditions of communication and available transmit power; the estimation accuracy for both IRCS and TRS can be improved through the iterative transmit and receive beamforming (ITRB) technique.

Fusion Rules for Distributed Detection in Clustered Wireless Sensor Networks With Imperfect Channels

In this paper, we investigate fusion rules for distributed detection in large random clustered wireless sensor networks with a three-tier hierarchy; the sensor nodes (SNs), the cluster heads (CHs), and the fusion center (FC). The CHs collect the SNs’ local decisions and relay them to the FC, which then fuses them to reach the ultimate decision. The SN–CH and the CH–FC channels suffer from additive white Gaussian noise. In this context, we derive the optimal log-likelihood ratio (LLR) fusion rule, which turns out to be intractable. So, we develop a sub-optimal linear fusion rule (LFR) that weighs the cluster's data according to both its local detection performance and the quality of the communication channels. In order to implement it, we propose an approximate maximum likelihood based LFR (LFR-aML), which estimates the required parameters for the LFR. We also derive Gaussian-tail upper bounds for the detection and false alarms probabilities for the LFR. Furthermore, an optimal CH transmission power allocation strategy is developed by solving the Karush–Kuhn–Tucker conditions for the related optimization problem. Extensive simulations show that the LFR attains a detection performance near to that of the optimal LLR and confirms the validity of the proposed upper bounds. Moreover, when compared to equal power allocation, simulations show that our proposed power allocation strategy achieves a significant power saving at the expense of a small reduction in the detection performance.

On Power Allocation for Distributed Detection With Correlated Observations and Linear Fusion

We consider a binary hypothesis testing problem in a wireless sensor network, where a fusion center (FC) makes a final decision about the underlying hypothesis. We assume under hypothesis ${\mathcal H}_0$ , sensors’ observations are uncorrelated and identically distributed Gaussian, however, under ${\mathcal H}_1$ they are correlated and nonidentically distributed Gaussian and sensors are unaware of this correlation when making decisions. Sensors send their local binary decisions over power constrained fading channels. We consider both parallel access channel (PAC) and multiple access channel (MAC) models. To obtain the detection statistic in PAC, we assume that the FC utilizes a linear fusion rule, which linearly combines the signals received from all sensors, while in MAC, the signal received at the FC is naturally the coherent sum of the transmitted signals. For both PAC and MAC, we derive modified deflection coefficient (MDC) of the detection statistic at the FC with coherent reception. Choosing MDC as the detection performance metric, we formulate several constrained transmit power optimization problems. In these problems, MDC is the objective function to be maximized and there are three different sets of transmit power constraints: total power constraint, individual and total power constraints, and individual power constraints only. We refer to the solutions of these constrained optimization problems as MDC-based transmit power allocation . When analytical solutions to these constrained optimization problems are elusive, we discuss how these problems can be converted to convex ones. Our results show that, compared with equal power allocation, detection performance improvement provided by our proposed MDC-based power allocation is more significant in MAC. We quantify the improvement in terms of several factors, including the degree of correlation among sensors’ observations, reliability of local decisions (local detection performance indices), communication channel properties, and the type of transmit power constraint. We also study how the power allocation among sensors varies as these factors change.

Efficient linear fusion of partial estimators

Many signal processing applications require performing statistical inference on large datasets, where computational and/or memory restrictions become an issue. In this big data setting, computing an exact global centralized estimator is often either unfeasible or impractical. Hence, several authors have considered distributed inference approaches, where the data are divided among multiple workers (cores, machines or a combination of both). The computations are then performed in parallel and the resulting partial estimators are finally combined to approximate the intractable global estimator. In this paper, we focus on the scenario where no communication exists among the workers, deriving efficient linear fusion rules for the combination of the distributed estimators. Both a constrained optimization perspective and a Bayesian approach (based on the Bernstein–von Mises theorem and the asymptotic normality of the estimators) are provided for the derivation of the proposed linear fusion rules. We concentrate on finding the minimum mean squared error (MMSE) global estimator, but the developed framework is very general and can be used to combine any type of unbiased partial estimators (not necessarily MMSE partial estimators). Numerical results show the good performance of the algorithms developed, both in problems where analytical expressions can be obtained for the partial estimators, and in a wireless sensor network localization problem where Monte Carlo methods are used to approximate the partial estimators.

Random Interruptions in Cooperation for Spectrum Sensing in Cognitive Radio Networks

In this paper, a new cooperation structure for spectrum sensing in cognitive radio networks is proposed, which outperforms the existing commonly used ones in terms of energy efficiency. The efficiency is achieved in the proposed design by introducing random interruptions in the cooperation process between the sensing nodes and the fusion center, along with a compensation process at the fusion center. Regarding the hypothesis testing problem concerned, first, the proposed system behavior is thoroughly analyzed and its associated likelihood-ratio test is provided. Next, based on a general linear fusion rule, the statistics of the global test summary are derived and the sensing quality is characterized in terms of the probability of false alarm and probability of detection. Then, the optimization of the overall detection performance is formulated according to the Neyman–Pearson criterion (NPC) and it is discussed that the optimization required is indeed a decision-making process with uncertainty which incurs prohibitive computational complexity. The NPC is then modified to achieve a good affordable solution by using semidefinite programming (SDP) techniques and it is shown that this new solution is nearly optimal according to the deflection criterion . Finally, the effectiveness of the proposed architecture and its associated SDP is demonstrated by simulation results.

Massive MIMO Channel-Aware Decision Fusion

In this paper, we provide a study of channel-aware decision fusion (DF) over a “virtual” multiple-input multiple-output (MIMO) channel in the large-array regime at the DF center (DFC). The considered scenario takes into account channel estimation and inhomogeneous large-scale fading between the sensors and the DFC. The aim is the development of (widely) linear fusion rules, as opposed to the unsuitable optimum log-likelihood ratio (LLR). The proposed rules can effectively benefit from performance improvement via a large array, differently from existing suboptimal alternatives. Performance evaluation, along with theoretical achievable performance and complexity analysis, is presented. Simulation results are provided to confirm the findings. Analogies and differences with uplink communication in a multiuser (massive) MIMO scenario are underlined.

Super-resolution images fusion via compressed sensing and low-rank matrix decomposition

Most of available image fusion approaches cannot achieve higher spatial resolution than the multisource images. In this paper we propose a novel simultaneous images super-resolution and fusion approach via the recently developed compressed sensing and multiscale dictionaries learning technology. Under the sparse prior of image patches and the framework of compressed sensing, multisource images fusion is reduced to a task of signal recovery from the compressive measurements. Then a set of multiscale dictionaries are learned from some groups of example high-resolution (HR) image patches via a nonlinear optimization algorithm. Moreover, a linear weights fusion rule is advanced to obtain the fused high-resolution image at each scale. Finally the high-resolution image is derived by performing a low-rank decomposition on the recovered high-resolution images at multiple scales. Some experiments are taken to investigate the performance of our proposed method, and the results prove its superiority to the counterparts.

Compressed Sensing and Low-Rank Matrix Decomposition in Multisource Images Fusion

We propose a novel super-resolution multisource images fusion scheme via compressive sensing and dictionary learning theory. Under the sparsity prior of images patches and the framework of the compressive sensing theory, the multisource images fusion is reduced to a signal recovery problem from the compressive measurements. Then, a set of multiscale dictionaries are learned from several groups of high-resolution sample image’s patches via a nonlinear optimization algorithm. Moreover, a new linear weights fusion rule is proposed to obtain the high-resolution image. Some experiments are taken to investigate the performance of our proposed method, and the results prove its superiority to its counterparts.

On Linear Estimation Fusion under Unknown Correlations of Estimator Errors

The linear fusion of estimators is widely used in decentralised state estimation. Because the maintaining of estimation error cross-correlations between local estimators is not affordable in large-scale problems, approaches dealing with unknown correlations were developed. The Covariance Intersection fusion is a linear fusion of estimators and it provides a fused estimator quality matrix that does not undervalue the mean square error matrix. This paper derives the matrix of the fused estimator quality for arbitrary weights of the linear fusion rule that considers the unknown correlations. It also shows that there can exist better matrices of the fused estimator quality than the ones proposed by the Covariance Intersection fusion rule.

Optimal multiband joint detection in cognitive radio networks with the Taguchi method

SUMMARYThis paper studies multiband joint detection in cognitive radio networks with the Taguchi method. At the fusion center, linear fusion rule is adopted to collect the observations from distributed secondary users. We aim to achieve maximum aggregate throughput with limited aggregate interference. The problem is challenging due to its nonconvexity and high computational complexity to find the global optimum. Existing works attempt to convert the problem into convex optimization problem. In this paper, the Taguchi method is employed to estimate the gradient of the aggregate throughput, determine the optimal thresholds of the energy detectors and combination weights of the linear fusion rule regardless of convexity. To optimize thresholds and linear weights simultaneously, we employ newly defined variables to represent the changing ranges of detector thresholds when we search for optimal linear weights. In addition, the sensing duration is another factor to be optimized in the Taguchi method. The simulation results show that the proposed method is efficient and applicable for all classes of cognitive radio without considering convexity. The optimization performance is considerably improved. Moreover, the Taguchi method is insensitive to parameter initialization, which provides a relatively robust output. Copyright © 2013 John Wiley & Sons, Ltd.

Optimal Training and Data Power Allocation in Distributed Detection With Inhomogeneous Sensors

We consider a binary distributed detection problem in a wireless sensor network with inhomogeneous sensors, in which sensors send their binary phase shift keying (BPSK) modulated decisions to the fusion center (FC) over orthogonal channels that are subject to pathloss, Rayleigh fading, and Gaussian noise. Assuming training based channel estimation, we consider a linear fusion rule which employs imperfect channel state information (CSI) to form the global decision at the FC. Under the constraint that the total transmit power of training and decision symbols at each sensor is fixed, we analytically derive the optimal power allocation between training and data at each sensor such that the deflection coefficient at the FC is maximized. Our analysis shows that the proposed optimal power allocation scheme is a function of signal-to-noise (SNR) and local detection indices, and at high SNR regime, the proposed scheme outperforms the uniform power allocation.

Optimal Linear Fusion of Smoothed State Estimates

The work presented here deals with estimation fusion of smoothed state estimates. Two problems of fusion for smoothing are considered: fixed point and fixed interval. Optimal linear fusion rules in the sense of the optimal weighted least squares (OWLS) and the linear minimum mean-square error (LMMSE) are obtained. These rules are in recursive forms convenient for implementation. We also propose a more practical method for real-time smoothing, which in essence is fusing smoothed and filtered estimates. Illustrative numerical results are provided to verify the performance and credibility of the fusion rules.

An efficient sensor quantization algorithm for decentralized estimation fusion

In this paper, we consider the design problem of optimal sensor quantization rules (quantizers) and an optimal linear estimation fusion rule in bandwidth-constrained decentralized random signal estimation fusion systems. First, we derive a fixed-point-type necessary condition for both optimal sensor quantization rules and an optimal linear estimation fusion rule: a fixed point of an integral operation. Then, we can motivate an iterative Gauss–Seidel algorithm to simultaneously search for both optimal sensor quantization rules and an optimal linear estimation fusion rule without Gaussian assumptions on the joint probability density function (pdf) of the estimated parameter and observations. Moreover, we prove that the algorithm converges to a person-by-person optimal solution in the discretized scheme after a finite number of iterations. It is worth noting that the new method can be applied to vector quantization without any modification. Finally, several numerical examples demonstrate the efficiency of our method, and provide some reasonable and meaningful observations how the estimation performance is influenced by the observation noise power and numbers of sensors or quantization levels.

Optimal Linear Fusion for Distributed Detection Via Semidefinite Programming

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> Consider the problem of signal detection via multiple distributed noisy sensors. We study a linear decision fusion rule of [Z. Quan, S. Cui, and A. H. Sayed, “Optimal Linear Cooperation for Spectrum Sensing in Cognitive Radio Networks,” <emphasis emphasistype="boldital">IEEE J. Sel. Topics Signal Process.</emphasis>, vol. 2, no. 1, pp. 28–40, Feb. 2008] to combine the local statistics from individual sensors into a global statistic for binary hypothesis testing. The objective is to maximize the probability of detection subject to an upper limit on the probability of false alarm. We propose a more efficient solution that employs a divide-and-conquer strategy to divide the decision optimization problem into two subproblems. Each subproblem is a nonconvex program with a quadratic constraint. Through a judicious reformulation and by employing a special matrix decomposition technique, we show that the two nonconvex subproblems can be solved by semidefinite programs in a globally optimal fashion. Hence, we can obtain the optimal linear fusion rule for the distributed detection problem. Compared with the likelihood-ratio test approach, optimal linear fusion can achieve comparable performance with considerable design flexibility and reduced complexity. </para>

Theory of distributed estimation using multiple asynchronous sensors

This work solves a practical sensor to sensor track fusion problem when the sensors used are asynchronous, communication delays exist between sensor platforms and track fusion center, and tracks may arrive out-of-sequence. For the proposed linear fusion rule, the solution is shown to be optimal in the minimum mean square sense. The new fusion rule does not require the synchronization of tracks for the purpose of track fusion. The batch processing of incoming tracks provides an elegant solution for the out-of-sequence and latent tracks without special processing of such data. As an example, it is shown that the Bar-Shalom-Campo fusion rule is a special case of the new asynchronous track fusion algorithm.

A distributed bearings-only tracking algorithm using reduced sufficient statistics

A distributed parameter estimation algorithm is presented for a general nonlinear measurement model with additive Gaussian noise. We show that the Bayes-closed estimation algorithm developed by Kulhavy, when extended to the multisensor case leads to a linear fusion rule, regardless of the form of the local a posteriori densities. Specifically, the Kulhavy algorithm generates a set of reduced sufficient statistics (RSS) representing the local sensor densities, which are simply added and subtracted at the global processor to obtain optimum fusion. We discuss various approximations to the Bayes-closed algorithm which lead to a practical parameter estimator for the nonlinear measurement model, and apply such an approximate technique to the bearings-only tracking problem. The performance of the distributed tracker is compared with an alternative algorithm based on the extended Kalman filter (EKF) implemented in modified polar coordinates (MPC). It is shown that the Bayes-closed estimator does not diverge in the sense of an ordinary EKF, and hence the Bayes-closed technique can be employed in both a unidirectional and bidirectional transmission mode.