Low-Complexity One-Dimensional Edge Detection in Wireless Sensor Networks
© Marco Martalò and Gianluigi Ferrari. 2010
Received: 16 February 2010
Accepted: 26 May 2010
Published: 27 June 2010
In various wireless sensor network applications, it is of interest to monitor the perimeter of an area of interest. For example, one may need to check if there is a leakage of a dangerous substance. In this paper, we model this as a problem of one-dimensional edge detection, that is, detection of a spatially nonconstant one-dimensional phenomenon, observed by sensors which communicate to an access point (AP) through (possibly noisy) communication links. Two possible quantization strategies are considered at the sensors: (i) binary quantization and (ii) absence of quantization. We first derive the minimum mean square error (MMSE) detection algorithm at the AP. Then, we propose a simplified (suboptimum) detection algorithm, with reduced computational complexity. Noisy communication links are modeled either as (i) binary symmetric channels (BSCs) or (ii) channels with additive white Gaussian noise (AWGN).
1. Introduction and Related Work
In , the authors consider a scenario with a single phenomenon status change (denoted, in the following, as edge) and propose a framework, based on minimum mean square error (MMSE) estimation, to determine the position of this edge. In , under the assumption of proper regularity of the observed edge, a reduced complexity MMSE decoder is proposed. In , the authors show that an MMSE decoder is unfeasible for large-scale sensor networks, due to its computational complexity, and propose a distributed detection strategy based on factor graphs and the sum product algorithm. Moreover, MMSE-based distributed detection schemes have also been investigated in scenarios with (i) a common binary phenomenon under observation and (ii) bandwidth constraints . In [11, 12], the authors examine the problem of determining edges of natural phenomena through proper processing of data collected by sensor networks. In these papers, particular attention is devoted to the estimation accuracy, given in terms of the confidence interval of the results obtained with the proposed framework.
The problem of edge detection is also well known in the realm of image processing, where it may be of interest to characterize the intensity changes in the processed image. In , the authors characterize, from a theoretical point of view, the types of possible intensity changes. In , using numerical optimization, optimal operators are preliminary derived for ridge and roof edges, and then specialized for step edges. In , the edge detection problem is tackled as a statistical inference problem. Other interesting approaches to edge detection, especially for noisy information fusion scenarios, are proposed in [16, 17].
In , we have proposed a preliminary analytical approach to the design of decentralized detection schemes for scenarios with spatially nonconstant binary phenomena, that is, phenomena with status (either "0" or "1") which may vary from sensor to sensor. We have also derived MMSE detection algorithms at the access point (AP), considering different quantization strategies at the sensors. In order to make our approach practical, a simplified detection algorithm, with a computational complexity much lower than that of the MMSE detection rule, has been proposed.
In this paper, we extend the approach presented in  to network scenarios where the communication links between the sensors and the AP may be noisy. These links are modeled either as binary symmetric channels (BSCs) or as additive white Gaussian noise (AWGN) channels. In particular, we study the relative impacts of communication and observation noises on the system performance, evaluated in terms of (i) distance between estimated and true phenomena and (ii) probability of local status estimation error (LSEE). As will be shown in the following, the proposed simplified detection algorithm incurs a limited performance loss with respect to the MMSE algorithm, yet guaranteeing a remarkable complexity reduction. Finally, the robustness and complexity of the proposed algorithms are investigated.
The structure of this paper is the following. In Section 2, we give preliminaries on decentralized detection. In Section 3, we derive the optimum MMSE detection rules at the AP in a scenario with noisy communication links and multiedge phenomena. In Section 4, we propose a simplified detection algorithm in order to reduce the computational complexity of the proposed decentralized detection scheme. In Section 5, numerical results on the performance of the proposed detection algorithms are presented. Finally, concluding remarks are given in Section 6.
2. Preliminaries on Decentralized Detection
As anticipated in Section 1, we focus on a network scenario where the status of the phenomenon under observation is characterized by a number of "edges," that is, sensor positions where the phenomenon changes its status from "0" (e.g., absence of a critical gas) to "1" (e.g., presence of a critical gas) or vice versa. For the sake of simplicity, we assume that the status of the phenomenon is independent from sensor to sensor. The proposed approach, however, can be extended to take into account the presence of correlation between sensors. In general, the presence of correlation would limit the number of edges and, if properly exploited at the AP, improve the performance with respect to that obtained in the following. A pictorial description of the proposed scenario is given in Figure 1(c). In particular, we investigate the performance when the communication links between the sensors and the AP are noisy, that is, errors may be introduced during data transmission. Note that, under the assumption that the geographical positions are known, from the estimated edges' positions the real geographic structure of the phenomenon (e.g., area with gas leakage) can be immediately determined.
where the notation stands for bit-by-bit EX-OR and is the estimated phenomenon. Given that are the true edges' positions, can be directly derived from the estimated edges' positions . Therefore, our goal is to accurately estimate . The particular expression for depends on the chosen distributed detection strategy, as will be shown in the following. We will also consider, as a meaningful performance indicator, the probability of LSEE, that is, the probability that the estimated phenomenon status at a sensor is wrong. In Section 5.2, it will be shown how the probability of LSEE is related to .
3. MMSE One-Dimensional Edge Detection
In the remainder of this section, we derive the MMSE detection rules depending on the quantization strategy at the sensors.
3.1. Binary Quantization
In this scenario, the th sensor makes a decision comparing its observation with a threshold value , and computes a local binary decision , where is the unit step function. To optimize the system performance, the thresholds need to be properly selected. In this paper, regardless of the value of , a common value at all sensors is considered .
(For ease of notational simplicity, in (8) we use the same symbol to denote both the random variable (in the second term) and its realization (in the third and fourth terms). This simplified notational approach will be considered in the remainder of Section 3. The context should eliminate any ambiguity.)
Taking into account the constraint (6), the upper and lower limits of the sum in (10) can be further refined, obtaining the right-hand side expression in (8).
The computation of the conditional probabilities appearing at the right-hand side of (8) can be carried out as outlined in Appendix A.1.
3.2. Absence of Quantization
with . Therefore, imposing that the BER in (12) is equal to the cross-over probability of the equivalent BSC, the corresponding value of can be obtained. This makes the performance comparison between the cases with binary quantization and without quantization consistent.
The computation of the conditional probabilities appearing at the right-hand side of (13) can be carried out as outlined in Appendix A.2.
We would like to remark that the MMSE strategy outlined above is based, regardless of the quantization strategy, on the assumption of knowledge of the number of edges at the AP. However, in the scenario of interest, for example, monitoring of a gas leakage, this knowledge may not be a priori available and should be properly estimated. In this case, by averaging over all possible realizations of , the average performance, with respect to the number of edges, could be determined. This extension goes beyond the scope of this paper. In fact, the performance of the MMSE algorithm with knowledge of at the AP will be used as a benchmark for the performance of the simplified (and feasible) one-dimensional edge detection algorithms introduced in Section 4.
4. Simplified One-Dimensional Edge Detection
Since the computational complexity of the MMSE detection strategy increases very quickly with the number of phenomenon edges (see Section 5.4 for more details), the derivation of a simplified distributed detection algorithm with low complexity (but limited performance loss) is crucial. As considered in Section 3 for MMSE detection, we distinguish between scenarios with binary quantization and without quantization.
4.1. Binary Quantization
4.2. Absence of Quantization
One should observe that, unlike the MMSE strategy, our simplified edge detection algorithm (with binary quantization and no quantization, resp.) does not require knowledge of the number of edges at the AP. Therefore, the simplified algorithm is suitable for area monitoring applications, since in this scenario is not a priori known. Obviously, we expect that the proposed algorithm will incur a performance degradation with respect to the MMSE algorithm. However, this loss will be limited, as shown with simulation results in Section 5.
5. Numerical Results
5.1. Performance Analysis: Distance
the number of edges is randomly generated—the AP is assumed to know this number in the MMSE case;
for a selected number of edges, their positions are randomly generated (From an operative viewpoint, in a scenario where the number of edges is larger than one, after the position of an edge is extracted, the following edge position is randomly chosen among the remaining positions. After all edges' positions are extracted, they are ordered.);
either the sensors' decisions or the PDFs of the observables, according to the chosen quantization strategy at the sensors, are transmitted to the AP;
a noisy version of the transmitted data is received at the AP;
the AP detects the edges' positions through either MMSE or simplified detection algorithms;
In order to better understand the impacts of the communication and observation noises, it is expedient to normalize, sensor SNR by sensor SNR, by . In this way, the normalized distance , denoted as , assumes values in and allows to directly compare scenarios with different numbers of sensors. The normalized versions of the distance curves of Figure 7(a) are shown in Figure 7(b). Obviously, when the distance goes to the same value (i.e., 1), regardless of the values of and . As expected, for a given value of (i.e., the communication quality), the higher the sensor SNR is (i.e., the observation quality) the more pronounced is the performance degradation for increasing values of .
5.2. Performance Analysis: Probability of Local Status Estimation Error
5.3. System Robustness
In the observation phase, we assume that there could be an error in the decision threshold used at each sensor. More precisely, denoting by the optimized decision threshold ( ), we assume that each sensor makes use of an actual decision threshold which is uniformly distributed in , where . The decision thresholds at different sensors are supposed to be independent.
In the communication phase, we assume that, while the detection algorithm at the AP assumes a constant cross-over probability (denoted as ) for all communication links, the actual cross-over probabilities in the various links are independent and uniformly distributed in the , where .
In case (a), one can observe that, for sufficiently high values of the communication noise intensity , there is a performance degradation (i.e., the distance increases) for increasing observation threshold mismatch (i.e., for increasing values of ), regardless of the sensor SNR. On the other hand, for low values of the communication noise intensity and very high values of the sensor SNR (e.g., dB), for increasing values of the distance slightly increases. Finally, for low values of the communication noise intensity and low/medium values of the sensor SNR, the distance slightly decreases for increasing values of —this is due to the fact that in the presence of strong observation noise, the considered local decision strategy is no longer optimized. In all possible situations, the distance saturates at . In other words, the proposed simplified detection strategy is robust against local decision threshold mismatches.
In case (b), instead, the decision threshold at the sensors is fixed to . As one can see, for high values of , for increasing variability of the communication link quality (i.e., for increasing values of ) the performance rapidly degrades. For low values of , for increasing there is a slight decrease of the distance, that is, a slight performance improvement—as previously commented, this depends on the fact that local sensor decision and AP detection strategies are no longer optimized. As in case (a), the performance with any scheme saturates at , that is, the proposed simplified detection rule is robust also against mismatches in the communication phase.
As a final remark, we point out that the fact that the proposed simplified detection rule is insensitive to strong fluctuations of the observation and communication qualities means that the system performance basically depends on the average observation and communication conditions.
5.4. Computational Complexity
Finally, we evaluate the improvement, in terms of computational complexity reduction with respect to the MMSE detection rule, brought by the use of the simplified detection algorithms. As complexity indicators, we choose the numbers of additions and multiplications (referred to as and , resp.) required by the considered detection algorithms, evaluated as functions of the number of sensors . In a scenario with noisy communication links, the same considerations carried out in  for a scenario with ideal communication links still hold. In fact, the structures of the proposed detection algorithms are the same in both scenarios, since only the expressions of the used probabilities and PDFs change. Therefore, it can be shown that the numbers of additions and multiplications required by the MMSE detection algorithm would be and . On the other hand, the computational complexity of the proposed simplified detection algorithm is characterized by and , showing a significant complexity reduction with respect to the MMSE detection algorithm—this also justifies the performance loss at small values of the sensor SNR.
6. Concluding Remarks
In this paper, we have analyzed the problem of one-dimensional edge detection in wireless sensor networking scenarios with noisy communication links. This situation arises in many practical applications such as those where an area of interest needs to be actively monitored to detect the presence of a phenomenon, for example, the presence of a gas leakage. We have proposed an analytical framework considering two quantization strategies at the sensors: (i) no quantization at the sensors and (ii) binary quantization. In each case, the MMSE detection algorithm at the AP has been derived and the impacts of relevant network parameters (e.g., the sensor SNR, the communication noise level, and the number of sensors) have been investigated. Then, a low-complexity and feasible detection algorithm, which does not require any a priori information on the number of edges, has been derived. We have shown that the performance penalty induced by the use of the simplified detection algorithms is asymptotically (for high sensor SNR and low communication noise level) negligible. Moreover, the simplified detection algorithm has proved to be robust against system parameters' variations. Finally, we have quantified the relevant computational complexity reduction brought by the use of the simplified detection algorithms with respect to the MMSE ones.
A. Details on the MMSE Distributed Detection Strategy
A.1. Binary Quantization
The last term at the right-hand side of (A.2) (i.e., the denominator) can be easily computed by observing that it is composed of terms similar to those evaluated in (A.3) and (A.8).
A.2. Absence of Quantization
One can notice that the effects of observation and communication AWGNs add directly.
B. Details on the Simplified One-Dimensional Edge Detection Strategy
B.1. Binary Quantization
B.2. Absence of Quantization
C. Limiting Distance for High Communication Noise
The computation of the above expression is analytically very cumbersome. However, as shown in Figure 7(a) , it can be obtained out through simulations. In particular, our results show that an accurate approximation (through interpolation) is given by , where —in this case, the relative error between and is lower than 2.4% for .
The authors would like to thank Marco Sarti (Elettric 80 S.p.A., Viano, Reggio Emilia, Italy) for his help in the derivation of part of the simulator.
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