 Research
 Open Access
Online rate adjustment for adaptive random access compressed sensing of timevarying fields
 Naveen Kumar^{1}Email author,
 Fatemeh Fazel^{2},
 Milica Stojanovic^{2} and
 Shrikanth S. Naryanan^{1}
https://doi.org/10.1186/s1363401603489
© Kumar et al. 2016
 Received: 20 July 2015
 Accepted: 7 April 2016
 Published: 19 April 2016
Abstract
We develop an adaptive sensing framework for tracking timevarying fields using a wireless sensor network. The sensing rate is iteratively adjusted in an online fashion using a scheme that relies on an integrated sensing and communication architecture. As a result, this scheme allows for an implementation that is both energy efficient and robust. The objective is to promote an “active" framework which uses the information extracted from the network data and iteratively adjusts the monitoring process to capture the temporal variations in the monitored field. We propose two metrics based on target detection/tracking for this feedback scheme that seek to trade off between energy efficiency and accuracy of the detection/tracking tasks. Our simulation results suggest that tying target detection with the rate adjustment algorithm ensures that the robustness to changes in the field can be achieved simultaneously with the end goal of accurate target detection. Compared to a baseline method that uses the correlation of the acquired field over time, our method exhibits better performance when the targets of interest have a smaller spatial spread.
Keywords
 Sensor networks
 Adaptive sensing
 Detection
 Random access
 Compressed sensing
 Joint communication and detection
1 Introduction
The emergence of compressed sensing framework presents significant potential for efficient sensing and sampling systems [1–3] by helping to reduce sample complexity under realistic communication constraints. Sensor network technology greatly benefits from the compressed sensing paradigm [4–12]. Consider for example, largescale networks that are deployed for longterm monitoring of dynamic fields such as the ocean bed that typically need to account for power consumption considerations. An efficient scheme in such cases requires performance optimization that jointly considers both the sensing and communication constraints.
To address this issue, Fazel et al. proposed a random access compressed sensing (RACS) scheme in [13, 14] for energyefficient reconstruction of sensing fields. The proposed sensing scheme depends on integrating information from the communication and channel access modules into the data acquisition process. The method is simple to implement under realistic communication constraints and requires minimal assumptions on the field. However, the proposed RACS scheme is designed for stationary^{1} sensing fields, where the field being monitored is assumed to remain static during sensing. In order to monitor timevarying fields, in [15], the authors employed lowrank matrix recovery to reconstruct the spacetime map of the field. However, this approach is offline and entails considerable delays since full recovery can be attained only after the data have been collected over multiple time segments. Moreover, it is assumed that the coherence properties of the underlying field are known a priori and remain unchanged throughout the full sensing duration. This assumption might be justified for the monitoring of natural phenomena, where the field is assumed to be either stationary or changing at a fixed rate. It is also common to find similar assumptions of stationarity in related works in object detection or classification in underwater fields [16, 17]. However, when the field being monitored undergoes a varying rate of change (e.g., when the process is impacted by a target that is moving at an unknown or variable speed), such assumptions may not hold.
In a more recent work, Kerse et al. [18] addressed this problem by proposing to unify target detection with reconstruction using a standard sparse identification technique. Targets are tracked from frame to frame and the authors further suggest that the tracking error could be used as a measure to adjust the sensing rate in turn. While their method does not require any a prioi knowledge of the number of targets or coherence properties of the underlying field, it relies on knowledge of the exact target signatures for both target localization and tracking. In this paper, we propose a rate adjustment method that does not require explicit knowledge of the target signature. Rather, knowledge of the family of models to which the field might belong is adequate as the field model parameters can be jointly estimated.
Similar to the work in [18], we consider that the end goal in sensing is to detect or track targets and incorporate these data processing aspects into the joint sensing and communication scheme. We propose a framework to adjust the sensing rate by estimating different attributes of the field to make an informed decision. Our adaptive rate adjustment procedure for compressed sensing iteratively adjusts the pernode sensing rate to capture the variations in the underlying field. First, we treat the field as piecewise stationary and apply random access compressed sensing within each sensing period. Second, we compute two heuristic metrics that seek to tie in the end goal of target detection/tracking with the rate adjustment scheme. Using the data collected in each segment, the fusion center (FC) relies on a detection algorithm to first determine the current state it is in. Finally, depending on the current state, a control algorithm instructs the FC to change the sensing rate if required.
A high rate of sensing would typically bode well for target detection/tracking, but it is not energy efficient. On the other hand, a low rate of sensing may not necessarily lead to poor performance in detection. Thus, there is a possible tradeoff between energy efficiency and the target detection accuracy which the rate adjustment seeks to exploit. We perform simulation experiments using the proposed method and present results that suggest that the proposed sensing rate adjustment method exhibits better performance compared to the baseline method when compared on the following evaluation criteria: (a) meansquared error of tracking the underlying coherence time and (b) Fscore of the target detection accuracy.
The paper is organized as follows. In Section 2, we explain the basic sensing model for the stationary case, and in Section 3, we relax the stationarity assumptions. In Section 4, we describe the sample field model for simulation. In Sections 5 and 6, we describe an adaptive strategy for sensing the timevarying field. Finally, in Sections 7 and 8, we provide results for the simulation.
2 Random sensing network over stationary fields
Consider a grid network consisting of N=P×Q sensors, with P and Q sensors in the x and y directions, respectively. The underlying assumption is that most signals of interest (natural or manmade) vary smoothly spatially and hence are compressible in the spatial discrete Fourier transform (DFT) basis. We denote the sparsity of the signal by S. The data from the distributed sensors is conveyed to the FC, where a full map of the sensing field is reconstructed. This map can be used for target detection as will be shown in Section 5.
Inspired by the theory of compressed sensing, the architecture proposed in [13, 14] employs random sensing, i.e., transmission of sensor data from only a random subset of all the nodes. For a stationary field, each sensor node measures the signal of interest at random time instants—independently of the other nodes—at a rate of λ _{1} measurements per second. It then encodes each measurement along with the node’s location tag into a packet, which is digitally modulated and transmitted to the FC in a random access fashion. Owing to the random nature of channel access, packets from different nodes may collide, creating interference at the FC, or they may be distorted as a result of the communication noise. A packet is declared erroneous if it does not pass the cyclic redundancy check or a similar verification procedure. Since the recovery is achieved using a randomly selected subset of all the nodes’ measurements, we let the FC discard the erroneous packets as long as there are sufficiently many packets remaining to allow for the reconstruction of the field.
where z represents the sensing noise, Ψ is the inverse DFT matrix, v is the sparse vector of Fourier coefficients, and R is a K×N matrix—with K corresponding to the number of useful packets collected during T—which models the selection of correct packets. Each row consists of a single one in the position corresponding to the sensor contributing the useful packet. The FC can form R from the correctly received packets, since they carry the location tag. We emphasize the distinction between the sensing noise z, which arises due to the limitations in the sensing devices, and the communication noise, which is a characteristic of the transmission system. The sensing noise appears as an additive term in Eq. 1, whereas the communication noise results in packet errors and its effect is captured in the matrix R.
3 Adaptive sensing
For a given collection interval T, the corresponding pernode sensing rate can be determined using Eq. (2), as shown in Fig. 1. The proper choice of T however depends on the rate of variations in the field and is adaptively tuned. In particular, we use an approach based on target detection, where we assume that an object/target model of interest is known beforehand. This is a common assumption in most supervised pattern recognition tasks. Given a reconstructed map, the location of targets is first estimated. Using this knowledge, we then estimate the parameters of the object model from the map. These parameters describe the detection system’s understanding of the field and can be used to generate the map of the field. Comparing this modelbased map with the observed one reconstructed using sparse approximation algorithms provides us with a reliability metric for reconstruction. In other words, if there is a difference in what the algorithm expects to see and what it sees, it indicates an error in either the acquisition of the field or the algorithm’s understanding of the map. In either case, the FC needs to adapt its sensing rate. We specifically discuss our methods in the context of the following two cases:
3.1 Oversensing
This situation corresponds to the case when there occurs redundant sensing because the pernode sensing rate is much larger than the rate of change of the field, i.e., T<<T _{coh}. Although this case favors reconstruction using the RACS architecture, it leads to a wastage of communication resources. Thus, we seek to lower the sensing rate in this case to an optimal point such that the accuracy of our end goal is not affected. Figure 7 shows the result for of oversensing for the example discussed in Section 5. In this case, we devise a scheme to estimate the motion of targets using multiple frames. We use the term frame here and elsewhere in the paper to refer to the field reconstructed from samples acquired within a single sensing time duration.
3.2 Undersensing
4 Simulation of a sample dynamic field
In this paper, we demonstrate the adaptive monitoring procedure using a sample model for the field. We demonstrate our adaptive scheme to adjust the per node sensing rate in RACS using a simulated example field in this paper. Using a realistic example allows us to control the coherence parameter T _{coh} and monitor the effect of changing the collection interval T in accordance with the adaptive algorithm.
where a _{ m }(t) and b _{ m }(t) are the coordinates of the mth target at time t, A _{ m } is its strength, and p is the decay rate of the sources. The process then evolves over time as the sources move along random trajectories. Similar models are commonly found in the energybased localization literature for static sensor networks [20–22] when the targets being detected are not moving.
Initially, the FC has no knowledge of the location of targets or the rate of variation in the field, i.e., the speed at which the targets are moving. It thus instructs the data collection to begin with an initial sensing rate \(\lambda _{1s}^{\text {init}}\). The initial sensing rate is determined using historical data by setting the desired parameters in Eq. (2). Once the map of the field G ^{ s a } is recovered using sparse approximation techniques, the FC may now use the rate adjustment algorithm described later to decide if the sensing duration T needs to be adjusted. The sensors employ the adjusted sensing rate in the ensuing sensing duration. In this paper, we discuss metrics for the proposed rate adjustment algorithm for this family of field models, although the framework itself is quite general.
For simulating a map of the field as defined above, we start with a set of randomly chosen parameters. In addition, each target is assigned a random velocity and direction of movement. To simulate the undersensing case, we consider that collection occurs over N _{ b } coherence time intervals (referred to as frames) where T≈N _{ b } T _{coh}. The randomly sampled packets are then collected uniformly over the last N _{ b } frames (Fig. 3). This effectively leads to motion blurring of the targets as mentioned before. To simulate the oversensing case, the collection interval is reduced to T ^{′}=T _{coh} while each target’s velocity is scaled by T/T _{coh} to make them appear to be moving slower. To deal with the issue of a finite field size, targets moving out of the field are replaced by new targets starting from the same location assigned a new random velocity and direction.
5 Adaptive field monitoring scheme
5.1 Target localization
Before taking any decisions about the current sensing state, we first detect and localize the targets in the image. Similar to [18], by involving target detection in the feedback process, we would like to ensure that the rate of sensing is optimized for the end goal of target detection.
Traditionally, most energybased localization methods [20–22] assume that the number of targets is known in advance. In addition, they assume a field model for target signature decay allowing for parameter estimation by model fitting. More recently, Kerse et al. proposed a method for direct target localization based on a standard sparse identification technique [18]. The advantage in their method is that target localization can be performed in a single step without first having to reconstruct the field. The number of targets can also be jointly estimated in this process using a sparsity constraint. However, the method still depends on exact knowledge of the field model.
To overcome this drawback, we propose a target localization algorithm based on local gradient ascent. The proposed method only requires that the target signatures be monotonically decaying away from the target. This technique is adapted from the mode seeking mean shift algorithm [23] which is frequently used for unsupervised clustering of data. Consequently, it can be interpreted as searching local peaks in the data histogram. In the task at hand, we are instead interested in finding local peaks in field intensity. We also modify the algorithm slightly to adapt to the discrete search space for this problem. Specifically, we start from a random initial point X on the map. A meanshiftbased gradient ascent technique is then used to update the location of this point in each iteration till a peak is found. Given the current location X, we estimate an intensity weighted mean X _{ c } for locations around X in a window of size W as shown in Eq. (4). Note that the direction from X to X _{ c } gives the direction of gradient ascent, along which X should be updated in the next iteration.
5.2 Parameter estimation
Once the positions a _{ m } and b _{ m } for the targets have been identified using the target localization algorithm described above, we try to obtain a minimum mean square error (MMSE) estimate of the field parameters defined in Eq. (3) using a method of comparison by synthesis to match the acquired map G ^{ s a } against a model. The parameters of model denoted by G ^{model} comprise the target locations (a _{ m },b _{ m }), their respective strengths A _{ m }, and the decay parameters of the target signature (p).
5.3 Reliability metric for reconstruction
5.4 Measuring motion by target tracking
In addition to compensating for the dynamic nature of the field in the undersensing case, we would also like to eliminate any redundant sensing. Recall, that in the oversensing case, the field changes very slowly, yielding perfect reconstruction and detection. This makes it necessary to monitor the field over multiple frames for detecting any signs of oversensing.
To quantify motion in the field, we track the location of detected targets over the last L frames. Note that this is not trivial since the number of targets detected is not guaranteed to be consistent from one frame to another. Moreover, the target indices assigned by the detection algorithm are nonunique. To deal with this issue, we use a tracking approach based on dynamic programming that ensures tracking even if the object is not detected in some of the intermediate frames. More specifically, we recursively minimize the total distance moved by a target over L frames. If the targets are spaced sufficiently apart, this ensures tracking of the slowest moving target over multiple frames.
6 Feedback algorithm for rate adjustment
After computing the error metrics e _{ r } and e _{ m }, a control algorithm is used to determine if the current sensing rate needs to be changed. This information is fed back to the FC which makes any necessary changes to the sensing rate, thereby establishing a closedloop control. The objective of control is to minimize the reconstruction errorbased metric e _{ r } at the same time ensuring that the targets move significantly from frame to frame as indicated by the tracking error metric e _{ r }. This ensures that the system is in a sweet spot between undersensing and oversensing.
Control feedback rules (“adjust” mode: A, D ; “calibrate” mode: B, C)

In practice, the thresholds can be initialized to extreme values such that the buffer region is wide enough and the system is guaranteed to start in state B. In subsequent intervals, the thresholds are updated in the “calibration” mode to shrink this buffer region. The ideal operating zone, centered at (t h _{ r },t h _{ m }), is shown by a shaded region in Fig. 9. If the system steps out of this operating zone, the thresholds are adjusted in a direction such that the current system state is contained within the operating zone. In the calibration mode, the thresholds are adjusted by predefined steps α and β such that the system stays within the operating zone. In other words, the aim of this adaptive control is to obtain the tightest stable region of control for T.
To provide an intuition of how the scheme works, consider the graphical illustration of the proposed feedback algorithm in Fig. 9. At any time instant, the system state can be represented as a point on the e _{ r } e _{ m } plane. The relative position of this point with respect to the thresholds t h _{ r } and t h _{ m } is then used to decide the next course of action. In general, the algorithm tries to maintain a narrow buffer region of control by keeping the operating point (e _{ r },e _{ m }) close to (t h _{ r },t h _{ m }). This can be achieved by either updating the sensing time period T (states A and D) or by adapting the thresholds (states B and C).
By observing how the states of the algorithm transition, we present an argument to show that the proposed algorithm is boundedinput boundedoutput (BIBO) stable. This means that for a finite coherence time T _{coh}, the controlled variable viz. the sensing time period T is always bounded. This can be shown by considering each of the four states of operation A, B, C, and D of the proposed algorithm above.
7 System operation
We presented an example of the control system in an openloop operation earlier in Fig. 8 which showed the value of metrics e _{ r } and e _{ m } for a particular T _{coh} profile. Figure 8 c, d suggests that a larger L might be related to a slower response time of e _{ m } as can be seen around t=80, 100 when the field’s T _{coh} changes. The error metrics based on model fitting (Fig. 8 a, b) are computed from the current frame and thus respond instantaneously to the changes in field coherence. It is also worth noting here that the HF errors (spatial highfrequency component) typically correspond to the reconstruction error in the undersensing case, while the LF error serves as a sanity check for detection.
8 Simulation results
8.1 Baseline algorithm
We compare the results against a baseline method based on correlation of the currently acquired field G ^{ s a } with the field acquired in the previous cycle as suggested in [15]. Control decisions are taken depending on two fixed high (δ _{ h }) and low (δ _{ l }) thresholds on the correlation value. T is incremented if correlation exceeds δ _{ h } and decremented if correlation falls below δ _{ l }. T is not updated if correlation lies between these two thresholds. The sampling duration T is changed using a scaling parameter κ as earlier.
Figure 12 a shows the results for the baseline method using correlation as a feedback metric. From the MSE and Fscore plots, it is evident that the baseline algorithm performance drops with increase in decay parameter p. This is expected since the targets with a lesser spread do not affect the correlation metric significantly. In contrast, for our proposed feedback algorithm, the performance improves with increase in the decay parameter p since targets with lesser spread are easier to localize and hence field reconstruction is more accurate (Fig. 12 b). A similar trend can be seen for average MSE which is related inversely to the target detection Fscore. In other words, this might indicate that poor tracking of the underlying T _{coh} is related to poor target detection, thereby making a case for the proposed sensing rate adjustment in this paper.
Since velocity for each target is assigned uniformly at random in the simulation, we control the velocity range that can be assigned to each target as a parameter instead. We note that increase in velocity has a positive influence on the performance of both the algorithms. This can be easily explained by considering that an increase in velocity of the targets makes it easier to discriminate between the oversensing, adequate sensing, and undersensing states. This holds for both the algorithms, and hence, we notice a general increase in Fscore and decrease in MSE with increase in velocity. Finally, we also note that our proposed detectionbased feedback mechanism outperforms the baseline correlationbased method in terms of the accuracy of target detection as indicated by the higher Fscore. The average target detection Fscore obtained for the baseline algorithm was 0.32 while our proposed method achieved an average Fscore of 0.45.
8.2 Simulations with measurement noise
The results show a clear degradation in Fscore for both the proposed detectionbased and baseline correlationbased methods, as the SNR decreases. However, we note that for the baseline algorithm, the Fscore initially increases with decrease in SNR hinting at suboptimality in processing. On further investigation, we found this peculiar trait to be an artifact of the control algorithm and the choice of metric used in the baseline scheme. As noise is added to the field value of the correlation metric drops, which leads the FC to confuse the current state as undersensing. This inference leads to a monotonic decrease in sampling duration T, temporarily increasing performance at the cost of energy efficiency. The target detection Fscore peaks around σ _{noise}=0.05 beyond which the accuracy of targetdetection algorithm is significantly affected. On the other hand, this artifact does not affect the proposed method significantly, since the metric used for our control algorithm is based directly on target detection. As a result, we note that our proposed method exhibits a more graceful degradation in performance with increase in noise without compromising on energy efficiency.
9 Conclusions
In this work, we present a rate adjustment scheme for random access compress sensing (RACS) to monitor and compensate for the rate of change in timevarying fields. Adjustment of sensing rate for RACS is motivated by the tradeoff between energy efficiency and the accuracy of tracking targets of interest. Although direct estimation of the coherence time might seem to be the best approach for sensing a time varying field, we note that it is not directly related to our end goal of target detection in the field. We also observe that the reconstruction error in RACS has a complex relation to the current coherence time of the field and factors like the position of targets, their velocity, etc. Thus, by making the algorithm depend on the detection and localization of targets, we ensure that the rate adjustment is tied in to our actual objective.
In this paper, we propose two unsupervised metrics to inform the rate adjustment scheme. A modelbased field reconstruction is done after target localization for each acquired frame, and the model fit error e _{ r } is used as a metric to detect cases when the FC is undersensing. To account for oversensing in operation, we define a measure for the motion of detected targets in the past L frames. This motion based metric e _{ m } indicates any redundancy in sensing and can be used to decrease the collection time interval T if needed.
We proposed a dualthresholdbased feedback mechanism using these error metrics for rate sensing adjustments. The technique assumes minimum prior knowledge and adapts the threshold in an online fashion using reasonable assumptions about the field. We also show that the proposed control mechanism is BIBO stable. In addition, we compare our results against a baseline algorithm that uses temporal correlation of the acquired fields and show that the proposed rate adjustment mechanism performs better on an average in terms of target detection accuracy even in the presence of noise.
10 Endnote
^{1} We use the term stationary in this paper to refer to fields that are temporally static during the sensing duration.
Declarations
Acknowledgements
This work was supported by ONR and NSF.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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