Exploring the Laplace Prior in Radio Tomographic Imaging with Sparse Bayesian Learning towards the Robustness to Multipath Fading

Radio tomographic imaging (RTI) is a technology for target localization by using radio frequency (RF) sensors in a wireless network. The change of the attenuation field caused by the target is represented by a shadowing image, which is then used to estimate the target’s position. The shadowing image can be reconstructed from the variation of the received signal strength (RSS) in the wireless network. However, due to the interference from multi-path fading, not all the RSS variations are reliable. If the unreliable RSS variations are used for image reconstruction, some artifacts will appear in the shadowing image, which may cause the target’s position being wrongly estimated. Due to the sparse property of the shadowing image, sparse Bayesian learning (SBL) can be employed for signal reconstruction. Aiming at enhancing the robustness to multipath fading, this paper explores the Laplace prior to characterize the shadowing image under the framework of SBL. Bayesian modeling, Bayesian inference and the fast algorithm are presented to achieve the maximum-a-posterior (MAP) solution. Finally, imaging, localization and tracking experiments from three different scenarios are conducted to validate the robustness to multipath fading. Meanwhile, the improved computational efficiency of using Laplace prior is validated in the localization-time experiment as well.


Introduction
Target localization has many applications in practice, such as terrorist localization for military and police surveillance [1,2], intruder detection [3], traffic flow and roadside monitoring [4], residential monitoring [5,6] and ambient assisted living [7,8] etc. There are many types of sensors that can be employed for target localization, such as optical cameras [9], ultrasonic sensors [10], light sensors [11], pyroelectric infrared (PIR) sensors [12,13] etc. Among them, RF sensors have the advantages of penetrating walls, smoke and other opaque obstructions, rendering it more appropriate for working in obstructed environments [14]. For RF sensing, narrow-band RF sensing strategy can locate the target by using the ubiquitous and inexpensive wireless devices compared with ultra-wide band (UWB) [1]. This advantage facilities the deployment of dense wireless networks for target localization.
In narrow-band RF sensing, received signal strength (RSS) and channel state information (CSI) are two types of information that can be used for target localization, but the CSI information is obtained • The image quality and localization accuracy are improved by enhancing the robustness to multipath fading. • The computational cost in localization becomes reduced as well, this is owning to the adoption of the fast algorithm.
The remainder of this paper is organized as follows. Section 2 gives some preliminary knowledge about RTI and presents the problems that the paper will address. Section 3 describes the proposed method in detail. In Section 4, the experiments are conducted to validate the effectiveness of the proposed approach. Finally, the conclusion is given in Section 5.

Radio Tomographic Imaging
Before performing RTI, a wireless network consisting of several RF nodes should be deployed firstly in the monitored area. As illustrated in Figure 1, 16 RF nodes build 16*(16-1)/2 = 120 bidirectional RF links. In RTI, the monitored area is considered as an image by dividing into several pixels according to an user-defined pixel-size δ. Moreover, a coordinate system needs to be established to determine the coordinate for each pixel and each RF node. Usually, a calibration procedure is required to obtain the RSS measurements in the empty scenario, which are denoted byr = [r 1 ,r 2 , · · · ,r M ] T , where M is the number of bidirectional links in the wireless network. Thereafter, the RSS measurements when the target enters into the area are recorded, which are represented by r = [r 1 , r 2 , · · · , r M ] T . RTI takes the RSS variation y =r − r = [y 1 , y 2 , · · · , y M ] T as the observation [1]. Seeking for online method to acquirer is not the focus of this paper, the readers can refer to [5][6][7]44] and the references therein.
According to the derivation in [1], RTI can be modeled as the following linear equation where x = [x 1 , x 2 , · · · , x N ] T represents the vector of the shadowing image, denoting the variation of the attenuation image (also called shadowing image) between the empty scenario and the scenario after the target's presence. The pixel value in the shadowing image can indicate where the target locates. Here N equals to the number of pixels. Usually, large pixel values occur near the position where the target locates. n = [n 1 , n 2 , · · · , n M ] T is the measurement noise due to interference from multi-path fading and sensor accuracy. Φ ∈ R M×N is the measurement matrix determined by the measurement model. In this paper, the ellipse measurement model [1] is adopted, which is expressed as follows where dis is the distance between the transmitter node and receiver node, d TX i,j and d RX i,j denote the distances from the centre of the j-th pixel to the transmitter and receiver of the i-th link, and η is a tunable parameter characterizing the width of the ellipse.
In RTI, the target's position needs to be estimated from the shadowing image, thus the first step of target localization is to reconstruct the shadowing image x from the RSS variation y. After the image vector x is obtained, it is re-arranged to the format of 2D image colum by colum, and the target is then located at the coordinate of the pixel with maximum pixel value. Therefore, the main task of RTI is to reconstruct the shadowing image x from the RSS variation y.

Problem Statement
The most challenging problem in RTI is that not all the RSS variations (i.e., y =r − r) are reliable due to multi-path fading [1,17,45]. Specifically, because the RSS measurement from the narrow-band RF sensor is a mixed RSS from both the LOS and NLOS paths, the RSS variation may not show decrease even if the target shadows the LOS of that link. The NLOS RSS components from reflection and diffraction caused by the target may overwhelm the LOS component from shadowing effect, thus the RSS variations may increase or remain unchanged. Figure 2 illustrates the LOS path and two NLOS paths from reflection and diffraction.  Furthermore, for the link that is not obstructed by the target, its corresponding RSS measurement may also be possible to show a decrease. This is because the target may introduce reflection on the NLOS path and the reflection could lead to decreased RSS variation because of small-scale fading. This kind of RSS variation is taken as fake-shadowing RSS variation. The more cluttered the environment is, the more severely the multipath fading exhibits on the RSS measurement. If the unreliable RSS variation is used for image reconstruction, some artifacts will emerge in the shadowing image, which will deteriorate the localization accuracy [1,24,41].
As analyzed in the third paragraph of Section 1, the shadowing image is sparse. In dealing with the sparse signal, SBL [46] provides one effective paradigm for the signal recovery due to its sufficient consideration on the signal and noise prior. By dynamically adjusting the objective function in each iteration, SBL could make use of the collected data more effectively. In RTI, it means that SBL could give superior treatment to the informative data (i.e., the RSS variations from shadowing effect) in its dynamic learning process if proper prior is provided in advance [42,43,47]. Therefore, the key problem in applying SBL to RTI is to build appropriate priors for the shadowing image and observed noise.
Existing studies on the application of SBL to RTI put more emphasis on the noise prior [42,43], while for the signal (i.e., the shadowing image), the student-t prior is still adopted. One drawback of the student-t prior is that it does not produce sufficiently sparse solutions, i.e., the recovered shadowing image is not sparse enough and still contains certain number of artifacts especially in the cluttered environment. Also it is usually computationally intensive. In this paper, to reduce the artifacts in the shadowing image caused by multipath fading, we explore the sparsity-promoting Laplace prior to characterize the shadowing image, and develop the fast algorithm to improve the computational efficiency. Meanwhile, the Jeffrey's prior is also discussed in this paper.

Laplace Prior used to Characterize the Shadowing Image
Under the framework of SBL, all unknowns are taken as random variables with certain probabilistic distributions. Here in RTI, the signal x (i.e., the shadowing image) is constrained by the Laplace prior, which is presented as follows where λ is the scale parameter of the Laplace distribution. One problem for the Laplace distribution in Equation (3) is that it is not conjugate to the Gaussian distribution, which causes the intractability for Bayesian analysis. To circumvent this problem, inspired by the work in [48], a 3-layer hierarchical prior with similar property is employed to approximate the Laplace distribution. Specifically, a zero-mean Gaussian distribution is imposed on the first layer shown as where γ = [γ 1 , γ 2 , · · · , γ N ] T . In the second layer, each γ i is considered to be i.i.d (independent identically distributed) with the following Gamma distribution Moreover, the third layer constrains λ using a Gamma distribution as well where ν denotes the shape and scale parameters of the Gamma distribution, which is defined by the user in practice.
In SBL, the conditional distribution for the RSS variation y also requires to be assigned, here the following Gaussian distribution is given to describe this conditional distribution where I stands for the identity matrix. Based on Equation (7) and y = Φx + n, one can derive that the noise n is indeed assumed to satisfy a zero-mean Gaussian distribution shown as follows where β −1 represents the noise variance. To extend the generality of the noise, following the framework of SBL in [46], β is then taken as a random variable and constrained by a Gamma distribution where c and d are the user-defined parameters, characterizing the shape parameter and scale parameter of the Gamma distribution respectively. To understand the hierarchical priors more vividly, Figure 3 presents the graphical explanation for the signal and noise priors.  In Figure 3, the left side depicts the signal with a 3-layer hierarchical prior, namely Equation (4) to Equation (6). The right side illustrates the noise structure with a 2-layer hierarchical prior, corresponding to Equation (8) to Equation (9). The directional arrow indicates that there is a dependency between two variables. The variable at the end of the arrow is determined by the variable at the start of the arrow.

Bayesian Inference to Obtain the Sparse Maximum-a-Posterior Solution
The Bayesian inference is to seek for the maximum-a-posterior (MAP) sparse estimation about the shadowing image x. The inference is based on the following posterior distribution As can be seen, the posterior distribution follows a multivariate Gaussian distribution with posterior mean µ and covariance Σ expressed as follows where A = diag(γ 1 , γ 2 , · · · , γ N ), B = βI and I is the M x M identity matrix. The detailed derivation of Equations (10)- (12) can be found in [46] and the references therein.
γ, λ and β are called hyper-parameters, their optimal solutions are approximated using the type-II maximum likelihood method. In which, the logarithm of the likelihood function is formulated as follows where Taking the partial derivatives of Equation (13) with respect to γ i , λ, β and setting them equal to zero, then the optimal solutions are obtained as follows In Equations (14) and (16), µ i is the i-th element of µ, Σ ii represents the i-th diagonal element in Σ and Σ i is the summation notation.
As can be seen from Equations (11) and (12) and Equations (14)- (16), the updates of µ and Σ depend on the hyper-parameters γ i , λ, β, meanwhile the updates of γ i , λ, β depend on µ and Σ. Thus an iterative procedure needs to be performed to obtain the solutions. The main computational cost is the matrix inversion in Equation (12), which requires O(N 3 ) operation in each iteration [48][49][50], where N stands for the total pixel number. The above presented non-fast algorithm is impractical in practice, particularly in the large wireless area where N (i.e., the pixel number) is huge.
Here, we present the fast algorithm to improve the computational efficiency. In which, only one γ i is updated at each iteration instead of updating the whole vector γ. Following the method in [48,49], the signal hyper-parameter γ i is updated as follows where ∆ = (s i + 2λ) 2 − 4λ(s i − q 2 i + λ) with s i (called 'sparsity factor' variable [49]) and q i (called 'quality factor' variable [49]) expressed as follows where Φ and Σ include only the columns that are selected in current iteration. Suppose the sparsity of the shadowing image is K, the computational complexity of Laplace prior is O(NK 2 ) in the fast algorithm [48,50]. The fast realization when using student-t prior has the same computational complexity O(NK 2 ), however, as analyzed in [48], student-t prior has higher sparsity level than Laplace prior (i.e., K student−t > K Laplace ). Therefore, the student-t prior requires more time than Laplace prior for target localization. For the Jeffrey's prior, it demands intensive O(N 3 ) operations [51,52] to obtain the final solution.
The fast algorithm of RTI using SBL with Laplace prior is presented in Algorithm 1. To note, in practice, we find that the update of β is not necessary in each iteration, because the estimates at the early iterations are quite inaccurate, which deteriorate the quality of the shadowing images.
while change of the likelihood function is greater than ε thresh and iteration number is less than Iter do Choose a candidate φ i that has maximum change to the likelihood function. if q 2 i − s i > λ and γ i = 0 then Add φ i and γ i to the model. end if q 2 i − s i > λ and γ i > 0 then Retain φ i and re-estimate γ i according to Equation (14). end if q 2 i − s i < λ then Prune φ i from the model and set γ i = 0. end Update posterior mean µ and covariance Σ according to Equations (11) and (12). Update 'sparsity factor' variable s i and 'quality factor' variable q i according to Equations (18) and (19). Update variable λ according to Equation (15). end Implement x ⇐ µ (x denotes the shadowing image vector). Change x to its 2-Dimentional matrix representation.

Experimental Setup
The experiments are carried out in three different environments and Figure 4 illustrates the layouts and photos of the experiments. The MICAz made by Crossbow is taken as the RF node in the experiments. The pixel size is set equal to 0.2 m. In the wireless network, each node is allocated its unique ID number. Like in many other studies about RTI [1,5,14,17,24,25,27], the low-cost narrow band IEEE 802.15.4 standard with 2.4 GHz frequency is used for communication. The low hardware cost renders the deployment of large and dense wireless network possible used for device-free target localization. Moreover, this dense deployment guarantees certain number of RF links are obstructed by the target and provides enough shadowing RSS measurements for target localization. Following the method in other studies about RTI, the token ring protocol is adopted to control the time-division sequence in the network. At one instant, only one node that catches the token can transmit the packet, other nodes are in the receiving state and store the received signal strength (RSS) in their memories. After the transmission, this node passes the token to the next node according to the pre-defined sequence. The link from the node to itself does not contain the informative shadowing RSS component used for target localization. Thus, the node is disabled to receive the RF signal from itself in our experimental settings.
The packet transmitted by each node contains three parts: ID of the transmitting node, RSS information that the packet carries in the middle part, and the end identifier. Figure 5 illustrates one example packet transmitted by the 2nd node in scenario 2. The start of the packet is the node number, which is assigned with 0x02 because the ID of the transmitting node is #2. The middle part of the packet comprises 20 bytes, because there are 20 nodes in scenario 2. Each byte in the middle part denotes the previously stored RSS received from other nodes when this node is in the receiving stage. Specifically, the first byte in the middle part represents the received RSS from node #1, the second byte representing RSS from itself (invalid and is assigned with zero decibel in practice), the third byte representing RSS from #3, and so forth, until the last byte denoting RSS received from node #20. The last part of the packet is 0x7F, which serves as the end identifier. The readers can appoint other characters as the end identifier in their experiments.  In addition, there is an extra node outside the wireless network, which is MIB520CB made by Crowsbow in our experiment. This node plays the role of gateway which listens to the packets transmitted by each node in the wireless network and sends them to the computer via USB port. On the computer, an application program developed based on Visual C++ is used to read data from USB serial port and stores the packets to a file with CSV format. Then the CSV file is imported to MATLAB and pre-processed by a function of mean filter on MATLAB. Thereafter, the averaged RSS measurements are taken as the input to the reconstruction algorithm. After that, the shadowing image is obtained, which is then used to estimate the target's location. The framework of the experimental system is presented in Figure 6. In the following experiments, the Laplace prior is compared with student-t prior and Jeffrey's prior. The robustness to multipath fading will be validated via imaging, localization and tracking experiments. The experimental evaluation parameters are given in Table 1.

Imaging Experiment
The robustness to multipath fading can be reflected on the improved image quality in RTI. Before the imaging experiment, a calibration procedure is performed as described in the 2nd paragraph of Section 2.1. In the experiments, the target stands at 66 different positions in scenario 1, 33 positions in scenario 2 and 30 positions in scenario 3. In each case, the shadowing image is reconstructed under Laplace prior, student-t prior and Jeffrey's prior respectively based on SBL.
Inspired by the work in [1], the cylindrical model is used as the "true" shadowing image x c for reference, which is formulated as where x cj is the pixel value of the j-th pixel in x c . P j , P true and R denote the center coordinate of the j-th pixel, the target's true position and the radius of the cylinder respectively. Considering that the target has the waist width of about 0.76 m, thus for simplicity, R is set to 0.4 m in the following experiment. Moreover, the pixels that have non-zero pixel values outside the circular region are taken as artifacts. Figure 7 shows three groups of images. All the images are normalized and are vividly shown in the colorful illustration. Different color stands for different attenuation value, ranging from zero to one. The larger the attenuation-value (or pixel value) is, the more possible the target locates there. The top row, middle row and bottom row represent the images from scenarios 1-3 correspondingly. From left to right on each row, they denote the results from the cylindrical model, student-t prior, Jeffrey's prior and Laplace prior respectively. As can be seen, both the Laplace prior and Jeffrey's prior can reduce the artifacts in the reconstructed images compared to student-t prior, and Laplace prior achieves the best performance among them. From an overall perspective, we record the number of artifacts contained in the reconstructed image in each experiment and compute the average as one metric for comparison. The results are shown in Table 2. As can be seen, Laplace prior could obtain the shadowing image with least artifacts and this advantage is more obvious in the cluttered scenario 3 compared with that in scenarios 1 and 2. Following the method in [1], the MSE criterion defined in Equation (23) is also used to evaluate the quality of the reconstructed images where N is the total number of pixels in the image.
x and x c represent the vectorial representation of the normalized reconstructed image and the "true" shadowing image from the cylindrical model. Table 3 reports the comparative results of the reconstructed images in MSE. As can be seen, Laplace prior achieves the least MSE compared with Jeffrey's prior and student-t prior. This result consists with the result shown in Table 2, which leads us to draw the conclusion that Laplace prior can improve the quality of the shadowing image via weakening the interference from multipath fading.

Device-free Localization Experiment
After the shadowing imagex is obtained, the next step is to estimate the target's position from the image. Usually the target is assumed to be located at the center coordinate of the pixel with maximum pixel value, which is described as followŝ where P k denotes the central coordinate of the k-th pixel andx i stands for the pixel value of the i-th pixel in the estimated shadowing imagex. The estimated position of the target is denoted as a cyan circle in each image of Figure 7. The localization error is defined as the Euclidean 2 norm of the distance between the estimated positionP and the true position P true . e loc = P − P true 2 (25) In the device-free localization experiment, also the target stands at 66 different pre-defined positions in scenario 1, 33 positions in scenario 2 and 30 positions in scenario 3. In each position, the RSS variation y =r − r is computed as the observation for image reconstruction and target localization. Thereafter, the shadowing image is reconstructed using different priors based on SBL, and the target's location and localization error are computed following Equations (24) and (25). The localization error is recorded in each localization experiment and we take the average for comparison between different priors. The results of the mean localization error are summarized in Table 4. Meanwhile, the CDF (Cumulative Distribution Function) curves of the localization errors in each scenario are shown in Figure 8.  As can be seen from Table 4 and Figure 8, both Laplace prior and Jeffrey's prior have less localization errors than student-t prior, and Laplace prior is more advantageous, particularly in the cluttered scenario 3. The above results in localization also reveal that RTI using SBL with Laplace prior is robust to multipath fading.

Experiment in Localization Time
To validate the improved computational efficiency of using Laplace prior in localization, the time consumption experiment is conducted. As the same procedure described in Section 4.3, 66 different pre-defined positions in scenario 1, 33 positions in scenario 2 and 30 positions in scenario 3 are used as the samples. The localization algorithm is implemented on the platform of MATLAB. For each sample, the localization time (recorded by the CPU execution time) using different priors under SBL is recorded. Then the average time consumption is computed as the result for comparison, which is reported in Table 5. The results are obtained on a computer with 3.7 GHz CPU and 4G RAM. As can be seen from Table 5, a localization system using Laplace prior can reduce the localization time compared with student-t prior and Jeffrey's prior. Although Jeffrey's prior has lower localization error as reported in Table 4, it demands much more time as illustrated in Table 5. By contrast, the localization system using Laplace prior under SBL can achieve better localization accuracy with lower time consumption as well.

Device-free Tracking Experiment
To further validate the effectiveness of the proposed Laplace prior in reducing the interference from multipath fading, we conduct the tracking experiment in scenario 2. Before the tracking experiment, the RSS measurements for all the links in the empty scenario are collected in advance, which is denoted byr. Thereafter, as shown in Figure 9, the target moves from the start point to the end point following the directions as the arrows indicate. During this procedure, the RSS measurements are recorded which is denoted by r. Then we take y =r − r as the observation used for tracking. Start End Figure 9. True trajectory and Estimated trajectories under different priors.
In Figure 9, the estimated trajectory is connected by 36 position samples. Each position sample is obtained via the localization algorithm presented in Section 4.3. The mean tracking error is defined as the average of the localization errors from each position sample.
The mean tracking errors for different priors are reported in Table 6. As can be seen, the Laplace prior can achieve the least tracking error because of its robustness to multipath interference.

Conclusions and Future Research
To enhance the robustness to multipath fading in narrow-band RF localization, this paper explores the Laplace prior to characterize the shadowing image. Bayesian modeling and Bayesian inference especially the fast algorithm are presented to obtain the MAP solution in this paper. Experiments from three different scenarios are conducted to validate the advantages of the proposed approach in dealing with the multipath fading. Imaging, localization and tracking results reveal that RTI using SBL with Laplace prior is robust to multipath fading. Meanwhile, the proposed strategy improves the computational efficiency in localization as well.
Future research will be necessary to establish the prior that is suitable for multi-target case. Because each target may only occupy few pixels near the position where he stands, the whole image can be separated into several blocks or groups according equaling to the target number. Perhaps the structured priors could be explored to extend the proposed strategy to multi-target situation.