Practically, the energy consumption metric φ_Cost contains three basic types: sensing energyφ_s, signal processing energy φ_p and communication energyφ_c.

The sensing energy φ_s is determined by the sensing power p_s and sensing time t_s: (2) φ s = p s t s

The signal processing energy φ_p is shown with the following equations [12]: (3) φ p = ∫ t 0 t 0 + t p p p ( t ) d twhere t_p is the time taken for signal processing and p_p (t) is the instantaneous power of the processor.

However, the overhead energy caused by fusion is much lower than the energy caused by local preprocessing, so each wireless sensor node almost consumes processing energy at same level. Because of this, the signal processing energy φ_p can be ignored in sensor selection. For the same reason, the sensing energy φ_s can be ignored too.

For communication energy φ_c, given minimum communication power p₀ along a standard distance l₀, instantaneous communication power is in direct proportion to the square of communication distance l_c between the source sensor node and destination sensor node [11] (4) p c = l c 2 l 0 2 ( 4 π ) 2 β G t G r λ 2 p 0where G_t, G_r are transmitting gain and receiving gain respectively. λ is the wavelength and β is the system loss factor. So the communication energy is (5) φ c = p 0 ( 4 π ) 2 β G t G r λ 2 l c 2 t c l 0 2

Obviously, the first term in Eq. (5) is a constant. Moreover, in the incremental data fusion procedure, the amount of transmitted data is almost constant, so the communication time t_c can be also considered as a constant. So, l c 2 l 0 2 can be used as a dimensionless metric of communication energy φ_c.

Furthermore, for ensuring the sensing performance of WSNs, energy should be consumed evenly among all sensor nodes. Entropy theory is adopted here to measure the randomness of reserved energy in each sensor node. The entropy of reserved energy is as follow.

(6) H ( S i ( t ) ) = − ∑ k p ( E ˜ k i ( t ) ) log p ( E ˜ k i ( t ) )where E ˜ k i ( t ) is the estimated amount of energy reserved in kth sensor node when selecting ith sensor node for progressive signal processing at instant t. The bigger the entropy is, the more evenly the reserved energy is, so 1 H ( S i ( t ) ) is used to scale the impact of energy consumption. The combined resource consumption metric is as follow.

(7) φ C o s t ( S i ( t ) ) = ( l c i ) 2 l 0 2 H ( S i ( t ) )where l c i is the communication distance between current sensor node and ith sensor node.

Information utility is another important metric for measuring the contribution of selected sensor nodes. In practice, the contribution of sensor nodes can be ideally predicted by analyzing the theoretical model of the specific application and the characteristics of sensor nodes. This metric can be called φ_Predicted. Besides the predicted contribution, the information utility is always impacted by some other factors in the environment, such as obstacles and noise. However, it is difficult to measure their impact in practice. A more practical alternative is to estimate the confidence degree φ_Confidence of the wireless sensor node, which can be measured by the contribution of the wireless sensor node in previous instants.

In conclusion, the information utility metric function can be defined as follows: (8) φ Utility ( S i ( t ) ) = φ Predicted ( S i ( t ) ) φ Confidence ( S i ( t ) )

And then the objective function of routing in distributed P2P signal processing is: (9) M ( S i ( t ) ) = ( 1 − α ) φ Predicted ( S i ( t ) ) φ Confidence ( S i ( t ) ) − α [ ( l c i ) 2 l 0 2 H ( S i ( t ) ) ]

Target tracking is one of the essential capabilities in WSNs. Because distributed P2P signal processing framework has advantages in dynamic, robust, energy efficiency and data transmission, target tracking in a distributed P2P signal processing framework can improve tracking performance by purposefully selecting proper set of wireless sensor nodes for progressive signal processing and data fusion. Moreover, a distributed framework decreases the energy consumption and reduces the execution time of signal processing, and the P2P architecture can improve the robustness and reduce the congestion and energy consumption in WSNs. Besides signal processing framework, in some ways, specific signal processing algorithms also have significant impact on the performance of target tracking, which will be discussed in the following section.

Background subtraction is demonstrated as a low-cost, simple, but efficient method for target detection. It is successfully used in W⁴ real time visual surveillance system for tracking multiple targets [13]. Let Vⁱ (x) be the intensity of location x in the ith image of N consecutive images array V. σ(x) and η(x) are the standard deviation and median value of intensities. The initial background model [γ_m (x), γ_n (x), γ_d (x)] is obtained as (10) [ γ m ( x ) γ n ( x ) γ d ( x ) ] = [ min i { V i ( x ) } max i { V i ( x ) } max i { V i ( x ) − V i − 1 ( x ) } ] , where | V i ( x ) − η ( x ) | < 2 * σ ( x ) .

After initialization, three change maps, the detection support map (gS), motion support map (mS) and change history map (hS), are adopted to represent the number of times a pixel location is classified as a background pixel, moving pixel and elapsed foreground pixel respectively.

(11) g S ( x , t ) = { g S ( x , t − 1 ) + 1 if x is background pixel g S ( x , t − 1 ) if x is foreground pixel (12) m S ( x , t ) = { m S ( x , t − 1 ) + 1 if ( | I ( x , t ) − I ( x , t + 1 ) | ) > 2 * σ ) ∧ ( | I ( x , t − 1 ) − I ( x , t ) | > 2 * σ ) m S ( x , t − 1 ) otherwise (13) h S ( x , t ) = { 255 if x is foreground pixel h S ( x , t − 1 ) − 255 N otherwise

Change maps are set to zero after the background model is updated.

For updating the background model, the new background model [γ_m (x), γ_n (x), γ_d (x)] is determined as follows.

(14) [ γ m ( x ) , γ n ( x ) , γ d ( x ) ] = { [ γ m b ( x ) , γ n b ( x ) , γ d b ( x ) ] if ( g S ( x ) > r 1 * N ) ( pixel − based ) [ γ m f ( x ) , γ n f ( x ) , γ d f ( x ) ] if ( g S ( x ) < r 1 * N ∧ m S ( x ) < r 2 * N ) ( object − based ) [ γ m c ( x ) , γ n c ( x ) , γ d c ( x ) ] otherwisewhere [ γ m c ( x ) , γ n c ( x ) , γ d c ( x ) ] is the current background model; [ γ m b ( x ) , γ n b ( x ) , γ d b ( x ) ] presents that the pixel is classified as background pixel, while [ γ m f ( x ) , γ n f ( x ) , γ d f ( x ) ] presents the pixel is classified as foreground pixel. As presented in [13], r₁ and r₂ are scaling factors to adjust the effect of number N, they are typically 0.8 and 0.1, respectively.

Giving the minimum γ_m (x), maximum γ_n (x), and the median of the largest interframe absolute difference γ_d' images, a pixel x from image I^t is a foreground pixel if: (15) B ( x ) = { 0 background ( ( I t ( x ) − γ m ( x ) ) < r 3 γ d ′ ) ∨ ( ( γ n ( x ) − I t ( x ) ) < r 3 γ d ′ ) 1 foreground otherwise .where r₃ is set to 2 in our system. According to the complexity of calculation, the computational cost of this algorithm is low, and the detailed discussion can be found in [13]. It means that background subtraction is practically feasible in WSNs.

Normally, the output result of target detection is the minimum boundary rectangle (MBR) of target. Target detection is always the basic and preliminary item for target classification and tracking which will be detailedly discussed in the following sections.

With the MBR results of target detection, the target information is acquired. Next, target classification is desired for further signal processing. Target classification contains two key elements: feature extraction and division. Feature extraction refers to a transformation for reducing the number of effective features of an original data set by retaining most of the intrinsic information. For images, compression can be considered an effective technique for feature extraction.

The use of the discrete wavelet transform (DWT) for embedded lossy image compression is now well established [14]. The wavelet transform of a function f (x)∈ L₂ (R) is defined as (1-D case): (16) W T { f ( x ) ; δ a , δ b } = ∫ f ( x ) ψ δ a , δ b ( x ) d xwhere ψ_δa,δb (x) defines the family of the wavelet functions: (17) ψ δ a , δ b ( x ) = 1 | δ a | ψ ( x − δ b δ a )where δ_a ≠ 0 is the scale of the transform and δ_b is the parameter of spatial location.

With the discrete translation of function f(x) on a dyadic scale δ_a =2^j and discrete translation δ_b = 2^jk, the discrete wavelet transform (DWT) can be presented as follow [15]: (18) DSWT { f ( x ) ; 2 j , 2 j k } = C j , k = W T { f ( x ) ; δ a = 2 j , δ b = 2 j k }where C_j,k is the wavelet coefficients of the function f (x), and the wavelet function forms a orthogonal and complete dyadic family: (19) ψ j , k ( x ) = 2 − 1 2 j ψ ( 2 − j x − k )

Thus, the function f (x) may be obtained from its wavelet coefficients C_j,k: (20) f ( x ) = ∑ j ∈ Z ∑ k ∈ Z C j , k ψ j , k ( x )

For discrete signals f (n), n∈Z, the DWT is defined as: (21) D W T { f ( n ) ; 2 i , 2 j k } = { c j , k } = ∑ n ∈ Z f ( n ) g j * ( n − 2 j k )where g j * ( n − 2 j k ) is the discrete equivalent of the 2 − 1 2 j ψ ( 2 − j ( x − 2 j k ) ). With DWT, the f (n) can be written as a multiresolution decomposition on J levels, j =1,…, J, given by [15]: (22) f ( n ) = ∑ j = 1 J ∑ k ∈ Z c j , k g ˜ j ( n − 2 j k ) + ∑ k ∈ Z e j , k h ˜ J ( n − 2 J k )where g̰_jj (n − 2^jk) is the synthesis wavelets and discretely equal to ψ_j,_k, and the scaling coefficients e_j,_k is defined as: (23) e j , k = ∑ n f ( n ) h J * ( n − 2 J k )where h J * ( n − 2 J k ) is the scaling sequences.

For images, a wavelet descomposition is applied first to image rows and then to columns. The two dimensional wavelet transform leads to a descomposition in four sets of coefficients: approximation and details in three different orientations, vertical, horizontal and diagonal. The full descomposition is obtained by iterating the filtering on the approximation set, where the approximation at j+1th level can be approximately considered as the compressed image information [14].

However, in WSNs, the limited processing abilities of wireless sensor nodes limit the possibility to employ DWT algorithm. To simplify the computation, a lifting scheme (LS) is introduced in DWT [16]. LS exploits the redundancy between the high pass and low pass filters necessary for perfect reconstruction and reduces the number of arithmetic operations up to a factor of two, compared to the filter-bank implementation, The LS based wavelet transform seems to be an ideal candidate for embedded image compression issues, due to its simple structure and good decorrelation properties [17].

As shown in Figure 2, in LS, the input data are split into two signals with evenly and oddly indexed samples respectively. One signal is convolved with a primal lifting filter pr₁ (z), and then the role of the two signals is then reversed and dual lifting filter du₁ (z) is applied, where pr₁ (z) and du₁ (z) are simple and short FIR filters. After k iterations of primal and dual lifting, the input signal χ^(n-1) is split into d⁽ⁿ⁾ which corresponds to the details of χ^(n-1), and χ⁽ⁿ⁾ which corresponds to the approximation. In reconstruction, the reconstruction filter p r i ′ ( z )and d u i ′ ( z ) are exactly equivalent to its decomposition counterpart pr_i (z) and du_i (z), without its sign, i.e., (24) p r i ′ ( z ) = − p r i ( z ) (25) p u i ′ ( z ) = − d u i ( z )

For conquering the disadvantage of lossy image compression in LWT, a further improvement is achieved by combining the benefits offered by the integer wavelet transform (IWT) and integer LS, because IWT enables true lossless reversible transformations [18]. The structure of ILWT is illustrated in Figure 3. Because of completeness, the performance of IWT is slightly worse than DWT. However, ILWT structure can greatly reduce the computational complexity which implies that ILWT is an ideal choice for low energy systems. In practice, ILWT has been successfully used in embedded processing of wireless sensor nodes for image compression in wireless sensor networks [18]. Then the compressed image can be considered as the compact representation of original image for target classification, which can decreasing the computational complexity in the polynomial computation in each node and ensuring this selected subspace can retain enough intrinsic information of the original space.

After image compression, SVM is used to construct a classifier for target classification. A SVM is essentially a linear classifier operating in a higher dimensional space. For reducing computational load, a kernel function K(·) is used to perform input-feature transformation φ(·), which is defined as follows: (26) K ( m ⋅ n ) = ϕ ( m ) ⋅ ϕ ( n ) .

With the kernel functions, the basic form of SVM is as follow.

(27) f ( m ) = sign ( ∑ i = 1 1 q 1 ( ϕ ( m ) ⋅ ϕ ( m i ) ) + b ) .where q_i is the weighting factors (0 < q_i< ∞), b is a scalar threshold for adjusting the results of classification, here, it is set to 0. In this paper, a third degree polynomial kernel is used: (28) K ( m , n ) = ( m T n + 1 ) 3

Unfortunately, the training phase of SVM algorithm may take a long time. However, once the classifier is trained, SVM just need to estimate the posterior probability for each class, so the computational complexity is rather low. For multi-category classification, multiple classifiers can be trained and combined for more complicated classification. Moreover, some distributed training methods for SVM algorithms were proposed in many literatures. These methods are demonstrated feasible in WSNs [19]. After training, the prerequisite parameters of kernel functions are transferred to each node for classification. Each node just needs to perform some polynomial operation, which is reasonable for the processing ability of wireless sensor nodes.

After target detection, wireless sensor node uses historical information of target location for target tracking and estimation. Target tracking is always performed by Kalman filter. However, it is extremely challenging to implement a Kalman filter to track a maneuvering target if the dynamic model of target is highly nonlinear [20]. Although a standard particle filter can solve the nonlinear non-Gaussian problem [21], it cannot solve the estimation error of cumulating problems when manoeuvers occur. Furthermore, some algorithms have been proposed for maneuvering target tracking, such as unscented particle filter [20], radial basis function based particle filter [21]. But these algorithms are computation-expensive for wireless sensor node with simple embedded data processing capabilities.

When the target is moving in Cartesian coordinates, the target motion can be described by the following state space model, which is expressed along the X-axis for simplicity.

(29) x ( k + 1 ) = Fx ( k ) + Gw ( k )

And the measurement model is given by (30) y ( k + 1 ) = Hx ( k ) + v ( k )where the process noise w (k) and the measurement noise v (k) are assumed to be zero mean, independent white Gaussian sequences. In the case of a maneuvering target, the state vector x is defined as x = [x ẋ ẍ] ^T, where x is the target position, ẋ is the velocity and ẍ] is the acceleration. The corresponding matrices are (31) F = [ 1 T T 2 2 0 1 T 0 0 T ] , G = [ T 3 6 T 2 2 T ] , H = [ 1 0 0 ]where T is the measuring interval.

Here, a new state vector x′ = [x₁x2 x3] is given, where x′ and x are related by the following equation (32) x ′ = P 0 − 1 xand the transformation matrix P 0 − 1 is given by (33) P 0 − 1 = [ H HF HF 2 ] = [ 1 0 0 1 T T 2 2 1 2 T 2 T 2 ]

Therefore x_i, i =1,2,3, in vector x′ are related to the target position, velocity and acceleration by (34) [ x 1 x 2 x 3 ] = P 0 − 1 [ x x ˙ x ¨ ] = [ x x + T x ˙ + T 2 2 x ¨ x + 2 T x ˙ + 2 T 2 x ¨ ]

The new state vector x′ contains only the target position information at each subsequent time instant, and it is equivalent to x in presenting the motion of target since they are related by a linear transformation. This implies that the problem of maneuvering target tracking can be viewed as a problem of adaptive time series prediction [22].

Here, the ARMA model is adopted due to its outstanding performance in model fitting and forecasting and its light computational cost [23]. ARMA model is a widely-used model for the prediction of future values. It contains two terms, the auto regressive (AR) term and the moving average (MA) term. The AR term is a linear regression of current value against one or more prior values. It captures the dependency of current value and its nearest prior values. And the MA term is introduced to capture the influence of random shocks to the future. In general, a linear system can be derived as follow.

(35) s ( n ) = ∑ i = 1 p a i s ( n − i ) + ∑ i = 0 q b i ω ( n − i )where the system input is ω(n), output is s(n), a_i and b_i are the AR coefficients and MA coefficients respectively, p is the order of the system and q ≤ p. The measurement equation is (36) y ( n ) = s ( n ) + υ ( n ) , 0 ≤ n ≤ N − 1where υ(n) is the measurement noise and N is the number of measurements.

Normally, the MA coefficients are estimated by minimizing the error between the actual measurements and the weighted impulse response sequence generated by the estimated denominator coefficients.

For reducing the computation complexity in estimating the AR and MA coefficients, the AR coefficients a_i is estimated by using the robust singular value decomposition (SVD) based linear predictive coding algorithm (LPCA) method, while the MA coefficients b_i are estimated by minimizing the error between the actual measurements and the weighted impulse response sequence generated by the estimated denominator coefficients as usual [22]. The SVD based LPCA method is presented as follows.

The forward prediction in LPCA is described by (37) y ^ ( n ) = ∑ i = 1 M a ^ i y ( n − i )where M is the order of the filter, ŷ(n) is the estimate, y(n) is the measured data, and the above equation can be written as (38) Y ^ = A a ^where â = [â₁,â₂,…â_M]^T, ŷ is a (N−M)×1 vector and A is a (N−M)×M matrix, p ≤ M ≤ N−M. The optimal vector a is given as follows: (39) A T A a ^ = A T Y

For decreasing the computational complexity of computing A^T A, the robust SVD method is applied. Let (40) A = U [ ρ 0 0 0 ] V Twhere U = [u₁,u₂,…u_N−M] and V = [v₁,v₂,…v_M] are unitary matrices of dimension (N−M)×(N−M) and M×N, ρ = diag [ρ₁, ρ₂,…,ρ_r] is an r×r matrix where and r is the rank of A. So the optimal weights â as (41) a ^ = ∑ k = 1 r ρ k − 1 v k u k T Y

According to the target state model, a third order filter is ideally sufficient to track a maneuvering target. However, it must be noted that the order of the proposed ARMA filter should be higher than the order of the target state model, especially for target tracking with noise. In this paper, the order of the system is set to p = 4, q = 3. Furthermore, tracking target with ARMA model requires a delay of N steps for the parameter estimation. However, this problem only occurs in the initialization. When sufficient measurements are available, no delay is required.

For integrating the bearing information of each wireless sensor node and achieve target localization, a simple multi-view localization method is proposed to conquer the negative effects of occlusion and obstacles by fusing MBR results of several sensor nodes. It is assumed that the measuring uncertainty of target position can be effectively approximated by a 2-D Gaussian distribution in MBR. As illustrated in Figure 4, after projecting the Gaussian distribution to the ground, multi-view localization fuses Gaussian distribution results by (42) E ( x ∣ τ 1 , … τ n ) = ξ ( ∑ i = 1 n p ( x ∣ τ i ) × p ( τ i ) )where p (x | τ_i) denotes Gaussian distribution acquired by ith wireless sensor node, p(τ_i) is the scale factor. Here, it's given by distance between wireless sensor node and target. ξ presents the normalizing operator. x is the fused result. Then the approximate position of moving target can be calculated as expectation of the distribution X: (43) E ( x ) = ∑ i x i p ( x i )

The fused results are at most as large as the uncertainty of the most accurate individual sensor node, if all sensor nodes work normally. But if some sensor nodes are broken or blocked, the corresponding results are not credible. So, if the probability distribution becomes zero or even low after fusing the MBR results of some sensor nodes, the relative sensor nodes will be considered as useless ones, the results will not be fused and the original results will be transmitted to other nodes instead.

Furthermore, a simple voting mechanism, utilizing the status consistency and the target tracking information, is applied. For each node, one vote is cast for the resulting target class, and votes are counted for 5 neighboring frames at each step, the target class receiving the largest number of votes decides the target's class in this node. This voting mechanism reduces the number of random misclassifications in video sequence caused by the temporal occlusion.

Because of the inherent characteristic of the multi-view localization method, it can be incrementally carried out in distributed P2P signal processing framework. In this application, the resource consumption retains the general form which is described in Section II, and the application-specific information utility metric is discussed in here.

Image sensor node is a kind of bearing-only sensor node. For bearing-only sensor nodes, Wang [24] has proposed an entropy-based sensor selection heuristic, which selects an informative sensor such that the fusion of the selected sensor observation with the prior target location distribution would yield on average the greatest or nearly the greatest reduction in the entropy of the target location distribution.

The entropy of the probability distribution of the view of wireless sensor node i, H i v, is (44) H i v = − ∫ p ( z i v ) log p ( z i v ) d z i vwhere Z i v is the view of sensor i about the sensor location. For computing the distribution in numerical method, the discrete representation p ( Z i v ) with a grid δ z i v, H i v is computed as (45) H i v = − ∑ p ( z i v ) log p ( z i v ) δ z i v

And the entropy of the sensing model of wireless sensor node i for the actual target location x^t is approximated as (46) H i s = − ∫ p ( z i | x ^ ) log p ( z i | x ^ ) d z iwhere x̂ is the maximum likelihood estimate of the target location in the prior target location distribution in the partially fused results.

Because the wireless sensor node i with larger entropy difference H i v − H i s yields on average larger reduction in the uncertainty of the posterior target location distribution [24], the sensor selection can be considered as the following problem (47) i ^ = arg max ( H i v − H i s )which implies that the H i v − H i s can be used as the predicted contribution metric φ_Predicted.

Furthermore, the confidence degree φ_Confidence of the wireless sensor nodes can be evaluated by the Mahalanobis distance between tracking result of the wireless sensor node and the final fusion result at last instant, because Mahalanobis distance can easily determine the similarity between two probability distribution sets [25]. The confidence degree is defined as: (48) φ Confidence ( S i ( t ) ) = − ( X s i ( t − 1 ) ′ − X ^ t − 1 ′ ) T Σ ^ t − 1 ′ − 1 ( X s i ( t − 1 ) ′ − X ^ t − 1 ′ )where X s i ( t ) ′ is the probability distribution of sensor at instant t−1, X ′ ^ t − 1 and Σ ^ t − 1 ′ is the mean and covariance of probability distribution of final fusion result at instant t−1 respectively.

Then the objective function of sensor node selection during information fusion is as follow.

(49) M ( S i ( t ) ) = − ( 1 − α ) ( H s i ( t ) v − H s i ( t ) s ) ( X s i ( t − 1 ) ′ − X ^ t − 1 ′ ) T Σ ^ t − 1 ′ − 1 ( X s i ( t − 1 ) ′ − X ^ t − 1 ′ ) − α ( ( l c i ) 2 l 0 2 H ( S i ( t ) ) )where the relative weight is set to α= 0.5 in this paper.

The criterion for finalizing fusion is evaluated by the entropy of probability distribution of target position which is defined as (50) H p ( X ) = − ∑ i p ( x i ) log p ( x i )

If the entropy exceeds the predefined value, the localization is considered to be finished, then the current wireless sensor node will transmit the result back.

For multi-target tracking, multiple leader sensor nodes will be dynamically and autonomously established and trigger the distributed P2P signal processing based multi-view localization for each target. The inherent characteristic of P2P network gives many benefits on distributed multithreading signal processing, which implies that distributed P2P signal processing framework will perform better in multi-target tracking.

Because the collision often occurs in multi-target tracking, it is difficult to associate the measurements of each sensor node with individual targets. The assignment between tracks and measurements is formulated as a discrete optimization problem to maximize a dimensionless global likelihood ratio of the measurements-to-tracks associations. The nearest neighbor approach [26] is adopted to associate the measurements to tracks by determining which measurement is closest to the predicted target-originated measurement. For simplifying the computation, Mahalanobis distance is also used to determine the optimal associated results for each target: (51) i ^ = arg min 1 < i < N ( ( P Predicted − X ^ i ) T Σ ^ i − 1 ( ( P Predicted − X ^ i ) ) )where P_predicted is the target position predicted by ARMA model, X̂_i and ∑̂_i are the mean and covariance of probability distribution of ith potential position, N is the number of potential position.

Actually, the multi-view localization algorithm can also be carried out in traditional centralized client/server framework and distributed client/server framework. For comparing the performance of centralized and distributed client/server frameworks and the proposed distributed P2P framework and studying the tracking performance of the combined tracking system with several specific algorithms, a real world target tracking experiment is described and the results are analyzed in the following section.