Multi-target Localization Based on an Improved FOCUSS Algorithm with Missing Information

The situation where the observation stations or the sensors cannot fully detect the targets is called missing information, which makes it difficult to locate targets. Due to its sparsity, Compressed Sensing (CS) can be applied to multi-target localization with missing information. An improved Focal Undetermined System Solver (IFOCUSS) algorithm is proposed, by dynamically updating the regularized factor that has a significant impact on the convergence of iteration processes. What’s more, a re-weighting method is adopted for the weighting matrix to improve convergence property and accelerate convergence, and a new method of choosing correct items from a sparse estimation vector is proposed to improve the positioning accuracy. Several simulations corroborate its notable reconstruction performance, which is illustrated by correctly estimating five targets with only 11 sensors on a moving station, achieving better convergence at lower signal-to-noise ratios (such as 12 dB) with about half the run time of Regularized FOCUSS (RFOCUSS) and being able to estimate two targets even with missing information up to 60%. Under the same conditions of missing information, IFOCUSS can estimate more targets with smaller positioning errors than RFOCUSS, achieving better convergence performance by eliminating almost all interference terms.


Introduction
In the actual multi-target localization environment, due to some interference factors, observation stations can not obtain all target information at some observation points, which is called missing information.Considering locating multiple targets with missing information with the target number unknown, Direct Positioning Determination (DPD) corrects the disadvantage that most "two-step" passive location algorithms may lose information during two processing stages, with strong resolution and relatively higher positioning accuracy [1] .The monitoring area of the observation station can be equidistantly divided into multiple grids.If the number of targets is much smaller than that of grids, the problem of multi-target passive localization can be transformed into the recovery of a sparse signal in the spatial domain.The Nyquist sampling theorem has a high bandwidth requirement.Subsequently, the Compressed Sensing(CS) theory was proposed, which can sample at a frequency much lower than the Nyquist sampling rate [2] .This paper combines DPD with a CS algorithm to solve the problem of multi-target localization with missing information.
Once proposed, CS has attracted attention and has been widely used in many fields such as biosensors [3] , wireless communications [4] , and optical imaging [5] .CS algorithms are mainly divided into three categories: greedy algorithms that directly optimize the 0  norm, algorithms based on the Bayesian framework, and convex optimization algorithms that approximate the 0  norm through the 1 norm.Greedy algorithms mainly address the minimum norm problem of the recovery process after the underdetermined sampling of sparse or compressible signals.The Match Tracking (MP) algorithm [6] is put forward based on a greedy idea, using an overcomplete dictionary atom to represent signals in Hilbert space.For faster convergence speed, the Orthogonal Matching Tracing (OMP) algorithm [7] was proposed by Schmidt to orthogonalize the selected atoms.Then, the Gaussian matrix is introduced as a measurement matrix to Stagewise Orthogonal Matching Pursuit (SWOMP) algorithm [8] but has a poor reconstruction accuracy.To solve this problem, researchers filtered atoms again in matching tracking, optimized the index set, and proposed the Regularized Orthogonal Matching Pursuit (ROMP) algorithm [9]   .In recent years, the Potential Orthogonal Matching Pursuit (POMP) is proposed for moving target localization with better recovery performance than OMP [10] .Multipath orthogonal matching pursuit (MPOMP) is developed from MP and OMP, as an equivalent source method (ESM) based on near-field acoustic holography (NAH) methods with higher efficiency than one norm convex optimization (L1CVX) and iteratively reweighted least squares (IRLS) [11] .The Bayesian algorithm was originally derived from Tipping's Relevance Vector Machine (RVM) theory [12] and then was developed into a Sparse Bayesian Learning (SBL) method based on RVM [13] .Until 2008, sparse Bayesian theory was first used to solve the problem of CS reconstruction and then improved by introducing a Laplacian before obtaining high-resolution reconstructed signals through the multi-task learning mode [14] .To improve the accuracy and robustness of passive location, the Variational Bayesian Inference (VBI) algorithm is proposed by a layered Gaussian mixture prior model and the posterior distribution of hidden variables [15] .Then, the Concurrent Variational Bayesian Inference (CVBI) algorithm is developed to mitigate the impact of noise on RSS-based CS methods [16] .Convex optimization algorithms can find globally optimal solutions, which are superior to the MP algorithm at the expense of computational speed.Non-convex problems can be converted into convex problems to seek signal approximation, which are optimization approximation algorithms, such as Focal Undetermined System Solver (FOCUSS) [17] algorithm, Iterative Reweighted Least Squares (IRLS) algorithm.Currently, FOCUSS and its derived algorithms are mostly used in the field of sparse imaging [18][19][20] , choosing Gaussian random matrix as dictionary matrix or sampling randomly, which are not suitable for multitarget localization.In the field of Direction Of Arrival (DOA) estimation, a new method is proposed to improve the estimation performance for non-coherent sources [21] .A new DOA estimation algorithm was proposed [22] , reducing the amount of computation through compressed sampling and obtaining better estimation accuracy by orthogonalizing the Gaussian matrix.It has been confirmed by several simulations that the regularization factor and norm p have a significant effect on the estimation performance of RFOCUSS [23] .In most cases, it is best to set it p as 0.8, which is also applicable to the problem studied here.
As sparsity must be known to greedy algorithms and it costs a large amount of computation for Bayesian algorithms, the FOCUSS algorithm is used.The reconstruction performance of the original FOCUSS algorithm is too poor [24] for complex positioning environments.In the derived FOCUSS algorithms, some are not suitable for the Multiple Measurement Vectors (MMV) models [25] , and some involve the conversion from the time domain to the Doppler frequency domain [26,27] , which makes improvements to RFOCUSS suitable for multi-target localization.For better convergence, a quadratic weighted FOCUSS algorithm with an improved regularization factor is proposed with three improvements on RFOCUSS, which is named IFOCUSS algorithm.Firstly, the regularization factor is dynamically updated in iteration processes to improve the balance, instead of taking a fixed initialization value.Secondly, a new vector is introduced to re-weight the weighting matrix for faster convergence and better convergence properties.Thirdly, an improved method of selecting the correct terms from the sparse estimation vector is proposed for better positioning performance.

Positioning environment
We consider a scene of K far-field targets monitored by an airborne observation station with an array consisting of N sensors, where N K  .The station moves at a constant speed v with an initial position 1 ξ .We take an observation point every  second to detect targets and H observation points in all.Then the h -th observation point is at   and no target is on the projection of the motion trajectory.

Two cases of missing information
We regard a situation where the observation stations or the sensors cannot fully detect the targets as missing information.Two main cases of missing information are considered in the above positioning environment.One is that there are obstacles in the positioning environment, and the observation station cannot detect some targets at certain observation points regardless of multipath effects.In this case, the amount of target information that can be observed at each observation point is different.Figure 1 shows an example in the first case.The solid dots represent the selected observation points on the trajectory, and the quadrangular stars represent the sensors arranged on the station.Triangles represent targets on the ground, and the gray three-dimensional square represents an obstacle.The line between an observation point and a target indicates that the target can be detected by the station at this point.We can observe that the observation station is unable to detect the targets K P and 1 P , respectively, at the observation points 1 ξ h ξ and in this example.That is, it occurs when some target information is lost at the two observation points while the targets are located by the station.

Figure 1. A vertical view of multi-target localization in the first case of missing information
The other is that some sensors are damaged and cannot detect all target information and feed it back to the observation station.In this case, the same amount of target information is missing at each observation point, resulting in a decrease in the total amount of target information obtained by the observation station.

Related expressions of RFOCUSS
We assume that the CS solution model is: where d is the sparse observation vector,  is the dictionary matrix, ε is the estimation error, and g is the sparse estimation vector we need to reconstruct.Let m denote the number of the current iteration.Mathematically, the FOCUSS algorithm of 1 m  the -th iteration can be expressed by three steps: , where W Where W is the weighting diagonal matrix, and     represents the pseudo inverse of the argument.
When there is noise, it brings unstable results by the calculation of the inverse matrix, leading to poor reconstruction accuracy.RFOCUSS introduces a regularization term reconstruction and the cost function can be rewritten as: where  is the regularized factor related to noise, and p  indicates the p  norm of the argument.Thus, RFOCUSS is given by: Where   T  represents the transpose of the argument and I is an identity matrix.

Analysis of reconstruction performance of RFOCUSS
The reconstruction performance of RFOCUSS is analyzed by a simulation of multiple targets observed by an observation station at a constant velocity, where the target number is unknown.We consider that five targets are scattered in a 100 100  square kilometers area which is sampled into 40 40  grids with side length 2 5 .km s.We assume that the station is initially located at   , where  is the wavelength.We take 100 Monte Carlo simulations in all and SNR is set to be 5dB .The targets may exist on either side of the motion trajectory of the observation station when azimuth is observed as target information for positioning.Thus, the real target location has a symmetric point (if within the monitoring area) about the station trajectory.That is, if the amplitude of a grid corresponding to a real target is  , there must be two items * g with an amplitude equal to  .The calculation result shows that the five groups of symmetrical grids corresponding to the above five targets are     for the first six iterations in a Monte Carlo simulation, with the grid number as the X-axis and the corresponding amplitude as the Y-axis, which demonstrates the gradual iterative process of RFOCUSS.It is clear that the sparse estimation vector barely changes after the third iteration, with many interference terms retained, which means that the convergence of RFOCUSS is not quite well.
Figure 3 is the sparse estimation vector m g at the sixth iteration, with five groups of correct grids corresponding to these targets and an incorrect grid marked.It shows that the amplitudes of the grids corresponding to the five targets are close to each other, except for the target 3 P .We can also observe that the amplitude of the interference grid   1148 is 1.5 times more than that of the grid corresponding to the target 3 P .In other words, the interference items cannot be filtered out only by decreasing the threshold, resulting in many erroneous locations in the positioning result based on RFOCUSS.

The IFOCUSS Algorithm
For better convergence and fewer interference items of sparse estimation vector, a quadratic weighted FOCUSS algorithm with an improved regularization factor is proposed with three improvements.

Updated regularized factors dynamically
The regularized factor  has a significant impact on RFOCUSS, which is generally taken as a fixed value.We consider updating it dynamically during iterations according to the result obtained from the previous iteration.Initially, it is related to the residual, that is, the sparse observation vector.We make 0 H 00  d as the initial value.From a formula perspective, we denote improving the balance since it only exists in Formula (6) during iterations.

00
 and 01  are both in the order of 2 10  , where   H  denotes the conjugate transposition of the argument.

Re-weight the weighting matrix
As the number of iterations increases, amplitudes of the non-zero terms in the weighting matrix W become larger, which means that it increasingly converges to the corresponding terms of the real targets. norm minimization during iterations [28] .This can be accomplished by multiplying the elements that may be zero by smaller weighting coefficients and multiplying nonzero elements by larger weighting coefficients.Based on this, we can re-weight W it to improve its convergence performance.A weighting vector c is introduced to construct W by selecting only the maximum amplitudes of c as the diagonal elements of  m W : where  norm of m c , and is the set of base coordinates corresponding to the first 2m element with the largest amplitudes in indicates that the argument is related to the first weighting.Thus, the first sparse estimation vector  m g can be obtained according to Formula (6).After that,  m g is used instead of m c re-weight  m W , and then the final sparse estimation vector m g can be solved according to Formula (6) at the m iteration.

A new method of processing the reconstruction vector
A new method of avoiding the selection of interference ones is proposed when we select grids corresponding to the real targets from the reconstruction vector * g .We have mentioned that the real target location has symmetric points about the station trajectory.Based on this point, we can process * g as follows: Firstly, we solve and obtain the corresponding index index g max g .
Secondly, we find out if there are two items * g with an amplitude equal to max g .If not, max g is an interference item.The index g -th element * g and its adjacent two grid points are as follows: We re-calculate   * max g  g max and repeat the above steps until index g has a symmetric point, which corresponds to an estimated target.
Thirdly, we assume that there are  grids whose amplitudes are greater than the threshold max g  * g , where  is the threshold coefficient.Then the number of estimated targets in the multi-target positioning results by IFOCUSS should be; Finally, we make the following loop: we assume that the number of the remaining estimated targets is  .We calculate the maximum amplitude term and its corresponding grid number * g and determine whether it has a symmetric grid.If so, we consider it as a correct grid and calculate its threedimensional coordinate.We process Formula (8) and decrease  , then repeat the above steps until 0   .If not, the grid calculated is invalid.We process Formula (8) and recalculate the maximum amplitude term and its corresponding grid number * g .The above is the optimized process of locating multiple targets by the IFOCUSS algorithm.Figure 5 is the sparse estimation vector for the first five iterations.Compared with Figure 2, we can observe that IFOCUSS converges faster than RFOCUSS, as the reconstruction result at the third iteration is well enough, with fewer interference items and more energy focused on the correct grids.Furthermore, IFOCUSS has a better convergence performance and fewer interference items, embodied in the significantly larger amplitudes of symmetric grids in the selected five groups than that of invalid grids, with a difference of approximately four times.

Comparison of the performance of the two algorithms under different signal-to-noise ratios
The following simulations were performed to more intuitively compare the reconstruction performance and speed of the two algorithms under different signal-to-noise ratios (SNRs).Table 1 shows the positioning errors of five targets by IFOCUSS and RFOCUSS under different SNRs.Errors exceeding 10% are marked in red.than 10%, or 100%.This means that it completely filters the interference terms of the sparse estimation vector, where the amplitudes of the grids that are independent of actual targets are very close to zero.In contrast, RFOCUSS has a large number of interference terms in its sparse estimation vector, resulting in many positions in the estimation results that are unrelated to the actual targets.In summary, IFOCUSS has a higher convergence performance than RFOCUSS when  2 shows the run time of the two algorithms under different SNRs.The run time of multi-target localization by RFOCUSS is approximately twice that of IFOCUSS, which means IFOCUSS has a faster convergence speed than RFOCUSS regardless of SNR.Both algorithms achieve the fastest convergence speed when SNR is around 5dB , and the more the SNR deviate 5dB s, the longer the algorithm reconstruction time generally is.
In sum, compared with RFOCUSS, IFOCUSS has better convergence performance and convergence speed, reflected in more accurate estimation results at very low SNRs and less run time when locating multiple targets regardless of SNR.

Positioning Performance of IFOCUSS With Missing Information
What needs to be focused on is the positioning performance of IFOCUSS compared to RFOCUSS when it occurs to missing information.Simulations and analysis of positioning results by IFOCUSS and ROCUSS in the two cases of missing information are performed as follows.Positioning conditions are the same as before, except for the modifications indicated below:

Location of targets in the first case of missing information
Table 3 provides four examples of the first case of missing information, indicating the subscripts of targets that can be observed at every observation point.Table 4 illustrates the positioning errors by IFOCUSS and RFOCUSS under the conditions of Table 3. Errors exceeding 10% are indicated by "-".
As shown, in Examples 1, 3, and 4, the number of targets that can be estimated by IFOCUSS is one to two (20% to 40% proportionally) more than that by RFOCUSS.In the first three examples, the positioning errors of IFOCUSS are much less than that of RFOCUSS, indicating better reconstruction accuracy of IFOCUSS.In conclusion, RFOCUSS is more sensitive to missing information and the number of targets that can be estimated by IFOCUSS is always more than or equal to it when several obstacles exist between some observation points and some targets.Besides, the positioning errors of IFOCUSS are smaller in general.

Location of targets in the second case of missing information
An invalid sensor will not be able to detect the target information and feed it back to the observation station.Because the change in sensor position on the linear array is almost negligible, we consider the failure of sensors as a decrease in the number of valid sensors.We decrease the number of valid sensors from 15 to simulate the second case of missing information.Table 5 shows the positioning errors of the two algorithms when more and more sensors are invalid.We can see from the comparison: IFOCUSS only requires 11 sensors to locate five targets under the above positioning conditions, while RFOCUSS requires 13 sensors.When the target information detected by three or four sensors (more than 20% of the target information proportionally) is missing, only three targets can be estimated by RFOCUSS, of which several positioning errors are much greater than that of IFOCUSS.As the number of valid sensors decreases to 8 to 10, the targets that can be estimated by both algorithms are the same, and the positioning errors based on IFOCUSS are almost smaller than half of the corresponding errors based on RFOCUSS.RFOCUSS will be invalid when the number of valid sensors continues to decrease, reaching up to 60% of information lost.Under this condition, the sensors can not process the information of almost the same number of targets via RFOCUSS, resulting in no target that can be estimated.However, 1 to 2 targets still can be estimated via IFOCUSS within the allowable error range, as a more balanced regularized factor and a quadratic weighted weighting matrix are used for better convergence.In short, the figures in Table 3 lead us to the conclusion that RFOCUSS is more sensitive than the improved algorithm to missing information when several sensors are damaged.
The above simulation results indicate that the reconstruction performance of IFOCUSS is superior to that of RFOCUSS regardless of whether there is missing information.For IFOCUSS, it has a faster convergence speed, with fewer iterations required to achieve the ideal convergence effect; it has better convergence performance, reflected in fewer interference items of the reconstruction result, smaller positioning errors of estimated targets, fewer sensors required to locate the five targets and more targets that can be estimated or smaller errors, when it occurs to the two cases of missing information, especially when there is a lot of information lost.

Conclusion
CS algorithm is combined with DPD to locate multiple targets with missing information.Considering locating multiple targets with an observation station moving at a constant velocity, two cases of missing information are introduced.For better positioning performance, IFOCUSS is proposed with three improvements.First, the regularization factor is dynamically updated in iteration processes to improve the balance, instead of taking a fixed initialization value.Secondly, the closer the column in the dictionary matrix is to the sparse observation vector, the greater its contribution to the weighting matrix is, and the greater the corresponding weight value is.According to this feature, a vector is introduced to re-weight the weight matrix for faster convergence to the corresponding items of real target location in the dictionary matrix, improving convergence property.Thirdly, the monitoring area is symmetric, so the grid where the real targets are located must have a symmetric grid concerning the trajectory of the station.Based on this, an improved method of selecting the correct terms from the sparse estimation vector is proposed in this paper, greatly eliminating most interference terms in the sparse estimation vector and achieving better positioning performance.
Simulation results show that IFOCUSS exhibits a superior convergence performance at low SNRs, embodied in much smaller positioning errors and fewer interference items in the sparse estimation vector, with approximately half the run time of RFOCUSS.It also overperforms RFOCUSS regardless of whether it is in a missing information environment, reflected in fewer sensors required to estimate the same targets and more targets that can be estimated or smaller positioning errors in the same conditions of missing information, with faster convergence performance.

.
on it.We take observation points every 20 second10 , The linear array arrangement is consistent with the trajectory, and the distance between adjacent sensors is T groups are the correct ones corresponding to the five targets, which are determined based on which side of the observation station the real targets are located.

Figure 2 .
Figure 2. Sparse estimation vector for the first six iterations by RFOCUSS Figure 2 shows the sparse estimation vector ,

Figure 3 .
Figure 3. Sparse estimation vector at the sixth iteration by RFOCUSS

Figure 4 .
Figure 4. Positioning result by RFOCUSS of 100 Monte Carlo simulations Figure 4 is the positioning result by RFOCUSS of 100 Monte Carlo simulations, with the initial position of the observation station and the actual target positions marked.The blue pentagrams are the estimated positions, of which three are interference terms, supporting the analysis of Figure 3.If this algorithm is used to locate multiple targets, it is highly likely to misjudge the number of targets and increase processing costs.
sparsity of the reconstruction results.It has been confirmed that the reconstruction results closer to the original sparse signal can be obtained by the reweighted 1

4 Figure 5 .
Figure 5. Sparse estimation vector for the first five iterations by IFOCUSSSimulation of multi-target localization by IFOCUSS under the same conditions as before is performed.Figure5is the sparse estimation vector for the first five iterations.Compared with Figure2, we can observe that IFOCUSS converges faster than RFOCUSS, as the reconstruction result at the third iteration is well enough, with fewer interference items and more energy focused on the correct grids.Furthermore, IFOCUSS has a better convergence performance and fewer interference items, embodied in the significantly larger amplitudes of symmetric grids in the selected five groups than that of invalid grids, with a difference of approximately four times.

Table 1 .
Positioning errors of IFOCUSS and RFOCUSS under different SNRsOverall, both algorithms can locate multiple targets when SNR is below 0, which is one of the great advantages of the CS algorithm.Estimating the five targets accurately is required by IFOCUSS, while

Table 2 .
Run time of IFOCUSS and RFOCUSS under different SNRs

Table 3 .
Subscripts of targets observed at every observation point

Table 4 .
Positioning errors of IFOCUSS and RFOCUSS

Table 5 .
Positioning errors of IFOCUSS and RFOCUSS