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This section presents the details of our scheme to detect moving foreground objects. Time-based sliding window approach is implemented to process the raw input data, as shown in Fig. 3. We fix size of the window to M and process the window matrix Y∈R^N×M. In this approach, the initial window matrix M is first accumulated via raw data vectors. All processing step is applied on the initial window matrix.

Fig. 3. 
Time-based sliding window approach

In the next cycle, the new data frame are added to matrix M, and the oldest data frame is discarded. We re-compute on new window matrix M.

Each window matrix is processed via three main steps, as illustrated in Fig. 4. The input data frame is calibrated and compensated based on experiments in pre-processing step. The received signal power from target decreases when the target mover farther away from radar called Path Loss Compensation. Based on practical experiments, we select the compensation factor as log₁₀(4πR²). The received signal at each point is weighted of log₁₀(4πR²). RPCA-GD is used to model the background. A novel method to updates the initial background for RPCA when each new frame has been accumulated is proposed. Finally, moving foreground objects’ positions are estimated using AMPD peak detection algorithm [14].

Fig. 4. 
Illustration of proposed method

In this section, we describe our novel algorithm for modeling the background based on RPCA in real-time. The basic idea is updating method to generate initial matrix based on the previous processing. The Online RPCA via GD is presented in Algorithm 1. First, we collect data for processing until t = M. When t = M, we utilize the method in [10] to extract the low-rank component.

S_init is generated based on the input window matrix Y using a sorting-based sparse estimator algorithm which basically conducted as follows for any matrix A∈R^d1×d2.

where Ai,.k and A.,jk denote the elements of A_(i,.) and A_(.,j) that have k-th magnitude, respectively.

After generating S_init Single Value Decomposition (SVD) method is applied on (Y - S_init) to obtain the matrix rank r.

Let’s U_t←HΣ^1/2, V_t←KΣ^1/2, we consider the following optimization problem.

With a given U_t,V_t, an iterative method is implemented to produce the sparse matrix S based on sparse estimator on St=JγαY-UtVtT. The projected gradient descent is computed as following equations to produce U_nw, V_nw.

where γ and η the algorithmic tuning parameters. The loss function L(U,V,S) is defined as follows

Finally, the low-rank and sparse matrices is updated in Phase 3. More details can be found in [10].

Continuously, Algorithm 2 is proposed to update and initialize U_t, V_t from frame t = M+1. The oldest column is removed based on a sliding window approach. The basis update algorithm includes three steps. First, last vectors are extracted from previous calculation U_t-1, V_t-1. These vectors are added to initial matrix U_t, V_t. Finally, we discard the oldest vectors U_t, V_t. The updated matrices U_t, V_t are utilized as initial matrices for optimization based on the Gradient Descent method. This critical initial step saves the computation cost to optimize U_t, V_t. Other steps are re-implemented as in phase 2 and phase 3 of Algorithm 1.

To examine the performance of proposed method on FMCW radar signal, we execute sliding window approach on RPCA via IALM [8], online RPCA [9] and RPCA via GD [10] and the proposed method with a window size M = 50, number of iteration of 100, rank r = 10, and stop criteria of 0.02 for evaluation. An illustration of four moving-target detections is depicted in Fig. 6. The original data includes noise which is reflected from other objects in the first row. There are 4 moving targets in this experimental scenario. Two targets move from radar to further away and come back, which make the V shape. Two other objects just go further for detection range, which make I shape. The experimental environment includes noises, as marked in Fig. 6, which make the detection task more challenge. The detection results are shown in the second row for comparison. From the original data in the first row, the power of noise is very high in range of 4 to 6 meters. When the object move further away over 6 meters, due to the noise, other RPCA-based methods are not able to detect the object.

Fig. 6. 
Illustration of detection performance.

However, our proposed method is still able to extract the moving objects. The blue points in Fig. 6 are the extracted position of moving objects. The detection performances are presented in Table 4 and Table 5. The proposed method reaches up to 89.5% in precision, 80.0% and 84.4% in recall and F1-measure, respectively. RPCA via IALM approach obtains the highest accuracy. However, this method is too slow for real-time application with processing speed of only 0.6 frames per second. Compared to the Online RPCA, the proposed method significant outperform in performance term at 6.7%, 5.0% and 5.9% of precision, recall and F1-score.

Furthermore, the baseline RPCA via GD obtains lower results in compare with the proposed method in both processing speed and accuracy. Our proposed method improve the baseline RPCA via GD [10] method up to 4.2%, 3.2% and 3.7% of precision, recall and F1-score, respectively. The original RPCA via GD is unable to apply in real-time applications due to processing speed is only 1.1 frame per second. Conversely, with the proposed updating algorithm, our system can work in real-time applications with processing speed up to 19 frames per second. There are three main reasons.

First, the initial matrix in our approach is based on a previous observation, which leads to reducing the processing time to obtain optimal results. Second, in other approaches, the initial matrix based on an SVD approach is re-computed when each new data was added which bring out higher consumption cost. Third, the low rank matrix in the current frame is not significantly different from the previous one when the new vector is added. Therefore, we do not have to re-compute the low rank matrix from beginning.

The impact of different the window size is evaluated in this section. We select the rank r = 10. The window size is varied from 30 to 90. The system performances are shown in Fig. 7. The system achieves 88.46% of precision, 75.79% and 81.64% of recall and F1-measure, respectively, with M = 50. When the siding window size is 30, there is too little data to model the low-rank background.

Fig. 7. 
Impact of the sliding window size

Therefore, the system performance get worse at M = 30 with 82.69% of precision, 76.42% of recall, and 79.43% of F1-measure. When the processing matrix size increases, the system performance also enhances due to the dissimilarity between the low-rank and the sparse matrix. When the matrix size was too large, the optimization step reached the maximum iteration value before obtaining optimization matrix. For that reason, the performance decreases when the matrix size increases, such as M = 70 or M = 90. On the other hand, the computational cost is high with a large input matrix. Due to real-time processing requirements, the window size is selected as 50 for other evaluations.

Another critical parameter which has an effect on the system performance is the number of iteration. In this experiment, we investigate the detection accuracy by changing the number of iteration from 10 to 150. The processing matrix size M is equal to 50, and the rank of the processing matrix was 10. The selected number of iteration includes 10, 30, 50, 70, 90, 110, 130, and 150 iterations. The system performance is sketched in Fig 8.

Fig. 8. 
Impact of the number of iteration

The system obtains up to 89.07% in precision, 75.04% of recall and 81.46% of recall with 110 iterations. The detection accuracy is significantly enhanced when the number of iteration changed from 10 to 30. The main reason is that with 30 iterations, the initialized matrix does not achieve optimized values. There is still existing space to improve the system performance based on an increase in the number of iterations. From 30 to 150, the system performance almost do not change because the system reaches the optimal solution at the first initialization.

We also analyze the impact of different updating methods in Algorithm 2 to investigate the efficiency of the Basis update algorithm. This evaluation is conducted based on three different update methods: average method, weighted method and update from the latest low-rank vector.

In the average method, we compute the average value vector from a previous observation of the U_t, V_t matrix. We add these vectors to the last column and discard the oldest vector, as in Algorithm 2. The average method is a specific case of a weighted method with all values having the same weight. In the weighted mean method, we insert the weight to each column vector based on its position as follows.

where w_i is the weight for the i-th column, and r is the rank of the matrix as in the initialization.

In the last case, we simply add the previous vector and remove the oldest vector when the new frame is inserted, as mentioned in Algorithm 2. Experimental results are shown in Fig. 9. According to Fig. 9, the updating method using a previous frame obtained the best performance with up to 88.5% of precision, 75.8 % and 81.6% of recall and F1-measure, respectively. The worst case use an average updating method with 88.2% of precision, 75.5% of recall and 81.3% of F1-measure. The similarity in the low-rank matrix between the previous and current update increases in the following order: average method, weighted method and previous method. Based on this property, the system performance increase in this order from the average method to the previous method. The difference between these updating methods is very small, and it mainly depends on the initial low-rank matrix in the first step.

Fig. 9. 
Detection performance across different update methods

The robust of proposed algorithm is proofed by several scenarios containing multiple moving objects with different velocities. Two objects move with different speeds (slow walking and normal walking) and distance are called scenario 1 and scenario 2, accordingly. Furthermore, three and four moving objects with different speeds (slow walking, walking and running) and various tracks are presented in scenario 3 and scenario 4, respectively. The numerical results of each scenario and average performance are shown in Table 6. We compare the performance of proposed method with other RPCA-based method such as RPCA via IALM [8] and RPCA via GD [10] and Online RPCA [9]. Our proposed method achieves comparable results with RPCA via IALM [8] with sliding window approach. In comparison to the baseline RPCA via GD [10], our method obtains better performance in all precision, recall and F1-measure metrics. Online RPCA [9] can not remove noises perfectly. As a result, Online RPCA [9] shows worst performance. As mentioned in Section 5.1. Our proposed method is able to works at real-time speed. From these experiments, our proposed approach is the best solution for practical applications due to high accuracy and good processing speed.