**Multi-targets example:**

E6: [0,10]; [0,10]; Main Target = [6,4]. Secondary targets = [5,3;6,5]; WN = 1;

E7: [0,50]; [0,50]; Main Target = [26,27]. Secondary targets = [24,30;31,27;28,25]; WN = 2;

E8: [0,100]; [0,100]; Main Target = [65,44]. Secondary targets = [60,39; 71,46; 74,45; 62,40]; WN = 3;

E9: [0,1000]; [0,1000]; Main Target = [654,598]. Secondary Targets = [514,597;735,525;486, 562;753,556;579,499; 425,786]; WN = 4;

E10: [0,10,000]; [0,10,000]; Main Target = [5149,4809]. Secondary targets = [6514, 5897;7035,5035;4986,5562;6013,5026;5179,5099;4125,4986;6056,5165;7022,5983; 7022,5983]; WN = 5.

#### *6.2. Comparison Test with Different Search Methods*

This paper compares the minimum search step number, mean search step number, and standard deviation searched by *Traditional Carpet* search, *Uniform* distribution, and *Lévy* distribution search run 32 times. The experimental parameters and results comparison on each target search are listed in Table 1. In the following table, the *Tradition Carpet* search, *Uniform* distribution, and *Lévy* distribution search are called *T*, *U*, and *L*, respectively.

**Remark 1.** *In Table 1, Ex is the search example; Abbreviation is the search methods shortened form; Min*/*Average*/*Std denote that search target used minimum run step, average step, and step standard deviation in run 32 times; e* + *n mean* ×10*n. Solution denotes whether to find the target, 1 denotes found, and 0 is not found. Succeed times is the successful search times in run 32 times.* Successful rate *is a successful search rate in 32 run times. The better solutions found in different search methods are illustrated with bold letters, and the best minimum average is shown with an underline, respectively.*

**Remark 2.** *In* Lévy *distribution search α* ∈ [0, 2]; *β* ∈ [−1, 1]; Γ = *X*; *δ* = Γ/2*. If the* Lévy *distribution generation number is negative, we use its absolute value or re-generate with Uni f orm distribution.*

From Table 1, the *Lévy* distribution search has the highest performance since using it has smaller minimum and arithmetic mean search steps found target compared to solutions obtained by others, and especially for bigger size target search problems, it has better search efficiency. In above test examples, the *Lévy* distribution search can obtain a higher search successful rate. If the target is in the search area center, the *Traditional Carpet* search method using less half steps of search area grid cannot find the target, but using *U* or *L* search method with the probability of more 40% and 60% can find the target. From the above experiments, we can find that scale is larger, the performance of *L* is more outstanding compared with *T* and *U* search. For a huge size search problem experiment, because of the experimental conditions limited, the data storage runs out of memory. In the real environment, because the search range is very large, the *Traditional Carpet* search is difficult for finding the target. From simulation results, we can obtain that the *Lévy* distribution search as being clearly better than *Traditional Carpet* search and *Uni f orm* distribution search for one target search problem. Especially for large size problem, the *L* search has strong ability.


**Table 1.** The comparison results of *T*, *U*, and *L* search for one target search.

Search paths, the convergence and succeed search times distribution figures of most effective of *Traditional Carpet* search, *Uni f orm* distribution, and *Lévy* distribution search run 32 rounds for five one-target search instances are as in Figure 5. For E4 and E5, the search paths are very complex, drawing a figure that almost cannot be seen clearly; therefore, we do not show their search paths.

From Figure 5, in search target path picture we can see *U* and *T* can quickly find target. The *Traditional Carpet* search method is the enumeration method, gradually slow search, efficiency is very low. From surface view *Traditional Carpet* search method, it can find target with a finite step, but the actual search area is too big, and it cannot use large search steps greatly to search. Within finite steps to search the target, using the random search method is more effective than the *Traditional Carpet* search. In addition, when the search area is huge, the *Uni f orm* distribution search compared to the *Lévy* distribution search efficiency is obviously much lower. In the following section, we test the multi-targets search using different search methods.

**Figure 5.** *Cont*.

**Figure 5.** *Cont*.

**Figure 5.** *Cont*.

**Figure 5.** *T*, *U*, and *L* search for one-target.

In order for more scientific evaluation search methods for a multi-targets search problem, this paper compares the minimum search step number, mean search step number,and standard deviation searched by *Traditional carpet* search, *Uni f orm* distribution, and *Lévy* distribution search run 20 times. The solutions of multi-targets test examples are listed in Table 2, and the about parameters are the same in Table 1.


**Table 2.** The comparison results of *T*, *U*, and *L* search for multi-targets.

From Table 2, the *Lévy* distribution search clearly has the highest performance, and especially for bigger size targets search problem, it has better search efficiency. In above test examples, the *Lévy* distribution search can obtain higher main and secondary search successful rate; at the same time, it can find more secondary targets, as well as use less search steps. Especially for large size problems, *L* optimal has strong search ability.

The search paths, the convergence, and succeed search times distribution figures of most effective of *Traditional Carpet* search, *Uni f orm* distribution, and *Lévy* distribution search run 20 rounds for five multi-targets search instances are in Figure 6. At the same time, for E8∼E10, the search paths are very complex and cannot be seen clearly; therefore, we do not show their search paths.

**Figure 6.** *Cont*.

**Figure 6.** *Cont*.

**Figure 6.** *Cont*.

**Convergence Curve of Different Search Methods for Multi−Targets 10000\*10000**

From Figure 6 we can see that the convergence rate of *Lévy* distribution search is clearly faster than *Traditional Carpet* search and *Uni f orm* distribution search. *Lévy* distribution search can not only fast find the main target but also find more secondary targets than *T* and *U* search method. Due to the particularity of this kind of target search problem, in general secondary targets stochastically scattered on the main target around, it is more suitable for *Lévy* distribution search which can quickly find the targets.

Targets can be found by only processing a few images (search steps) using *Lévy* distribution search. In the scenario under discussion, the advantages lie in that the convergence speed of *Lévy distribution* search being clearly faster than *Traditional Carpet* search and random search based on some other distributions. In limited search steps, *Lévy distribution* search can not only quickly find the main target but also find more secondary targets than the aforementioned search methods.

Accordingly, we can state that the *Lévy* distribution search is more effective than *Traditional Carpet* search and *Uni f orm* distribution search method for discrete targets search. In limited search steps, the *Lévy* distribution search can, with higher probability, quickly and accurately find the main target and more secondary targets. *Lévy* distribution search is one better search tool.

#### *6.3. Experiment of Lévy Distribution Search*

From the above experiment, we find that the *Lévy* distribution search is efficient for discrete targets search. *Lévy* distribution search has many parameters, in order to find their different search performance, we simulate and obtain optimal parameters. *Lévy* distributions solve the test problems, in which *α* from 0 to 2 with 0.1 step changes and *β* from −1 to 1 with 0.1 step changes for the *X* axis and for *Y* axis, respectively. The *L* uses a group of parameters *α* and *β* combination run 10 times, using the average to search results. The best parameters configuration of *L* for search one target and multi-targets are in the following Table 3.


**Table 3.** The optimum parameters of *L* for target search.

From above Table 3, we can see when *β* approximately equal to 0.9 have a good search performance. Especially for large-scale one target and multi-targets search problems, *β* equal to 0.9 is effective and stable. About *α* for *L* is not obvious influence and is not stable, relatively speaking in most cases, *α* is equal to 1.5 has a better search efficiency. As a result of the experiment and paper length limit, this paper do not analyze testing results for large problems E4 and E5.

With the *Lévy* distributions search run 10 times with *X* and *Y* each parameters combination, and use the mean main target and the mean secondary targets number of round 10 times, respectively, as *Z* axis, the three-dimensional graphics are as following (Figures 7 and 8):

 **Figure 7.** The 3-*D Lévy* distribution random search with different parameters for one target.

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/6IRU0DLQ7DUJHWRI0XOWL7DUJHWV( /6IRU6HFRQGDU\7DUJHWVRI0XOWL7DUJHWV(

/6IRU0DLQ7DUJHWRI0XOWL7DUJHWV( /6IRU6HFRQGDU\7DUJHWVRI0XOWL7DUJHWV(

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**Figure 8.** The 3-*D Lévy* distribution random search with different parameters for multi-targets.

From the above figures we can discover that the parameters have greater influence on performance of *L* algorithm for target search problem. Clearly, we can see when *β* ≈ 0.9 *L* has better performance for bigger size target search problem and *α* influence the performance of *L* is not stable. Experiments show that *L* search performance is strong, with fewer steps to find the target.

#### **7. Conclusions and Perspectives**

The plane crashed, the ship sank, and desert missing, landslide spot search, forest fire spot search and rescue, grassland search, enemy warships group search, etc., are all called discrete target search problem. According to production and living needs and shortcoming of the tradition search method, this paper offers comprehensive and systematic research on the discrete targets search problem.

This paper analyzes the particularity and complexity of the discrete target search problem, and the paper simulates and analyzes the different and performance characteristics of *Tradition Carpet* search, *Uni f orm* distribution search, and *Lévy* distribution search for one target and multi-targets. We give that the computer quickly selects, grabs the satellite images and image analysis method to determine the target coordinate position by software system, and greatly improves the target recognition speed. Preliminary experiments show

that the efficiency of *Lévy* distribution search is better than *Tradition Carpet* search and *Uni f orm* distribution search. Using *Lévy* distribution search only process, a few images (search steps) can find targets. We find the *Lévy* distribution search is one better search tool, which can, with higher probability, quickly and accurately find targets for discrete targets search problem. The simulations experiment prove that *DTSP* is faster to search for discrete single target or multiple targets in a wide area. It provides a new method for solving the discrete target search problem.

There are a number of research directions that can be considered as useful extensions of this research. Although the proposed search method is tested with 10 instances, a more comprehensive computational research should be made to test the efficiency of proposed new method. This paper is the only research on the target of rectangular areas search and study a new method to search target in circular area and irregular graphics area. Recently, 48 robust M-estimators were presented for regression analysis [22]. In addition to image processing, M-estimators have been used in several areas of knowledge such as astronomy, pharmaceutics, medical, econometric, finance, geodesy, and engineering (electrical, telecommunications, civil, mechanical, chemical, and nuclear). In future we will research on other more effective search methods, different distribution search such Mittag-Leffler [23] or multi-population pattern searching algorithm [24] for one target and multi-targets search problem. For future work, as well as in data regression analysis, it is possible to use a function that measures the distance from each location estimated point to the correct locations as performance criteria, principally for multi-targets.

**Author Contributions:** Conceptualization, Y.C. and Z.L.; Methodology, Y.C. and Z.L.; Investigation, Z.L., D.L. and B.D.; Writing— Z.L., D.L., B.D. and Z.L.; All authors have read and agreed to the published version of the manuscript.

**Funding:** This work was supported by Shanghai Innovation Project(20AZ10,20AZ11), the Shanghai Educational Science Research Project(Grant No.C2022120), the National Natural Science Foundation of China (Grant No.11661009), the High Level Innovation Teams and Excellent Scholars Program in Guangxi institutions of higher education (Grant No.[2019]52)), the Guangxi Natural Science Key Fund (Grant No.2017GXNSFDA198046), and the Special Funds for Local Science and Technology Development Guided by the Central Government (Grant No.ZY20198003).

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** All authors know this paper and agree to submit it.

**Acknowledgments:** The authors would like to thank the referee for his or her careful reading and some comments on improving the presentation of this paper. Thanks Shengming Jiang of Shanghai Maritime University, Tomas Oppenheim and Zhuo Li, School of Engineering, University of California, for their comprehensive revision and some discussion of value.

**Conflicts of Interest:** We have no conflict of interest in the paper.
