**1. Introduction**

This paper is motivated by a world-wide focused case, the lost connection of the Malaysia Airlines Flight 370 (MH370). It has been missing up to now since then. Dozens of countries around the world have attempted the search using different kinds of most advanced technology such as radio "pings" between the aircraft and an Inmarsat satellite based on military radar data. Unfortunately, there has been neither confirmation of flight debris nor indication of crash sites. This paper defines the sea target search, grassland target search, desert target search, etc., in a large area as *Discrete Targets Search Problem* which is *DTSP* for short. *DTSP* has a wide range of applications, such as in the case of war, how to quickly search the enemy warships group. In addition, the rescue of airplane crashes, shipwrecks, and the desert search, etc. can be conversed with *DTSP* to be solved. Because search goal area is very large, the search target is a decentralized point, and characteristics of targets are not clear, *DTSP* is very difficult, and the research report on it is more visible. In a large area, see in Figure 1 for how to quickly search target. Research *DTSP* has grea<sup>t</sup> practical meaning.

There are some reports on the study of target recognition search, mainly through radar, infrared, robot, and other tracking. Some pioneer researchers reported about target recognition, target acquisition, and target tracking. For example, references [1,2] considered the case of heuristic search where the location of the goal may change during the course of the search. They introduced ideas from the area of resource-bounded planning into a Moving Target Search (*MTS*) algorithm, including commitment to god and deliberation for selecting plans. J. R. R. Uijlings et al. [3] addresses the problem of generating possible object

**Citation:** Lian, Z.; Luo, D.; Dai, B.; Chen, Y. A Lévy Distribution Based Searching Scheme for the Discrete Targets in Vast Region. *Symmetry* **2022**, *14*, 272. https://doi.org/ 10.3390/sym14020272

Academic Editor: Alexander Shelupanov

Received: 7 November 2021 Accepted: 12 December 2021 Published: 29 January 2022

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locations for the use in object recognition. They introduce selective search by combining the strength of both an exhaustive search and segmentation. Grimaldo Silva et al. [4] proposed the use of saliency information to organize regions based on their probability of containing objects. Xin, ZH et al. introduced a new ground moving-target-focused method for airborne synthetic aperture radar in [5]. Compared to conventional *SAR* imaging methods, it can eliminate the effects of Doppler ambiguity and azimuth spectrum split. Mohamed Abd Allah El-Hadidy in [6] presented a search model that finds *n*-dimensional randomly moving target in which no information of the target position is available to the searchers. Zhen Shi et al. in [7] proposed a parallel search strategy based on kernel sparse representation to perform the tracking task in the FLIR sequences. The kernel method is used to deal with complex appearance variations. A variety of different target search problems were reported with different technical strategies, such as *UAVs* [8] and robots [9–11]. However, there are a few references about the discrete point object search problem. This paper discusses different strategies to study how to search discrete point object efficiently. In addition, there are more researches on the problem of discrete target search from the aspects of infrared and radar technology but fewer research reports from the perspective of image processing and algorithms.

**Figure 1.** Discrete point target search. \* is the label of the search target location.

The discrete point search problem is very different from the extreme value of function search problem (see Figure 2). In this case, the objects are isolated, and they are not related to each other. However, in the extreme value of function search problem, extreme points and other points are related, such that intelligent iterative calculation can be used to find their extreme point. Aiming at the particularity of discrete target search problem, this article proposes an efficient and reliable search method to achieve rapid searching target in a wide range of areas. We study the effectiveness of *Traditional Carpet* search, random search, and *Lévy distribution* random search for discrete point object, with the following differences from existing ones: Firstly, based on the practical problems of searching for discrete targets in people's production and life, *DTSP* is proposed. From an actual target search problem analysis, it summarized and extracted a common technical problem. From the actual target search problem analysis, it summed up the common technical solutions to the problem; secondly, it formulates a new search with different evolution mode, for discrete point object problem; thirdly, it strictly analyzes and compares different search method performance. We show that the proposed search method is more efficient than tradition search and discuss the implication of the results and suggestion for further research.

The remainder of this paper is organized as follows: In Section 2, we sum up the search method for discrete targets in the current reality. In Section 3, we develop the general framework of the search solution. In Section 4, we formulate some dominance properties and model mathematical of *DTSP*. In Section 5, we discuss the *Traditional Carpet* search, random search, and *Lévy distribution* random search for discrete point object. The *Lévy distribution* random search was developed to enhance the search efficiency for the *DTSP*. The results of a computational experiment are provided in Section 6. Some conclusions are given in the last section.

**Figure 2.** Extreme value of function search.

#### **2. The Current Search Method**

The traditional search methods are usually used for search in plane crash, ship sank, and desert missing, as well as the search of landslide spot, forest fire spot and rescue, grassland, and enemy warships group, etc. For example, sending aircraft flights to the target area to search for suspicious targets or sending ground and rescue personnel for on-site search. Most of the current searches are *Traditional Carpet* search, which divide the target area into many lattices to be searched one by one in turn, as shown by Figure 3:


**Figure 3.** The carpet search model. \* is the label of the search target location.

In the past, most actual discrete point object search problems use the *Traditional Carpet* search to look for target. Rescuers use any possible means; aircraft and ships deploy equipment to listen for signals from the underwater locator beacons attached to the aircraft's "black box" flight recorders. However the batteries of the beacons become exhausted in a short time. For example in MH370 search, the search operation continued in the area of the signals using a robotic submarine for a long time but without obtaining any evidence of debris.

The efficiency of *Traditional Carpet* search method is low due to its enumeration nature. It is almost invalid when the target is in a big area. The paper [12] presented a FogLight approach for searching metabolic networks utilizing Boolean (AND-OR) operations represented in matrix notation to efficiently reduce the search space. This paper presents a new idea and method for discrete point object search.

#### **3. New Search Solution Based on Satellite Image**

Currently the image processing technology is very mature, and satellite images can be used to look for targets. Methods for image processing include the so-called "M-estimators" (from robust statistics) [13], among others machine learning techniques. This reinforces

the credibility of satellite images than can be used in discrete search problems. Robust Mestimators, which are generalizations of maximum-likelihood estimators, tend to look at the bulk of the data/image information and ignore atypical values (outliers or multiple outliers) during estimation procedure (pattern recognition), due to their mathematical structure.

The new algorithm based on satellite images processing operations is as follows:

**(1)** The search command center informs the satellite control station to adjust satellite attitude and direction to target search area;

**(2)** The satellite takes a picture of search target area and then promptly sends the image to the processing center;

**(3)** The image processing center first amplifies satellite photographs and cuts them into the lattice images. Secondly, the algorithm chooses the lattice images to identify which image is the target. Thirdly, it determines the actual position of the target image and then quickly identifies the suspected target coordinates;

**(4)** It sends the coordinates of the suspected target to the *UAV*, search aircraft, ship, or ground rescuers to affirm the target and obtain more detailed information;

**(5)** All information of targets is sent to the search center.

The Discrete Targets search diagram based on the new algorithm with satellite image processing is illustrated in Figure 4:

**Figure 4.** The search logic diagram.

#### **4. Discrete Target Search Problem**

#### *4.1. Target (Black Box) Search Problem*

In extreme value of function search problem, the extreme point and other points are related and can share useful information for looking for object. However, with the discrete point target search problem, its target is isolated, and all points are not relative each other. With *DTSP*, other points are not able to provide the useful information to help find the target point. For the special case of *DTSP*, it is impossible to use a traditional method such as intelligent iterative calculation to solve it. Here, we present a new method to solve *DTSP*.

Put the target into the coordinate system. If the coordinate value of target is focused, it is a successful search. The problem of discrete point search is translated into a search problem of a target element in the matrix to look for the coordinate of 1 in one target search matrix (1).

$$
\begin{pmatrix}
0 & 0 & \cdots & 0 & 0 \\
0 & 0 & \cdots & 0 & 0 \\
\cdots & \cdots & \cdots & \cdots & \cdots & \cdots \\
0 & 0 & \cdots & 1 & 0 \\
0 & 0 & \cdots & 0 & 0
\end{pmatrix}
\tag{1}
$$

**Code rule/0-1:** The target area image is divided into lattice images. If a grid image is an object, the element of the grid image is set to 1 in the matrix. Put all lattice images into matrix with the target image denoted by truth/1 and others by false/0. In the search process, if the coordinate of the 1 point is found the search is successful.

#### *4.2. Multi-Targets (Black Box and Wreckage) Search*

Put the main target(black box) and wreckage into the coordinate system. If the main target and some wreckage coordinate value are found, it is a successful search. The problem of some discrete target search, is translated into a search problem of target elements, i.e., the coordinates of targets *B*(black box) and *wi* (0 < *wi* < *A* wreckage), in multi-targets matrix (2), in which the *B* and *wi* denote the black box and wreckage, respectively.


**Code rule/0-***B***-***wi***:** Put all lattice images into a matrix. Main target/black box image are denoted by *B*, wreckage is *wi*, and others are *false/0*. In matrix (2), *B* is the goal/black box, *wi* is the wreckage, and 0 is blank. A successful search aims to find the black box image or most of wreckage coordinates or the black box and a small part of the wreckage coordinates. If the coordinates of black box *B* point or wreckage *wi* point with the total number more than *X* are found, the search is successful.

In the following section, we further discuss how to use minimum step to look for the coordinate of targets, how image processing center uses the least number processing picture, and how to judge and look for a target and determine its coordinates.

#### **5. Screening/Search Target Picture**

#### *5.1. A Uniform Distribution Search Method*

A uniform random search target image method is as follows:

**Step 1:** Based on *Uni f orm* distribution, randomly generate two integers (*<sup>x</sup>*, *y*) as the ordinate and abscissa of the candidate target;

**Step 2:** It is judged whether or not the generated random number (*<sup>x</sup>*, *y*) is within the search area, and if it is not, using *Uni f orm* distribution, reproduce the number of random integers (*a*, *b*) as coordinates;

**Step 3:** Take the satellite grid image in the (*<sup>x</sup>*, *y*) coordinates, judge whether is it real target. If it is not a target, go to step 1.

**Step 4:** If the target is searched, the coordinates of the target and its image are output;

Above using uniform distribution, the search random extracts the satellite picture judgment target, which is the *Uni f orm* distribution search method. Of course, one can also use other random distribution methods to generate coordinates to extract the satellite picture judgment search target.

#### *5.2. A Lévy Distribution Search Method*

*Lévy* stable distributions are a rich class of probability distributions that have many intriguing mathematical properties. *Lévy* flights are a class of random walks in which the step lengths are drawn from a probability distribution with a power law tail. The application of *Lévy* fight pattern in the *Cuckoo* search algorithm [14] greatly improves the algorithm search speed to solve the design optimization [15], selected engineering applications of gradient free optimization [16], traveling salesman problem [17,18], and the selection of optimal machine parameters in milling operations [19]. Tarik Ljouad et al. [20] presented a new tracking approach. Gurjit S. W and Rajiv K [21] proposed an evolutionary particle filter based on improved cuckoo search which overcomes the sample impoverishment problem of generic particle filter. *Lévy* flights are commonly used in physics to model a variety of processes including diffusion. Broadly speaking, *Lévy* flights are a random walk by step size following *Lévy* distribution, and the walking direction is *Uniform* distribution.

In the *Mantegna* rule, steps size *s* design is as follows:

$$s = \frac{\mu}{|\upsilon|^{\frac{1}{\beta}}} \tag{3}$$

The *μ*, *υ* follows *Uniform* distribution, i.e.,

$$
\mu \sim N(0, \sigma\_{\mu}^2), \upsilon \sim N(0, \sigma\_{\upsilon}^2) \tag{4}
$$

Here,

$$\sigma\_v = \{ \frac{\Gamma(1+\beta)\sin(\frac{\pi\beta}{2})}{\Gamma[(1+\beta)/2]\beta \* 2^{(\beta-1)/2}} \}^{\frac{1}{\beta}} \tag{5}$$

$$
\sigma\_{\mu} = 1 \tag{6}
$$

This paper presents the *Lévy* distribution search method for *DTSP*. Based on *Lévy* distribution with the *Mantegna* rule, it generates the ordinate and abscissa (*<sup>x</sup>*, *y*) of the candidate target, in turn grabbing the picture of coordinate's points (*<sup>x</sup>*, *y*) and then determining whether it is real target. The *Lévy* distribution generation coordinates (*<sup>x</sup>*, *y*) of candidate target must be in the search area. This random extracts satellite picture judgment method, that is the *Lévy* distribution search method.

#### *5.3. Pseudo Code of One Target Search*

**Step 1:** Let parameters; including maximum generation *Endgen*; **Step 2:** Iteration process;


While (*t* < *Endgen*) or (Other stop criterion)


**Step 3:** Output last step search point coordinates and its image.

#### *5.4. Pseudo Code of Multi-Targets Search*

**Step 1:** Let parameters; including maximum generation *Endgen*; the total *num* of wreckages searched.

**Step 2:** Iteration process


While (*t* < *Endgen* and secondary target ≥ *num*) or (black box target==1) or (Other stop criterion)

• t := t + 1;


**Step 3:** Output secondary target number and their coordinates, last step search point coordinate, and their images.

#### **6. Experiment for Target Search Problem**
