1 Introduction

Radioactive sources are widely used in many nuclear technologies in industry [1], health care [2], nuclear research [3], and isotope production [4]. According to the International Atomic Energy Agency [5], more than 3000 radioactive source incidents have occurred globally, 10% of which were related to trafficking or malicious use. Most applications that use radiation sources are conducted in fixed places, such as hospitals, factories, and laboratories. If radioactive material is lost, a group of technicians searches for the lost source by using handheld detectors. This procedure is time-consuming and may adversely affect the health of the technicians. Hence, developing effective systems for localizing radiation sources in fixed areas is crucial. Distributed sensor networks are commonly used for localization [6, 7]. The network consists of several stationary radiation detectors, in which a radiation source can be localized through a data-processing algorithm using the readings of the detectors.

In this study, localization was performed through a sensor network, where the sensors measured the radiation level and then sent the measured data to a base station. The collected data, including detector readings and positions, were processed to estimate the radiation source parameters. The position of each detector (sensor) is normally provided through a global positioning system receiver [8]. Typically, the radiation source parameters are represented by the source’s location and intensity (strength) [9]. In order to perform effective localization, deployed detectors measure gamma-ray radiation, which can travel greater distances than other radiation types. The type of detector used depends primarily on the detection material and the surface area, which affects the detection efficiency [10].

Normally, without the presence of a radiation source, radiation levels can still be detected because of the presence of naturally occurring radioactive materials (NORM) in the surrounding environment [11]. The radiation detected by NORM is referred to as background radiation. Hence, a detection process must be applied before applying the localization method to determine whether the detector reading is due to an anomalous source or NORM. Traditionally, a source is detected if the measured radiation level exceeds a certain value, referred to as the detection threshold. In some sensor networks, data transmission is only performed when a source is detected to reduce the power consumption of each sensor [12]. However, the detection of radiation sources may result in high false positive rates if the background radiation fluctuates significantly according to the concentration of NORM in the surrounding area [13].

The localization of radiation sources is typically applied in environments with uniform background radiation [14] and low variance. Accordingly, background radiation is measured periodically, and the average value of the measured data is used later in the detection or localization process. Many studies have investigated radiation detection and localization while considering different background radiation conditions. For radiation monitoring in large geographic areas, background radiation can be estimated in regions not covered by detectors [15, 16]. In [16], the missing values in the measured radiation data were obtained using the Kriging interpolation method, which is a geo-statistical technique used for predicting spatial attributes. Similarly [15], used the Kriging method to estimate the radiation source parameters and background radiation. Localization using the Kriging interpolation method [15, 16] requires statistical parameters for the predicted data, which may not be available. Furthermore, in Kriging interpolation, an assumption is that the joint probability distribution is fixed in all the studied spaces, which does not hold for all areas. Some methods have been proposed to localize the radiation source in environments with high variance background radiation [13] evaluated using estimation techniques. Usually, estimation methods are used for background radiation and radiation source parameters [13, 17]. However, many observation samples are required for effective estimation [13].

Most of the localization methods are applied in estimating the parameters of unknown radiation sources [18]. However, the source type must be known to perform the localization process accurately. Some methods are used to estimate only the location of the source and can be applied without the knowledge of the source type [19]. In [19], source localization was performed using the ratio of square distances (ROSD), which can only be used to estimate the location parameter. This method provides an accurate estimation using only four sensors in an ideal environment (without any source of randomness). However, more than four sensors are required when background radiation and measurement randomness are considered in practical situations. Localization using ROSD provides many locations where only one location is considered the true location. Accordingly, an additional process is essential to select the true location from the estimated group of locations. Similarly, in [20], the location of the source was estimated using the relationship between the readings of each sensor and the difference between their distances from the source. The intensity of the source was then calculated using the estimated location and average values of the sensor readings. Hence, the accuracy of the estimated intensity depends on the estimated location.

Among the many estimation methods, maximum likelihood estimation (MLE) and Bayesian estimation are commonly applied for radiation source localization [13, 21]. In Bayesian-based estimation methods [18, 21], estimated parameters are assumed to follow a known prior distribution. Hence, the estimation performance relies on the accuracy of the prior distribution, which represents the stochastic process of the source parameters. The authors of [22, 23] and [9] have studied two commonly employed Bayesian estimation methods for localization using particle and Kalman filters, respectively. However, the methods considered are unreliable in practical situations. However, MLE provides reliable estimates without requiring a prior distribution. It is a statistical method that provides the most probable values of the parameters to be estimated according to a likelihood function. MLE has been employed in many radiation-related applications. In [24], background radiation was modeled using MLE via a mobile sensor network for radiation monitoring. In [25], MLE was used to estimate the parameters of multiple radiation sources in a given area where the number of sources is unknown. This method is based on the generalized maximum likelihood rule to estimate the number of sources by exploring different numbers until the best fit with the observation data is achieved. In [13], a localization method for estimating radiation source parameters in an environment with highly fluctuating background radiation was conducted using MLE to localize the source and estimate the background radiation.

Implementing MLE is challenging regarding the radiation localization problem because of the difficulty in finding a closed form solution for the corresponding likelihood function. Normally, MLE is performed numerically through a grid search. This process can be time-consuming, and the duration increases with the size of the search space. Hence, studies have attempted to solve the MLE time consumption problem. Multi-resolution MLE is the most commonly used approach to accelerate the grid search process [11, 26], where the search is performed iteratively. The process is based on decreasing the search boundaries per iteration according to the evaluated estimates provided by the previous iterations. In multi-resolution MLE, the estimation accuracy depends on the initial solution. Accordingly, the search may provide a local maximum rather than a global maximum, referred to as premature convergence.

In this study, we aimed to overcome the time consumption problem of MLE in the localization of radioactive sources. To achieve this objective, we applied alternative search methods using heuristic techniques that reduced the time consumption and likelihood of premature convergence more than multi-resolution MLE has in the literature. According to our review of the literature, heuristic techniques have not been employed for the localization of radiation sources by using MLE. Heuristic methods are inspired by nature, for example, genetic algorithm [27], firefly algorithm [28], particle swarm optimization [29], artificial ant colony [30], artificial bee colony [28], prairie dog optimization [31], gazelle optimization algorithm [32], dwarf mongoose optimization [33], reptile search algorithm [34], and aquila optimizer algorithm [35]. In this comparative study, we investigated the performance of the most commonly used and most effective heuristic methods for solving localization problems. The contributions of this study are as follows:

  1. 1.

    Four common heuristic techniques are used for the maximum likelihood localization of a radiation source, which are: (1) firefly algorithm (FFA), (2) particle swarm optimization (PSO), (3) ant colony optimization (ACO), and (4) artificial bee colony (ABC).

  2. 2.

    The performance of the considered heuristic methods, in terms of estimation error and time consumption, is compared.

  3. 3.

    Verification through experimental results using real data is applied to show that heuristic methods can significantly reduce the time consumption of the MLE localization process.

2 Localization of a radioactive source using MLE

This section describes the localization process in detail. The radiation source parameters, location and intensity, were estimated using the MLE algorithm according to the readings collected from stationary detectors (sensors). For each detector, the gamma-ray radiation was measured in counts per second according to a Poisson distribution model [9] as follows:

$$P\left({R}_{i,t}={c}_{i,t}\right)=\frac{{e}^{-{\lambda }_{i}}{\cdot \left({\lambda }_{i}\right)}^{{c}_{i,t}}}{{c}_{i,t}!} , 1\le t\le {T}_{\mathrm{W}},$$
(1)

where P(Ri,t = ci,t) is the probability that the reading of the ith detector Ri,t at the tth time instant is equal to ci,t counts/s within the measured time window TW. The reading Ri,t depends on the average count rate denoted by λi, which can be expressed as follows [36]:

$${\lambda }_{i}=\delta \cdot \frac{I}{{{dis}_{i}}^{2}} +BG,$$
(2)

where δ is a constant that depends on the type of radiation isotope and detector parameters, such as the detector area and efficiency. The Euclidean distance between the radiation source and ith detector is referred to as disi. The value of BG represents the background radiation emitted from NORM in the environment surrounding the detector. Typically, in localization problems, an assumption is that all the detectors are identical and affected by the same background radiation.

In estimating the radiation source parameters, denoted by Ɵ, the MLE can be used as follows:

$${\Theta }_{\mathrm{ML}}=\left[{X}_{\mathrm{ML}}, {Y}_{\mathrm{ML}}, {I}_{\mathrm{ML}}\right]= \underset{\theta }{\mathrm{argmax}}L\left(C;\theta \right),$$
(3)

where ƟML is the estimated parameter that indicates the estimated location (XML, YML) and intensity value IML. For M detectors, the likelihood function L(C; Ɵ) represents the joint probability of the count rates C and the source parameter Ɵ, where C = [c1, c2, …, cM] is the average count rates of each detector. According to the Poisson model in (1), the likelihood function can be calculated as [13]

$$L\left(C;\Theta \right)= \sum_{i=1}^{M}\sum_{t=1}^{{T}_\text{W}}\left({c}_{i,t}\cdot \mathrm{ln}{\lambda }_{i}-{\lambda }_{i}\right).$$
(4)

In practice, a detection process must be performed before estimating the radiation source parameters. In other words, all the readings of the M detectors considered in the localization procedure indicate a radiation source in the surrounding area. A simple technique detects a radiation source if the corresponding reading exceeds a threshold value. In this study, the detection threshold value, thrd, was calculated using the k-sigma method as follows:

$$th{r}_{\mathrm{d}}= {\mu }_{\mathrm{BG}}+k\cdot {\sigma }_{\mathrm{BG}}.$$
(5)

where μBG and σBG are the mean and standard deviation of the readings corresponding to background radiation, respectively. k is a user-defined value that can affect the true-positive rate of detection accuracy. For each detector, the average count rate was calculated and compared with thrd to verify the existence of a radiation source as follows:

$${D}_{i}= \left\{\begin{array}{ll}1 &\quad \text{if} \,\,{c}_{i}>th{r}_{\mathrm{d}}\\ 0 &\quad \text{if}\, \,{c}_{i}\le th{r}_\text{d}\end{array},\right.$$
(6)

where Di is a value from the set [0, 1] that indicates the detection status of the ith detector. If Di is 1, the detector’s readings correspond to the radiation source. However, the readings of the ith detector are considered measurements of the background radiation if Di = 0. ci represents the average count rate measured by the ith detector and is calculated as follows:

$${c}_{i}= \frac{1}{{T}_{\mathrm{W}}}\sum_{t=1}^{{T}_{\mathrm{W}}}{c}_{i,t}.$$
(7)

For N detectors deployed in a given area, the k-sigma method was applied periodically to the readings of each detector. All readings collected from the network of detectors are provided in the list R, such as

$$R= \left[{R}_{1}, {R}_{2}, \dots , {R}_{N}\right],$$
(8)

where Ri denotes the reading of the ith detector. Ri consists of all readings measured within TW as follows:

$${R}_{i}= \left[\begin{array}{c}{c}_{i,1}\\ \vdots \\ {c}_{i,{T}_{\mathrm{W}}}\end{array}\right],$$
(9)

where ci,t denotes the measured count rate at the tth time instant for the ith detector. If a radiation source is detected by M detectors, where M ≤ N, the localization algorithm can be performed using the corresponding readings. Figure 1 shows the process of selecting the readings of the detectors that detect a radioactive source according to the k-sigma method. As shown in Fig. 1, the selected readings are stored in a list referred to as list_R such that ||list_R||= M. According to [37], MLE can be applied for radiation source localization using M detectors such that M ≥ 3.

Fig. 1
figure 1

Process of selecting the readings of the detectors that detected a radiation source

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After selecting the M detectors that detect a radiation source, the corresponding readings in list_R are used for the estimation process using MLE. Due to the nonlinearity relation between the average count rate λ and the source’s parameters (X, Y, I), there is no closed form solution to the likelihood function presented in (5). Hence, a search method is used to solve the problem numerically. Search methods were used to find the best solution within the parameters’ ranges: [Xmin, Xmax], [Ymin, Ymax], and [Imin, Imax]. The traditional search method is a midpoint grid search [13], where the range of each parameter is divided into grids of equal size, and the midpoint of each grid is explored. Algorithm 1 presents the MLE algorithm using a midpoint grid search. As shown in Algorithm 1, the parameters’ ranges, [Xmin, Xmax], [Ymin, Ymax], and [Imin, Imax], were divided into NX, NY, and NI grids, respectively. Hence, the total number of points explored was NX × NY × NI. Usually, an equal number of grids is used for each range, that is, NX = NY = NI.

In this study, computational complexity is defined as the number of times the likelihood function (L) is calculated. Accordingly, the computational complexity of the MLE using grid search can be provided in terms of the big O notation O(Ng3), where Ng = NX = NY = NI. Notably, localization accuracy is affected by the number of grids such that better estimates can be obtained using a larger number of grids. However, the time consumption increases as the number of grids increases. The estimation time of the grid search-based MLE, denoted as TMLE, can be calculated according to the following relation:

$${T}_{\mathrm{MLE}}= {N}_{\mathrm{g}}^{3}\cdot {T}_{\mathrm{L}},$$
(10)

where TL is the time for calculating the likelihood function (L).

figure a

Another search method for MLE is multi-resolution MLE [11, 36, 38], which aims to reduce the time consumed by the grid search. In multi-resolution MLE, the search is performed through an iterative process over a parameter range that decreases per iteration. Algorithm 2 presents an MLE algorithm using a multi-resolution grid search. The algorithm applies the conventional grid search using the number of grids NX_MR, NY_MR, and NI_MR, where NX_MR≪ NX, NY_MR≪ NY, and NI_MR NI, respectively. At each iteration, the resulting estimates XML, YML, and IML were used to calculate the new ranges [Xmin, Xmax], [Ymin, Ymax], and [Imin, Imax], respectively, to evaluate the new estimated values using Algorithm 1. A significant time reduction can be achieved using multi-resolution MLE without significantly reducing the estimation accuracy compared with traditional grid search MLE. The computational complexity of the multi-resolution MLE can be represented by \(O({N}_{\text{g}\_\text{MR}}^{3}\cdot {N}_{\text{itr}\_\text{MR}})\) for Ng_MR = NX_MR = NY_MR = NI_MR, where Nitr_MR is the number of iterations. Accordingly, the estimation time of the multi-resolution grid search MLE can be formulated as.

$${T}_{\mathrm{MR}\_\mathrm{MLE}}\approx {N}_{{\mathrm{g}}_{\mathrm{MR}}}^{3}{\cdot N}_{{\mathrm{itr}}_{\mathrm{MR}}}\cdot {T}_{L},$$
(11)

where TMR_MLE is the time consumed by the MLE using a multi-resolution grid search. One limitation of the multi-resolution MLE is that the search method can provide a local maximum solution rather than the global maximum. In other words, the algorithm does not present a procedure for avoiding convergence toward a local maximum solution in the search space.

figure b

3 Localization using heuristic-based MLE

Heuristic techniques can be used to reduce search time and guarantee an optimum or near optimum solution. We studied the effect of heuristic methods on the performance of maximum likelihood localization of radioactive sources. Figure 2 presents the basic procedure for MLE using a heuristic-based search applied for estimating the radiation source parameter Ɵ. As shown in Fig. 2, the optimum solution was selected from a solution space bounded by Ɵmin = [Xmin, Ymin, Imin] and Ɵmax = [Xmax, Ymax, Imax]. Initially, Np solutions are randomly selected to represent the search population as follows:

$${\Theta }_{n,d}= {\Theta }_{\mathrm{min},\mathrm{d}}+ \epsilon \cdot \left({\Theta }_{\mathrm{max},\mathrm{d}}- {\Theta }_{\mathrm{min},\mathrm{d}}\right) , 1\le n\le {N}_{\mathrm{p}} , 1\le d\le {Dim}_{\theta },$$
(12)

where Ɵn,d is the dth dimension of the nth solution in the Np solutions, and DimƟ is the dimension of the parameter Ɵ. A random selection is conducted using a random value ϵ that follows a uniform distribution over the range [0, 1]. After the initialization step, the Np solutions are stored in the list ƟP as follows:

Fig. 2
figure 2

Heuristic-based search MLE for estimating radiation source parameters XML, YML, and IML

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$${\Theta }_{\mathrm{P}}= \left[{\Theta }_{1}, {\Theta }_{2}, \cdots , {\Theta }_{{N}_{\mathrm{p}}}\right].$$
(13)

Each parameter (solution) Ɵn was evaluated using the likelihood function L(Ɵn) that represents the fitness function, where 1 ≤ n ≤ Np. The search was then performed in a stochastic manner through an iterative process such that solutions with high fitness values could be found. In the heuristic method, a solution update technique is used to generate new solutions from randomly selected solutions. Hence, each heuristic method has a different computational complexity. In the next subsections, four heuristic methods are presented for the maximum likelihood localization of the radiation sources.

3.1 FFA-based MLE

The FFA is a population-based optimization method [28] that simulates the behavior of fireflies to attract mating partners. It is easily implemented to search for optimal solutions from a continuous range of values. The selection is performed, where a firefly moves toward another firefly that produces the brightest light. For the two fireflies, the observed light intensity (brightness) decreases as the distance between them increases according to the inverse square law. Algorithm 3 presents the pseudo code of the FFA-based MLE used to localize a radiation source. The algorithm simulates the behavior of fireflies to select the maximum likelihood estimates such that.

  • The solution space represents the locations of all possible fireflies in a given area.

  • The fitness value L(Ɵn) indicates the light intensity of the nth firefly according to its location represented by the parameter value Ɵn.

  • The solution update represents the movement of a firefly at location Ɵn toward another firefly located at Ɵm according to the following relation [39]:

    $${{\Theta }_{n}}^{*}= {\Theta }_{n}+ {\beta }_{0} {e}^{- \gamma {\cdot dis}_{n,m}^{2}}\left({\Theta }_{n}-{\Theta }_{m}\right)+ \alpha \epsilon , 1\le n\le {N}_\text{p},$$
    (14)

    where Ɵn* is the updated value for the nth solution Ɵn. The values of β0 and γ indicate the attractiveness and the light absorption coefficient, respectively. To prevent a premature convergence toward local optima, the term αϵ is added for further randomization. Parameter α is referred to as the randomization parameter. The distance between Ɵn and Ɵm is denoted by disn,m, which is calculated as follows:

    $${dis}_{n,m}= \sqrt{\sum_{d=1}^{{D}_{\Theta }}{\left({\Theta }_{n,d}-{\Theta }_{m,d}\right)}^{2},}$$
    (15)

    where Ɵn,d and Ɵm,d are the values of the dth dimension of Ɵn and Ɵm, respectively.

  • The optimum solution represents the location of the firefly with the highest brightness.

According to the pseudo code in Algorithm 3, the computational complexity of the FFA-based MLE can be expressed as O(Np2 Nitr). Hence, the estimation time for FFA-based MLE can be calculated as follows:

$${T}_{\text{FFA}\_\text{MLE}}\approx {N}_\text{p}^{2}{\cdot N}_\text{itr}\cdot {T}_\text{L},$$
(16)

where TFFA_MLE is the time required by the MLE algorithm using an FFA-based search.

figure c

3.2 PSO-based MLE

In PSO, the search procedure is inspired by the foraging behavior of animals, for example, birds or fish [29]. Each particle represents a location in the search space and has a memory that stores its best location per iteration, referred to as pbest (personal best). Similarly, the overall best location of the swarm (population) is stored in memory as the gbest (global best). The algorithm defined the swarm, position, and velocity of each particle. Initially, the positions and velocities of the population were randomly evaluated. Next, the position and velocity were updated per iteration according to the values of pbest and gbest as follows:

$${v}_{n}^{*}=w\cdot {v}_{n}+ {a}_{1}{\epsilon }_{1}\left({pbest}_{n}- {x}_{n}\right)+{a}_{2}{\epsilon }_{2}\left(gbest- {x}_{n}\right),$$
(17)
$${x}_{n}^{*}= {x}_{n}+{v}_{n}^{*},$$
(18)

where xn* and vn* denote the updated values for the nth position xn and velocity vn, respectively. The parameters a1 and a2 are the acceleration constants used to guide the search toward the best local and global locations, respectively. Furthermore, є1 and є2 are random variables, and each follows a uniform distribution over the interval [0, 1]. A weight value denoted by w is used to control the exploration and exploitation of the swarm and is referred to as the inertia weight [40]. The values of pbest and gbest are the positions corresponding to the best fitness values for a single particle and the overall population, respectively. The pseudo code for the PSO-based MLE is presented in Algorithm 4. The particles’ positions are represented by the radiation source parameters ƟP, which are initialized according to (13). However, the velocity of each particle is initialized to a small random value according to [41], where r is a small constant value:

$${v}_{n}=r\cdot \epsilon , 1\le n\le {N}_{\mathrm{p}},$$
(19)

As shown in Algorithm 4, the computational complexity of PSO-based MLE is O(Np Nitr). Accordingly, the time consumption of the PSO-based MLE can be expressed as

$${T}_{\text{PSO}\_\text{MLE}}\approx {N}_\text{p}{\cdot N}_\text{itr}\cdot {T}_\text{L},$$
(20)

where TPSO_MLE is the estimation time of the PSO-based MLE.

figure d

3.3 ACO-based MLE

ACO was first presented by Colorni et al. [30] as an ant system that simulates ant foraging behavior. During food searching, each ant deposits its pheromone on the path it follows to find the food. Consequently, ants select a path according to the amount of pheromones deposited on that path. A path with more pheromones indicates that it was selected by more ants, representing the shortest path to food. The ACO was first used for discrete optimization problems. Many studies have been conducted on the application of ACO to continuous domains [42]. In this study, ACO was applied to estimate the radiation source parameter by using the MLE according to the following representations:

  • The solution space \(({\Theta }_{\mathrm{min}}\to {\Theta }_{\mathrm{max}})\) represents all feasible paths for the population of ants.

  • The pheromone level of a selected path is represented by its fitness value, which is calculated using the likelihood function L.

  • A new path is selected based on the pheromone levels of the explored paths, according to [43], as follows:

    $${\Theta }_{n}^{*}={R}_{\mathrm{G}} , {R}_{\mathrm{G}}\sim N\left({\Theta }_{\mathrm{best}}, {\upsigma }_{\mathrm{col}}\right), 1\le n\le {N}_{\mathrm{p}},$$
    (21)

where RG is a random number that follows a Gaussian (normal) distribution of mean Ɵbest and standard deviation σcol. The value of Ɵbest represents the best explored path in terms of fitness value. σcol is the standard deviation of all explored paths.

The pseudocode for the ACO-based MLE is presented in Algorithm 5. The computation complexity of the ACO-based MLE can be expressed as O(Np Nitr). Furthermore, the time consumption of the ACO-based MLE compared with that of the grid-search-based MLE can be formulated as

$${T}_{\mathrm{ACO}\_\mathrm{MLE}}\approx {N}_{\mathrm{p}}{\cdot N}_{\mathrm{itr}}\cdot {T}_{\mathrm{L}},$$
(22)

where TACO_MLE is the estimation time of the ACO-based MLE.

figure e

3.4 ABC-based MLE

The ABC algorithm is one of the effective nature-inspired optimization techniques. It was introduced by Karaboga, in [44], as a population-based optimization algorithm that simulates the foraging process of a bee swarm. According to [44], bees search for the highest quality food source by dividing the bee swarm into three groups:

  1. 1.

    Employed bees: explore selected food sources.

  2. 2.

    Onlooker bees: select new food sources for employed bees to explore based on the quality of previously explored sources.

  3. 3.

    Scout bees: randomly explore new food sources.

In ABC, the solution space represents all possible food sources for the bee swarm. Notably, the population represents the location of the food sources to be explored. The solution-updating procedure simulated the behavior of the bees through three stages:

  1. 1.

    Employed stage:

Each solution is updated according to the following relation:

$${{\Theta }_{n,d}}^{*}={\Theta }_{n,d}+\mathrm{\varnothing }\cdot \left({\Theta }_{n,d}-{\Theta }_{r,d}\right), 1\le n,r\le {N}_{\mathrm{p}},$$
(23)

where Ɵn,d* is the updated value for the dth dimension of the nth solution Ɵn,d, and Ɵr,d is the value of the dth dimension of the rth solution. For each solution, only the value of the dth dimension is updated such that d is selected randomly from the range [1, DimƟ]. The rth solution is selected randomly from the range [1, Np] under the condition that r ≠ n. The update equation is based on a random value denoted by Φ, which follows a uniform distribution over the range [− 1, 1].

  1. 2.

    Onlooker stage:

A group of solutions is selected for further updating based on the selection probability, denoted by Pn, which is calculated as

$${P}_{n}= \frac{{L}\left({\Theta }_{\mathrm{n}}\right)}{\sum_{m=1}^{{N}_{\mathrm{p}}}{L}\left({\Theta }_{\mathrm{m}}\right)}, 1\le n\le {N}_{\mathrm{p}}$$
(24)
  1. 3.

    Scout stage:

In ABC, for each solution, the number of unsuccessful updates (those not providing a better fitness value) is counted and stored in a counter, which is referred to as trials. If the value of trials exceeds a predefined value, referred to as Limit, the solution is randomly updated as follows:

$${\Theta }_{n}^{*}=\left\{\begin{array}{ll}{\Theta }_{\mathrm{min}}+ \epsilon \cdot \left({\Theta }_{\mathrm{max}}- {\Theta }_{\mathrm{min}}\right) &\quad \text{if}\,\, {trials}_{n}>Limit \\ {\Theta }_{n} &\quad \text{Otherwise} \end{array}\right., 1\le n\le {N}_{\mathrm{p}},$$
(25)

where trialsn is the value of trials for the nth solution.

Algorithm 6 presents the pseudo code of the ABC-based MLE method. In the onlooker phase, the selection probability is evaluated and compared with rand(0,1), which generates a uniformly distributed random value in the interval [0, 1]. Accordingly, a solution is selected for updating. The onlooker phase is conducted to provide Np solutions for exploration. Hence, an iterative process is applied until Np solutions are selected. A number referred to as Nmax_look is defined to limit the number of onlooker iterations, where Np < Nmax_look. In other words, the number of onlooker iterations is bounded between the minimum and maximum values, Np and Nmax_look, respectively. However, in the scout phase, updating is performed only for solutions whose corresponding trials values exceed the value of Limit. Hence, the calculation of the fitness function (likelihood function) is conducted NL times per iteration, where 2Np ≤ NL ≤ 2Np + Nmax_look. Accordingly, the time consumption of the ABC-based MLE can be calculated as

$${T}_{\mathrm{ABC}\_\mathrm{MLE}}\approx {\left(\sum_{i=1}^{{N}_{\mathrm{itr}}}{N}_{\text{L}_{i}}\right)\cdot }T_\text{L},$$
(26)

where TABC_MLE is the estimation time for the ABC-based MLE, and NLi is the number of times that the likelihood function is calculated at the ith iteration.

figure f

In the next section, we discuss the performance of each search method. Figure 3 shows the procedure performed to provide the estimated values for the evaluation. Each source parameter was read from a given dataset that consists of detector readings from sources at different locations, where the total number of locations is denoted by Nloc. The detection process was performed using the k-sigma method, and the estimated parameters were then calculated using the MLE method with different searching techniques.

Fig. 3
figure 3

Procedure for calculating the estimated values

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4 Experimental results

This section presents the performance of the maximum likelihood localization of a radioactive source by using traditional and heuristic search-based methods. In this study, real data were used to evaluate the performance of the MLE methods. The data were collected using the Low Scatter Irradiator facility at the Savannah River National Laboratory as part of the Intelligent Radiation Sensing System program [45, 46]. The count rate measurements were read using 22 stationary radiation detectors scattered in a room with an area of 8 m × 8 m (Fig. 4). First, the detector readings were collected without a radiation source inside the room to measure the background radiation. For the evaluation of the detection and localization methods, radiation sources were placed at different locations inside the room (Fig. 5). The experimental results were calculated using Python 3.9 on a core-i7 PC of 2 GHz and 16 GB RAM. The Python programming language was used for the following: (1) reading the data from the dataset text file, (2) implementing all considered methods, and (3) evaluating the performance of each method. Table 1 lists the experimental parameters.

Fig. 4
figure 4

Locations of radiation detectors inside an area of size 8 m × 8 m, used for collecting the dataset

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Fig. 5
figure 5

Considered positions of the radiation source and radiation detectors

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Table 1 Parameters’ values used to conduct the experiment
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In this study, we have presented a performance comparison among four heuristic-based search techniques and traditional grid search-based methods applied to MLE. The comparison was conducted in terms of the most significant factors that affect MLE performance: estimation accuracy and time consumption. First, radiation source detection was applied through the k-sigma method using μBG = 2.4 cps and σBG = 1.5 cps, calculated from the radiation background dataset. Traditionally, the value of k has been set to 1.645 to control the false positive rate to 5% [47]. Consequently, the detection threshold thrd was 5.4 cps. In the localization process, the likelihood function was evaluated according to the detectors’ readings and the average count rate λ by using δ = 1.6 s−1 m2/μCi, calculated via calibration [19]. Notably, the background radiation count rate BG was set to its average value μBG. Table 2 presents the values of the parameters used for the grid search-based MLE and heuristic-based search MLE. As shown in Table 2, various population sizes and numbers of iterations were used to examine their effect on the performance of heuristic-based search MLE methods. According to the dataset parameters listed in Table 1, a suitable range for the source intensity and location was determined. The number of grids used for grid search-based MLE and multi-resolution MLE were selected experimentally. Additionally, different values were selected for the parameters of the heuristic techniques (i.e., FFA, PSO, and ABC) to study their effect on localization performance. Some parameters, such as β0 and w, were set to their most suitable values, according to the literature, for FFA [39] and PSO [48], respectively.

Table 2 Values of parameters used for calculating MLE estimates
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To evaluate localization performance, we measured the estimation error for the radiation source location and intensity. The location error, denoted as errloc, was represented by the Euclidean distance between the estimated and actual source locations as follows:

$${err}_{\mathrm{loc}}=\frac{\sum_{m=1}^{{N}_{\mathrm{Loc}}}{Dis}_{m}}{{N}_\text{Loc}},$$
(27)

where Dism is the Euclidian distance between the actual and estimated locations of the mth source as follows:

$$Di{s}_{m}= \sqrt{{\left({X}_{m}-{\widehat{X}}_{m}\right)}^{2}+{\left({Y}_{m}-{\widehat{Y}}_{m}\right)}^{2},}$$
(28)

where \(\left({X}_{m},{Y}_{m}\right)\) and (\({\widehat{X}}_{m}, {\widehat{Y}}_{m}\)) are the coordinates of the mth actual and estimated source locations, respectively. The intensity error, denoted by errI, was measured using the normalized root mean square error (NRMSE) between the actual and estimated intensities as follows:

$${err}_{\mathrm{I}}=\frac{1}{({I}_{\mathrm{max}}-{I}_{\mathrm{min}})}\cdot \sqrt{\frac{\sum_{m=1}^{{N}_{\mathrm{loc}}}{\left({I}_{m}-{\widehat{I}}_{m}\right)}^{2}}{{N}_\text{loc}},}$$
(29)

where \({I}_{m}\) and \({\widehat{I}}_{m}\) are the actual and estimated intensities of the radiation source at the mth location, respectively. The minimum and maximum NRMSE values are 0 and 1, respectively. In evaluating the estimation time, the number of detectors (M) affects the time consumed (tL) for calculating the likelihood function. In the considered experiment, the number of detectors M varied according to the detection threshold thrd and the reading of each detector, such that 3 ≤ M ≤ N. Hence, a different estimation time was provided according to the source’s parameters (location and intensity). The average time consumption was used to represent the estimation time for each localization method over the considered source locations and intensities and was calculated as follows:

$${T}_\text{est}=\frac{1}{{N}_\text{Loc}}\sum_{m=1}^{{N}_\text{Loc}}{T}_{m},$$
(30)

where Test is the average estimation time, and Tm is the consumed time to localize the radiation source at the mth location. The time consumed (Tm) by each method was measured using the time module of the Python language.

Usually, in localization problems, the accuracy of the estimated location is considered the most significant factor for performance evaluation [36]. Hence, in this study, heuristic parameters were selected according to their effects on the location error errloc. Figure 6 shows the location error at different values of α and γ for the FFA-based MLE. Notably, the minimum error was achieved at α = 1 and γ = 15. Similarly, for the PSO-based MLE, Fig. 7 presents the location error at different values of parameters a1 and a2, where the best performance can be provided at a1 = 2 and a2 = 2. In Fig. 8, the location error was measured for the ABC-based MLE at different values of the user-defined parameter Limit. Accordingly, the Limit parameter was set to 30 to minimize the errors.

Fig. 6
figure 6

Location error errloc using FFA-based MLE at different values for parameters α and γ

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Fig. 7
figure 7

Location error errloc using PSO-based MLE at different values for parameters a1 and a2

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Fig. 8
figure 8

Location error errloc using ABC-based MLE at different values for the Limit parameter

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A performance comparison of the heuristic methods was conducted after setting the parameters of each heuristic method to their best values in terms of the achieved location error. Tables 3 and 4 present a comparison between the heuristic-based MLE methods in terms of the location and intensity estimation errors, respectively, for different population sizes and numbers of iterations. As shown in Table 3, the ACO and ABC algorithms provide a more accurate location estimate than the FFA and PSO do. As aforementioned, the accuracy of the estimation increases with the population size. However, the improvement in the estimation accuracy decreases as the population size increase. According to the results, the difference in performance between the FFA-based MLE using Np < 10 and Np > 10 was relatively large. Similarly, the number of iterations affected the estimation accuracy. As shown in Table 3, the difference between ACO and ABC was not significant in terms of the location error for Nitr values greater than 10. Furthermore, the estimation accuracy of the FFA-based MLE was approximately the same for Nitr greater than 5. Moreover, population size Np and number of iterations Nitr had a slight effect on the intensity estimation error (Table 4). The ABC-based MLE has the least estimation error of the heuristic methods for most of the considered population sizes and numbers of iterations.

Table 3 Comparison of heuristic-based MLE methods: location error errloc (m) at different population sizes Np and number of iterations Nitr
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Table 4 Comparison of heuristic-based MLE methods: intensity error errI at different population sizes Np and number of iterations Nitr
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According to the estimation time, we expected the time consumption of heuristic-based methods to increase as the population size or the number of iterations increased. Table 5 compares the heuristic-based MLE methods in terms of the estimation time (Test) for different population sizes and numbers of iterations. The time consumption for the FFA-based MLE increased significantly with an increase in population size. Furthermore, as concluded in the previous section, the PSO-based MLE and ACO-based MLE provide approximately the same estimation time, which is less than that of the ABC-based MLE. Accordingly, the FFA-based MLE is the most time-consuming among the considered heuristic-based MLE methods, especially for large population sizes.

Table 5 Comparison of heuristic-based MLE methods: estimation time Test (s) at different population sizes Np and number of iterations Nitr
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To illustrate the effectiveness of the heuristic-based MLE methods, we compared them with traditional grid search-based MLE methods. The performance value, denoted by Pv, was calculated to indicate the overall performance by using the weighted sum approach as follows:

$$P_\text{v}= {w}_{\mathrm{i}}\cdot {err}_{\mathrm{I}}+ {w}_{\mathrm{l}}\cdot {err}_{\mathrm{loc}}+ {w}_{\mathrm{t}}\cdot {T}_{\mathrm{est}},$$
(31)

where wi, wl, and wt are the weights of the intensity error, location error, and estimation time, respectively. The weights were calculated for normalization as follows:

$${w}_{\mathrm{i}}=\frac{1}{\mathrm{max}\left({err}_\text{I}\right)} , {w}_{\mathrm{l}}=\frac{1}{\mathrm{max}\left({err}_\text{loc}\right)} , {w}_{\mathrm{t}}=\frac{1}{\mathrm{max}\left({T}_\text{est}\right)},$$
(32)

where max(errI), max(errloc), and max(Test) are the maximum values of the intensity error, location error, and estimation time, respectively, calculated for the considered population sizes and the number of iterations. Based on these results, weights were calculated such that wi = 5.208, wl = 0.757, and wt = 0.713. Table 6 compares the heuristic-based MLE methods in terms of performance value Pv. The minimum Pv value was 0.732, obtained using the ABC-based MLE method with the population size and number of iterations equal to 20 and 10, respectively. Hence, a performance comparison of the traditional grid search MLE methods and heuristic-based MLE methods using Np = 20 and Nitr = 10 was presented.

Table 6 Comparison of heuristic-based MLE methods: performance value Pv at different population sizes and number of iterations
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Table 7 shows the estimated parameters (source intensity and location) at some of the considered source positions in the dataset by using the grid search MLE and the heuristic-based MLE methods. Table 8 compares the performance of traditional grid search-based MLE methods and heuristic-based search MLE methods in terms of estimation error, time consumption, and performance value. The results demonstrate that the multi-resolution grid search significantly reduced the time consumption of MLE and provided estimation accuracy approximately equal to that of a fixed resolution grid search. However, heuristic-based MLE methods present different estimation times and accuracies depending on the solution update procedure of each heuristic technique. The FFA-based MLE method consumes more time and provides lower estimation accuracy than the multi-resolution MLE method does. However, other heuristic techniques (i.e., PSO, ACO, and ABC) provide better accuracy than the FFA and lower estimation time than the multi-resolution MLE. The results in Table 8 verify that the PSO and ACO methods require less time than the ABC method does. In addition, better localization accuracy can be achieved using the ABC-based MLE than by using the considered heuristic-based MLE localization methods.

Table 7 Comparison of grid search-based MLE methods and heuristic-based MLE methods according to the estimated parameters (IML, XML, YML)T
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Table 8 Comparison of grid search-based MLE methods and heuristic-based MLE methods: location error (errloc), intensity error (errI), time consumption (Test), and performance value (Pv)
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5 Conclusion

In this study, heuristic techniques were investigated for estimating the parameters of a radiation source using MLE. Detection and localization of the radiation source were performed using a sensor network. First, the detection process was conducted using the k-sigma method for each radiation detector. Next, the localization of the radiation source was performed using only the readings and positions of the detectors that detected the source. This study considered four effective heuristic techniques that are commonly used: FFA, PSO, ACO, and ABC. A performance comparison was conducted between the heuristic-based MLE and the traditional MLE by using fixed resolution and multi-resolution grid searches. To provide reliable results, we used real data to evaluate the performance of the considered methods in terms of estimation accuracy and time consumption. The evaluation was conducted by estimating 65 different locations of a 137CS radiation source with intensities of 7.6 μCi and 16 μCi, inside an 8 m × 8 m area, using 22 stationary radiation detectors. According to the results, the considered methods were able to estimate the location of the radiation source with an estimation error of approximately 0.4 m. However, the time consumption varies for each search method. The estimation time using fixed and multi-resolution grid search MLE was 3.27 and 0.59 s, respectively. On the other hand, MLE using FFA, PSO, ACO, and ABC provided maximum likelihood estimates of 0.75, 0.03, 0.02, and 0.059 s, respectively. Hence, the results of this study imply that heuristic-based MLE can provide approximately the same estimation accuracy and less estimation time compared to both the fixed and multi-resolution MLE methods. Furthermore, among the heuristic algorithms considered, the most accurate estimates were obtained using the ACO and ABC methods.

Although heuristic-based MLE methods are less time-consuming than conventional MLE methods, they may not be suitable for real-time applications. In further research, machine learning can be utilized to further reduce the time consumption of the MLE-based localization process. Moreover, additional general localization problems can be considered, such as the localization of (1) more than one radiation source, (2) a radiation source in high variance background radiation, (3) a shielded radiation source, and (4) a moving radiation source.