Three intelligent optimization algorithms, namely, the standard Particle Swarm Optimization (PSO), the Stochastic Particle Swarm Optimization (SPSO), and the hybrid Differential EvolutionParticle Swarm Optimization (DEPSO), were applied to solve the inverse transient radiation problem in twodimensional (2D) turbid media irradiated by the short pulse laser. The timeresolved radiative intensity signals simulated by finite volume method (FVM) were served as input for the inverse analysis. The sensitivities of the timeresolved radiation signals to the geometric parameters of the circular inclusions were also investigated. To illustrate the performance of these PSO algorithms, the optical properties, the size, and location of the circular inclusion were retrieved, respectively. The results showed that all these radiative parameters could be estimated accurately, even with noisy data. Compared with the PSO algorithm with inertia weights, the SPSO and DEPSO algorithm were demonstrated to be more effective and robust, which had the potential to be implemented in 2D transient radiative transfer inverse problems.
The problem of transient radiative transfer in participating media has attracted increasing interest over the last two decades due to the emergence of ultrashort pulse laser and its application to the picosecond level or higher accuracy level of timeresolved techniques [
For solving the inverse transient radiation problems in the noninvasive reconstruction, the first step is to solve the direct TRTE model accurately and efficiently. Nowadays, several numerical strategies have been developed to solve the forward model, including the discrete ordinate method (DOM) [
To solve the inverse problems, a wide variety of inverse techniques have been successfully employed in the inverse radiation analyses which can be grouped into two categories, that is, the traditional algorithm based on gradient and the intelligent optimizations [
The objective of present work is to apply the PSObased algorithms to solve the inverse transient radiation problem in twodimensional (2D) participating medium. The optical properties, the size, and location of the circular inclusion were retrieved, respectively. The remainder of the paper is organized as follows. The detailed mathematical formulation and computational steps of the direct model are described in Section
In the present work, the thermal effect caused by the incident laser was ignored and only the transient radiative transfer was considered in the numerical model. Thus, the transient radiative transfer equation can be expressed as [
For the square pulse laser, the intensity within the participating media can be expressed as [
For the 2D participating media, the discrete equation could be obtained using the FVM model:
In (
The total source term
Commonly, there are three ways to obtain the internal information of media, that is, the continuous wave method, the timedomain method, and the frequencydomain method [
The PSO algorithm was first introduced by Eberhart and Kennedy [
In order to avoid premature convergence of the standard PSO and ensure the overall convergence, a considerable number of modified PSO algorithms were proposed in recent years, like the Stochastic PSO (SPSO), the Multiphase PSO (MPPSO), the QuantumBehaved PSO (QPSO), the hybrid Ant Colony Optimization and PSO (ACOPSO), and so forth [
The differential evolution algorithm (DEA) is a novel parallel, direct, and stochastic searching method which was proposed to solve the continuous domain problems in 1995 [
The schematic diagram of vectorgeneration process for DEA.
In the process of DEPSO, the differential evolution operator is introduced to the position updating equation, which can be expressed as [
The main function of the third term on the right side of (
Initialize the position
Check if the value of the objective function is smaller than the preset parameter
Update the velocity and position of each particle according to (
Calculate the objective function of each newly generated particle. If the value of the objective function is smaller than that of the corresponding particle in the last generation, then the new particle can be maintained. Otherwise, regenerate the particle according to (
Update the generation from
Output the global best position and its corresponding value of objective function.
To illustrate the difference between PSO, SPSO, and DEPSO algorithms more clearly, the flowcharts of the three algorithms are shown in Figure
The flowcharts of (a) PSO, (b) SPSO, and (c) DEPSO algorithms.
The transient radiative inverse problems for estimating the internal radiative properties of the 2D participating media are solved by minimizing the objective function which can be defined as
To demonstrate the effects of measurement errors on the predicted terms, the random errors were considered. In the present paper, the simulated measured signals with random errors were obtained by adding normally distributed errors to exact reflectance and transmittance as follows:
Consider a 2D homogeneous semitransparent medium with length 1.0 m × 1.0 m whose left side exposes to a collimated square pulse laser beam in the
Validation of the FVM solution for transient radiative transfer of a 2D homogeneous rectangle medium. (a) Transmittance and (b) reflectance.
The estimation in this paper is expected to be applied in the nondestructive detection of human or small animals. Figure
The control parameters of the four cases.
Parameters  Case 1  Case 2  Case 3  Case 4  

Size [m] 









 


Location [m] 

(0.3, 0.45)  (0.3, 0.45)  (0.7, 0.65)  (0.3, 0.45) 

(0.75, 0.65)  (0.75, 0.65)  (0.35, 0.25)  (0.75, 0.65)  


Optical properties [m^{−1}] 

(1.0, 7.5)  (1.0, 7.5)  (1.0, 7.5)  (4.0, 5.5) 

(0.2, 9.8)  (0.2, 9.8)  (0.2, 9.8)  (1.5, 9.0) 
The simplification of actual models.
The physical model contains two inclusions.
The (a) reflectance and (b) transmittance signals in the function of
To test the reconstruction ability of the three swarm intelligence optimization algorithms, two cases for estimating the optical parameters were considered. For the laser incidence, two different conditions were included (see Figure
The retrieving results of the optical properties for Case 1 using PSO, SPSO, and DEPSO without measurement error.
Algorithm  Condition  True values or inversion results [m^{−1}]  Relative errors [%]  




 
PSO  PI 




FI 



 


SPSO  PI 




FI 



 


DEPSO  PI 




FI 




The retrieving results of the optical properties for Case 2 using PSO, SPSO, and DEPSO of different generations.
Generations  True value [m^{−1}]  Algorithm  Retrieving results [m^{−1}]  Relative errors [%] 

10 

PSO  (0.9750, 9.3216)  (2.4962, 3.5738) 
(7.4025, 5.9205)  (64.5009, 7.6456)  
SPSO  (0.9803, 8.9818)  (1.9718, 0.2021)  
(5.4861, 0.3347)  (21.9131, 93.9149)  
DEPSO  (0.9844, 8.8901)  (1.5635, 1.2207)  
(3.3674, 3.7704)  (25.1700, 31.4477)  


50 

PSO  (0.9790, 9.0331)  (2.1025, 0.3679) 
(6.4496, 7.5222)  (43.3244, 36.7678)  
SPSO  (0.9995, 8.9996)  (0.0478, 0.0040)  
(3.5513, 4.6093)  (21.0814, 16.1946)  
DEPSO  (1.0021, 9.0036)  (0.2071, 0.0397)  
(3.3049, 4.0275)  (26.5584, 26.7718)  


100 

PSO  (0.9790, 9.0331)  (2.1025, 0.3679) 
(6.449, 7.5222)  (43.3244, 36.7678)  
SPSO  (0.9998, 8.9995)  (0.0234, 0.0054)  
(3.5119, 4.8832)  (21.9588, 11.2147)  
DEPSO  (1.0007, 9.0042)  (0.0696, 0.0472)  
(3.6868, 4.8677)  (18.0717, 11.4957) 
The retrieving results of the optical properties for Case 2 using PSO, SPSO, and DEPSO without measurement error.
Algorithm  Time [s]  True values [m^{−1}]  Retrieving results [m^{−1}]  Relative errors [%] 

PSO  25476 

(0.9985, 9.0019)  (0.1486, 0.0207) 

(6.9803, 7.5019)  (55.118, 36.398)  


SPSO  32775 

(1.0000, 8.9996)  (0.0034, 0.0046) 

(4.4444, 5.4556)  (1.2349, 0.8077)  


DEPSO  49554 

(1.0005, 9.0036)  (0.0544, 0.0397) 

(4.5248, 5.4454)  (0.5508, 0.9925) 
The incident types of pulse laser.
The schematic of pulse laser irradiation and the measurement position of timedomain signals of the two test cases.
It can be seen from Table
The objective function of PSO, DEPSO, and SPSO in Case 2.
In the inverse procedure, the long sampling span will reduce the computational efficiency. Hence, it is of great significance to select the proper sampling span. Thus, the sensitivity of radiative signal to the inverse parameters needs to be analyzed. The sensitivity coefficient is one of the most important characteristic parameters in the sensitivity analysis, which is the first derivative of the radiative signals to a certain inverse parameter. The sensitivity coefficient is defined as
The sensitivities of the inclusion location (
The sensitivity of the radiation signals for the geometric parameters of the inclusion.
In many conditions, the shapes and optical properties of inclusions in the media are known in the actual application cases of reconstructing the medium’s inside information. For example, a tumor or cyst in the biological tissue is spherical or sphericallike, and the defects in the specimen, such as the air pore, are mainly the shape of a cube, sphere, or ellipsoid. In addition, the optical properties of human and animals tissues, in vitro or in vivo, are studied thoroughly [
In this section, the inclusion location (
The inverse results of size and location of circular inclusion using PSO, SPSO, and DEPSO without measurement error under FI condition.
Algorithm  True values or inverse results [m]  Relative errors [%]  






 
True value 



—  —  — 
PSO 






SPSO 






DEPSO 








True value 



—  —  — 
PSO 






SPSO 






DEPSO 








True value 



—  —  — 
PSO 






SPSO 






DEPSO 






The inverse results of size and location of circular inclusion using PSO, SPSO, and DEPSO with 10% measurement error under FI condition.
Algorithm  True values or inverse results [m]  Relative errors [%]  






 
True value 



—  —  — 
PSO 






SPSO 






DEPSO 








True value 



—  —  — 
PSO 






SPSO 






DEPSO 








True value 



—  —  — 
PSO 






SPSO 






DEPSO 






The comparison of the exact and reconstructed reflectance signals.
It can be seen that the size and location can be estimated accurately using PSO, SPSO, and DEPSO without measurement errors under FI condition. Furthermore, the results of DEPSO are more accurate than those of the SPSO and PSO. The relative errors are reasonable even with 10% measurement error. However, it is worth noting that the relative errors of DEPSO with 10% measurement error are bigger than those of the SPSO, which means the SPSO algorithm is more robust than the DEPSO.
In the present study, the standard PSO, SPSO, and DEPSO algorithms were applied to estimate the size, location, and optical parameters of the circular inclusions in a 2D rectangular medium. The conclusion was obtained that the retrieving results of absorption coefficient and scattering coefficient under FI condition are more accurate than those under PI condition, the reason of which is that, under FI condition, the reflectance and transmittance signals are stronger and are carrying more information available. In addition, the DEPSO algorithm shows its priority in reconstructing the inclusion properties. When only one inclusion is considered, under both laser incident conditions, the results’ relative errors of DEPSO are smaller than those of the SPSO. However, when estimating the size and location of inclusion media, the SPSO is more robust than DEPSO. Furthermore, compared with the performance of standard PSO, both the SPSO and DEPSO are proved to be more accurate and robust, which have the potential to be applied in the field of 2D inverse transient radiative problems.
The vector of estimated properties
The speed of light, m/s
The acceleration coefficients of PSO
The control variable of DEA
The objective function
The radiation intensity,
The number of the population
The number of grids
The number of polar angles and azimuthal angles
The global best position discovered by all particles at generation
The local best position of particle
The radius of the inclusions
The uniformly distributed random numbers
The sensitivity coefficient
Time or generation in PSO algorithm
The incident pulse width, s
The start and end point of sampling span
The time step, s
The trial vector of DEA
The velocity array of the
The inertia weight coefficient
The position array of the
The coordinate values of the inclusions’ center.
The extinction coefficient, m^{−1}
The tolerance for minimizing the objective function
The relative error, %
The scattering phase function
The measured error
Absorption coefficient, m^{−1}
The direction cosine
The measured timeresolved reflectance signals with noisy data
The measured timeresolved reflectance signals without noisy data
The estimated timeresolved reflectance signals
The scattering coefficient, m^{−1}
The direction
The solid angle in the direction
The average relative error of the reflectance signals
The global best position in PSO
The searching index
The pulse width
The relative error.
The inverse of the matrix
Dimensionless.
The authors declare that there is no conflict of interests regarding the publication of this paper.
The support of this work by the National Natural Science Foundation of China (no. 51476043), the Major National Scientific Instruments and Equipment Development Special Foundation of China (no. 51327803), and the Technological Innovation Talent Research Special Foundation of Harbin (no. 2014RFQXJ047) is gratefully acknowledged. A very special acknowledgement is made to the editors and referees who make important comments to improve this paper.