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Article

Parameter Optimization Method for Metal Surface pBRDF Model Based on Improved Strawberry Algorithm

by
Xue Gong
1,2,
Fangbin Wang
1,2,*,
Darong Zhu
1,2,
Feng Wang
3,
Weisong Zhao
1,
Song Chen
1,2,
Ping Wang
1 and
Shu Zhang
1
1
School of Mechanical and Electrical Engineering, Anhui Jianzhu University, Hefei 230601, China
2
Key Laboratory of Construction Machinery Fault Diagnosis and Early Warning Technology of Anhui Jianzhu University, Hefei 230601, China
3
Polarized Light Imaging Detection Technology Anhui Provincial Key Laboratory, Hefei 230031, China
*
Author to whom correspondence should be addressed.
Appl. Sci. 2024, 14(14), 6022; https://doi.org/10.3390/app14146022
Submission received: 11 June 2024 / Revised: 30 June 2024 / Accepted: 5 July 2024 / Published: 10 July 2024
Browse Figures
Figure 1
<p>BRDF coordinate system.</p> "> Figure 2
<p>Strawberry plant.</p> "> Figure 3
<p>The initial population generated by the random method.</p> "> Figure 4
<p>Initial population generated by chaotic mapping.</p> "> Figure 5
<p>Comparison of Levy flight and random walk.</p> "> Figure 6
<p>Flowchart of parameters retrieval.</p> "> Figure 7
<p>SBA convergence curve.</p> "> Figure 8
<p>ISBA convergence curve.</p> "> Figure 9
<p>Curves fitted with predicted and referred DOP values under the incident angle of 60° (<b>a</b>)copper; (<b>b</b>) green paint; (<b>c</b>) black paint.</p> "> Figure 10
<p>Schematic diagram of the experimental setup.</p> "> Figure 11
<p>Fitting curves by LM, PSO, SBA and ISBA at different incident angles (<b>a</b>) <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>θ</mi> </mrow> <mrow> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mn>30</mn> </mrow> </semantics></math>°; (<b>b</b>) <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>θ</mi> </mrow> <mrow> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mn>40</mn> <mo>°</mo> </mrow> </semantics></math>; and (<b>c</b>) <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>θ</mi> </mrow> <mrow> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mn>50</mn> <mo>°</mo> </mrow> </semantics></math>.</p> ">
Versions Notes

Abstract

:
To study the polarization reflection characteristics of metal surfaces, a parameter optimization method for the polarization bidirectional reflection distribution function (pBRDF) model of metal surfaces based on the improved strawberry algorithm has been proposed. Firstly, the light scattering characteristics of metal surfaces were analyzed and a multi-parameter pBRDF model was constructed. Then, the working mechanism of the strawberry optimization algorithm was investigated and improved by introducing the chaotic mapping and Levy flight strategy to overcome the shortcomings, such as low convergence rate and easily falling into local optimum. Finally, the method proposed in this paper was validated by simulating open-source data from references and the obtained ones with a self-built experimental platform. The results show that the proposed method outperforms those by nonlinear least squares, particle swarm optimization and the original strawberry algorithm in fitting the detected degree of polarization (DOP) data, indicating the modeling accuracy was significantly improved and better suited to characterize the polarized reflection properties of metal surfaces.

1. Introduction

Polarization is a fundamental property of light [1]. Using the polarization characteristics of reflected light from the target surface, the multi-dimensional optical information of the target can be estimated, and the accuracy of target recognition and characterization features can be significantly improved [2]. It has become a hot topic in remote sensing [3], industrial production [4], biomedical [5] and other fields. Among the most common materials in industrial production, metal is often used for various equipment and instruments. So, it is of great theoretical and practical significance to study the polarization reflection characteristics of metal surfaces.
The bidirectional reflectance distribution function (BRDF) describes specular and diffuse reflection from a target surface, which is usually described by establishing mathematical models, such as T-S [6], Phong [7], Maxwell Bear [8], C-T [9] models, etc. However, BRDF only reflects the intensity distribution of reflected light. In order to better present the valuable target information about the material features and the responding physical and chemical properties, polarization characteristics have been integrated into BRDF, and the polarization bidirectional reflection distribution function (pBRDF) has been proposed. For example, Priest et al. proposed that combining BRDF with a Muller matrix based on the T-S model provided a complete definition and corresponding application solution, while the difference between model calculations and experimental results seems slightly large [10]. From then, many improved pBRDF models have been proposed by supplementing or modifying the shadowing function, microfacet distribution and reflection component separation. Zhan H et al. established an improved pBRDF model by introducing a diffuse component into the reflecting part to retrieve metal characteristic parameters with better accuracy, while the impact of multiple reflections was not considered [11]. To solve this issue, Zhu DR et al. comprehensively investigated the shadowing effect and diffuse reflection on a target surface and used the Minnaert model to characterize the diffuse component, proposing a six-parameter pBRDF model, by which the fitting accuracy is further improved [12].
From a deeper analysis of the above-mentioned pBRDF, it can be seen that the traditional methods mainly start from a model whose parameters are usually retrieved with a least squares method. For example, Ingmar G. E. Renhorn et al. proposed a four-parameter pBRDF model whose parameters were retrieved just using least squares, by which the real and imaginary parts of the effective refractive index, grazing incidence absorption and angular scattering were obtained [13]. Pan JH et al. used the least square method to retrieve the real and imaginary parts of the target surface refractive index, as well as the surface slope parameters [14]. However, these methods did not pay enough attention to the possible influence of extreme points and missing data values on the retrieved model parameters. So, Feng WW et al. improved the model and proposed a pBRDF hybrid model based on the microfacet theory and obtained key model parameters from the experimental data by a genetic algorithm; the extraction ability for target polarization features was enhanced [15]. Meanwhile, Liu YY proposed a five-parameter model through a hybrid particle swarm genetic parameter optimization algorithm, by which the fitting accuracy and convergence speed were significantly improved compared to the genetic or particle swarm algorithm under the same conditions [16]. The above-mentioned algorithms not only obtained good local search ability but also improved global search capacity in the parameter retrieval process, though the retrieval results still depend on the initial selected values to a large extent, which is prone to falling into local optimal solutions for multi-dimensional problems.
To better characterize the light polarization reflection from metal surfaces, in this paper, the strawberry optimization algorithm was improved by introducing chaos mapping and Levy flight strategy into and applied to estimate the parameters of the metal surface pBRDF model. The performance of the proposed method was verified through the data from open source data and experiments by a self-established platform.

2. Polarization Bidirectional Reflection Distribution Function Model

BRDF is defined as the ratio of the reflected radiance along the outgoing to the incident irradiance along the incoming direction, expressed [17] as follows:
f r θ i , φ i ; θ r , φ r ; λ = d L r θ i , φ i ; θ r , φ r ; λ d E i θ i , φ i ; λ ( s r − 1 )
where d L r is the reflected radiance, and d E i is the incident irradiance. The parameters θ i and φ i denote the incident polar and azimuth angles, respectively, and θ r , φ r denote the reflected polar and azimuth angles, respectively, as shown in Figure 1. The parameter λ is the transferring light wavelength. Considering the fact that the microstructure of metal surfaces is usually in the micrometer scale and random distribution pattern, the BRDF model can be established using microfacet theory as shown in Figure 1.
In order to better analyze the optical properties of metal surfaces, the polarized bidirectional reflectance distribution function (pBRDF), usually reduced for BRDF with the help of the Muller matrix, is applied on many occasions.
Assuming the pBRDF of a metal surface contains specular reflection, diffuse reflection and volume scattering denoted F j , k s , F j , k d and F j , k v , it can be represented as shown in the following:
F = k s F j , k s + k d F j , k d + k v F j , k v
where k s , k d   a n d   k v denote the coefficients of the three components, respectively.
Specular reflection, the main component of polarized reflecting light, represents the single reflection from the interaction between incident light and a metal surface, it can be denoted as [18]:
F j , k s = ( q − 1 ) ( 2 σ ) 2 q − 2 ⋅ m i n ⁡ ( 1.0 , 2 cos ⁡ α cos ⁡ θ r cos ⁡ β , 2 cos ⁡ α cos ⁡ θ i cos ⁡ β ) 4 π cos ⁡ θ i cos ⁡ θ r cos 4 ⁡ α [ tan 2 ⁡ α + ( 2 σ ) 2 ] q × M j , k s
where σ is the surface roughness, α is the angle between the normals of the microfacet and the macroscopic surface, β is the angle between the incident light and the normal of the microfacet, and q is the related constant about the properties of the metal material, reflecting the scattering inclusivity of the metal surface.
Diffuse reflection and volume scattering are related to the texture and roughness of metal surfaces and are generally thought to be non-polarized for the randomness of the texture distribution. In this paper, the diffuse reflection from a metal surface was represented by the Minnaert model considering the light reversibility as follows [12]:
F j , k d = 1 π ( cos ⁡ θ i cos ⁡ θ r ) c M j , k d = 1 π ( cos ⁡ θ i cos ⁡ θ r ) c 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
where ( cos ⁡ θ i cos ⁡ θ r ) c reflects the variation in diffuse intensity with incident and reflection angles, and c is a constant in the range (−1, 0).
Volume scattering is the multiple reflection result of incident light on a metal surface among adjacent microfacets with fine structure, whose magnitude is related to the reflection angle and generally considered to follow a Gaussian distribution [19]. In this paper, the volume scattering component was represented as follows:
F j , k v = 1 2 π σ v e x p − θ r 2 2 σ v 2 M j . k v = 1 2 π σ v e x p − θ r 2 2 σ v 2 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
where σ v is the undetermined coefficient.
Based on the above analysis, the multi-parameter pBRDF model of metal surfaces can be expressed as:
F j , k = ( q − 1 ) ( 2 σ ) 2 q − 2 ⋅ m i n ⁡ ( 1.0 , 2 cos ⁡ α cos ⁡ θ r cos ⁡ β , 2 cos ⁡ α cos ⁡ θ i cos ⁡ β ) 4 π cos ⁡ θ i cos ⁡ θ r   cos 4 ⁡ α [ tan 2 ⁡ α + ( 2 σ ) 2 ] q M j , k s + k d π ( cos ⁡ θ i cos ⁡ θ r ) c M j , k d + k v 1 2 π σ v e x p − θ r 2 2 σ v 2 M j . k v
where n , k , σ , k s , k d , k v , q , c   and   σ v are the unknown parameters to be retrieved.
Generally speaking, the polarization degree of the reflected light from metal surfaces can be expressed by the degree of polarization (DOP). Assuming that the Stokes vector of the incident light is an unpolarized natural light and can be represented as E i = 1 0 0 T , where the circularly polarized component of natural light is extremely small and can be ignored [20], and then, the Stokes vector of the outgoing from a metal surface E r can be calculated as follows:
E r = S 0 S 1 S 2 = F j , k ⋅ E i = I Q U = F 00 s + F 00 d + F 00 v F 10 s F 20 s
The DOP value can be derived as follows:
D O P = Q 2 + U 2 I = F 10 s 2 + F 20 s 2 F 00 s + F 00 d + F 00 v
When the incident direction and observation direction are coplanar, DOP can be rewritten as:
D O P = F 10 s F 00 s + F 00 d + F 00 v

3. Improved Strawberry Optimization Algorithm

3.1. Strawberry Optimization

The strawberry algorithm (SBA) is a heuristic algorithm first proposed by F. Merrikh-Bayat [21]. The mathematical model of SBA describes how the strawberry plants migrate from one to another place and generates the number of runners and roots. Strawberry plants use both roots and runners to perform local and global searches and discover life resources, such as minerals and water, which present an optimization effect from roots and runners. For strawberry plants, roots and runners are usually generated randomly where there is a large amount of water resources and are pregnant with many child plants. The child plants also can generate more roots and runners affecting the growth of the strawberry forest.
From the growth process of strawberries, it can be seen that
(1)
Each strawberry plant performs local searches through roots and global searches through runners randomly to find life resources [22].
(2)
The runners generate new roots through global search and produce child plants, as shown in Figure 2.
(3)
Strawberry child plants grow faster and generate more roots and runners when they approach more affluent resources, and on the other hand, strawberry child plants are more likely to die in the case of inadequate resources.
Applsci 14 06022 g002
Figure 2. Strawberry plant.
Figure 2. Strawberry plant.
Applsci 14 06022 g002
The basic idea of the strawberry algorithm is that each position of the root and runner in the strawberry reproduction process represents the solution of an objective function, and whether the location is rich in resources means the quality of the objective function solution. Moving the root and runner to a new position with abundant resources to produce child plants is seen as finding an optimization solution. Through continuous iteration, the optimal solution of the objective function can be obtained. The steps of the algorithm are as follows:
(1)
Initialization
First, N points representing an initial mother plant population in the feasible domain of the problem to be solved are randomly generated, where each mother plant has m variables to optimize and is denoted by an m × N matrix. Then, each mother plant randomly generates a closer root and a farther runner based on the initial population at each iteration, by which the new roots and runners were produced. To obtain an optimization solution, the runners jump around. When the runners jump over the local minimum well, the algorithm will achieve a faster speed and obtain better global search ability. Through continuous iteration, each variable can be searched to be a possible optimization solution x p ( t + 1 ) at (t + 1)th iteration. The process can be represented as follows:
x p ( t + 1 ) = x r o o t ( t + 1 )   x r u n n e r ( t + 1 ) = [ x i ( t ) + d r o o t r 1   x i ( t ) + d r u n n e r r 2 ]
where x p ( t + 1 ) = x m , 1 , x m , 2 , ⋯ x m , 2 N , x i ( t ) = x m , 1 , x m , 2 , ⋯ x m , N , t is the number of iterations, m signifies the number of variables to be optimized and N denotes the number of mother plants. The parameter x p ( t + 1 ) denotes that the plant propagation matrix at iteration ( t + 1 ) is of dimension m × 2 N , indicating both runners and roots in one matrix. The parameters x r o o t ( t + 1 ) and x r u n n e r ( t + 1 ) contain the locations of roots and runners at this iteration, both of them are of size m × N . The parameter x i ( t ) is the optimal solution for the roots’ and runners’ positions at the t-th iteration. The parameter d r o o t and d r u n n e r are two scalars representing the distance of roots and runners from the mother plant, respectively, (often we have d r u n n e r > d r o o t ); r 1 and r 2 are random matrices whose members are uniformly distributed in the range [−0.5, 0.5].
(2)
Calculation of fitness value
The fitness values f i t ( x p ) of all optimization solutions based on the objective function can be calculated as:
f i t ( x p ) = 1 f x p                   f x p > 0 | f x p |                 f x p ≤ 0
where f x p is the objective function to be minimized.
(3)
Update of optimal solutions
First, the calculated objective function values f i t ( x p ) are sorted in ascending order, and the optimization solution with the first N/2 smaller objective function values is selected. Then the fitness value and roulette wheel are used to select N/2 of the remaining optimal solutions, and finally, the N optimal solutions combined by the two selections are used as the new mother plant population for the next iteration.
(4)
Iteration
The objective function values of the mother plant population are compared. If the objective function value after iterative updates is smaller, the updated mother plant population will be used as the initial population for the new iteration, which can make the solution further approach to the optimal one. This procedure is iterated until the predetermined termination condition is satisfied, and the final optimal solution is obtained.

3.2. Improved Algorithm

Usually, there exist many microstructures with a certain random distribution pattern and many grooves on a metal surface after mechanical treatment. In order to accurately retrieve the parameters of the metal surface pBRDF model, it is necessary to make, as much as possible, the initial population more diverse and distribution more uniform and balance the global search and local search ability to avoid the algorithm falling into local optima. Therefore, Tent mapping and Levy flight were introduced into SBA.

3.2.1. Chaotic Mapping to Initialize the Mother Population

Due to the use of random operators to initialize the position of the mother plant in the strawberry optimization algorithm, it is easy to encounter the problem of an uneven distribution of mother plant positions, weak global search ability, and low population diversity, causing a fall into local optima and affecting the overall optimization efficiency.
In the field of optimization, chaotic mapping is usually used to replace pseudo-random number generators and generate chaotic sequences of [0, 1], which can simulate complex and random dynamic behaviors. Tent chaotic mapping has the characteristics of randomness and traversal. To ensure the distribution of the initial population is uniform and diverse, Tent chaotic mapping was introduced into SBA to initialize the population of the mother plants to maintain the population diversity when improving the global search ability of the algorithm. In this paper, The Tent chaotic mapping sequence is defined as follows:
T n + 1 = T n u ,     0 ≤ T n < u 1 − T n 1 − u , u ≤ T n < 1
where n represents the number of populations, and u is a random number ranging [0, 1]. Combining with Tent chaotic mapping, the initial group position of the mother plants in the feasible domain is redefined:
x i = x l b + ( x u b − x l b ) T i
Figure 3 and Figure 4 show the initial population of the mother plants generated by the random method and the introduced chaotic mapping, respectively.
From Figure 3 and Figure 4, it can be seen that the initial population distribution generated by Tent chaotic mapping is more uniform than that by the random method, which is more conducive to expanding the search range and increasing the diversity of population positions. To some extent, it can overcome the defect of SBA easily falling into local extremum and improve the convergence accuracy and optimization efficiency.

3.2.2. Update with Levy Flight

By analyzing the root and runner position update process of the original strawberry algorithm, it can be seen that the root and runner are usually updated by random perturbation, and the distances from the mother plant are fixed, which makes the optimal solution easily fall into local optima. In this paper, the Levy flight strategy was used to update the position, which can make some individuals spread to a wider search space.
The Levy flight strategy is a random search method proposed by mathematician Paul Levy, according to the phenomenon that the trajectory of many biological activities in nature has long periods of random walks with small steps and occasionally jumps with larger steps [23]. The characteristic of the Levy flight, which alternates between long and short distances, plays an important role in balancing the local exploration and global search capabilities of optimization algorithms, where smaller random walks are beneficial to conduct local exploration, and the occasional larger jump is beneficial to break out of local optima and improve global search capabilities. Therefore, the introduction of the Levy flight strategy is expected to compensate for the shortcomings of the strawberry algorithm in falling into local optima and reduce the number of iterations.
Figure 5 shows the simulation comparison between random walking and the Levy flight path trajectory on a two-dimensional plane with both paths set to 200 steps.
From Figure 5, it can be seen that the Levy flight has a wider search range and stronger search ability compared to the normal distribution. In this paper, the random step length L e v y ( β ) was generated by the Mantegna algorithm and represented as [24]:
L e v y ( β ) = u | v | 1 β
where the value β is in the range of [0, 2] and usually taken as 1.5; u   and   v are random variables obeying a normal distribution and can be represented as follows:
u ~ N 0 , σ u 2 ,         σ u = [ Γ ( 1 + β ) sin   ( π β 2 ) β Γ ( 1 + β 2 ) 2 β − 1 2 ] 1 β v ~ N 0 , σ v 2 ,                                     σ v = 1                            
where Γ ( · ) is a Gamma function.
In order to expand the location search range of the strawberry algorithm, the runner position update formula of the long-distance search is modified in this paper as follows:
X r u n n e r i = X i + d r u n n e r r 2 + α · L e v y ( β )
where α usually takes an empirical value of 0.01.

3.2.3. Optimization of Metal Surface pBRDF Model Parameters

As mentioned above, the Tent chaotic mapping and Levy flight strategy are introduced into SBA, and an improved strawberry algorithm (ISBA) has been proposed, which is applied to the parameter retrieval of the metal surface pBRDF model on metal surfaces to obtain the optimal solution.
The algorithm flowchart is shown in Figure 6.
To avoid errors caused by randomness in the optimization, the SBA and ISBA were run 50 times continuously in the same environment. The minimum, average and maximum values of the objective function values were calculated. The extreme value in the optimization process was compared between ISBA and SBA and is shown in Figure 7 and Figure 8. It can be seen that the optimization efficiency and accuracy of ISBA are enhanced compared to those of SBA.

4. Simulation and Experiment

4.1. Simulation

To validate the performance of the proposed method and its applicability, open-source data of copper, black paint and green paint materials from reference [1] were fitted.
According to the analysis mentioned above, the pBRDF model of metal surface established in this paper has nine unknown parameters, namely n ,     k ,     σ ,     k s ,   k d ,   k v ,     q ,     c   and   σ v , shown in Equation (6). In this paper, the reflected polarization feature of the metal surface is characterized by a degree of polarization, and the optimization target is to select the optimal model parameters, minimizing the mean squared error ∆ E of the degree of polarization between the model predicted values and the experimentally measured ones. Therefore, the following optimization objective function can be constructed:
m i n ∆ E n , k , σ , k s , k d , k v , q , c , σ v = ∑ θ i ∑ θ r [ D θ i , θ r , ∆ φ − D m θ i , θ r , ∆ φ ] 2 ∑ θ i ∑ θ r [ D m θ i , θ r , ∆ φ ] 2
where D m θ i , θ r , ∆ φ and D θ i , θ r , ∆ φ are the DOP values measured in the experiment and predicted by the proposed model, respectively.
Table 1 shows the retrieved complex refractive index of copper, black paint and green paint surfaces with different models under the incident light at the wavelength of 650 nm.
It can be seen from Table 1 that the retrieved complex refractive index by the proposed method in this paper is closer to the true value taken from reference [1].
By bringing the retrieved complex refractive index into Equation (9), the predicted degree of polarization, which is the function of the reflection angle, can be calculated. The curves fitted with the degree of polarization predicted by the proposed method and that taken from reference [1] were redrawn in Figure 9.
From Figure 9, it can be seen that the referred DOP values are slightly higher than that predicted by the proposed method under small incident angles and reversed when the incident angle is beyond a certain value, while the fitting curve by the proposed method in this paper is closer to values in reference [1]. Compared to the model proposed in the reference, the multi-parameter pBRDF model optimized by ISBA takes into account the depolarization effect caused by the volume scattering by the roughness of the target surface and avoids obtaining a DOP value at virtual height. The inversion process of the pBRDF model with the improved strawberry algorithm can keep from falling into local optimum, while the accuracy and reliability of the pBRDF model can be enhanced.

4.2. Experimental Verification

In order to further validate the performance of the proposed method for parameters retrieval of the metal pBRDF model, several experiments were conducted in a dark room and the results by the proposed method were compared with a nonlinear least squares algorithm (LM), particle swarm optimization algorithm (PSO) and original strawberry algorithm (SBA).
A standard 45# steel gauge block with a roughness of 0.67 μ m was selected and irradiated with incoming light at a wavelength of 546 nm at the incident angles of 30°, 40° and 50°. The outgoing light from the block was detected at different observing angles ranging from 20° to 65° by a step of 5° with an SALSA liquid crystal polarization camera and the polarization reflection images of the target were obtained. Then the DOP values of the reflected beam were calculated, which was repeated 10 times and the average was selected as the final measured value. The schematic diagram of the experimental setup is shown in Figure 10.
The retrieving pBRDF parameters for the 45# steel surface with three algorithms of PSO, LM, SBA and the proposed algorithm ISBA are compared and shown in Figure 11. The simulation accuracy of DOP is evaluated using the root mean square error (RMSE), and the results are shown in Table 2.
From Figure 11 and Table 2, it can be seen that under different incident angles, the pBRDF model established by the proposed method agrees better with the experimental data with higher accuracy; the fitting accuracy of the proposed method in this paper is significantly higher than those by SBA, LM and PSO. It can be explained through the fact that the weakness of the original SBA’s insufficient search ability at the early stage was overcome by introducing the strategy of chaotic initialization and the Levy flight, and the global search and local exploration capabilities were balanced, which can accelerate the convergence and improve the data fitting accuracy. From the retrieving process of the pBRDF parameters, it can be concluded that the ISBA algorithm is more efficient and accurate in solving the parameter optimization problem of the pBRDF model for metal surfaces.

5. Conclusions

To study the polarization reflection characteristics of metal surfaces, an improved strawberry optimization algorithm (ISBA) has been proposed, and it is applied to parameters retrieval of the pBRDF model for metal surfaces. By introducing Tent chaotic mapping and the Levy flight strategy, the ISBA has a more diverse initial population and a large search range and makes up for the shortcomings of being prone to falling into local extreme points. Simulation and experimental results show that the retrieved model parameters by the proposed method have higher fitting accuracy and better consistency with the measured data compared to the SBA, LM and PSO.

Author Contributions

Conceptualization, X.G. and F.W. (Fangbin Wang); methodology, X.G., S.Z., S.C. and P.W.; software, X.G. and D.Z.; validation, X.G. and D.Z.; investigation, X.G. and F.W. (Fangbin Wang); data curation, X.G., W.Z. and S.Z.; writing—original draft preparation, X.G. and D.Z.; writing—review and editing, X.G. and F.W. (Feng Wang). All authors have read and agreed to the published version of the manuscript.

Funding

This research was funded by Anhui Natural Science Foundation (2008085UD09); Anhui University Collaborative Innovation Project (GXXT-2021-010); Anhui Construction Plan Project (2022-YF016, 2022-YF065, 2023-YF050); Anhui Province Higher Education Science Research Project (2022AH040044); Anhui Province University Outstanding Youth Research Project (2022AH020025); Anhui Province University Outstanding Young Talents Support Program (gxyq2017025).

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

The raw data supporting the conclusions of this article will be made available by the authors on request.

Acknowledgments

We would like to acknowledge the support from the Anhui Natural Science Foundation, the Anhui Construction Plan Project, and the Anhui Provincial Department of Housing and Urban Rural Development.

Conflicts of Interest

The authors declare no conflicts of interest.

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Figure 1. BRDF coordinate system.
Figure 1. BRDF coordinate system.
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Figure 3. The initial population generated by the random method.
Figure 3. The initial population generated by the random method.
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Figure 4. Initial population generated by chaotic mapping.
Figure 4. Initial population generated by chaotic mapping.
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Figure 5. Comparison of Levy flight and random walk.
Figure 5. Comparison of Levy flight and random walk.
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Figure 6. Flowchart of parameters retrieval.
Figure 6. Flowchart of parameters retrieval.
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Figure 7. SBA convergence curve.
Figure 7. SBA convergence curve.
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Figure 8. ISBA convergence curve.
Figure 8. ISBA convergence curve.
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Figure 9. Curves fitted with predicted and referred DOP values under the incident angle of 60° (a)copper; (b) green paint; (c) black paint.
Figure 9. Curves fitted with predicted and referred DOP values under the incident angle of 60° (a)copper; (b) green paint; (c) black paint.
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Figure 10. Schematic diagram of the experimental setup.
Figure 10. Schematic diagram of the experimental setup.
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Figure 11. Fitting curves by LM, PSO, SBA and ISBA at different incident angles (a) θ i = 30 °; (b) θ i = 40 ° ; and (c) θ i = 50 ° .
Figure 11. Fitting curves by LM, PSO, SBA and ISBA at different incident angles (a) θ i = 30 °; (b) θ i = 40 ° ; and (c) θ i = 50 ° .
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Table 1. Retrieved complex refractive index.
Table 1. Retrieved complex refractive index.
SampleTruth ValueReference ValueEstimated Value
nknknk
Copper0.402.950.543.190.512.90
Black Paint1.400.221.461.321.260.27
Green Paint1.390.341.470.471.370.34
Table 2. RMSE values by different methods with different incident light.
Table 2. RMSE values by different methods with different incident light.
Incident AngleLMPSOSBAISBA
30°0.86%0.16%0.23%0.09%
40°0.70%0.39%0.21%0.02%
50°0.26%0.40%0.23%0.14%
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MDPI and ACS Style

Gong, X.; Wang, F.; Zhu, D.; Wang, F.; Zhao, W.; Chen, S.; Wang, P.; Zhang, S. Parameter Optimization Method for Metal Surface pBRDF Model Based on Improved Strawberry Algorithm. Appl. Sci. 2024, 14, 6022. https://doi.org/10.3390/app14146022

AMA Style

Gong X, Wang F, Zhu D, Wang F, Zhao W, Chen S, Wang P, Zhang S. Parameter Optimization Method for Metal Surface pBRDF Model Based on Improved Strawberry Algorithm. Applied Sciences. 2024; 14(14):6022. https://doi.org/10.3390/app14146022

Chicago/Turabian Style

Gong, Xue, Fangbin Wang, Darong Zhu, Feng Wang, Weisong Zhao, Song Chen, Ping Wang, and Shu Zhang. 2024. "Parameter Optimization Method for Metal Surface pBRDF Model Based on Improved Strawberry Algorithm" Applied Sciences 14, no. 14: 6022. https://doi.org/10.3390/app14146022

APA Style

Gong, X., Wang, F., Zhu, D., Wang, F., Zhao, W., Chen, S., Wang, P., & Zhang, S. (2024). Parameter Optimization Method for Metal Surface pBRDF Model Based on Improved Strawberry Algorithm. Applied Sciences, 14(14), 6022. https://doi.org/10.3390/app14146022

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