System simulation of the optical imaging model was conducted. An 800 mm-aperture FZP diffraction imaging system was constructed using simulation software. The imaging system comprised two FZP elements, three double bonded lenses, and a filter. The parameters of each element in the system are used as design parameters for the FZP diffraction imaging system, which include the radius of curvature, effective aperture, distance between optical surfaces, and corresponding materials. The FZP is parameterized. Simulation of the FZP is achieved by combining a binary surface with a standard optical surface. The setting of the binary surface requires not only setting the basic surface parameters but also setting the corresponding phase coefficients. The phase coefficient is determined by Equation 1. ϕ = M ∑ i = 1 N a i ρ 2 i (1) Where ϕ denotes the phase of the optical surface, M denotes the diffraction order used by the optical surface, N corresponds to the index of the polynomial coefficient in the order, and a i is the coefficient of the 2i-th power of ρ (the phase coefficient). The phase coefficients a 1 to a 5 are to be set, and the phase coefficients are obtained by setting the optimization function with respect to the system wavefront aberration and the system focal length. Finally, quartz was selected as the substrate material for FZP, which determined all parameters of the FZP.

Subsequently, the double bond lens is designed and calculated in the system. Unlike traditional lens calculations, the radius of curvature for a double bond lens is calculated by selecting material combinations. The combination of crown glass in front and flint glass in back is commonly used in double bond lens design. Material refractive indices are obtained by searching the corresponding combinations. Parameters are substituted into equations Equations 2–4 to determine the optical surface curvature, which includes the radius of curvature, refractive index, focal length, and other parameters. φ = 1 f (2) φ 1 = n 1 − 1   ∗   1 r 1 − 1 r 2 (3) φ 2 = n 2 − 1   ∗   1 r 2 − 1 r 3 (4) Where φ represents the optical power between adjacent optical surfaces, which is inversely proportional to the focal length f . Among them, φ 1 represents the optical power from the first optical surface to the second optical surface, and φ 2 represents the optical power from the second optical surface to the third optical surface. n represents the refractive index between adjacent optical surfaces. r is the radius of curvature of each optical surface.

The FZP imaging system, as shown in Figure 1a, is analyzed in terms of its imaging model. The analysis of the imaging model primarily involves the analysis of point target imaging and extended target imaging, such as the imaging system’s PSF, spot diagram, wavefront, MTF, and extended target imaging analysis.

As shown in Figure 1b, the inconsistent sizes of light spots focused at different wavelengths on the same image plane, which shows that chromatic aberration still exists in the system. Moreover, chromatic aberration causes image quality degradation.

As shown in Figure 1c, the PSF exhibits no distortion, the energy is concentrated, and the Airy disk size is 3.027 μ m, which shows the system has excellent focusing performance.

In the wavefront Figure 1d, the wave trough is approximately −0.031 wavelengths, and the wave crest is approximately 0.102 wavelengths. The wavefront aberration was calculated to be 0.132 wavelengths, which is less than one-quarter wavelength. This result is consistent with the Rayleigh criterion, which indicates that although wavefront distortion exists in the system, but wavefront distortion has little effect on imaging, which indicates the system image quality is good.

In Figure 1e, the optical transfer function (OTF) approaches zero when the spatial frequency reaches 414 cycles/mm. The system MTF closely matches the diffraction-limited MTF curve, which indicates that the system’s transmission capability approaches that of an ideal imaging system. However, the mid-frequency component of the system MTF differs by 9.2 cycles/mm from the diffraction-limited mid-frequency component, which indicates the presence of aberrations within the system. System chromatic aberration leads to reduced imaging quality.

After completing the point target imaging analysis, the extended target imaging is analyzed at 550 nm primary wavelength. The extended target imaging plane sampling is set to 1024 × 1024 . The single-wavelength imaging is acquired as shown in Figures 1f–i. In the extended target imaging, the image exhibits not distortion, but the imaging has the problem of edge blurring and loss of imaging details. Combined with PSF analysis, the imaging blur was attributed to multiple order diffraction from the diffractive elements. To address the imaging blur issue, the imaging model was derived based on the imaging characteristics of the FZP.

During the process of imaging optimization research, the imaging model is established for the system, which is used for subsequent further development of imaging optimization studies. In 2023, Li Jiaxun et al. proposed an imaging restoration method based on the FZP imaging model, performing mathematical derivations for the FZP diffraction imaging system. An imaging model for an FZP imaging system was derived, with the expression shown in Equation 5; Li et al. (2023). I = I 0 ∗ P S F 1 × η 1 + P S F n × 1 − η 1 (5) Where I denotes the extended target image directly obtained by the FZP imaging system, and I 0 denotes the optimized result of the extended target image from the FZP system. η 1 and P S F 1 represent the diffraction efficiency and point spread function (PSF) of the designed imaging order, respectively. P S F n denotes the effective point spread function of stray light and noise in the imaging system. The imaging model is represented by the interaction between the designed-order light and scattered light and noise, where the designed-order light is expressed as I 0 ∗ ( P S F 1 × η 1 ) , and the non-designed-order light and noise are expressed as I 0 ∗ [ P S F n × ( 1 − η 1 ) ] .

However, Equation 5 cannot directly solve for numerical solutions, which only solves the analytical solutions in practical applications. Meanwhile, analytical solutions cannot intuitively reflect imaging effects. To solve the problem, data is input into imaging models for specific calculations. However, the imaging quality collected by the system cannot meet the requirements for direct solutions. Therefore, to solve the problem of degraded imaging quality, the input image is optimized. Optimizing the image actually constitutes a multi-objective optimization problem, which can be summarized as a multi-objective optimization expression, as shown in Equation 6. max  or min  f x = f 1 x , f 2 x , … , f n x  s.t  x ∈ Ω (6) Where f ( x ) usually is the evaluation function of the optimization objective, which is also called the objective space. Meanwhile, Ω is the space where ( x 1 , x 2 , … , x n ) is located, which is called the decision space. For multi-objective optimization problems related to image optimization, image metrics are usually selected as evaluation functions. By combining the imaging model, the imaging of the FZP imaging system is converted into a mathematical expression, and the iterative optimization problem related to the image is converted into a multi-objective optimization problem. The expression is shown in Equation 7. Z = max f P S F = f 1 P S F , f 2 P S F , … , f n P S F  s.t  P S F ∈ Ω (7) Where image gradient and contrast are selected as optimization metrics. Parameters are adjusted while changes in gradient and contrast are observed throughout the optimization process. The decision space is populated through iterative updates, while the decision space values are continuously input to the target space. The Z-value is calculated, which corresponds to the PSF value in the decision space representing the optimized result. Finally, the optimized PSF is input to the imaging model to obtain the iteratively optimized image I.

However, some problems still exist in the specific research of PSF iterative optimization. Optimization algorithms such as Stochastic Parallel Gradient Descent (SPGD), Simulated Annealing (SA), and Cuckoo Search are applied to imaging optimization, which have some drawbacks including high iteration counts, slow iteration speeds, numerous parameters, and suboptimal optimization results. PSO effectively addresses these issues due to its simple behavior, fast iteration speed, and minimal parameter requirements.

The particle swarm optimization achieves global search optimization primarily through collaboration and competition among individuals. First, a group of random solutions must be generated through initialization, which are equivalently regarded as particles that have no mass or volume. Global optimization is completed by particles flying in the search space. This flight pattern references the flying behavior of flocks of birds or the foraging behavior of schools of fish, and the behavior function is shown in Equation 8. x i j + 1 = x i j + v i j + 1 (8)

The PSO iteration method is simple: particles move based on the velocity generated by weighting their current positions, and the velocity is continuously updated during the iteration process. The iteration formula for velocity is shown in Equation 9. v i j + 1 = ω × v i j + c 1 × r a n d n d , n d × p b e s t i − x i j + c 2 × r a n d n d , n d × g b e s t i − x i j (9)

Equation 9 is primarily used to update the velocity of particles in each movement within the PSO. The velocity iteration formula includes a memory term, an individual learning term, and a social learning term. Where ω is the inertia factor, which is required to be non-negative and it can be used to adjust the strength of the algorithm’s optimization capability. c is the learning factor, where c 1 is the cognitive learning factor, c 2 is the social learning factor. The learning factor can effectively adjust the search capability of the algorithm. Where nd represents the matrix size. where pbest denotes the local optimum of the particle swarm, which is the optimal population. Gbest denotes the global optimum of the particle swarm, which is the optimal particle. During velocity iteration, the first memory term is used to relate the previous generation’s velocity to the post-iteration velocity, because the velocity itself has no memory. If the memory term is absent, the iterative optimization process assumes that a particle at the globally optimal position remains stationary, while other particles move toward a weighted center between the individual optimal and global optimal positions. In this case, the optimal particle will only be found at the current best position. To induce particle movement and achieve global optimization, the inertia weight is added to the memory term. The inertia weight originates from the concept proposed by Clerc, which uses a decay factor to ensure algorithm convergence. However, subsequent research discovered that setting a maximum velocity limit can enhance algorithm performance. This method eliminates the link between the inertia factor and the learning factor, which ensures the effectiveness and independence of the memory term Clerc (1999). The second term represents the individual cognitive component, which indicates the particle’s own thinking, providing iterative particles with powerful global search capabilities and avoiding local minimal. Finally, the third term is the population cognitive component, embodying information sharing among particles. The PSO’s optimization is effectively achieved through the synergistic interaction of these three components.

Before the iteration begins, the input population, velocity, and optimal population must be initialized. To accommodate optimization requirements, each particle in the solution space is two dimensional data, which needs to be fitted. The input population, velocity, and initialized optimal population must be converted into two dimensional matrices. The initialization formula is shown in Equation 10. v i = D L .   ∗   o n e s n d , n d + U L .   ∗   o n e s n d , n d − D L .   ∗   o n e s n d , n d .   ∗   0.01 .   ∗   r a n d n d , n d (10) Where DL denotes the minimum value of a single pixel in the matrix, and UL denotes the maximum value. ‘ones ()’ is used to generate a matrix with all values set to 1, while ‘rand ()’ is used to generate a random matrix.

After outputting the new velocity, the particle update iteration is achieved: the new velocity is input into the PSO particle update behavior, which makes the particle move. Subsequently, the output particle swarm is evaluated. The gradient of the expanded target image is selected as the criterion for algorithm termination, which is also used as the judgment condition for iterative optimization. The calculation expression is shown in Equation 11. G = I x , y − 1 M × N ∑ x = 1 , y = 1 M , N I x , y 2 / M × N (11) Where I (x, y) is the restored image obtained through a single iteration of the optimization algorithm, (x, y) denotes the pixel coordinates in the Cartesian coordinate system, and M and N respectively represent the dimensions of the image.

Based on the above calculations, the operation flow of the PSO-based image optimization algorithm is shown in Figure 2.

In the optimization process of the target image, initial parameters are firstly input. The upper and lower limits are determined by collecting the maximum and minimum values of the PSF in the simulation system. Because of the simulation model operating under ideal conditions, the background noise of collected images is not considered in the input PSF and target image expansion process. After the initial parameters are input, the initial velocity and a random population are generated. The generated random population and random velocities are input to the evaluation function. The gradient and contrast are calculated, which are compared with the next-generation’s results. Subsequently, the corresponding parameters are input to the PSO for iterative optimization. Firstly, the velocity values for the next particle movement are calculated in the optimization process. Subsequently, the output velocity values are input to the particle movement behavior, and a new particle swarm is obtained. The new particle swarm is input to the evaluation function for assessment, which results in corresponding image metrics. Finally, the image metrics are evaluated to determine whether convergence conditions are met, thereby deciding whether to terminate the algorithm.

This paper combines imaging models with the PSO algorithm to propose a PSO-based image optimization method. This approach logically simplifies the optimization iteration process, thereby reducing iteration time, and improving iterative optimization efficiency. To further validate the theoretical effectiveness, the expanded target image from the imaging model is input into the PSO-based optimization algorithm.

Randomly generated PSFs are treated as particles in the iterative optimization process, while PSFs and extended target images collected from the FZP system image model are input to the PSO optimization algorithm. Subsequently, PSO initial parameters are set, including the population size, upper and lower velocity bounds, filtering radius, learning factor, and inertia factor. The parameters are configured and adjusted, which ensures stable algorithm operation. Subsequently, to evaluate optimization effectiveness, the image gradient G is selected to assess convergence of the optimization iterations and serves as the algorithm’s termination condition. The algorithm then stops automatically upon meeting the specified optimization conditions. The corresponding pseudocode is shown in Algorithm 1.

Algorithm 1

Input: The inertia factor ω , the diffraction efficiency η 0 , the learning factors c 1 and c 2 , the image size M and N, filter radius R, minimum value DL and maximum value UL, and number of population n

Where the diffraction efficiency η 0 is calculated through the simulation system. The inertia factor is set to a small value, which ensures the accuracy of optimization by reducing the influence of the previous generation’s particle positions on the next-generation. The cognitive learning factor c 1 and the social learning factor c 2 are set to the same value, which balances local and global optimization capabilities. After inputting the corresponding parameters, optimization results are obtained through continuous iterative optimization until the algorithm terminates. Unlike traditional PSO algorithms, the termination condition is explicitly defined, because the PSO applied to image optimization cannot complete optimization within a determined number of iterations. so, the termination condition is set: the algorithm stops when the image gradient G ceases to increase over a period of time.

At every iteration, the updated PSF, the PSF of the imaging system under consideration, and the expanded target image are input to the imaging model for computation, which results in the corresponding optimized image. The results including the P S F n after optimization iterative, images before and after optimization, and the gradient enhancement curve of the optimization iteration. The relevant optimization results are shown in Figure 3.

As shown in Figure 3, four images are optimized through iterative processing, which results in enhanced imaging details and improved image resolution. The result confirms the feasibility of the PSO algorithm in image restoration.

The comparative analysis of SPGD optimization results and PSO optimization results, to systematically evaluate PSO’s image optimization capabilities and mitigate algorithmic randomness and chance effects. The optimization results comparison is shown in Figure 4.

By comparing the optimization results of the PSO and SPGD algorithms, the PSO-optimized images have richer and clearer details compared to those optimized by SPGD.

To quantitatively evaluate the optimization performance of both algorithms, the contrast, gradient, and number of iterations for each algorithm were compared, which further validates the feasibility of PSO for system image optimization. The iterative optimization comparison is shown in Table 1.

The relevant metrics for iterative optimization between the two methods were compared, which include contrast, gradient, mean absolute error, normalized mean square error, and number of iterations. The relevant data from the iterative optimization of both methods were compared, leading to the conclusion: PSO achieves superior iterative optimization results for images, requiring fewer iterations and demonstrating higher optimization efficiency. PSO algorithm is particularly suitable for diffraction imaging systems based on FZP was further demonstrated.

To obtain a fast restoration model, the imaging model is derived. First, the imaging model equation is transformed using a two-dimensional Fourier transform on both sides, obtaining the expression shown in Equation 12. F F T 2 I = F F T 2 I 0 × F F T 2 P S F 1 × η 1 + P S F n × 1 − η 1 (12) Where F F T 2 denotes the two-dimensional Fourier transform. If the data for P S F n , η 1 , and P S F 1 can be obtained, images acquired by the FZP imaging system can be rapidly reconstructed. The analytical expression for the optimized image is derived through formula manipulation, as shown in Equation 13. I 0 = I F F T 2 F F T 2 I F F T 2 P S F opt (13) The I F F T 2 denotes the two-dimensional inverse Fourier transform. The analytical solutions for the imaging model parameters P S F n , η 1 , and P S F 1 are derived through analytical methods, which are substituted into Equation 13 to obtain the optimized image. This paper obtains the optimal numerical solution by using the particle swarm optimization algorithm, which achieves image optimization. The workflow is shown in Figure 5.

To achieve fast restoration of acquired images, P S F n for different images is extracted after PSO optimization iterations. P S F opt is obtained by averaging multiple P S F n . Subsequently, P S F opt and other images are input into the fast restoration model for reconstruction, which completes the PSO-based fast image reconstruction method.

To further evaluate the optimization performance of the PSO algorithm, while avoiding the randomness and unpredictability of reconstruction, the reconstruction results from PSO-based imaging were compared with those from SPGD. Since fast reconstruction without iteration, the comparative analysis primarily included the image details, image gradients, and image resolution of the extended target imaging.

Restored images based on the PSO algorithm and SPGD algorithm are obtained through fast optimization. Comparison of the two algorithms is used to validate the feasibility and accuracy of the fast optimization. The comparison of the fast-restored images is shown in Figure 6.

Four images are selected for comparison, and when compared with the original images, the conclusion is reached: The P S F opt obtained through iterations of both algorithms is input to the fast restoration model. The optimized images are locally enlarged and compared, which reveals that the fast-optimized images exhibit increased imaging details and improved image resolution. However, compared to the SPGD algorithm, the restoration results obtained by PSO iteration when P S F opt is input into the imaging model output demonstrate superior performance. However, the comparison of images cannot quantitatively describe the restoration effect. Therefore, the restoration metrics of SPGD and PSO are compared with the original image metrics, as shown in Table 2.

Through the comparison, the conclusion that PSO achieves superior optimization results in imaging is verified. PSO optimizes image contrast and gradient more effectively. In the optimization of extended target imaging, PSO significantly enhances image detail compared to SPGD optimization. Consequently, the conclusion can be reached that PSO effectively optimizes imaging with superior results.