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Abstract
Accurate, fast, and reliable modeling and optimization methods play a crucial role in designing light field display (LFD) system. Here, an automatic co-design method of LFD system based on simulated annealing and visual simulation is proposed. The process of LFD content acquisition and optical reconstruction are modeled and simulated, the objective function for evaluating the display effect of the LFD system is established according to the simulation results. In case of maximum objective function, the simulated annealing optimization method is used to find the optimal parameters of the LFD system. The validity of the proposed method is confirmed through optical experiments.
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1. Introduction
Light field display technology is considered as a promising method to provide audience with real and natural 3D perception, because it can reproduce all depth cues and light field distribution of the real 3D scene [1–10]. In LFDs, the relative direction and intensity information of the light field originated from 3D images is recorded and then the light field information is recovered by generating the beams with the same relative direction and intensity based on the recorded information. Researchers have made a lot of efforts to design and improve light field 3D displays [11–14]. However, the design and improvement of LFD systems is a time-consuming process, especially when its 3D display effect cannot be obtained directly. Commercial optical software such as Zemax and Lighttools pay attention to optical characteristics analysis rather than graphic visualization. Several simulation models have been used to describe the optical phenomena for barrier or lenticular array-based autostereoscopic displays [15–18]. Nevertheless, these numerical simulation method is not intuitive enough for designers to predict display effect. Our previous research introduced a high-speed LFD simulation method based on backward ray tracing (BRT) technique, which can be more intuitive to get the display effect of LFD [19]. It is very helpful for the design and improvement of LFD systems.
The LFD system is complex from acquisition to display. Traditional optical design software can only optimize the parameters of the lens locally, although the software offers important guidelines, it is still not trivial to find the right structures to realize the desired lens, especially when the geometry and spatial arrangements of the lens become complicated. Optimization techniques have been indispensable for designing high-performance meta-devices targeted to a wide range of applications. Unless simple geometric features are used to form optical, RF, and meta-device designs and the relationship between the input parameters and costs or fitness functions is simple and well understood, it is generally beneficial to employ a global optimization strategy to obtain the best performance available within the design constraints [20]. Now, there have been some studies on lens design and optimization using global optimization technology [21–28]. D. Pinchera, et al. proposed a method based on the minimization of a properly defined global cost function named Quantized Lexicographic Weighted Sum (QLWS), which can improve the lens architecture [27]. J. Budhu, et al. introduced a novel algorithm hybridizing geometrical optics ray tracing techniques and particle swarm optimization, which can achieve a high-efficiency lens design [28]. However, there is no effective method to co-design the global parameters of LFD.
Here, an automatic co-design method of LFD system based on simulated annealing and visual simulation is proposed. Firstly, the content acquisition and optical reconstruction process of light field are simulated based on BRT. Each component is modeled, and the acquisition and reproduction processes are designed and optimized at the same time. Due to the rendering ability of BRT, the speed of content acquisition and optical reconstruction can be considered as real-time, which makes simulation-based intelligent design optimization feasible. According to the simulation results, the optimization environment including decision variables, constraints, and the objective function is established. In the optimization environment, a fast optimization strategy based on simulated annealing is used to find the appropriate parameters. Finally, a new LFD is constructed, the effectiveness of the proposed method is verified by optical experiments.
This paper is organized into five sections. In Section 2 the basic flow of simulation based on BRT is introduced, and the process of content acquisition and optical reconstruction is described in detail. In Section 3, the construction of the optimization environment and the working principle of the specific optimization algorithm are proposed. Section 4 discusses the results of the whole design optimization experiment. Section 5 is the conclusion.
2. Simulation-based on BRT
A typical LFD system consists of two processes: content acquisition and optical reconstruction. Such a two-process procedure often complicates both the hardware and the software and results in the design and optimization of the LFD system being more difficult. In the previous paper, the researchers are often only concerned about the optimization of the optical system while ignoring the effect of content acquisition. However, the parameters (such as the shape of the lens, the distance between components, etc.) are one-to-one correspondence in the process of acquisition and display. When optimizing the whole light field system, it inevitably affects the change of acquisition process parameters. Therefore, in the process of simulation and design optimization, parameters should be optimized and updated both in content acquisition and optical reconstruction at the same time, that is, the shape and relative position of LFD components should be consistent. In the case of multiple iterations in the optimization process, an efficient simulation method based on BRT is used. A basic backward ray tracer includes three parts [29,30]. The first part is ray generation, where the origin and the direction of each pixel's view light ray based on the camera geometry are calculated. The second part is a ray intersection, where the closest object intersecting the view ray is found. Shading processing is the final part, which computes the pixel color based on the results of the ray intersection.
The whole process of simulation is shown in Fig. 1. The process of content acquisition and optical reconstruction are simulated respectively. The simulation of the content acquisition process is described in section 2.1. In the virtual scene, animation model, lens array model, and elemental image array(EIA) screen model are established, according to lens model and pixel position, the BRT method is used to simulate the light sample, and the obtained 3D is saved in EIA as the simulation result of content acquisition. The simulation of the optical reconstruction process is described in section 2.2. The models of Liquid crystal display (LCD), lens array, and diffuser models are constructed in the virtual scene. And the simulation result is obtained by a program based on BRT.
Fig. 1. The ray-tracing based simulation processed of content acquisition and optical reconstruction.
2.1 Simulation of content acquisition
The simulation method of content acquisition is actually the process of the spatio-angular light ray distribution, i.e., EIA, generation by computer. Considering the accuracy, the lens model is used to simulate the imaging process instead of the traditional method of simplifying it as a pinhole [31]. To make the simulation results more precise and realistic, the sub-pixel based rendering method is used [32]. In the virtual scene, the virtual camera array, lens array, animation model, and the LCD with EIA are constructed, and the EIA is generated by the BRT technique, as shown in Fig. 2(a). A pixel is composed of three sub-pixels: red, green, and blue. A path is traced from a virtual camera through each sub-pixel in a virtual screen, and the color of the object visible through it is calculated. After the collisions between the virtual view ray and all objects in the virtual space, the RGB value corresponding to the color of the nearest collision point is used as the color of the sub-pixel. In this way, the EIA is obtained.
The lens array consists of many lenses of the same structure, and a single lens is the basis of the lens array. The lens is modeled by the method of constructive solid geometry (CSG). The core of CSG is to define all the surface boundaries with implicit equations, which makes the model easy to express and has the characteristics of simple structure and high precision. Besides, the CSG modeling method can get the new model by adjusting the implicit equations used to define the model, instead of reconstructing the whole model. So the model with CSG is easy to generate, easy to maintain, and has good scalability. In addition, the optimization of the LFD system based on simulation needs to move the viewpoint and change the lens shape. This requires the lens model to have high precision and strong model maneuverability. Based on the above considerations, CSG is used to construct the lens.
The surface of the typical lens in the LFD system includes 3 parts; a top sphere surface, a partial cylinder surface, and a bottom sphere surface. The geometric shape of the lens is shown in Fig. 2(b). The refractive index of the lens determines the propagation path of the view light in the lens. nlens represents the refractive of the lens, and nair represents the refractive of air. i and t are the incident and transmitted ray vectors, and n is the unit normal vector at the incident point of the lens surface. r is the radius of the cylinder surfaces, h is the height of the cylinder (the thickness of the lens), and R1 and R2 are the radius of the top and bottom surface of the lens respectively. The CSG surface implicit equations of the lens can be expressed as Eq. (1).
In the backward ray tracker that generates the EIA, the backward view rays should start from one of the sub-pixels, and their directions should be from the start point to the center position of its corresponding lens. It is assumed that the EIA on the LCD panel consists of 3N×N sub-pixels. Each sub-pixel has its own Index(k, m)${\in} ${(0,3W], (0,H]} in whole LCD panel. Given that a point(x, y, 0) is on the LCD panel, its Index(k, m)${\in} ${(0,3W], (0, H]} in the whole LCD panel can be expressed as Eq. (2):
Here, u, v, and w represent unit vectors in the 3D coordinate system respectively. The Index(i, j) of the elemental image (EI) on which the point(x, y, 0) is in the EIA can be expressed as Eq. (3). Since there is a one-to-one mapping between the lens and the EI in space, the index of the corresponding lens Lensij is the same as the Index(i, j) in the lens array. Then the center position Plens_center of the corresponding lens can be expressed as Eq. (4). Where d is the distance between adjacent lenses, z is the relative depth of the lens array from the virtual panel.
Through the above discussion, the origin RO and the direction $\overrightarrow R $ of the point (x, y, 0)’s view ray can be described in the following equation.
After obtaining the origin and direction of the point(x, y, 0)’s view ray $\overrightarrow R $, the intersection position of the ray $\overrightarrow R $ and the lens can be solved by substituting the parameter expression of the ray $\overrightarrow R $ into the surface implicit equations S of the lens. Taking the direction of the view ray $\overrightarrow R $ and the intersection point of the lens into the expression of the optical relationship between the incident ray and refracted ray on the given surface of the lens, as shown in Eq. (6), the real view ray expression refracted by lens model can be derived.
After deriving the expression of view rays, parallel computing method is adapted to generate the view rays. Every ray has its dependent thread, and its ray-object intersection, shading happens in the same thread. When the backward ray tracing is finished, the RGB value corresponding to the color of the nearest collision point is used for the sub-pixel color in the virtual screen, and the EIA is obtained, as shown in Fig. 3.
2.2 Simulation of optical reconstruction
The simulation method of optical reconstruction is actually the process of 3D display result generation by computer. In LFD system, the LCD panel is used as a basic output to show the EIA. The lens array is composed of many lenses of the same structure arranged uniformly. Through the modulation effect of the lens array, the position information of different pixels is converted into the direction information of the light intensity of different viewpoints, to realize the purpose of displaying different viewpoints images at different positions in the space. Finally, the light distribution from the lens array is reconstructed by the directional diffuser to approximate the light field distribution of a real 3D scene, thereby uniform 3D images are formed in 3D space. To simulate this process, the components of the LFD system are constructed in the virtual scene, including LCD, lens array, and diffuser. The display result images are obtained based on the BRT, as shown in Fig. 4.
The geometric model of the LCD panel is rectangular, and its texture is the EIA obtained in the process of LFD content acquisition as shown in Fig. 5(a). The resolution of the LCD model is set by the resolution of the EIA image. On the basis of the original visual simulation work, the improvement is realized, the RGB sub-pixel based LCD simulation replaces the previous pixel-based LCD simulation. The chromatic aberration can be simulated by the improved LCD simulation due to the different wavelengths of each RGB color, which is more accurate and closer to the real effect than the previous pixel simulation. The model and position of the lens array in the optical reconstruction part are the same as the lens array in the content acquisition part. Since the components of content acquisition and optical reconstruction are completely consistent, the optical path refracted by the lenses on both sides of the LCD is symmetrical, as shown in Fig. 5(b).
Fig. 5. (a) The texture of LCD plane model, (b) Symmetrical light path diagram on both sides of LCD, (c) diffuser with scattering effect.
The geometric model of the diffuser panel is also rectangular. As shown in Fig. 5(c), when the incident rays hit the diffuser surface, the intensity of beams is evenly distributed in the range of w in the horizontal and vertical directions [1]. N rays are sampled from the exit beam, and the accuracy increases with the increase of N. The direction of the nth exit ray tn can be expressed as Eq. (7).
After constructing the components of LFD in the virtual scene, BRT works by tracing a path of light from a virtual camera through each pixel in a virtual screen. Then the 3D display result image of LFD is obtained, and the simulation of optical reconstruction is realized.
3. Optimization method
3.1 Construction environment of optimization
Before looking for optimal designs, it is important to construct the environment of optimization including objective function, decision variable, and constraints. The objective function can be regarded as a mathematical expression to evaluate the performance indexes of the design system. In practical engineering design, the designer hopes to optimize these performance indexes at the same time. The display quality of LFD typically involves multiple evaluation parameters, such as field angle of view(FOV), depth of field(DOF), structural similarity index (SSIM). Hence, the optimization design of the LFD system can be regarded as a multi-objective optimization problem. The objective of the optimization is to optimize these evaluation parameters as much as possible. However, these sub-objective functions are not independent of each other, and there are certain contradictions. The improvement of one sub-objective may cause the performance degradation of another or several sub-objectives, and it is impossible to achieve the optimal value of multiple sub-objectives at the same time.
Therefore, on the basis of synthesizing all kinds of evaluation parameters, an objective function Q for 3D display quality evaluation is presented. Thus, the multi-objective optimization problem of LFD is transformed into a single objective optimization problem. The Q is calculated based on the SSIM and Peak Signal to Noise Ratio (PSNR) between the simulation image and the photograph of a 3D object captured by a virtual camera in a different location expressed as Eq. (9). The SSIM and PSNR are methods for predicting the perceived quality of digital television and cinematic pictures, as well as other kinds of digital images and videos [23]. The similarity between two images is measured by SSIM equally from brightness, contrast, and structure. PSNR is an objective image evaluation index based on the error between corresponding pixels. Here, SSIM and PSNR δ are used to quantify the difference between the simulation image and the parallax image (ideally observed image) to objectively evaluate the final imaging quality of LFD. The arrangement of reference viewpoint positions used for Q-value calculation is as shown in Fig. 6.
Here, n is the number of the reference viewpoints, which is 9. The reference viewpoints are arranged at equal intervals; the spacing is 50cm. SSIMi is the SSIM value of each reference viewpoint. For Eq. (9), the Q value needs to be as small as possible. The Q value is used to evaluate the system display effect after each optimization iteration as the objective function, and the LFD system is optimized by changing the Q value.
The LFD system is shown in Fig. 1 and includes three parts: LCD, LLA, and diffuser. The surface of the typical lens in the LFD system includes 3 parts; a top sphere surface, a partial cylinder surface, and a bottom sphere surface, as shown in Fig. 2(b). The radius R1 of the top surface of the lens, the radius R2 of the bottom surface of the lens, the thickness h of the lens, the diffusion angle ω of the diffuser, and the distance Lll between the lens and the LCD are the variable parameters in the optimization process. For simplicity, the parameters of each lens in the lens array are consistent. Therefore, there are five optimization variables in the optimization process, and the boundary conditions of the optimization variables are given physically. The parameters that affect the objective function are shown in Table 1.
Table 1. Parameters affecting the objective function
There is a certain correlation between the parameters. These relationships can be used as constraints for optimization, which can greatly improve the speed of optimization. Since the reconstructed 3D images are the results of each sub-image imaging and merging through the unit lens, the position relationship between the whole LFD system can be expressed by the Gauss theorem.
Where Lll is the distance between the lens array and the LCD, Lld is the distance between the lens array and the diffuser. To get the highest image quality and reduce the influences of crosstalk, the diffuser of LFD must be located on the center plane of the reconstructed images. And f is the focal length of a single lens in the lens array which can be calculated using a simple equation, as follows:Here, nlens and nair are the refractive index of the lens and the air, R1 and R2 are the radius of the top and bottom surface of the lens respectively, and h is the thickness of the lens. To reduce the process difficulty and optimization complexity, R1 and R2 values are set to be equal.
In the optimization process, the Gauss theorem and the focal length formula are used as parameter constraints at the same time. When the optimization parameters are adjusted, the position coordinates of the image plane (the diffuser coordinates), are kept unchanged. In this way, the focal length and position of the evaluation camera do not need to be changed during the optimization iteration.
3.2 Optimization method based on simulated annealing
The LFD system is composed of many components and contains many variable parameters. The influence of these variables on the display effect is complex and nonlinear. In the case of all other variables being kept unchanged, the method of design optimization and combination of individual components to observe the impact on the display effect, is often time-consuming while it only results in partial improvement. In contrast to this design approach, the joint optimization of an optical system is considered. The optimal design of the LFD system can be attributed to the global optimization problem of multi-dimension and multi-extremum of continuous variables. The lack of an effective global optimization method is the key to solving this problem. A framework for the design of the LFD system with the simulated annealing algorithm and visual simulation is developed. The optimization variables in the LFD system are optimized with respect to the objective function Q based on SSIM and PSNR.
The simulated annealing algorithm is a simple and universal global optimization algorithm [33]. It does not require the continuity and convexity of objective function and constraints, and has strong adaptability to the uncertainty of data in the calculation. It can find the best point when the analysis of function is unknown, and it is suitable for high-dimensional nonlinear optimization problems. The simulated annealing algorithm is derived from the solid annealing process. Its essential difference from the ordinary algorithm is that under the control of the temperature parameter T, in addition to accepting the optimal solution, it also accepts the deteriorating solution according to the Metropolis algorithm with a certain probability. This makes the simulated backtracking algorithm not only have the ability of local optimization, but also have the ability to jump out of the local optimal “trap”.
Because the LFD system cannot be simply reduced to a direct function containing independent variables, and the variables to be optimized are too many and continuous, the search space is too large to traverse all cases, this paper adopts the optimization method based on simulated annealing. One solution of the optimization problem is set to W = [ R1, R2, h, ω, Lll] T, and the objective function is Q. The initial temperature and termination temperatures are T0 and Tf. The temperature attenuation coefficient is α and the temperature attenuation function is Tk+1=α*Tk. The length of the Markov Chain is Lmax, which is the maximum number of iterations at each fixed temperature.
The implementation process of the simulated annealing algorithm shown in Fig. 7 is as follows :
- Step1: Initialization. Set the iteration index k= 0, and Tk= T0. An initial solution Wx= [ R1x, R2x, hx, ωx, Lxll] T, ${\in} $[Wmin, Wmax] is generated randomly and brought into the simulation-optimization system to obtain the simulation results and calculate the objective function Qx.
- Step2: A new solution Wy= [ R1y, R2y, hy, ωy, Lyll] T, ${\in} $ [Wmin, Wmax] is generated randomly and brought into the simulation-optimization system to obtain the simulation results and calculate the objective function Qy. Where Wy= Wx+ random * δ * (Wmax –Wmin), δ is the step of change factor, and random is the uniform random number between [- 1,1].
- Step3: Transition probability at temperature T according to the Metropolis algorithm:$${P_{xy}} = \left\{ \begin{array}{ll} {1,} &{{Q^y} \le {Q^x}}\\ {\exp \left( {\frac{{{Q^y} - {Q^x}}}{{KT}}} \right),} &{{Q^y} > {Q^x}} \end{array} \right..$$Where K is the Boltzmann’s constant, T is the temperature at present. If the new objective function value is greater than or equal to the old objective function value, accept the new solution. If not, generate a random number randomnum in the range of [0, 1], If Pxy > randomnum, still accept the new solution, otherwise, reject.
- Step4: If the heat balance is reached (The maximum number of iterations is reached), go to step 5, otherwise go to step 2.
- Step5: If stopping conditions are met, end and output the optimization results. Else, k = k+1and Tk+1=α*Tk, L = 1, and go to step 2.
The whole optimization process, therefore, consists of two circulations: the internal circulation and the external circulation; the internal circulation generates the trial points in a Markov chain and the external circulation reduces the “temperature”. The internal circulation ensures that the parameter space is searched sufficiently at a given “temperature” and the external circulation models the cooling speed of the “temperature”, both of which are necessary to guarantee that the algorithm finds a global optimum. The simulated annealing algorithm allows arbitrary initial sequences and random numbers. Then other basic parameters are determined as shown in Table 2.
Table 2. Basic parameters of the simulated annealing algorithm
4. Simulation and optimization result
The parameters and values used for simulation are given in Table 1, and the relative position of each component is shown in Fig. 1. The simulations and design optimizations are carried out using a PC with Intel Core i7-4790 CPU @ 3.6GHZ, 8Gb RAM, and NVidia GeForce GTX 1660Ti (4 GB / NVidia) graphic card. The simulation program is implemented via CUDA SDK 8.0 and OpenGL4.0. The design optimization program is implemented via Python3.6.
Based on the simulation described in Section 2, the process of LFD content generation and light field reconstruction are simulated at the same time. The obtained EIA image, the simulation of the reconstructed images, and the experimental results of the reconstructed images are shown in Fig. 8. The simulation results are consistent with the experimental results obviously, which also proves that it is feasible to co-design the LFD system based on the simulation results. The reason for the obvious lens gap in the experimental results is that under the existing technology, the diffusion of the diffuser is Gaussian distribution, which cannot achieve the theoretical uniform diffusion. However, in the simulation experiment, the ideal uniform diffusion effect can be achieved, which leads to no lens gap in the simulation results of light field reconstruction.
Here, a simulated annealing LFD design framework based on the evaluation of simulation results is constructed, the parameters of the optimization algorithm are set, and the LFD system is co-designed by using this algorithm. Figure 9(a) shows the variation in the objective function with the number of iterations in three optimization co-design processes. The x-axis is the number of iterations in the whole optimization process, and the y-axis is the value of the objective function. We can see that although the initial values are different in the three optimization processes, the overall basic trend is the same. With the increase of the number of iterations, the system temperature gradually decreases and the objective function curves gradually tend to be stable. After 7000 iterations, the objective function is no longer significantly optimized and is in a state of slight fluctuation. The reason is that the simulated annealing algorithm adopts the Metropolis criterion in order to jump out of the local optimal solution and get the global optimal solution, and there is a certain probability to accept the result with poor objective function as the current solution. Figure 9(b) shows the variation of the LFD parameters with the number of iterations. The x-axis is the number of iterations in the whole optimization process, and the y-axis is the value of the parameters. It can be observed from the figure that with the increase of the number of iterations, the variation range of optimization variables is from large to small, and finally tend to be stable. As the number of iterations increases, the system temperature decreases, and the step size of the optimization variable decreases. This indicates that after 7000 iterations, no significant improvement is observed, and the parameters basically reach the optimal effect. Different initial conditions are used to achieve the same convergence value, which proves that our model building and 3D images quality evaluation methods have high robustness. At the same time, in order to find the global optimal solution, simulated annealing can jump out of the optimal solution domain after finding the optimal solution, so the historical optimal solution is additionally recorded. Table 3 shows the corresponding co-design parameter results and LFD indicators.
Fig. 9. (a) The variation in the objective function with the number of iterations. (b)The change of three-dimensional optical field parameters with the number of iterations.
Table 3. Co-Design results
According to the automatically optimized parameters, a new LFD system is constructed. The real reconstructed 3D images (Monkey model) with the traditional parameters and our proposed method captured from different directions at 2 m from the designed LFD are shown in Fig. 10. With our proposed method, the quality of the 3D image is higher than that of the traditional parameters. It can be seen that the LFD designed by our proposed method can achieve continuous and uniform 3D images with high quality and provide detailed information for the 3D model. The effectiveness of the automatic co-design method is verified by simulation experiments and optical experiments.
Fig. 10. The reconstructed image of the LFD with the traditional parameters and our proposed method captured from different perspectives
5. Conclusion
In summary, an automatic co-design method of LFD parameters based on simulated annealing and visual simulation is presented. A simulation tool based on BRT is combined with constructive solid geometry and bidirectional scattering distribution function modeling method to enable fast simulations of content acquisition and optical reconstruction process. According to the simulation results, the quality evaluation function of the 3D images is established, and under this function, the light field parameters are automatically co-designed by using simulated annealing. Experiments demonstrate that the designed LFD system using our approach produces clear and full-parallax 3D images, which can provide natural 3D visual effect to observers with high quality.
Funding
National Natural Science Foundation of China (62075016).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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