# Fast Numerical Reconstruction of Integral Imaging Based on a Determined Interval Mapping
## 3 Results  
### Numerical Experiment  
To verify the validity of the proposed method, a numerical experiment was carried out. Figure 5 shows the 3D object used in the numerical experiment and its acquired EIA. As depicted in Figure 5a, three letters with 'W', 'K', and 'U' utilized by the 3 ds Max are used as the virtual 3D object. Each letter is located at a distance of 150 mm, 350 mm, and 550 mm from the virtual lenslet array, respectively; 96 \(\times\) 54 lenslets are used, and the interval of each elemental lenslet and its FOV is 10 mm and 10\({}^{\circ}\). The EIA is acquired by the computational acquisition procedure as shown in Figure 5b. The number of pixels of each EI is 45 \(\times\) 45. Thus, the number of pixels of the acquired EIA is 4320 \(\times\) 2430. Figure 4: Process for the compensation of the position error and the use of adaptive normalization. For the numerical reconstruction procedure, hardware with an Intel(R) Core(TM) i9-9900 KF CPU, 32.0 GB ram, and solid-state drive (SSD) is used. Visual studio 2019 and Open CV are used to implement the conventional and proposed method. The number of the virtual pinholes is 96 \(\times\) 54 to match the number of the virtual lenslets, and the interval of each pinhole is 10 mm, which is the same as the pickup lenslet. The value of \(g\) between the virtual pinhole array and the displayed EIA is 57.15 mm. In the numerical experiment, each reconstruction plane of three letters with 'W', 'K', and 'U' is set up as 150 mm, 350 mm, and 550 mm. In the conventional numerical reconstruction, each value of \(M\) related to 'W', 'K', and 'U' is calculated as 2.62, 6.12, and 9.62, respectively. Thus, each EI will be magnified as much as the calculated corresponding value of \(M\). In addition, the number of pixels of each reconstructed image by the conventional method is 4394 \(\times\) 2504, 4551 \(\times\) 2661, and 4709 \(\times\) 2819 in order of the reconstruction plane. In the proposed method, each value of the determined interval \(\Delta\tau\) is 3.81 mm, 1.63 mm, and 1.04 mm by Equation (1). Thus, the number of pixels of the numerical reconstruction by the proposed method is 1660 \(\times\) 946, 710 \(\times\) 416, and 425 \(\times\) 257, respectively. Figure 6 shows three kinds of the refocused images reconstructed by the conventional and proposed method to use the nearest, linear, and bicubic interpolation. When the reconstruction plane is set up as 150 mm, the focused 'W' is reconstructed by whole numerical reconstruction methods. In addition, the refocused letters 'K' and 'U' are visualized when the reconstructed plane is 350 mm and 550 mm, respectively. From these numerical experiments, the proposed method can numerically reconstruct the 3D image depending on the reconstruction plane of the objects 'W', 'K', and 'U'. From the experimental result in Figure 6, we can confirm that the proposed method can numerically reconstruct the 3D image similar to the conventional method. Additionally, each reconstructed image has no grid noise in the proposed method because the grid noise is removed by adaptive normalization. To analyze the performance of computation time in whole methods, the comparison of the measured computation time was carried out, as shown in Figure 7.  
The \(x\)-axis means the value of the reconstruction plane ranging from 100 mm to 700 mm, and the \(y\)-axis is the computation time (milliseconds, ms). The blue-colored bar is the computation time by the conventional method, and the orange-, gray-, and yellow-colored bars indicate the individual computation time of the proposed method to employ three different kinds of interpolation methods. As shown in Figure 7, the computation time represented by the blue-colored bar steadily increases in compliance with the increasing reconstruction plane because each EI is magnified as much as \(M\) times. Moreover, the increasing reconstruction plane causes a huge memory requirement related to the resolution of the refocused image.