# Fast Numerical Reconstruction of Integral Imaging Based on a Determined Interval Mapping
## 2 Numerical Reconstruction of the Integral Imaging
### Proposed Numerical Reconstruction  
For the acceleration of computation time, the proposed method employs the determined interval mapping in which the interval of each EI is variable depending on the reconstruction plane. Additionally, the proposed method does not require the interpolation process for the magnifying EI during the reconstruction procedure. Since interpolation is not used, the computation time for the numerical reconstruction of II can be reduced. Figure 2 shows the process of the proposed numerical reconstruction. Figure 1: The conceptual diagram of both procedures in the conventional numerical reconstruction method based on the single point \(O\). (**a**) Acquisition procedure; (**b**) numerical reconstruction procedure. The EIA from the 3D object is acquired by the acquisition procedure in the same way as the conventional method. The acquired EIA is virtually displayed at the EIA plane, and the virtual pinhole array is located in front of the display EIA. Rays from each displayed EI are passed through the corresponding pinhole, and the flipped EIs are formed. Then, the determined interval \(\Delta\tau\) is calculated depending on the reconstruction plane. Each flipped EI is shifted to match \(\Delta\tau\) at the reconstruction plane and overlaid with the others to form the superimposed image. However, this superimposed image involves the grid noise caused by the different occurrences of the superposition of each shifted EI. Thus, the superimposed image is divided by the number of occurrences of the superposition to remove the grid noise. Then, the position error between the location of the shifted EI and the pixel position of the reconstruction plane is corrected by inter.polation. Consequently, the refocused image without the grid noise and the position error is numerically reconstructed according to the reconstruction plane. In the proposed method, the determined interval \(\Delta\tau\) is an important factor to reduce the computation time. Thus, the determined interval \(\Delta\tau\) can be induced from the image formation model, as shown in Figure 3.  
As shown in Figure 3, \(g\) is the distance between the displayed EIA plane and the plane of the virtual pinhole array. \(L\) is the distance of the reconstruction plane from the virtual pinhole array. The total number of EIs is \(K\), and these EIs arranged from \((0)^{th}\) EI to \((K-1)^{th}\) EI are displayed. In addition, the position difference of the red-colored pixels in adjacent EIs constantly remains as \(d\), as shown in Figure 1.  
In the image formation model, each plane of the virtual pinhole array and displayed EIA is located at 0 and \(-g\), respectively. Thus, the position of the red-colored pixel in the displayed \((K-1)^{th}\) and \((K-2)^{th}\) EI is expressed as {\(-g\), \((K-1)P\)+\((K-1)d\)} and {\(-g\), \((K-2)P\) + \((K-2)d\)}, respectively. When rays propagated from each red-colored pixel are inversely mapped at \(+g\), each flipped EI is formed, and its interval remains as \(P\). Hence, the red-colored pixel position of the flipped \((K-1)^{th}\) and \((K-2)^{th}\) EI can be expressed as {\(g\), \((K-1)P\) \(-\)\((K-1)d\)} and {\(g\), \((K-2)P\) \(-\)\((K-2)d\)}. When each flipped EI is shifted according to \(\Delta\tau\) at the reconstruction plane, each red-colored pixel position in the shifted \((K-1)^{th}\) and \((K-2)^{th}\) EI becomes {\(L\), \((K-1)\Delta\tau\) \(-\)\((K-1)d\)} and {\(L\), \((K-2)\Delta\tau\) \(-\)\((K-2)d\)}. By the reconstruction condition, the relation of {\(L\), \((K-1)\Delta\tau\) \(-\)\((K-1)d\)} = {\(L\), \((K-2)\Delta\tau\) \(-\)\((K-2)d\)}  
Figure 2: Conceptual configuration of the proposed method. must be satisfied for the numerical reconstruction. Through this relation, the determined interval \(\Delta\tau\) depending on the reconstruction plane can be induced in Equation (1). \[\Delta\tau=\frac{g}{L}P \tag{1}\]  
From Equation (1), we can realize that the determined interval \(\Delta\tau\) is determined by the ratio between the \(g\) and \(L\).