# Fast Numerical Reconstruction of Integral Imaging Based on a Determined Interval Mapping
## 2 Numerical Reconstruction of the Integral Imaging  
### Conventional Numerical Reconstruction Method  
To understand the conventional numerical reconstruction of II, the conceptual diagram of both procedures is described in Figure 1.  
Figure 1a shows the acquisition procedure of EIA to use the single point \(O\). Here, \(Z\) is the distance between a single point \(O\) and the lenslet array plane. \(g\) denotes the distance between the lenslet array and the capturing device. In addition, the parameter \(P\) is an interval between neighborhood lenslets. \(L\) is the distance between the plane of the virtual pinhole array and the reconstruction plane. \(s_{x}\) is the the number of pixels (resolution) of each EI.  
In the acquisition procedure, rays emanating from a single point \(O\) are passed through each lenslet and recorded on the red-colored pixel. When whole recorded rays through the corresponding lenslet are formed as an individual projection image, this image becomes the EI, and the EIA is formed by the group of EIs. However, each position of the red-colored pixel in each EI is different according to the relative position of the lenslet. This position difference is called the disparity \(d\), which is expressed as \(d=gP/Z\). The feature of this disparity is generally analyzed in the stereo camera-based system [39]. In addition, the EIA capturing procedure based on the lenslet array can be considered as the multiple cameraswith \(P\). Therefore, the position difference of the red-colored pixel in arbitrary neighboring EIs constantly remains as \(d\), as shown in Figure 1a.  
In the numerical reconstruction procedure of the conventional method, the virtual pinhole array and the acquired EIA are used, as shown in Figure 1b. Here, the acquired EIA is displayed at the EIA plane, and each EI is inversely propagated through the corresponding virtual pinhole along the longitudinal direction. Each inversely propagated EI is magnified according to the magnification factor \(M\), which is expressed as \(M=L/g\)[11]. In other words, the number of pixels of the magnified EI becomes \(M\)s\({}_{x}\) by the interpolation since the number of pixels of each EI is \(s_{x}\). After these magnified EIs are superimposed at the reconstruction plane \(L\) together, the single point located at the reconstruction plane is numerically reconstructed.  
In the conceptual diagram, the reconstructed single point means that the position difference of each red-colored pixel in every magnified EI becomes 0. In other words, \(d\) in two arbitrary adjacent magnified EIs becomes 0. Thus, the condition of \(d\) becoming 0 can be treated as the reconstruction condition for the numerical reconstruction in the conventional method. Furthermore, the relation of \(M=L/g\) can be derived based on the reconstruction condition.  
In the conventional method, the interpolation for the magnification is employed and applied to each EI during the numerical reconstruction. However, the interpolation process for every EI increases the computation time. To overcome this problem, we will explain how to accelerate the computation time of the numerical reconstruction of II under the reconstruction condition in the next section.