# Fast Numerical Reconstruction of Integral Imaging Based on a Determined Interval Mapping
## 1 Introduction  
Integral imaging (II) has been researched to implement a realistic three-dimensional (3D) display for a providing full-color 3D images with full-parallax and continuous-viewing points [1; 2; 3; 4; 5; 6; 7; 8; 9; 10]. II has consisted of two procedures of acquisition and reconstruction. In the acquisition procedure, a lenslet array and a capturing device are used. When rays from the 3D object are acquired by the individual lenslet and the capturing device, these acquired rays form an elemental image (EI). Furthermore, the array-formed EIs become an elemental image array (EIA). Through the acquired EIA, the 3D image can be reconstructed by a lenslet array and flat-panel display (FPD). This is called the optical reconstruction procedure. However, optical reconstruction can degrade the quality of the reconstructed image due to the optical limitation of the lenslet array, such as lens aberrations and a wide diameter of the Airy disk by a diffraction limit [11]. In addition, optical reconstruction cannot be applied in imaging-processing-based applications. To overcome these drawbacks, the numerical reconstruction of II was proposed [11]. This technique allows the refocused image, depending on the reconstruction plane, to be numerically reconstructed from optically or digitally acquired EIA without optical limitations, such as lens aberrations. From this technique, various kinds of research have been conducted according to specific purposes of the imaging-processing-based application [12; 13; 14; 15; 16; 17; 18; 19; 20; 21; 22; 23; 24; 25; 26; 27; 28; 29; 30; 31; 32; 33]. For example, some approaches for improving image quality have been proposed to adopt the analysis of the behavior of interpolation, the pixel arrangement, etc. [12; 13; 14; 15; 16; 17; 18; 19; 20; 21; 22; 23]. Other approaches for synthesizing the intermediate view or reconstructing thespecific view image have been presented based on the disparity estimation [24; 25; 26]. Additionally, various attempts to extract the depth information or remove the occlusion from the acquired EIA have been carried out based on the digital processing of II [27; 28; 29; 30; 31; 32; 33; 34; 35; 36]. However, these methods are basically time-consuming since each EI needs to be magnified through interpolation during their processing. This drawback restricts the practical application of conventional methods. To reduce the computation time, fast numerical reconstruction of the II based on parallel processing has been proposed [37; 38]. However, these methods are not enough to implement the real-time numerical reconstruction because each EI needs to be magnified by interpolation. Thus, a different approach without interpolation is required to improve the computation time for numerical reconstruction. In this paper, a fast numerical reconstruction of the integral imaging based on a determined interval mapping is proposed. Since this method does not require interpolation, the computation time for numerical reconstruction can be significantly reduced. To achieve the proposed method, the EIA acquired from the 3D object is first displayed, and a virtual pinhole array is positioned in front of the displayed EIA plane. Each displayed EI is transformed into a flipped EI by the propagated rays through the corresponding virtual pinhole. In addition, the determined interval is calculated depending on the distance to the reconstruction plane. These flipped EIs are shifted to match the determined interval at the reconstruction plane and superimposed together. This superimposed image is divided by the number of the superposition, and the position error in the superimposed image is corrected by interpolation. Consequently, the refocused image depending on the reconstruction plane is reconstructed in a shorter computation time than the conventional method. To validate the feasibility of the proposed method, two numerical experiments related to the computation time and the quality comparison of each refocused image were carried out.