## 2 Calibration Inaccuracies  
In order to calibrate a projection based light field display system, the DDHOE screen is placed at the projector's focus and is lit by the projector as shown in Figure 2a. A camera is placed at the position where the user will be located during playback. The camera captures light bouncing back from the DDHOE screen (Figure 2a). By projecting a structured light pattern (graycodes in this case), the relation between projector pixels and micromirror position can be determined. The main assumption used in the calibration reported previously is that the brightest spots of the micromirrors were at the centers of the micromirrors (shown as rays of peak intensity in Figure 2a). In reality, this assumption does not hold and can result in a shift of the projected scene with relation to the real-world objects. Figure 1: (**a**) schematic of the light-field display system; (**b**) photograph of the top-view of the real display system; (**c**) photograph of the side view of the real display system. Figure 2: (**a**) schematic of the calibration configuration; (**b**) difference between the observed point \(C^{\prime}\) and the actual center \(C\) of the mirror. The position of the observed point is dependent on the focal length of the mirror and the direction of the projected ray, as well as the surface normal \(\hat{n}\). In fact, as can be seen in Figure 2b, the observed point \(C^{\prime}\) corresponds to the location at which the light ray that hits the camera sensor, is reflected. This means that there is an offset between the observed point \(C^{\prime}\) and the actual center \(C\) of the mirror. It is to be noted that the curved surface represented in Figure 2b corresponds to a virtual curvature, whereas the real hologram surface is flat. In other words, the light rays that are reflected off the hologram surface behaves as if they are getting reflected off of a curved surface. Thus, the surface normal \(\hat{n}\) here is the normal to the virtual curvature at the point of incidence. In addition to that, the optical axis of each optical mirrors are tilted by a small angle [11]. Hence, the offset is dependent on several parameters such as the focal length and tilt of the corresponding micromirror, as well as the positioning of both the projector and the camera used for calibration. By ignoring this shift and assuming that the centers of the projected elemental images should be mapped onto the observed brightest point for the corresponding micromirror, two issues are introduced:  
1. View Shift: since the observed points can only be noticed at the position of the calibration camera, the calibration procedure will map the central view of the scene to the position of the camera. This means that, if the camera is located to the left of the viewing zone's center the views of the scene will all be shifted to the left as well. As a result, some views cannot be perceived at the left side of the viewing zone but will be visible on the right side instead. 2. Distortion: although, the screen handles the divergence of the projector [11] (if positioned correctly), the camera will still observe a converging set of rays being reflected from the screen. Because of this, the observed points will not be located at the same position within each micromirror. This introduces a distortion since the calibration will try to minimize the reprojection error as much as possible, resulting in an overcompensation of the lens distortion parameters. Assuming that the projector is located precisely on a designated position, one can mitigate these issues by precise camera placement. To remove the shift of the view, one can place the camera at the center of the viewing zone as precisely as possible. This makes the system reflect the center view towards the center of the zone. In order to reduce distortion, one can place the camera at a larger distance from the screen.