*3.1. The Selection of the Hogel on Virtual H*<sup>1</sup> *Plate*

First, hogel number of the H1 plate should be fixed. As shown in Figure 6, to make full use of the LCD resolution, choosing the appropriate field of view (FOV) of the hogel on the H2 plate and *L*<sup>2</sup> makes the display area of the LCD completely contained by the FOV. The display area dimensions of the LCD panel is denoted as *L*LCD. Using *l*<sup>1</sup> and *l*<sup>2</sup> to represent the dimension of hogel on H1 plate, we denote the dimension of effective perspective images segments as *l*LCD. According to the geometric relationship, there is *l*<sup>1</sup> = *l*<sup>2</sup> = *l*LCD = *l*. As shown in Figure 7, to determine the number of the hogel on

the virtual H1 plate, we first determine the number of segments on the LCD panel, which is presented by *n*LCD. Choosing the hogel on H1 plate corresponds to the hogel on the H2 plate, then

$$m\_{\rm LCD} = \begin{cases} \frac{\frac{L\_{\rm LCD}}{I}}{\frac{L\_{\rm LCD}}{I} - 1}, & \text{if } \frac{\frac{L\_{\rm LCD}}{I}}{\frac{L\_{\rm LCD}}{I}} \text{ is odd} \\\frac{\frac{L\_{\rm LCD}}{I} - 1}{I}, & \text{if } \frac{\frac{L\_{\rm LCD}}{I}}{\frac{1}{I}} \text{ is even} \end{cases} \tag{1}$$

where *n*LCD should be odd here, *n*hogel represents the segment number on both sides of the center segment on the LCD panel, *<sup>n</sup>*hogel = *<sup>n</sup>*LCD−<sup>1</sup> <sup>2</sup> .

**Figure 7.** Determination of the hogel number for holographic element-based EPISM methods at *k* = 1.

Introducing an index *k*, *k* represents the ratio of the distance from the LCD panel to virtual H1 plate and H2 plate, and *k* = *<sup>L</sup>*<sup>1</sup> *L*2 . According to the geometric relationship, when *k* = 1, the hogel corresponding to the adjacent segment of the LCD panel is separated by one hogel. When *L*<sup>1</sup> and *L*<sup>2</sup> are not equal, let *k* be a positive integer; this means that the adjacent segment of the LCD panel has corresponding hogels on H1 plate. The interval hogel number between adjacent hogels is exactly equal to *k*.

The number of hogels on the H1 plate can be represented as *n*H1= 2 × *n*hogel × (*k* + 1)+*n*H2 .

#### *3.2. Effective Perspective Image Segmentation and Mosaicking*

As shown in Figure 8, for the *i*th hogel on the H2 plate, the order of the hogel on the virtual H1 plate that corresponds to the *i*th hogel on the H2 plate should be *i*+*n*hogel × (*k* + 1). For the *i*th hogel on the H2 plate, the segment on the center of the LCD panel is denoted as the 0*th* segment, and let *m* = <sup>−</sup>*n*hogel,*n*hogel , the order of hogel on the virtual H1 plate that corresponds to the *m*th segment of the LCD panel and *i*th hogel on the H2 plate is *i*+*n*hogel × (*k* + 1)+*m* × (*k* + 1), and this is the hogel we are looking for. Left endpoint of the hogel on the virtual H1 plate that corresponds to the 0*th* segment of the LCD panel is denoted as *x*0; according to the previous results, we have *x*<sup>0</sup> = *i*+*n*hogel × (*k* + 1) −1 × *l*1, left end the hogel on the H1 plate that corresponds to the *m*th segment of LCD denoted as *s*, *s* = *i* + *m* × (*k* + 1) +*n*hogel × (*k* + 1) −1 × *l*1, *h* = (*x*0−*s*) × (1<sup>−</sup> <sup>1</sup> *<sup>k</sup>*+<sup>1</sup> ) × *l*<sup>1</sup> represents the distance between the corresponding effective perspective images segments and the location of hogel on H1 plate. The left end coordinates of the segment are denoted as *e*,

$$\sigma = \left[ \left( h - \frac{1}{2} \right) \times l\_1 + \frac{L\_{\rm LCD}}{2} \right] \times 100 + 1 \tag{2}$$

and the right end coordinates of the segment denote as *f* ,

$$f = \left[ \left( h - \frac{1}{2} \right) \times l\_1 + \frac{L\_{\rm LCD}}{2} \right] \times 100 + e\_p \tag{3}$$

where *ep* = *l*<sup>1</sup> × 100, according to the left and right ends of the segment, determine the *m*th effective perspective images segments. By mosaicking all the selected effective perspective images segments in sequence, we obtain the synthetic perspective image corresponding to the *i*th hogel on H2 plate. By exposing the hogel on H2 plate in sequence, the holographic stereogram is obtained.

**Figure 8.** Diagram of effective perspective images segments.

#### **4. Experimental Verification**

An LCD panel was used for exposing holographic stereogram, for convenience in calculation, the corresponding resolution on 1 cm × 1 cm display area of LCD panel was 100 × 100 pixels. To this end, the Panasonic LCD panel (VVX09F035M20) had been selected, its size was 8.9 inches, and the resolution was 1920 × 1200. Select 10 cm × 10 cm area in the center of LCD panel as the effective display area. Thus, the resolution of the synthetic perspective image for exposure is 1000 × 1000 pixels.

A mapped teapot model is used as the 3D scene. The depth was 4.8 cm, the height was 3 cm, the width was 4.2 cm, and it was tipped 40◦. Let *k* = 1, *l*<sup>1</sup> = *l*<sup>2</sup> = 0.2 cm, *L*<sup>1</sup> = *L*<sup>2</sup> = 18.6 cm. A camera was set in front of the teapot, and the FOV was 30◦. The size of H2 plate was 6 cm × 6 cm, *nH*<sup>2</sup> = 30, and the hogel number of H2 plate was 30 × 30 = 900. According to the previous formula, *nH*<sup>1</sup> = 126, the number of the camera position should be 126 × 126 = 15,876.

As shown in Figure 9, the synthetic perspective images of the holographic element-based EPISM method are given, image(6, 16) is the synthetic perspective image corresponding to the order number of the sixth row and the sixteenth column's hogel on H2 plate. As EPISM methods, the synthetic perspective images are pseudoscopic images, and the reconstructed scene can reproduce the recording scene truthfully in correct depth.

**Figure 9.** Synthetic perspective image of the holographic element-based EPISM methods.

As shown in Figure 10, the optical setup using for holographic-based EPISM method holographic stereogram is presented. A 639 nm custom-made red laser was used as the laser source for holographic stereogram printing. The max power of laser source was 1.2w. An electric shutter was used to control the exposure time. Its model was Sigma Koki SSH-C2B. The direction of laser beam was changed by a reflector. A non-polarizing beam splitter(NPBS) was set in the new direction of the laser beam, and the laser beam was divided into two laser beams perpendicular to each other. The laser beam vertical to the original direction was denoted as signal beam, and the laser beam in the same direction as the original direction was denoted as reference beam. An attenuator was set in the same direction of signal beam to adjust the intensities, and a spatial filter was used to expand the signal beam. The LCD panel introduced earlier was used to load the synthetic perspective image, a diffuser was used to scatter the signal beam to the hogel aperture, and the diffuser was placed close to the LCD panel. The reference beam direction was changed by a reflector, and an attenuator was used to adjust the intensities of the reference beam. Another reflector was used to change the reference beam to the backside of holographic plate, and a spatial filter was used to expand the reference beam. Then, the expanded beam was changed into a uniform plane wave by a collimating lens; the focal length of the lens is 75 mm. A manual ultrafine silver-halide plate for He-Ne laser was used as holographic plate; its grain size was about 10–12 nm. The holographic plate placed 18.6 cm away from LCD panel, two diaphragms with apertures are placed on both sides of the LCD panel, the signal beam and reference beam passing through the aperture interfere on the holographic plate to generate hogel. A X-Y motorized stage(KSA300, MC600) was used to carry the holographic plate; it can move both on the horizontal and vertical rail. The step of the X-Y motorized stage was *l*2. A computer was used to time-synchronously control the shutter, the LCD panel, and the motorized stage.

**Figure 10.** The optical setup of the synthetic holographic stereogram printing system.

The synthetic perspective image used to expose the hogel on the H2 plate needs to be flipped horizontally, because the LCD panel corresponding to the virtual H1 plate and the LCD panel corresponding to the H2 plate were in the opposite direction.

Equation *Te* = *E*/(*Ps* + *Pr*) was used to express the exposure time, where *Ps* is the intensity of signal beam energy, *Pr* is the intensity of reference beam energy, and *E* denotes as the light sensitivity of holographic plate. In this experiment, *E* = 1250 μJ/cm2, *Ps* = 10 μJ/cm2, *Pr* = 300 μJ/cm2. The energy ratio between *Ps* and *Pr* was selected as 1:30; this ratio can greatly reduce the printing time on the premise of guaranteeing the image quality, and the exposure time was *Te* = 4 s. The waiting time was 4 times as much as exposure time to reduce the vibration result, and the waiting time *Tw* = 16 s. The printing time was the sum of exposure time and waiting time, the hogel's printing time *Th* = *Te* + *Tw* = 20 s, and total printing time *Tt* = *Th* × *NH*<sup>2</sup> × *NH*<sup>2</sup> = 18,000 s.

Figure 11 shows the reconstruction images of an optical experiment result. A conjugate beam of reference beam was used to illuminate the holographic plate, and a camera used to record the reconstruction image of holographic plate. Its model is Canon EOS 5D-mark3 DS126091, the focal length of camera lens is 100 mm, the camera was set 50 cm away from holographic plate, and a real image of 3D scene was captured by camera.

**Figure 11.** Optical reconstruction images in different viewing position obtained by experiments.

As shown in Figure 11, by the effective display size and the distance *L*2, the FOV of H2 plate is about 30◦. The details of the original scene are perfectly reproduced, and full parallax information can be presented by the holographic element-based EPISM method. When the camera position is close to the limited FOV, due to the use of simple camera sampling, the reconstructed scene is as incomplete as the original scene, with the observation area unable to achieve 30◦ FOV.

To confirm the position of reconstructed image, we put two rulers together with the holographic plate. A camera was put 50 cm away from holographic plate, and the result is shown in Figure 12. The position relation of the rulers and holographic plate are shown in Figure 12a; the ruler on the left side is on the same plane as the holographic plate, and the ruler on the right side is on the position of the reconstruction image, away from the holographic plate by 18.6 cm. In Figure 12b, focusing on the left ruler, both the figure on the ruler and the hogel on holographic plate are clear, and the figure on the right ruler is blurred. In Figure 12c, focusing on the right ruler, the figure on the ruler and the map on the reconstruction image are clear, and the left ruler is blurred.

**Figure 12.** (**a**)The spatial position relation of two rulers and the holographic plate. (**b**) Focused on the left ruler, both rulers the holographic plate are clear. (**c**) Focused on the right ruler, both ruler and the surface of the teapot is clear.

As shown in Figure 13, the camera was also put in the position 50 cm away from the holographic plate; an EPISM method holographic stereogram was made according to the same configuration. Comparing the reconstructed images obtained by the EPISM method and the holographic element-based EPISM method, Figure 13a shows the position relation of rulers and the holographic plate; the farther ruler is placed in the position of the reconstruction image, with the ruler as far from the holographic plate as 18.6 cm, and the closer ruler is placed 1.5 cm behind the position of the farther ruler. Figure 13b,c show the reconstruction images of the EPISM method. Both the figure on the ruler and the map on the teapot are clear when focusing on the farther ruler in Figure 13b; as shown in Figure 13c, the figure on the ruler is clear and the map on the teapot is blurred when focusing on the closer ruler, and there is a mosaic misplacement on the map of the teapot. The reconstruction images of the holographic element-based EPISM method are shown in Figure 13d,e. As shown in Figure 13d, the figure on the ruler and the map of the teapot are clear when focusing on the farther ruler; the figure on the ruler is clear and the map on the teapot is only blurred when focusing on the closer ruler in Figure 13e.

Figure 14 shows the detail of Figure 13c,e, Figure 14a magnifies part of the Figure 13c, and the mosaic misplacement of the map of the teapot is revealed; Figure 14b magnifies the same part on the Figure 13e, and this part of the surface map of the teapot is blurred without mosaic misplacement. The brightness and contrast of the reconstructed image will be affected by the efficiency of the developer and the slight adjustment of the camera when shooting, but under the same conditions, the effective perspective images segments in EPISM method is smaller than in the proposed method, whether this difference will affect the quality of the reconstruction image still needs further study.

**Figure 13.** Comparison of reconstructed images between the EPISM method and the holographic element-based EPISM method. (**b**,**c**) are the reconstructed images of the EPISM method, (**d**,**e**) are the reconstruction images of the holographic element-based EPISM method. (**a**) The spatial position relation of two rulers and the holographic plate. (**b**) The reconstruction image of the EPISM method when focused on the farther ruler. (**c**) The reconstruction image of the EPISM method when focused on the closer ruler. (**d**) The reconstruction image of the holographic element-based EPISM method when focused on the farther ruler. (**e**) The reconstruction image of the holographic element-based EPISM method when focused on the closer ruler.

**Figure 14.** Comparison of details between reconstructed images of the EPISM method and reconstructed images of the holographic element-based EPISM method (**a**) Details of reconstructed images of the EPISM method (**b**) Details of reconstructed images of the holographic element-based EPISM method.

The experimental results show that the holographic element-based EPISM method can reproduce the 3D scene as well as the EPISM method, and solve the problem of mosaic misplacement in the EPISM method, but there are some restrictions on the EPISM method. The condition *l*<sup>1</sup> = *l*<sup>2</sup> and *L*<sup>1</sup> = *L*<sup>2</sup> must be satisfied so that there will be a hogel on virtual H1 plate corresponding to the area formed by hogel correspondence between H2 plate and LCD panel. In the future, we will consider how to generate effective perspective image segments when we relax this restriction appropriately.
