1. Introduction

Past research has demonstrated that the Fresnel hologram of a three-dimensional scene can be generated numerically by computing the fringe patterns emerged from each object point to the hologram plane. In brief, given a scene is of self-illuminating object points$O=\left[o_0\left(x_0,y_0,z_0\right),o_1\left(x_1,y_1,z_1\right),\mathrm{.....},o_{N-1}\left(x_{N-1},y_{N-1},z_{N-1}\right)\right]$,the diffraction pattern D(x,y) on the hologram plane can be derived as

2. Background of the wavefront-recording plane (WRP) method

For clarity of explanation, a brief outline of the method in [13] is summarized in this section. To begin with, the following terminology is adopted. The hologram $u\left(x,y\right)$ is a vertical, 2D image that is positioned at the origin. A virtual WRP, $u_w\left(x,y\right)$, is placed at a depth $z_w$ from $u\left(x,y\right)$. The object scene is composed of a set of self-illuminating pixels, each having an intensity value of $a_j$ and located at a perpendicular distance of $d_j$from the WRP. Without loss of generality we assume that both the hologram, the WRP, and the object scene have the same horizontal and vertical extents of X and Y pixels, respectively, as well as identical sampling pitch p. The hologram generation process can be divided into two stages. In the first stage, the complex wavefront contributed by the object points is computed as

In Eq. (3), the computation of the WRP for each object point is only confined to the region of the virtual window on the WRP. As W is much smaller than X and Y, the computation load is significantly reduced as compared with Eq. (2). In [13], the calculation is further simplified by pre-computing the exponential terms for all combinations of $\left(x_j,y_j,d_j\right)$, and the estimated computational amount is $2\alpha N{\overline{L}}^2$. $\overline{L}$ is the mean perpendicular distance of the object points to the WRP, and α is the arithmetic operations involved in computing the wavefront contributed by each object point. In the second stage, the WRP is expanded to the hologram as

3. Proposed computational-free interpolated wavefront-recording plane (IWRP) method

Our proposed method is described as follows. First, we note that the resolution of the scene image is generally smaller than that of the hologram. Hence, it is unnecessary to convert every object point of the scene to its wavefront on the WRP. On this basis, we propose to sub-sample the scene image evenly by M times (where $M>0$) along the horizontal and the vertical directions. Let $I\left(m,n\right)$ and $d\left(m,n\right)$ represent the intensity and distance from the WRP of the sample object point located at the $nth$ row and the $mth$ column of the object scene. With the sample pitch set to $Mp$, the physical horizontal and vertical positions of the sample point are located at $x_m=mMp+Mp/2$ and $y_n=nMp+Mp/2$, respectively, where m and n are integer values. A square support, as shown in Fig. 1a , is defined for each sample point, with the left side $l_m$, and the right side $r_m$ given by$l_m=x_m-Mp/2$, and $r_m=x_m+\left(M-1\right)p/2$. Similarly, the bottom side $t_n$ and top side $b_n$ of the square support are given by $b_n=y_n-Mp/2$, and $t_n=y_n+\left(M-1\right)p/2$. We point out that the square supports of adjacent sample points are non-overlapping and just touching each other at their boundaries. Next, we assume the contribution of each sample point is contributing to a square virtual window in the WRP with side length equals to $Mp$ as shown in Fig. 1b.

 

Fig. 1 a. Sampling lattice and square support Fig. 1b. A pair of sample points, each associate with a square support and a virtual window on the scene image and the WRP, respectively. Note that the depth (i.e., z position) of the object points are not shown in the diagram.

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The virtual window is aligned with the square support of the object point, and the wavefront within the virtual window is only contributed by the object point in the square support. Under this approximation, Eq. (2) therefore can be re-written as

It can be inferred from Eq. (6) that the function $G\left(x-x_m,y-x_n,I\left(m,n\right),d\left(m,n\right)\right)$ represents a Fresnel zone plate $G\left(x,y,I\left(m,n\right),d\left(m,n\right)\right)$, within the virtual window, which is shifted to the position $\left(x_m,y_n\right)$. The Fresnel zone plate is contributed by an object point of intensity $I\left(m,n\right)$ and at distance $d\left(m,n\right)$ from the wavefront plane. Hence for a finite variations of $d\left(m,n\right)$ and$I\left(m,n\right)$, all the possible combinations of $G\left(x,y,I\left(m,n\right),d\left(m,n\right)\right)$ can be pre-computed in advance, and store in a look up table (LUT). For example, if the depth $d\left(m,n\right)$ and $I\left(m,n\right)$ are quantized into $N_d$ and $N_I$ levels, respectively, there will be a total of $N_d\times N_I$ combinations. As a result, in the generation of $u_w\left(x,y\right)$, each of its constituting virtual window can be retrieved from the corresponding entries in the LUT. In another words, the process is computational-free.

Although the decimated has effectively reduced the computation time, as will be shown later, the reconstructed images obtained with the WRP derived from Eq. (6) are weak, noisy, and difficult to observe. This is caused by the sparse distribution of the object points caused by the sub-sampling of the scene image. To overcome this problem, we propose the interpolated WRP (IWRP) to interpolate the associated support of each object point with padding, i.e., the object point is duplicated to all the pixels within each square support. After the interpolation, the wavefront of a virtual window will be contributed by all the object points (which are identical in intensity and depth) within the support as given by

Similar to Eq. (6), the wavefront function $=G_A\left(x-x_m,y-y_n,I\left(m,n\right),d\left(m,n\right)\right)$ in Eq. (8) is simply a shifted version of $G_A\left(x,y,I\left(m,n\right),d\left(m,n\right)\right)$ which can be pre-computed and stored in a LUT for different combinations of $I\left(m,n\right)$ and$d\left(m,n\right)$. Consequently, each virtual window in the IWRP can be generated in a computation-free manner by retrieving, from the LUT, the wavefront corresponding to the intensity and depth of the corresponding object point. Comparing Eq. (6) and Eq. (8), it can also be inferred that the number of combination on the values of $G\left(x,y,I\left(m,n\right),d\left(m,n\right)\right)$ and $G_A\left(x,y,I\left(m,n\right),d\left(m,n\right)\right)$, and hence the size of the corresponding LUTs, are identical. After $u_w\left(x,y\right)|{}_{l_m\le x<r_m,t_n\le y<b_n}$ is generated within the IWRP, Eq. (4) is applied to generate the hologram $u\left(x,y\right)$.

4. Experimental results

Our proposed method is evaluated with the test image Fig. 2a . The horizontal and vertical extents of the hologram, the IWRP, and the test image are identical, comprising of 2048 by 2048 square pixels each with a size of 9 by 9 um and quantized with 8bits. The test image is divided into a left and a right part located at distances of $z_1=0.005m$ and $z_s=0.01m$ from the IWRP. Each pixel in the image is taken to generate the hologram, constituting to a total of around $4\times {10}^6$ object points. λ and the distance z_w between the WRP/IWRP and the hologram are set to 650nm and 0.4m, respectively. $N_d$ and $N_d$ are both set to 256, and $M=8$, resulting in a LUT of around 4.1Mb. We decimated the source image by 8 times in the horizontal and vertical direction (i.e., M = 8 and size of virtual window = 8x8), and applied Eqs. (5) and 6 to derive a WRP. The latter is then expand it into a hologram with Eq. (4). A real, off-axis hologram $H\left(x,y\right)$ is generated by adding a planar reference wave $R\left(y\right)$ (illuminating at an inclined angle ${1.2}^o$ on the hologram) to $u\left(x,y\right)$, and taking the real part of the result as given by

 

Fig. 2 a. An image divided (dotted line) into a left and right parts, positioned at z₁ = 0.005m (left) and z₂ = 0.01m (right) from the IWRP, respectively. The IWRP is positioned at z_w = 0.4m from the hologram.Fig. 2b. Optical reconstructed image of the upper part of the hologram corresponding to the WRP derived from Eqs. (5) and 6. Fig. 2c. Optical reconstructed image of the lower part of the hologram corresponding to the WRP derived from Eqs. (5) and 6. Fig. 2d. Optical reconstructed image of the upper part of the hologram corresponding to the proposed IWRP derived from Eqs. (7) and 8. Fig. 2e. Optical reconstructed image of the lower part of the hologram corresponding to the proposed IWRP derived from Eqs. (7) and 8. Fig. 2f. Single-frame excerpts from the optical reconstructed animation clip of the hologram sequence representing a rotating earth globe located at 0.3m from the hologram. The hologram sequence is generated from the IWRP derived from Eqs. (7) and (8) (Media 1).

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Table 1. Computation Time and Equivalent Frame Rate in Generating the IWRP, and Expanding the Latter to a Fresnel Hologram

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5. Conclusion

In this paper, we propose a method for real-time generation of Fresnel holograms. A proposed interpolated wavefront recording plane (IWRP) is first constructed with a computation-free process. Subsequently, the IWRP is expanded into a Fresnel hologram via a pair of fast Fourier transform operations that are realized with the GPU. Based on our method a hologram size of 2048x2048, representing an image scene comprising of over $4\times {10}^6$ points, can be generated in less than 25ms, equivalent to 40 frames per second. These results correspond to state-of-the-art speed in the calculation of CGH.