1. Introduction

Holographic display [1], regarded as the ultimate 3D display technology, has attracted more and more attentions in recent years. Computer generated hologram (CGH) as the digital method has been considered to be the most promising method to realize the real-time 3D display. There are two mainstream approaches for the CGH computation: one is the point-based method [2–9 ], and the other is the polygon-based method [10–14 ]. Since the point-based method treats the 3D object as a set of points and the propagation is based on distribution of spherical wavefront, it is simple and flexible. However, the big challenge for CGH technique by employing point based method is the large computational complexity.

The computational complexity reduction of the point-based method is widely investigated in recent years. Coherent ray trace (CRT) [2] method is a simple and common method to generate CGH, where the reconstructed images are high quality, but the calculation is quite slow because of point-to-point inline computation. Look up table (LUT) [3] method has been proposed to speed up the calculation, where all the possible CGH patterns of points are pre-computed and stored in a table. When inline computation, generating hologram by reading out the CGH patterns from the table. Though it is faster than CRT, LUT requires huge memory to store the pre-computed CGH patterns. Novel look up table (N-LUT) [4] and split look up table (S-LUT) [5] method have been proposed to reduce the memory usage of LUT, where the CGH pattern of the center point or the light modulation factors is pre-computed and stored in the table for each 2D image plane with different depth, but they still need gigabytes (GBs) of memory because 3D image is composed of large amount of 2D image planes. Therefore, generating the table and reading out the data from the table still cost plenty of computational time. Compressed look up table (C-LUT) [6] method has been developed to reduce the large memory usage of S-LUT, where the CGH of the horizontal and vertical light modulation factors are pre-computed and stored in only one two dimensional (2D) image plane in the table. However, this method is based on an approximation where the size of the reconstructed image is much smaller than the distance between the object and the hologram, which will cause the distortion of the reconstructed image, and the distortion increases with the large object depth. That could greatly affect the quality of 3D reconstructed image. For 3D display, when the reconstructed images match within the resolution of human eyes, that is, human eyes cannot tell the difference between the reconstructed images and prototype, we call it ‘relative accurate’ reconstruction.

In holographic display, the ultimate goal is to realize the dynamic 3D holographic display. Therefore, the method used to generate holograms must has little memory usage and fast speed. For LUT and N-LUT method, they either require large memory usage or have slowly computational speed and cannot match with S-LUT and C-LUT method. S-LUT and C-LUT method are just a little bit more appropriate to realize the dynamic 3D holographic display than CRT, LUT or N-LUT method, and they still have problems, like memory usage, computational speed, and reconstructed image quality and so on.

In this paper, we proposed a simple computation method based on Fresnel diffraction and look up table to reduce the large memory usage of S-LUT and alleviate the distortion of C-LUT without sacrificing the computational speed. Compared with S-LUT and C-LUT method, we can get accurate reconstructed images with lower memory usage, so we called our method as accurate compressed look up table method (AC-LUT). Numerical simulations and optical experiments are performed, and they are in nice agreement. The advantage of our proposed method is demonstrated by the comparison with various related methods.

2. New method based on AC-LUT

In conventional CRT method, a 3D object is considered as being consisted of many point sources, and field distribution in the hologram after all point sources transmitting the free space can be written as [15,16 ]:

In Fresnel region [17], Eq. (1) can be written as

Using complex form, it is described as

In mathematics, $\exp (iab)=[exp(a)].^(ib),a\in R^{m\times n},m\in Z^{+},n\in Z^{+},b\in R$.

So we can write Eq. (3) in a simple form as

Now define$H(x_h,x_j)=\exp (\frac{{(x_h-x_j)}^2}{2})$as the horizontal light modulation factor,$V(y_h,y_j)=\exp (\frac{{(y_h-y_j)}^2}{2})$ as the vertical light modulation factor, and $L_1(z_j)=\exp \left[ik(d-z_j)\right]$, $L_2(z_j)=\frac{ik}{d-z_j}$as the longitudinal light modulation factors. Here, $H(x_h,x_j)$and $V(y_h,y_j)$ are real numbers.

Then Eq. (4) can be written as

For $N_{xy}$object points falling on the same layer of the 3D object, they have the same longitudinal light modulation factors$L_1(z_j)$and$L_2(z_j)$. So Eq. (5) can be written as

For $N_x$object points falling on the same vertical line of each 2D image plane, they have the same vertical light modulation factor$V(y_h,y_j)$, so Eq. (6) equals

Then our method can be divided into two steps.

The first step is to calculate the basic light modulation factors$H(x_h,x_j)$and $V(y_h,y_j)$, and store them in table during offline time (see Table 1 ).

Table 1. Offline pre-computation.

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The second step is to read out the light modulation factors from table and generate the hologram in inline computation (see Table 2 ).

Table 2. Inline computation.

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According to the computation program, the number of the loop to compute the light modulation factors$H(x_h,x_j)$and $V(y_h,y_j)$ is$N_x\times p+N_y\times q$. And the total size of table is$(N_x\times p+N_y\times q)\times M$, where$M$denotes the memory usage of one pixel of the CGH pattern in offline table. The light modulation factors are real numbers, so here ‘M’ only denotes the memory usage of real numbers. The number of the loop about inline computation is$N_z\times (N_y\times (N_x\times p+p\times q)+p\times q)$. The number of the loop for computing a hologram on inline computation or building a table on offline computation is called the computational complexity.

We compare the computational complexity and the memory usage among the S-LUT, C-LUT and AC-LUT method, as listed in Table 3 .

Table 3. Comparison among AC-LUT, S-LUT and C-LUT

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From Table 3 one can see that the three methods have the same inline computational complexity. And AC-LUT method has the lowest memory usage among the three method. Because the light modulation factors$H(x_h,x_j)$and $V(y_h,y_j)$ in AC-LUT are real numbers, the offline calculation of AC-LUT is simpler than C-LUT and S-LUT, and the time costs of AC-LUT is less than S-LUT and C-LUT to build the table, as shown in Fig. 1 . Our program is run by a computer with CPU of 2.6 GHz clock frequency under MATLAB. The parameters used in comparison are listed in Table 4 .

 

Fig. 1 Comparison of offline calculation time among AC-LUT, C-LUT, S-LUT method.

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Table 4. CGH computation parameters

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Figure 1 shows that the AC-LUT method has the least time costs for building the table among the three methods and keeps unchanged when the object depth layers increase. One can also see that the time cost has a linear relationship with the object depth layers in S-LUT method. For AC-LUT method, the offline calculation time is 0.2 seconds, about 3800 times faster than S-LUT method and two times faster than C-LUT method to build the table including $200\times 200\times 1000$points.

The comparison of inline computational speed among these three methods is shown in Fig. 2 . Here the 3D objects we used in inline calculation consist of depth map and intensity map, so the object depth layers equal 256. And for S-LUT, the table has 256 layers.

 

Fig. 2 Comparison of inline computational speed among AC-LUT, C-LUT and S-LUT method.

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From Fig. 2 it can be seen that the new method, AC-LUT, has a faster computational speed than S-LUT method and a bit slower than C-LUT method. But large memory usage problem for S-LUT method or the distortion problem for C-LUT method is very critical and even fatal. Table 5 and Fig. 3 have a detailed description about the problems of S-LUT and C-LUT method.

Table 5. Memory usage and reconstructed image distortion

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Fig. 3 (a) Memory usage among AC-LUT, C-LUT and S-LUT (b) distortion among AC-LUT, C-LUT and S-LUT.

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From Table 5 one can see that the AC-LUT method has a lowest memory usage among these three methods and keeps unchanged for different object depth layers. For S-LUT method, the memory usage increases with the number of object depth layers. For C-LUT method, the distortion increases with the number of object depth layers though who has a lower memory usage compared with S-LUT, where the distortion cannot be ignored compared with the acuity of the human visual system 0.06 mm [6].

From Fig. 3 one can see that for S-LUT, whose memory usage increases linearly as the object depth layers increased. For C-LUT, reconstructed image distortion increases linearly with the increase of the object depth layers, and for S-LUT method, the distortion in close to zero, but they use the approximation that $\text{2}\pi {\text{z}}_j/\lambda$ is multiple of $2\pi$, which may be not accurate and can also cause large distortion in certain conditions [5]. For our new method AC-LUT, the memory usage keeps unchanged without distortion.

To better illustrate the comparison of distortion, simulation and optical experiments are performed. The parameters we used are listed in Table 4. The results are shown in Fig. 4 .

 

Fig. 4 3D scenes reconstruction results, where (a) is the model, (b), (c),(f) and (g)are simulation and experimental results using AC-LUT method; (d), (e), (h) and (i) are simulation and experimental results using C-LUT method. Here, (b), (d), (f) and (h) are focused on 500mm, (c), (e), (g) and (i) are focused on 600mm.

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As shown in Fig. 4, Fig. 4 a) is the model, where the white line in the model on the same height and the two rhombus are same too. The size of rhomb is 7.15 mm. Figs. 4(b), 4(c), 4(f), and 4(g) are simulation and experimental results using AC-LUT method; Figs. 4(d), 4(e), 4(h), and 4(i) are simulation and experimental results using C-LUT method. Here, Figs. 4(b), 4(d), 4(f), and 4(h) are focused on 500mm, Figs. 4(c), 4(e), 4(g), and 4(i) are focused on 600mm.

In C-LUT paper [6], the authors give two distortion formulas$(x_j,y_j,z_j)$and $(x_j,y_j,z_j)$, where the $(x_j,y_j,z_j)$is the object point $j$’s coordinates and$d$is the distance between the 3D object and hologram plane. From the formulas, we can get that the C-LUT method has no distortion only when or and .

The reconstructed image is projected by the reflective phase-only SLM (HOLOEYE LETO), where the SLM device has 1920 × 1080 pixels, its pixels size is 6.4μm.The CGH type we used is phase-only CGH. And the image is captured by a CCD (Lumenera camera INFINITY 4-11C). In the optical experiments, the zero-order beam elimination method [18] and the multiplexing encoding method [19] are adopted to improve the image quality of the 3D reconstructed image. Because CCD can only capture 2D image, the images on different distances are focused on different 2D image planes that is when one is clear, the other is blurred.

From Fig. 4 we can see that the size of the reconstructed images using AC-LUT method is 7.15 mm without distortion. And for C-LUT, the distortion is obvious that can be seen from Figs. 4(d) and 4(e). Using the two distortion formulas, we can get that the distortion is mm and mm, where the mm, = 500mm, mm, mm. So, for direction or direction, the total distortion of C-LUT is 2 mm in this condition.

3. Numerical and optical experimental results

Numerical simulations and optical experiments are performed to demonstrate the feasibility of AC-LUT method. The parameters we used are listed in Table 4. The results are shown in the next.

We reconstruct 3D objects located at different positions, as shown in Fig. 5 and Fig. 6 . In Fig. 5, the distance between each objects is 50mm. The characters are reconstructed on the different planes, as shown in Figs. 5(a)-5(f), where Figs. 5(a), 5(b), and 5(c) are the numerical simulation results and Figs. 5(d), 5(e), and 5(f) are the optical experimental results. In Fig. 6, the distance between the two objects is 100mm. The objects are reconstructed on the different planes, as shown in Figs. 6(a)-6(d), where Figs. 6(a) and 6(b) are the numerical simulation results and Figs. 6(c) and 6(d) are the optical experimental results.

 

Fig. 5 3D scenes reconstruction results, where (a),(b)and (c) are simulation results while (d),(e) and (f) are experimental results.(a) and (d) are focused on 500mm,(b) and (e) are focused on 550mm, (c) and (f) are focused on 600mm.

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Fig. 6 3D scenes reconstruction results, where (a) and (b) are simulation results while (c) and (d) are experimental results.(a) and (c) are focused on 400mm,(b) and (d) are focused on 500 mm.

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Figs. 5 and 6 show that our proposed method can achieve 3D image with long-depth cue and without the distortion. And the simulation results match well with the optical experimental results.

Figure 7 shows the other 3D scenes reconstruction results used our method. The prototype of 3D image is a horse with two stars and a moon. Figure 7(a) is the numerical simulation result and Fig. 7(b) is the optical experimental result. From Fig. 7 one can see that the reconstructed image matches well with the prototype when the prototype is a complex objects.

 

Fig. 7 3D scenes reconstruction results (a) simulation result (b) experimental result.

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4. Conclusion

We have developed an AC-LUT computation method based on Fresnel diffraction theory and LUT to reduce the memory usage of S-LUT and alleviate the distortion of C-LUT without sacrificing the computational speed. Numerical simulations and optical experiments are performed to verify this method, and the results match well with each other. It is believed AC-LUT method is a promising method for realizing dynamic 3D holographic display with the low memory usage, high speed, and high reconstructed image quality. This method can also be used in the calculation of various digitalized optical transmission where the speed and the accuracy are required at the same time.