## Abstract

Several approaches for increasing the speed in computation of the digital holograms of three-dimensional objects have been presented with applications to real-time display of holographic images. Among them, a look-up table (LUT) approach, in which the precalculated principal fringe patterns for all possible image points of the object are provided, has gained a large speed increase in generation of computer-generated holograms. But the greatest drawback of this method is the enormous memory size of the LUT. A novel approach to dramatically reduce the size of the conventional LUT, still keeping its advantage of fast computational speed, is proposed, which is called here a novel LUT (N-LUT) method. A three-dimensional object can be treated as a set of image planes discretely sliced in the *z* direction, in which each image plane having a fixed depth is approximated as some collection of self-luminous object points of light. In the proposed method, only the fringe patterns of the center points on each image plane are precalculated, called principal fringe patterns (PFPs) and stored in the LUT. Then, the fringe patterns for other object points on each image plane can be obtained by simply shifting this precalculated PFP according to the displaced values from the center to those points and adding them together. Some experimental results reveal that the computational speed and the required memory size of the proposed approach are found to be 69.5 times faster than that of the ray-tracing method and 744 times smaller than that of the conventional LUT method, respectively.

© 2008 Optical Society of America

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### Equations (11)

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(1)
$$O(x,y)={\displaystyle \sum _{p=1}^{N}\frac{{a}_{p}}{{r}_{p}}\text{\hspace{0.17em}}\mathrm{exp}\left[j\right(k{r}_{p}+{\phi}_{p}\left)\right]},$$
(2)
$${r}_{p}=\sqrt{{(x-{x}_{p})}^{2}+{(y-{y}_{p})}^{2}+{z}_{p}^{2}}\text{.}$$
(3)
$$R(x,y)={a}_{R}\text{\hspace{0.17em}}\mathrm{exp}\left[j\right(kx\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}{\theta}_{R}\left)\right]\text{.}$$
(4)
$$I(x,y)={\left|R\right(x,y)+O(x,y\left)\right|}^{2}={\left|R\right(x,y\left)\right|}^{2}+{\left|O\right(x,y\left)\right|}^{2}+2\left|R\right(x,y\left)\right|\left|O\right(x,y\left)\right|\times \mathrm{cos}[k{r}_{p}+kx\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}{\theta}_{R}+{\phi}_{p}]\mathrm{.}$$
(5)
$$I(x,y)=2{\displaystyle \sum _{p=1}^{N}\frac{{a}_{p}}{{r}_{p}}\text{\hspace{0.17em}}\mathrm{cos}(k{r}_{p}+kx\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}{\theta}_{R}+{\phi}_{p})}\mathrm{.}$$
(6)
$$T(x,y;\text{\hspace{0.17em}}{x}_{p},{y}_{p},{z}_{p})\equiv \frac{1}{{r}_{p}}\text{\hspace{0.17em}}\mathrm{cos}\left[k{r}_{p}+kx\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}{\theta}_{R}+{\phi}_{p}\right]\mathrm{.}$$
(7)
$$I(x,y)={\displaystyle \sum _{p=1}^{N}{a}_{p}T\left(x,y;\text{\hspace{0.17em}}{x}_{p},{y}_{p},{z}_{p}\right)}\mathrm{.}$$
(8)
$$T(x,y;\text{\hspace{0.17em}}{z}_{p})\equiv \frac{1}{{r}_{p}}\text{\hspace{0.17em}}\mathrm{cos}\left[k{r}_{p}+kx\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}{\theta}_{R}+{\phi}_{p}\right]\mathrm{.}$$
(9)
$$I(x,y)={\displaystyle \sum _{p=1}^{N}{a}_{p}T\left(x-{x}_{p},y-{y}_{p};\text{\hspace{0.17em}}{z}_{p}\right)}\text{,}$$
(10)
$$\mathrm{H}\mathrm{o}\mathrm{r}\mathrm{i}\mathrm{z}\mathrm{o}\mathrm{n}\mathrm{t}\mathrm{a}\mathrm{l}\text{\hspace{0.17em} \hspace{0.17em} \hspace{0.17em}}\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{o}\mathrm{l}\mathrm{u}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\text{\hspace{0.17em} \hspace{0.17em}}\mathrm{o}\mathrm{f}\text{\hspace{0.17em} \hspace{0.17em}}\mathrm{t}\mathrm{h}\mathrm{e}\text{\hspace{0.17em} \hspace{0.17em}}\mathrm{P}\mathrm{F}\mathrm{P}:\text{\hspace{0.17em}}[{h}_{x}+(\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{k}\times {O}_{x})]\text{,}$$
(11)
$$\mathrm{V}\mathrm{e}\mathrm{r}\mathrm{t}\mathrm{i}\mathrm{c}\mathrm{a}\mathrm{l}\text{\hspace{0.17em} \hspace{0.17em}}\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{o}\mathrm{l}\mathrm{u}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\text{\hspace{0.17em} \hspace{0.17em}}\mathrm{o}\mathrm{f}\text{\hspace{0.17em} \hspace{0.17em}}\mathrm{t}\mathrm{h}\mathrm{e}\text{\hspace{0.17em} \hspace{0.17em}}\mathrm{P}\mathrm{F}\mathrm{P}:\text{\hspace{0.17em}}[{h}_{y}+(\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{k}\times {O}_{y})]\text{,}$$