## Abstract

We propose a method that achieves efficient texture mapping in fully-analytic computer generated holograms based on triangular meshes. In computer graphics, the texture mapping is commonly used to represent the details of objects without increasing the number of the triangular meshes. In fully-analytic triangular-mesh-based computer generated holograms, however, those methods cannot be directly applied because each mesh cannot have arbitrary amplitude distribution inside the triangular mesh area in order to keep the analytic representation. In this paper, we propose an efficient texture mapping method for fully-analytic mesh-based computer generated hologram. The proposed method uses an adaptive triangular mesh division to minimize the increase of the number of the triangular meshes for the given texture image data. The geometrical similarity relationship between the original triangular mesh and the divided one is also exploited to obtain the angular spectrum of the divided mesh from pre-calculated data for the original one. As a result, the proposed method enables to obtain the computer generated hologram of high details with much smaller computation time in comparison with the brute-force approach. The feasibility of the proposed method is confirmed by simulations and optical experiments.

© 2016 Optical Society of America

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### Equations (10)

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(1)
$$\text{u}=\text{0}\text{.5}+\frac{\phi}{\text{2}\pi}\text{,}\text{v}=\text{0}\text{.5}+\frac{\theta}{\pi},$$
(2)
$$U\left({r}_{x,y}\right)\text{=}{\displaystyle \int {\displaystyle {\int}_{-\infty}^{\infty}G\left({f}_{x,y}^{}\right)\mathrm{exp}\left[j2\pi {f}_{x,y}^{T}{r}_{x,y}\right]}d{f}_{x,y}},$$
(3)
$$G\left({f}_{x,y}\right)\text{=}{\text{G}}_{l}\left({f}_{xl,yl}\right)\text{exp}\left[j2\pi {f}_{xl,yl,zl}^{T}c\right]\frac{{f}_{zl}}{{f}_{z}},$$
(4)
$${G}_{l}\left({f}_{xl,yl}\right)\text{=}\frac{{G}_{o}\left({A}^{-T}\left\{{f}_{xl,yl}-\frac{1}{\lambda}\left[\begin{array}{c}{u}_{xl}^{T}\\ {u}_{yl}^{T}\end{array}\right]{u}_{c}\right\}\right)}{\text{det}(A)}=\frac{{G}_{o}\left({A}^{-T}{{f}^{\prime}}_{xl,yl}\right)}{\text{det}(A)}=\frac{{G}_{o}\left({{f}^{\prime}}_{xr,yr}\right)}{\text{det}(A)},$$
(5)
$${g}_{l}^{(n)}\left({r}_{xl,yl}\right)\text{=}{g}_{l,bare}\left({\left(-1\right)}^{k}{2}^{n}\left({r}_{xl,yl}-{r}_{gl}^{(n)}\right)+{r}_{gl}\right)\mathrm{exp}\left\{j\frac{2\pi}{\lambda}{\left(\left[\begin{array}{c}{u}_{xl}^{T}\\ {u}_{yl}^{T}\end{array}\right]{u}_{c}\right)}^{T}{r}_{xl,yl}\right\},$$
(6)
$${G}_{l}^{(n)}\left({f}_{xl,yl}\right)\text{=}\frac{1}{{4}^{n}}\mathrm{exp}\left[-j2\pi {{f}^{\prime}}_{xl,yl}^{T}\left({r}_{gl}^{(n)}-\frac{{r}_{gl}}{{\left(-1\right)}^{k}{2}^{n}}\right)\right]{G}_{l,bare}\left(\frac{{{f}^{\prime}}_{xl,yl}^{}}{{\left(-1\right)}^{k}{2}^{n}}\right),$$
(7)
$${G}_{l}^{(n)}\left({f}_{xl,yl}\right)\text{=}\frac{1}{{4}^{n}}\mathrm{exp}\left[-j2\pi {{f}^{\prime}}_{xl,yl}^{T}\left({r}_{gl}^{(n)}-\frac{{r}_{gl}}{{\left(-1\right)}^{k}{2}^{n}}\right)\right]\frac{{G}_{o}\left(\frac{1}{{\left(-1\right)}^{k}{2}^{n}}{{f}^{\prime}}_{xr,yr}^{}\right)}{\mathrm{det}\left(A\right)},$$
(8)
$$\begin{array}{l}{G}_{o}^{}\left({{f}^{\prime}}_{xr,yr}^{}\right)\text{=}\frac{\mathrm{exp}\left[-j2\pi {{f}^{\prime}}_{xr}\right]-1}{{\left(2\pi \right)}^{2}{{f}^{\prime}}_{xr}{{f}^{\prime}}_{yr}}+\frac{1-\mathrm{exp}\left[-j2\pi \left({{f}^{\prime}}_{xr}+{{f}^{\prime}}_{yr}\right)\right]}{{\left(2\pi \right)}^{2}{{f}^{\prime}}_{yr}\left({{f}^{\prime}}_{xr}+{{f}^{\prime}}_{yr}\right)}\\ \text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{1em}}=\left({T}_{2}-{T}_{1}\right)+{T}_{1}\mathrm{exp}\left(-j{T}_{3}\right)-{T}_{2}\mathrm{exp}\left(-j{T}_{4}\right),\end{array}$$
(9)
$$\begin{array}{l}{T}_{1}=\frac{1}{{\left(2\pi \right)}^{2}{{f}^{\prime}}_{xr}{{f}^{\prime}}_{yr}},\text{\hspace{1em}}{T}_{2}=\frac{1}{{\left(2\pi \right)}^{2}{{f}^{\prime}}_{yr}\left({{f}^{\prime}}_{xr}+{{f}^{\prime}}_{yr}\right)},\\ {T}_{3}=2\pi {{f}^{\prime}}_{xr},\text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{T}_{4}=2\pi \left({{f}^{\prime}}_{xr}+{{f}^{\prime}}_{yr}\right).\end{array}$$
(10)
$$\begin{array}{l}{G}_{l}^{(n)}\left({f}_{xl,yl}\right)\text{=}\frac{1}{\mathrm{det}\left(A\right)}\mathrm{exp}\left[-j2\pi {{f}^{\prime}}_{xl,yl}^{T}\left({r}_{gl}^{(n)}-\frac{{r}_{gl}}{{\left(-1\right)}^{k}{2}^{n}}\right)\right]\\ \text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{1em}}\times \left\{\left({T}_{2}-{T}_{1}\right)+{T}_{1}\mathrm{exp}\left[{(-1)}^{k+1}j\frac{{T}_{3}}{{2}^{n}}\right]-{T}_{2}\mathrm{exp}\left({(-1)}^{k+1}j\frac{{T}_{4}}{{2}^{n}}\right)\right\}.\end{array}$$