## Abstract

A three-dimensional optical correlator using a lens array is proposed and demonstrated. The proposed method captures three-dimensional objects using the lens array and transforms them into sub-images. Through successive two-dimensional correlations between the sub-images, a three-dimensional optical correlation is accomplished. As a result, the proposed method is capable of detecting out-of-plane rotations of three-dimensional objects as well as three-dimensional shifts.

© 2005 Optical Society of America

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### Equations (6)

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(1)
$${\theta}_{\mathrm{sub},y-z,i}={\mathrm{tan}}^{-1}\left(\frac{{y}_{i}}{f}\right),$$
(2)
$$\Delta {u}_{r,i,s,i}={u}_{r,i}-{u}_{s,i}=\frac{{y}_{r}-{y}_{s}+\left({z}_{r}-{z}_{s}\right)\mathrm{tan}{\theta}_{\mathrm{sub},y-z,i}}{\phi}.$$
(3)
$$\Delta {u}_{r,i,s,j}=\frac{{y}_{r}-{y}_{s}+{z}_{r}\mathrm{tan}{\theta}_{\mathrm{sub},y-z,i}-{z}_{s}\mathrm{tan}{\theta}_{\mathrm{sub},y-z,j}}{\phi}$$
(4)
$$=\frac{{y}_{r}-{y}_{s}+{z}_{r}\mathrm{tan}{\theta}_{\mathrm{sub},y-z,i}-{z}_{s}\mathrm{tan}\left({\theta}_{\mathrm{sub},y-z,j}+{\theta}_{y-z}\right)}{\phi}.$$
(5)
$$\Delta \theta ={\mathrm{tan}}^{-1}\left(\frac{{y}_{i+1}}{f}\right)-{\mathrm{tan}}^{-1}\left(\frac{{y}_{i}}{f}\right)\approx \frac{{y}_{i+1}-{y}_{i}}{f}=\frac{s}{f},$$
(6)
$$\Omega =2\phantom{\rule{.2em}{0ex}}{\mathrm{tan}}^{-1}\left(\frac{\phi}{2f}\right)\approx \frac{\phi}{f},$$