## Abstract

A novel method of three-dimensional (3-D) object recognition is proposed. Several projections of a 3-D target are recorded under white-light illumination and fused into a single complex two-dimensional function. After proper filtering, the resulting function is coded into a computer-generated hologram. When this hologram is coherently illuminated, a correlation space is reconstructed such that light peaks indicate the existence and locations of true targets in the observed 3-D scene. Experimental results and comparisons with results of another 3-D object recognition technique are presented.

© 2002 Optical Society of America

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### Equations (13)

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(1)
$$({x}_{i},{y}_{i})=M(xcos{\theta}_{i}+zsin{\theta}_{i},y),$$
(2)
$${o}_{3}(u,v)\propto \iint {o}_{2}({x}_{i},{y}_{i},\mathit{au})exp[-j2\pi \times ({\mathit{ux}}_{i}+{\mathit{vy}}_{i})/\mathrm{\lambda}f]\mathrm{d}{x}_{i}\mathrm{d}{y}_{i}.$$
(3)
$${o}_{3}(u,v)\propto \iint {o}_{2}({x}_{i},{y}_{i},\mathit{au})exp[-j2\pi M\times (\mathit{ux}+\mathit{vy}+{\mathit{au}}^{2}z)/\mathrm{\lambda}f]\mathrm{d}{x}_{i}\mathrm{d}{y}_{i}.$$
(4)
$${o}_{3}(u,v)\propto {o}_{1}(x,y,z)exp[-j2\pi M\times (\mathit{ux}+\mathit{vy}+{\mathit{au}}^{2}z)/\mathrm{\lambda}f]\mathrm{\Delta}x\mathrm{\Delta}y\mathrm{\Delta}z.$$
(5)
$${o}_{3}(u,v)\propto \iiint {o}_{1}(x,y,z)exp[-j2\pi M\times (\mathit{ux}+\mathit{vy}+{\mathit{au}}^{2}z)/\mathrm{\lambda}f]\mathrm{d}x\mathrm{d}y\mathrm{d}z.$$
(6)
$$F(u,v)\propto \iiint {f}^{*}(-x,-y,-z)exp[-j2\pi M\times (\mathit{ux}+\mathit{vy}+{\mathit{au}}^{2}z)/\mathrm{\lambda}f]\mathrm{d}x\mathrm{d}y\mathrm{d}z,$$
(7)
$$T(u,v)={o}_{3}(u,v)F(u,v)\propto \iiint {o}_{1}(x,y,z)exp[-j2\pi M\times (\mathit{ux}+\mathit{vy}+{\mathit{au}}^{2}z)/\mathrm{\lambda}f]\mathrm{d}x\mathrm{d}y\mathrm{d}z\times \iiint {f}^{*}(-\xi ,-\eta ,-\zeta )exp[-j2\pi M\times (u\xi +v\eta +{\mathit{au}}^{2}\zeta )/\mathrm{\lambda}f]\mathrm{d}\xi \mathrm{d}\eta \mathrm{d}\zeta =\iiint \iiint {o}_{1}(x,y,z){f}^{*}(-\xi ,-\eta ,-\zeta )\times exp-\{j2\pi M[u(x+\xi )+v(y+\eta )+{\mathit{au}}^{2}(z+\zeta )/\mathrm{\lambda}f]\}\mathrm{d}x\mathrm{d}y\mathrm{d}z\mathrm{d}\xi \mathrm{d}\eta \mathrm{d}\zeta =\iiint \iiint {o}_{1}(x,y,z){f}^{*}(x-\stackrel{\u02c6}{x},y-\stackrel{\u02c6}{y},z-\stackrel{\u02c6}{z})\mathrm{d}x\mathrm{d}y\mathrm{d}z\times exp[-j2\pi M(\stackrel{\u02c6}{\mathit{ux}}+\stackrel{\u02c6}{\mathit{vy}}+{\mathit{au}}^{2}\stackrel{\u02c6}{z})/\mathrm{\lambda}f]\mathrm{d}\stackrel{\u02c6}{x}\mathrm{d}\stackrel{\u02c6}{y}\mathrm{d}\stackrel{\u02c6}{z}=\iiint g(\stackrel{\u02c6}{x},\stackrel{\u02c6}{y},\stackrel{\u02c6}{z})exp[-j2\pi M(\stackrel{\u02c6}{\mathit{ux}}+\stackrel{\u02c6}{\mathit{vy}}+{\mathit{au}}^{2}\stackrel{\u02c6}{z})/\mathrm{\lambda}f]\mathrm{d}\stackrel{\u02c6}{x}\mathrm{d}\stackrel{\u02c6}{y}\mathrm{d}\stackrel{\u02c6}{z},$$
(8)
$$g(\stackrel{\u02c6}{x},\stackrel{\u02c6}{y},\stackrel{\u02c6}{z})=\iiint {o}_{1}(x,y,z){f}^{*}(x-\stackrel{\u02c6}{x},y-\stackrel{\u02c6}{y},z-\stackrel{\u02c6}{z})\mathrm{d}x\mathrm{d}y\mathrm{d}z,$$
(9)
$$\stackrel{\u02c6}{x}=x+\xi ,\hspace{1em}\hspace{1em}\stackrel{\u02c6}{y}=y+\eta ,\hspace{1em}\hspace{1em}\stackrel{\u02c6}{z}=z+\zeta .$$
(10)
$$\tilde{T}(u,v)=\iiint g(\stackrel{\u02c6}{x},\stackrel{\u02c6}{y},\stackrel{\u02c6}{z})exp\{-j2\pi M\times [\stackrel{\u02c6}{\mathit{ux}}+\stackrel{\u02c6}{\mathit{vy}}+a({u}^{2}+{v}^{2})\stackrel{\u02c6}{z}]/\mathrm{\lambda}f\}\mathrm{d}\stackrel{\u02c6}{x}\mathrm{d}\stackrel{\u02c6}{y}\mathrm{d}\stackrel{\u02c6}{z}.$$
(11)
$${T}_{r}(u,v)=0.5\left(1+\mathrm{Re}\left\{T(u,v)exp\left[\frac{j2\pi}{\mathrm{\lambda}f}({\mathrm{d}}_{x}u+{\mathrm{d}}_{y}v)\right]\right\}\right),$$
(12)
$$\mathrm{SNR}=\frac{\mathrm{maximum}\hspace{0.5em}\mathrm{correlation}\hspace{0.5em}\mathrm{peak}\hspace{0.5em}\mathrm{intensity}\hspace{0.5em}\mathrm{of}\hspace{0.5em}\mathrm{the}\hspace{0.5em}\mathrm{true}\hspace{0.5em}\mathrm{target}}{\mathrm{maximum}\hspace{0.5em}\mathrm{noise}\hspace{0.5em}\mathrm{intensity}}.$$
(13)
$$\mathrm{PCE}=\frac{\mathrm{maximum}\hspace{0.5em}\mathrm{correlation}\hspace{0.5em}\mathrm{peak}\hspace{0.5em}\mathrm{intensity}\hspace{0.5em}\mathrm{of}\hspace{0.5em}\mathrm{the}\hspace{0.5em}\mathrm{true}\hspace{0.5em}\mathrm{target}}{\mathrm{average}\hspace{0.5em}\mathrm{correlation}\hspace{0.5em}\mathrm{plane}\hspace{0.5em}\mathrm{energy}},$$