## Abstract

Generally, the reconstruction of an object image from its diffraction field requires both the amplitude and the phase information of this field. We systematically investigated the effects of using only the real part, the imaginary part, or the phase information of the diffraction field to reconstruct the original image for both the binary and the gray-level images. We show that the phase information can yield a better result of image retrieval than the real or imaginary part and that the recovered image from the phase information is satisfactory especially for binary input. On the basis of this idea, a new technique of image encryption and watermarking by use of only one delivered image—the phase map of the diffraction field of the original image—through double random-phase encoding is proposed and verified by computer simulations with phase-shifting interferometry. This method can greatly cut down the communication load and is suitable for Internet transmission.

© 2005 Optical Society of America

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### Equations (10)

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(1)
$${U}_{2}({x}_{2},{y}_{2})=\frac{exp(ik{d}_{1})}{i\mathrm{\lambda}{d}_{1}}\mathit{\iint}t({x}_{1},{y}_{1})exp[i2\mathrm{\pi}p({x}_{1},{y}_{1})]\times exp\left\{\frac{i\mathrm{\pi}}{\mathrm{\lambda}{d}_{1}}[{({x}_{2}-{x}_{1})}^{2}+{({y}_{2}-{y}_{1})}^{2}]\right\}\text{d}{x}_{1}\text{d}{y}_{1},$$
(2)
$$U(x,y)=\frac{exp(ik{d}_{2})}{i\mathrm{\lambda}{d}_{2}}\mathit{\iint}{U}_{2}({x}_{2},{y}_{2})exp[i2\mathrm{\pi}q({x}_{2},{y}_{2})]\times exp\left\{\frac{i\mathrm{\pi}}{\mathrm{\lambda}{d}_{2}}[{(x-{x}_{2})}^{2}+{(y-{y}_{2})}^{2}]\right\}\text{d}{x}_{2}\text{d}{y}_{2}.$$
(3)
$$\begin{array}{ll}{I}_{1}(x,y)\hfill & ={A}^{2}(x,y)+{{A}_{r}}^{2}+2{A}_{r}A(x,y)cos\mathrm{\varphi}(x,y),\hfill \\ {I}_{2}(x,y)\hfill & ={A}^{2}(x,y)+{{A}_{r}}^{2}+2{A}_{r}A(x,y)sin\mathrm{\varphi}(x,y),\hfill \\ {I}_{3}(x,y)\hfill & ={A}^{2}(x,y)+{{A}_{r}}^{2}-2{A}_{r}A(x,y)cos\mathrm{\varphi}(x,y),\hfill \\ {I}_{4}(x,y)\hfill & ={A}^{2}(x,y)+{{A}_{r}}^{2}-2{A}_{r}A(x,y)sin\mathrm{\varphi}(x,y),\hfill \end{array}$$
(4)
$$U(x,y)=\frac{1}{4{A}_{r}}[({I}_{1}-{I}_{3})+i({I}_{2}-{I}_{4})].$$
(5)
$${{I}_{1}}^{\prime}={I}_{1}-{I}_{3}+{I}_{0},$$
(6)
$${I}_{1}-{I}_{3}=cAcos\mathrm{\varphi}=\frac{1}{2}c[Aexp(i\mathrm{\varphi})+Aexp(-i\mathrm{\varphi})],$$
(7)
$$\mathrm{\varphi}(x,y)=-iln\frac{({I}_{1}-{I}_{3})+i({I}_{2}-{I}_{4})}{4{A}_{r}A(x,y)},$$
(8)
$${\text{I}}^{\u2033}(x,y)={I}^{\prime}(x,y)+\mathrm{\alpha}C(x,y)$$
(9)
$${\mathrm{\varphi}}^{\prime}(x,y)=\mathrm{\varphi}(x,y)+\mathrm{\alpha}C(x,y),$$
(10)
$$\text{MSE}=\frac{\text{\u2211}_{x=0}^{M-1}\text{\u2211}_{y=0}^{N-1}[{|f(x,y)-{f}_{r}(x,y)|}^{2}]}{\text{\u2211}_{x=0}^{M-1}\text{\u2211}_{y=0}^{N-1}[{|f(x,y)|}^{2}]},$$