## Abstract

We discuss quantization effects of hologram recording on the quality of reconstructed images in phase-shifting digital holography. We vary bit depths of phase-shifted holograms in both numerical simulation and experiments and then derived the complex amplitude, which is subjected to Fresnel transformation for the image reconstruction. The influence of bit-depth limitation in quantization has been demonstrated in a numerical simulation for spot-array patterns with linearly varying intensities and a continuous intensity object. The objects are provided with uniform and random phase modulation. In experiments, digital holograms are originally recorded at 8 bits and the bit depths are changed to deliver holograms at bit depths of 1 to 8 bits for the image reconstruction. The quality of the reconstructed images has been evaluated for the different quantization levels.

© 2005 Optical Society of America

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### Equations (11)

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(1)
$${U}_{O}({x}^{\prime},{y}^{\prime})=\sum _{n=0}^{N-1}\sum _{m=0}^{M-1}\sqrt{{I}_{O}({x}^{\prime},{y}^{\prime})}\text{exp}[i{\mathrm{\varphi}}_{O}({x}^{\prime},{y}^{\prime})]\times \mathrm{\delta}({x}^{\prime}-n\mathrm{\Delta}{x}^{\prime},{y}^{\prime}-m\mathrm{\Delta}{y}^{\prime}),$$
(2)
$$U(x,y)={\int}_{-\infty}^{\infty}{\int}_{-\infty}^{\infty}{U}_{O}({x}^{\prime},{y}^{\prime})\text{exp}\left\{\frac{ik}{2{z}_{O}}[{(x-{x}^{\prime})}^{2}+{(y-{y}^{\prime})}^{2}]\right\}\text{d}{x}^{\prime}\text{d}{y}^{\prime},$$
(3)
$$\begin{array}{l}{I}_{H}(x,y;\mathrm{\delta})=\mid {U}_{R}(x,y;\mathrm{\delta})+U(x,y){\mid}^{2}\\ =\mid {U}_{R}{\mid}^{2}+\mid U{\mid}^{2}+2\mathfrak{R}[{U}_{R}{U}^{*}\hspace{0.17em}\text{exp}(i\mathrm{\delta})].\end{array}$$
(4)
$$U(x,y)=\frac{1}{4\sqrt{{I}_{R}}}\{[{I}_{H}(x,y;0)-{I}_{H}(x,y;\mathrm{\pi})]+i[{I}_{H}(x,y;\mathrm{\pi}/2)-{I}_{H}(x,y;3\mathrm{\pi}/2)]\},$$
(5)
$$U(x,y)=\frac{1-i}{4\sqrt{{I}_{R}}}\{[{I}_{H}(x,y;0)-{I}_{H}(x,y;\hspace{0.17em}\mathrm{\pi}/2)]+i[{I}_{H}(x,y;\mathrm{\pi}/2)-{I}_{H}(x,y;\mathrm{\pi})]\},$$
(6)
$${U}_{I}(X,Y;Z)={\int}_{-\infty}^{\infty}{\int}_{-\infty}^{\infty}U(x,y)\text{exp}\left\{\frac{ik}{2Z}[{(X-x)}^{2}+{(Y-y)}^{2}]\right\}\text{d}x\text{d}y.$$
(7)
$${U}_{I}(x,y;-{z}_{O})={U}_{O}(x,y).$$
(8)
$${U}_{I}(X,Y)=\text{exp}\left[\frac{ik}{2Z}({X}^{2}+{Y}^{2})\right]{\int}_{-\infty}^{\infty}{\int}_{-\infty}^{\infty}U(x,y)\times \text{exp}\left[\frac{ik}{2Z}({x}^{2}+{y}^{2})\right]\times \text{exp}\left[-\frac{ik}{Z}(Xx+Yy)\right]\text{d}x\text{d}y.$$
(9)
$$\frac{\mathrm{\lambda}{z}_{O}}{{N}_{O}\mathrm{\Delta}{x}^{\prime}\mathrm{\Delta}x}=1,\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\frac{\mathrm{\lambda}{z}_{O}}{{M}_{O}\mathrm{\Delta}{y}^{\prime}\mathrm{\Delta}y}=1.$$
(10)
$${I}_{HQ}(\mathrm{\delta})=[f\{Q{I}_{H}(\mathrm{\delta}){{I}_{H\text{max}}}^{-1}\}]{I}_{H\text{max}}{Q}^{-1},$$
(11)
$$D(b;8)={\left(\left\{\sum _{n=1}^{512}\sum _{m=1}^{512}{[\mid {I}_{b}(m,n){\mid}^{2}-\mid {I}_{8}(m,n){\mid}^{2}]}^{2}\right\}\right)}^{1/2}\times {\left\{\sum _{n=1}^{512}\sum _{m=1}^{512}{[\mid {I}_{8}(m,n){\mid}^{2}]}^{2}\right\}}^{-1/2},$$