## Abstract

Free-viewpoint images obtained from phase-shifting synthetic aperture digital holography are given for scenes that include multiple objects and a concave object. The synthetic aperture technique is used to enlarge the effective sensor size and to make it possible to widen the range of changing perspective in the numerical reconstruction. The lensless Fourier setup and its aliasing-free zone are used to avoid aliasing errors arising at the sensor edge and to overcome a common problem in digital holography, namely, a narrow field of view. A change of viewpoint is realized by a double numerical propagation and by clipping the wave field by a given pupil. The computational complexity for calculating an image in the given perspective from the base complex-valued image is estimated at a double fast Fourier transform. The experimental results illustrate the natural change of appearance in cases of both multiple objects and a concave object.

© 2008 Optical Society of America

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### Equations (14)

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(1)
$$R({x}_{s},{y}_{s})=\mathrm{exp}\left[ik\frac{{x}_{s}^{2}+{y}_{s}^{2}}{2{d}_{R}}\right],$$
(2)
$$O{R}^{*}({x}_{s},{y}_{s})=\int \int f(x,y,0)\mathrm{exp}[i\frac{k}{2{d}_{R}}({x}^{2}+{y}^{2})]\mathrm{exp}[-i\frac{2\pi}{\lambda {d}_{R}}(x{x}_{s}+y{y}_{s})]\text{d}x\text{d}y=\mathcal{F}\{f(x,y,0){\varphi}^{*}(x,y){\}}_{u={x}_{s}/\lambda {d}_{R},v={y}_{s}/\lambda {d}_{R}},$$
(3)
$$\varphi (x,y)=\mathrm{exp}[-i\frac{k}{2{d}_{R}}({x}^{2}+{y}^{2})],$$
(4)
$$f(x,y,0)=\mathcal{F}\{O{R}^{*}({x}_{s},{y}_{s}){\}}_{{u}_{s}=-x/\lambda {d}_{R},{v}_{s}=-y/\lambda {d}_{R}}\times \varphi (x,y),$$
(5)
$${\mathrm{\Delta}}_{x}=\frac{\lambda {d}_{R}}{{N}_{x}\delta x},\phantom{\rule[-0.0ex]{2em}{0.0ex}}{\mathrm{\Delta}}_{y}=\frac{\lambda {d}_{R}}{{N}_{y}\delta y},$$
(6)
$$f(x,y,{d}_{P})={\mathcal{P}}_{{d}_{P}}\{f(x,y,0)\}\phantom{\rule{0ex}{0ex}}={\mathcal{F}}^{-1}\{{F}_{0}(u,v)\mathrm{exp}[i2\pi ({\lambda}^{-2}-{u}^{2}-{v}^{2}{)}^{1/2}{d}_{P}]\},$$
(7)
$$\widehat{f}(x,y,{d}_{P};\text{\hspace{0.17em}}{x}_{e},{y}_{e})=f(x,y,{d}_{P})p(x,y;\text{\hspace{0.17em}}{x}_{e},{y}_{e}).$$
(8)
$$\widehat{f}(x,y,0;\text{\hspace{0.17em}}{x}_{e},{y}_{e})={\mathcal{P}}_{-{d}_{B}}\{\widehat{f}(x,y,{d}_{P};\text{\hspace{0.17em}}{x}_{e},{y}_{e})\},$$
(9)
$$\mathrm{\Lambda}=\frac{2\pi}{|{k}_{P}-{k}_{R}|},$$
(10)
$${\mathrm{\Lambda}}_{\mathrm{min}}=\frac{\lambda}{2\hspace{0.17em}\mathrm{sin}(\mathrm{\Delta}\theta /2)},$$
(11)
$$\mathrm{sin}(\mathrm{\Delta}\theta /2)=\frac{w/4}{\sqrt{{d}^{2}+(w/4{)}^{2}}}.$$
(12)
$$w\le \frac{4\lambda d}{\sqrt{16{\delta}^{2}-{\lambda}^{2}}}.$$
(13)
$${p}_{x}^{\prime}\times {p}_{y}^{\prime}=({p}_{x}\times {p}_{y}){d}_{P}^{\prime 2}/{d}_{P}^{2},$$
(14)
$${d}_{P}\ge \frac{2{x}_{e,\mathrm{max}}+{w}_{x}}{2\hspace{0.17em}\mathrm{tan}{\theta}_{x}^{\prime}},\phantom{\rule[-0.0ex]{1em}{0.0ex}}\phantom{\rule[-0.0ex]{1em}{0.0ex}}\frac{2{y}_{e,\mathrm{max}}+{w}_{y}}{2\hspace{0.17em}\mathrm{tan}{\theta}_{y}^{\prime}},$$