## Abstract

Multiple view projection holography is a method to obtain a digital hologram by recording different views of a 3D scene with a conventional digital camera. Those views are digitally manipulated in order to create the digital hologram. The method requires a simple setup and operates under white light illuminating conditions. The multiple views are often generated by a camera translation, which usually involves a scanning effort. In this work we apply a compressive sensing approach to the multiple view projection holography acquisition process and demonstrate that the 3D scene can be accurately reconstructed from the highly subsampled generated Fourier hologram. It is also shown that the compressive sensing approach, combined with an appropriate system model, yields improved sectioning of the planes of different depths.

© 2011 OSA

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### Equations (12)

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(1)
$${u}_{i}(x,y)={\mathcal{F}}^{-1}\left\{h\left({\upsilon}_{x},{\upsilon}_{y}\right)\cdot \mathrm{exp}\left[-j\pi \lambda {z}_{i}\left({\upsilon}_{x}^{2}+{\upsilon}_{y}^{2}\right)\right]\right\},$$
(2)
$${u}_{i}(p,q)={\displaystyle \sum _{m}{\displaystyle \sum _{n}h(m,n)\mathrm{exp}\left\{-j\pi \lambda {z}_{i}\left[{\left(\Delta {\upsilon}_{x}m\right)}^{2}+{\left(\Delta {\upsilon}_{y}n\right)}^{2}\right]\right\}\mathrm{exp}\left\{j2\pi \left(\frac{mp}{{N}_{x}}+\frac{nq}{{N}_{y}}\right)\right\}}},$$
(3)
$${u}_{i}={F}^{-1}{Q}_{\text{-}{\lambda}^{2}{z}_{i}}h,$$
(4)
$$\mathrm{min}\left\{{\Vert {u}_{i}-F{Q}_{{\text{-\lambda}}^{\text{2}}{\text{z}}_{\text{i}}}{h}^{M}\Vert}_{2}^{2}+\gamma {\Vert {\Psi}_{i}{u}_{i}\Vert}_{1}\right\},$$
(5)
$$h({\upsilon}_{x}^{},{\upsilon}_{y}^{})={\displaystyle \sum _{i=1}^{{N}_{z}}\mathrm{exp}\left[j\pi \lambda {z}_{i}\left({\upsilon}_{x}^{2}+{\upsilon}_{y}^{2}\right)\right]\mathcal{F}\left\{{u}_{i}\right\}}={\displaystyle \sum _{i=1}^{{N}_{z}}\mathcal{F}\left\{{u}_{i}\ast \mathrm{exp}\left[\frac{-j\pi}{\lambda {z}_{i}}\left({x}^{2}+{y}^{2}\right)\right]\right\}},$$
(6)
$$h(m,n)={\displaystyle \sum _{i=1}^{{N}_{z}}{Q}_{{\lambda}^{2}{z}_{i}}F{u}_{i}}.$$
(7)
$$h=\left[{Q}_{{\lambda}^{2}{z}_{1}}F;\mathrm{...};{Q}_{{\lambda}^{2}{z}_{{N}_{z}}}F\right]\phantom{\rule{.2em}{0ex}}{\left[{u}_{1};\mathrm{...};{u}_{{N}_{Z}}\right]}^{T}=\Phi {u}^{T}.$$
(8)
$$\mathrm{min}\left\{{\Vert {h}^{M}-\Phi {u}^{T}\Vert}_{2}^{2}+\tau {\Vert u\Vert}_{TV}\right\},$$
(9)
$${\Vert u\Vert}_{TV}={\displaystyle \sum _{l}{\displaystyle \sum _{i,j}\sqrt{{\left({u}_{i+1,j,l}-{u}_{i,j,l}\right)}^{2}+{\left({u}_{i,j+1,l}-{u}_{i,j,l}\right)}^{2}}}}$$
(10)
$$\Delta x=\mathrm{max}\left\{\Delta {x}_{optical},\Delta {x}_{geometrical},\Delta {x}_{ho\mathrm{log}ram}\right\}=\mathrm{max}\left\{\frac{\lambda {z}_{o}}{A},\frac{\Delta s}{{M}_{T}},\frac{{N}_{p}\Delta s\sqrt{{L}^{2}+{z}_{o}^{2}}}{bL{M}_{T}}\right\}$$
(11)
$$\Delta {z}_{MA}=\frac{{z}_{0}}{L}\Delta x.$$
(12)
$$\frac{\Delta {z}_{gain}}{Number\text{\hspace{0.17em}}of\text{\hspace{0.17em}}projections}=\frac{L/A}{K\mathrm{log}N}=\frac{N}{K\mathrm{log}N}.$$