## Abstract

We present an approach that provides superresolution beyond the classical limit as well as image restoration in the presence of aberrations; in particular, the ability to obtain superresolution while extending the depth of field (DOF) simultaneously is tested experimentally. It is based on an approach, recently proposed, shown to increase the resolution significantly for in-focus images by speckle encoding and decoding. In our approach, an object multiplied by a fine binary speckle pattern may be located anywhere along an extended DOF region. Since the exact magnification is not known in the presence of defocus aberration, the acquired low-resolution image is electronically processed via a parallel-branch decoding scheme, where in each branch the image is multiplied by the same high-resolution synchronized time-varying binary speckle but with different magnification. Finally, a hard-decision algorithm chooses the branch that provides the highest-resolution output image, thus achieving insensitivity to aberrations as well as DOF variations. Simulation as well as experimental results are presented, exhibiting significant resolution improvement factors.

© 2007 Optical Society of America

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### Equations (9)

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(1)
$$I(x,\xi )=\kappa \int {I}_{obj}({x}^{\prime}\u2215{M}_{t}){S}_{obj}(({x}^{\prime}-\xi )\u2215{M}_{t}){\mid h(x-{x}^{\prime})\mid}^{2}\mathrm{d}{x}^{\prime},$$
(2)
$$I(x,t)=\kappa \int {I}_{obj}({x}^{\prime}\u2215{M}_{t}){S}_{obj}({x}^{\prime}\u2215{M}_{t};t){\mid h(x-{x}^{\prime})\mid}^{2}\mathrm{d}{x}^{\prime}.$$
(3)
$$I\left(x\right)=\kappa \sum _{n}{S}_{obj}((x-{\xi}_{n})\u2215{M}_{t})\int {I}_{obj}({x}^{\prime}\u2215{M}_{t}){S}_{obj}(({x}^{\prime}-{\xi}_{n})\u2215{M}_{t}){\mid h(x-{x}^{\prime})\mid}^{2}\mathrm{d}{x}^{\prime},$$
(4)
$$I\left(x\right)=\kappa \sum _{n}{S}_{obj}(x\u2215{M}_{t};{t}_{n})\int {I}_{obj}({x}^{\prime}\u2215{M}_{t}){S}_{obj}({x}^{\prime}\u2215{M}_{t};{t}_{n}){\mid h(x-{x}^{\prime})\mid}^{2}\mathrm{d}{x}^{\prime}.$$
(5)
$$\underset{N\to \infty}{\mathrm{lim}}\frac{{\xi}_{n+1}-{\xi}_{n}}{N}\sum _{n=1}^{N}{S}_{obj}((x-{\xi}_{n})\u2215{M}_{t}){S}_{obj}(({x}^{\prime}-{\xi}_{n})\u2215{M}_{t})=\int {S}_{obj}((x-\xi )\u2215{M}_{t}){S}_{obj}(({x}^{\prime}-\xi )\u2215{M}_{t})\mathrm{d}\xi =\u27e8I{\u27e9}^{2}(1+{\mid \mu (x-{x}^{\prime})\mid}^{2}),$$
(6)
$$\underset{N\to \infty}{\mathrm{lim}}\frac{1}{N}\sum _{n=1}^{N}{S}_{obj}(x\u2215{M}_{t};{t}_{n}){S}_{obj}({x}^{\prime}\u2215{M}_{t};{t}_{n})=p[p+(1-p)\Lambda \left(\frac{x-{x}^{\prime}}{{M}_{t}L}\right)],$$
(7)
$$I\left(x\right)=\kappa p\int {I}_{obj}({x}^{\prime}\u2215{M}_{t}){\mid h(x-{x}^{\prime})\mid}^{2}[p+(1-p)\Lambda \left(\frac{x-{x}^{\prime}}{{M}_{t}L}\right)]\mathrm{d}{x}^{\prime}=\kappa {p}^{2}\int {I}_{obj}({x}^{\prime}\u2215{M}_{t}){\mid h(x-{x}^{\prime})\mid}^{2}\mathrm{d}{x}^{\prime}+\kappa (p-{p}^{2})\int {I}_{obj}({x}^{\prime}\u2215{M}_{t}){\mid h(x-{x}^{\prime})\mid}^{2}\Lambda \left(\frac{x-{x}^{\prime}}{{M}_{t}L}\right)\mathrm{d}{x}^{\prime}.$$
(8)
$${I}_{blur}\left(x\right)=\underset{N\to \infty}{\mathrm{lim}}\frac{\kappa}{N}\sum _{n=1}^{N}\int {I}_{obj}({x}^{\prime}\u2215{M}_{t}){S}_{obj}({x}^{\prime}\u2215{M}_{t};{t}_{n}){\mid h(x-{x}^{\prime})\mid}^{2}\mathrm{d}{x}^{\prime}.$$
(9)
$${I}_{blur}\left(x\right)=\kappa p\int {I}_{obj}({x}^{\prime}\u2215{M}_{t}){\mid h(x-{x}^{\prime})\mid}^{2}\mathrm{d}{x}^{\prime}.$$