## Abstract

Objects that have slow temporal variations may be superresolved
with two moving masks such as pinhole or grating. The first mask is
responsible for encoding the input image, and the second one performs
the decoding operation. This approach is efficient for exceeding
the resolving capability beyond Abbe’s limit of
resolution. However, the proposed setup requires two physical
gratings that should move in a synchronized manner. We propose what
is believed to be a novel configuration in which the second grating
responsible for the information decoding is replaced with a detector
array and some postprocessing digital procedures. In this way the
synchronization problem that exists when two gratings are used is
simplified. Experimental results are provided for illustrating the
utility of the new approach.

© 1999 Optical Society of America

Full Article |

PDF Article
### Equations (12)

Equations on this page are rendered with MathJax. Learn more.

(1)
$$a\left(\mathit{\nu}\right)=\left[{\tilde{U}}_{0}\left(\mathit{\nu}\right)\mathrm{rect}\left(\mathit{\nu}/\mathrm{\Delta}\mathit{\nu}\right)\right],b\left(\mathit{\nu}\right)=\left[{\tilde{U}}_{0}\left(\mathit{\nu}+{v}_{0}\right)\mathrm{rect}\left(\mathit{\nu}/\mathrm{\Delta}\mathit{\nu}\right)\right],c\left(\mathit{\nu}\right)=\left[{\tilde{U}}_{0}\left(\mathit{\nu}-{v}_{0}\right)\mathrm{rect}\left(\mathit{\nu}/\mathrm{\Delta}\mathit{\nu}\right)\right],$$
(2)
$$\mathrm{FT}\left[{U}_{0}\left(x\right)\right]={\tilde{U}}_{0}\left(\mathit{\nu}\right)=\left[b\left(\mathit{\nu}\right)\otimes \mathrm{\delta}\left(\mathit{\nu}+{\mathit{\nu}}_{0}\right)+a\left(\mathit{\nu}\right)\otimes \mathrm{\delta}\left(\mathit{\nu}\right)+c\left(\mathit{\nu}\right)\otimes \mathrm{\delta}\left(\mathit{\nu}-{\mathit{\nu}}_{0}\right)\right],$$
(3)
$${\tilde{U}}_{0}\left(\mathit{\nu}\right)*{\tilde{U}}_{0}\left(\mathit{\nu}\right)=\left\{\begin{array}{l}\left[c\left(\mathit{\nu}\right)*b\left(\mathit{\nu}\right)\right]\otimes \mathrm{\delta}\left(\mathit{\nu}+2{\mathit{\nu}}_{0}\right)\\ +\left[a\left(\mathit{\nu}\right)*b\left(\mathit{\nu}\right)+c\left(\mathit{\nu}\right)*a\left(\mathit{\nu}\right)\right]\otimes \mathrm{\delta}\left(\mathit{\nu}+{\mathit{\nu}}_{0}\right)\\ +\left[a\left(\mathit{\nu}\right)*a\left(\mathit{\nu}\right)+b\left(\mathit{\nu}\right)*b\left(\mathit{\nu}\right)+c\left(\mathit{\nu}\right)*c\left(\mathit{\nu}\right)\right]\otimes \mathrm{\delta}\left(\mathit{\nu}\right)\\ +\left[b\left(\mathit{\nu}\right)*a\left(\mathit{\nu}\right)+a\left(\mathit{\nu}\right)*c\left(\mathit{\nu}\right)\right]\otimes \mathrm{\delta}\left(\mathit{\nu}-{\mathit{\nu}}_{0}\right)\\ +\left[b\left(\mathit{\nu}\right)*c\left(\mathit{\nu}\right)\right]\otimes \mathrm{\delta}\left(\mathit{\nu}-2{\mathit{\nu}}_{0}\right)\end{array}\right\},$$
(4)
$$\frac{1}{\mathrm{\tau}}{\int}_{-\mathrm{\tau}/2}^{\mathrm{\tau}/2}{U}_{0}\left({x}_{1},t\right){U}_{0}^{*}\left({x}_{2},t\right)\mathrm{d}t.$$
(5)
$${\tilde{U}}_{1}\left(\mathit{\nu}\right)*{\tilde{U}}_{1}\left(\mathit{\nu}\right)=\left\{\begin{array}{l}\left[c\left(\mathit{\nu}\right)*b\left(\mathit{\nu}\right)\right]\otimes \mathrm{\delta}\left(\mathit{\nu}+2{\mathit{\nu}}_{0}\right)\\ +2\left[a\left(\mathit{\nu}\right)*b\left(\mathit{\nu}\right)+c\left(\mathit{\nu}\right)*a\left(\mathit{\nu}\right)\right]\otimes \mathrm{\delta}\left(\mathit{\nu}+{\mathit{\nu}}_{0}\right)\\ +3\left[a\left(\mathit{\nu}\right)*a\left(\mathit{\nu}\right)+b\left(\mathit{\nu}\right)*b\left(\mathit{\nu}\right)+c\left(\mathit{\nu}\right)*c\left(\mathit{\nu}\right)\right]\otimes \mathrm{\delta}\left(\mathit{\nu}\right)\\ +2\left[b\left(\mathit{\nu}\right)*a\left(\mathit{\nu}\right)+a\left(\mathit{\nu}\right)*c\left(\mathit{\nu}\right)\right]\otimes \mathrm{\delta}\left(\mathit{\nu}-{\mathit{\nu}}_{0}\right)\\ +\left[b\left(\mathit{\nu}\right)*c\left(\mathit{\nu}\right)\right]\otimes \mathrm{\delta}\left(\mathit{\nu}-2{\mathit{\nu}}_{0}\right)\end{array}\right\}.$$
(6)
$${\tilde{U}}_{1}\left(\mathit{\nu}\right)*{\tilde{U}}_{1}\left(\mathit{\nu}\right)=\left\{\begin{array}{l}7\left[c\left(\mathit{\nu}\right)*b\left(\mathit{\nu}\right)\right]\otimes \mathrm{\delta}\left(\mathit{\nu}+2{\mathit{\nu}}_{0}\right)\\ +8\left[a\left(\mathit{\nu}\right)*b\left(\mathit{\nu}\right)+c\left(\mathit{\nu}\right)*a\left(\mathit{\nu}\right)\right]\otimes \mathrm{\delta}\left(\mathit{\nu}+{\mathit{\nu}}_{0}\right)\\ +9\left[a\left(\mathit{\nu}\right)*a\left(\mathit{\nu}\right)+b\left(\mathit{\nu}\right)*b\left(\mathit{\nu}\right)+c\left(\mathit{\nu}\right)*c\left(\mathit{\nu}\right)\right]\otimes \mathrm{\delta}\left(\mathit{\nu}\right)\\ +8\left[b\left(\mathit{\nu}\right)*a\left(\mathit{\nu}\right)+a\left(\mathit{\nu}\right)*c\left(\mathit{\nu}\right)\right]\otimes \mathrm{\delta}\left(\mathit{\nu}-{\mathit{\nu}}_{0}\right)\\ +7\left[b\left(\mathit{\nu}\right)*c\left(\mathit{\nu}\right)\right]\otimes \mathrm{\delta}\left(\mathit{\nu}-2{\mathit{\nu}}_{0}\right)\end{array}\right\}.$$
(7)
$$\frac{1}{\mathrm{\tau}}{\int}_{-\mathrm{\tau}/2}^{\mathrm{\tau}/2}{U}_{0}\left({x}_{1},t\right){U}_{0}^{*}\left({x}_{2},t\right)\mathrm{d}t=|{U}_{0}\left({x}_{1}\right){|}^{2}\mathrm{\delta}\left({x}_{1}-{x}_{2}\right).$$
(8)
$$G\left(x,y\right)=\sum _{{n}_{x}}\sum _{{n}_{y}}{A}_{{n}_{x},{n}_{y}}exp\left[2\mathrm{\pi}j\left({\mathit{\nu}}_{{0}_{x}}x+{\mathit{\nu}}_{{0}_{y}}y\right)\right]$$
(9)
$$I\left(x,y,t\right)={u}_{0}\left(x,y,t\right){u}_{0}\left(x,y,t\right)*.$$
(10)
$${\mathit{\nu}}_{{0}_{x}}=\mathrm{\Delta}{\mathit{\nu}}_{x},{\mathit{\nu}}_{{0}_{y}}=\mathrm{\Delta}{\mathit{\nu}}_{y},$$
(11)
$$I\left(x,y\right)=\frac{1}{\mathrm{\tau}}{\int}_{-\mathrm{\tau}/2}^{\mathrm{\tau}/2}I\left(x,y,t\right)G\left(x-{V}_{x}t,y-{V}_{y}t\right){|}^{2}\mathrm{d}t,$$
(12)
$${I}_{c}\left(x,y\right)=\sum _{k=1}^{M}I\left(x,y,{t}_{k}\right)G\left(x-{V}_{x}{t}_{k},y-{V}_{y}{t}_{k}\right){|}^{2},$$