## Abstract

Hereby we describe an improved aberration-free super resolution imaging system that uses binary speckle patterns (BSP) for encoding and decoding. According to the scheme, the object is multiplied by a fine random speckle pattern and subsequently the poorly resolved acquired image is multiplied by the same pattern, stored digitally. Summing many samples of similar operations restores a high quality image. In this paper we define encoding and decoding BSP that form complementary sets. Contrary to using totally random BSP, the use of complementary sets reduces the noise of the resulting image significantly. Moreover, they allow reducing the number of operations necessary to restore the image. We found out that using sparse BSP followed by a stochastic noise reduction algorithm, fitted for the decoding process, reduces the noise significantly and allows use of even a single set of complementary BSP for obtaining excellent results. Simulation results are presented both for binary images as well as for gray level objects.

© 2007 Optical Society of America

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### Equations (34)

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(1)
$${I}_{\mathit{en}}(x;n)={I}_{\mathit{obj}}\left(x\right){S}_{n}\left(x\right)$$
(2)
$${I}_{\mathit{F}}(x;n)=\kappa {\mathrm{\int}I}_{\mathit{en}}{(\frac{x\xb4}{M};n)\mid h\left(x-x\xb4\right)\mid}^{2}\mathit{dx}\xb4+{\xi}_{n}\left(x\right)$$
(3)
$${I}_{\mathit{out}}\left(x\right)=\frac{1}{N}\sum _{n=1}^{N}{I}_{F}(x;n){S}_{n}\left(\frac{x}{M}\right)$$
(4)
$${I}_{\mathit{out}}\left(x\right)=\kappa {\mathrm{\int}I}_{\mathit{obj}}{\left(\frac{x\xb4}{M}\right)\mid h\left(x-x\xb4\right)\mid}^{2}\left[\frac{1}{N}\sum _{n=1}^{N}{S}_{n}{\left(\frac{x\xb4}{M}\right)S}_{n}\left(\frac{x}{M}\right)\right]\mathit{dx}\xb4+r\left(x\right)$$
(5)
$$r\left(x\right)\equiv \frac{1}{N}\sum _{n=1}^{N}{\xi}_{n}\left(x\right){S}_{n}\left(\frac{x}{M}\right)$$
(6)
$${lim}_{N\to \infty}=\left(\frac{1}{N}\sum _{n=1}^{N}{S}_{n}{\left(\frac{x\xb4}{M}\right)S}_{n}\left(\frac{x}{M}\right)\right)=\mathrm{\Gamma}\left(x-x\xb4\right)={p}^{2}+p\left(1-p\right)\mathrm{\Lambda}\left(\frac{x-x\xb4}{B}\right)$$
(7)
$${I}_{\mathit{blur}}\left(x\right)={lim}_{N\to \infty}\frac{1}{N}\sum _{n=1}^{N}{I}_{F}(x,n)=\mathit{p\kappa}\int {I}_{\mathit{obj}}\left(\frac{x\xb4}{M}\right){\mid h\left(x-x\xb4\right)\mid}^{2}\mathit{dx}\xb4$$
(8)
$${I}_{\mathit{hr};\infty}\left(x\right)={I}_{\mathit{out}}\left(x\right)-\mathit{p}{I}_{\mathit{blur}}\left(x\right)=\left(1-p\right)\mathit{p\kappa}\int {I}_{\mathit{obj}}\left(\frac{x\xb4}{M}\right){\mid h\left(x-x\xb4\right)\mid}^{2}\mathrm{\Lambda}\left(\frac{x-x\xb4}{B}\right)\mathit{dx}\xb4+r\left(x\right)$$
(9)
$$\u3008{r}^{2}\left(x\right)\u3009=\frac{1}{{N}^{2}}\sum _{n}\sum _{n\xb4}\u3008{S}_{n}\left(\frac{x}{M}\right){S}_{n\xb4}\left(\frac{x}{M}\right)\u3009\u3008{\xi}_{n}\left(x\right){\xi}_{n\xb4}\left(x\right)\u3009=$$
(10)
$$=\frac{1}{{N}^{2}}\sum _{n=n\xb4}\u3008{S}_{n}\left(\frac{x}{M}\right){S}_{n\xb4}\left(\frac{x}{M}\right)\u3009\u3008{\xi}_{n}\left(x\right){\xi}_{n\xb4}\left(x\right)\u3009\le \frac{1}{{N}^{2}}\sum _{n}\u3008{\xi}_{n}^{2}\left(x\right)\u3009=\frac{{\sigma}^{2}}{N}$$
(11)
$${I}_{\mathit{hr};\infty}\left(x\right)={I}_{\mathit{out}}\left(x\right)-\mathit{p}{I}_{\mathit{blur}}\left(x\right)=\left(1-p\right)\mathit{p\kappa}\int {I}_{\mathit{obj}}\left(\frac{x\xb4}{M}\right){\mid h\left(x-x\xb4\right)\mid}^{2}\mathrm{\Lambda}\left(\frac{x-x\xb4}{B}\right)\mathit{dx}\xb4$$
(12)
$${S}_{N}\left(x\right)=\frac{1}{N}\sum _{n=1}^{N}{S}_{n}\left(x\right);\mathrm{n}\in \left\{1\dots \mathrm{N}\right\}$$
(13)
$${p}_{k}=\left(\genfrac{}{}{0ex}{}{N}{k}\right){p}^{k}{\left(1-p\right)}^{N-k};\mathrm{where}\left(\genfrac{}{}{0ex}{}{N}{k}\right)\equiv \frac{N!}{\left(N-k\right)!k!}\phantom{\rule{.2em}{0ex}}\mathrm{and}\phantom{\rule{.2em}{0ex}}k\in \{0\u2025\mathrm{N}\}$$
(14)
$$\u3008{S}_{N}\left(x\right)\u3009=\sum _{\mathit{k}=0}^{N}{A}_{k}{p}_{k}=p$$
(15)
$$\u3008{S}_{\mathit{N}}^{2}\left(x\right)\u3009=\sum _{\mathit{k}=0}^{N}{A}_{k}^{2}{p}_{k}={p}^{2}\frac{N-1}{N}+\frac{p}{N}$$
(16)
$${V}_{\mathit{N}}=\u3008{S}_{\mathit{N}}^{2}\left(x\right)\u3009-{\u3008{S}_{N}\left(x\right)\u3009}^{2}=\frac{p-{p}^{2}}{N}=\frac{p\left(1-p\right)}{N}$$
(17)
$${\sigma}_{N}=\sqrt{{V}_{N}}=\sqrt{\frac{p\left(1-p\right)}{N}}$$
(18)
$$\mathrm{\Gamma}(x;x\xb4)=\frac{1}{N}\sum _{n=1}^{N}{S}_{n}\left(\frac{x\xb4}{M}\right){S}_{n}\left(\frac{x}{M}\right)=\mathrm{\Delta}{(x,x\xb4)}_{\mid x-x\xb4\mid >B}+{\mathrm{\Delta}}_{\mid x-x\xb4\mid \le \mathit{B}}\mathrm{\Lambda}\left(\frac{x-x\xb4}{B}\right)$$
(19)
$$\mathrm{\Delta}{(x,x\xb4)}_{\mid x-x\xb4\mid \le B}\equiv \frac{1}{N}{\sum _{n=1}^{N}S}_{n}\left(\frac{x}{M}\right){S}_{n}\left(\frac{x}{M}\right),$$
(20)
$$\mathrm{\Delta}{(x,x\xb4)}_{\mid x-x\xb4\mid >B}\equiv \frac{1}{N}{\sum _{n=1}^{N}S}_{n}\left(\frac{x\xb4}{M}\right){S}_{n}\left(\frac{x}{M}\right),$$
(21)
$$\u3008\mathrm{\Delta}{(x,x\xb4)}_{\mid x-x\xb4\mid >B}\u3009={p}^{2}$$
(22)
$$\sigma {\left(x\ne x\xb4\right)}_{N}=\sqrt{\frac{{p}^{2}\left(1-{p}^{2}\right)}{N}}$$
(23)
$${S}_{N}\left(x\right)=\frac{1}{{L}_{0}{N}_{1}}\sum _{n=1}^{{L}_{0}}\sum _{k=1}^{{N}_{1}}{S}_{n;k}\left(x\right)$$
(24)
$$\sum _{k=1}^{{N}_{1}}{S}_{n;k}\left(x\right)=1\left(x\right)\equiv 1\mathrm{everywhere}.$$
(25)
$${S}_{N}\left(x\right)=\frac{1}{{L}_{0}{N}_{1}}\sum _{n=1}^{{L}_{0}}1\left(x\right)=\frac{1}{{N}_{1}}\phantom{\rule{.2em}{0ex}}=p$$
(26)
$${I}_{\mathit{blur}}\left(x\right)=\frac{1}{N}\sum _{n=1}^{N}{I}_{F}(x,n)=\mathit{p\kappa}\int {I}_{\mathit{obj}}\left(\frac{x\xb4}{M}\right){\mid h\left(x-x\xb4\right)\mid}^{2}\mathit{dx}\xb4$$
(27)
$$\mathrm{\Gamma}(x;x\xb4)=\Delta {(x,x\xb4)}_{\mid x-x\xb4\mid >B}+p\mathrm{\Lambda}\left(\frac{x-x\xb4}{B}\right)$$
(28)
$${I}_{\mathit{hr}}\left(x\right)=\mathit{\kappa}\int {I}_{\mathit{obj}}\left(\frac{x\xb4}{M}\right){\mid h\left(x-x\xb4\right)\mid}^{2}\left[\left(1-p\right)\mathit{p}\mathrm{\Lambda}\right(\frac{x-x\xb4}{B}\left)+\mathit{spn}{(x,x\xb4)}_{\mid x-x\xb4\mid >B}\right]\mathit{dx}\xb4$$
(29)
$${I}_{\mathit{hr}}\left(x\right)={I}_{\mathit{hr};\infty}\left(x\right)+\mathit{spn}\left(x\right)$$
(30)
$${I}_{\mathit{hr};\infty}\left(x\right)\equiv \mathit{\kappa}\left(1-p\right)p\int {I}_{\mathit{obj}}\left(\frac{x\xb4}{M}\right){\mid h\left(x-x\xb4\right)\mid}^{2}\mathrm{\Lambda}\left(\frac{x-x\xb4}{B}\right)\mathit{dx}\xb4$$
(31)
$$\mathit{spn}\left(x\right)\equiv \mathit{\kappa}\mathit{\int}{I}_{\mathit{obj}}\left(\frac{x\xb4}{M}\right){\mathit{spn}{(x,x\xb4)}_{\mid x-x\xb4\mid >B}\mid h\left(x-x\xb4\right)\mid}^{2}\mathit{dx}\xb4$$
(32)
$${S}_{F}(x,n)\equiv \mathit{\kappa}\int {S}_{\mathit{n}}\left(\frac{x\xb4}{{M}_{t}}\right){\mid h\left(x-x\xb4\right)\mid}^{2}\mathit{dx}\xb4$$
(33)
$${S\xb4}_{F}(x,n)=\{\begin{array}{c}\begin{array}{cc}\frac{{S}_{n}\left(\frac{x}{M}\right)}{{S}_{F}(x,n)}& \mathrm{if}\phantom{\rule{.2em}{0ex}}{S}_{n}\left(\frac{x}{M}\right)\end{array}\ne 0\\ \begin{array}{cc}0& \mathrm{otherwise}\end{array}\end{array}$$
(34)
$${I}_{\mathit{out}}\left(x\right)=\frac{1}{N}\sum _{n=1}^{N}{I}_{F}(x;n){S\xb4}_{F}(x,n)$$