High-quality quantum-imaging algorithm and experiment based on compressive sensing

Liu Jiying; Zhu Jubo; Lu Chuan; Huang Shisheng

doi:10.1364/OL.35.001206

Optics Letters
Vol. 35,
Issue 8,
pp. 1206-1208
(2010)
•https://doi.org/10.1364/OL.35.001206

High-quality quantum-imaging algorithm and experiment based on compressive sensing

Liu Jiying, Zhu Jubo, Lu Chuan, and Huang Shisheng

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Liu Jiying, Zhu Jubo, Lu Chuan, and Huang Shisheng, "High-quality quantum-imaging algorithm and experiment based on compressive sensing," Opt. Lett. 35, 1206-1208 (2010)

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Abstract

Quantum imaging (QI) has some unique advantages, such as nonlocal imaging and enhanced space resolution. However, the quality of the reconstructed images and the time of data acquisition leave much to be desired. Based on the framework of compressive sensing, we propose an optimization criterion for high-quality QI whereby total variation restriction is specifically utilized for noise suppression. The corresponding reported algorithm uses a combination of a greedy strategy and the interactive reweight least-squares method. The simulation and the actual imaging experiment both show a significant improvement of the proposed algorithm the over previous imaging method.

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Initialize: $tol, ϵ, η, S, γ, I$ for certain values, and $k = 0, α_{k} = 0, Γ = \emptyset$
While ${‖ α_{k} - α_{k - 1} ‖}_{2} > tol$
$r_{k} = a - M \cdot D \cdot α_{k}$
$p_{k} = D^{T} \cdot M^{T} \cdot r_{k}$
$Γ_{add} = {τ : \| p (τ) \|$ is one of the S largest entries in $\| p \|}$
$Γ = Γ \cup (Γ_{add} ∕ Γ)$
$α_{k, 1} = {({(M D_{Γ})}^{T} M D_{Γ} + γ G_{Γ}^{T} G_{Γ})}^{- 1} {(M D_{Γ})}^{T} a$
$D_{Γ}$ and $G_{Γ}$ are submatrixes of D and G, whose columns are indexed by Γ
For $i = 1$ to I
$W_{D} = diag (f_{D} (M \cdot D_{Γ} \cdot α_{k, i} - a))$ ; $W_{G} = diag (f_{G} (G_{Γ} \cdot D_{Γ} \cdot α_{k, i}))$

where: $f_{D} (x) = {\begin{matrix} {\| x \|}^{- 1}, if \| x \| > ϵ \\ ϵ^{- 1}, if \| x \| ⩽ ϵ \end{matrix}$ ; $f_{G} (x) = {\begin{matrix} {\| x \|}^{- 1}, if \| x \| > η \\ 0, if \| x \| ⩽ η \end{matrix}$

$α_{k, i} = {({(M D_{Γ})}^{T} W_{F} M D_{Γ} + γ G_{Γ}^{T} W_{R} G_{Γ})}^{- 1} {(M D_{Γ})}^{T} W_{F} a$
End
$α_{k} = {α_{k} (τ) : α_{k} (τ) = {\begin{matrix} α_{k, I} (τ), if τ ∊ Γ \\ 0, if τ \notin Γ \end{matrix}}$
End
Output: $x_{k} = D \cdot α_{k}$ .

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