From Fienup’s phase retrieval techniques to regularized inversion for in-line holography: tutorial

Fabien Momey; Loïc Denis; Thomas Olivier; Corinne Fournier

doi:10.1364/JOSAA.36.000D62

Journal of the Optical Society of America A
Vol. 36,
Issue 12,
pp. D62-D80
(2019)
•https://doi.org/10.1364/JOSAA.36.000D62

From Fienup’s phase retrieval techniques to regularized inversion for in-line holography: tutorial

Fabien Momey, Loïc Denis, Thomas Olivier, and Corinne Fournier

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Fabien Momey, Loïc Denis, Thomas Olivier, and Corinne Fournier, "From Fienup’s phase retrieval techniques to regularized inversion for in-line holography: tutorial," J. Opt. Soc. Am. A 36, D62-D80 (2019)

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Abstract

This paper includes a tutorial on how to reconstruct in-line holograms using an inverse problems approach, starting with modeling the observations, selecting regularizations and constraints, and ending with the design of a reconstruction algorithm. A special focus is placed on the connections between the numerous alternating projections strategies derived from Fienup’s phase retrieval technique and the inverse problems framework. In particular, an interpretation of Fienup’s algorithm as iterates of a proximal gradient descent for a particular cost function is given. Reconstructions from simulated and experimental holograms of micrometric beads illustrate the theoretical developments. The results show that the transition from alternating projections techniques to the inverse problems formulation is straightforward and advantageous.

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Fig. 1. Hologram reconstruction based on inverse problems approaches: the direct model connects the sample of interest to the measurements (the holograms); inverting this model leads to a reconstruction and/or estimation of the optical parameters of the sample. The case chosen as an illustration in this paper is highlighted in yellow.

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Fig. 2. In-line holography principle: a plane wave illuminates the sample plane with normal incidence. The presence of inhomogeneity in the spatial distribution of the complex-valued refractive index, in the sample plane, distorts the illumination wave. A diffraction pattern is recorded on the sensor plane.

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Fig. 3. Illustration of the behavior of several regularization terms on reconstructions (in a region of interest) of a simulated hologram (a) using a regularized inverse problems approach. Details on the method, the simulation, and reconstruction parameters are given in Section 5 and Tables 1 and 2. (a) Data (in-line hologram). The yellow frame indicates the region of interest that is extracted from the field of view for visualization. (b) Ground truth phase. (c)–(f) Reconstructed images using the following regularizations: (c) squared

$ {l_2} $

norm of the gradient

$ {{\cal J}_{{{l}_{2}}\nabla }} $

[Eq. (15)], (d) edge-preserving smoothness

$ {{\cal J}_{{{\text{TV}}_ \epsilon }}} $

[Eq. (16)], (e)

$ {l_1} $

sparsity constraint

$ {{\cal J}_{{l_1}}} $

[Eq. (17)], and (f) composition of the

$ {l_1} $

sparsity constraint and edge-preserving smoothness

$ {{\cal J}_{\textsf{reg}}} = {{\cal J}_{{l_1}}} + {{\cal J}_{{{\text{TV}}_ \epsilon }}} $

[Eq. (18)]. Red curves show a line profile passing through two particles of the image.

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Fig. 4. Principle that underlies proximal gradient methods is the iterative minimization of a local approximation of the original cost function. If the parameter

$ t $

is chosen small enough, the approximation is majorant, and minimizing the approximation improves the current solution until convergence. When the non-smooth component

$ {\cal H} $

is separable (as is the case of the

$ {\ell _1} $

norm), minimizing the local approximation is easily done [see Fig. (5)].

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Fig. 5. Computation of the proximal operator: an illustration with the case of the

$ {l_1} $

norm under positivity constraints. Values that are smaller than

$ \mu t $

are mapped to zero by the proximal operator. Larger values of the regularization weight

$ \mu $

therefore lead to more values being set to zero, i.e., a sparser solution.

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Fig. 6. Phase image reconstructed from simulated data with different methods and regularization terms. (a) Data (in-line hologram). The yellow frames indicate the regions of interest that are extracted from the field of view for visualization. (b) Ground truth phase. (c) Evolution of the normalized cost criterion (minimum and maximum values ranged between zero and one) for each reconstruction. (d)–(i) Reconstructed images using (d) Fienup method, (e) Fienup method + soft-thresholding, (f) Fienup method + soft-thresholding, stopped at 10 iterations, (g) RI method using FISTA with soft-thresholding, (h) RI method using FISTA with soft-thresholding + edge-preserving, (i) RI method using FISTA with soft-thresholding + edge-preserving, stopped at 10 iterations. For each reconstruction, a positivity constraint is imposed on the solution. The values of the hyperparameters are indicated in Table 2. Black dots delimit the contour of the ground truth image. Red curves show line profiles passing through particles of the image.

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Fig. 7. Phase image reconstructed from experimental data with different methods and regularization terms. (a) Data (in-line hologram). The yellow frame indicates the region of interest that is extracted from the field of view for visualization. (b) Evolution of the normalized cost criterion (minimum and maximum values ranged between zero and one) for each reconstruction. (c)–(h) Reconstructed images using (c) Fienup method, (d) Fienup method + soft-thresholding, (e) Fienup method + soft-thresholding, stopped at 10 iterations, (f) RI method using FISTA with soft-thresholding, (g) RI method using FISTA with soft-thresholding + edge-preserving, (h) RI method using FISTA with soft-thresholding + edge-preserving, stopped at 10 iterations. For each reconstruction, a positivity constraint is imposed on the solution. The values of the hyperparameters are indicated in Table 2. Red curves show a line profile passing through two particles of the image.

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Fig. 8. Reconstruction (detection) of an out-of-field particle using RI method and field-of-view extension. (a) Data in-line hologram highlighting (in yellow) a part of the out-of-field hologram. (b) Ground truth image. (c), (d) Reconstructions with two different sets of regularization weights values [cf. Table 2]. The red frames show the initial data field of view (pixels seen by the detector). The green frames are zooms on the out-of-field particle.

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Fig. 9. Derivation of a reconstruction algorithm from a cost function.

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Tables (6)

Algorithm 1. Fienup’s ER algorithm [4]

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Algorithm 2. FISTA algorithm [61] for solving problem Eq. (19)

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Table 1. Experimental and Simulation Parameters of In-Line Holograms Data

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Table 2. Reconstruction Parameters for Simulations and Experiments in Figs. 3,6–8

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Algorithm 3. Detailed implementation of Algorithm 1

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Algorithm 4. Detailed implementation of Algorithm 2

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Equations (29)

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\underline{t} (r) = 1 + \underline{o} (r) = ρ (r) e^{i φ (r)} .

\begin{aligned} I (r) = {| \iint_{R^{2}} {\underline{h}}_{z} (r - r^{'}) {\underline{a}}_{0} (r^{'}) \underline{t} (r^{'}) d r^{'} |}^{2} = | {\underline{h}}_{z} \underset{r}{*} ({\underline{a}}_{0} \cdot \underline{t}) |^{2}, \end{aligned}

{\underline{h}}_{z} (r) = \frac{1}{i λ z} \exp (\frac{i π}{λ z} ∥ r ∥^{2}) .

\begin{aligned} I (r) = | {\underline{a}}_{0} |^{2} | 1 + {\underline{h}}_{z} \underset{r}{*} \underline{o} |^{2} = | {\underline{a}}_{0} |^{2} \underset{model m (\underline{o})}{\underset{⏟}{(1 + 2 R e ({\underline{h}}_{z} \underset{r}{*} \underline{o}) + | {\underline{h}}_{z} \underset{r}{*} \underline{o} |^{2})}} . \end{aligned}

d = c m (\underline{o}) + η = c | 1 + {\underline{H}}_{z} \underline{o} |^{2} + η,

\underline{t} (r) = 1 + \underline{o} (r) = e^{i φ (r)} \approx 1 + i φ (r) = 1 + i o (r);

\tilde{m} (o) = 1 - 2 J m ({\underline{h}}_{z}) * o .

d = c \tilde{m} (o) + η with \tilde{m} (o) = (1 + G_{z} o),

\begin{aligned} J_{fid} (c, o, d) & = {‖ c \tilde{m} (o) - d ‖}_{W}^{2} \\ = {(c \tilde{m} (o) - d)}^{T} W (c \tilde{m} (o) - d) \\ = \sum_{q} w_{q} {({\tilde{m}}_{q} (o) - d_{q})}^{2} (if errors are
uncorrelated,i.e., W is diagonal) \\ = \sum_{q} w_{q} (1 + {[g_{z} * o]}_{q} - d_{q})^{2}, \end{aligned}

{o^{*}, c^{*}} = \underset{o, c}{argmin} J_{fid} (c, o, d)

c^{*} (o) = \frac{\tilde{m} {(o)}^{T} W d}{\tilde{m} {(o)}^{T} W \tilde{m} (o)} .

c^{*} (o) = \frac{\sum_{q} w_{q} {\tilde{m}}_{q} (o) d_{q}}{\sum_{q} w_{q} {\tilde{m}}_{q} {(o)}^{2}} .

o^{*} = \underset{o}{arg min} J_{fid} (c^{*} (o), o, d) .

o^{*} = \underset{o \in O}{arg min} J_{fid} (c^{*} (o), o, d) + J_{reg} (o, θ),

J_{l_{2} \nabla} (o, μ) = μ \sum_{q} | | \nabla_{q} o | |_{2}^{2},

J_{{TV}_{ϵ}} (o, μ, ϵ) = μ \sum_{q} \sqrt{| | \nabla_{q} o | |_{2}^{2} + ϵ^{2}},

J_{l_{1}} (o, μ) = μ {‖ o ‖}_{1} = μ \sum_{q} | o_{q} | .

J_{reg} (o, θ) = J_{l_{1}} (o, μ_{l_{1}}) + J_{{TV}_{ϵ}} (o, μ_{TV}, ϵ_{TV}),

\begin{aligned} o^{*} & = \underset{o \geq 0}{arg min} \underset{smooth part G}{\underset{⏟}{J_{fid} (c^{*} (o), o, d)}} + \underset{non - smooth part H}{\underset{⏟}{J_{l_{1}} (o, μ)}} \\ = \underset{o \geq 0}{arg min} {‖ c^{*} (o) \tilde{m} (o) - d ‖}_{W}^{2} + μ {‖ o ‖}_{1} . \end{aligned}

\begin{aligned} o^{(i + 1)} & = T_{μ t} (o^{(i)} - t \nabla G (o^{(i)})) \\ = T_{μ t} (o^{(i)} - t \nabla J_{fid} (c^{*} (o^{(i)}), o^{(i)}, d)) \\ = T_{μ t} (o^{(i)} - 2 t c^{*} (o^{(i)}) {G_{z}}^{T} W (c^{*} (o^{(i)}) \tilde{m} (o^{(i)}) - d)), \end{aligned}

T_{α} (o)_{q} = max (0, o_{q} - α) .

{\underline{o}}^{(i + 1)} = P_{O} ({\underline{H}}_{z}^{†} (\sqrt{\bar{d}} ⊙ \frac{{\underline{a}}_{z}^{(i + 1 / 2)}}{| {\underline{a}}_{z}^{(i + 1 / 2)} |} - 1)),

\begin{aligned} {\underline{o}}^{(i + 1)} \approx P_{O} [{\underline{o}}^{(i)} - {\underline{H}}_{z}^{†} (\frac{{\underline{a}}_{z}^{(i + 1 / 2)}}{| {\underline{a}}_{z}^{(i + 1 / 2)} |} ⊙ (| {\underline{a}}_{z}^{(i + 1 / 2)} | - \sqrt{\bar{d}}))] . \end{aligned}

\begin{aligned} J_{fid} (o, \bar{d}) = {‖ | \underline{a_{z}} (\underline{o}) | - \sqrt{\bar{d}} ‖}_{2}^{2} = {‖ | 1 + {\underline{H}}_{z} \underline{o} | - \sqrt{\bar{d}} ‖}_{2}^{2} . \end{aligned}

ι_{O} (\underline{o}) = {\begin{array}{l} 0 & if \underline{o} \in O \\ + \infty & otherwise \end{array} .

{\underline{o}}^{*} = \underset{\underline{o}}{a r g m i n} \underset{s m o o t h p a r t G}{\underset{⏟}{{‖ | 1 + {\underline{H}}_{z} \underline{o} | - \sqrt{\bar{d}} ‖}_{2}^{2}}} + \underset{n o n - s m o o t h p a r t H}{\underset{⏟}{ι_{O} (\underline{o})}},

\begin{aligned} o^{*} & = \underset{o \geq 0}{a r g m i n} \underset{s m o o t h p a r t G}{\underset{⏟}{J_{fid} (c^{*} (o), o, d) + J_{{T V}_{ϵ}} (o, μ_{T V})}} \\ + \underset{n o n - s m o o t h p a r t H}{\underset{⏟}{J_{l_{1}} (o, μ_{l_{1}})}} \\ = \underset{o \geq 0}{a r g m i n} {‖ c^{*} (o) \tilde{m} (o) - d ‖}_{W}^{2} \\ + μ_{T V} \sum_{q} \sqrt{| | \nabla_{q} o | |_{2}^{2}} + ϵ^{2} + μ_{l_{1}} {‖ o ‖}_{1} . \end{aligned}

\begin{aligned} o^{(i + 1)} & = T_{μ_{l_{1}} t} (o^{(i)} - t (\nabla J_{fid} (c^{*} (o^{(i)}), o^{(i)}, d) + \nabla J_{{T V}_{ϵ}} (o^{(i)}, μ_{TV}))) \\ = T_{μ_{l_{1}} t} (o^{(i)} - 2 t c^{*} (o^{(i)}) G_{z}^{T} W (c^{*} \tilde{m} (o^{(i)}) - d) \\ - t μ_{TV} \sum_{q} \nabla_{q}^{T} (\frac{\nabla_{q} o^{(i)}}{\sqrt{| | \nabla_{q} o^{(i)} | |_{2}^{2} + ϵ^{2}}})) . \end{aligned}

\tilde{m} (o) = T (1 + G_{z} o) .

		Simulation	Experiment
Illumination wavelength (in nm)	$λ$	532
Propagation distance (in µm)	$z$	12.5	7.3
Refractive index of the medium	$n_{0}$	1.52
Refractive index of the beads	$n_{bead}$	1.59	1.59 (commercial data)
Diameter of the beads (in µm)	$d_{bead}$	1.0	1.0 (commercial data)
Magnification factor (microscope objective)	$m a g$	56.7
Size of the image (in pixels)	$N = M$	$512 \times 512$	$1920 \times 1080$
Pixel size (in µm)	$Δ_{x} = Δ_{y}$	4.4	2.2
Noise type	$η$	Gaussian i.i.d.	—
Signal-to-noise ratio	$⟨ d ⟩ / σ_{η}$	200	—

		Reconstruction Parameters
		Gradient Descent Step		Gradient Smoothness	$l_{1}$ -Sparsity	Edge-Preserving Smoothness		Max. nb. of
Figure	Method	$t$	Positivity	$μ_{l_{2} \nabla}$	$μ_{l_{1}}$	$μ_{TV}$	$ϵ_{TV}$	Iterations
Fig. 3(b); Fig. 6(b)	Ground Truth
Fig. 6(d)	Fienup	1.0	Yes	—	—	—	—	1000
Fig. 6(e)	Fienup	1.0	Yes	—	0.01	—	—	1000
Fig. 6(f)	Fienup	1.0	Yes	—	0.01	—	—	10
Fig. 3(c)	RI \| VMLMB	—	Yes	1.0	—	—	—	1000
Fig. 3(d)	RI \| VMLMB	—	Yes	—	—	0.1	0.01	1000
Fig. 3(e); Fig. 6(g)	RI \| FISTA	0.1	Yes	—	0.1	—	—	1000
Fig. 3(f); Fig. 6(h)	RI \| FISTA	0.05	Yes	—	0.1	0.1	0.01	1000
Fig. 6(i)	RI \| FISTA	0.05	Yes	—	0.1	0.1	0.01	10
Fig. 7(c)	Fienup	1.0	Yes	—	—	—	—	100
Fig. 7(d)	Fienup	1.0	Yes	—	0.1	—	—	100
Fig. 7(e)	Fienup	1.0	Yes	—	0.1	—	—	10
Fig. 7(f)	RI \| FISTA	0.1	Yes	—	0.5	—	—	100
Fig. 7(g)	RI \| FISTA	0.1	Yes	—	0.5	0.01	0.01	100
Fig. 7(h)	RI \| FISTA	0.1	Yes	—	0.5	0.01	0.01	10
Fig. 8(c)	RI \| FISTA	0.01	Yes	—	0.1	0.001	0.01	100
Fig. 8(d)	RI \| FISTA	0.01	Yes	—	0.1	0.1	0.01	100

Input:
$d \in R^{N_{x} \times N_{y}}$ ;	{intensity measurements}
${\underline{h}}_{z} \in C^{N_{x} \times N_{y}}$ ;	{propagation kernel ${\underline{h}}_{z}$ }
${\underline{h}}_{- z} \in C^{N_{x} \times N_{y}}$ ;	{backpropagation kernel ${\underline{h}}_{- z}$ }
$Λ r m i n \in R^{N_{x} \times N_{y}}$ ;	{minimum bound constraint for each pixel of $R e (o)$ (feasible domain $O$ )}
$Λ r m a x \in R^{N_{x} \times N_{y}}$ ;	{maximum bound constraint for each pixel of $R e (o)$ (feasible domain $O$ )}
$Λ i m i n \in R^{N_{x} \times N_{y}}$ ;	{minimum bound constraint for each pixel of $J m (o)$ (feasible domain $O$ )}
$Λ i m a x \in R^{N_{x} \times N_{y}}$ ;	{maximum bound constraint for each pixel of $J m (o)$ (feasible domain $O$ )}
maxiter;	{maximum number of iterations}
Output:
$\underline{o} \in C^{N_{x} \times N_{y}}$ ;	{unknown deviation from unit transmittance $\underline{t}$ }
Allocations:
$\underline{a} \in C^{N_{x} \times N_{y}}$ ;	{array for storing the simulated diffracted wave $a_{z}$ }
begin
	{Step 0: initializations}
1 $d \leftarrow normalize (d)$ ;	{normalize the data hologram so that the background equals 1}
	{N.B.: for instance, divide $d$ by its mean or its median (almost valid in the case of a low-density distribution of objects).}
2 $\underline{o} \leftarrow {\underline{h}}_{- z} * (\sqrt{d} - 1)$ ;	{first guess: for instance, direct backpropagation of the data (or initialization with random values).}
	{N.B.: the square root $\sqrt{}$ is applied pixelwise.}
for $i \leftarrow 1 : m a x i t e r$ do
3 $\underline{a} \leftarrow 1 + {\underline{h}}_{z} * \underline{o}$ ;	{Step 1: propagation to the sensor plane}
for $q \leftarrow 1 : N_{x} \cdot N_{y}$ do	{Step 2: enforce the measured amplitude at sensor plane}
if ${\underline{a}}_{q} \neq 0$ then
4 ${\underline{a}}_{q} \leftarrow \sqrt{d_{q}} \cdot ({\underline{a}}_{q} / \| {\underline{a}}_{q} \|)$ ;
else
5 ${\underline{a}}_{q} \leftarrow 0$ ;
end
end
6 $\underline{o} \leftarrow {\underline{h}}_{- z} * (\underline{a} - 1);$	{Step 3: backpropagation to the sample plane}
for $q \leftarrow 1 : N_{x} \cdot N_{y}$ do	{Step 4: projection on the domain $O$ }
7 $R e {(\underline{o})}_{q} \leftarrow m i n (R e {(\underline{o})}_{q}, Λ {r m a x}_{q});$
8 $R e {(\underline{o})}_{q} \leftarrow m a x (R e {(\underline{o})}_{q}, Λ {r m i n}_{q});$
9 $J m {(\underline{o})}_{q} \leftarrow m i n (J m {(\underline{o})}_{q}, Λ {r m a x}_{q});$
10 $J m {(\underline{o})}_{q} \leftarrow m a x (J m {(\underline{o})}_{q}, Λ {r m i n}_{q});$
	{N.B.: the operators $R e ()$ and $J m ()$ , giving, respectively, the real and imaginary parts (arrays) of the array $\underline{o}$ , are applied pixelwise.}
end
end
end

Input:
$d \in R^{N_{x} \times N_{y}}$ ;	{intensity measurements}
$g_{z} \in R^{N_{x} \times N_{y}}$ ;	{propagation kernel $g_{z}$ }
$w \in R^{N_{x} \times N_{y}}$ ;	{confidence weights applied to each pixel data (these arrays’ values constitute the diagonal of the matrix $W$ in Eq. (9))}
$f l a g_{p o s}$ ;	{flag ( $t r u e$ or $f a l s e$ ) for enforcing a positivity constraint}
$μ$ ;	{hyperparameter value for the soft-thresholding (sparsity contraint)}
$t$ ;	{steepest gradient descent step length}
Output:
$o \in R^{N_{x} \times N_{y}}$ ;	{unknown deviation from unit transmittance $\underline{t}$ }
Allocations:
$o_{p r e v} \in R^{N_{x} \times N_{y}}$ ;	{estimate at previous iterate}
$\tilde{m} \in R^{N_{x} \times N_{y}}$ ;	{array for storing the direct model}
$r \in R^{N_{x} \times N_{y}}$ ;	{array for storing the residues (difference between the model and the data)}
$u \in R^{N_{x} \times N_{y}}$ ;	{intermediate array}
$c \in R$ ;	{intensity hologram scaling factor}
$s \in R$ ;	{scalar factor for the acceleration step}
$s_{p r e v} \in R$ ;	{previous scalar factor for the acceleration step}
begin
	{Step 0: initializations}
1 $o_{p r e v} \leftarrow zeros (N_{x}, N_{y})$ ;	{first guess: for instance, $zeros (N_{x}, N_{y})$ returns a $N_{x} \times N_{y}$ array of zeros.}
2 $u \leftarrow o_{p r e v}$ ;
3 $s_{p r e v} \leftarrow 1$ ;
repeat
4 $\tilde{m} \leftarrow 1 + [g_{z} * u]$ ;	{Step 1: calculate the direct model}
5 $c \leftarrow \sum_{q} w_{q} {\tilde{m}}_{q} d_{q} / \sum_{q} w_{q} {\tilde{m}}_{q}^{2}$ ;	{Step 2: get optimal current scaling factor}
for $q \leftarrow 1 : N_{x} \cdot N_{y}$ do
6 $r_{q} \leftarrow w_{q} (c {\tilde{m}}_{q} - d_{q})$ ;	{Step 3: get the weighted residual pixel values}
end
7 $r \leftarrow g_{z} * r$ ;	{Step 4: backpropagation of the residues}
	{N.B.: the backpropagation kernel is equal to the propagation kernel (cf. Section 4.1)}
for $q \leftarrow 1 : N_{x} \cdot N_{y}$ do
8 $o_{q} \leftarrow u_{q} - 2 t c r_{q}$ ;	{Step 5: steepest gradient descent step}
if $f l a g_{p o s} = t r u e$ then	{Step 6: soft-thresholding}
9 $o_{q} \leftarrow max (0, o_{q} - μ t)$ ;
else
10 $o_{q} \leftarrow sign (o_{q}) max (0, \| o_{q} \| - μ t)$ ;
end
end
11 $s \leftarrow \frac{1}{2} (1 + \sqrt{1 + 4 {(s_{p r e v})}^{2}})$ ;	{Step 7: get new scalar factor $s$ for the acceleration step}
12 $u \leftarrow o + \frac{s_{p r e v} - 1}{s} (o - o_{p r e v})$ ;	{Step 8: new intermediate array $u$ according to previous iterate}
	{N.B.: this step is applied pixelwise.}
13 $s_{p r e v} \leftarrow s$ ;
14 $o_{p r e v} \leftarrow o$ ;	{Step 9: update memory of previous values}
until convergence;
end

		Simulation	Experiment
Illumination wavelength (in nm)	$λ$	532
Propagation distance (in µm)	$z$	12.5	7.3
Refractive index of the medium	$n_{0}$	1.52
Refractive index of the beads	$n_{bead}$	1.59	1.59 (commercial data)
Diameter of the beads (in µm)	$d_{bead}$	1.0	1.0 (commercial data)
Magnification factor (microscope objective)	$m a g$	56.7
Size of the image (in pixels)	$N = M$	$512 \times 512$	$1920 \times 1080$
Pixel size (in µm)	$Δ_{x} = Δ_{y}$	4.4	2.2
Noise type	$η$	Gaussian i.i.d.	—
Signal-to-noise ratio	$⟨ d ⟩ / σ_{η}$	200	—

Abstract

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Equations (29)

Journal of the Optical Society of America A