Numerical optimization algorithms for wavefront phase retrieval from multiple measurements

Ji Li; Tie Zhou

doi:10.3934/ipi.2017034

Article Contents

2017, Volume 11, Issue 4: 721-743. Doi: 10.3934/ipi.2017034

This issue Previous Article Convergence of the gradient method for ill-posed problems Next Article Increasing stability for the inverse source scattering problem with multi-frequencies

Numerical optimization algorithms for wavefront phase retrieval from multiple measurements

Ji Li and
Tie Zhou^,

LMAM, School of Mathematical Sciences, Peking University, Beijing 100871, China

* Corresponding author: Ji Li, Tie Zhou
* Corresponding author: Ji Li, Tie Zhou

Received: February 29, 2016

Revised: February 28, 2017

Published: October 2017

This work was supported by NSFC grants (61421062,11471024).

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Abstract

Wavefront phase retrieval from a set of intensity measurements can be formulated as an optimization problem. Two nonconvex models (MLP and its variant LS) based on maximum likelihood estimation are investigated in this paper. We derive numerical optimization algorithms for real-valued function of complex variables and apply them to solve the wavefront phase retrieval problem efficiently. Numerical simulation is given with application to three test examples. The LS model shows better numerical performance than that of the MLP model. An explanation for this is that the distribution of the eigenvalues of Hessian matrix of the LS model is more clustered than that of the MLP model. We find that the LBFGS method shows more robust performance and takes fewer calculations than other line search methods for this problem.

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Mathematics Subject Classification: Primary: 49N45; Secondary: 49N30.

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Figure 1. Three test examples: (a) Zernike (size $128\times 128$), (b) von Karman (size $128\times 128$) and (c) JWST (size $1024\times 1024$). (d) is the colorbar used through this paper, if not specified. The pointwise angle (defined in interval $(-\pi,\pi]$) of the complex wavefront is plotted

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Figure 2. Comparison of algorithms (SD, NCG, LBFGS, TN) for (a) Zernike and (b) von Karman examples in LS model with noiseless data. Top row plots RMS versus iterations, bottom row shows the change of misfit function versus iterations

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Figure 3. Comparison of the MLP, LS and LSI models for two examples: (a) Zernike, (b) von Karman with noiseless data. RMSs of the solution versus iterations are plotted

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Figure 4. Reconstructed wavefront for three examples: (a) Zernike, (b) von Karman, (c) JWST with noisy data in different SNR levels. Top row is without noise, then the SNR level decreasing from $30$dB to $10$dB

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Figure 5. Difference between reconstructed and original wavefront: (a) Zernike, (b) von Karman, (c) JWST for SNR levels $20$dB (top row), $10$dB (bottom row), respectively. (d) is the corresponding colorbar

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Algorithm 1 LBFGS two-loop recursion
Input: $\boldsymbol{g}_k$, $\boldsymbol{s}_i=\boldsymbol{z}_{i+1}-\boldsymbol{z}_i$, $\boldsymbol{y}_i=\boldsymbol{g}_{i+1}-\boldsymbol{g}_i$, $\rho_i =\frac{1}{{\rm{Re}}(\boldsymbol{y}_i^*\boldsymbol{s}_i)}$, for $i=k-m,\ldots,k-1$,
Output: $\boldsymbol{d}$, such that $\boldsymbol{d}^{\mathcal{C}}=-B_k^{\mathcal{C}}\boldsymbol{g}_k^{\mathcal{C}}$
$\boldsymbol{d}\leftarrow -\boldsymbol{g}_k$
for $i=k-1,k-2,\ldots,k-m$ do
$\alpha_i = \rho_i{\rm{Re}}(\boldsymbol{s}_i^*\boldsymbol{d})$
$\boldsymbol{d}\leftarrow \boldsymbol{d}-\alpha_i \boldsymbol{y}_i$
end for
$\boldsymbol{d}\leftarrow \gamma\boldsymbol{d}$, where $\gamma=\frac{{\rm{Re}}(\boldsymbol{y}_{k-1}^\boldsymbol{s}_{k-1})}{\boldsymbol{y}_{k-1}^\boldsymbol{y}_{k-1}}$
for $i=k-m,k-m+1,\ldots,k-1$ do
$\beta\leftarrow \rho_i{\rm{Re}}(\boldsymbol{y}_i^*\boldsymbol{d})$
$\boldsymbol{d}\leftarrow \boldsymbol{d}+(\alpha_i-\beta)\boldsymbol{s}_i$
end for

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Table 1. Total average number of FFT calls for different methods in 10 independent runs

Case	SD	NCG	LBFGS	TN
Zernike	1309	545	299	1559
von Karman	1868	713	418	2767

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