# American Institute of Mathematical Sciences

August  2017, 11(4): 721-743. doi: 10.3934/ipi.2017034

## Numerical optimization algorithms for wavefront phase retrieval from multiple measurements

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

* Corresponding author: Ji Li, Tie Zhou

Received  March 2016 Revised  March 2017 Published  June 2017

Fund Project: This work was supported by NSFC grants (61421062,11471024)

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.

Citation: Ji Li, Tie Zhou. Numerical optimization algorithms for wavefront phase retrieval from multiple measurements. Inverse Problems & Imaging, 2017, 11 (4) : 721-743. doi: 10.3934/ipi.2017034
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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
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
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
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
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
 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
 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
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
 Case SD NCG LBFGS TN Zernike 1309 545 299 1559 von Karman 1868 713 418 2767
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