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
References:
[1]

E. J. Akutowicz, On the determination of the phase of a Fourier integral, Ⅰ, Trans. Am. Math. Soc., 83 (1956), 179-192. doi: 10.2307/1992910. Google Scholar

[2]

E. J. Akutowicz, On the determination of the phase of a Fourier integral, Ⅱ, Proc. Am. Math. Soc., 8 (1957), 234-238. doi: 10.2307/2033718. Google Scholar

[3]

R. Barakat and G. Newsam, Necessary conditions for a unique solution to two-dimensional phase recovery, J. Math. Phys., 25 (1984), 3190-3193. doi: 10.1063/1.526089. Google Scholar

[4]

R. H. T. Bates, Fourier phase problems are uniquely solvable in more than one dimension, Ⅰ, Underlying Theory, 61 (1982), 247-262. Google Scholar

[5]

E. J. CandésY. C. EldarT. Strohmer and V. Voroninski, Phase retrieval via matrix completion, SIAM Rev., 57 (2015), 225-251. doi: 10.1137/151005099. Google Scholar

[6]

E. J. Candés and X. Li, Solving quadratic equations via phaselift when there are about as many equations as unknowns, Found. Comput. Math., 14 (2014), 1017-1026. doi: 10.1007/s10208-013-9162-z. Google Scholar

[7]

E. J. CandésT. Strohmer and V. Voroninski, PhaseLift: Exact and stable signal recovery from magnitude measurements via convex programming, Comm. Pure Appl. Math., 66 (2012), 1241-1274. doi: 10.1002/cpa.21432. Google Scholar

[8]

H. Chang, Y. Lou, M. Ng and T. Zeng, Phase retrieval from incomplete magnitude information via total variation regularization, SIAM J. Sci. Comput., 38 (2016), A3672-A3695, ftp://ftp.math.ucla.edu/pub/camreport/cam16-39.pdf. doi: 10.1137/15M1029357. Google Scholar

[9]

A. Fannjiang, Absolute uniqueness of phase retrieval with random illumination, Inverse Problems, 28 (2012), 075008, 20pp. doi: 10.1088/0266-5611/28/7/075008. Google Scholar

[10]

J. R. Fienup and C. C. Wackerman, Phase-retrieval stagnation problems and solutions, J. Opt. Soc. Am. A, 3 (1986), 1897-1907. doi: 10.1364/JOSAA.3.001897. Google Scholar

[11]

J. R. Fienup, Phase retrieval algorithms: A comparison, Appl. Opt., 21 (1982), 2758-2769. doi: 10.1364/AO.21.002758. Google Scholar

[12]

B. A. Frigyik, S. Srivastava and M. R. Gupta, An Introduction to Functional Derivatives, UWEE Tech Report, https://www.ee.washington.edu/techsite/papers/documents/UWEETR-2008-0001.pdf.Google Scholar

[13]

R. A. Gonsalves, Phase retrieval from modulus data, J. Opt. Soc. Am., 66 (1976), 961-964. doi: 10.1364/JOSA.66.000961. Google Scholar

[14]

J. W. Goodman, Introduction to Fourier Optics, Roberts & Company Publishers, 2005.Google Scholar

[15]

M. H. Hayes, The reconstruction of a multidimensional sequence from the phase or magnitude of its Fourier transform, IEEE Trans. Acoust., Speech, Signal Process., 30 (1982), 140-154. doi: 10.1109/TASSP.1982.1163863. Google Scholar

[16]

T. IserniaG. LeoneR. Pierri and F. Soldovieri, Role of support information and zero locations in phase retrieval by a quadratic approach, J. Opt. Soc. Am. A, 16 (1999), 1845-1856. doi: 10.1364/JOSAA.16.001845. Google Scholar

[17]

K. Kreutz-Delgado, The complex gradient operator and the cr-calculus, arXiv: 0906.4835v1.Google Scholar

[18]

T. I. Kuznetsova, On the phase retrieval problem in optics Sov. Phys. Usp. , 31 (1988), p364. doi: 10.1070/pu1988v031n04abeh005755. Google Scholar

[19]

J. Li and T. Zhou, On gradient descent algorithm for generalized phase retrieval problem, arXiv: 1607.01121v1.Google Scholar

[20]

X. Li and V. Voroninski, Sparse signal recovery from quadratic measurements via convex programming, SIAM J. Math. Anal., 45 (2013), 3019-3033. doi: 10.1137/120893707. Google Scholar

[21]

D. R. Luke, Relaxed averaged alternating reflections for diffraction imaging, Inverse Problems, 21 (2004), 37-50. doi: 10.1088/0266-5611/21/1/004. Google Scholar

[22]

D. R. LukeH. H. Bauschke and P. L. Combettes, Hybrid projection-reflection method for phase retrieval, J. Opt. Soc. Am. A, 20 (2003), 1025-1034. Google Scholar

[23]

D. R. LukeJ. V. Burke and R. G. Lyon, Optical wavefront reconstruction: Theory and numerical methods, SIAM Rev., 44 (2002), 169-224. doi: 10.1137/S003614450139075. Google Scholar

[24]

V. N. Mahajan and G.-M. Dai, Orthonormal polynomials in wavefront analysis: Analytical solution, J. Opt. Soc. Am. A, 24 (2007), 2994-3016. doi: 10.1364/FIO.2006.FWX1. Google Scholar

[25]

S. Maretzke, A uniqueness result for propagation-based phase contrast imaging from a single measurement Inverse Problems, 31 (2015), 065003, 16pp. doi: 10.1088/0266-5611/31/6/065003. Google Scholar

[26]

J. MiaoJ. Kirz and D. Sayre, The oversampling phasing method, Acta Crystallogr D Biol Cryst, 56 (2000), 1312-1315. doi: 10.1107/S0907444900008970. Google Scholar

[27]

R. P. Millane, Phase retrieval in crystallography and optics, J. Opt. Soc. Am. A, 7 (1990), 394-411. doi: 10.1364/JOSAA.7.000394. Google Scholar

[28]

D. L. Misell, A method for the solution of the phase problem in electron microscopy, J. Phys. D: Appl. Phys., 6 (1973), L6.Google Scholar

[29]

J. J. Moré and D. J. Thuente, Line search algorithms with guaranteed sufficient decrease, ACM Transactions on Mathematical Software (TOMS), 20 (1994), 286-307. doi: 10.1145/192115.192132. Google Scholar

[30]

J. Nocedal and S. Wright, Numerical Optimization, Springer Science & Business Media, 2006. Google Scholar

[31]

A. Pascarella and A. Sorrentino, Statistical approaches to the inverse problem, INTECH Open Access Publisher, 5 (2011), 93-112. Google Scholar

[32]

J. QianC. YangA. SchirotzekF. Maia and S. Marchesini, Efficient algorithms for ptychographic phase retrieval, Contemporary Mathematics, American Mathematical Society (AMS), 615 (2014), 261-279. doi: 10.1090/conm/615/12259. Google Scholar

[33]

J. L. C. Sanz, Mathematical considerations for the problem of Fourier transform phase retrieval from magnitude, SIAM J. Appl. Math., 45 (1985), 651-664. doi: 10.1137/0145038. Google Scholar

[34]

J. L. C. SanzT. S. Huang and F. Cukierman, Stability of unique Fourier-transform phase reconstruction, J. Opt. Soc. Am., 73 (1983), 1442-1445. doi: 10.1364/josa.73.001442. Google Scholar

[35]

Y. ShechtmanY. C. EldarO. CohenH. N. ChapmanJ. Miao and M. Segev, Phase retrieval with application to optical imaging: A contemporary overview, IEEE Signal Process. Mag., 32 (2015), 87-109. doi: 10.1109/MSP.2014.2352673. Google Scholar

[36]

D. L. SnyderR. L. White and A. M. Hammoud, Image recovery from data acquired with a charge-coupled-device camera, J. Opt. Soc. Am. A, 10 (1993), 1014-1023. doi: 10.1364/JOSAA.10.001014. Google Scholar

[37]

T. Steihaug, The conjugate gradient method and trust regions in large scale optimization, SIAM J. Numer. Anal., 20 (1983), 626-637. doi: 10.1137/0720042. Google Scholar

[38]

A. van den Bos, Complex gradient and hessian, IEE Proc., Vis. Image Signal Process., 141 (1994), 380-382. doi: 10.1049/ip-vis:19941555. Google Scholar

show all references

References:
[1]

E. J. Akutowicz, On the determination of the phase of a Fourier integral, Ⅰ, Trans. Am. Math. Soc., 83 (1956), 179-192. doi: 10.2307/1992910. Google Scholar

[2]

E. J. Akutowicz, On the determination of the phase of a Fourier integral, Ⅱ, Proc. Am. Math. Soc., 8 (1957), 234-238. doi: 10.2307/2033718. Google Scholar

[3]

R. Barakat and G. Newsam, Necessary conditions for a unique solution to two-dimensional phase recovery, J. Math. Phys., 25 (1984), 3190-3193. doi: 10.1063/1.526089. Google Scholar

[4]

R. H. T. Bates, Fourier phase problems are uniquely solvable in more than one dimension, Ⅰ, Underlying Theory, 61 (1982), 247-262. Google Scholar

[5]

E. J. CandésY. C. EldarT. Strohmer and V. Voroninski, Phase retrieval via matrix completion, SIAM Rev., 57 (2015), 225-251. doi: 10.1137/151005099. Google Scholar

[6]

E. J. Candés and X. Li, Solving quadratic equations via phaselift when there are about as many equations as unknowns, Found. Comput. Math., 14 (2014), 1017-1026. doi: 10.1007/s10208-013-9162-z. Google Scholar

[7]

E. J. CandésT. Strohmer and V. Voroninski, PhaseLift: Exact and stable signal recovery from magnitude measurements via convex programming, Comm. Pure Appl. Math., 66 (2012), 1241-1274. doi: 10.1002/cpa.21432. Google Scholar

[8]

H. Chang, Y. Lou, M. Ng and T. Zeng, Phase retrieval from incomplete magnitude information via total variation regularization, SIAM J. Sci. Comput., 38 (2016), A3672-A3695, ftp://ftp.math.ucla.edu/pub/camreport/cam16-39.pdf. doi: 10.1137/15M1029357. Google Scholar

[9]

A. Fannjiang, Absolute uniqueness of phase retrieval with random illumination, Inverse Problems, 28 (2012), 075008, 20pp. doi: 10.1088/0266-5611/28/7/075008. Google Scholar

[10]

J. R. Fienup and C. C. Wackerman, Phase-retrieval stagnation problems and solutions, J. Opt. Soc. Am. A, 3 (1986), 1897-1907. doi: 10.1364/JOSAA.3.001897. Google Scholar

[11]

J. R. Fienup, Phase retrieval algorithms: A comparison, Appl. Opt., 21 (1982), 2758-2769. doi: 10.1364/AO.21.002758. Google Scholar

[12]

B. A. Frigyik, S. Srivastava and M. R. Gupta, An Introduction to Functional Derivatives, UWEE Tech Report, https://www.ee.washington.edu/techsite/papers/documents/UWEETR-2008-0001.pdf.Google Scholar

[13]

R. A. Gonsalves, Phase retrieval from modulus data, J. Opt. Soc. Am., 66 (1976), 961-964. doi: 10.1364/JOSA.66.000961. Google Scholar

[14]

J. W. Goodman, Introduction to Fourier Optics, Roberts & Company Publishers, 2005.Google Scholar

[15]

M. H. Hayes, The reconstruction of a multidimensional sequence from the phase or magnitude of its Fourier transform, IEEE Trans. Acoust., Speech, Signal Process., 30 (1982), 140-154. doi: 10.1109/TASSP.1982.1163863. Google Scholar

[16]

T. IserniaG. LeoneR. Pierri and F. Soldovieri, Role of support information and zero locations in phase retrieval by a quadratic approach, J. Opt. Soc. Am. A, 16 (1999), 1845-1856. doi: 10.1364/JOSAA.16.001845. Google Scholar

[17]

K. Kreutz-Delgado, The complex gradient operator and the cr-calculus, arXiv: 0906.4835v1.Google Scholar

[18]

T. I. Kuznetsova, On the phase retrieval problem in optics Sov. Phys. Usp. , 31 (1988), p364. doi: 10.1070/pu1988v031n04abeh005755. Google Scholar

[19]

J. Li and T. Zhou, On gradient descent algorithm for generalized phase retrieval problem, arXiv: 1607.01121v1.Google Scholar

[20]

X. Li and V. Voroninski, Sparse signal recovery from quadratic measurements via convex programming, SIAM J. Math. Anal., 45 (2013), 3019-3033. doi: 10.1137/120893707. Google Scholar

[21]

D. R. Luke, Relaxed averaged alternating reflections for diffraction imaging, Inverse Problems, 21 (2004), 37-50. doi: 10.1088/0266-5611/21/1/004. Google Scholar

[22]

D. R. LukeH. H. Bauschke and P. L. Combettes, Hybrid projection-reflection method for phase retrieval, J. Opt. Soc. Am. A, 20 (2003), 1025-1034. Google Scholar

[23]

D. R. LukeJ. V. Burke and R. G. Lyon, Optical wavefront reconstruction: Theory and numerical methods, SIAM Rev., 44 (2002), 169-224. doi: 10.1137/S003614450139075. Google Scholar

[24]

V. N. Mahajan and G.-M. Dai, Orthonormal polynomials in wavefront analysis: Analytical solution, J. Opt. Soc. Am. A, 24 (2007), 2994-3016. doi: 10.1364/FIO.2006.FWX1. Google Scholar

[25]

S. Maretzke, A uniqueness result for propagation-based phase contrast imaging from a single measurement Inverse Problems, 31 (2015), 065003, 16pp. doi: 10.1088/0266-5611/31/6/065003. Google Scholar

[26]

J. MiaoJ. Kirz and D. Sayre, The oversampling phasing method, Acta Crystallogr D Biol Cryst, 56 (2000), 1312-1315. doi: 10.1107/S0907444900008970. Google Scholar

[27]

R. P. Millane, Phase retrieval in crystallography and optics, J. Opt. Soc. Am. A, 7 (1990), 394-411. doi: 10.1364/JOSAA.7.000394. Google Scholar

[28]

D. L. Misell, A method for the solution of the phase problem in electron microscopy, J. Phys. D: Appl. Phys., 6 (1973), L6.Google Scholar

[29]

J. J. Moré and D. J. Thuente, Line search algorithms with guaranteed sufficient decrease, ACM Transactions on Mathematical Software (TOMS), 20 (1994), 286-307. doi: 10.1145/192115.192132. Google Scholar

[30]

J. Nocedal and S. Wright, Numerical Optimization, Springer Science & Business Media, 2006. Google Scholar

[31]

A. Pascarella and A. Sorrentino, Statistical approaches to the inverse problem, INTECH Open Access Publisher, 5 (2011), 93-112. Google Scholar

[32]

J. QianC. YangA. SchirotzekF. Maia and S. Marchesini, Efficient algorithms for ptychographic phase retrieval, Contemporary Mathematics, American Mathematical Society (AMS), 615 (2014), 261-279. doi: 10.1090/conm/615/12259. Google Scholar

[33]

J. L. C. Sanz, Mathematical considerations for the problem of Fourier transform phase retrieval from magnitude, SIAM J. Appl. Math., 45 (1985), 651-664. doi: 10.1137/0145038. Google Scholar

[34]

J. L. C. SanzT. S. Huang and F. Cukierman, Stability of unique Fourier-transform phase reconstruction, J. Opt. Soc. Am., 73 (1983), 1442-1445. doi: 10.1364/josa.73.001442. Google Scholar

[35]

Y. ShechtmanY. C. EldarO. CohenH. N. ChapmanJ. Miao and M. Segev, Phase retrieval with application to optical imaging: A contemporary overview, IEEE Signal Process. Mag., 32 (2015), 87-109. doi: 10.1109/MSP.2014.2352673. Google Scholar

[36]

D. L. SnyderR. L. White and A. M. Hammoud, Image recovery from data acquired with a charge-coupled-device camera, J. Opt. Soc. Am. A, 10 (1993), 1014-1023. doi: 10.1364/JOSAA.10.001014. Google Scholar

[37]

T. Steihaug, The conjugate gradient method and trust regions in large scale optimization, SIAM J. Numer. Anal., 20 (1983), 626-637. doi: 10.1137/0720042. Google Scholar

[38]

A. van den Bos, Complex gradient and hessian, IEE Proc., Vis. Image Signal Process., 141 (1994), 380-382. doi: 10.1049/ip-vis:19941555. Google Scholar

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
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
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
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
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
Table1 
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
Table 1.  Total average number of FFT calls for different methods in 10 independent runs
CaseSDNCGLBFGSTN
Zernike13095452991559
von Karman18687134182767
CaseSDNCGLBFGSTN
Zernike13095452991559
von Karman18687134182767
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