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.

[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.

[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.

[4]

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

[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.

[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.

[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.

[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.

[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.

[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.

[11]

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

[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.

[13]

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

[14]

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

[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.

[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.

[17]

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

[18]

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

[19]

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

[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.

[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.

[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.

[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.

[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.

[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.

[26]

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

[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.

[28]

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

[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.

[30]

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

[31]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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.

[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.

[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.

[4]

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

[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.

[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.

[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.

[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.

[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.

[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.

[11]

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

[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.

[13]

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

[14]

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

[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.

[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.

[17]

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

[18]

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

[19]

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

[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.

[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.

[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.

[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.

[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.

[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.

[26]

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

[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.

[28]

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

[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.

[30]

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

[31]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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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