Abstract

A rank-constrained reformulation of the blind deconvolution problem on images taken with coherent illumination is proposed. Since in the reformulation the rank constraint is imposed on a matrix that is affine in the decision variables, we propose a novel convex heuristic for the blind deconvolution problem. The proposed heuristic allows for easy incorporation of prior information on the decision variables and the use of the phase diversity concept. The convex optimization problem can be iteratively re-parameterized to obtain better estimates. The proposed methods are demonstrated on numerically illustrative examples.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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2019 (2)

S. Ling and T. Strohmer, “Regularized gradient descent: a non-convex recipe for fast joint blind deconvolution and demixing,” Information and Inference: A Journal of the IMA 8, 1–49 (2019).
[Crossref]

H. Chang, P. Enfedaque, and S. Marchesini, “Blind ptychographic phase retrieval via convergent alternating direction method of multipliers,” SIAM J. Imaging Sci. 12, pp. 153–185 (2019).

2018 (4)

2017 (2)

D. Wilding, O. Soloviev, P. Pozzi, G. Vdovin, and M. Verhaegen, “Blind multi-frame deconvolution by tangential iterative projections (tip),” Opt. Express 25, 32305–32322 (2017).
[Crossref]

Y. S. Nashed, T. Peterka, J. Deng, and C. Jacobsen, “Distributed automatic differentiation for ptychography,” Procedia Comput. Sci. 108, 404–414 (2017).
[Crossref]

2016 (1)

M. Tofighi, O. Yorulmaz, K. Köse, D. C. Yıldırım, R. Çetin-Atalay, and A. E. Cetin, “Phase and TV based convex sets for blind deconvolution of microscopic images,” IEEE J. Sel. Top. Signal Process. 10, 81–91 (2016).
[Crossref]

2015 (5)

D. Pastor, T. Stefaniuk, P. Wróbel, C. J. Zapata-Rodríguez, and R. Kotyński, “Determination of the point spread function of layered metamaterials assisted with the blind deconvolution algorithm,” Opt. Quantum Electron. 47, 17–26 (2015).
[Crossref]

Y. Zhang, W. Jiang, and Q. Dai, “Nonlinear optimization approach for Fourier ptychographic microscopy,” Opt. Express 23, 33822–33835 (2015).
[Crossref]

L.-H. Yeh, J. Dong, J. Zhong, L. Tian, M. Chen, G. Tang, M. Soltanolkotabi, and L. Waller, “Experimental robustness of Fourier ptychography phase retrieval algorithms,” Opt. Express 23, 33214–33240 (2015).
[Crossref]

R. Hesse, D. R. Luke, S. Sabach, and M. K. Tam, “Proximal heterogeneous block implicit-explicit method and application to blind ptychographic diffraction imaging,” SIAM J. Imaging Sci. 8, 426–457(2015).
[Crossref]

R. Horstmeyer, R. Y. Chen, X. Ou, B. Ames, J. A. Tropp, and C. Yang, “Solving ptychography with a convex relaxation,” New J. Phys. 17, 053044 (2015).
[Crossref]

2014 (4)

2013 (3)

F. Jian and L. Peng, “A general phase retrieval algorithm based on a ptychographical iterative engine for coherent diffractive imaging,” Chin. Phys. B 22, 014204 (2013).
[Crossref]

M. Foreman, C. Giusca, P. Török, and R. Leach, “Phase-retrieved pupil function and coherent transfer function in confocal microscopy,” J. Microsc. 251, 99–107 (2013).
[Crossref]

Y. Hu, D. Zhang, J. Ye, X. Li, and X. He, “Fast and accurate matrix completion via truncated nuclear norm regularization,” IEEE Trans. Pattern Anal. Mach. Intell. 35, 2117–2130 (2013).
[Crossref]

2012 (1)

P. Thibault and M. Guizar-Sicairos, “Maximum-likelihood refinement for coherent diffractive imaging,” New J. Phys. 14, 063004(2012).
[Crossref]

2011 (2)

A. M. Maiden, M. J. Humphry, F. Zhang, and J. M. Rodenburg, “Superresolution imaging via ptychography,” J. Opt. Soc. Am. A 28, 604–612 (2011).
[Crossref]

S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein, “Distributed optimization and statistical learning via the alternating direction method of multipliers,” Found. Trends Mach. Learn. 3, 1–122 (2011).
[Crossref]

2010 (2)

J.-F. Cai, E. J. Candès, and Z. Shen, “A singular value thresholding algorithm for matrix completion,” SIAM J. Optim. 20, 1956–1982 (2010).
[Crossref]

B. Recht, M. Fazel, and P. A. Parrilo, “Guaranteed minimum-rank solutions of linear matrix equations via nuclear norm minimization,” SIAM Rev. 52, 471–501 (2010).
[Crossref]

2009 (5)

Z. Liu and L. Vandenberghe, “Interior-point method for nuclear norm approximation with application to system identification,” SIAM J. Matrix Anal. Appl. 31, 1235–1256 (2009).
[Crossref]

G. R. Brady, M. Guizar-Sicairos, and J. R. Fienup, “Optical wavefront measurement using phase retrieval with transverse translation diversity,” Opt. Express 17, 624–639 (2009).
[Crossref]

M. Guizar-Sicairos and J. R. Fienup, “Measurement of coherent x-ray focused beams by phase retrieval with transverse translation diversity,” Opt. Express 17, 2670–2685 (2009).
[Crossref]

P. Thibault, M. Dierolf, O. Bunk, A. Menzel, and F. Pfeiffer, “Probe retrieval in ptychographic coherent diffractive imaging,” Ultramicroscopy 109, 338–343 (2009).
[Crossref]

A. M. Maiden and J. M. Rodenburg, “An improved ptychographical phase retrieval algorithm for diffractive imaging,” Ultramicroscopy 109, 1256–1262 (2009).
[Crossref]

2006 (1)

P. Sarder and A. Nehorai, “Deconvolution methods for 3-D fluorescence microscopy images,” IEEE Signal Process. Mag. 23(3), 32–45 (2006).
[Crossref]

2004 (1)

J. M. Rodenburg and H. M. Faulkner, “A phase retrieval algorithm for shifting illumination,” Appl. Phys. Lett. 85, 4795–4797 (2004).
[Crossref]

1999 (1)

J. Miao, P. Charalambous, J. Kirz, and D. Sayre, “Extending the methodology of X-ray crystallography to allow imaging of micrometre-sized non-crystalline specimens,” Nature 400, 342–344 (1999).
[Crossref]

1995 (1)

1993 (1)

1992 (1)

1988 (1)

1982 (1)

R. A. Gonsalves, “Phase retrieval and diversity in adaptive optics,” Opt. Eng. 21, 215829 (1982).
[Crossref]

Ahmed, A.

A. Ahmed, A. Cosse, and L. Demanet, “A convex approach to blind deconvolution with diverse inputs,” in 6th International Workshop on Computational Advances in Multi-Sensor Adaptive Processing (CAMSAP) (IEEE, 2015), pp. 5–8.

Ames, B.

R. Horstmeyer, R. Y. Chen, X. Ou, B. Ames, J. A. Tropp, and C. Yang, “Solving ptychography with a convex relaxation,” New J. Phys. 17, 053044 (2015).
[Crossref]

Ayers, G.

Ba, J.

D. P. Kingma and J. Ba, “Adam: a method for stochastic optimization,” arXiv:1412.6980 (2014).

Becker, J.-M.

R. Mourya, L. Denis, J.-M. Becker, and E. Thiébaut, “A blind deblurring and image decomposition approach for astronomical image restoration,” in 23rd European Signal Processing Conference (EUSIPCO) (IEEE, 2015), pp. 1636–1640.

Boyd, S.

S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein, “Distributed optimization and statistical learning via the alternating direction method of multipliers,” Found. Trends Mach. Learn. 3, 1–122 (2011).
[Crossref]

Brady, G. R.

Brinicombe, A.

Bunk, O.

P. Thibault, M. Dierolf, O. Bunk, A. Menzel, and F. Pfeiffer, “Probe retrieval in ptychographic coherent diffractive imaging,” Ultramicroscopy 109, 338–343 (2009).
[Crossref]

Cai, J.-F.

J.-F. Cai, E. J. Candès, and Z. Shen, “A singular value thresholding algorithm for matrix completion,” SIAM J. Optim. 20, 1956–1982 (2010).
[Crossref]

Candès, E. J.

J.-F. Cai, E. J. Candès, and Z. Shen, “A singular value thresholding algorithm for matrix completion,” SIAM J. Optim. 20, 1956–1982 (2010).
[Crossref]

Cetin, A. E.

M. Tofighi, O. Yorulmaz, K. Köse, D. C. Yıldırım, R. Çetin-Atalay, and A. E. Cetin, “Phase and TV based convex sets for blind deconvolution of microscopic images,” IEEE J. Sel. Top. Signal Process. 10, 81–91 (2016).
[Crossref]

Çetin-Atalay, R.

M. Tofighi, O. Yorulmaz, K. Köse, D. C. Yıldırım, R. Çetin-Atalay, and A. E. Cetin, “Phase and TV based convex sets for blind deconvolution of microscopic images,” IEEE J. Sel. Top. Signal Process. 10, 81–91 (2016).
[Crossref]

Chang, H.

H. Chang, P. Enfedaque, and S. Marchesini, “Blind ptychographic phase retrieval via convergent alternating direction method of multipliers,” SIAM J. Imaging Sci. 12, pp. 153–185 (2019).

Chang, S.

G. Liu, S. Chang, and Y. Ma, “Blind image deblurring using spectral properties of convolution operators,” IEEE Trans. Image Process. 23, 5047–5056 (2014).
[Crossref]

Charalambous, P.

J. Miao, P. Charalambous, J. Kirz, and D. Sayre, “Extending the methodology of X-ray crystallography to allow imaging of micrometre-sized non-crystalline specimens,” Nature 400, 342–344 (1999).
[Crossref]

Chen, M.

Chen, R. Y.

R. Horstmeyer, R. Y. Chen, X. Ou, B. Ames, J. A. Tropp, and C. Yang, “Solving ptychography with a convex relaxation,” New J. Phys. 17, 053044 (2015).
[Crossref]

Chu, E.

S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein, “Distributed optimization and statistical learning via the alternating direction method of multipliers,” Found. Trends Mach. Learn. 3, 1–122 (2011).
[Crossref]

Chung, J.

Cosse, A.

A. Ahmed, A. Cosse, and L. Demanet, “A convex approach to blind deconvolution with diverse inputs,” in 6th International Workshop on Computational Advances in Multi-Sensor Adaptive Processing (CAMSAP) (IEEE, 2015), pp. 5–8.

Dai, Q.

Dainty, J. C.

Demanet, L.

A. Ahmed, A. Cosse, and L. Demanet, “A convex approach to blind deconvolution with diverse inputs,” in 6th International Workshop on Computational Advances in Multi-Sensor Adaptive Processing (CAMSAP) (IEEE, 2015), pp. 5–8.

Deng, J.

Y. S. Nashed, T. Peterka, J. Deng, and C. Jacobsen, “Distributed automatic differentiation for ptychography,” Procedia Comput. Sci. 108, 404–414 (2017).
[Crossref]

Denis, L.

R. Mourya, L. Denis, J.-M. Becker, and E. Thiébaut, “A blind deblurring and image decomposition approach for astronomical image restoration,” in 23rd European Signal Processing Conference (EUSIPCO) (IEEE, 2015), pp. 1636–1640.

Dierolf, M.

P. Thibault, M. Dierolf, O. Bunk, A. Menzel, and F. Pfeiffer, “Probe retrieval in ptychographic coherent diffractive imaging,” Ultramicroscopy 109, 338–343 (2009).
[Crossref]

Doelman, R.

R. Doelman, N. H. Thao, and M. Verhaegen, “Solving large-scale general phase retrieval problems via a sequence of convex relaxations,” J. Opt. Soc. Am. A 35, 1410–1419 (2018).
[Crossref]

R. Doelman and M. Verhaegen, “Sequential convex relaxation for convex optimization with bilinear matrix equalities,” in European Control Conference (ECC) (IEEE, 2016), pp. 1946–1951.

Dong, J.

Eckstein, J.

S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein, “Distributed optimization and statistical learning via the alternating direction method of multipliers,” Found. Trends Mach. Learn. 3, 1–122 (2011).
[Crossref]

Enfedaque, P.

H. Chang, P. Enfedaque, and S. Marchesini, “Blind ptychographic phase retrieval via convergent alternating direction method of multipliers,” SIAM J. Imaging Sci. 12, pp. 153–185 (2019).

Faulkner, H. M.

J. M. Rodenburg and H. M. Faulkner, “A phase retrieval algorithm for shifting illumination,” Appl. Phys. Lett. 85, 4795–4797 (2004).
[Crossref]

Fazel, M.

B. Recht, M. Fazel, and P. A. Parrilo, “Guaranteed minimum-rank solutions of linear matrix equations via nuclear norm minimization,” SIAM Rev. 52, 471–501 (2010).
[Crossref]

Fienup, J. R.

Fish, D.

Foreman, M.

M. Foreman, C. Giusca, P. Török, and R. Leach, “Phase-retrieved pupil function and coherent transfer function in confocal microscopy,” J. Microsc. 251, 99–107 (2013).
[Crossref]

Giusca, C.

M. Foreman, C. Giusca, P. Török, and R. Leach, “Phase-retrieved pupil function and coherent transfer function in confocal microscopy,” J. Microsc. 251, 99–107 (2013).
[Crossref]

Gonsalves, R. A.

R. A. Gonsalves, “Phase retrieval and diversity in adaptive optics,” Opt. Eng. 21, 215829 (1982).
[Crossref]

Goodman, J.

J. Goodman, Introduction to Fourier Optics (McGraw-Hill, 2008).

Guizar-Sicairos, M.

Gupta, A.

He, X.

Y. Hu, D. Zhang, J. Ye, X. Li, and X. He, “Fast and accurate matrix completion via truncated nuclear norm regularization,” IEEE Trans. Pattern Anal. Mach. Intell. 35, 2117–2130 (2013).
[Crossref]

Hesse, R.

R. Hesse, D. R. Luke, S. Sabach, and M. K. Tam, “Proximal heterogeneous block implicit-explicit method and application to blind ptychographic diffraction imaging,” SIAM J. Imaging Sci. 8, 426–457(2015).
[Crossref]

Holmes, T. J.

Horstmeyer, R.

R. Horstmeyer, R. Y. Chen, X. Ou, B. Ames, J. A. Tropp, and C. Yang, “Solving ptychography with a convex relaxation,” New J. Phys. 17, 053044 (2015).
[Crossref]

R. Horstmeyer, X. Ou, J. Chung, G. Zheng, and C. Yang, “Overlapped Fourier coding for optical aberration removal,” Opt. Express 22, 24062–24080 (2014).
[Crossref]

Hu, Y.

Y. Hu, D. Zhang, J. Ye, X. Li, and X. He, “Fast and accurate matrix completion via truncated nuclear norm regularization,” IEEE Trans. Pattern Anal. Mach. Intell. 35, 2117–2130 (2013).
[Crossref]

Humphry, M. J.

Jacobsen, C.

Y. S. Nashed, T. Peterka, J. Deng, and C. Jacobsen, “Distributed automatic differentiation for ptychography,” Procedia Comput. Sci. 108, 404–414 (2017).
[Crossref]

Jian, F.

F. Jian and L. Peng, “A general phase retrieval algorithm based on a ptychographical iterative engine for coherent diffractive imaging,” Chin. Phys. B 22, 014204 (2013).
[Crossref]

Jiang, W.

Jung, P.

P. Jung, F. Krahmer, and D. Stöger, “Blind demixing and deconvolution at near-optimal rate,” IEEE Trans. Information Theory 64, 704–727 (2018).
[Crossref]

D. Stöger, P. Jung, and F. Krahmer, “Blind deconvolution and compressed sensing,” in 4th International Workshop on Compressed Sensing Theory and Its Applications to Radar, Sonar and Remote Sensing (CoSeRa) (IEEE, 2016), pp. 24–27.

Kingma, D. P.

D. P. Kingma and J. Ba, “Adam: a method for stochastic optimization,” arXiv:1412.6980 (2014).

Kirz, J.

J. Miao, P. Charalambous, J. Kirz, and D. Sayre, “Extending the methodology of X-ray crystallography to allow imaging of micrometre-sized non-crystalline specimens,” Nature 400, 342–344 (1999).
[Crossref]

Köse, K.

M. Tofighi, O. Yorulmaz, K. Köse, D. C. Yıldırım, R. Çetin-Atalay, and A. E. Cetin, “Phase and TV based convex sets for blind deconvolution of microscopic images,” IEEE J. Sel. Top. Signal Process. 10, 81–91 (2016).
[Crossref]

Kotynski, R.

D. Pastor, T. Stefaniuk, P. Wróbel, C. J. Zapata-Rodríguez, and R. Kotyński, “Determination of the point spread function of layered metamaterials assisted with the blind deconvolution algorithm,” Opt. Quantum Electron. 47, 17–26 (2015).
[Crossref]

Krahmer, F.

P. Jung, F. Krahmer, and D. Stöger, “Blind demixing and deconvolution at near-optimal rate,” IEEE Trans. Information Theory 64, 704–727 (2018).
[Crossref]

D. Stöger, P. Jung, and F. Krahmer, “Blind deconvolution and compressed sensing,” in 4th International Workshop on Compressed Sensing Theory and Its Applications to Radar, Sonar and Remote Sensing (CoSeRa) (IEEE, 2016), pp. 24–27.

Leach, R.

M. Foreman, C. Giusca, P. Török, and R. Leach, “Phase-retrieved pupil function and coherent transfer function in confocal microscopy,” J. Microsc. 251, 99–107 (2013).
[Crossref]

Li, X.

Y. Hu, D. Zhang, J. Ye, X. Li, and X. He, “Fast and accurate matrix completion via truncated nuclear norm regularization,” IEEE Trans. Pattern Anal. Mach. Intell. 35, 2117–2130 (2013).
[Crossref]

Ling, S.

S. Ling and T. Strohmer, “Regularized gradient descent: a non-convex recipe for fast joint blind deconvolution and demixing,” Information and Inference: A Journal of the IMA 8, 1–49 (2019).
[Crossref]

Liu, G.

G. Liu, S. Chang, and Y. Ma, “Blind image deblurring using spectral properties of convolution operators,” IEEE Trans. Image Process. 23, 5047–5056 (2014).
[Crossref]

Liu, Z.

Z. Liu and L. Vandenberghe, “Interior-point method for nuclear norm approximation with application to system identification,” SIAM J. Matrix Anal. Appl. 31, 1235–1256 (2009).
[Crossref]

Luke, D. R.

R. Hesse, D. R. Luke, S. Sabach, and M. K. Tam, “Proximal heterogeneous block implicit-explicit method and application to blind ptychographic diffraction imaging,” SIAM J. Imaging Sci. 8, 426–457(2015).
[Crossref]

Ma, Y.

G. Liu, S. Chang, and Y. Ma, “Blind image deblurring using spectral properties of convolution operators,” IEEE Trans. Image Process. 23, 5047–5056 (2014).
[Crossref]

MacDonald, A.

A. MacDonald, “Blind deconvolution of anisoplanatic images collected by a partially coherent imaging system,” Ph.D. thesis (Air Force Institute of Technology, Wright-Patterson Air Force Base Ohio, 2004).

Maiden, A. M.

A. M. Maiden, M. J. Humphry, F. Zhang, and J. M. Rodenburg, “Superresolution imaging via ptychography,” J. Opt. Soc. Am. A 28, 604–612 (2011).
[Crossref]

A. M. Maiden and J. M. Rodenburg, “An improved ptychographical phase retrieval algorithm for diffractive imaging,” Ultramicroscopy 109, 1256–1262 (2009).
[Crossref]

Marchesini, S.

H. Chang, P. Enfedaque, and S. Marchesini, “Blind ptychographic phase retrieval via convergent alternating direction method of multipliers,” SIAM J. Imaging Sci. 12, pp. 153–185 (2019).

McNulty, I.

Menzel, A.

M. Odstrčil, A. Menzel, and M. Guizar-Sicairos, “Iterative least-squares solver for generalized maximum-likelihood ptychography,” Opt. Express 26, 3108–3123 (2018).
[Crossref]

P. Thibault, M. Dierolf, O. Bunk, A. Menzel, and F. Pfeiffer, “Probe retrieval in ptychographic coherent diffractive imaging,” Ultramicroscopy 109, 338–343 (2009).
[Crossref]

Miao, J.

J. Miao, P. Charalambous, J. Kirz, and D. Sayre, “Extending the methodology of X-ray crystallography to allow imaging of micrometre-sized non-crystalline specimens,” Nature 400, 342–344 (1999).
[Crossref]

Mourya, R.

R. Mourya, L. Denis, J.-M. Becker, and E. Thiébaut, “A blind deblurring and image decomposition approach for astronomical image restoration,” in 23rd European Signal Processing Conference (EUSIPCO) (IEEE, 2015), pp. 1636–1640.

Nashed, Y. S.

Y. S. Nashed, T. Peterka, J. Deng, and C. Jacobsen, “Distributed automatic differentiation for ptychography,” Procedia Comput. Sci. 108, 404–414 (2017).
[Crossref]

Nehorai, A.

P. Sarder and A. Nehorai, “Deconvolution methods for 3-D fluorescence microscopy images,” IEEE Signal Process. Mag. 23(3), 32–45 (2006).
[Crossref]

Odstrcil, M.

Ou, X.

Parikh, N.

S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein, “Distributed optimization and statistical learning via the alternating direction method of multipliers,” Found. Trends Mach. Learn. 3, 1–122 (2011).
[Crossref]

Parrilo, P. A.

B. Recht, M. Fazel, and P. A. Parrilo, “Guaranteed minimum-rank solutions of linear matrix equations via nuclear norm minimization,” SIAM Rev. 52, 471–501 (2010).
[Crossref]

Pastor, D.

D. Pastor, T. Stefaniuk, P. Wróbel, C. J. Zapata-Rodríguez, and R. Kotyński, “Determination of the point spread function of layered metamaterials assisted with the blind deconvolution algorithm,” Opt. Quantum Electron. 47, 17–26 (2015).
[Crossref]

Peleato, B.

S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein, “Distributed optimization and statistical learning via the alternating direction method of multipliers,” Found. Trends Mach. Learn. 3, 1–122 (2011).
[Crossref]

Peng, L.

F. Jian and L. Peng, “A general phase retrieval algorithm based on a ptychographical iterative engine for coherent diffractive imaging,” Chin. Phys. B 22, 014204 (2013).
[Crossref]

Peterka, T.

Y. S. Nashed, T. Peterka, J. Deng, and C. Jacobsen, “Distributed automatic differentiation for ptychography,” Procedia Comput. Sci. 108, 404–414 (2017).
[Crossref]

Pfeiffer, F.

P. Thibault, M. Dierolf, O. Bunk, A. Menzel, and F. Pfeiffer, “Probe retrieval in ptychographic coherent diffractive imaging,” Ultramicroscopy 109, 338–343 (2009).
[Crossref]

Pike, E.

Piscaer, P.

Pozzi, P.

Recht, B.

B. Recht, M. Fazel, and P. A. Parrilo, “Guaranteed minimum-rank solutions of linear matrix equations via nuclear norm minimization,” SIAM Rev. 52, 471–501 (2010).
[Crossref]

Rodenburg, J. M.

A. M. Maiden, M. J. Humphry, F. Zhang, and J. M. Rodenburg, “Superresolution imaging via ptychography,” J. Opt. Soc. Am. A 28, 604–612 (2011).
[Crossref]

A. M. Maiden and J. M. Rodenburg, “An improved ptychographical phase retrieval algorithm for diffractive imaging,” Ultramicroscopy 109, 1256–1262 (2009).
[Crossref]

J. M. Rodenburg and H. M. Faulkner, “A phase retrieval algorithm for shifting illumination,” Appl. Phys. Lett. 85, 4795–4797 (2004).
[Crossref]

Sabach, S.

R. Hesse, D. R. Luke, S. Sabach, and M. K. Tam, “Proximal heterogeneous block implicit-explicit method and application to blind ptychographic diffraction imaging,” SIAM J. Imaging Sci. 8, 426–457(2015).
[Crossref]

Sarder, P.

P. Sarder and A. Nehorai, “Deconvolution methods for 3-D fluorescence microscopy images,” IEEE Signal Process. Mag. 23(3), 32–45 (2006).
[Crossref]

Sayre, D.

J. Miao, P. Charalambous, J. Kirz, and D. Sayre, “Extending the methodology of X-ray crystallography to allow imaging of micrometre-sized non-crystalline specimens,” Nature 400, 342–344 (1999).
[Crossref]

Schulz, T. J.

Shen, Z.

J.-F. Cai, E. J. Candès, and Z. Shen, “A singular value thresholding algorithm for matrix completion,” SIAM J. Optim. 20, 1956–1982 (2010).
[Crossref]

Shpyrko, O. G.

Soloviev, O.

Soltanolkotabi, M.

Stefaniuk, T.

D. Pastor, T. Stefaniuk, P. Wróbel, C. J. Zapata-Rodríguez, and R. Kotyński, “Determination of the point spread function of layered metamaterials assisted with the blind deconvolution algorithm,” Opt. Quantum Electron. 47, 17–26 (2015).
[Crossref]

Stöger, D.

P. Jung, F. Krahmer, and D. Stöger, “Blind demixing and deconvolution at near-optimal rate,” IEEE Trans. Information Theory 64, 704–727 (2018).
[Crossref]

D. Stöger, P. Jung, and F. Krahmer, “Blind deconvolution and compressed sensing,” in 4th International Workshop on Compressed Sensing Theory and Its Applications to Radar, Sonar and Remote Sensing (CoSeRa) (IEEE, 2016), pp. 24–27.

Strohmer, T.

S. Ling and T. Strohmer, “Regularized gradient descent: a non-convex recipe for fast joint blind deconvolution and demixing,” Information and Inference: A Journal of the IMA 8, 1–49 (2019).
[Crossref]

Tam, M. K.

R. Hesse, D. R. Luke, S. Sabach, and M. K. Tam, “Proximal heterogeneous block implicit-explicit method and application to blind ptychographic diffraction imaging,” SIAM J. Imaging Sci. 8, 426–457(2015).
[Crossref]

Tang, G.

Thao, N. H.

Thibault, P.

P. Thibault and M. Guizar-Sicairos, “Maximum-likelihood refinement for coherent diffractive imaging,” New J. Phys. 14, 063004(2012).
[Crossref]

P. Thibault, M. Dierolf, O. Bunk, A. Menzel, and F. Pfeiffer, “Probe retrieval in ptychographic coherent diffractive imaging,” Ultramicroscopy 109, 338–343 (2009).
[Crossref]

Thiébaut, E.

R. Mourya, L. Denis, J.-M. Becker, and E. Thiébaut, “A blind deblurring and image decomposition approach for astronomical image restoration,” in 23rd European Signal Processing Conference (EUSIPCO) (IEEE, 2015), pp. 1636–1640.

Tian, L.

Tofighi, M.

M. Tofighi, O. Yorulmaz, K. Köse, D. C. Yıldırım, R. Çetin-Atalay, and A. E. Cetin, “Phase and TV based convex sets for blind deconvolution of microscopic images,” IEEE J. Sel. Top. Signal Process. 10, 81–91 (2016).
[Crossref]

Török, P.

M. Foreman, C. Giusca, P. Török, and R. Leach, “Phase-retrieved pupil function and coherent transfer function in confocal microscopy,” J. Microsc. 251, 99–107 (2013).
[Crossref]

Tripathi, A.

Tropp, J. A.

R. Horstmeyer, R. Y. Chen, X. Ou, B. Ames, J. A. Tropp, and C. Yang, “Solving ptychography with a convex relaxation,” New J. Phys. 17, 053044 (2015).
[Crossref]

Turkington, D. A.

D. A. Turkington, Generalized Vectorization, Cross-Products, and Matrix Calculus (Cambridge University, 2013).

Tuy, H.

H. Tuy, “DC optimization: theory, methods and algorithms,” in Handbook of Global Optimization (Springer, 1995), pp. 149–216.

Vandenberghe, L.

Z. Liu and L. Vandenberghe, “Interior-point method for nuclear norm approximation with application to system identification,” SIAM J. Matrix Anal. Appl. 31, 1235–1256 (2009).
[Crossref]

Vdovin, G.

Verhaegen, M.

Walker, J.

Waller, L.

Wilding, D.

Wróbel, P.

D. Pastor, T. Stefaniuk, P. Wróbel, C. J. Zapata-Rodríguez, and R. Kotyński, “Determination of the point spread function of layered metamaterials assisted with the blind deconvolution algorithm,” Opt. Quantum Electron. 47, 17–26 (2015).
[Crossref]

Yang, C.

Ye, J.

Y. Hu, D. Zhang, J. Ye, X. Li, and X. He, “Fast and accurate matrix completion via truncated nuclear norm regularization,” IEEE Trans. Pattern Anal. Mach. Intell. 35, 2117–2130 (2013).
[Crossref]

Yeh, L.-H.

Yildirim, D. C.

M. Tofighi, O. Yorulmaz, K. Köse, D. C. Yıldırım, R. Çetin-Atalay, and A. E. Cetin, “Phase and TV based convex sets for blind deconvolution of microscopic images,” IEEE J. Sel. Top. Signal Process. 10, 81–91 (2016).
[Crossref]

Yorulmaz, O.

M. Tofighi, O. Yorulmaz, K. Köse, D. C. Yıldırım, R. Çetin-Atalay, and A. E. Cetin, “Phase and TV based convex sets for blind deconvolution of microscopic images,” IEEE J. Sel. Top. Signal Process. 10, 81–91 (2016).
[Crossref]

Zapata-Rodríguez, C. J.

D. Pastor, T. Stefaniuk, P. Wróbel, C. J. Zapata-Rodríguez, and R. Kotyński, “Determination of the point spread function of layered metamaterials assisted with the blind deconvolution algorithm,” Opt. Quantum Electron. 47, 17–26 (2015).
[Crossref]

Zhang, D.

Y. Hu, D. Zhang, J. Ye, X. Li, and X. He, “Fast and accurate matrix completion via truncated nuclear norm regularization,” IEEE Trans. Pattern Anal. Mach. Intell. 35, 2117–2130 (2013).
[Crossref]

Zhang, F.

Zhang, Y.

Zheng, G.

Zhong, J.

Appl. Phys. Lett. (1)

J. M. Rodenburg and H. M. Faulkner, “A phase retrieval algorithm for shifting illumination,” Appl. Phys. Lett. 85, 4795–4797 (2004).
[Crossref]

Chin. Phys. B (1)

F. Jian and L. Peng, “A general phase retrieval algorithm based on a ptychographical iterative engine for coherent diffractive imaging,” Chin. Phys. B 22, 014204 (2013).
[Crossref]

Found. Trends Mach. Learn. (1)

S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein, “Distributed optimization and statistical learning via the alternating direction method of multipliers,” Found. Trends Mach. Learn. 3, 1–122 (2011).
[Crossref]

IEEE J. Sel. Top. Signal Process. (1)

M. Tofighi, O. Yorulmaz, K. Köse, D. C. Yıldırım, R. Çetin-Atalay, and A. E. Cetin, “Phase and TV based convex sets for blind deconvolution of microscopic images,” IEEE J. Sel. Top. Signal Process. 10, 81–91 (2016).
[Crossref]

IEEE Signal Process. Mag. (1)

P. Sarder and A. Nehorai, “Deconvolution methods for 3-D fluorescence microscopy images,” IEEE Signal Process. Mag. 23(3), 32–45 (2006).
[Crossref]

IEEE Trans. Image Process. (1)

G. Liu, S. Chang, and Y. Ma, “Blind image deblurring using spectral properties of convolution operators,” IEEE Trans. Image Process. 23, 5047–5056 (2014).
[Crossref]

IEEE Trans. Information Theory (1)

P. Jung, F. Krahmer, and D. Stöger, “Blind demixing and deconvolution at near-optimal rate,” IEEE Trans. Information Theory 64, 704–727 (2018).
[Crossref]

IEEE Trans. Pattern Anal. Mach. Intell. (1)

Y. Hu, D. Zhang, J. Ye, X. Li, and X. He, “Fast and accurate matrix completion via truncated nuclear norm regularization,” IEEE Trans. Pattern Anal. Mach. Intell. 35, 2117–2130 (2013).
[Crossref]

Information and Inference: A Journal of the IMA (1)

S. Ling and T. Strohmer, “Regularized gradient descent: a non-convex recipe for fast joint blind deconvolution and demixing,” Information and Inference: A Journal of the IMA 8, 1–49 (2019).
[Crossref]

J. Microsc. (1)

M. Foreman, C. Giusca, P. Török, and R. Leach, “Phase-retrieved pupil function and coherent transfer function in confocal microscopy,” J. Microsc. 251, 99–107 (2013).
[Crossref]

J. Opt. Soc. Am. A (6)

Nature (1)

J. Miao, P. Charalambous, J. Kirz, and D. Sayre, “Extending the methodology of X-ray crystallography to allow imaging of micrometre-sized non-crystalline specimens,” Nature 400, 342–344 (1999).
[Crossref]

New J. Phys. (2)

R. Horstmeyer, R. Y. Chen, X. Ou, B. Ames, J. A. Tropp, and C. Yang, “Solving ptychography with a convex relaxation,” New J. Phys. 17, 053044 (2015).
[Crossref]

P. Thibault and M. Guizar-Sicairos, “Maximum-likelihood refinement for coherent diffractive imaging,” New J. Phys. 14, 063004(2012).
[Crossref]

Opt. Eng. (1)

R. A. Gonsalves, “Phase retrieval and diversity in adaptive optics,” Opt. Eng. 21, 215829 (1982).
[Crossref]

Opt. Express (9)

D. Wilding, O. Soloviev, P. Pozzi, G. Vdovin, and M. Verhaegen, “Blind multi-frame deconvolution by tangential iterative projections (tip),” Opt. Express 25, 32305–32322 (2017).
[Crossref]

A. Tripathi, I. McNulty, and O. G. Shpyrko, “Ptychographic overlap constraint errors and the limits of their numerical recovery using conjugate gradient descent methods,” Opt. Express 22, 1452–1466 (2014).
[Crossref]

G. R. Brady, M. Guizar-Sicairos, and J. R. Fienup, “Optical wavefront measurement using phase retrieval with transverse translation diversity,” Opt. Express 17, 624–639 (2009).
[Crossref]

M. Guizar-Sicairos and J. R. Fienup, “Measurement of coherent x-ray focused beams by phase retrieval with transverse translation diversity,” Opt. Express 17, 2670–2685 (2009).
[Crossref]

M. Odstrčil, A. Menzel, and M. Guizar-Sicairos, “Iterative least-squares solver for generalized maximum-likelihood ptychography,” Opt. Express 26, 3108–3123 (2018).
[Crossref]

Y. Zhang, W. Jiang, and Q. Dai, “Nonlinear optimization approach for Fourier ptychographic microscopy,” Opt. Express 23, 33822–33835 (2015).
[Crossref]

L.-H. Yeh, J. Dong, J. Zhong, L. Tian, M. Chen, G. Tang, M. Soltanolkotabi, and L. Waller, “Experimental robustness of Fourier ptychography phase retrieval algorithms,” Opt. Express 23, 33214–33240 (2015).
[Crossref]

R. Horstmeyer, X. Ou, J. Chung, G. Zheng, and C. Yang, “Overlapped Fourier coding for optical aberration removal,” Opt. Express 22, 24062–24080 (2014).
[Crossref]

X. Ou, G. Zheng, and C. Yang, “Embedded pupil function recovery for Fourier ptychographic microscopy,” Opt. Express 22, 4960–4972 (2014).
[Crossref]

Opt. Lett. (1)

Opt. Quantum Electron. (1)

D. Pastor, T. Stefaniuk, P. Wróbel, C. J. Zapata-Rodríguez, and R. Kotyński, “Determination of the point spread function of layered metamaterials assisted with the blind deconvolution algorithm,” Opt. Quantum Electron. 47, 17–26 (2015).
[Crossref]

Procedia Comput. Sci. (1)

Y. S. Nashed, T. Peterka, J. Deng, and C. Jacobsen, “Distributed automatic differentiation for ptychography,” Procedia Comput. Sci. 108, 404–414 (2017).
[Crossref]

SIAM J. Imaging Sci. (2)

R. Hesse, D. R. Luke, S. Sabach, and M. K. Tam, “Proximal heterogeneous block implicit-explicit method and application to blind ptychographic diffraction imaging,” SIAM J. Imaging Sci. 8, 426–457(2015).
[Crossref]

H. Chang, P. Enfedaque, and S. Marchesini, “Blind ptychographic phase retrieval via convergent alternating direction method of multipliers,” SIAM J. Imaging Sci. 12, pp. 153–185 (2019).

SIAM J. Matrix Anal. Appl. (1)

Z. Liu and L. Vandenberghe, “Interior-point method for nuclear norm approximation with application to system identification,” SIAM J. Matrix Anal. Appl. 31, 1235–1256 (2009).
[Crossref]

SIAM J. Optim. (1)

J.-F. Cai, E. J. Candès, and Z. Shen, “A singular value thresholding algorithm for matrix completion,” SIAM J. Optim. 20, 1956–1982 (2010).
[Crossref]

SIAM Rev. (1)

B. Recht, M. Fazel, and P. A. Parrilo, “Guaranteed minimum-rank solutions of linear matrix equations via nuclear norm minimization,” SIAM Rev. 52, 471–501 (2010).
[Crossref]

Ultramicroscopy (2)

P. Thibault, M. Dierolf, O. Bunk, A. Menzel, and F. Pfeiffer, “Probe retrieval in ptychographic coherent diffractive imaging,” Ultramicroscopy 109, 338–343 (2009).
[Crossref]

A. M. Maiden and J. M. Rodenburg, “An improved ptychographical phase retrieval algorithm for diffractive imaging,” Ultramicroscopy 109, 1256–1262 (2009).
[Crossref]

Other (9)

R. Mourya, L. Denis, J.-M. Becker, and E. Thiébaut, “A blind deblurring and image decomposition approach for astronomical image restoration,” in 23rd European Signal Processing Conference (EUSIPCO) (IEEE, 2015), pp. 1636–1640.

D. Stöger, P. Jung, and F. Krahmer, “Blind deconvolution and compressed sensing,” in 4th International Workshop on Compressed Sensing Theory and Its Applications to Radar, Sonar and Remote Sensing (CoSeRa) (IEEE, 2016), pp. 24–27.

A. Ahmed, A. Cosse, and L. Demanet, “A convex approach to blind deconvolution with diverse inputs,” in 6th International Workshop on Computational Advances in Multi-Sensor Adaptive Processing (CAMSAP) (IEEE, 2015), pp. 5–8.

A. MacDonald, “Blind deconvolution of anisoplanatic images collected by a partially coherent imaging system,” Ph.D. thesis (Air Force Institute of Technology, Wright-Patterson Air Force Base Ohio, 2004).

J. Goodman, Introduction to Fourier Optics (McGraw-Hill, 2008).

R. Doelman and M. Verhaegen, “Sequential convex relaxation for convex optimization with bilinear matrix equalities,” in European Control Conference (ECC) (IEEE, 2016), pp. 1946–1951.

D. P. Kingma and J. Ba, “Adam: a method for stochastic optimization,” arXiv:1412.6980 (2014).

H. Tuy, “DC optimization: theory, methods and algorithms,” in Handbook of Global Optimization (Springer, 1995), pp. 149–216.

D. A. Turkington, Generalized Vectorization, Cross-Products, and Matrix Calculus (Cambridge University, 2013).

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Figures (7)

Fig. 1.
Fig. 1. Convergence of the constraint violations | | y | g i ^ | 2 | | F 2 (measurement fit) and | | g i ^ g o ^ B v ^ | | F 2 (convolution fit) through updates in Algorithm 1 for three different sets of tuning parameters.
Fig. 2.
Fig. 2. Top: the three 12 × 12 (noiseless) measured intensities y = | g o h | 2 . Bottom: the three 5 × 5 intensity impulse response functions (point spread functions) s = | h | 2 corresponding to the three different diversities that generate the different h .
Fig. 3.
Fig. 3. Left: the amplitude of the object. Right: the phase in radians of the object.
Fig. 4.
Fig. 4. Measurement fit generated by the two algorithms. Black: Algorithm 1. Red: gradient descent. Solid lines show the case with noisy measurements (SNR: 20 dB); dashed lines show the noiseless case.
Fig. 5.
Fig. 5. Frobenius norm of the residual between the true variables g o , the complex-valued object, and v , the radial basis function coefficients, and the (ambiguity removed) estimated variables g o ^ and v ^ . Top figure: residuals for g o . Bottom figure: residuals for v . Black: Algorithm 1. Red: gradient descent. Solid lines show the case with noisy measurements (SNR: 20 dB); dashed lines show the noiseless case.
Fig. 6.
Fig. 6. Estimates and residuals of g o using Algorithm 1 and gradient descent with identical initialization. Top left: the estimated amplitude of g o and the residual for the noiseless case. Top right: the estimated angle of g o and the residual for the noiseless case. Bottom left: the estimated amplitude of g o and the residual for the noisy case. Bottom right: the estimated angle of g o and the residual for the noisy case. Since estimates of very small complex values can have a radically different complex angle, the angle is only plotted for the nonzero pixels in the original object of Fig. 3.
Fig. 7.
Fig. 7. Estimates of | h | . The maximum absolute value of h ^ is scaled to 1.

Tables (1)

Tables Icon

Algorithm 1. Convex optimization-based blind deconvolution for images taken with coherent illumination

Equations (36)

Equations on this page are rendered with MathJax. Learn more.

y = | g o h | 2 ,
find g o , h subject to y = | g o h | 2 , g o K g o , h K h ,
find f , s subject to y = f s , f K f , s K s ,
P ( ρ , θ ) = A ( ρ , θ ) exp ( j ϕ ( ρ , θ ) ) ,
P d ( ρ , θ ) = A ( ρ , θ ) exp ( j ϕ ( ρ , θ ) ) exp ( j ϕ d ( ρ , θ ) ) .
G i = χ ( x , y ) exp ( λ i ( ( x x i ) 2 + ( y y i ) 2 ) ) , P ( x , y ) P ˜ ( x , y , v ) = i = 1 s v i G i ( x , y ) ,
P ˜ d ( x , y , v ) = i = 1 s v i G i ( x , y ) exp ( j ϕ d ( x , y ) ) .
h d ( u , v ) = i = 1 s v i F 1 { G i ( x , y ) exp ( j ϕ d ( x , y ) ) } = i = 1 s v i B d , i ( u , v ) .
g i ( u , v ) = g o ( u , v ) h d ( u , v ) .
y ( u , v ) = | g i ( u , v ) | 2 .
find g o , v , h , g i subject to y = | g i | 2 , g i = g o h , h = Bv , g o K g o , h K h .
M ( C , A , B , Q , X , Y , W 1 , W 2 ) = ( W 1 0 0 I ) ( C + A Q Y + X Q B + X Q Y ( A + X ) Q Q ( B + Y ) Q ) ( W 2 0 0 I ) ,
rank ( M ( C , A , B , Q , X , Y , W 1 , W 2 ) ) = rank ( Q )
C = A Q B .
vect ( g i ) = ( vect ( g o ) T I r t ) V vect ( Bv )
C = vect ( g i ) , A = ( vect ( g o ) T I r t ) , Q = V , B = vect ( Bv ) .
rank ( M ( vect ( g i ) , vect ( g o ) T I r t , vect ( Bv ) , V , X , Y , W 1 , W 2 ) ) = rank ( V ) .
X = vect ( g o ^ ) T I r t , Y = vect ( B v ^ ) , W 1 = I , W 2 = I ,
M c ( g i , g o , v , V , g o ^ , v ^ ) = M ( vect ( g i ) , vect ( g o ) T I r t , vect ( Bv ) , V , X , Y , W 1 , W 2 ) ,
y = | g i | 2 d ( vect ( y ) ) = d ( vect ( g i ) ) H d ( vect ( g i ) ) .
rank ( M ( d ( vect ( y ) ) , d ( vect ( g i ) ) H , d ( vect ( g i ) ) , I r t , X , Y , W 1 , W 2 ) ) = r t .
X = d ( vect ( g i ^ ) ) H , Y = d ( vect ( g i ^ ) ) , W 1 = I , W 2 = I ,
M m ( y , g i , g i ^ ) = M ( d ( vect ( y ) ) , d ( vect ( g i ) ) H , d ( vect ( g i ) ) , I r t , X , Y , W 1 , W 2 ) ,
find g o , v , g i ,
subject to rank ( M m ( y , g i , g i ^ ) ) = r t ,
rank ( M c ( g i , g o , v , V , g o ^ , v ^ ) ) = rank ( V ) ,
g o K g o , h = Bv K h .
X * = i σ i ( X ) ,
min g o , v , g i μ M m ( y , g i , g i ^ ) * + M c ( g i , g o , v , V , g o ^ , v ^ ) * , g o K g o , h = Bv K h ,
μ M m ( y , g i * , g i * ) * + M c ( g i * , g o * , v * , V , g o * , v * ) * = μ ( 0 0 0 I r t ) * + ( 0 0 0 V ) * = μ r t + V * ,
min c C c v ^ v 2 .
vect ( g i ) = L vect ( vect ( g o ) vect ( Bv ) T ) .
vect ( A X B ) = ( B T A ) vect ( X )
L ( I p q vect ( g o ) T ) vect ( Bv ) .
l i ( I p q vect ( g o ) T ) = vect ( g o ) T L i ,
vect ( g i ) = ( I r t vect ( g o ) T ) ( L 1 L r t ) vect ( Bv ) .

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