Abstract

We propose a new approach to robustly retrieve the exit wave of an extended sample from its coherent diffraction pattern by exploiting sparsity of the sample’s edges. This approach enables imaging of an extended sample with a single view, without ptychography. We introduce nonlinear optimization methods that promote sparsity, and we derive update rules to robustly recover the sample’s exit wave. We test these methods on simulated samples by varying the sparsity of the edge-detected representation of the exit wave. Our tests illustrate the strengths and limitations of the proposed method in imaging extended samples.

© 2016 Optical Society of America

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References

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2016 (1)

2015 (1)

A. Tripathi, S. Leyffer, T. Munson, and S. M. Wild, “Visualizing and improving the robustness of phase retrieval algorithms,” Procedia Comput. Sci. 51, 815–824 (2015).
[Crossref]

2014 (2)

Y. S. G. Nashed, D. J. Vine, T. Peterka, J. Deng, R. Ross, and C. Jacobsen, “Parallel ptychographic reconstruction,” Opt. Express 22, 32082–32097 (2014).
[Crossref]

R. Fan, Q. Wan, F. Wen, H. Chen, and Y. Liu, “Iterative projection approach for phase retrieval of semi-sparse wave field,” EURASIP J. Adv. Signal Process. 2014, 1–13 (2014).
[Crossref]

2013 (2)

Y. Zhang, B. Dong, and Z. Lu, “ℓ0 minimization for wavelet frame based image restoration,” Math. Comput. 82, 995–1015 (2013).
[Crossref]

P. Thibault and A. Menzel, “Reconstructing state mixtures from diffraction measurements,” Nature 494, 68–71 (2013).
[Crossref]

2012 (3)

L. Sorber, M. Barel, and L. Lathauwer, “Unconstrained optimization of real functions in complex variables,” SIAM J. Optim. 22, 879–898 (2012).
[Crossref]

J. J. Moré and S. M. Wild, “Estimating derivatives of noisy simulations,” ACM Trans. Math. Software 38, 1–21 (2012).
[Crossref]

B. Dong and Y. Zhang, “An efficient algorithm for ℓ0 minimization in wavelet frame based image restoration,” J. Sci. Comput. 54, 350–368 (2012).
[Crossref]

2011 (3)

X. Zhang, Y. Lu, and T. Chan, “A novel sparsity reconstruction method from Poisson data for 3D bioluminescence tomography,” J. Sci. Comput. 50, 519–535 (2011).
[Crossref]

F. Bach, R. Jenatton, J. Mairal, and G. Obozinski, “Optimization with sparsity-inducing penalties,” Found. Trends Mach. Learn. 4, 1–106 (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 (4)

2009 (2)

S. Yi, D. Labate, G. R. Easley, and H. Krim, “A shearlet approach to edge analysis and detection,” IEEE Trans. Image Process. 18, 929–941 (2009).
[Crossref]

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

2008 (5)

E. J. Candés, M. B. Wakin, and S. P. Boyd, “Enhancing sparsity by reweighted ℓ1 minimization,” J. Fourier Anal. Appl. 14, 877–905 (2008).
[Crossref]

T. Blumensath and M. E. Davies, “Iterative thresholding for sparse approximations,” J. Fourier Anal. Appl. 14, 629–654 (2008).
[Crossref]

B. Abbey, K. Nugent, G. Williams, J. Clark, A. Peele, M. Pfeifer, M. de Jonge, and I. McNulty, “Keyhole coherent diffractive imaging,” Nat. Phys. 4, 394–398 (2008).
[Crossref]

E. Candés and M. Wakin, “An introduction to compressive sampling,” IEEE Signal Proc. Mag. 25, 21–30 (2008).
[Crossref]

P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, and F. Pfeiffer, “High-resolution scanning X-ray diffraction microscopy,” Science 321, 379–382 (2008).
[Crossref]

2007 (2)

M. L. Moravec, J. K. Romberg, and R. G. Baraniuk, “Compressive phase retrieval,” Proc. SPIE 6701, 670120 (2007).
[Crossref]

D. Lazzaro and L. Montefusco, “Edge-preserving wavelet thresholding for image denoising,” J. Comput. Appl. Math. 210, 222–231 (2007).
[Crossref]

2006 (2)

2004 (1)

H. M. L. Faulkner and J. M. Rodenburg, “Movable aperture lensless transmission microscopy: A novel phase retrieval algorithm,” Phys. Rev. Lett. 93, 023903 (2004).
[Crossref]

2003 (2)

S. Marchesini, H. He, H. N. Chapman, S. P. Hau-Riege, A. Noy, M. R. Howells, U. Weierstall, and J. C. H. Spence, “X-ray image reconstruction from a diffraction pattern alone,” Phys. Rev. B 68, 140101 (2003).
[Crossref]

R. H. Chan, T. F. Chan, L. Shen, and Z. Shen, “Wavelet algorithms for high-resolution image reconstruction,” SIAM J. Sci. Comput. 24, 1408–1432 (2003).
[Crossref]

2002 (1)

J.-L. Starck, E. J. Candés, and D. L. Donoho, “The curvelet transform for image denoising,” IEEE Trans. Image Process. 11, 670–684 (2002).
[Crossref]

2000 (1)

R. Neutze, R. Wouts, D. van der Spoel, E. Weckert, and J. Hajdu, “Potential for biomolecular imaging with femtosecond x-ray pulses,” Nature 406, 752–757 (2000).
[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]

1992 (1)

L. I. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica D 60, 259–268 (1992).
[Crossref]

1982 (1)

Abbey, B.

B. Abbey, K. Nugent, G. Williams, J. Clark, A. Peele, M. Pfeifer, M. de Jonge, and I. McNulty, “Keyhole coherent diffractive imaging,” Nat. Phys. 4, 394–398 (2008).
[Crossref]

Bach, F.

F. Bach, R. Jenatton, J. Mairal, and G. Obozinski, “Optimization with sparsity-inducing penalties,” Found. Trends Mach. Learn. 4, 1–106 (2011).
[Crossref]

Baraniuk, R. G.

M. L. Moravec, J. K. Romberg, and R. G. Baraniuk, “Compressive phase retrieval,” Proc. SPIE 6701, 670120 (2007).
[Crossref]

Barel, M.

L. Sorber, M. Barel, and L. Lathauwer, “Unconstrained optimization of real functions in complex variables,” SIAM J. Optim. 22, 879–898 (2012).
[Crossref]

Barty, A.

Beetz, T.

Blumensath, T.

T. Blumensath and M. E. Davies, “Iterative thresholding for sparse approximations,” J. Fourier Anal. Appl. 14, 629–654 (2008).
[Crossref]

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]

Boyd, S. P.

E. J. Candés, M. B. Wakin, and S. P. Boyd, “Enhancing sparsity by reweighted ℓ1 minimization,” J. Fourier Anal. Appl. 14, 877–905 (2008).
[Crossref]

Bunk, O.

P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, and F. Pfeiffer, “High-resolution scanning X-ray diffraction microscopy,” Science 321, 379–382 (2008).
[Crossref]

Candés, E.

E. Candés and M. Wakin, “An introduction to compressive sampling,” IEEE Signal Proc. Mag. 25, 21–30 (2008).
[Crossref]

Candés, E. J.

E. J. Candés, M. B. Wakin, and S. P. Boyd, “Enhancing sparsity by reweighted ℓ1 minimization,” J. Fourier Anal. Appl. 14, 877–905 (2008).
[Crossref]

J.-L. Starck, E. J. Candés, and D. L. Donoho, “The curvelet transform for image denoising,” IEEE Trans. Image Process. 11, 670–684 (2002).
[Crossref]

Chan, R. H.

R. H. Chan, T. F. Chan, L. Shen, and Z. Shen, “Wavelet algorithms for high-resolution image reconstruction,” SIAM J. Sci. Comput. 24, 1408–1432 (2003).
[Crossref]

Chan, T.

X. Zhang, Y. Lu, and T. Chan, “A novel sparsity reconstruction method from Poisson data for 3D bioluminescence tomography,” J. Sci. Comput. 50, 519–535 (2011).
[Crossref]

Chan, T. F.

R. H. Chan, T. F. Chan, L. Shen, and Z. Shen, “Wavelet algorithms for high-resolution image reconstruction,” SIAM J. Sci. Comput. 24, 1408–1432 (2003).
[Crossref]

Chapman, H. N.

H. N. Chapman, A. Barty, S. Marchesini, A. Noy, S. P. Hau-Riege, C. Cui, M. R. Howells, R. Rosen, H. He, J. C. H. Spence, U. Weierstall, T. Beetz, C. Jacobsen, and D. Shapiro, “High-resolution ab initio three-dimensional x-ray diffraction microscopy,” J. Opt. Soc. Am. A 23, 1179–1200 (2006).
[Crossref]

S. Marchesini, H. He, H. N. Chapman, S. P. Hau-Riege, A. Noy, M. R. Howells, U. Weierstall, and J. C. H. Spence, “X-ray image reconstruction from a diffraction pattern alone,” Phys. Rev. B 68, 140101 (2003).
[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, B.

Chen, H.

R. Fan, Q. Wan, F. Wen, H. Chen, and Y. Liu, “Iterative projection approach for phase retrieval of semi-sparse wave field,” EURASIP J. Adv. Signal Process. 2014, 1–13 (2014).
[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]

Clark, J.

B. Abbey, K. Nugent, G. Williams, J. Clark, A. Peele, M. Pfeifer, M. de Jonge, and I. McNulty, “Keyhole coherent diffractive imaging,” Nat. Phys. 4, 394–398 (2008).
[Crossref]

Clark, J. N.

Cui, C.

David, C.

P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, and F. Pfeiffer, “High-resolution scanning X-ray diffraction microscopy,” Science 321, 379–382 (2008).
[Crossref]

Davies, M. E.

T. Blumensath and M. E. Davies, “Iterative thresholding for sparse approximations,” J. Fourier Anal. Appl. 14, 629–654 (2008).
[Crossref]

de Jonge, M.

B. Abbey, K. Nugent, G. Williams, J. Clark, A. Peele, M. Pfeifer, M. de Jonge, and I. McNulty, “Keyhole coherent diffractive imaging,” Nat. Phys. 4, 394–398 (2008).
[Crossref]

Deng, J.

Dierolf, M.

P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, and F. Pfeiffer, “High-resolution scanning X-ray diffraction microscopy,” Science 321, 379–382 (2008).
[Crossref]

Dong, B.

Y. Zhang, B. Dong, and Z. Lu, “ℓ0 minimization for wavelet frame based image restoration,” Math. Comput. 82, 995–1015 (2013).
[Crossref]

B. Dong and Y. Zhang, “An efficient algorithm for ℓ0 minimization in wavelet frame based image restoration,” J. Sci. Comput. 54, 350–368 (2012).
[Crossref]

Donoho, D. L.

J.-L. Starck, E. J. Candés, and D. L. Donoho, “The curvelet transform for image denoising,” IEEE Trans. Image Process. 11, 670–684 (2002).
[Crossref]

Easley, G. R.

S. Yi, D. Labate, G. R. Easley, and H. Krim, “A shearlet approach to edge analysis and detection,” IEEE Trans. Image Process. 18, 929–941 (2009).
[Crossref]

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]

Fan, R.

R. Fan, Q. Wan, F. Wen, H. Chen, and Y. Liu, “Iterative projection approach for phase retrieval of semi-sparse wave field,” EURASIP J. Adv. Signal Process. 2014, 1–13 (2014).
[Crossref]

Fatemi, E.

L. I. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica D 60, 259–268 (1992).
[Crossref]

Faulkner, H. M. L.

H. M. L. Faulkner and J. M. Rodenburg, “Movable aperture lensless transmission microscopy: A novel phase retrieval algorithm,” Phys. Rev. Lett. 93, 023903 (2004).
[Crossref]

Fienup, J. R.

Fullagar, W.

Gilbert, A.

K. Herrity, A. Gilbert, and J. Tropp, “Sparse approximation via iterative thresholding,” in Proceedings of IEEE International Conference on Acoustics, Speech and Signal Processing (IEEE, 2006), pp. 624–627.

Hajdu, J.

R. Neutze, R. Wouts, D. van der Spoel, E. Weckert, and J. Hajdu, “Potential for biomolecular imaging with femtosecond x-ray pulses,” Nature 406, 752–757 (2000).
[Crossref]

Hall, C.

Harder, R.

M. A. Pfeifer, G. J. Williams, I. A. Vartanyants, R. Harder, and I. K. Robinson, “Three-dimensional mapping of a deformation field inside a nanocrystal,” Nature 442, 63–66 (2006).
[Crossref]

Hau-Riege, S. P.

H. N. Chapman, A. Barty, S. Marchesini, A. Noy, S. P. Hau-Riege, C. Cui, M. R. Howells, R. Rosen, H. He, J. C. H. Spence, U. Weierstall, T. Beetz, C. Jacobsen, and D. Shapiro, “High-resolution ab initio three-dimensional x-ray diffraction microscopy,” J. Opt. Soc. Am. A 23, 1179–1200 (2006).
[Crossref]

S. Marchesini, H. He, H. N. Chapman, S. P. Hau-Riege, A. Noy, M. R. Howells, U. Weierstall, and J. C. H. Spence, “X-ray image reconstruction from a diffraction pattern alone,” Phys. Rev. B 68, 140101 (2003).
[Crossref]

He, H.

H. N. Chapman, A. Barty, S. Marchesini, A. Noy, S. P. Hau-Riege, C. Cui, M. R. Howells, R. Rosen, H. He, J. C. H. Spence, U. Weierstall, T. Beetz, C. Jacobsen, and D. Shapiro, “High-resolution ab initio three-dimensional x-ray diffraction microscopy,” J. Opt. Soc. Am. A 23, 1179–1200 (2006).
[Crossref]

S. Marchesini, H. He, H. N. Chapman, S. P. Hau-Riege, A. Noy, M. R. Howells, U. Weierstall, and J. C. H. Spence, “X-ray image reconstruction from a diffraction pattern alone,” Phys. Rev. B 68, 140101 (2003).
[Crossref]

Herrity, K.

K. Herrity, A. Gilbert, and J. Tropp, “Sparse approximation via iterative thresholding,” in Proceedings of IEEE International Conference on Acoustics, Speech and Signal Processing (IEEE, 2006), pp. 624–627.

Howells, M. R.

H. N. Chapman, A. Barty, S. Marchesini, A. Noy, S. P. Hau-Riege, C. Cui, M. R. Howells, R. Rosen, H. He, J. C. H. Spence, U. Weierstall, T. Beetz, C. Jacobsen, and D. Shapiro, “High-resolution ab initio three-dimensional x-ray diffraction microscopy,” J. Opt. Soc. Am. A 23, 1179–1200 (2006).
[Crossref]

S. Marchesini, H. He, H. N. Chapman, S. P. Hau-Riege, A. Noy, M. R. Howells, U. Weierstall, and J. C. H. Spence, “X-ray image reconstruction from a diffraction pattern alone,” Phys. Rev. B 68, 140101 (2003).
[Crossref]

Huang, X.

Jacobsen, C.

Jenatton, R.

F. Bach, R. Jenatton, J. Mairal, and G. Obozinski, “Optimization with sparsity-inducing penalties,” Found. Trends Mach. Learn. 4, 1–106 (2011).
[Crossref]

Kang, H. C.

Kim, C.

Kim, S.

Kim, S. N.

Kim, S. S.

Kirz, J.

X. Huang, J. Nelson, J. Steinbrener, J. Kirz, J. J. Turner, and C. Jacobsen, “Incorrect support and missing center tolerances of phasing algorithms,” Opt. Express 18, 26441–26449 (2010).
[Crossref]

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]

Krim, H.

S. Yi, D. Labate, G. R. Easley, and H. Krim, “A shearlet approach to edge analysis and detection,” IEEE Trans. Image Process. 18, 929–941 (2009).
[Crossref]

Labate, D.

S. Yi, D. Labate, G. R. Easley, and H. Krim, “A shearlet approach to edge analysis and detection,” IEEE Trans. Image Process. 18, 929–941 (2009).
[Crossref]

Lathauwer, L.

L. Sorber, M. Barel, and L. Lathauwer, “Unconstrained optimization of real functions in complex variables,” SIAM J. Optim. 22, 879–898 (2012).
[Crossref]

Lazzaro, D.

D. Lazzaro and L. Montefusco, “Edge-preserving wavelet thresholding for image denoising,” J. Comput. Appl. Math. 210, 222–231 (2007).
[Crossref]

Leyffer, S.

A. Tripathi, S. Leyffer, T. Munson, and S. M. Wild, “Visualizing and improving the robustness of phase retrieval algorithms,” Procedia Comput. Sci. 51, 815–824 (2015).
[Crossref]

Liu, Y.

R. Fan, Q. Wan, F. Wen, H. Chen, and Y. Liu, “Iterative projection approach for phase retrieval of semi-sparse wave field,” EURASIP J. Adv. Signal Process. 2014, 1–13 (2014).
[Crossref]

Loock, S.

Lu, Y.

X. Zhang, Y. Lu, and T. Chan, “A novel sparsity reconstruction method from Poisson data for 3D bioluminescence tomography,” J. Sci. Comput. 50, 519–535 (2011).
[Crossref]

Lu, Z.

Y. Zhang, B. Dong, and Z. Lu, “ℓ0 minimization for wavelet frame based image restoration,” Math. Comput. 82, 995–1015 (2013).
[Crossref]

Maiden, A. M.

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

Mairal, J.

F. Bach, R. Jenatton, J. Mairal, and G. Obozinski, “Optimization with sparsity-inducing penalties,” Found. Trends Mach. Learn. 4, 1–106 (2011).
[Crossref]

Marathe, S.

Marchesini, S.

McNulty, I.

Menzel, A.

P. Thibault and A. Menzel, “Reconstructing state mixtures from diffraction measurements,” Nature 494, 68–71 (2013).
[Crossref]

P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, and F. Pfeiffer, “High-resolution scanning X-ray diffraction microscopy,” Science 321, 379–382 (2008).
[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]

Montefusco, L.

D. Lazzaro and L. Montefusco, “Edge-preserving wavelet thresholding for image denoising,” J. Comput. Appl. Math. 210, 222–231 (2007).
[Crossref]

Moravec, M. L.

M. L. Moravec, J. K. Romberg, and R. G. Baraniuk, “Compressive phase retrieval,” Proc. SPIE 6701, 670120 (2007).
[Crossref]

Moré, J. J.

J. J. Moré and S. M. Wild, “Estimating derivatives of noisy simulations,” ACM Trans. Math. Software 38, 1–21 (2012).
[Crossref]

Munson, T.

A. Tripathi, S. Leyffer, T. Munson, and S. M. Wild, “Visualizing and improving the robustness of phase retrieval algorithms,” Procedia Comput. Sci. 51, 815–824 (2015).
[Crossref]

Nashed, Y. S. G.

Nelson, J.

Neutze, R.

R. Neutze, R. Wouts, D. van der Spoel, E. Weckert, and J. Hajdu, “Potential for biomolecular imaging with femtosecond x-ray pulses,” Nature 406, 752–757 (2000).
[Crossref]

Nickles, P. V.

Noh, D. Y.

Noy, A.

H. N. Chapman, A. Barty, S. Marchesini, A. Noy, S. P. Hau-Riege, C. Cui, M. R. Howells, R. Rosen, H. He, J. C. H. Spence, U. Weierstall, T. Beetz, C. Jacobsen, and D. Shapiro, “High-resolution ab initio three-dimensional x-ray diffraction microscopy,” J. Opt. Soc. Am. A 23, 1179–1200 (2006).
[Crossref]

S. Marchesini, H. He, H. N. Chapman, S. P. Hau-Riege, A. Noy, M. R. Howells, U. Weierstall, and J. C. H. Spence, “X-ray image reconstruction from a diffraction pattern alone,” Phys. Rev. B 68, 140101 (2003).
[Crossref]

Nugent, K.

B. Abbey, K. Nugent, G. Williams, J. Clark, A. Peele, M. Pfeifer, M. de Jonge, and I. McNulty, “Keyhole coherent diffractive imaging,” Nat. Phys. 4, 394–398 (2008).
[Crossref]

Nugent, K. A.

Obozinski, G.

F. Bach, R. Jenatton, J. Mairal, and G. Obozinski, “Optimization with sparsity-inducing penalties,” Found. Trends Mach. Learn. 4, 1–106 (2011).
[Crossref]

Osher, S.

L. I. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica D 60, 259–268 (1992).
[Crossref]

Paganin, D.

D. Paganin, Coherent X-ray Optics (Oxford University, 2006).
[Crossref]

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]

Peele, A.

B. Abbey, K. Nugent, G. Williams, J. Clark, A. Peele, M. Pfeifer, M. de Jonge, and I. McNulty, “Keyhole coherent diffractive imaging,” Nat. Phys. 4, 394–398 (2008).
[Crossref]

Peele, A. G.

Pein, A.

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]

Peterka, T.

Pfeifer, M.

B. Abbey, K. Nugent, G. Williams, J. Clark, A. Peele, M. Pfeifer, M. de Jonge, and I. McNulty, “Keyhole coherent diffractive imaging,” Nat. Phys. 4, 394–398 (2008).
[Crossref]

Pfeifer, M. A.

J. N. Clark, C. T. Putkunz, M. A. Pfeifer, A. G. Peele, G. J. Williams, B. Chen, K. A. Nugent, C. Hall, W. Fullagar, S. Kim, and I. McNulty, “Use of a complex constraint in coherent diffractive imaging,” Opt. Express 18, 1981–1993 (2010).
[Crossref]

M. A. Pfeifer, G. J. Williams, I. A. Vartanyants, R. Harder, and I. K. Robinson, “Three-dimensional mapping of a deformation field inside a nanocrystal,” Nature 442, 63–66 (2006).
[Crossref]

Pfeiffer, F.

P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, and F. Pfeiffer, “High-resolution scanning X-ray diffraction microscopy,” Science 321, 379–382 (2008).
[Crossref]

Plonka, G.

Putkunz, C. T.

Robinson, I. K.

M. A. Pfeifer, G. J. Williams, I. A. Vartanyants, R. Harder, and I. K. Robinson, “Three-dimensional mapping of a deformation field inside a nanocrystal,” Nature 442, 63–66 (2006).
[Crossref]

Rodenburg, J. M.

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

H. M. L. Faulkner and J. M. Rodenburg, “Movable aperture lensless transmission microscopy: A novel phase retrieval algorithm,” Phys. Rev. Lett. 93, 023903 (2004).
[Crossref]

Romberg, J. K.

M. L. Moravec, J. K. Romberg, and R. G. Baraniuk, “Compressive phase retrieval,” Proc. SPIE 6701, 670120 (2007).
[Crossref]

Rosen, R.

Ross, R.

Rudin, L. I.

L. I. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica D 60, 259–268 (1992).
[Crossref]

Salditt, T.

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]

Shapiro, D.

Shen, L.

R. H. Chan, T. F. Chan, L. Shen, and Z. Shen, “Wavelet algorithms for high-resolution image reconstruction,” SIAM J. Sci. Comput. 24, 1408–1432 (2003).
[Crossref]

Shen, Z.

R. H. Chan, T. F. Chan, L. Shen, and Z. Shen, “Wavelet algorithms for high-resolution image reconstruction,” SIAM J. Sci. Comput. 24, 1408–1432 (2003).
[Crossref]

Sorber, L.

L. Sorber, M. Barel, and L. Lathauwer, “Unconstrained optimization of real functions in complex variables,” SIAM J. Optim. 22, 879–898 (2012).
[Crossref]

Spence, J. C. H.

H. N. Chapman, A. Barty, S. Marchesini, A. Noy, S. P. Hau-Riege, C. Cui, M. R. Howells, R. Rosen, H. He, J. C. H. Spence, U. Weierstall, T. Beetz, C. Jacobsen, and D. Shapiro, “High-resolution ab initio three-dimensional x-ray diffraction microscopy,” J. Opt. Soc. Am. A 23, 1179–1200 (2006).
[Crossref]

S. Marchesini, H. He, H. N. Chapman, S. P. Hau-Riege, A. Noy, M. R. Howells, U. Weierstall, and J. C. H. Spence, “X-ray image reconstruction from a diffraction pattern alone,” Phys. Rev. B 68, 140101 (2003).
[Crossref]

Starck, J.-L.

J.-L. Starck, E. J. Candés, and D. L. Donoho, “The curvelet transform for image denoising,” IEEE Trans. Image Process. 11, 670–684 (2002).
[Crossref]

Steinbrener, J.

Thibault, P.

P. Thibault and A. Menzel, “Reconstructing state mixtures from diffraction measurements,” Nature 494, 68–71 (2013).
[Crossref]

P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, and F. Pfeiffer, “High-resolution scanning X-ray diffraction microscopy,” Science 321, 379–382 (2008).
[Crossref]

Tripathi, A.

A. Tripathi, S. Leyffer, T. Munson, and S. M. Wild, “Visualizing and improving the robustness of phase retrieval algorithms,” Procedia Comput. Sci. 51, 815–824 (2015).
[Crossref]

Tropp, J.

K. Herrity, A. Gilbert, and J. Tropp, “Sparse approximation via iterative thresholding,” in Proceedings of IEEE International Conference on Acoustics, Speech and Signal Processing (IEEE, 2006), pp. 624–627.

Turner, J. J.

van der Spoel, D.

R. Neutze, R. Wouts, D. van der Spoel, E. Weckert, and J. Hajdu, “Potential for biomolecular imaging with femtosecond x-ray pulses,” Nature 406, 752–757 (2000).
[Crossref]

Vartanyants, I. A.

M. A. Pfeifer, G. J. Williams, I. A. Vartanyants, R. Harder, and I. K. Robinson, “Three-dimensional mapping of a deformation field inside a nanocrystal,” Nature 442, 63–66 (2006).
[Crossref]

Vine, D. J.

Wakin, M.

E. Candés and M. Wakin, “An introduction to compressive sampling,” IEEE Signal Proc. Mag. 25, 21–30 (2008).
[Crossref]

Wakin, M. B.

E. J. Candés, M. B. Wakin, and S. P. Boyd, “Enhancing sparsity by reweighted ℓ1 minimization,” J. Fourier Anal. Appl. 14, 877–905 (2008).
[Crossref]

Wan, Q.

R. Fan, Q. Wan, F. Wen, H. Chen, and Y. Liu, “Iterative projection approach for phase retrieval of semi-sparse wave field,” EURASIP J. Adv. Signal Process. 2014, 1–13 (2014).
[Crossref]

Weckert, E.

R. Neutze, R. Wouts, D. van der Spoel, E. Weckert, and J. Hajdu, “Potential for biomolecular imaging with femtosecond x-ray pulses,” Nature 406, 752–757 (2000).
[Crossref]

Weierstall, U.

H. N. Chapman, A. Barty, S. Marchesini, A. Noy, S. P. Hau-Riege, C. Cui, M. R. Howells, R. Rosen, H. He, J. C. H. Spence, U. Weierstall, T. Beetz, C. Jacobsen, and D. Shapiro, “High-resolution ab initio three-dimensional x-ray diffraction microscopy,” J. Opt. Soc. Am. A 23, 1179–1200 (2006).
[Crossref]

S. Marchesini, H. He, H. N. Chapman, S. P. Hau-Riege, A. Noy, M. R. Howells, U. Weierstall, and J. C. H. Spence, “X-ray image reconstruction from a diffraction pattern alone,” Phys. Rev. B 68, 140101 (2003).
[Crossref]

Wen, F.

R. Fan, Q. Wan, F. Wen, H. Chen, and Y. Liu, “Iterative projection approach for phase retrieval of semi-sparse wave field,” EURASIP J. Adv. Signal Process. 2014, 1–13 (2014).
[Crossref]

Wild, S. M.

A. Tripathi, S. Leyffer, T. Munson, and S. M. Wild, “Visualizing and improving the robustness of phase retrieval algorithms,” Procedia Comput. Sci. 51, 815–824 (2015).
[Crossref]

J. J. Moré and S. M. Wild, “Estimating derivatives of noisy simulations,” ACM Trans. Math. Software 38, 1–21 (2012).
[Crossref]

Williams, G.

B. Abbey, K. Nugent, G. Williams, J. Clark, A. Peele, M. Pfeifer, M. de Jonge, and I. McNulty, “Keyhole coherent diffractive imaging,” Nat. Phys. 4, 394–398 (2008).
[Crossref]

Williams, G. J.

J. N. Clark, C. T. Putkunz, M. A. Pfeifer, A. G. Peele, G. J. Williams, B. Chen, K. A. Nugent, C. Hall, W. Fullagar, S. Kim, and I. McNulty, “Use of a complex constraint in coherent diffractive imaging,” Opt. Express 18, 1981–1993 (2010).
[Crossref]

M. A. Pfeifer, G. J. Williams, I. A. Vartanyants, R. Harder, and I. K. Robinson, “Three-dimensional mapping of a deformation field inside a nanocrystal,” Nature 442, 63–66 (2006).
[Crossref]

Wouts, R.

R. Neutze, R. Wouts, D. van der Spoel, E. Weckert, and J. Hajdu, “Potential for biomolecular imaging with femtosecond x-ray pulses,” Nature 406, 752–757 (2000).
[Crossref]

Xie, L.

Z. Yang, C. Zhang, and L. Xie, “Robust compressive phase retrieval via L1 minimization with application to image reconstruction,” Tech. Rep. 1302.0081, arXiv (2013).

Yang, Z.

Z. Yang, C. Zhang, and L. Xie, “Robust compressive phase retrieval via L1 minimization with application to image reconstruction,” Tech. Rep. 1302.0081, arXiv (2013).

Yi, S.

S. Yi, D. Labate, G. R. Easley, and H. Krim, “A shearlet approach to edge analysis and detection,” IEEE Trans. Image Process. 18, 929–941 (2009).
[Crossref]

Zhang, C.

Z. Yang, C. Zhang, and L. Xie, “Robust compressive phase retrieval via L1 minimization with application to image reconstruction,” Tech. Rep. 1302.0081, arXiv (2013).

Zhang, X.

X. Zhang, Y. Lu, and T. Chan, “A novel sparsity reconstruction method from Poisson data for 3D bioluminescence tomography,” J. Sci. Comput. 50, 519–535 (2011).
[Crossref]

Zhang, Y.

Y. Zhang, B. Dong, and Z. Lu, “ℓ0 minimization for wavelet frame based image restoration,” Math. Comput. 82, 995–1015 (2013).
[Crossref]

B. Dong and Y. Zhang, “An efficient algorithm for ℓ0 minimization in wavelet frame based image restoration,” J. Sci. Comput. 54, 350–368 (2012).
[Crossref]

ACM Trans. Math. Software (1)

J. J. Moré and S. M. Wild, “Estimating derivatives of noisy simulations,” ACM Trans. Math. Software 38, 1–21 (2012).
[Crossref]

Appl. Opt. (1)

EURASIP J. Adv. Signal Process. (1)

R. Fan, Q. Wan, F. Wen, H. Chen, and Y. Liu, “Iterative projection approach for phase retrieval of semi-sparse wave field,” EURASIP J. Adv. Signal Process. 2014, 1–13 (2014).
[Crossref]

Found. Trends Mach. Learn. (2)

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]

F. Bach, R. Jenatton, J. Mairal, and G. Obozinski, “Optimization with sparsity-inducing penalties,” Found. Trends Mach. Learn. 4, 1–106 (2011).
[Crossref]

IEEE Signal Proc. Mag. (1)

E. Candés and M. Wakin, “An introduction to compressive sampling,” IEEE Signal Proc. Mag. 25, 21–30 (2008).
[Crossref]

IEEE Trans. Image Process. (2)

S. Yi, D. Labate, G. R. Easley, and H. Krim, “A shearlet approach to edge analysis and detection,” IEEE Trans. Image Process. 18, 929–941 (2009).
[Crossref]

J.-L. Starck, E. J. Candés, and D. L. Donoho, “The curvelet transform for image denoising,” IEEE Trans. Image Process. 11, 670–684 (2002).
[Crossref]

J. Comput. Appl. Math. (1)

D. Lazzaro and L. Montefusco, “Edge-preserving wavelet thresholding for image denoising,” J. Comput. Appl. Math. 210, 222–231 (2007).
[Crossref]

J. Fourier Anal. Appl. (2)

E. J. Candés, M. B. Wakin, and S. P. Boyd, “Enhancing sparsity by reweighted ℓ1 minimization,” J. Fourier Anal. Appl. 14, 877–905 (2008).
[Crossref]

T. Blumensath and M. E. Davies, “Iterative thresholding for sparse approximations,” J. Fourier Anal. Appl. 14, 629–654 (2008).
[Crossref]

J. Opt. Soc. Am. A (1)

J. Sci. Comput. (2)

X. Zhang, Y. Lu, and T. Chan, “A novel sparsity reconstruction method from Poisson data for 3D bioluminescence tomography,” J. Sci. Comput. 50, 519–535 (2011).
[Crossref]

B. Dong and Y. Zhang, “An efficient algorithm for ℓ0 minimization in wavelet frame based image restoration,” J. Sci. Comput. 54, 350–368 (2012).
[Crossref]

Math. Comput. (1)

Y. Zhang, B. Dong, and Z. Lu, “ℓ0 minimization for wavelet frame based image restoration,” Math. Comput. 82, 995–1015 (2013).
[Crossref]

Nat. Phys. (1)

B. Abbey, K. Nugent, G. Williams, J. Clark, A. Peele, M. Pfeifer, M. de Jonge, and I. McNulty, “Keyhole coherent diffractive imaging,” Nat. Phys. 4, 394–398 (2008).
[Crossref]

Nature (4)

R. Neutze, R. Wouts, D. van der Spoel, E. Weckert, and J. Hajdu, “Potential for biomolecular imaging with femtosecond x-ray pulses,” Nature 406, 752–757 (2000).
[Crossref]

M. A. Pfeifer, G. J. Williams, I. A. Vartanyants, R. Harder, and I. K. Robinson, “Three-dimensional mapping of a deformation field inside a nanocrystal,” Nature 442, 63–66 (2006).
[Crossref]

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]

P. Thibault and A. Menzel, “Reconstructing state mixtures from diffraction measurements,” Nature 494, 68–71 (2013).
[Crossref]

Opt. Express (6)

Phys. Rev. B (1)

S. Marchesini, H. He, H. N. Chapman, S. P. Hau-Riege, A. Noy, M. R. Howells, U. Weierstall, and J. C. H. Spence, “X-ray image reconstruction from a diffraction pattern alone,” Phys. Rev. B 68, 140101 (2003).
[Crossref]

Phys. Rev. Lett. (1)

H. M. L. Faulkner and J. M. Rodenburg, “Movable aperture lensless transmission microscopy: A novel phase retrieval algorithm,” Phys. Rev. Lett. 93, 023903 (2004).
[Crossref]

Physica D (1)

L. I. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica D 60, 259–268 (1992).
[Crossref]

Proc. SPIE (1)

M. L. Moravec, J. K. Romberg, and R. G. Baraniuk, “Compressive phase retrieval,” Proc. SPIE 6701, 670120 (2007).
[Crossref]

Procedia Comput. Sci. (1)

A. Tripathi, S. Leyffer, T. Munson, and S. M. Wild, “Visualizing and improving the robustness of phase retrieval algorithms,” Procedia Comput. Sci. 51, 815–824 (2015).
[Crossref]

Science (1)

P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, and F. Pfeiffer, “High-resolution scanning X-ray diffraction microscopy,” Science 321, 379–382 (2008).
[Crossref]

SIAM J. Optim. (1)

L. Sorber, M. Barel, and L. Lathauwer, “Unconstrained optimization of real functions in complex variables,” SIAM J. Optim. 22, 879–898 (2012).
[Crossref]

SIAM J. Sci. Comput. (1)

R. H. Chan, T. F. Chan, L. Shen, and Z. Shen, “Wavelet algorithms for high-resolution image reconstruction,” SIAM J. Sci. Comput. 24, 1408–1432 (2003).
[Crossref]

Ultramicroscopy (1)

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

Other (4)

K. Herrity, A. Gilbert, and J. Tropp, “Sparse approximation via iterative thresholding,” in Proceedings of IEEE International Conference on Acoustics, Speech and Signal Processing (IEEE, 2006), pp. 624–627.

Z. Yang, C. Zhang, and L. Xie, “Robust compressive phase retrieval via L1 minimization with application to image reconstruction,” Tech. Rep. 1302.0081, arXiv (2013).

D. Paganin, Coherent X-ray Optics (Oxford University, 2006).
[Crossref]

S. Marchesini, “Ab initio compressive phase retrieval,” Tech. Rep. 0809.2006, arXiv (2008).

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Figures (8)

Fig. 1
Fig. 1 CDI experimental setup: a complex-valued exit wave ρ results in a real-valued data D collected by the detector.
Fig. 2
Fig. 2 Simulation results when using the approximate diameter of the probe as the support for the exit wave ρ. (a) The probe function generated by solving the Fresnel integral using a simulated circular pinhole as input. (b) The ground truth exit wave ρtrue generated by using the projection approximation with the probe shown in (a) and a simulated sample transmission function (not shown) defined as a Voronoi pattern. A typical boundary, where inside (outside) the red dotted line the support �� �� ( k ) takes on values of 1 (0), of the binary support mask ���� generated by using shrinkwrap is shown as the red dotted line. (c) The unsuccessfully recovered exit wave ρ(k) after k = 105 iterations of combined HIO, ER, and shrinkwrap. (d) A linecut of the exit wave ρtrue through the main diagonal. The boundary of the support (red dotted line) in (b) is shown here also as the red dotted lines. The support boundary shown potentially can zero out features of |ρtrue| below approximately 0.05|ρtrue |max. (e) The measurement metric Eq. (3) versus iteration number during the HIO+ER+shrinkwrap sequence.
Fig. 3
Fig. 3 (a) The ground truth exit wave ρtrue. (b) A support ���� generated by hard thresholding |ρtrue| by 2% of the maximum value of |ρtrue|, with r �� �� ( r ) / M N = 0.15 being the resulting sparsity ratio and ����(r) defined in Eq. (6). In (c,e), xρtrue and yρtrue, respectively, the x and y components of the forward-difference gradient of ρtrue, are shown. Shown in (d,f) are the supports ����x and ����y generated by hard thresholding 2% of the maximum value of |xρtrue| and |yρtrue|, respectively; 1 M N �� �� y 0 = 0.0141 and 1 M N �� �� y 0 = 0.0133 are the respective sparsity ratios.
Fig. 4
Fig. 4 At an illustrative iteration k: (a) Numerical evaluation of ψ(k)(τx, τy) over the box defined by (τx, τy) = (0.882, 0.884) to (7.232, 7.168), which corresponds to the sparsity ratio box (κx, κy) = (0.01, 0.01) to (0.1, 0.1). At this scale in (τx, τy), ψ(k) looks well behaved. A single global minimum is indicated by the magenta × marker; the black arrows denote the gradient (rescaled to be a unit vector) of ψ(k)(τx, τy) with respect to (τx, τy). (b) Evaluation of ψ(k)(τx, τy) in the red box subregion shown in (a). At this scale in (τx, τy), multiple local minima and discontinuity of the derivatives become apparent.
Fig. 5
Fig. 5 (a) Sparsity ratios for the sparse representation jρ for j ∈ {x, y} of the test problems we benchmark to evaluate performance of Eq. (27). To illustrate what is changing in (a), the exit wave ρtrue for the polygon densities 16 M N, 3072 M N, and 12288 M N are shown in (b), (c), and (d), respectively. The supports on the sparse representation xρtrue (i.e., ����x) for each of these respective cases are shown in (e), (f), and (g). The probe function used to generate the exit waves is the same used in Fig. 2(a), and for all the transmission functions corresponding to these exit waves the modulus contrast varies between 0.1 and 0.9, while the phase contrast varies between ±π.
Fig. 6
Fig. 6 (a–b) Fourier modulus measurement metric Eq. (3) vs a mesh of fixed sparsity ratios (κx, κy). (a) Use of hard thresholding and the ADMM updates over the box of fixed sparsity ratios from (0.005, 0.005) to (0.255, 0.255). In this case, a minimum (located by the magenta circle) is clearly defined at a small highly constraining sparsity ratio pair allowing prompt convergence to the ground truth exit wave ρtrue. (b) Use of soft thresholding and the ADMM updates over the box of fixed sparsity ratios from (0.005, 0.005) to (0.505, 0.505). (c) Recovered exit wave using hard thresholding corresponding to the minimum (red circle/cross) in (a) at sparsity ratio pair (κx, κy) = (0.0175, 0.03). (d) Recovered exit wave using soft thresholding corresponding to the minimum (blue circle/cross) in (a) at sparsity ratio pair (κx, κy) = (0.105, 0.105) (e) Recovered exit wave using soft thresholding corresponding to the minimum (magenta circle/cross) in (a) at sparsity ratio pair (κx, κy) = (0.505, 0.505).
Fig. 7
Fig. 7 (a) Sorted ε true 2 = ρ true ρ ( k ) F 2 values vs different sparsity ratios in jρ for j ∈ {x, y} after k = 2.5 × 104 iterations and using 50 independent trials with different starting random exit waves. For 16 M N, (b) the ρtrue, (c) ρ(k) with lowest ε true 2 value, and (d) ρ(k) with largest ε true 2 value. For 128 M N, (e) the ρtrue and (f) ρ(k) with lowest ε true 2 value, and (g) ρ(k) with largest ε true 2 value.
Fig. 8
Fig. 8 Effects of Poisson degraded diffraction intensity on the sparsity of the supports for the sparse representations jρ. (a) Azimuthal average of diffraction patterns without noise D (black curve) and with noise Dn (green curve). (b) Azimuthal average of the noisy diffraction pattern filtered in Fourier space using the forward difference x (black curve) and Sobel Ẽ′x (green curve) edge detection convolution matrices. (c) |x|, (d) |Ẽ′x|. (e) An example noise-free support for |xρtrue|, (f) the support of |xΠn [ρtrue]| using forward difference edge detection, and (g) the |∂′xΠn [ρtrue]| using Sobel edge detection. For (e–g), hard thresholding is used to generate the supports with an enforced sparsity ratio of κx = 0.05.

Equations (37)

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a 0 = r 𝕀 { a 0 } ( r ) ,
𝕀 { a 0 } ( r ) = { 1 if a ( r ) 0 0 if a ( r ) = 0 ,
ε 2 ( ρ ( k ) ) = q | | ρ ˜ ( k ) ( q ) | D ( q ) | 2 ,
E x = [ 1 1 0 0 0 0 0 0 0 0 0 0 ] , E y = E x T = [ 1 0 0 1 0 0 0 0 0 0 0 0 ] ,
j ρ = E j * ρ = 1 [ [ E j ] [ ρ ] ] = 1 [ E ˜ j ρ ˜ ] ,
𝕀 𝒮 j ( r ) = { 1 if | j ρ | > 0.02 | j ρ | max 0 if | j ρ | 0.02 | j ρ | max , for j = x , y .
min u x , u y , ρ u x + u y subject to u x = x ρ u y = y ρ ρ ,
= { ρ : ρ = 1 [ D e i ϕ ˜ ] }
Π [ ρ ] = 1 [ D e i ϕ ˜ ] ,
a 1 = r | a ( r ) | .
( u x , u y , ρ , λ x , λ y , τ x , τ y , ) = j { x , y } j ( u j , ρ , λ j , τ j , ) = j { x , y } [ τ j u j + u j j ρ F 2 + 2 Re [ r λ j * ( u j j ρ ) ] ] = j { x , y } [ τ j u j + u j j ρ + λ j F 2 λ j F 2 ]
= j { x , y } [ τ j u j + u ˜ j E ˜ j ρ ˜ + λ ˜ j F 2 λ j F 2 ] ,
a F 2 = r | a ( r ) | 2 = q | a ˜ ( q ) | 2 ,
τ j ( k + 1 ) = arg min τ j ψ j ( u j ( k ) , ρ ( k ) , λ j ( k ) , τ j , ) for j { x , y }
u j ( k + 1 ) = arg min u j j ( u j , ρ ( k ) , λ j ( k ) , τ j ( k + 1 ) , ) for j { x , y }
ρ ( k + 1 ) = arg min ρ ( u x ( k + 1 ) , u y ( k + 1 ) , ρ , λ x ( k ) , λ y ( k ) , τ x ( k + 1 ) , τ y ( k + 1 ) , )
λ j ( k + 1 ) = λ j ( k ) + β j ( u j ( k + 1 ) j ρ ( k + 1 ) ) for j { x , y }
u j ( k + 1 ) = arg min u j τ j u j + u j b F 2 ,
T 1 ( τ , b ) = sgn ( b ) ( | b | τ ) 𝕀 { | b | > τ } ,
T 0 ( τ , b ) = b 𝕀 { | b | > τ } .
𝕀 { | b | > μ } ( r ) = { 1 if | b ( r ) | > μ 0 if | b ( r ) | μ
u j ( k + 1 ) = arg min u j τ j ( k ) u j + u j j ρ ( k ) + λ j ( k ) F 2 = T ( τ j ( k ) , j ρ ( k ) λ j ( k ) ) , for j { x , y }
ρ ˜ j { x , y } u ˜ j ( k + 1 ) E ˜ j ρ ˜ + λ ˜ j ( k ) F 2 = 0 .
ρ ˜ = E ˜ x * u ˜ x ( k + 1 ) + E ˜ x * λ ˜ x ( k ) + E ˜ y * u ˜ y ( k + 1 ) + E ˜ y * λ ˜ y ( k ) | E ˜ y | 2 + | E ˜ x | 2 ,
ρ ( k + 1 ) = Π [ 1 [ E ˜ x * u ˜ x ( k + 1 ) + E ˜ x * λ ˜ x ( k ) + E ˜ y * u ˜ y ( k + 1 ) + E ˜ y * λ ˜ y ( k ) | E ˜ y | 2 + | E ˜ x | 2 ] ] ,
ψ j ( u j ( k ) , ρ ( k ) , λ j ( k ) , τ j , ) = | E ˜ j | D | [ z ( k ) ( τ j ) ] | F 2 + w j ( k ) z ( k ) ( τ j ) ,
ψ ( k ) ( τ x , τ y ) = j { x , y } ψ j ( u j ( k ) , ρ ( k ) , λ j ( k ) , τ j , ) .
w j ( k ) = ς | E ˜ j | D | [ u j ( k ) ] | F 2 u j ( k ) ,
h x ( k ) = arg min h x { 0 , h ± 1 x , h ± 2 x , } ψ ( k ) ( τ x ( k ) + h x , τ y ( k ) )
h y ( k ) = arg min h y { 0 , h ± 1 y , h ± 2 y , } ψ ( k ) ( τ x ( k + 1 ) , τ y ( k ) + h y ) ,
τ j ( k + 1 ) = { τ j ( k ) + h j ( k ) if k mod L = 0 τ j ( k ) if k mod L 0 for j { x , y } ,
τ j ( k + 1 ) = { τ j ( k ) + h j ( k ) if k mod L = 0 τ j ( k ) if k mod L 0 for j { x , y }
u j ( k + 1 ) = T ( τ j ( k + 1 ) , j ρ ( k ) λ j ( k ) ) for j { x , y }
ρ ( k + 1 ) = Π [ 1 [ E ˜ x * u ˜ x ( k + 1 ) + E ˜ x * λ ˜ x ( k ) + E ˜ y * u ˜ y ( k + 1 ) + E ˜ y * λ ˜ y ( k ) | E ˜ y | 2 + | E ˜ x | 2 ] ]
λ j ( k + 1 ) = λ j ( k ) + β j ( u j ( k + 1 ) j ρ ( k + 1 ) ) for j { x , y } .
F ( q r , q θ ) q θ = 1 N ( q r ) q F ( q r , q θ ) A ( q r , q θ )
E x = 1 4 [ 1 0 1 0 0 2 0 2 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 ] M × N ,

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