Abstract

We present a convex relaxation-based algorithm for large-scale general phase retrieval problems. General phase retrieval problems include, e.g., the estimation of the phase of the optical field in the pupil plane based on intensity measurements of a point source recorded in the image (focal) plane. The non-convex problem of finding the complex field that generates the correct intensity is reformulated into a rank constraint problem. The nuclear norm is used to obtain the convex relaxation of the phase retrieval problem. A new iterative method referred to as convex optimization-based phase retrieval (COPR) is presented, with each iteration consisting of solving a convex problem. In the noise-free case and for a class of phase retrieval problems, the solutions of the minimization problems converge linearly or faster towards a correct solution. Since the solutions to nuclear norm minimization problems can be computed using semidefinite programming, and this tends to be an expensive optimization in terms of scalability, we provide a fast algorithm called alternating direction method of multipliers (ADMM) that exploits the problem structure. The performance of the COPR algorithm is demonstrated in a realistic numerical simulation study, demonstrating its improvements in reliability and speed with respect to state-of-the-art methods.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2018 (1)

2017 (1)

A. Martinez-Finkelshtein, D. Ramos-Lopez, and D. Iskander, “Computation of 2D Fourier transforms and diffraction integrals using Gaussian radial basis functions,” Appl. Comput. Harmon. Anal. 43, 424–448 (2017).
[Crossref]

2015 (3)

E. J. Candes, X. Li, and M. Soltanolkotabi, “Phase retrieval via Wirtinger flow: theory and algorithms,” IEEE Trans. Inf. Theory 61, 1985–2007 (2015).
[Crossref]

J. Antonello and M. Verhaegen, “Modal-based phase retrieval for adaptive optics,” J. Opt. Soc. Am. A 32, 1160–1170 (2015).
[Crossref]

Y. Shechtman, Y. C. Eldar, O. Cohen, H. N. Chapman, J. Miao, and M. Segev, “Phase retrieval with application to optical imaging: a contemporary overview,” IEEE Signal Process. Mag. 32(3), 87–109 (2015).
[Crossref]

2014 (1)

Y. Shechtman, A. Beck, and Y. C. Eldar, “GESPAR: efficient phase retrieval of sparse signals,” IEEE Trans. Signal Process. 62, 928–938 (2014).
[Crossref]

2013 (1)

E. J. Candes, T. Strohmer, and V. Voroninski, “Phaselift: exact and stable signal recovery from magnitude measurements via convex programming,” Commun. Pure Appl. Math. 66, 1241–1274 (2013).
[Crossref]

2011 (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).

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]

2007 (1)

C.-C. Chen, J. Miao, C. Wang, and T. Lee, “Application of optimization technique to noncrystalline x-ray diffraction microscopy: guided hybrid input-output method,” Phys. Rev. B 76, 064113 (2007).
[Crossref]

2003 (1)

V. Elser, “Solution of the crystallographic phase problem by iterated projections,” Acta Crystallogr. A 59, 201–209 (2003).
[Crossref]

2002 (4)

1986 (1)

H. Hauptman, “The direct methods of X-ray crystallography,” Science 233, 178–183 (1986).
[Crossref]

1982 (1)

1972 (2)

R. Gerchberg and W. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

1952 (1)

D. Sayre, “Some implications of a theorem due to Shannon,” Acta Crystallogr. 5, 843 (1952).
[Crossref]

Antonello, J.

Balakrishnan, V. R.

L. Vandenberghe, V. R. Balakrishnan, R. Wallin, A. Hansson, and T. Roh, “Interior-point algorithms for semidefinite programming problems derived from the KYP lemma,” in Positive Polynomials in Control (2005), p. 579.

Bauschke, H.

Bauschke, H. H.

H. H. Bauschke and P. L. Combettes, Convex Analysis and Monotone Operator Theory in Hilbert Spaces, CMS Books Math./Ouvrages Math. SMC (Springer, 2011).

Beck, A.

Y. Shechtman, A. Beck, and Y. C. Eldar, “GESPAR: efficient phase retrieval of sparse signals,” IEEE Trans. Signal Process. 62, 928–938 (2014).
[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).

Boyd, S. P.

M. Fazel, H. Hindi, and S. P. Boyd, “A rank minimization heuristic with application to minimum order system approximation,” in Proceedings of the 2001 American Control Conference (IEEE, 2001), Vol. 6, pp. 4734–4739.

Braat, J.

Burke, J. V.

D. R. Luke, J. V. Burke, and R. G. Lyon, “Optical wavefront reconstruction: theory and numerical methods,” SIAM Rev. 44, 169–224 (2002).
[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]

Candes, E. J.

E. J. Candes, X. Li, and M. Soltanolkotabi, “Phase retrieval via Wirtinger flow: theory and algorithms,” IEEE Trans. Inf. Theory 61, 1985–2007 (2015).
[Crossref]

E. J. Candes, T. Strohmer, and V. Voroninski, “Phaselift: exact and stable signal recovery from magnitude measurements via convex programming,” Commun. Pure Appl. Math. 66, 1241–1274 (2013).
[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]

Chapman, H. N.

Y. Shechtman, Y. C. Eldar, O. Cohen, H. N. Chapman, J. Miao, and M. Segev, “Phase retrieval with application to optical imaging: a contemporary overview,” IEEE Signal Process. Mag. 32(3), 87–109 (2015).
[Crossref]

Chen, C.-C.

C.-C. Chen, J. Miao, C. Wang, and T. Lee, “Application of optimization technique to noncrystalline x-ray diffraction microscopy: guided hybrid input-output method,” Phys. Rev. B 76, 064113 (2007).
[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).

Cohen, O.

Y. Shechtman, Y. C. Eldar, O. Cohen, H. N. Chapman, J. Miao, and M. Segev, “Phase retrieval with application to optical imaging: a contemporary overview,” IEEE Signal Process. Mag. 32(3), 87–109 (2015).
[Crossref]

Combetters, P.

Combettes, P. L.

H. H. Bauschke and P. L. Combettes, Convex Analysis and Monotone Operator Theory in Hilbert Spaces, CMS Books Math./Ouvrages Math. SMC (Springer, 2011).

Dirksen, P.

Doelman, R.

R. Doelman and M. Verhaegen, “Sequential convex relaxation for convex optimization with bilinear matrix equalities,” in Proceedings of the European Control Conference (2016).

Dong, R.

H. Ohlsson, A. Y. Yang, R. Dong, and S. S. Sastry, “Compressive phase retrieval from squared output measurements via semidefinite programming,” arXiv: 1111.6323 (2011).

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).

Eldar, Y. C.

Y. Shechtman, Y. C. Eldar, O. Cohen, H. N. Chapman, J. Miao, and M. Segev, “Phase retrieval with application to optical imaging: a contemporary overview,” IEEE Signal Process. Mag. 32(3), 87–109 (2015).
[Crossref]

Y. Shechtman, A. Beck, and Y. C. Eldar, “GESPAR: efficient phase retrieval of sparse signals,” IEEE Trans. Signal Process. 62, 928–938 (2014).
[Crossref]

Elser, V.

V. Elser, “Solution of the crystallographic phase problem by iterated projections,” Acta Crystallogr. A 59, 201–209 (2003).
[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]

M. Fazel, H. Hindi, and S. P. Boyd, “A rank minimization heuristic with application to minimum order system approximation,” in Proceedings of the 2001 American Control Conference (IEEE, 2001), Vol. 6, pp. 4734–4739.

Fienup, J. R.

Gerchberg, R.

R. Gerchberg and W. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Gerchberg, R. W.

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Goodman, J.

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

Gupta, A.

Hansson, A.

L. Vandenberghe, V. R. Balakrishnan, R. Wallin, A. Hansson, and T. Roh, “Interior-point algorithms for semidefinite programming problems derived from the KYP lemma,” in Positive Polynomials in Control (2005), p. 579.

Hauptman, H.

H. Hauptman, “The direct methods of X-ray crystallography,” Science 233, 178–183 (1986).
[Crossref]

Hindi, H.

M. Fazel, H. Hindi, and S. P. Boyd, “A rank minimization heuristic with application to minimum order system approximation,” in Proceedings of the 2001 American Control Conference (IEEE, 2001), Vol. 6, pp. 4734–4739.

Iskander, D.

A. Martinez-Finkelshtein, D. Ramos-Lopez, and D. Iskander, “Computation of 2D Fourier transforms and diffraction integrals using Gaussian radial basis functions,” Appl. Comput. Harmon. Anal. 43, 424–448 (2017).
[Crossref]

Janssen, A. J.

Lee, T.

C.-C. Chen, J. Miao, C. Wang, and T. Lee, “Application of optimization technique to noncrystalline x-ray diffraction microscopy: guided hybrid input-output method,” Phys. Rev. B 76, 064113 (2007).
[Crossref]

Li, X.

E. J. Candes, X. Li, and M. Soltanolkotabi, “Phase retrieval via Wirtinger flow: theory and algorithms,” IEEE Trans. Inf. Theory 61, 1985–2007 (2015).
[Crossref]

Löfberg, J.

J. Löfberg, “YALMIP: a toolbox for modeling and optimization in MATLAB,” in Proceedings of the CACSD Conference, Taipei, Taiwan (2004).

Luke, D.

Luke, D. R.

D. R. Luke, J. V. Burke, and R. G. Lyon, “Optical wavefront reconstruction: theory and numerical methods,” SIAM Rev. 44, 169–224 (2002).
[Crossref]

D. R. Luke, M. Teboulle, and N. H. Thao, “Necessary conditions for linear convergence of iterated expansive, set-valued mappings with application to alternating projections,” arXiv: 1704.08926v2.

D. R. Luke, N. H. Thao, and M. K. Tam, “Quantitative convergence analysis of iterated expansive, set-valued mappings,” arXiv: 1605.05725 (2016).

Lyon, R. G.

D. R. Luke, J. V. Burke, and R. G. Lyon, “Optical wavefront reconstruction: theory and numerical methods,” SIAM Rev. 44, 169–224 (2002).
[Crossref]

Martinez-Finkelshtein, A.

A. Martinez-Finkelshtein, D. Ramos-Lopez, and D. Iskander, “Computation of 2D Fourier transforms and diffraction integrals using Gaussian radial basis functions,” Appl. Comput. Harmon. Anal. 43, 424–448 (2017).
[Crossref]

Miao, J.

Y. Shechtman, Y. C. Eldar, O. Cohen, H. N. Chapman, J. Miao, and M. Segev, “Phase retrieval with application to optical imaging: a contemporary overview,” IEEE Signal Process. Mag. 32(3), 87–109 (2015).
[Crossref]

C.-C. Chen, J. Miao, C. Wang, and T. Lee, “Application of optimization technique to noncrystalline x-ray diffraction microscopy: guided hybrid input-output method,” Phys. Rev. B 76, 064113 (2007).
[Crossref]

Ohlsson, H.

H. Ohlsson, A. Y. Yang, R. Dong, and S. S. Sastry, “Compressive phase retrieval from squared output measurements via semidefinite programming,” arXiv: 1111.6323 (2011).

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).

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]

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).

Piscaer, P. J.

Ramos-Lopez, D.

A. Martinez-Finkelshtein, D. Ramos-Lopez, and D. Iskander, “Computation of 2D Fourier transforms and diffraction integrals using Gaussian radial basis functions,” Appl. Comput. Harmon. Anal. 43, 424–448 (2017).
[Crossref]

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]

Roddier, F.

F. Roddier, Adaptive Optics in Astronomy (Cambridge University, 1999).

Roh, T.

L. Vandenberghe, V. R. Balakrishnan, R. Wallin, A. Hansson, and T. Roh, “Interior-point algorithms for semidefinite programming problems derived from the KYP lemma,” in Positive Polynomials in Control (2005), p. 579.

Sastry, S. S.

H. Ohlsson, A. Y. Yang, R. Dong, and S. S. Sastry, “Compressive phase retrieval from squared output measurements via semidefinite programming,” arXiv: 1111.6323 (2011).

Saxton, W.

R. Gerchberg and W. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Saxton, W. O.

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Sayre, D.

D. Sayre, “Some implications of a theorem due to Shannon,” Acta Crystallogr. 5, 843 (1952).
[Crossref]

Segev, M.

Y. Shechtman, Y. C. Eldar, O. Cohen, H. N. Chapman, J. Miao, and M. Segev, “Phase retrieval with application to optical imaging: a contemporary overview,” IEEE Signal Process. Mag. 32(3), 87–109 (2015).
[Crossref]

Shechtman, Y.

Y. Shechtman, Y. C. Eldar, O. Cohen, H. N. Chapman, J. Miao, and M. Segev, “Phase retrieval with application to optical imaging: a contemporary overview,” IEEE Signal Process. Mag. 32(3), 87–109 (2015).
[Crossref]

Y. Shechtman, A. Beck, and Y. C. Eldar, “GESPAR: efficient phase retrieval of sparse signals,” IEEE Trans. Signal Process. 62, 928–938 (2014).
[Crossref]

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]

Soloviev, O.

Soltanolkotabi, M.

E. J. Candes, X. Li, and M. Soltanolkotabi, “Phase retrieval via Wirtinger flow: theory and algorithms,” IEEE Trans. Inf. Theory 61, 1985–2007 (2015).
[Crossref]

Strohmer, T.

E. J. Candes, T. Strohmer, and V. Voroninski, “Phaselift: exact and stable signal recovery from magnitude measurements via convex programming,” Commun. Pure Appl. Math. 66, 1241–1274 (2013).
[Crossref]

Tam, M. K.

D. R. Luke, N. H. Thao, and M. K. Tam, “Quantitative convergence analysis of iterated expansive, set-valued mappings,” arXiv: 1605.05725 (2016).

Teboulle, M.

D. R. Luke, M. Teboulle, and N. H. Thao, “Necessary conditions for linear convergence of iterated expansive, set-valued mappings with application to alternating projections,” arXiv: 1704.08926v2.

Thao, N. H.

D. R. Luke, M. Teboulle, and N. H. Thao, “Necessary conditions for linear convergence of iterated expansive, set-valued mappings with application to alternating projections,” arXiv: 1704.08926v2.

D. R. Luke, N. H. Thao, and M. K. Tam, “Quantitative convergence analysis of iterated expansive, set-valued mappings,” arXiv: 1605.05725 (2016).

Vandenberghe, L.

L. Vandenberghe, V. R. Balakrishnan, R. Wallin, A. Hansson, and T. Roh, “Interior-point algorithms for semidefinite programming problems derived from the KYP lemma,” in Positive Polynomials in Control (2005), p. 579.

Verhaegen, M.

Voroninski, V.

E. J. Candes, T. Strohmer, and V. Voroninski, “Phaselift: exact and stable signal recovery from magnitude measurements via convex programming,” Commun. Pure Appl. Math. 66, 1241–1274 (2013).
[Crossref]

Wallin, R.

L. Vandenberghe, V. R. Balakrishnan, R. Wallin, A. Hansson, and T. Roh, “Interior-point algorithms for semidefinite programming problems derived from the KYP lemma,” in Positive Polynomials in Control (2005), p. 579.

Wang, C.

C.-C. Chen, J. Miao, C. Wang, and T. Lee, “Application of optimization technique to noncrystalline x-ray diffraction microscopy: guided hybrid input-output method,” Phys. Rev. B 76, 064113 (2007).
[Crossref]

Yang, A. Y.

H. Ohlsson, A. Y. Yang, R. Dong, and S. S. Sastry, “Compressive phase retrieval from squared output measurements via semidefinite programming,” arXiv: 1111.6323 (2011).

Acta Crystallogr. (1)

D. Sayre, “Some implications of a theorem due to Shannon,” Acta Crystallogr. 5, 843 (1952).
[Crossref]

Acta Crystallogr. A (1)

V. Elser, “Solution of the crystallographic phase problem by iterated projections,” Acta Crystallogr. A 59, 201–209 (2003).
[Crossref]

Appl. Comput. Harmon. Anal. (1)

A. Martinez-Finkelshtein, D. Ramos-Lopez, and D. Iskander, “Computation of 2D Fourier transforms and diffraction integrals using Gaussian radial basis functions,” Appl. Comput. Harmon. Anal. 43, 424–448 (2017).
[Crossref]

Appl. Opt. (1)

Commun. Pure Appl. Math. (1)

E. J. Candes, T. Strohmer, and V. Voroninski, “Phaselift: exact and stable signal recovery from magnitude measurements via convex programming,” Commun. Pure Appl. Math. 66, 1241–1274 (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).

IEEE Signal Process. Mag. (1)

Y. Shechtman, Y. C. Eldar, O. Cohen, H. N. Chapman, J. Miao, and M. Segev, “Phase retrieval with application to optical imaging: a contemporary overview,” IEEE Signal Process. Mag. 32(3), 87–109 (2015).
[Crossref]

IEEE Trans. Inf. Theory (1)

E. J. Candes, X. Li, and M. Soltanolkotabi, “Phase retrieval via Wirtinger flow: theory and algorithms,” IEEE Trans. Inf. Theory 61, 1985–2007 (2015).
[Crossref]

IEEE Trans. Signal Process. (1)

Y. Shechtman, A. Beck, and Y. C. Eldar, “GESPAR: efficient phase retrieval of sparse signals,” IEEE Trans. Signal Process. 62, 928–938 (2014).
[Crossref]

J. Opt. Soc. Am. A (5)

Optik (2)

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

R. Gerchberg and W. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Phys. Rev. B (1)

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Figures (6)

Fig. 1.
Fig. 1. 16 radial basis functions with centers in a 4 × 4 grid, with circular aperture support.
Fig. 2.
Fig. 2. Convergence plot for four different combinations of COPR iterations and ADMM iterations. Denoted in the legend are first the number of COPR iterations and then the number of ADMM iterations used to solve Eq. (16) in each COPR iteration. Markers denote each new COPR iteration.
Fig. 3.
Fig. 3. Absolute values of 16 estimated coefficients according to four different algorithms.
Fig. 4.
Fig. 4. Computation time comparison between PhaseLift and COPR for different numbers of coefficients.
Fig. 5.
Fig. 5. Strehl ratio of the estimated phase aberration as a function of SNR.
Fig. 6.
Fig. 6. Example PSF and phase estimates of the COPR, Alternating Projection [12], Averaged Projection [30], and PhaseLift [12] algorithms for three PSF measurements with an SNR of approximately 36 dB.

Tables (2)

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Algorithm 1 Convex Optimization-Based Phase Retrieval (COPR)

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Algorithm 2 ADMM Algorithm for Solving Eq. (16)

Equations (58)

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find a C n a    such that    y i = | u i H a | 2 for    i = 1 , , n y ,
find    a C n a    such that    y = | U a | 2 ,
min a C n a y | U a | 2 ,
P ( ρ , θ ) = A ( ρ , θ ) e j ϕ ( ρ , θ ) ,
P d ( ρ , θ ) = A ( ρ , θ ) e j ϕ ( ρ , θ ) e j ϕ d ( ρ , θ ) .
y d = | F { A ( ρ , θ ) e j ϕ ( ρ , θ ) e j ϕ d ( ρ , θ ) } | 2 .
Z = A ( ρ , θ ) e j ϕ ( ρ , θ ) C m × m , a = vect ( Z ) C m 2 .
p d = vect ( e j ϕ d ( ρ , θ ) ) C m 2 ,
y d = | F { e j ϕ d ( ρ , θ ) Z } | 2 = | F { vect 1 ( D d a ) } | 2 .
y d = | U d a | 2 ,
G i ( x , y ) = χ ( x , y ) e λ i ( ( x x i ) 2 + ( y y i ) 2 ) , P ( x , y ) P ˜ ( x , y , a ) = i = 1 n a a i G i ( x , y ) ,
P ˜ d ( x , y , a , ϕ d ) = i = 1 n a a i G i ( x , y ) e j ϕ d ( x , y ) .
p d ( u , v ) = i = 1 n a a i F { G i ( x , y ) e j ϕ d ( x , y ) } = i = 1 n a a i U d , i ( u , v ) ,
p = U a .
L ( A , B , C , X , Y ) = ( C + A Y + X B + X Y A + X B + Y I ) ,
M ( U , a , b , y ) L ( d ( a H U H ) , d ( U a ) , d ( y ) , d ( b H U H ) , d ( U b ) ) .
rank ( M ( U , a , b , y ) ) = n y .
min a C n a rank ( M ( U , a , b , y ) ) .
min a C n a f ( a ) M ( U , a , b , y ) * ,
min a C n a f ( a ) + λ a 1 ,
M ( U , a , a , y ) * = y | U a | 2 1 + n y .
min X , a X * subject to    X = M ( U , a , b , y ) .
arg min a X M ( U , a , b , y ) + 1 ρ Y F 2
arg min X X * + ρ 2 X M ( U , a + , b , y ) + 1 ρ Y F 2
Z = X + 1 ρ Y , X = d ( b H U H ) .
min x u ADMM u COPR A B x 2 2 .
u ADMM = ( d ^ ( R ( Z 1 ) ) d ^ ( R ( Z 2 ) ) d ^ ( R ( Z 3 ) ) d ^ ( I ( Z 2 ) ) d ^ ( I ( Z 3 ) ) ) , u COPR = ( y + d ^ ( | X | 2 ) d ^ ( R ( X ) ) d ^ ( R ( X ) ) d ^ ( I ( X ) ) d ^ ( I ( X ) ) ) , A = ( 2 R ( X ) 2 I ( X ) I 0 I 0 0 I 0 I ) , B = ( R ( U ) I ( U ) I ( U ) R ( U ) ) ,
x * = ( B T A T A B ) 1 B T A T ( u ADMM u COPR ) .
arg min X X * + λ X C F 2 .
X * + λ X C F 2 = trace ( Σ X ) + λ ( X , X + C , C 2 X , C ) .
min X ( trace ( Σ X ) + λ ( X , X + C , C 2 X , C ) ) min X ( trace ( Σ X ) + λ ( X , X + C , C 2 trace ( Σ X Σ C ) ) ) ,
arg min σ X , i i = 1 2 n y ( σ X , i + λ ( σ X , i σ C , i ) 2 ) .
σ X , i = max ( 0 , σ C , i 1 2 λ ) , i = 1 , , 2 n y .
T ( a ) = arg min x C n a M ( U , x , a , y ) * .
{ a | y = | U a | 2 } Fix T { a C n a | a T ( a ) } .
P Ω ( x ) { ω Ω | x ω = dist ( x , Ω ) } , x .
a + a + * 2 a a * 2 1 α α ( a + a ) ( a + * a * ) 2 .
γ dist ( a , ψ 1 ( 0 ) ) dist ( 0 , ψ ( a ) ) , a W .
min c C , | c | = 1 c a ^ a * 2 2 10 5 ,
ϕ DM = H u .
y = max ( 0 , | F { P d ( ρ , θ ) } | 2 + ϵ ) , ϵ N ( 0 , σ I ) ,
10 log 10 y | F { P d ( ρ , θ ) } | 2 2 2 | F { P d ( ρ , θ ) } | 2 2 2 .
S e δ 2 ,
rank ( M ( U , a , a , y ) ) = rank ( 0 0 0 I n y ) = n y .
a arg min x C n a M ( U , x , a , y ) * .
M ( 1 , x i , a i , y i ) = ( y i 2 R ( x i a i ¯ ) + | a i | 2 x i a i x i ¯ a i ¯ 1 ) .
f i ( x i ) M ( 1 , x i , a i , y i ) * 2 = ( r s s 1 ) * 2 = r 2 + 2 s 2 + 1 + 2 | r s 2 | ,
r y i 2 R ( x i a i ¯ ) + | a i | 2 , s | x i a i | .
T i ( a i ) arg min x i C f i ( x i ) .
T i ( a i ) = { { z C | | z | y i } , if    a i = 0 , { y i | a i | a i } , if    0 < | a i | λ i , { y i + | a i | 2 + 1 2 ( | a i | 2 + 1 ) a i } , if    | a i | λ i ,
T ( a ) = arg min x C n a i = 1 n a f i ( x i ) , a C n a ,
T ( a ) = T 1 ( a 1 ) × T 2 ( a 2 ) × T n a ( a n a ) ,
dist ( w , S ) dist ( w , Fix T ) , w W .
T ^ ( a ) = U 1 T ( U a ) , a C n a .
T ^ ( a ) = arg min x C n a M ( U , x , a , y ) * = arg min x C n a M ( I n a , U x , a , y ) * = U 1 ( arg min x C n a M ( I n a , x , a , y ) * ) = U 1 ( T ( a ) ) = U 1 ( T ( U a ) ) .
Fix T ^ = { a C n a | a T ^ ( a ) } = { a C n a | a U 1 T ( U a ) } = { a C n a | U a T ( U a ) } = { a C n a | U a Fix T } = U 1 ( Fix T ) .
P U 1 ( S ) ( U 1 w ) = U 1 ( P S ( w ) ) ,
dist ( U 1 w , U 1 ( S ) ) = dist ( U 1 w , U 1 ( Fix T ) ) .

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