Abstract

Optical coherence tomography (OCT) reconstruction by using frequency measurements in the wavelength domain is presented in this paper. The method directly recovers the axial scan by formulating the frequency domain OCT (FD-OCT) into an algebraic reconstruction problem. In this way, the need for interpolation is removed. Then by solving the problem with ℓ1 optimization, the computational load is significantly reduced. It is demonstrated by experiment and simulation that the proposed method can achieve high resolution and longer imaging depth compared to the FD-OCT method.

© 2011 OSA

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  1. D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
    [CrossRef] [PubMed]
  2. O. P. Bruno and J. Chaubell, “Inverse scattering problem for optical coherence tomography,” Opt. Lett. 28, 2049–2051 (2003).
    [CrossRef] [PubMed]
  3. P. E Andersen, L. Thrane, H. T Yura, A. Tycho, T. M. Jrgensen, and M. H Frosz, “Advanced modelling of optical coherence tomography systems,” Phys. Med. Biol. 49, 1307–1327 (2004).
    [CrossRef] [PubMed]
  4. G. Häusler and M. W. Lindner, “‘Coherence radar’ and ‘spectral radar’–new tools for dermatological diagnosis,” J. Biomed. Opt. 3, 21–31 (1998).
    [CrossRef]
  5. M. E. Brezinski, Optical Coherence Tomography: Principles and Applications (Academic Press, 2006).
  6. C. Dorrer, N. Belabas, J. Likforman, and M. Joffre, “Spectral resolution and sampling issues in Fourier-transform spectral interferometry,” J. Opt. Soc. Am. B 17, 1795–1802 (2000).
    [CrossRef]
  7. S. S. Sherif, C. Flueraru, Y. Mao, and S. Change, “Swept source optical coherence tomography with nonuniform frequency domain sampling,” in Biomedical Optics, OSA Technical Digest (CD)(Optical Society of America, 2008), paper BMD86.
  8. D. Hillmann, G. Huttmann, and P. Koch, “Using Nonequispaced fast Fourier transformation to process optical coherence tomography signals,” Proc. SPIE 7372, 73730 (2009).
  9. S. Vergnole, D. Lvesque, and G. Lamouche, “Experimental validation of an optimized signal processing method to handle non-linearity in swept-source optical coherence tomography,” Opt. Express 18, 10446–10461 (2010).
    [CrossRef] [PubMed]
  10. A. C. Kak and M. Slaney, “Algebraic reconstruction algorithms,” in Principles of Computerized Tomographic Imaging, (IEEE Press, 1999).
  11. S. Vergnole, D. Levesque, G. Lamouche, M. Dufour, and B. Gauthier, “Characterization of thin layered structures using deconvolution techniques in time-domain and Fourier-domain optical coherence tomography,” Proc SPIE 6796, 67961 (2007).
    [CrossRef]
  12. S. A. Boppart, A. L. Oldenburg, C. Xu, and D. L. Marks, “Optical probes and techniques for molecular contrast enhancement in coherence imaging,” J. Biomed. Opt. 10, 041208 (2005).
    [CrossRef]
  13. M. Lustig, D. L. Donoho, J. M. Santos, and J. M. Pauly, “Compressed sensing MRI,” IEEE Signal Processing Mag. 25, 72–82 (2008).
    [CrossRef]
  14. E. J. Candès and T. Tao, “Decoding by linear programming,” IEEE Trans. Inf. Theory 51, 4203–4215 (2005).
    [CrossRef]
  15. M. Grant and S. Boyd, CVX: Matlab software for disciplined convex programming (web page and software). http://stanford.edu/$_{\mathaccent''0365\relax{~}}$boyd/cvx (2009).
  16. W. Qiu and E. Skafidas, “Robust estimation of GCD with sparse coefficients,” Signal Processing 90, 972–976 (2010).
    [CrossRef]
  17. G. H. Golub and C. F. Van Loan, Matrix computations, (Johns Hopkins University Press, 1984).
  18. M. S. Muller and J. M. Fraser, “Contrast improvement in Fourier-domain optical coherence tomography through time gating,” J. Opt. Soc. Am. A 26, 969–976 (2009).
    [CrossRef]
  19. K. Wang, Z. Ding, T. Wu, C. Wang, J. Meng, M. Chen, and L. Xu, “Development of a non-uniform discrete Fourier transform based high speed spectral domain optical coherence tomography system,” Opt. Express 17, 12121–12131 (2009).
    [CrossRef] [PubMed]

2010 (2)

2009 (3)

2008 (1)

M. Lustig, D. L. Donoho, J. M. Santos, and J. M. Pauly, “Compressed sensing MRI,” IEEE Signal Processing Mag. 25, 72–82 (2008).
[CrossRef]

2007 (1)

S. Vergnole, D. Levesque, G. Lamouche, M. Dufour, and B. Gauthier, “Characterization of thin layered structures using deconvolution techniques in time-domain and Fourier-domain optical coherence tomography,” Proc SPIE 6796, 67961 (2007).
[CrossRef]

2005 (2)

S. A. Boppart, A. L. Oldenburg, C. Xu, and D. L. Marks, “Optical probes and techniques for molecular contrast enhancement in coherence imaging,” J. Biomed. Opt. 10, 041208 (2005).
[CrossRef]

E. J. Candès and T. Tao, “Decoding by linear programming,” IEEE Trans. Inf. Theory 51, 4203–4215 (2005).
[CrossRef]

2004 (1)

P. E Andersen, L. Thrane, H. T Yura, A. Tycho, T. M. Jrgensen, and M. H Frosz, “Advanced modelling of optical coherence tomography systems,” Phys. Med. Biol. 49, 1307–1327 (2004).
[CrossRef] [PubMed]

2003 (1)

2000 (1)

1998 (1)

G. Häusler and M. W. Lindner, “‘Coherence radar’ and ‘spectral radar’–new tools for dermatological diagnosis,” J. Biomed. Opt. 3, 21–31 (1998).
[CrossRef]

1991 (1)

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Andersen, P. E

P. E Andersen, L. Thrane, H. T Yura, A. Tycho, T. M. Jrgensen, and M. H Frosz, “Advanced modelling of optical coherence tomography systems,” Phys. Med. Biol. 49, 1307–1327 (2004).
[CrossRef] [PubMed]

Belabas, N.

Boppart, S. A.

S. A. Boppart, A. L. Oldenburg, C. Xu, and D. L. Marks, “Optical probes and techniques for molecular contrast enhancement in coherence imaging,” J. Biomed. Opt. 10, 041208 (2005).
[CrossRef]

Brezinski, M. E.

M. E. Brezinski, Optical Coherence Tomography: Principles and Applications (Academic Press, 2006).

Bruno, O. P.

Candès, E. J.

E. J. Candès and T. Tao, “Decoding by linear programming,” IEEE Trans. Inf. Theory 51, 4203–4215 (2005).
[CrossRef]

Chang, W.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Change, S.

S. S. Sherif, C. Flueraru, Y. Mao, and S. Change, “Swept source optical coherence tomography with nonuniform frequency domain sampling,” in Biomedical Optics, OSA Technical Digest (CD)(Optical Society of America, 2008), paper BMD86.

Chaubell, J.

Chen, M.

Ding, Z.

Donoho, D. L.

M. Lustig, D. L. Donoho, J. M. Santos, and J. M. Pauly, “Compressed sensing MRI,” IEEE Signal Processing Mag. 25, 72–82 (2008).
[CrossRef]

Dorrer, C.

Dufour, M.

S. Vergnole, D. Levesque, G. Lamouche, M. Dufour, and B. Gauthier, “Characterization of thin layered structures using deconvolution techniques in time-domain and Fourier-domain optical coherence tomography,” Proc SPIE 6796, 67961 (2007).
[CrossRef]

Flotte, T.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Flueraru, C.

S. S. Sherif, C. Flueraru, Y. Mao, and S. Change, “Swept source optical coherence tomography with nonuniform frequency domain sampling,” in Biomedical Optics, OSA Technical Digest (CD)(Optical Society of America, 2008), paper BMD86.

Fraser, J. M.

Frosz, M. H

P. E Andersen, L. Thrane, H. T Yura, A. Tycho, T. M. Jrgensen, and M. H Frosz, “Advanced modelling of optical coherence tomography systems,” Phys. Med. Biol. 49, 1307–1327 (2004).
[CrossRef] [PubMed]

Fujimoto, J. G.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Gauthier, B.

S. Vergnole, D. Levesque, G. Lamouche, M. Dufour, and B. Gauthier, “Characterization of thin layered structures using deconvolution techniques in time-domain and Fourier-domain optical coherence tomography,” Proc SPIE 6796, 67961 (2007).
[CrossRef]

Golub, G. H.

G. H. Golub and C. F. Van Loan, Matrix computations, (Johns Hopkins University Press, 1984).

Gregory, K.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Häusler, G.

G. Häusler and M. W. Lindner, “‘Coherence radar’ and ‘spectral radar’–new tools for dermatological diagnosis,” J. Biomed. Opt. 3, 21–31 (1998).
[CrossRef]

Hee, M. R.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Hillmann, D.

D. Hillmann, G. Huttmann, and P. Koch, “Using Nonequispaced fast Fourier transformation to process optical coherence tomography signals,” Proc. SPIE 7372, 73730 (2009).

Huang, D.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Huttmann, G.

D. Hillmann, G. Huttmann, and P. Koch, “Using Nonequispaced fast Fourier transformation to process optical coherence tomography signals,” Proc. SPIE 7372, 73730 (2009).

Joffre, M.

Jrgensen, T. M.

P. E Andersen, L. Thrane, H. T Yura, A. Tycho, T. M. Jrgensen, and M. H Frosz, “Advanced modelling of optical coherence tomography systems,” Phys. Med. Biol. 49, 1307–1327 (2004).
[CrossRef] [PubMed]

Kak, A. C.

A. C. Kak and M. Slaney, “Algebraic reconstruction algorithms,” in Principles of Computerized Tomographic Imaging, (IEEE Press, 1999).

Koch, P.

D. Hillmann, G. Huttmann, and P. Koch, “Using Nonequispaced fast Fourier transformation to process optical coherence tomography signals,” Proc. SPIE 7372, 73730 (2009).

Lamouche, G.

S. Vergnole, D. Lvesque, and G. Lamouche, “Experimental validation of an optimized signal processing method to handle non-linearity in swept-source optical coherence tomography,” Opt. Express 18, 10446–10461 (2010).
[CrossRef] [PubMed]

S. Vergnole, D. Levesque, G. Lamouche, M. Dufour, and B. Gauthier, “Characterization of thin layered structures using deconvolution techniques in time-domain and Fourier-domain optical coherence tomography,” Proc SPIE 6796, 67961 (2007).
[CrossRef]

Levesque, D.

S. Vergnole, D. Levesque, G. Lamouche, M. Dufour, and B. Gauthier, “Characterization of thin layered structures using deconvolution techniques in time-domain and Fourier-domain optical coherence tomography,” Proc SPIE 6796, 67961 (2007).
[CrossRef]

Likforman, J.

Lin, C. P.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Lindner, M. W.

G. Häusler and M. W. Lindner, “‘Coherence radar’ and ‘spectral radar’–new tools for dermatological diagnosis,” J. Biomed. Opt. 3, 21–31 (1998).
[CrossRef]

Lustig, M.

M. Lustig, D. L. Donoho, J. M. Santos, and J. M. Pauly, “Compressed sensing MRI,” IEEE Signal Processing Mag. 25, 72–82 (2008).
[CrossRef]

Lvesque, D.

Mao, Y.

S. S. Sherif, C. Flueraru, Y. Mao, and S. Change, “Swept source optical coherence tomography with nonuniform frequency domain sampling,” in Biomedical Optics, OSA Technical Digest (CD)(Optical Society of America, 2008), paper BMD86.

Marks, D. L.

S. A. Boppart, A. L. Oldenburg, C. Xu, and D. L. Marks, “Optical probes and techniques for molecular contrast enhancement in coherence imaging,” J. Biomed. Opt. 10, 041208 (2005).
[CrossRef]

Meng, J.

Muller, M. S.

Oldenburg, A. L.

S. A. Boppart, A. L. Oldenburg, C. Xu, and D. L. Marks, “Optical probes and techniques for molecular contrast enhancement in coherence imaging,” J. Biomed. Opt. 10, 041208 (2005).
[CrossRef]

Pauly, J. M.

M. Lustig, D. L. Donoho, J. M. Santos, and J. M. Pauly, “Compressed sensing MRI,” IEEE Signal Processing Mag. 25, 72–82 (2008).
[CrossRef]

Puliafito, C. A.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Qiu, W.

W. Qiu and E. Skafidas, “Robust estimation of GCD with sparse coefficients,” Signal Processing 90, 972–976 (2010).
[CrossRef]

Santos, J. M.

M. Lustig, D. L. Donoho, J. M. Santos, and J. M. Pauly, “Compressed sensing MRI,” IEEE Signal Processing Mag. 25, 72–82 (2008).
[CrossRef]

Schuman, J. S.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Sherif, S. S.

S. S. Sherif, C. Flueraru, Y. Mao, and S. Change, “Swept source optical coherence tomography with nonuniform frequency domain sampling,” in Biomedical Optics, OSA Technical Digest (CD)(Optical Society of America, 2008), paper BMD86.

Skafidas, E.

W. Qiu and E. Skafidas, “Robust estimation of GCD with sparse coefficients,” Signal Processing 90, 972–976 (2010).
[CrossRef]

Slaney, M.

A. C. Kak and M. Slaney, “Algebraic reconstruction algorithms,” in Principles of Computerized Tomographic Imaging, (IEEE Press, 1999).

Stinson, W. G.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Swanson, E. A.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Tao, T.

E. J. Candès and T. Tao, “Decoding by linear programming,” IEEE Trans. Inf. Theory 51, 4203–4215 (2005).
[CrossRef]

Thrane, L.

P. E Andersen, L. Thrane, H. T Yura, A. Tycho, T. M. Jrgensen, and M. H Frosz, “Advanced modelling of optical coherence tomography systems,” Phys. Med. Biol. 49, 1307–1327 (2004).
[CrossRef] [PubMed]

Tycho, A.

P. E Andersen, L. Thrane, H. T Yura, A. Tycho, T. M. Jrgensen, and M. H Frosz, “Advanced modelling of optical coherence tomography systems,” Phys. Med. Biol. 49, 1307–1327 (2004).
[CrossRef] [PubMed]

Van Loan, C. F.

G. H. Golub and C. F. Van Loan, Matrix computations, (Johns Hopkins University Press, 1984).

Vergnole, S.

S. Vergnole, D. Lvesque, and G. Lamouche, “Experimental validation of an optimized signal processing method to handle non-linearity in swept-source optical coherence tomography,” Opt. Express 18, 10446–10461 (2010).
[CrossRef] [PubMed]

S. Vergnole, D. Levesque, G. Lamouche, M. Dufour, and B. Gauthier, “Characterization of thin layered structures using deconvolution techniques in time-domain and Fourier-domain optical coherence tomography,” Proc SPIE 6796, 67961 (2007).
[CrossRef]

Wang, C.

Wang, K.

Wu, T.

Xu, C.

S. A. Boppart, A. L. Oldenburg, C. Xu, and D. L. Marks, “Optical probes and techniques for molecular contrast enhancement in coherence imaging,” J. Biomed. Opt. 10, 041208 (2005).
[CrossRef]

Xu, L.

Yura, H. T

P. E Andersen, L. Thrane, H. T Yura, A. Tycho, T. M. Jrgensen, and M. H Frosz, “Advanced modelling of optical coherence tomography systems,” Phys. Med. Biol. 49, 1307–1327 (2004).
[CrossRef] [PubMed]

IEEE Signal Processing Mag. (1)

M. Lustig, D. L. Donoho, J. M. Santos, and J. M. Pauly, “Compressed sensing MRI,” IEEE Signal Processing Mag. 25, 72–82 (2008).
[CrossRef]

IEEE Trans. Inf. Theory (1)

E. J. Candès and T. Tao, “Decoding by linear programming,” IEEE Trans. Inf. Theory 51, 4203–4215 (2005).
[CrossRef]

J. Biomed. Opt. (2)

G. Häusler and M. W. Lindner, “‘Coherence radar’ and ‘spectral radar’–new tools for dermatological diagnosis,” J. Biomed. Opt. 3, 21–31 (1998).
[CrossRef]

S. A. Boppart, A. L. Oldenburg, C. Xu, and D. L. Marks, “Optical probes and techniques for molecular contrast enhancement in coherence imaging,” J. Biomed. Opt. 10, 041208 (2005).
[CrossRef]

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

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

Opt. Express (2)

Opt. Lett. (1)

Phys. Med. Biol. (1)

P. E Andersen, L. Thrane, H. T Yura, A. Tycho, T. M. Jrgensen, and M. H Frosz, “Advanced modelling of optical coherence tomography systems,” Phys. Med. Biol. 49, 1307–1327 (2004).
[CrossRef] [PubMed]

Proc SPIE (1)

S. Vergnole, D. Levesque, G. Lamouche, M. Dufour, and B. Gauthier, “Characterization of thin layered structures using deconvolution techniques in time-domain and Fourier-domain optical coherence tomography,” Proc SPIE 6796, 67961 (2007).
[CrossRef]

Proc. SPIE (1)

D. Hillmann, G. Huttmann, and P. Koch, “Using Nonequispaced fast Fourier transformation to process optical coherence tomography signals,” Proc. SPIE 7372, 73730 (2009).

Science (1)

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Signal Processing (1)

W. Qiu and E. Skafidas, “Robust estimation of GCD with sparse coefficients,” Signal Processing 90, 972–976 (2010).
[CrossRef]

Other (5)

G. H. Golub and C. F. Van Loan, Matrix computations, (Johns Hopkins University Press, 1984).

M. Grant and S. Boyd, CVX: Matlab software for disciplined convex programming (web page and software). http://stanford.edu/$_{\mathaccent''0365\relax{~}}$boyd/cvx (2009).

S. S. Sherif, C. Flueraru, Y. Mao, and S. Change, “Swept source optical coherence tomography with nonuniform frequency domain sampling,” in Biomedical Optics, OSA Technical Digest (CD)(Optical Society of America, 2008), paper BMD86.

M. E. Brezinski, Optical Coherence Tomography: Principles and Applications (Academic Press, 2006).

A. C. Kak and M. Slaney, “Algebraic reconstruction algorithms,” in Principles of Computerized Tomographic Imaging, (IEEE Press, 1999).

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Figures (5)

Fig. 1
Fig. 1

Fourier domain OCT setup.

Fig. 2
Fig. 2

Simulated imaging range of the proposed method.

Fig. 3
Fig. 3

Experiment results on the imaging depth of various OCT methods.

Fig. 4
Fig. 4

The reconstruction error as a function of SNR.

Fig. 5
Fig. 5

Cross-section images obtained by various methods using an OSA spectral resolution of 0.4 nm (The arrow in the figures indicates the imaging depth determined by Eq. (7); the vertical axis represents the depth and the range shown is 1 mm; the horizon axis represents lateral position and the range is 1 mm; the intensity in all the images is in logarithmic scale). (a) The proposed method; (b) The conventional FD-OCT method; (c) The non-uniform Fourier transform method [19]; (d) The conventional ART method.

Tables (1)

Tables Icon

Table 1 Comparison of computational time among the proposed method, the conventional ART method, and the conventional FD-OCT method

Equations (7)

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

S ( λ ) = R r G ( λ ) + G ( λ ) 0 z m a x 0 z o a ( z ) a ( z ) e i 2 ( z z ) 2 π λ d z d z + G ( λ ) 0 z max a ( z ) cos ( 2 z π λ ) d z ,
Ax = b ,
A = ( G ( λ 1 ) e i 4 π λ 1 z 1 G ( λ 1 ) e i 4 π λ 1 z 2 G ( λ 1 ) e i 4 π λ 1 z M 1 G ( λ 1 ) e i 4 π λ 1 z M G ( λ 2 ) e i 4 π λ 2 z 1 G ( λ 2 ) e i 4 π λ 2 z 2 G ( λ 2 ) e i 4 π λ 2 z M 1 G ( λ 2 ) e i 4 π λ 2 z M G ( λ N ) e i 4 π λ N z 1 G ( λ N ) e i 4 π λ N z 2 G ( λ N ) e i 4 π λ N z M 1 G ( λ N ) e i 4 π λ N z M ) , x = ( a ( z 1 ) a ( z 2 ) a ( z M ) ) ,
b = ( S ( λ 1 ) S ( λ 2 ) S ( λ N ) ) R r ( G ( λ 1 ) G ( λ 2 ) G ( λ N ) ) .
min x x 1 subject to Ax = b .
min x x 1 subject to Ax b 2 < ɛ ,
z max = 1 4 n λ 2 δ λ ,

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