Abstract

Systematic phase errors in Fourier transform spectroscopy can severely degrade the calculated spectra. Compensation of these errors is typically accomplished using post-processing techniques, such as Fourier deconvolution, linear unmixing, or iterative solvers. This results in increased computational complexity when reconstructing and calibrating many parallel interference patterns. In this paper, we describe a new method of calibrating a Fourier transform spectrometer based on the use of artificial neural networks (ANNs). In this way, it is demonstrated that a simpler and more straightforward reconstruction process can be achieved at the cost of additional calibration equipment. To this end, we provide a theoretical model for general systematic phase errors in a polarization birefringent interferometer. This is followed by a discussion of our experimental setup and a demonstration of our technique, as applied to data with and without phase error. The technique’s utility is then supported by comparison to alternative reconstruction techniques using fast Fourier transforms (FFTs) and linear unmixing.

© 2016 Optical Society of America

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References

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  1. L. Mertz, “Auxiliary computation for Fourier spectrometry,” Infrared Phys. 7(1), 17–23 (1967).
    [Crossref]
  2. M. L. Forman, W. H. Steel, and G. A. Vanasse, “Correction of Asymmetric Interferograms Obtained in Fourier Spectroscopy,” J. Opt. Soc. Am. 56(1), 59–61 (1966).
    [Crossref]
  3. A. Ben-David and A. Ifarraguerri, “Computation of a spectrum from a single-beam fourier-transform infrared interferogram,” Appl. Opt. 41(6), 1181–1189 (2002).
    [Crossref] [PubMed]
  4. V. Saptari, Fourier Transform Spectroscopy Instrumentation Engineering (SPIE press, 2004), Vol. 61.
  5. J. A. De Haseth, “Stability of Rapid Scanning Interferometers,” Appl. Spectrosc. 36(5), 544–552 (1982).
    [Crossref]
  6. M. W. Kudenov, M. E. L. Jungwirth, E. L. Dereniak, and G. R. Gerhart, “White light Sagnac interferometer for snapshot linear polarimetric imaging,” Opt. Express 17(25), 22520–22534 (2009).
    [Crossref] [PubMed]
  7. J. M. Harlander, H. Tran, F. L. Roesler, K. P. Jaehnig, S. M. Seo, W. T. Sanders, and R. J. Reynolds, “Field-widened spatial heterodyne spectroscopy: correcting for optical defects and new vacuum ultraviolet performance tests,” In 2280, 310–319 (1994).
  8. C. R. Englert, J. M. Harlander, J. G. Cardon, and F. L. Roesler, “Correction of phase distortion in spatial heterodyne spectroscopy,” Appl. Opt. 43(36), 6680–6687 (2004).
    [Crossref] [PubMed]
  9. M. W. Kudenov and E. L. Dereniak, “Compact real-time birefringent imaging spectrometer,” Opt. Express 20(16), 17973–17986 (2012).
    [Crossref] [PubMed]
  10. H. Yang, “A back-propagation neural network for mineralogical mapping from AVIRIS data,” Int. J. Remote Sens. 20(1), 97–110 (1999).
    [Crossref]
  11. T. Fearn, “REVIEW: Standardisation and calibration transfer for near infrared instruments: a review,” J. Near Infrared Spectrosc. 9(1), 229 (2001).
    [Crossref]
  12. L. Duponchel, C. Ruckebusch, J. P. Huvenne, and P. Legrand, “Standardisation of near-IR spectrometers using artificial neural networks,” J. Mol. Struct. 480, 551–556 (1999).
    [Crossref]
  13. C. A. Osorio-Gómez, E. Mejía-Ospino, and J. E. Guerrero-Bermúdez, “Spectral reflectance curves for multispectral imaging, combining different techniques and a neural network,” Rev. Mex. Fis. 55, 120–124 (2009).
  14. D. A. Naylor, T. R. Fulton, P. W. Davis, I. M. Chapman, B. G. Gom, L. D. Spencer, J. V. Lindner, N. E. Nelson-Fitzpatrick, M. K. Tahic, and G. R. Davis, “Data processing pipeline for a time-sampled imaging Fourier transform spectrometer,” in Optical Science and Technology, the SPIE 49th Annual Meeting (International Society for Optics and Photonics, 2004), pp. 61–72.
    [Crossref]
  15. J. Craven-Jones, M. W. Kudenov, M. G. Stapelbroek, and E. L. Dereniak, “Infrared hyperspectral imaging polarimeter using birefringent prisms,” Appl. Opt. 50(8), 1170–1185 (2011).
    [Crossref] [PubMed]
  16. P. Connes and G. Michel, “Astronomical Fourier spectrometer,” Appl. Opt. 14(9), 2067–2084 (1975).
    [Crossref] [PubMed]
  17. M. W. Kudenov, M. N. Miskiewicz, M. J. Escuti, and J. F. Coward, “Polarization spatial heterodyne interferometer: model and calibration,” Opt. Eng. 53(4), 044104 (2014).
    [Crossref]
  18. J. Kim, R. K. Komanduri, K. F. Lawler, D. J. Kekas, and M. J. Escuti, “Efficient and monolithic polarization conversion system based on a polarization grating,” Appl. Opt. 51(20), 4852–4857 (2012).
    [Crossref] [PubMed]
  19. S. W. B. Joseph and P. Rice, “A hyperspectral image projector for hyperspectral imagers,” (2007).
  20. L. J. Hornbeck, “Digital light processing and MEMS: Timely convergence for a bright future,” Proc. SPIE 2642, 2 (1995).
    [Crossref]
  21. C. Oh and M. J. Escuti, “Achromatic diffraction from polarization gratings with high efficiency,” Opt. Lett. 33(20), 2287–2289 (2008).
    [Crossref] [PubMed]
  22. S. Geman, E. Bienenstock, and R. Doursat, “Neural Networks and the Bias/Variance Dilemma,” Neural Comput. 4(1), 1–58 (1992).
    [Crossref]
  23. V. C. Chan, M. Kudenov, and E. Dereniak, “Phase correction algorithms for a snapshot hyperspectral imaging system,” 2015, vol. 9611, pp. 961111.
  24. P. R. Griffiths and J. A. De Haseth, Fourier Transform Infrared Spectrometry, Chemical Analysis ; v. 83 (Wiley, 1986).
  25. D. C. Heinz and C.-I. Chang, “Fully constrained least squares linear spectral mixture analysis method for material quantification in hyperspectral imagery,” IEEE Trans. Geosci. Rem. Sens. 39(3), 529–545 (2001).
    [Crossref]
  26. D. E. Rummelhart, “Learning representations by back-propagating errors,” Nature 323(6088), 533–536 (1986).
    [Crossref]
  27. H. Demuth, M. Beale, and M. Hagan, “Neural network toolbox user’s guide, The MathWorks,” Inc., Natrick, USA (2009).
  28. E. B. Baum and D. Haussler, “What Size Net Gives Valid Generalization?” Neural Comput. 1(1), 151–160 (1989).
    [Crossref]

2014 (1)

M. W. Kudenov, M. N. Miskiewicz, M. J. Escuti, and J. F. Coward, “Polarization spatial heterodyne interferometer: model and calibration,” Opt. Eng. 53(4), 044104 (2014).
[Crossref]

2012 (2)

2011 (1)

2009 (2)

M. W. Kudenov, M. E. L. Jungwirth, E. L. Dereniak, and G. R. Gerhart, “White light Sagnac interferometer for snapshot linear polarimetric imaging,” Opt. Express 17(25), 22520–22534 (2009).
[Crossref] [PubMed]

C. A. Osorio-Gómez, E. Mejía-Ospino, and J. E. Guerrero-Bermúdez, “Spectral reflectance curves for multispectral imaging, combining different techniques and a neural network,” Rev. Mex. Fis. 55, 120–124 (2009).

2008 (1)

2004 (1)

2002 (1)

2001 (2)

D. C. Heinz and C.-I. Chang, “Fully constrained least squares linear spectral mixture analysis method for material quantification in hyperspectral imagery,” IEEE Trans. Geosci. Rem. Sens. 39(3), 529–545 (2001).
[Crossref]

T. Fearn, “REVIEW: Standardisation and calibration transfer for near infrared instruments: a review,” J. Near Infrared Spectrosc. 9(1), 229 (2001).
[Crossref]

1999 (2)

L. Duponchel, C. Ruckebusch, J. P. Huvenne, and P. Legrand, “Standardisation of near-IR spectrometers using artificial neural networks,” J. Mol. Struct. 480, 551–556 (1999).
[Crossref]

H. Yang, “A back-propagation neural network for mineralogical mapping from AVIRIS data,” Int. J. Remote Sens. 20(1), 97–110 (1999).
[Crossref]

1995 (1)

L. J. Hornbeck, “Digital light processing and MEMS: Timely convergence for a bright future,” Proc. SPIE 2642, 2 (1995).
[Crossref]

1994 (1)

J. M. Harlander, H. Tran, F. L. Roesler, K. P. Jaehnig, S. M. Seo, W. T. Sanders, and R. J. Reynolds, “Field-widened spatial heterodyne spectroscopy: correcting for optical defects and new vacuum ultraviolet performance tests,” In 2280, 310–319 (1994).

1992 (1)

S. Geman, E. Bienenstock, and R. Doursat, “Neural Networks and the Bias/Variance Dilemma,” Neural Comput. 4(1), 1–58 (1992).
[Crossref]

1989 (1)

E. B. Baum and D. Haussler, “What Size Net Gives Valid Generalization?” Neural Comput. 1(1), 151–160 (1989).
[Crossref]

1986 (1)

D. E. Rummelhart, “Learning representations by back-propagating errors,” Nature 323(6088), 533–536 (1986).
[Crossref]

1982 (1)

1975 (1)

1967 (1)

L. Mertz, “Auxiliary computation for Fourier spectrometry,” Infrared Phys. 7(1), 17–23 (1967).
[Crossref]

1966 (1)

Baum, E. B.

E. B. Baum and D. Haussler, “What Size Net Gives Valid Generalization?” Neural Comput. 1(1), 151–160 (1989).
[Crossref]

Ben-David, A.

Bienenstock, E.

S. Geman, E. Bienenstock, and R. Doursat, “Neural Networks and the Bias/Variance Dilemma,” Neural Comput. 4(1), 1–58 (1992).
[Crossref]

Cardon, J. G.

Chang, C.-I.

D. C. Heinz and C.-I. Chang, “Fully constrained least squares linear spectral mixture analysis method for material quantification in hyperspectral imagery,” IEEE Trans. Geosci. Rem. Sens. 39(3), 529–545 (2001).
[Crossref]

Connes, P.

Coward, J. F.

M. W. Kudenov, M. N. Miskiewicz, M. J. Escuti, and J. F. Coward, “Polarization spatial heterodyne interferometer: model and calibration,” Opt. Eng. 53(4), 044104 (2014).
[Crossref]

Craven-Jones, J.

De Haseth, J. A.

Dereniak, E. L.

Doursat, R.

S. Geman, E. Bienenstock, and R. Doursat, “Neural Networks and the Bias/Variance Dilemma,” Neural Comput. 4(1), 1–58 (1992).
[Crossref]

Duponchel, L.

L. Duponchel, C. Ruckebusch, J. P. Huvenne, and P. Legrand, “Standardisation of near-IR spectrometers using artificial neural networks,” J. Mol. Struct. 480, 551–556 (1999).
[Crossref]

Englert, C. R.

Escuti, M. J.

Fearn, T.

T. Fearn, “REVIEW: Standardisation and calibration transfer for near infrared instruments: a review,” J. Near Infrared Spectrosc. 9(1), 229 (2001).
[Crossref]

Forman, M. L.

Geman, S.

S. Geman, E. Bienenstock, and R. Doursat, “Neural Networks and the Bias/Variance Dilemma,” Neural Comput. 4(1), 1–58 (1992).
[Crossref]

Gerhart, G. R.

Guerrero-Bermúdez, J. E.

C. A. Osorio-Gómez, E. Mejía-Ospino, and J. E. Guerrero-Bermúdez, “Spectral reflectance curves for multispectral imaging, combining different techniques and a neural network,” Rev. Mex. Fis. 55, 120–124 (2009).

Harlander, J. M.

C. R. Englert, J. M. Harlander, J. G. Cardon, and F. L. Roesler, “Correction of phase distortion in spatial heterodyne spectroscopy,” Appl. Opt. 43(36), 6680–6687 (2004).
[Crossref] [PubMed]

J. M. Harlander, H. Tran, F. L. Roesler, K. P. Jaehnig, S. M. Seo, W. T. Sanders, and R. J. Reynolds, “Field-widened spatial heterodyne spectroscopy: correcting for optical defects and new vacuum ultraviolet performance tests,” In 2280, 310–319 (1994).

Haussler, D.

E. B. Baum and D. Haussler, “What Size Net Gives Valid Generalization?” Neural Comput. 1(1), 151–160 (1989).
[Crossref]

Heinz, D. C.

D. C. Heinz and C.-I. Chang, “Fully constrained least squares linear spectral mixture analysis method for material quantification in hyperspectral imagery,” IEEE Trans. Geosci. Rem. Sens. 39(3), 529–545 (2001).
[Crossref]

Hornbeck, L. J.

L. J. Hornbeck, “Digital light processing and MEMS: Timely convergence for a bright future,” Proc. SPIE 2642, 2 (1995).
[Crossref]

Huvenne, J. P.

L. Duponchel, C. Ruckebusch, J. P. Huvenne, and P. Legrand, “Standardisation of near-IR spectrometers using artificial neural networks,” J. Mol. Struct. 480, 551–556 (1999).
[Crossref]

Ifarraguerri, A.

Jaehnig, K. P.

J. M. Harlander, H. Tran, F. L. Roesler, K. P. Jaehnig, S. M. Seo, W. T. Sanders, and R. J. Reynolds, “Field-widened spatial heterodyne spectroscopy: correcting for optical defects and new vacuum ultraviolet performance tests,” In 2280, 310–319 (1994).

Joseph, S. W. B.

S. W. B. Joseph and P. Rice, “A hyperspectral image projector for hyperspectral imagers,” (2007).

Jungwirth, M. E. L.

Kekas, D. J.

Kim, J.

Komanduri, R. K.

Kudenov, M. W.

Lawler, K. F.

Legrand, P.

L. Duponchel, C. Ruckebusch, J. P. Huvenne, and P. Legrand, “Standardisation of near-IR spectrometers using artificial neural networks,” J. Mol. Struct. 480, 551–556 (1999).
[Crossref]

Mejía-Ospino, E.

C. A. Osorio-Gómez, E. Mejía-Ospino, and J. E. Guerrero-Bermúdez, “Spectral reflectance curves for multispectral imaging, combining different techniques and a neural network,” Rev. Mex. Fis. 55, 120–124 (2009).

Mertz, L.

L. Mertz, “Auxiliary computation for Fourier spectrometry,” Infrared Phys. 7(1), 17–23 (1967).
[Crossref]

Michel, G.

Miskiewicz, M. N.

M. W. Kudenov, M. N. Miskiewicz, M. J. Escuti, and J. F. Coward, “Polarization spatial heterodyne interferometer: model and calibration,” Opt. Eng. 53(4), 044104 (2014).
[Crossref]

Oh, C.

Osorio-Gómez, C. A.

C. A. Osorio-Gómez, E. Mejía-Ospino, and J. E. Guerrero-Bermúdez, “Spectral reflectance curves for multispectral imaging, combining different techniques and a neural network,” Rev. Mex. Fis. 55, 120–124 (2009).

Reynolds, R. J.

J. M. Harlander, H. Tran, F. L. Roesler, K. P. Jaehnig, S. M. Seo, W. T. Sanders, and R. J. Reynolds, “Field-widened spatial heterodyne spectroscopy: correcting for optical defects and new vacuum ultraviolet performance tests,” In 2280, 310–319 (1994).

Rice, P.

S. W. B. Joseph and P. Rice, “A hyperspectral image projector for hyperspectral imagers,” (2007).

Roesler, F. L.

C. R. Englert, J. M. Harlander, J. G. Cardon, and F. L. Roesler, “Correction of phase distortion in spatial heterodyne spectroscopy,” Appl. Opt. 43(36), 6680–6687 (2004).
[Crossref] [PubMed]

J. M. Harlander, H. Tran, F. L. Roesler, K. P. Jaehnig, S. M. Seo, W. T. Sanders, and R. J. Reynolds, “Field-widened spatial heterodyne spectroscopy: correcting for optical defects and new vacuum ultraviolet performance tests,” In 2280, 310–319 (1994).

Ruckebusch, C.

L. Duponchel, C. Ruckebusch, J. P. Huvenne, and P. Legrand, “Standardisation of near-IR spectrometers using artificial neural networks,” J. Mol. Struct. 480, 551–556 (1999).
[Crossref]

Rummelhart, D. E.

D. E. Rummelhart, “Learning representations by back-propagating errors,” Nature 323(6088), 533–536 (1986).
[Crossref]

Sanders, W. T.

J. M. Harlander, H. Tran, F. L. Roesler, K. P. Jaehnig, S. M. Seo, W. T. Sanders, and R. J. Reynolds, “Field-widened spatial heterodyne spectroscopy: correcting for optical defects and new vacuum ultraviolet performance tests,” In 2280, 310–319 (1994).

Seo, S. M.

J. M. Harlander, H. Tran, F. L. Roesler, K. P. Jaehnig, S. M. Seo, W. T. Sanders, and R. J. Reynolds, “Field-widened spatial heterodyne spectroscopy: correcting for optical defects and new vacuum ultraviolet performance tests,” In 2280, 310–319 (1994).

Stapelbroek, M. G.

Steel, W. H.

Tran, H.

J. M. Harlander, H. Tran, F. L. Roesler, K. P. Jaehnig, S. M. Seo, W. T. Sanders, and R. J. Reynolds, “Field-widened spatial heterodyne spectroscopy: correcting for optical defects and new vacuum ultraviolet performance tests,” In 2280, 310–319 (1994).

Vanasse, G. A.

Yang, H.

H. Yang, “A back-propagation neural network for mineralogical mapping from AVIRIS data,” Int. J. Remote Sens. 20(1), 97–110 (1999).
[Crossref]

Appl. Opt. (5)

Appl. Spectrosc. (1)

IEEE Trans. Geosci. Rem. Sens. (1)

D. C. Heinz and C.-I. Chang, “Fully constrained least squares linear spectral mixture analysis method for material quantification in hyperspectral imagery,” IEEE Trans. Geosci. Rem. Sens. 39(3), 529–545 (2001).
[Crossref]

In (1)

J. M. Harlander, H. Tran, F. L. Roesler, K. P. Jaehnig, S. M. Seo, W. T. Sanders, and R. J. Reynolds, “Field-widened spatial heterodyne spectroscopy: correcting for optical defects and new vacuum ultraviolet performance tests,” In 2280, 310–319 (1994).

Infrared Phys. (1)

L. Mertz, “Auxiliary computation for Fourier spectrometry,” Infrared Phys. 7(1), 17–23 (1967).
[Crossref]

Int. J. Remote Sens. (1)

H. Yang, “A back-propagation neural network for mineralogical mapping from AVIRIS data,” Int. J. Remote Sens. 20(1), 97–110 (1999).
[Crossref]

J. Mol. Struct. (1)

L. Duponchel, C. Ruckebusch, J. P. Huvenne, and P. Legrand, “Standardisation of near-IR spectrometers using artificial neural networks,” J. Mol. Struct. 480, 551–556 (1999).
[Crossref]

J. Near Infrared Spectrosc. (1)

T. Fearn, “REVIEW: Standardisation and calibration transfer for near infrared instruments: a review,” J. Near Infrared Spectrosc. 9(1), 229 (2001).
[Crossref]

J. Opt. Soc. Am. (1)

Nature (1)

D. E. Rummelhart, “Learning representations by back-propagating errors,” Nature 323(6088), 533–536 (1986).
[Crossref]

Neural Comput. (2)

S. Geman, E. Bienenstock, and R. Doursat, “Neural Networks and the Bias/Variance Dilemma,” Neural Comput. 4(1), 1–58 (1992).
[Crossref]

E. B. Baum and D. Haussler, “What Size Net Gives Valid Generalization?” Neural Comput. 1(1), 151–160 (1989).
[Crossref]

Opt. Eng. (1)

M. W. Kudenov, M. N. Miskiewicz, M. J. Escuti, and J. F. Coward, “Polarization spatial heterodyne interferometer: model and calibration,” Opt. Eng. 53(4), 044104 (2014).
[Crossref]

Opt. Express (2)

Opt. Lett. (1)

Proc. SPIE (1)

L. J. Hornbeck, “Digital light processing and MEMS: Timely convergence for a bright future,” Proc. SPIE 2642, 2 (1995).
[Crossref]

Rev. Mex. Fis. (1)

C. A. Osorio-Gómez, E. Mejía-Ospino, and J. E. Guerrero-Bermúdez, “Spectral reflectance curves for multispectral imaging, combining different techniques and a neural network,” Rev. Mex. Fis. 55, 120–124 (2009).

Other (6)

D. A. Naylor, T. R. Fulton, P. W. Davis, I. M. Chapman, B. G. Gom, L. D. Spencer, J. V. Lindner, N. E. Nelson-Fitzpatrick, M. K. Tahic, and G. R. Davis, “Data processing pipeline for a time-sampled imaging Fourier transform spectrometer,” in Optical Science and Technology, the SPIE 49th Annual Meeting (International Society for Optics and Photonics, 2004), pp. 61–72.
[Crossref]

V. Saptari, Fourier Transform Spectroscopy Instrumentation Engineering (SPIE press, 2004), Vol. 61.

S. W. B. Joseph and P. Rice, “A hyperspectral image projector for hyperspectral imagers,” (2007).

V. C. Chan, M. Kudenov, and E. Dereniak, “Phase correction algorithms for a snapshot hyperspectral imaging system,” 2015, vol. 9611, pp. 961111.

P. R. Griffiths and J. A. De Haseth, Fourier Transform Infrared Spectrometry, Chemical Analysis ; v. 83 (Wiley, 1986).

H. Demuth, M. Beale, and M. Hagan, “Neural network toolbox user’s guide, The MathWorks,” Inc., Natrick, USA (2009).

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Figures (15)

Fig. 1
Fig. 1 The interferogram (l) contaminated with three different phase step ( ϕ s ) functions of dN at (a) 0π, (b) π/2, and (c) π.
Fig. 2
Fig. 2 (a) The RMS error plot of the reconstructed spectra calculated with and without phase error. Ideal and contaminated spectra are shown in (b-d) for dN equal to (b) 0π, (c) π/2, and (d) π.
Fig. 3
Fig. 3 Schematic of the BFTS used for validating the calibration procedures.
Fig. 4
Fig. 4 (a) The half wave plate PM configuration showing the “phase A” and “phase B” regions with periodic fast axis orientation along x. (b) The phase ( ϕ( m ) ) step function that is generated, by the geometric phase shift, within regions A and B. This creates monochromatic interferograms which appear as (c) I PMA and (d) I PMB for regions A and B, respectively. These are compared to the ideal interferogram I ideal and plotted alongside the PM’s fast axis orientation to illustrate its relation to the geometric phase.
Fig. 5
Fig. 5 The calibration system’s optical schematic. Light from a xenon light source illuminates a grating, which disperses light onto a DLP. This light is homogenized within an integrating sphere for generating calibration and testing spectra for the interferometer.
Fig. 6
Fig. 6 The Fourier Transform calibration procedure. (a) The artificial phase contaminated interferogram applied with the numerical phase correction technique on each section of the interferogram, (b) the numerical phase compensated interferogram, (c) the upsampled, zero-padded interferogram with three different apodization filters applied, and (d) the phase corrected as well as double-sided symmetric interferogram.
Fig. 7
Fig. 7 Representation of a monochromatic, dichromatic, and random continuous spectrum used to generate the NN training data set.
Fig. 8
Fig. 8 The Neural Network architecture showing one input-output pair. BFTS interferograms and OO spectra were matched between the input and output layers, respectively.
Fig. 9
Fig. 9 The NN and H-Matrix calibration procedure. The interferogram is depicted (a) after averaging and mean value removal; (b) after appending zeros, alongside the triangular apodization filter; and (c) after apodization.
Fig. 10
Fig. 10 The monochromatic, dichromatic, and random reconstructed spectra obtained by all the three calibration techniques and the calculated absolute error for case I.
Fig. 11
Fig. 11 RMS error for Case I (no phase error) for each spectral type.
Fig. 12
Fig. 12 The monochromatic, dichromatic, and random reconstructed spectra obtained by all the three calibration techniques and the calculated absolute error for case II.
Fig. 13
Fig. 13 RMS error for Case II (90 degree phase error) for each spectral type.
Fig. 14
Fig. 14 The monochromatic, dichromatic, and random reconstructed spectra obtained by all the three calibration techniques and the calculated absolute error for case III.
Fig. 15
Fig. 15 RMS error for Case III (180 degree phase error) for each spectral type.

Equations (19)

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I(x)= 1 2 m M [ 1+cos( 2πOPD(x) σ m + ϕ s (x) ) ] ,
ϕ s (x)= dN 2 { 1+sgn[ sin( πx/P ) ] },
I( σ )=| FFT( I( x ) ) |,
RMS= 1 N sim n=1 N ( I ideal ( σ )I cont ( σ ) ) 2 ,
S out = M A M QWP M WP M G S in ,
I= 1 2 [ 1+cos( 4πBtan( α )σx ) ],
S out = M A M HWP ( θ ) M QWP M WP M G S in ,
I( θ )= 1 2 [ 1sin( 2θ )cos( 4πBtan( α )σx )+cos( 4θ )sin( 4πBtan( α )σx ) ],
I( σ )= 1 2 [ δ( σ+ σ 0 ) e idN +δ( σ σ 0 ) e idN ],
I (σ)=I(σ) e ik ϕ p ( m ) ,{ k=1 if σ<0 k=1 if σ>0
I=Hf,
f=WI,
e= 1 N h k=1 N h ( t(k)s(k) ) 2 ,
y(v)=tanh(v),
y(v)=v.
Δσ=1/2OP D max .
I m ( σ n )={ I ( σ n ) if σ0, I ( σ n ) if σ<0.
T( λ )= I Sample ( λ ) I WhiteLight ( λ ) ,
RMS= 1 N T n=1 N ( T( λ n ) T OO ( λ n ) ) 2 ,

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