Abstract

Fizeau Fourier transform imaging spectroscopy yields both spatial and spectral information about an object. Spectral information, however, is not obtained for a finite area of low spatial frequencies. A nonlinear reconstruction algorithm based on a gray-world approximation is presented. Reconstruction results from simulated data agree well with ideal Michelson interferometer-based spectral imagery. This result implies that segmented-aperture telescopes and multiple telescope arrays designed for conventional imaging can be used to gather useful spectral data through Fizeau FTIS without the need for additional hardware.

©2008 Optical Society of America

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References

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  1. J. Kauppinen and J. Partanen, Fourier Transforms in Spectroscopy, (Wiley-VCH, Berlin, 2001).
    [Crossref]
  2. N. J. E. Johnson, “Spectral imaging with the Michelson interferometer,” in Infrared Imaging Systems Technology, W. L Wolfe and J. Zimmerman, eds., Proc. SPIE226, 2–9 (1980).
  3. C. L. Bennett, M. Carter, D. Fields, and J. Hernandez, “Imaging Fourier transform spectrometer,” in Imaging Spectrometry of the Terrestrial Environment, G. Vane, ed., Proc. SPIE1937, 191–200 (1993).
    [Crossref]
  4. M. R. Carter, C. L. Bennett, D. J. Fields, and F. D. Lee, “Livermore imaging Fourier transform infrared spectrometer,” in Imaging Spectrometry, M. R. Descour, J. M. Mooney, D. L. Perry, and L. R. Illing, eds., Proc. SPIE2480, 380–386 (1995).
    [Crossref]
  5. M. Frayman and J. A. Jamieson, “Scene imaging and spectroscopy using a spatial spectral interferometer,” in Amplitude and Intensity Spatial Interferometry, J. B. Breckingridge, ed., Proc. SPIE1237, 585–603 (1990).
    [Crossref]
  6. R. L. Kendrick, E. H. Smith, and A. L. Duncan, “Imaging Fourier transform spectrometry with a Fizeau interferometer,” in Interferometry in Space, M. Shao, ed., Proc. SPIE4852, 657–662 (2003).
    [Crossref]
  7. S. T. Thurman and J. R. Fienup, “Multi-aperture Fourier transform imaging spectroscopy: theory and imaging properties,” Opt. Express 13, 2160–2175 (2005).
    [Crossref] [PubMed]
  8. J. Goodman, Introduction to Fourier Optics 2nd ed., (McGraw-Hill, New York, 1996).
  9. Note that the additional factor of A/(λ2f2i) at the beginning of Eq. (9) was omitted from Eqs. (9), (16), and (17) of Ref. [6] in error.
  10. C. W. Helstrom, “Image restoration by the method of least squares,” J. Opt. Soc. Am. 57, 297–303 (1967).
    [Crossref]
  11. J. R. Fienup, “Invariant error metrics for image reconstruction,” Appl. Opt. 36, 8352–8357 (1997).
    [Crossref]
  12. G. J. Burton and I. R. Moorhead, “Color and spatial structure in natural scenes,” Appl. Opt. 26, 157–170 (1987).
    [Crossref] [PubMed]
  13. D. J. Field, “Relations between the statistics of natural images and the response properties of cortical cells,” J. Opt. Soc. Am. A 4, 2379–2394 (1987).
    [Crossref] [PubMed]
  14. D. J. Tolhurst, Y. Tadmor, and T. Chao, “Amplitude spectra of natural images,” Ophthalic. Physiol. Opt. 12, 229–232 (1992).
    [Crossref]
  15. D. L. Ruderman and W. Bialek, “Statistics of natural images: scaling in the woods,” Phys. Rev. Lett. 73, 814–817 (1994).
    [Crossref] [PubMed]
  16. L. P. Yaroslavsky and H. J. Caulfield, “Deconvolution of multiple images of the same object,” Appl. Opt. 33, 2157–2162 (1994).
    [Crossref] [PubMed]
  17. Provided through the courtesy of Jet Propulsion Laboratory, California Institute of Technology, Pasadena, California, http://aviris.jpl.nasa.gov/.
  18. D. A. Naylor, B. G. Gom, M. K. Tahic, and G. R. Davis, “Astronomical spectroscopy using an aliased, step-and-integrate, Fourier transform spectrometer,” in Millimeter and Submillimeter Detectors for Astronomy II, J. Zmuidzinas, W. S. Holland, and S. Withington, eds., Proc. SPIE5498-85 (2004).
    [Crossref]
  19. S. T. Thurman and J. R. Fienup, “Signal-to-noise ratio trade-offs associated with coarsely sampled Fourier transform spectroscopy,” J. Opt. Soc. Am. A 24, 2817–2821 (2007).
    [Crossref]
  20. http://eo1.gsfc.nasa.gov/miscPages/home.html
  21. R. O. Green, B. E. Pavri, and T. G. Chrien, “On-orbit radiometric and spectral calibration characteristics of EO-1 Hyperion derived with an underflight of AVIRIS and in situ measurements at Salar de Arizaro, Argentina,” IEEE Trans. in Geosci. Remote Sens.,  41, 1194–1203 (2003).
    [Crossref]
  22. B. R. Hunt, “Super-resolution of images: algorithms, principles, performance,” Int. J. Imaging Syst. Technol. 6, 297–304 (1995).
    [Crossref]

2007 (1)

2005 (1)

2003 (1)

R. O. Green, B. E. Pavri, and T. G. Chrien, “On-orbit radiometric and spectral calibration characteristics of EO-1 Hyperion derived with an underflight of AVIRIS and in situ measurements at Salar de Arizaro, Argentina,” IEEE Trans. in Geosci. Remote Sens.,  41, 1194–1203 (2003).
[Crossref]

1997 (1)

1995 (1)

B. R. Hunt, “Super-resolution of images: algorithms, principles, performance,” Int. J. Imaging Syst. Technol. 6, 297–304 (1995).
[Crossref]

1994 (2)

D. L. Ruderman and W. Bialek, “Statistics of natural images: scaling in the woods,” Phys. Rev. Lett. 73, 814–817 (1994).
[Crossref] [PubMed]

L. P. Yaroslavsky and H. J. Caulfield, “Deconvolution of multiple images of the same object,” Appl. Opt. 33, 2157–2162 (1994).
[Crossref] [PubMed]

1992 (1)

D. J. Tolhurst, Y. Tadmor, and T. Chao, “Amplitude spectra of natural images,” Ophthalic. Physiol. Opt. 12, 229–232 (1992).
[Crossref]

1987 (2)

1967 (1)

Bennett, C. L.

M. R. Carter, C. L. Bennett, D. J. Fields, and F. D. Lee, “Livermore imaging Fourier transform infrared spectrometer,” in Imaging Spectrometry, M. R. Descour, J. M. Mooney, D. L. Perry, and L. R. Illing, eds., Proc. SPIE2480, 380–386 (1995).
[Crossref]

C. L. Bennett, M. Carter, D. Fields, and J. Hernandez, “Imaging Fourier transform spectrometer,” in Imaging Spectrometry of the Terrestrial Environment, G. Vane, ed., Proc. SPIE1937, 191–200 (1993).
[Crossref]

Bialek, W.

D. L. Ruderman and W. Bialek, “Statistics of natural images: scaling in the woods,” Phys. Rev. Lett. 73, 814–817 (1994).
[Crossref] [PubMed]

Burton, G. J.

Carter, M.

C. L. Bennett, M. Carter, D. Fields, and J. Hernandez, “Imaging Fourier transform spectrometer,” in Imaging Spectrometry of the Terrestrial Environment, G. Vane, ed., Proc. SPIE1937, 191–200 (1993).
[Crossref]

Carter, M. R.

M. R. Carter, C. L. Bennett, D. J. Fields, and F. D. Lee, “Livermore imaging Fourier transform infrared spectrometer,” in Imaging Spectrometry, M. R. Descour, J. M. Mooney, D. L. Perry, and L. R. Illing, eds., Proc. SPIE2480, 380–386 (1995).
[Crossref]

Caulfield, H. J.

Chao, T.

D. J. Tolhurst, Y. Tadmor, and T. Chao, “Amplitude spectra of natural images,” Ophthalic. Physiol. Opt. 12, 229–232 (1992).
[Crossref]

Chrien, T. G.

R. O. Green, B. E. Pavri, and T. G. Chrien, “On-orbit radiometric and spectral calibration characteristics of EO-1 Hyperion derived with an underflight of AVIRIS and in situ measurements at Salar de Arizaro, Argentina,” IEEE Trans. in Geosci. Remote Sens.,  41, 1194–1203 (2003).
[Crossref]

Davis, G. R.

D. A. Naylor, B. G. Gom, M. K. Tahic, and G. R. Davis, “Astronomical spectroscopy using an aliased, step-and-integrate, Fourier transform spectrometer,” in Millimeter and Submillimeter Detectors for Astronomy II, J. Zmuidzinas, W. S. Holland, and S. Withington, eds., Proc. SPIE5498-85 (2004).
[Crossref]

Duncan, A. L.

R. L. Kendrick, E. H. Smith, and A. L. Duncan, “Imaging Fourier transform spectrometry with a Fizeau interferometer,” in Interferometry in Space, M. Shao, ed., Proc. SPIE4852, 657–662 (2003).
[Crossref]

Field, D. J.

Fields, D.

C. L. Bennett, M. Carter, D. Fields, and J. Hernandez, “Imaging Fourier transform spectrometer,” in Imaging Spectrometry of the Terrestrial Environment, G. Vane, ed., Proc. SPIE1937, 191–200 (1993).
[Crossref]

Fields, D. J.

M. R. Carter, C. L. Bennett, D. J. Fields, and F. D. Lee, “Livermore imaging Fourier transform infrared spectrometer,” in Imaging Spectrometry, M. R. Descour, J. M. Mooney, D. L. Perry, and L. R. Illing, eds., Proc. SPIE2480, 380–386 (1995).
[Crossref]

Fienup, J. R.

Frayman, M.

M. Frayman and J. A. Jamieson, “Scene imaging and spectroscopy using a spatial spectral interferometer,” in Amplitude and Intensity Spatial Interferometry, J. B. Breckingridge, ed., Proc. SPIE1237, 585–603 (1990).
[Crossref]

Gom, B. G.

D. A. Naylor, B. G. Gom, M. K. Tahic, and G. R. Davis, “Astronomical spectroscopy using an aliased, step-and-integrate, Fourier transform spectrometer,” in Millimeter and Submillimeter Detectors for Astronomy II, J. Zmuidzinas, W. S. Holland, and S. Withington, eds., Proc. SPIE5498-85 (2004).
[Crossref]

Goodman, J.

J. Goodman, Introduction to Fourier Optics 2nd ed., (McGraw-Hill, New York, 1996).

Green, R. O.

R. O. Green, B. E. Pavri, and T. G. Chrien, “On-orbit radiometric and spectral calibration characteristics of EO-1 Hyperion derived with an underflight of AVIRIS and in situ measurements at Salar de Arizaro, Argentina,” IEEE Trans. in Geosci. Remote Sens.,  41, 1194–1203 (2003).
[Crossref]

Helstrom, C. W.

Hernandez, J.

C. L. Bennett, M. Carter, D. Fields, and J. Hernandez, “Imaging Fourier transform spectrometer,” in Imaging Spectrometry of the Terrestrial Environment, G. Vane, ed., Proc. SPIE1937, 191–200 (1993).
[Crossref]

Hunt, B. R.

B. R. Hunt, “Super-resolution of images: algorithms, principles, performance,” Int. J. Imaging Syst. Technol. 6, 297–304 (1995).
[Crossref]

Jamieson, J. A.

M. Frayman and J. A. Jamieson, “Scene imaging and spectroscopy using a spatial spectral interferometer,” in Amplitude and Intensity Spatial Interferometry, J. B. Breckingridge, ed., Proc. SPIE1237, 585–603 (1990).
[Crossref]

Johnson, N. J. E.

N. J. E. Johnson, “Spectral imaging with the Michelson interferometer,” in Infrared Imaging Systems Technology, W. L Wolfe and J. Zimmerman, eds., Proc. SPIE226, 2–9 (1980).

Kauppinen, J.

J. Kauppinen and J. Partanen, Fourier Transforms in Spectroscopy, (Wiley-VCH, Berlin, 2001).
[Crossref]

Kendrick, R. L.

R. L. Kendrick, E. H. Smith, and A. L. Duncan, “Imaging Fourier transform spectrometry with a Fizeau interferometer,” in Interferometry in Space, M. Shao, ed., Proc. SPIE4852, 657–662 (2003).
[Crossref]

Lee, F. D.

M. R. Carter, C. L. Bennett, D. J. Fields, and F. D. Lee, “Livermore imaging Fourier transform infrared spectrometer,” in Imaging Spectrometry, M. R. Descour, J. M. Mooney, D. L. Perry, and L. R. Illing, eds., Proc. SPIE2480, 380–386 (1995).
[Crossref]

Moorhead, I. R.

Naylor, D. A.

D. A. Naylor, B. G. Gom, M. K. Tahic, and G. R. Davis, “Astronomical spectroscopy using an aliased, step-and-integrate, Fourier transform spectrometer,” in Millimeter and Submillimeter Detectors for Astronomy II, J. Zmuidzinas, W. S. Holland, and S. Withington, eds., Proc. SPIE5498-85 (2004).
[Crossref]

Partanen, J.

J. Kauppinen and J. Partanen, Fourier Transforms in Spectroscopy, (Wiley-VCH, Berlin, 2001).
[Crossref]

Pavri, B. E.

R. O. Green, B. E. Pavri, and T. G. Chrien, “On-orbit radiometric and spectral calibration characteristics of EO-1 Hyperion derived with an underflight of AVIRIS and in situ measurements at Salar de Arizaro, Argentina,” IEEE Trans. in Geosci. Remote Sens.,  41, 1194–1203 (2003).
[Crossref]

Ruderman, D. L.

D. L. Ruderman and W. Bialek, “Statistics of natural images: scaling in the woods,” Phys. Rev. Lett. 73, 814–817 (1994).
[Crossref] [PubMed]

Smith, E. H.

R. L. Kendrick, E. H. Smith, and A. L. Duncan, “Imaging Fourier transform spectrometry with a Fizeau interferometer,” in Interferometry in Space, M. Shao, ed., Proc. SPIE4852, 657–662 (2003).
[Crossref]

Tadmor, Y.

D. J. Tolhurst, Y. Tadmor, and T. Chao, “Amplitude spectra of natural images,” Ophthalic. Physiol. Opt. 12, 229–232 (1992).
[Crossref]

Tahic, M. K.

D. A. Naylor, B. G. Gom, M. K. Tahic, and G. R. Davis, “Astronomical spectroscopy using an aliased, step-and-integrate, Fourier transform spectrometer,” in Millimeter and Submillimeter Detectors for Astronomy II, J. Zmuidzinas, W. S. Holland, and S. Withington, eds., Proc. SPIE5498-85 (2004).
[Crossref]

Thurman, S. T.

Tolhurst, D. J.

D. J. Tolhurst, Y. Tadmor, and T. Chao, “Amplitude spectra of natural images,” Ophthalic. Physiol. Opt. 12, 229–232 (1992).
[Crossref]

Yaroslavsky, L. P.

Appl. Opt. (3)

IEEE Trans. in Geosci. Remote Sens. (1)

R. O. Green, B. E. Pavri, and T. G. Chrien, “On-orbit radiometric and spectral calibration characteristics of EO-1 Hyperion derived with an underflight of AVIRIS and in situ measurements at Salar de Arizaro, Argentina,” IEEE Trans. in Geosci. Remote Sens.,  41, 1194–1203 (2003).
[Crossref]

Int. J. Imaging Syst. Technol. (1)

B. R. Hunt, “Super-resolution of images: algorithms, principles, performance,” Int. J. Imaging Syst. Technol. 6, 297–304 (1995).
[Crossref]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (2)

Ophthalic. Physiol. Opt. (1)

D. J. Tolhurst, Y. Tadmor, and T. Chao, “Amplitude spectra of natural images,” Ophthalic. Physiol. Opt. 12, 229–232 (1992).
[Crossref]

Opt. Express (1)

Phys. Rev. Lett. (1)

D. L. Ruderman and W. Bialek, “Statistics of natural images: scaling in the woods,” Phys. Rev. Lett. 73, 814–817 (1994).
[Crossref] [PubMed]

Other (11)

Provided through the courtesy of Jet Propulsion Laboratory, California Institute of Technology, Pasadena, California, http://aviris.jpl.nasa.gov/.

D. A. Naylor, B. G. Gom, M. K. Tahic, and G. R. Davis, “Astronomical spectroscopy using an aliased, step-and-integrate, Fourier transform spectrometer,” in Millimeter and Submillimeter Detectors for Astronomy II, J. Zmuidzinas, W. S. Holland, and S. Withington, eds., Proc. SPIE5498-85 (2004).
[Crossref]

http://eo1.gsfc.nasa.gov/miscPages/home.html

J. Kauppinen and J. Partanen, Fourier Transforms in Spectroscopy, (Wiley-VCH, Berlin, 2001).
[Crossref]

N. J. E. Johnson, “Spectral imaging with the Michelson interferometer,” in Infrared Imaging Systems Technology, W. L Wolfe and J. Zimmerman, eds., Proc. SPIE226, 2–9 (1980).

C. L. Bennett, M. Carter, D. Fields, and J. Hernandez, “Imaging Fourier transform spectrometer,” in Imaging Spectrometry of the Terrestrial Environment, G. Vane, ed., Proc. SPIE1937, 191–200 (1993).
[Crossref]

M. R. Carter, C. L. Bennett, D. J. Fields, and F. D. Lee, “Livermore imaging Fourier transform infrared spectrometer,” in Imaging Spectrometry, M. R. Descour, J. M. Mooney, D. L. Perry, and L. R. Illing, eds., Proc. SPIE2480, 380–386 (1995).
[Crossref]

M. Frayman and J. A. Jamieson, “Scene imaging and spectroscopy using a spatial spectral interferometer,” in Amplitude and Intensity Spatial Interferometry, J. B. Breckingridge, ed., Proc. SPIE1237, 585–603 (1990).
[Crossref]

R. L. Kendrick, E. H. Smith, and A. L. Duncan, “Imaging Fourier transform spectrometry with a Fizeau interferometer,” in Interferometry in Space, M. Shao, ed., Proc. SPIE4852, 657–662 (2003).
[Crossref]

J. Goodman, Introduction to Fourier Optics 2nd ed., (McGraw-Hill, New York, 1996).

Note that the additional factor of A/(λ2f2i) at the beginning of Eq. (9) was omitted from Eqs. (9), (16), and (17) of Ref. [6] in error.

Supplementary Material (1)

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Figures (11)

Fig. 1.
Fig. 1. Pupil of six-telescope array optical system, where the telescopes in group q=1 are white and those in group q=2 are gray.

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Fig. 2.
Fig. 2. (a). Panchromatic view of object spectral density S o(x,y,ν) in units of W/m2/sr, and (b) narrowband view of reference imagery S ref(x,y,ν) in units of W/m2/sr/µm for ν=140 THz (λ=2.13 µm).

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Fig. 3.
Fig. 3. Plot of the average radiance for each spectral band of S o(x,y,ν). The errorbars represent +/− one standard deviation of the radiance for each spectral band.

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Fig. 4.
Fig. 4. Movie (668 KB) of Fizeau FTIS image intensity I(x,y,τ) [the still frame shows I(x,y,0)]. The visibility of the intensity modulation is very small for most of the movie, but is greatest near τ=0, which occurs halfway through the movie. [Media 1]

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Fig. 5.
Fig. 5. (a). Gray-world spectrum ψ(ν) (solid line) estimated from the spectral image data G i(fx ,fy ,ν) and the average spectrum of the reference spectral image ψ avg(ν) (dotted line), and (b) azimuthal average of the spatial-frequency domain normalized RMSE E(fx ,fy ) corresponding to each gray world spectrum.

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Fig. 6.
Fig. 6. Power spectrum fitting results for: (a) |G i (Re)(0,fy ,ν)|2 data (solid line) and the power spectrum fit (dotted line) for ν=140 THz and (b) |G avg(fx ,0)|2 (solid line) and the corresponding gray-world power spectrum model.

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Fig. 7.
Fig. 7. Spectral image data: (a) S i (Re)(x,y,ν), (b) S 1(x,y,ν), (c) S 2(x,y,ν), and (d) S 3(x,y,ν) for ν=140THz. The corresponding reference image S ref(x,y,ν) is shown in Fig. 2(b).

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Fig. 8.
Fig. 8. Normalized RMSE [calculated with respect to S ref(x,y,ν)] for each image point for: (a) S i (Re)(x,y,ν), (b) S 1(x,y,ν), (c) S 2(x,y,ν), and (d) S 3(x,y,ν).

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Fig. 9.
Fig. 9. Normalized RMSE [calculated with respect to G ref(fx ,fy ,ν)] for each spatial frequency for: (a) G i (Re)(fx ,fy ,ν), (b) G 1(fx ,fy ,ν), (c) G 2(fx ,fy ,ν), and (d) G 3(fx ,fy ,ν).

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Fig. 10.
Fig. 10. (a). Normalized RMSE and (b) standard RMSE [calculated with respect to S ref(x,y,ν)] for each spectral band for: S i (Re)(x,y,ν), S 1(x,y,ν), S 2(x,y,ν), and S 3(x,y,ν).

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Fig. 11.
Fig. 11. Comparison of Fizeau and Michelson FTIS reconstruction results: (a) RMSE and (b) fractional RMSE [calculated with respect to S ref(x,y,ν)] for each spectral band.

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Tables (4)

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Table 1. Optical System Parameters.

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Table 2. Detector Parameters.

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Table 3. FTIS parameters.

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Table 4. Overall Error.

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Equations (32)

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T ( ξ , η , v , τ ) = T 1 ( ξ , η , v ) + T 2 ( ξ , η , v ) exp ( i 2 π v τ ) ,
t q ( x , y , v ) = 1 λ 2 f 2 ∫ − ∞ ∞ ∫ − ∞ ∞ T q ( ξ , η , v ) exp [ − i 2 π ( x λ f ξ + y λ f η ) ] d ξ d η ,
h p , q ( x , y , v ) = t p * ( x , y , v ) t q ( x , y , v ) ,
h ( x , y , v , τ ) = h 1,1 ( x , y , v ) + h 1,2 ( x , y , v ) exp ( − i 2 π v τ )
+ h 2,2 ( x , y , v ) + h 2,1 ( x , y , v ) exp ( i 2 π v τ ) .
I ( x , y , τ ) = ∫ 0 ∞ ∫ − ∞ ∞ ∫ − ∞ ∞ κ S o ( x ′ , y ′ , v ) h ( x − x ′ , y − y ′ , v , τ ) d x ′ d y ′ d v ,
S i ( x , y , v ) = ∫ − ∞ ∞ [ I ( x , y , τ ) − I avg ( x , y ) ] exp ⁡ ( − i 2 π τ v ) d τ ,
I avg ( x , y ) = Lim T → ∞ 1 2 T ∫ − T T I ( x , y , τ ) d τ
S i ( Re ) ( x , y , v ) = κ 2 ∫ − ∞ ∞ ∫ − ∞ ∞ S o ( x ′ , y ′ , v ) [ h 1 , 2 ( x − x ′ , y − y ′ , v )
+ h 2 , 1 ( x − x ′ , y − y ′ , v ) d x ′ d y ′ ,
G i ( Re ) ( f x , f y , v ) = κ A pup 2 λ 2 f 2 [ H 1 , 2 ( f x , f y , v ) + H 2 , 1 ( f x , f y , v ) ] G o ( f x , f y , v ) ,
H p , q ( f x , f y , v ) = 1 A pup ∫ − ∞ ∞ ∫ − ∞ ∞ T p ( ξ , η , v ) T q * ( ξ + λ ff x , η + λ ff y , v ) d ξ d η .
G avg ( f x , f y ) = ∫ 0 ∞ κ A pup λ 2 f 2 [ H 1 , 1 ( f x , f y , v ) + H 2 , 2 ( f x , f y , v ) ] G o ( f x , f y , v ) d v .
S o ( x , y , v ) ≈ f ( x , y ) ψ ( v ) ,
E ψ = ∑ ( f x , f y , v ) ∣ H 2 , 1 ( f x , f y , v ) F 1 ( f x , f y ) ψ ( v ) − G i ( f x , f y , v ) ∣ 2 ,
F 1 ( f x , f y ) = ∑ v H 2 , 1 * ( f x , f y , v ) ψ ( v ) G i ( f x , f y , v ) ε + ∑ v ∣ H 2 , 1 ( f x , f y , v ) ψ ( v ) ∣ 2 ,
Φ o ( f x , f y , v ) = { A 0 2 ψ 2 ( v )   for ( f x , f y ) = ( 0 , 0 ) A 2 ψ 2 ( v ) ( f x 2 + f y 2 ) − α for ( f x , f y ) ≠ ( 0 , 0 )
E Φ = ∑ ( f x , f y , v ) 1 f y 2 + f y 2 { ln [ ∣ G i ( Re ) ( f x , f y , v ) ∣ ]
  − ln [ 0.5 ∣ H 1 , 2 ( f x , f y , v ) + H 2 , 1 ( f x , f y , v ) ∣ Φ o ( f x , f y , v ) + Φ n ] } 2 ,
A 0 = G avg ( 0 , 0 ) ∑ v ψ ( v ) .
W 1 ( f x , f y , v ) = 1 0.5 [ H 1 , 2 ( f x , f y , v ) + H 2 , 1 ( f x , f y , v ) ] SNR 1 ( f x , f y , v ) 1 + SNR 1 ( f x , f y , v ) ,
SNR 1 ( f x , f y , v ) = 0.25 ∣ H 1 , 2 ( f x , f y , v ) + H 2 , 1 ( f x , f y , v ) ∣ 2 Φ o ( f x , f y , v ) Φ n .
G 1 ( f x , f y , v ) = W 1 ( f x , f y , v ) G i ( Re ) ( f x , f y , v ) .
G avg ( f x , f y ) ≈ ∑ v [ H 1 , 1 ( f x , f y , v ) + H 2 , 2 ( f x , f y , v ) ] F 2 ( f x , f y ) ψ ( v )
SNR 2 ( f x , f y ) = ∣ ∑ v [ H 1 , 2 ( f x , f y , v ) + H 2 , 1 ( f x , f y , v ) ] Φ o ( f x , f y , v ) ∣ 2 Φ n .
F 2 ( f x , f y ) = G avg ( f x , f y ) ∑ v [ H 1,1 ( f x , f y , v ) + H 2,2 ( f x , f y , v ) ] ψ ( v ) SNR 2 ( f x , f y ) 1 + SNR 2 ( f x , f y ) ,
G 2 ( f x , f y , v ) = F 2 ( f x , f y ) ψ ( v ) .
G 3 ( f x , f y , v ) = 1 + SNR 1 ( f x , f y , v ) 1 + SNR 1 ( f x , f y , v ) + c − 1 SNR 2 ( f x , f y ) G 1 ( f x , f y , v )
+ 1 + SNR 2 ( f x , f y ) 1 + c SNR 1 ( f x , f y , v ) + SNR 2 ( f x , f y ) G 2 ( f x , f y , v ) ,
ψ avg ( v ) = 1 N x N y ∑ ( x , y ) S ref ( x , y , v ) .
E ( f x , f y ) = ∑ v ∣ G ref ( f x , f y , v ) − β ( f x , f y ) ψ ( v ) ∣ 2 ∑ v ∣ G ref ( f x , f y , v ) ∣ 2 ,
β ( f x , f y ) = [ ∑ v G ref ( f x , f y , v ) ψ ( v ) ] ∑ v ∣ ψ ( v ) ∣ 2 .

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