Abstract

Fizeau Fourier transform imaging spectroscopy (FTIS) is a technique for collecting both spatial and spectral information about an object with a Fizeau imaging interferometer and postprocessing. The technique possesses unconventional imaging properties due to the fact that the system transfer functions, including the imaging and spectral postprocessing operations, are given by cross correlations between subapertures of the optical system, in comparison with the conventional optical transfer function, which is given by the autocorrelation of the entire aperture of the system. The unconventional imaging properties of Fizeau FTIS can be exploited to form spatially dealiased spectral images from undersampled intensity measurements (obtain superresolution relative to the detector pixel spacing). We demonstrate this dealiasing technique through computer simulations and discuss the associated design and operational trade-offs.

© 2006 Optical Society of America

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References

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  1. 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. SPIE 1237, 585-603 (1990).
  2. 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. SPIE 4852, 657-662 (2003).
  3. S. T. Thurman and J. R. Fienup, 'Multi-aperture Fourier transform imaging spectroscopy: theory and imaging properties,' Opt. Express 13, 2160-2175 (2005), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-6-2160.
    [CrossRef] [PubMed]
  4. S. T. Thurman and J. R. Fienup, 'Fizeau Fourier transform imaging spectroscopy: direct nonlinear image reconstruction' (submitted to Opt. Express).
  5. A. Papoulis, 'Generalized sampling expansion,' IEEE Trans. Circuits Syst. 24, 652-654 (1977).
    [CrossRef]
  6. J. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts & Co., 2004).
  7. J. Schott, Remote Sensing: The Image Chain Approach (Oxford, 1996).
  8. R. D. Fiete, 'Image quality and lambdaFN/p for remote sensing systems,' Opt. Eng. 38, 1229-1240 (1999).
    [CrossRef]
  9. C. W. Helstrom, 'Image restoration by the method of least squares,' J. Opt. Soc. Am. 57, 297-303 (1967).
    [CrossRef]
  10. S. T. Thurman and J. R. Fienup, 'Dealiased spectral images from aliased Fizeau Fourier transform imaging spectroscopy measurements,' in Frontiers in Optics/Laser Science XXI (Optical Society of America, 2005), paper FTuM2.

2005 (1)

1999 (1)

R. D. Fiete, 'Image quality and lambdaFN/p for remote sensing systems,' Opt. Eng. 38, 1229-1240 (1999).
[CrossRef]

1977 (1)

A. Papoulis, 'Generalized sampling expansion,' IEEE Trans. Circuits Syst. 24, 652-654 (1977).
[CrossRef]

1967 (1)

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. SPIE 4852, 657-662 (2003).

Fienup, J. R.

S. T. Thurman and J. R. Fienup, 'Multi-aperture Fourier transform imaging spectroscopy: theory and imaging properties,' Opt. Express 13, 2160-2175 (2005), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-6-2160.
[CrossRef] [PubMed]

S. T. Thurman and J. R. Fienup, 'Fizeau Fourier transform imaging spectroscopy: direct nonlinear image reconstruction' (submitted to Opt. Express).

S. T. Thurman and J. R. Fienup, 'Dealiased spectral images from aliased Fizeau Fourier transform imaging spectroscopy measurements,' in Frontiers in Optics/Laser Science XXI (Optical Society of America, 2005), paper FTuM2.

Fiete, R. D.

R. D. Fiete, 'Image quality and lambdaFN/p for remote sensing systems,' Opt. Eng. 38, 1229-1240 (1999).
[CrossRef]

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. SPIE 1237, 585-603 (1990).

Goodman, J.

J. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts & Co., 2004).

Helstrom, C. W.

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. SPIE 1237, 585-603 (1990).

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. SPIE 4852, 657-662 (2003).

Papoulis, A.

A. Papoulis, 'Generalized sampling expansion,' IEEE Trans. Circuits Syst. 24, 652-654 (1977).
[CrossRef]

Schott, J.

J. Schott, Remote Sensing: The Image Chain Approach (Oxford, 1996).

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. SPIE 4852, 657-662 (2003).

Thurman, S. T.

S. T. Thurman and J. R. Fienup, 'Multi-aperture Fourier transform imaging spectroscopy: theory and imaging properties,' Opt. Express 13, 2160-2175 (2005), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-6-2160.
[CrossRef] [PubMed]

S. T. Thurman and J. R. Fienup, 'Fizeau Fourier transform imaging spectroscopy: direct nonlinear image reconstruction' (submitted to Opt. Express).

S. T. Thurman and J. R. Fienup, 'Dealiased spectral images from aliased Fizeau Fourier transform imaging spectroscopy measurements,' in Frontiers in Optics/Laser Science XXI (Optical Society of America, 2005), paper FTuM2.

IEEE Trans. Circuits Syst. (1)

A. Papoulis, 'Generalized sampling expansion,' IEEE Trans. Circuits Syst. 24, 652-654 (1977).
[CrossRef]

J. Opt. Soc. Am. (1)

Opt. Eng. (1)

R. D. Fiete, 'Image quality and lambdaFN/p for remote sensing systems,' Opt. Eng. 38, 1229-1240 (1999).
[CrossRef]

Opt. Express (1)

Other (6)

S. T. Thurman and J. R. Fienup, 'Fizeau Fourier transform imaging spectroscopy: direct nonlinear image reconstruction' (submitted to Opt. Express).

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. SPIE 1237, 585-603 (1990).

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. SPIE 4852, 657-662 (2003).

S. T. Thurman and J. R. Fienup, 'Dealiased spectral images from aliased Fizeau Fourier transform imaging spectroscopy measurements,' in Frontiers in Optics/Laser Science XXI (Optical Society of America, 2005), paper FTuM2.

J. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts & Co., 2004).

J. Schott, Remote Sensing: The Image Chain Approach (Oxford, 1996).

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Figures (10)

Fig. 1
Fig. 1

Fizeau imaging interferometer composed of an array of two telescopes.

Fig. 2
Fig. 2

Example pupil arrangements: (a) two concentric annular subapertures, where the different subapertures are indicated by shading, and (b) corresponding SOTF; (c) two rectangular subapertures and (d) corresponding SOTF; (e) two square subapertures and (f) corresponding SOTF; and (g) aliased version of the SOTF shown in (f).

Fig. 3
Fig. 3

Pupil configuration used for computer simulation. The subaperture group pupil functions, T q ( ξ , η , ν ) for q = 1 , 2, and 3, are indicated by shading.

Fig. 4
Fig. 4

SOTFs corresponding to the pupil configuration of Fig. 3: (a), (c), and (e) show unaliased ( 11.25 μ m detector sample spacing) versions of H 2 , 1 ( f x , f y , ν 0 ) , H 3 , 2 ( f x , f y , ν 0 ) , and H 3 , 1 ( f x , f y , ν 0 ) , respectively, and (b), (d), and (f) show aliased ( 22.5 μ m detector sample spacing) versions of the same SOTFs, for ν 0 = 387 THz .

Fig. 5
Fig. 5

Data from simulation: (a) panchromatic view of the DIRSIG scene used for the object (scaled to units of detector counts) and (b) simulated image measurement for τ = 0 .

Fig. 6
Fig. 6

Aliased spectral imagery from simulation: the real part of the complex-valued spectral image S i ( x , y , ν ) for (a) ν = ν 0 3 , (c) ν = 2 ν 0 3 , and (e) ν = ν 0 , and (b), (d), and (f) the corresponding Fourier magnitude of each of the complex-valued images represented in the left column, for ν 0 = 387 THz . The corresponding SOTFs are shown in Figs. 4b, 4d, 4f.

Fig. 7
Fig. 7

Illustration of the dealiasing procedure: (a) tile the aliased spatial-frequency data, G i ( f x , f y , ν ) , forming a data set with 256 × 256 spatial-frequency dimensions from the 128 × 128 aliased data set; (b) band limit the tiled spatial-frequency data to the support of the corresponding SOTF to form G d ( f x , f y , ν ) , the dealiased spatial-frequency data; and (c) inverse Fourier transform to yield the dealiased spectral image S d ( x , y , ν ) . The data shown in this figure correspond to ν = ν 0 3 . Thus Fig. 7c is the dealiased version of Fig. 6a.

Fig. 8
Fig. 8

Comparison of magnified portions of the composite spectral imagery with the object data for ν = ν 0 : (a) real part of aliased composite spectral image, (b) real part of dealiased composite spectral image, and (c) object truth data. The arrows point to specific features that are discussed in the text.

Fig. 9
Fig. 9

Comparison of reconstructed spectral imagery with the object data for ν = ν 0 : (a) linear Wiener–Helstrom reconstruction, (b) nonlinear reconstruction using the algorithm from Ref. [4] and (c) object truth data.

Fig. 10
Fig. 10

Panchromatic views of (a) nonlinear reconstruction of the spectral image, (b) object truth data band limited to spatial frequencies passed by the conventional OTF, (c) aliased version of (b), and (d) raw τ = 0 image.

Tables (2)

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Table 1 Optical System Parameters for Simulation

Tables Icon

Table 2 Detector Parameters for Simulation

Equations (7)

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T pup ( ξ , η , ν , τ ) = q = 1 Q T q ( ξ , η , ν ) exp ( i 2 π ν γ q τ ) ,
S i ( x , y , ν ) = p = 1 Q q = 1 Q p q κ γ p γ q M 2 S o ( x M , y M , ν γ p γ q ) × h p , q ( x x , y y , ν γ p γ q ) d x d y ,
h p , q ( x , y , ν ) = t p ( x , y , ν ) t q * ( x , y , ν ) ,
H p , q ( f x , f y , ν ) = 1 A pup T p ( λ f i f x , λ f i f y , ν ) T q ( λ f i f x , λ f i f y , ν ) ,
A pup = T pup ( ξ , η , ν , 0 ) 2 d ξ d η ,
Λ Nyquist = min ( λ ) f i 2 D = 11.25 μ m
H det ( f x , f y ) = sinc ( Λ det f det f x ) sinc ( Λ det f det f y ) .

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