Abstract

A high étendue static Fourier transform spectral imager has been developed for airborne use. This imaging spectrometer, based on a Michelson interferometer with rooftop mirrors, is compact and robust and benefits from a high collection efficiency. Experimental airborne images were acquired in the visible domain. The processing chain to convert raw images to hyperspectral data is described, and airborne spectral images are presented. These experimental results show that the spectral resolution is close to the one expected, but also that the signal to noise ratio is limited by various phenomena (jitter, elevation fluctuations, and one parasitic image). We discuss the origin of those limitations and suggest solutions to circumvent them.

© 2011 Optical Society of America

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    [CrossRef]
  3. R. F. Horton, C. A. Conger, and L. S. Pellegrino, “High etendue imaging Fourier transform spectrometer: initial results,” Proc. SPIE 3118, 380–390 (1997).
    [CrossRef]
  4. Y. Ferrec, J. Taboury, P. Fournet, H. Sauer, F. Goudail, P. Chavel, N. Guérineau, C. Coudrain, S. Thétas, P. Cymbalista, J. Primot, and J. Deschamps, “Design of a Fourier-transform spectral imager for airborne measurements,” in Fourier Transform Spectroscopy/ Hyperspectral Imaging and Sounding of the Environment, OSA Technical Digest Series (Optical Society of America, 2007), paper FThB3.
  5. C. Zhang, B. Xiangli, B. Zhao, and X. Yuan, “A static polarization imaging spectrometer based on a Savart polariscope,” Opt. Commun. 203, 21–26 (2002).
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    [CrossRef]
  8. A. Barducci, F. Castagnoli, G. Castellini, D. Guzzi, P. Marcoionni, and I. Pippi, “ALISEO on MIOSat: an aerospace imaging interferometer for Earth observation,” in Proceedings of IEEE Conference on Geoscience and Remote Sensing Symposium IGARSS 2009 (IEEE, 2009), pp. 464–467.
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    [CrossRef] [PubMed]
  10. X. Ma, J. Yang, W. Qiao, and B. Xue, “An improved Fourier-based sub-pixel image registration algorithm for raw image sequence of LASIS,” in Proc. SPIE 6623, 66230A (2007).
    [CrossRef]
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  12. This imaging system is two-dimensional, without an entrance slit as in , hence the name “high étendue Fourier transform spectral imager.” The French word étendue is sometimes translated as throughput.
  13. M. Françon, Optical Interferometry (Academic Press, 1966), pp. 62–86.
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    [CrossRef]
  16. G. Fortunato and P. Jacquinot, “Recherche de l’étendue maximale dans les interférometres,” Comptes Rendus de l’Académie des Sciences Paris 274-B, 688–691 (1972).
  17. P. Bouchareine and P. Connes, “Interféromètre à champ compensé pour spectroscopie par transformation de Fourier,” J. Phys. 24, 134–138 (1963).
  18. H. M. Harlander, F. Roesler, J. Cardon, C. Englert, and R. Conway, “Shimmer: a spatial heterodyne spectrometer for remote sensing of Earth’s middle atmosphere,” Appl. Opt. 41, 1343–1352 (2002).
    [CrossRef] [PubMed]
  19. R. G. Sellar and G. D. Boreman, “Comparison of relative signal-to-noise ratios of different classes of imaging spectrometer,” Appl. Opt. 44, 1614–1624 (2005).
    [CrossRef] [PubMed]
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    [CrossRef]
  21. A. T. Pritt Jr., P. N. Kupferman, S. J. Young, and R. A. Keller, “Imaging LWIR spectrometers for remote sensing applications,” Proc. SPIE 3063, 138–149 (1997).
    [CrossRef]
  22. Y. Ferrec and C. Coudrain, “CaHyD experiment on Sethi: a Fourier transform spectral imager in an aircraft pod,” in Proceedings of Optro 2010, 4th International Symposium on Optronics in Defence and Security (Association Aéronautique et Astronautique de France, 2010), paper 1783455.
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    [CrossRef] [PubMed]
  26. Q. Tian and M. Huhns, “Algorithms for subpixel registration,” Comput. Vis. Graph., Image Process. 35, 220–233 (1986).
    [CrossRef]
  27. H. Foroosh, J. Zerubia, and M. Berthod, “Extension of phase correlation to subpixel registration,” IEEE Trans. Image Process. 11, 188–200 (2002).
    [CrossRef]
  28. www.ittvis.com.
  29. P. Griffiths and J. de Haseth, Fourier Transform Infrared Spectrometry (Wiley, 2007), pp. 85–88.
  30. We call this noise “jitter noise,” but it should not to be confused with the sampling jitter noise, which describes the error in sampling the optical path differences of a Michelson interferometer with a moving mirror .
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    [CrossRef]
  32. J.-P. Allard, M. Chamberland, V. Farley, F. Marcotte, M. Rolland, A. Vallières, and A. Villemaire, “Airborne measurements in the longwave infrared using an imaging hyperspectral sensor,” Proc. SPIE 7086, 70860K (2008).
    [CrossRef]
  33. A. Kattnig and J. Primot, “Model of the second-order statistic of the radiance field of natural scenes, adapted to system conceiving,” Proc. SPIE 3074, 132–141 (1997).
    [CrossRef]
  34. M. Bryson, M. Johnson-Roberson, and S. Sukkarieh, “Airborne smoothing and mapping using vision and inertial sensors,” in Proceedings of IEEE Conference on Robotics and Automation ICRA ’09 (IEEE, 2009), pp. 2037–2042.
  35. W. H. Smith and P. D. Hammer, “Digital array scanned interferometer: sensors and results,” Appl. Opt. 35, 2902–2909(1996).
    [CrossRef] [PubMed]
  36. R. Meynart, “Sampling jitter in Fourier-transform spectrometers: spectral broadening and noise effects,” Appl. Opt. 31, 6383–6388 (1992).
    [CrossRef] [PubMed]
  37. P. Tremblay and M. Chamberland, “Continuous-scan imaging FTS with an integrating camera—contributions of sampling jitter noise to NESR,” in Fourier Transform Spectroscopy, OSA Technical Digest (CD) (Optical Society of America, 2009), paper FThA3.

2011

2008

J.-P. Allard, M. Chamberland, V. Farley, F. Marcotte, M. Rolland, A. Vallières, and A. Villemaire, “Airborne measurements in the longwave infrared using an imaging hyperspectral sensor,” Proc. SPIE 7086, 70860K (2008).
[CrossRef]

D. Cabib, “Performance and examples of measurements of a mid infrared interferometric hyperspectral imager,” Proc. SPIE 7113, 711310 (2008).
[CrossRef]

P. G. Lucey, K. A. Horton, and T. Williams, “Performance of a long-wave infrared hyperspectral imager using a Sagnac interferometer and an uncooled microbolometer array,” Appl. Opt. 47, F107–F113 (2008).
[CrossRef] [PubMed]

2007

X. Ma, J. Yang, W. Qiao, and B. Xue, “An improved Fourier-based sub-pixel image registration algorithm for raw image sequence of LASIS,” in Proc. SPIE 6623, 66230A (2007).
[CrossRef]

2005

R. G. Sellar and G. D. Boreman, “Comparison of relative signal-to-noise ratios of different classes of imaging spectrometer,” Appl. Opt. 44, 1614–1624 (2005).
[CrossRef] [PubMed]

R. G. Sellar and G. D. Boreman, “Classification of imaging spectrometers for remote sensing applications,” Opt. Eng. 44, 013602 (2005).
[CrossRef]

2002

H. Foroosh, J. Zerubia, and M. Berthod, “Extension of phase correlation to subpixel registration,” IEEE Trans. Image Process. 11, 188–200 (2002).
[CrossRef]

H. M. Harlander, F. Roesler, J. Cardon, C. Englert, and R. Conway, “Shimmer: a spatial heterodyne spectrometer for remote sensing of Earth’s middle atmosphere,” Appl. Opt. 41, 1343–1352 (2002).
[CrossRef] [PubMed]

C. Zhang, B. Xiangli, B. Zhao, and X. Yuan, “A static polarization imaging spectrometer based on a Savart polariscope,” Opt. Commun. 203, 21–26 (2002).
[CrossRef]

1997

R. F. Horton, C. A. Conger, and L. S. Pellegrino, “High etendue imaging Fourier transform spectrometer: initial results,” Proc. SPIE 3118, 380–390 (1997).
[CrossRef]

A. T. Pritt Jr., P. N. Kupferman, S. J. Young, and R. A. Keller, “Imaging LWIR spectrometers for remote sensing applications,” Proc. SPIE 3063, 138–149 (1997).
[CrossRef]

A. Kattnig and J. Primot, “Model of the second-order statistic of the radiance field of natural scenes, adapted to system conceiving,” Proc. SPIE 3074, 132–141 (1997).
[CrossRef]

1996

W. H. Smith and P. D. Hammer, “Digital array scanned interferometer: sensors and results,” Appl. Opt. 35, 2902–2909(1996).
[CrossRef] [PubMed]

K. Itoh, “Interferometric multispectral imaging,” in Prog. Opt. 35, 145–196 (1996).
[CrossRef]

D. Cabib, R. A. Buckwald, Y. Garini, and D. G. Soenksen, “Spatially resolved Fourier transform spectroscopy (spectral imaging): a powerful tool for quantitative analytical microscopy,” Proc. SPIE 2678, 278–291 (1996). In this reference, the instrument is not exactly static, since the fringe scrolling is due to a rotation of the whole interferometer, while the scene and the camera are fixed, but this is mainly due to the fact that it was developed for microscopy imaging.
[CrossRef]

1992

1987

P. Vermande, C. Buil, and F. Delbru, “Interferometric spectro-imager system (ISIS),” Proc. SPIE 810, 117–124 (1987).

1986

Q. Tian and M. Huhns, “Algorithms for subpixel registration,” Comput. Vis. Graph., Image Process. 35, 220–233 (1986).
[CrossRef]

1985

1972

G. Fortunato and P. Jacquinot, “Recherche de l’étendue maximale dans les interférometres,” Comptes Rendus de l’Académie des Sciences Paris 274-B, 688–691 (1972).

1971

1963

P. Bouchareine and P. Connes, “Interféromètre à champ compensé pour spectroscopie par transformation de Fourier,” J. Phys. 24, 134–138 (1963).

1958

P. Jacquinot, “Caractères communs aux nouvelles méthodes de spectroscopie interférentielle; facteur de mérite,” J. Phys. Radium 19, 223–229 (1958).
[CrossRef]

Allard, J.-P.

J.-P. Allard, M. Chamberland, V. Farley, F. Marcotte, M. Rolland, A. Vallières, and A. Villemaire, “Airborne measurements in the longwave infrared using an imaging hyperspectral sensor,” Proc. SPIE 7086, 70860K (2008).
[CrossRef]

Barducci, A.

A. Barducci, F. Castagnoli, G. Castellini, D. Guzzi, P. Marcoionni, and I. Pippi, “ALISEO on MIOSat: an aerospace imaging interferometer for Earth observation,” in Proceedings of IEEE Conference on Geoscience and Remote Sensing Symposium IGARSS 2009 (IEEE, 2009), pp. 464–467.

Berthod, M.

H. Foroosh, J. Zerubia, and M. Berthod, “Extension of phase correlation to subpixel registration,” IEEE Trans. Image Process. 11, 188–200 (2002).
[CrossRef]

Boreman, G. D.

R. G. Sellar and G. D. Boreman, “Classification of imaging spectrometers for remote sensing applications,” Opt. Eng. 44, 013602 (2005).
[CrossRef]

R. G. Sellar and G. D. Boreman, “Comparison of relative signal-to-noise ratios of different classes of imaging spectrometer,” Appl. Opt. 44, 1614–1624 (2005).
[CrossRef] [PubMed]

Bouchareine, P.

P. Bouchareine and P. Connes, “Interféromètre à champ compensé pour spectroscopie par transformation de Fourier,” J. Phys. 24, 134–138 (1963).

Bryson, M.

M. Bryson, M. Johnson-Roberson, and S. Sukkarieh, “Airborne smoothing and mapping using vision and inertial sensors,” in Proceedings of IEEE Conference on Robotics and Automation ICRA ’09 (IEEE, 2009), pp. 2037–2042.

Buckwald, R. A.

D. Cabib, R. A. Buckwald, Y. Garini, and D. G. Soenksen, “Spatially resolved Fourier transform spectroscopy (spectral imaging): a powerful tool for quantitative analytical microscopy,” Proc. SPIE 2678, 278–291 (1996). In this reference, the instrument is not exactly static, since the fringe scrolling is due to a rotation of the whole interferometer, while the scene and the camera are fixed, but this is mainly due to the fact that it was developed for microscopy imaging.
[CrossRef]

Buil, C.

P. Vermande, C. Buil, and F. Delbru, “Interferometric spectro-imager system (ISIS),” Proc. SPIE 810, 117–124 (1987).

Cabib, D.

D. Cabib, “Performance and examples of measurements of a mid infrared interferometric hyperspectral imager,” Proc. SPIE 7113, 711310 (2008).
[CrossRef]

D. Cabib, R. A. Buckwald, Y. Garini, and D. G. Soenksen, “Spatially resolved Fourier transform spectroscopy (spectral imaging): a powerful tool for quantitative analytical microscopy,” Proc. SPIE 2678, 278–291 (1996). In this reference, the instrument is not exactly static, since the fringe scrolling is due to a rotation of the whole interferometer, while the scene and the camera are fixed, but this is mainly due to the fact that it was developed for microscopy imaging.
[CrossRef]

Cardon, J.

Castagnoli, F.

A. Barducci, F. Castagnoli, G. Castellini, D. Guzzi, P. Marcoionni, and I. Pippi, “ALISEO on MIOSat: an aerospace imaging interferometer for Earth observation,” in Proceedings of IEEE Conference on Geoscience and Remote Sensing Symposium IGARSS 2009 (IEEE, 2009), pp. 464–467.

Castellini, G.

A. Barducci, F. Castagnoli, G. Castellini, D. Guzzi, P. Marcoionni, and I. Pippi, “ALISEO on MIOSat: an aerospace imaging interferometer for Earth observation,” in Proceedings of IEEE Conference on Geoscience and Remote Sensing Symposium IGARSS 2009 (IEEE, 2009), pp. 464–467.

Chamberland, M.

J.-P. Allard, M. Chamberland, V. Farley, F. Marcotte, M. Rolland, A. Vallières, and A. Villemaire, “Airborne measurements in the longwave infrared using an imaging hyperspectral sensor,” Proc. SPIE 7086, 70860K (2008).
[CrossRef]

P. Tremblay and M. Chamberland, “Continuous-scan imaging FTS with an integrating camera—contributions of sampling jitter noise to NESR,” in Fourier Transform Spectroscopy, OSA Technical Digest (CD) (Optical Society of America, 2009), paper FThA3.

Chavel, P.

Y. Ferrec, J. Taboury, H. Sauer, and P. Chavel, “Compactness of lateral shearing interferometers,” Appl. Opt. 50, 4656–4663(2011).
[CrossRef] [PubMed]

Y. Ferrec, J. Taboury, P. Fournet, H. Sauer, F. Goudail, P. Chavel, N. Guérineau, C. Coudrain, S. Thétas, P. Cymbalista, J. Primot, and J. Deschamps, “Design of a Fourier-transform spectral imager for airborne measurements,” in Fourier Transform Spectroscopy/ Hyperspectral Imaging and Sounding of the Environment, OSA Technical Digest Series (Optical Society of America, 2007), paper FThB3.

Y. Ferrec, J. Taboury, P. Fournet, H. Sauer, F. Goudail, P. Chavel, N. Guérineau, C. Coudrain, S. Thétas, P. Cymbalista, J. Primot, and J. Deschamps, “A Fourier transform interferometer for airborne spectral imaging in the visible and near infrared,” presented at International Conference on Optics and Laser Applications in Medicine and Environmental Monitoring for Sustainable Development, Cape Coast, Ghana, 19–24 November 2007, pp. 67–70.

Conger, C. A.

R. F. Horton, C. A. Conger, and L. S. Pellegrino, “High etendue imaging Fourier transform spectrometer: initial results,” Proc. SPIE 3118, 380–390 (1997).
[CrossRef]

Connes, P.

P. Bouchareine and P. Connes, “Interféromètre à champ compensé pour spectroscopie par transformation de Fourier,” J. Phys. 24, 134–138 (1963).

Conway, R.

Coudrain, C.

Y. Ferrec, J. Taboury, P. Fournet, H. Sauer, F. Goudail, P. Chavel, N. Guérineau, C. Coudrain, S. Thétas, P. Cymbalista, J. Primot, and J. Deschamps, “A Fourier transform interferometer for airborne spectral imaging in the visible and near infrared,” presented at International Conference on Optics and Laser Applications in Medicine and Environmental Monitoring for Sustainable Development, Cape Coast, Ghana, 19–24 November 2007, pp. 67–70.

Y. Ferrec, J. Taboury, P. Fournet, H. Sauer, F. Goudail, P. Chavel, N. Guérineau, C. Coudrain, S. Thétas, P. Cymbalista, J. Primot, and J. Deschamps, “Design of a Fourier-transform spectral imager for airborne measurements,” in Fourier Transform Spectroscopy/ Hyperspectral Imaging and Sounding of the Environment, OSA Technical Digest Series (Optical Society of America, 2007), paper FThB3.

Y. Ferrec and C. Coudrain, “CaHyD experiment on Sethi: a Fourier transform spectral imager in an aircraft pod,” in Proceedings of Optro 2010, 4th International Symposium on Optronics in Defence and Security (Association Aéronautique et Astronautique de France, 2010), paper 1783455.

Cymbalista, P.

Y. Ferrec, J. Taboury, P. Fournet, H. Sauer, F. Goudail, P. Chavel, N. Guérineau, C. Coudrain, S. Thétas, P. Cymbalista, J. Primot, and J. Deschamps, “Design of a Fourier-transform spectral imager for airborne measurements,” in Fourier Transform Spectroscopy/ Hyperspectral Imaging and Sounding of the Environment, OSA Technical Digest Series (Optical Society of America, 2007), paper FThB3.

Y. Ferrec, J. Taboury, P. Fournet, H. Sauer, F. Goudail, P. Chavel, N. Guérineau, C. Coudrain, S. Thétas, P. Cymbalista, J. Primot, and J. Deschamps, “A Fourier transform interferometer for airborne spectral imaging in the visible and near infrared,” presented at International Conference on Optics and Laser Applications in Medicine and Environmental Monitoring for Sustainable Development, Cape Coast, Ghana, 19–24 November 2007, pp. 67–70.

de Haseth, J.

P. Griffiths and J. de Haseth, Fourier Transform Infrared Spectrometry (Wiley, 2007), pp. 85–88.

Delbru, F.

P. Vermande, C. Buil, and F. Delbru, “Interferometric spectro-imager system (ISIS),” Proc. SPIE 810, 117–124 (1987).

Deschamps, J.

Y. Ferrec, J. Taboury, P. Fournet, H. Sauer, F. Goudail, P. Chavel, N. Guérineau, C. Coudrain, S. Thétas, P. Cymbalista, J. Primot, and J. Deschamps, “Design of a Fourier-transform spectral imager for airborne measurements,” in Fourier Transform Spectroscopy/ Hyperspectral Imaging and Sounding of the Environment, OSA Technical Digest Series (Optical Society of America, 2007), paper FThB3.

Y. Ferrec, J. Taboury, P. Fournet, H. Sauer, F. Goudail, P. Chavel, N. Guérineau, C. Coudrain, S. Thétas, P. Cymbalista, J. Primot, and J. Deschamps, “A Fourier transform interferometer for airborne spectral imaging in the visible and near infrared,” presented at International Conference on Optics and Laser Applications in Medicine and Environmental Monitoring for Sustainable Development, Cape Coast, Ghana, 19–24 November 2007, pp. 67–70.

Dohi, T.

Englert, C.

Farley, V.

J.-P. Allard, M. Chamberland, V. Farley, F. Marcotte, M. Rolland, A. Vallières, and A. Villemaire, “Airborne measurements in the longwave infrared using an imaging hyperspectral sensor,” Proc. SPIE 7086, 70860K (2008).
[CrossRef]

Ferrec, Y.

Y. Ferrec, J. Taboury, H. Sauer, and P. Chavel, “Compactness of lateral shearing interferometers,” Appl. Opt. 50, 4656–4663(2011).
[CrossRef] [PubMed]

Y. Ferrec and C. Coudrain, “CaHyD experiment on Sethi: a Fourier transform spectral imager in an aircraft pod,” in Proceedings of Optro 2010, 4th International Symposium on Optronics in Defence and Security (Association Aéronautique et Astronautique de France, 2010), paper 1783455.

Y. Ferrec, J. Taboury, P. Fournet, H. Sauer, F. Goudail, P. Chavel, N. Guérineau, C. Coudrain, S. Thétas, P. Cymbalista, J. Primot, and J. Deschamps, “A Fourier transform interferometer for airborne spectral imaging in the visible and near infrared,” presented at International Conference on Optics and Laser Applications in Medicine and Environmental Monitoring for Sustainable Development, Cape Coast, Ghana, 19–24 November 2007, pp. 67–70.

Y. Ferrec, J. Taboury, P. Fournet, H. Sauer, F. Goudail, P. Chavel, N. Guérineau, C. Coudrain, S. Thétas, P. Cymbalista, J. Primot, and J. Deschamps, “Design of a Fourier-transform spectral imager for airborne measurements,” in Fourier Transform Spectroscopy/ Hyperspectral Imaging and Sounding of the Environment, OSA Technical Digest Series (Optical Society of America, 2007), paper FThB3.

Foroosh, H.

H. Foroosh, J. Zerubia, and M. Berthod, “Extension of phase correlation to subpixel registration,” IEEE Trans. Image Process. 11, 188–200 (2002).
[CrossRef]

Fortunato, G.

G. Fortunato and P. Jacquinot, “Recherche de l’étendue maximale dans les interférometres,” Comptes Rendus de l’Académie des Sciences Paris 274-B, 688–691 (1972).

Fournet, P.

Y. Ferrec, J. Taboury, P. Fournet, H. Sauer, F. Goudail, P. Chavel, N. Guérineau, C. Coudrain, S. Thétas, P. Cymbalista, J. Primot, and J. Deschamps, “A Fourier transform interferometer for airborne spectral imaging in the visible and near infrared,” presented at International Conference on Optics and Laser Applications in Medicine and Environmental Monitoring for Sustainable Development, Cape Coast, Ghana, 19–24 November 2007, pp. 67–70.

Y. Ferrec, J. Taboury, P. Fournet, H. Sauer, F. Goudail, P. Chavel, N. Guérineau, C. Coudrain, S. Thétas, P. Cymbalista, J. Primot, and J. Deschamps, “Design of a Fourier-transform spectral imager for airborne measurements,” in Fourier Transform Spectroscopy/ Hyperspectral Imaging and Sounding of the Environment, OSA Technical Digest Series (Optical Society of America, 2007), paper FThB3.

Françon, M.

M. Françon, Optical Interferometry (Academic Press, 1966), pp. 62–86.

Garini, Y.

D. Cabib, R. A. Buckwald, Y. Garini, and D. G. Soenksen, “Spatially resolved Fourier transform spectroscopy (spectral imaging): a powerful tool for quantitative analytical microscopy,” Proc. SPIE 2678, 278–291 (1996). In this reference, the instrument is not exactly static, since the fringe scrolling is due to a rotation of the whole interferometer, while the scene and the camera are fixed, but this is mainly due to the fact that it was developed for microscopy imaging.
[CrossRef]

Goudail, F.

Y. Ferrec, J. Taboury, P. Fournet, H. Sauer, F. Goudail, P. Chavel, N. Guérineau, C. Coudrain, S. Thétas, P. Cymbalista, J. Primot, and J. Deschamps, “Design of a Fourier-transform spectral imager for airborne measurements,” in Fourier Transform Spectroscopy/ Hyperspectral Imaging and Sounding of the Environment, OSA Technical Digest Series (Optical Society of America, 2007), paper FThB3.

Y. Ferrec, J. Taboury, P. Fournet, H. Sauer, F. Goudail, P. Chavel, N. Guérineau, C. Coudrain, S. Thétas, P. Cymbalista, J. Primot, and J. Deschamps, “A Fourier transform interferometer for airborne spectral imaging in the visible and near infrared,” presented at International Conference on Optics and Laser Applications in Medicine and Environmental Monitoring for Sustainable Development, Cape Coast, Ghana, 19–24 November 2007, pp. 67–70.

Griffiths, P.

P. Griffiths and J. de Haseth, Fourier Transform Infrared Spectrometry (Wiley, 2007), pp. 85–88.

Guérineau, N.

Y. Ferrec, J. Taboury, P. Fournet, H. Sauer, F. Goudail, P. Chavel, N. Guérineau, C. Coudrain, S. Thétas, P. Cymbalista, J. Primot, and J. Deschamps, “A Fourier transform interferometer for airborne spectral imaging in the visible and near infrared,” presented at International Conference on Optics and Laser Applications in Medicine and Environmental Monitoring for Sustainable Development, Cape Coast, Ghana, 19–24 November 2007, pp. 67–70.

Y. Ferrec, J. Taboury, P. Fournet, H. Sauer, F. Goudail, P. Chavel, N. Guérineau, C. Coudrain, S. Thétas, P. Cymbalista, J. Primot, and J. Deschamps, “Design of a Fourier-transform spectral imager for airborne measurements,” in Fourier Transform Spectroscopy/ Hyperspectral Imaging and Sounding of the Environment, OSA Technical Digest Series (Optical Society of America, 2007), paper FThB3.

Guzzi, D.

A. Barducci, F. Castagnoli, G. Castellini, D. Guzzi, P. Marcoionni, and I. Pippi, “ALISEO on MIOSat: an aerospace imaging interferometer for Earth observation,” in Proceedings of IEEE Conference on Geoscience and Remote Sensing Symposium IGARSS 2009 (IEEE, 2009), pp. 464–467.

Hammer, P. D.

Hariharan, P.

P. Hariharan, Optical Interferometry (Academic Press, 1986), pp. 18–24.

Harlander, H. M.

Horton, K. A.

Horton, R. F.

R. F. Horton, C. A. Conger, and L. S. Pellegrino, “High etendue imaging Fourier transform spectrometer: initial results,” Proc. SPIE 3118, 380–390 (1997).
[CrossRef]

Huhns, M.

Q. Tian and M. Huhns, “Algorithms for subpixel registration,” Comput. Vis. Graph., Image Process. 35, 220–233 (1986).
[CrossRef]

Itoh, K.

K. Itoh, “Interferometric multispectral imaging,” in Prog. Opt. 35, 145–196 (1996).
[CrossRef]

Jacquinot, P.

G. Fortunato and P. Jacquinot, “Recherche de l’étendue maximale dans les interférometres,” Comptes Rendus de l’Académie des Sciences Paris 274-B, 688–691 (1972).

P. Jacquinot, “Caractères communs aux nouvelles méthodes de spectroscopie interférentielle; facteur de mérite,” J. Phys. Radium 19, 223–229 (1958).
[CrossRef]

Johnson-Roberson, M.

M. Bryson, M. Johnson-Roberson, and S. Sukkarieh, “Airborne smoothing and mapping using vision and inertial sensors,” in Proceedings of IEEE Conference on Robotics and Automation ICRA ’09 (IEEE, 2009), pp. 2037–2042.

Kattnig, A.

A. Kattnig and J. Primot, “Model of the second-order statistic of the radiance field of natural scenes, adapted to system conceiving,” Proc. SPIE 3074, 132–141 (1997).
[CrossRef]

Keller, R. A.

A. T. Pritt Jr., P. N. Kupferman, S. J. Young, and R. A. Keller, “Imaging LWIR spectrometers for remote sensing applications,” Proc. SPIE 3063, 138–149 (1997).
[CrossRef]

Kupferman, P. N.

A. T. Pritt Jr., P. N. Kupferman, S. J. Young, and R. A. Keller, “Imaging LWIR spectrometers for remote sensing applications,” Proc. SPIE 3063, 138–149 (1997).
[CrossRef]

Lucey, P. G.

Ma, X.

X. Ma, J. Yang, W. Qiao, and B. Xue, “An improved Fourier-based sub-pixel image registration algorithm for raw image sequence of LASIS,” in Proc. SPIE 6623, 66230A (2007).
[CrossRef]

Marcoionni, P.

A. Barducci, F. Castagnoli, G. Castellini, D. Guzzi, P. Marcoionni, and I. Pippi, “ALISEO on MIOSat: an aerospace imaging interferometer for Earth observation,” in Proceedings of IEEE Conference on Geoscience and Remote Sensing Symposium IGARSS 2009 (IEEE, 2009), pp. 464–467.

Marcotte, F.

J.-P. Allard, M. Chamberland, V. Farley, F. Marcotte, M. Rolland, A. Vallières, and A. Villemaire, “Airborne measurements in the longwave infrared using an imaging hyperspectral sensor,” Proc. SPIE 7086, 70860K (2008).
[CrossRef]

Meynart, R.

Minnett, P. J.

P. J. Minnett and R. G. Sellar, “The high efficiency hyperspectral imager—a new instrument for measurements of the Arctic surface,” presented at the 8th Conference on Polar Meteorology and Oceanography of the American Meteorological Society, San Diego, California, 9–13 January 2005, poster P1.2.

Okamoto, T.

Pellegrino, L. S.

R. F. Horton, C. A. Conger, and L. S. Pellegrino, “High etendue imaging Fourier transform spectrometer: initial results,” Proc. SPIE 3118, 380–390 (1997).
[CrossRef]

Pippi, I.

A. Barducci, F. Castagnoli, G. Castellini, D. Guzzi, P. Marcoionni, and I. Pippi, “ALISEO on MIOSat: an aerospace imaging interferometer for Earth observation,” in Proceedings of IEEE Conference on Geoscience and Remote Sensing Symposium IGARSS 2009 (IEEE, 2009), pp. 464–467.

Primot, J.

A. Kattnig and J. Primot, “Model of the second-order statistic of the radiance field of natural scenes, adapted to system conceiving,” Proc. SPIE 3074, 132–141 (1997).
[CrossRef]

Y. Ferrec, J. Taboury, P. Fournet, H. Sauer, F. Goudail, P. Chavel, N. Guérineau, C. Coudrain, S. Thétas, P. Cymbalista, J. Primot, and J. Deschamps, “Design of a Fourier-transform spectral imager for airborne measurements,” in Fourier Transform Spectroscopy/ Hyperspectral Imaging and Sounding of the Environment, OSA Technical Digest Series (Optical Society of America, 2007), paper FThB3.

Y. Ferrec, J. Taboury, P. Fournet, H. Sauer, F. Goudail, P. Chavel, N. Guérineau, C. Coudrain, S. Thétas, P. Cymbalista, J. Primot, and J. Deschamps, “A Fourier transform interferometer for airborne spectral imaging in the visible and near infrared,” presented at International Conference on Optics and Laser Applications in Medicine and Environmental Monitoring for Sustainable Development, Cape Coast, Ghana, 19–24 November 2007, pp. 67–70.

Pritt, A. T.

A. T. Pritt Jr., P. N. Kupferman, S. J. Young, and R. A. Keller, “Imaging LWIR spectrometers for remote sensing applications,” Proc. SPIE 3063, 138–149 (1997).
[CrossRef]

Qiao, W.

X. Ma, J. Yang, W. Qiao, and B. Xue, “An improved Fourier-based sub-pixel image registration algorithm for raw image sequence of LASIS,” in Proc. SPIE 6623, 66230A (2007).
[CrossRef]

Roesler, F.

Rolland, M.

J.-P. Allard, M. Chamberland, V. Farley, F. Marcotte, M. Rolland, A. Vallières, and A. Villemaire, “Airborne measurements in the longwave infrared using an imaging hyperspectral sensor,” Proc. SPIE 7086, 70860K (2008).
[CrossRef]

Satoshi, K.

Sauer, H.

Y. Ferrec, J. Taboury, H. Sauer, and P. Chavel, “Compactness of lateral shearing interferometers,” Appl. Opt. 50, 4656–4663(2011).
[CrossRef] [PubMed]

Y. Ferrec, J. Taboury, P. Fournet, H. Sauer, F. Goudail, P. Chavel, N. Guérineau, C. Coudrain, S. Thétas, P. Cymbalista, J. Primot, and J. Deschamps, “A Fourier transform interferometer for airborne spectral imaging in the visible and near infrared,” presented at International Conference on Optics and Laser Applications in Medicine and Environmental Monitoring for Sustainable Development, Cape Coast, Ghana, 19–24 November 2007, pp. 67–70.

Y. Ferrec, J. Taboury, P. Fournet, H. Sauer, F. Goudail, P. Chavel, N. Guérineau, C. Coudrain, S. Thétas, P. Cymbalista, J. Primot, and J. Deschamps, “Design of a Fourier-transform spectral imager for airborne measurements,” in Fourier Transform Spectroscopy/ Hyperspectral Imaging and Sounding of the Environment, OSA Technical Digest Series (Optical Society of America, 2007), paper FThB3.

Sellar, R. G.

R. G. Sellar and G. D. Boreman, “Comparison of relative signal-to-noise ratios of different classes of imaging spectrometer,” Appl. Opt. 44, 1614–1624 (2005).
[CrossRef] [PubMed]

R. G. Sellar and G. D. Boreman, “Classification of imaging spectrometers for remote sensing applications,” Opt. Eng. 44, 013602 (2005).
[CrossRef]

P. J. Minnett and R. G. Sellar, “The high efficiency hyperspectral imager—a new instrument for measurements of the Arctic surface,” presented at the 8th Conference on Polar Meteorology and Oceanography of the American Meteorological Society, San Diego, California, 9–13 January 2005, poster P1.2.

Shigeo, M.

Smith, W. H.

Soenksen, D. G.

D. Cabib, R. A. Buckwald, Y. Garini, and D. G. Soenksen, “Spatially resolved Fourier transform spectroscopy (spectral imaging): a powerful tool for quantitative analytical microscopy,” Proc. SPIE 2678, 278–291 (1996). In this reference, the instrument is not exactly static, since the fringe scrolling is due to a rotation of the whole interferometer, while the scene and the camera are fixed, but this is mainly due to the fact that it was developed for microscopy imaging.
[CrossRef]

Sukkarieh, S.

M. Bryson, M. Johnson-Roberson, and S. Sukkarieh, “Airborne smoothing and mapping using vision and inertial sensors,” in Proceedings of IEEE Conference on Robotics and Automation ICRA ’09 (IEEE, 2009), pp. 2037–2042.

Suzuki, T.

Taboury, J.

Y. Ferrec, J. Taboury, H. Sauer, and P. Chavel, “Compactness of lateral shearing interferometers,” Appl. Opt. 50, 4656–4663(2011).
[CrossRef] [PubMed]

Y. Ferrec, J. Taboury, P. Fournet, H. Sauer, F. Goudail, P. Chavel, N. Guérineau, C. Coudrain, S. Thétas, P. Cymbalista, J. Primot, and J. Deschamps, “Design of a Fourier-transform spectral imager for airborne measurements,” in Fourier Transform Spectroscopy/ Hyperspectral Imaging and Sounding of the Environment, OSA Technical Digest Series (Optical Society of America, 2007), paper FThB3.

Y. Ferrec, J. Taboury, P. Fournet, H. Sauer, F. Goudail, P. Chavel, N. Guérineau, C. Coudrain, S. Thétas, P. Cymbalista, J. Primot, and J. Deschamps, “A Fourier transform interferometer for airborne spectral imaging in the visible and near infrared,” presented at International Conference on Optics and Laser Applications in Medicine and Environmental Monitoring for Sustainable Development, Cape Coast, Ghana, 19–24 November 2007, pp. 67–70.

Thétas, S.

Y. Ferrec, J. Taboury, P. Fournet, H. Sauer, F. Goudail, P. Chavel, N. Guérineau, C. Coudrain, S. Thétas, P. Cymbalista, J. Primot, and J. Deschamps, “A Fourier transform interferometer for airborne spectral imaging in the visible and near infrared,” presented at International Conference on Optics and Laser Applications in Medicine and Environmental Monitoring for Sustainable Development, Cape Coast, Ghana, 19–24 November 2007, pp. 67–70.

Y. Ferrec, J. Taboury, P. Fournet, H. Sauer, F. Goudail, P. Chavel, N. Guérineau, C. Coudrain, S. Thétas, P. Cymbalista, J. Primot, and J. Deschamps, “Design of a Fourier-transform spectral imager for airborne measurements,” in Fourier Transform Spectroscopy/ Hyperspectral Imaging and Sounding of the Environment, OSA Technical Digest Series (Optical Society of America, 2007), paper FThB3.

Tian, Q.

Q. Tian and M. Huhns, “Algorithms for subpixel registration,” Comput. Vis. Graph., Image Process. 35, 220–233 (1986).
[CrossRef]

Tremblay, P.

P. Tremblay and M. Chamberland, “Continuous-scan imaging FTS with an integrating camera—contributions of sampling jitter noise to NESR,” in Fourier Transform Spectroscopy, OSA Technical Digest (CD) (Optical Society of America, 2009), paper FThA3.

Vallières, A.

J.-P. Allard, M. Chamberland, V. Farley, F. Marcotte, M. Rolland, A. Vallières, and A. Villemaire, “Airborne measurements in the longwave infrared using an imaging hyperspectral sensor,” Proc. SPIE 7086, 70860K (2008).
[CrossRef]

Vermande, P.

P. Vermande, C. Buil, and F. Delbru, “Interferometric spectro-imager system (ISIS),” Proc. SPIE 810, 117–124 (1987).

Villemaire, A.

J.-P. Allard, M. Chamberland, V. Farley, F. Marcotte, M. Rolland, A. Vallières, and A. Villemaire, “Airborne measurements in the longwave infrared using an imaging hyperspectral sensor,” Proc. SPIE 7086, 70860K (2008).
[CrossRef]

Williams, T.

Xiangli, B.

C. Zhang, B. Xiangli, B. Zhao, and X. Yuan, “A static polarization imaging spectrometer based on a Savart polariscope,” Opt. Commun. 203, 21–26 (2002).
[CrossRef]

Xue, B.

X. Ma, J. Yang, W. Qiao, and B. Xue, “An improved Fourier-based sub-pixel image registration algorithm for raw image sequence of LASIS,” in Proc. SPIE 6623, 66230A (2007).
[CrossRef]

Yang, J.

X. Ma, J. Yang, W. Qiao, and B. Xue, “An improved Fourier-based sub-pixel image registration algorithm for raw image sequence of LASIS,” in Proc. SPIE 6623, 66230A (2007).
[CrossRef]

Young, S. J.

A. T. Pritt Jr., P. N. Kupferman, S. J. Young, and R. A. Keller, “Imaging LWIR spectrometers for remote sensing applications,” Proc. SPIE 3063, 138–149 (1997).
[CrossRef]

Yuan, X.

C. Zhang, B. Xiangli, B. Zhao, and X. Yuan, “A static polarization imaging spectrometer based on a Savart polariscope,” Opt. Commun. 203, 21–26 (2002).
[CrossRef]

Zerubia, J.

H. Foroosh, J. Zerubia, and M. Berthod, “Extension of phase correlation to subpixel registration,” IEEE Trans. Image Process. 11, 188–200 (2002).
[CrossRef]

Zhang, C.

C. Zhang, B. Xiangli, B. Zhao, and X. Yuan, “A static polarization imaging spectrometer based on a Savart polariscope,” Opt. Commun. 203, 21–26 (2002).
[CrossRef]

Zhao, B.

C. Zhang, B. Xiangli, B. Zhao, and X. Yuan, “A static polarization imaging spectrometer based on a Savart polariscope,” Opt. Commun. 203, 21–26 (2002).
[CrossRef]

Appl. Opt.

Comptes Rendus de l’Académie des Sciences Paris

G. Fortunato and P. Jacquinot, “Recherche de l’étendue maximale dans les interférometres,” Comptes Rendus de l’Académie des Sciences Paris 274-B, 688–691 (1972).

Comput. Vis. Graph., Image Process.

Q. Tian and M. Huhns, “Algorithms for subpixel registration,” Comput. Vis. Graph., Image Process. 35, 220–233 (1986).
[CrossRef]

IEEE Trans. Image Process.

H. Foroosh, J. Zerubia, and M. Berthod, “Extension of phase correlation to subpixel registration,” IEEE Trans. Image Process. 11, 188–200 (2002).
[CrossRef]

J. Phys.

P. Bouchareine and P. Connes, “Interféromètre à champ compensé pour spectroscopie par transformation de Fourier,” J. Phys. 24, 134–138 (1963).

J. Phys. Radium

P. Jacquinot, “Caractères communs aux nouvelles méthodes de spectroscopie interférentielle; facteur de mérite,” J. Phys. Radium 19, 223–229 (1958).
[CrossRef]

Opt. Commun.

C. Zhang, B. Xiangli, B. Zhao, and X. Yuan, “A static polarization imaging spectrometer based on a Savart polariscope,” Opt. Commun. 203, 21–26 (2002).
[CrossRef]

Opt. Eng.

R. G. Sellar and G. D. Boreman, “Classification of imaging spectrometers for remote sensing applications,” Opt. Eng. 44, 013602 (2005).
[CrossRef]

Proc. SPIE

J.-P. Allard, M. Chamberland, V. Farley, F. Marcotte, M. Rolland, A. Vallières, and A. Villemaire, “Airborne measurements in the longwave infrared using an imaging hyperspectral sensor,” Proc. SPIE 7086, 70860K (2008).
[CrossRef]

A. Kattnig and J. Primot, “Model of the second-order statistic of the radiance field of natural scenes, adapted to system conceiving,” Proc. SPIE 3074, 132–141 (1997).
[CrossRef]

A. T. Pritt Jr., P. N. Kupferman, S. J. Young, and R. A. Keller, “Imaging LWIR spectrometers for remote sensing applications,” Proc. SPIE 3063, 138–149 (1997).
[CrossRef]

D. Cabib, “Performance and examples of measurements of a mid infrared interferometric hyperspectral imager,” Proc. SPIE 7113, 711310 (2008).
[CrossRef]

P. Vermande, C. Buil, and F. Delbru, “Interferometric spectro-imager system (ISIS),” Proc. SPIE 810, 117–124 (1987).

D. Cabib, R. A. Buckwald, Y. Garini, and D. G. Soenksen, “Spatially resolved Fourier transform spectroscopy (spectral imaging): a powerful tool for quantitative analytical microscopy,” Proc. SPIE 2678, 278–291 (1996). In this reference, the instrument is not exactly static, since the fringe scrolling is due to a rotation of the whole interferometer, while the scene and the camera are fixed, but this is mainly due to the fact that it was developed for microscopy imaging.
[CrossRef]

R. F. Horton, C. A. Conger, and L. S. Pellegrino, “High etendue imaging Fourier transform spectrometer: initial results,” Proc. SPIE 3118, 380–390 (1997).
[CrossRef]

X. Ma, J. Yang, W. Qiao, and B. Xue, “An improved Fourier-based sub-pixel image registration algorithm for raw image sequence of LASIS,” in Proc. SPIE 6623, 66230A (2007).
[CrossRef]

Prog. Opt.

K. Itoh, “Interferometric multispectral imaging,” in Prog. Opt. 35, 145–196 (1996).
[CrossRef]

Other

Y. Ferrec, J. Taboury, P. Fournet, H. Sauer, F. Goudail, P. Chavel, N. Guérineau, C. Coudrain, S. Thétas, P. Cymbalista, J. Primot, and J. Deschamps, “A Fourier transform interferometer for airborne spectral imaging in the visible and near infrared,” presented at International Conference on Optics and Laser Applications in Medicine and Environmental Monitoring for Sustainable Development, Cape Coast, Ghana, 19–24 November 2007, pp. 67–70.

This imaging system is two-dimensional, without an entrance slit as in , hence the name “high étendue Fourier transform spectral imager.” The French word étendue is sometimes translated as throughput.

M. Françon, Optical Interferometry (Academic Press, 1966), pp. 62–86.

P. Hariharan, Optical Interferometry (Academic Press, 1986), pp. 18–24.

Y. Ferrec, J. Taboury, P. Fournet, H. Sauer, F. Goudail, P. Chavel, N. Guérineau, C. Coudrain, S. Thétas, P. Cymbalista, J. Primot, and J. Deschamps, “Design of a Fourier-transform spectral imager for airborne measurements,” in Fourier Transform Spectroscopy/ Hyperspectral Imaging and Sounding of the Environment, OSA Technical Digest Series (Optical Society of America, 2007), paper FThB3.

A. Barducci, F. Castagnoli, G. Castellini, D. Guzzi, P. Marcoionni, and I. Pippi, “ALISEO on MIOSat: an aerospace imaging interferometer for Earth observation,” in Proceedings of IEEE Conference on Geoscience and Remote Sensing Symposium IGARSS 2009 (IEEE, 2009), pp. 464–467.

P. J. Minnett and R. G. Sellar, “The high efficiency hyperspectral imager—a new instrument for measurements of the Arctic surface,” presented at the 8th Conference on Polar Meteorology and Oceanography of the American Meteorological Society, San Diego, California, 9–13 January 2005, poster P1.2.

Y. Ferrec and C. Coudrain, “CaHyD experiment on Sethi: a Fourier transform spectral imager in an aircraft pod,” in Proceedings of Optro 2010, 4th International Symposium on Optronics in Defence and Security (Association Aéronautique et Astronautique de France, 2010), paper 1783455.

P. Tremblay and M. Chamberland, “Continuous-scan imaging FTS with an integrating camera—contributions of sampling jitter noise to NESR,” in Fourier Transform Spectroscopy, OSA Technical Digest (CD) (Optical Society of America, 2009), paper FThA3.

M. Bryson, M. Johnson-Roberson, and S. Sukkarieh, “Airborne smoothing and mapping using vision and inertial sensors,” in Proceedings of IEEE Conference on Robotics and Automation ICRA ’09 (IEEE, 2009), pp. 2037–2042.

www.ittvis.com.

P. Griffiths and J. de Haseth, Fourier Transform Infrared Spectrometry (Wiley, 2007), pp. 85–88.

We call this noise “jitter noise,” but it should not to be confused with the sampling jitter noise, which describes the error in sampling the optical path differences of a Michelson interferometer with a moving mirror .

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Figures (14)

Fig. 1
Fig. 1

Schematic description of three narrow band filters (left) and of three sinusoidal filters (right) associated with three columns of the focal plane array in a wedge filter spectral imager (left) and in a high étendue Fourier transform spectral imager (right). The focal plane arrays are the gray areas.

Fig. 2
Fig. 2

Principle for obtaining a spectral image using a high étendue Fourier transform hyperspectral imager: first, a sequence of images are acquired by scanning the scene (only three images are shown on the diagram). These images are then registered, so that the signal associated to one ground pixel can be retrieved in all the images of the sequence. This signal is the interferogram of the pixel; its spectrum is deduced by a Fourier transform.

Fig. 3
Fig. 3

Diagram of CaHyD. The two double arrows symbolize the translation adjustments for the dihedrons. The interferometer (dotted box) is made of the assembly of the beam splitter and the two dihedra.

Fig. 4
Fig. 4

Transmission and reflection of the beam splitter for the two polarization states.

Fig. 5
Fig. 5

Division of the sequence of images between reference and secondary images. The reference images are in red, and the thick lines indicate the secondary image associated with the first reference image. Note that the second reference image belongs to these secondary images, and that one image can be associated with several reference images.

Fig. 6
Fig. 6

Result of registration between a reference image (left) and a secondary image (right). The crosses on the reference image indicate the centers of the subimages. On the right, the arrows indicate the estimated positions of these centers in the secondary image, if we assume that locally the deformation can be treated as a mere translation. It is because the displacement between two consecutive frames is small that we can gradually estimate deformations of high amplitude.

Fig. 7
Fig. 7

Strategy for interpolating the images. An image is a slice in the ( x , y , δ ) space (the y axis is not represented on this figure), with a sampling at determined values for δ. The measured points are symbolized by red filled circles for the reference image, and by red hollow circles for the other images of the sequence. These secondary images are resampled on the spatial reference grid (blue squares). The solid arrows represent the neighboring points used for interpolation, if the latter is made within a single image. The dotted arrows show the other neighboring points that could also be used. It would also be possible to resample the images on the nodes of the dotted mesh, which is regular along both x and δ.

Fig. 8
Fig. 8

Pseudo and false color images from the first flight. See the text for the spectral intervals used.

Fig. 9
Fig. 9

Pseudo and false color images from the second flight. See the text for the spectral intervals used.

Fig. 10
Fig. 10

Three spectra from CaHyD (nonradiometrically calibrated).

Fig. 11
Fig. 11

Left: Atmospheric transmission over a 2 km vertical path (in black) and normalized spectrum delivered by CaHyD (in red). Right: details of the O 2 absorption line of the CaHyD spectrum.

Fig. 12
Fig. 12

Effects of a change in elevation of the scene. At the beginning, the two pixels are both from the roof (bottom left image: image number 65). Because of errors in registration, the pixel indicated by a red cross slips gradually toward the front of the house (top image: image number 435). On the right are the interferograms associated with these two pixels.

Fig. 13
Fig. 13

Example of parasitic image: the parasitic image of the bank stands out against the dark river (the lookup table is logarithmic and was hand adjusted to outline the parasitic image).

Fig. 14
Fig. 14

Origin of the parasitic image. The nominal rays are in red solid lines, and the parasitic rays in blue dotted lines. The parasitic reflection comes from the output face of the cube beam splitter (point E). E, origin of the parasitic ray; L, lens; I, image plane; N, nominal image; P, parasitic image.

Tables (1)

Tables Icon

Table 1 Features of the Dalsa 1M15 Camera (from the Datasheet)

Equations (6)

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

t opt = argmax { TF [ W · I ˜ ref · I ˜ sec * | I ˜ ref · I ˜ sec * | ] ( t ) } ,
( x y 1 ) ref = ( a b t x c d t y 0 0 1 ) · ( x y 1 ) sec .
I ω ( r ) = I 0 ( r ω ) ,
I ˜ ω ( ν ) = I ˜ 0 ( ν ) · e 2 i π ω · ν ,
I ˜ ω ( ν ) = I ˜ 0 ( ν ) · P ˜ Ω ( ν )
| I ˜ ω ( ν ) I ˜ ω ( ν ) | 2 = | I ˜ 0 ( ν ) | 2 · [ 1 | P ˜ Ω ( ν ) | 2 ] .

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