Abstract

A temporally and spatially nonscanning imaging spectrometer is described in terms of computed-tomography concepts, specifically the central-slice theorem. A sequence of three transmission sinusoidal-phase gratings rotated in 60° increments achieves dispersion in multiple directions and into multiple orders. The dispersed images of the system’s field stop are interpreted as two-dimensional projections of a three-dimensional (x, y, λ) object cube. Because of the size of the finite focal-plane array, this imaging spectrometer is an example of a limited-view-angle tomographic system. The imaging spectrometer’s point spread function is measured experimentally as a function of wavelength and position in the field of view. Reconstruction of the object cube is then achieved through the maximum-likelihood, expectation-maximization algorithm under the assumption of a Poisson likelihood law. Experimental results indicate that the instrument performs well in the case of broadband and narrow-band emitters.

© 1995 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
  8. F. V. Bulygin, G. N. Vishnyakov, G. G. Levin, D. V. Karpukhin, “Spectrotomography—a new method of obtaining spectrograms of 2-D objects,” Opt. Spectrosc. (USSR) 71, 561–563 (1991).
  9. J. M. Mooney, “Spectral imaging via computed tomography,” in Proceedings of Infrared Information Symposia (IRIS) Specialty Group on Passive Sensors [Environmental Research Institute of Michigan (ERIM), Ann Arbor, Mich., 1994], Vol. 1, pp. 203–215.
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    [CrossRef]
  11. H. H. Barrett, W. Swindell, Radiological Imaging/The Theory of Image Formation, Detection, and Processing (Academic, New York, 1981), Vol. 2.
  12. J. N. Aarsvold, “Multiple-pinhole transaxial tomography: a model and analysis,” Ph.D. dissertation (University of Arizona, Tucson, Ariz., 1993), Chap. 2.
  13. M. Y. Chiu, H. H. Barrett, R. G. Simpson, C. Chou, J. W. Arendt, G. R. Rindi, “Three-dimensional radiographic imaging with a restricted view angle,” J. Opt. Soc. Am. 69, 1323–1333 (1979).
    [CrossRef]
  14. H. H. Barrett, “Editorial: limited-angle tomography for the nineties,” J. Nucl. Med. 31, 1688–1692 (1990).
    [PubMed]
  15. H. H. Barrett, “Image reconstruction and the solution of inverse problems in medical imaging,” in The Formation, Handling, and Evaluation of Medical Images, A. Todd-Pokropek, M. A. Viergever, eds. (Springer-Verlag, Berlin, 1991), pp. 3–42.
  16. J. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 4, pp. 69– 70.
  17. M. R. Descour, “Non-scanning imaging spectrometry,” Ph.D. dissertation (University of Arizona, Tucson, Ariz., 1994), Chap. 2.
  18. R. K. Rowe, “A system for three-dimensional SPECT without motion,” Ph.D. dissertation (University of Arizona, Tucson, Ariz., 1991), Chap. 5.
  19. L. A. Shepp, Y. Vardi, “Maximum likelihood reconstruction for emission tomography,” IEEE Trans. Med. Imaging MI-1, 113–122 (1982).
    [CrossRef]
  20. B. R. Frieden, Probability, Statistical Optics, and Data Testing (Springer-Verlag, Berlin, 1991), Chap. 17.
    [CrossRef]
  21. D. L. Say, R. A. Hedler, L. L. Maninger, R. A. Momberger, J. D. Robbins, “Monochrome and color image-display devices,” in Television Engineering Handbook, K. B. Benson, ed. (McGraw-Hill, New York, 1992), Chap. 12.

1993 (2)

T. Okamoto, A. Takahashi, I. Yamaguchi, “Simultaneous acquisition of spectral and spatial intensity distribution,” Appl. Spectrosc. 47, 1198–1202 (1993).
[CrossRef]

Y. Bétrémieux, T. A. Cook, D. M. Cotton, S. Chakrabarti, “SPINR: two-dimensional spectral imaging through tomographic reconstruction,” Opt. Eng. 32, 3133–3138 (1993).
[CrossRef]

1991 (2)

T. Okamoto, I. Yamaguchi, “Simultaneous acquisition of spectral image information,” Opt. Lett. 16, 1277–1279 (1991).
[CrossRef] [PubMed]

F. V. Bulygin, G. N. Vishnyakov, G. G. Levin, D. V. Karpukhin, “Spectrotomography—a new method of obtaining spectrograms of 2-D objects,” Opt. Spectrosc. (USSR) 71, 561–563 (1991).

1990 (1)

H. H. Barrett, “Editorial: limited-angle tomography for the nineties,” J. Nucl. Med. 31, 1688–1692 (1990).
[PubMed]

1988 (1)

G. Vane, A. F. H. Goetz, “Terrestrial imaging spectroscopy,” Remote Sensing Environ. 24, 1–29 (1988).
[CrossRef]

1982 (1)

L. A. Shepp, Y. Vardi, “Maximum likelihood reconstruction for emission tomography,” IEEE Trans. Med. Imaging MI-1, 113–122 (1982).
[CrossRef]

1979 (1)

1973 (1)

1969 (1)

Aarsvold, J. N.

J. N. Aarsvold, “Multiple-pinhole transaxial tomography: a model and analysis,” Ph.D. dissertation (University of Arizona, Tucson, Ariz., 1993), Chap. 2.

Arendt, J. W.

Barrett, H. H.

H. H. Barrett, “Editorial: limited-angle tomography for the nineties,” J. Nucl. Med. 31, 1688–1692 (1990).
[PubMed]

M. Y. Chiu, H. H. Barrett, R. G. Simpson, C. Chou, J. W. Arendt, G. R. Rindi, “Three-dimensional radiographic imaging with a restricted view angle,” J. Opt. Soc. Am. 69, 1323–1333 (1979).
[CrossRef]

H. H. Barrett, “Image reconstruction and the solution of inverse problems in medical imaging,” in The Formation, Handling, and Evaluation of Medical Images, A. Todd-Pokropek, M. A. Viergever, eds. (Springer-Verlag, Berlin, 1991), pp. 3–42.

H. H. Barrett, W. Swindell, Radiological Imaging/The Theory of Image Formation, Detection, and Processing (Academic, New York, 1981), Vol. 2.

Bétrémieux, Y.

Y. Bétrémieux, T. A. Cook, D. M. Cotton, S. Chakrabarti, “SPINR: two-dimensional spectral imaging through tomographic reconstruction,” Opt. Eng. 32, 3133–3138 (1993).
[CrossRef]

Blyleven, W.

W. Blyleven, DALSA CCD Image Sensors Inc. Waterloo, Ontario, Canada N2V 2E9 (personal communication, 1994).

Bulygin, F. V.

F. V. Bulygin, G. N. Vishnyakov, G. G. Levin, D. V. Karpukhin, “Spectrotomography—a new method of obtaining spectrograms of 2-D objects,” Opt. Spectrosc. (USSR) 71, 561–563 (1991).

Chakrabarti, S.

Y. Bétrémieux, T. A. Cook, D. M. Cotton, S. Chakrabarti, “SPINR: two-dimensional spectral imaging through tomographic reconstruction,” Opt. Eng. 32, 3133–3138 (1993).
[CrossRef]

Chiu, M. Y.

Chou, C.

Cook, T. A.

Y. Bétrémieux, T. A. Cook, D. M. Cotton, S. Chakrabarti, “SPINR: two-dimensional spectral imaging through tomographic reconstruction,” Opt. Eng. 32, 3133–3138 (1993).
[CrossRef]

Cotton, D. M.

Y. Bétrémieux, T. A. Cook, D. M. Cotton, S. Chakrabarti, “SPINR: two-dimensional spectral imaging through tomographic reconstruction,” Opt. Eng. 32, 3133–3138 (1993).
[CrossRef]

Descour, M. R.

M. R. Descour, “Non-scanning imaging spectrometry,” Ph.D. dissertation (University of Arizona, Tucson, Ariz., 1994), Chap. 2.

Fine, T.

Frieden, B. R.

B. R. Frieden, Probability, Statistical Optics, and Data Testing (Springer-Verlag, Berlin, 1991), Chap. 17.
[CrossRef]

Goetz, A. F. H.

G. Vane, A. F. H. Goetz, “Terrestrial imaging spectroscopy,” Remote Sensing Environ. 24, 1–29 (1988).
[CrossRef]

Goodman, J.

J. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 4, pp. 69– 70.

Harwit, M.

Hedler, R. A.

D. L. Say, R. A. Hedler, L. L. Maninger, R. A. Momberger, J. D. Robbins, “Monochrome and color image-display devices,” in Television Engineering Handbook, K. B. Benson, ed. (McGraw-Hill, New York, 1992), Chap. 12.

Karpukhin, D. V.

F. V. Bulygin, G. N. Vishnyakov, G. G. Levin, D. V. Karpukhin, “Spectrotomography—a new method of obtaining spectrograms of 2-D objects,” Opt. Spectrosc. (USSR) 71, 561–563 (1991).

Levin, G. G.

F. V. Bulygin, G. N. Vishnyakov, G. G. Levin, D. V. Karpukhin, “Spectrotomography—a new method of obtaining spectrograms of 2-D objects,” Opt. Spectrosc. (USSR) 71, 561–563 (1991).

Maninger, L. L.

D. L. Say, R. A. Hedler, L. L. Maninger, R. A. Momberger, J. D. Robbins, “Monochrome and color image-display devices,” in Television Engineering Handbook, K. B. Benson, ed. (McGraw-Hill, New York, 1992), Chap. 12.

Momberger, R. A.

D. L. Say, R. A. Hedler, L. L. Maninger, R. A. Momberger, J. D. Robbins, “Monochrome and color image-display devices,” in Television Engineering Handbook, K. B. Benson, ed. (McGraw-Hill, New York, 1992), Chap. 12.

Mooney, J. M.

J. M. Mooney, “Spectral imaging via computed tomography,” in Proceedings of Infrared Information Symposia (IRIS) Specialty Group on Passive Sensors [Environmental Research Institute of Michigan (ERIM), Ann Arbor, Mich., 1994], Vol. 1, pp. 203–215.

Okamoto, T.

Phillips, P. G.

Radon, J.

J. Radon, “Über die Bestimmung von Funktionen durch ihre Integralwerte längs gewisser Mannigfaltigkeiten,” in Gesammelte Abhandlungen/Collected Works (Birkhäuser Verlag, Vienna, 1987), Vol. 2, pp. 11–26.

Rindi, G. R.

Robbins, J. D.

D. L. Say, R. A. Hedler, L. L. Maninger, R. A. Momberger, J. D. Robbins, “Monochrome and color image-display devices,” in Television Engineering Handbook, K. B. Benson, ed. (McGraw-Hill, New York, 1992), Chap. 12.

Rowe, R. K.

R. K. Rowe, “A system for three-dimensional SPECT without motion,” Ph.D. dissertation (University of Arizona, Tucson, Ariz., 1991), Chap. 5.

Say, D. L.

D. L. Say, R. A. Hedler, L. L. Maninger, R. A. Momberger, J. D. Robbins, “Monochrome and color image-display devices,” in Television Engineering Handbook, K. B. Benson, ed. (McGraw-Hill, New York, 1992), Chap. 12.

Shepp, L. A.

L. A. Shepp, Y. Vardi, “Maximum likelihood reconstruction for emission tomography,” IEEE Trans. Med. Imaging MI-1, 113–122 (1982).
[CrossRef]

Simpson, R. G.

Sloane, N. J. A.

Swindell, W.

H. H. Barrett, W. Swindell, Radiological Imaging/The Theory of Image Formation, Detection, and Processing (Academic, New York, 1981), Vol. 2.

Takahashi, A.

Vane, G.

G. Vane, A. F. H. Goetz, “Terrestrial imaging spectroscopy,” Remote Sensing Environ. 24, 1–29 (1988).
[CrossRef]

Vardi, Y.

L. A. Shepp, Y. Vardi, “Maximum likelihood reconstruction for emission tomography,” IEEE Trans. Med. Imaging MI-1, 113–122 (1982).
[CrossRef]

Vishnyakov, G. N.

F. V. Bulygin, G. N. Vishnyakov, G. G. Levin, D. V. Karpukhin, “Spectrotomography—a new method of obtaining spectrograms of 2-D objects,” Opt. Spectrosc. (USSR) 71, 561–563 (1991).

Wyant, J. C.

Yamaguchi, I.

Appl. Opt. (2)

Appl. Spectrosc. (1)

IEEE Trans. Med. Imaging (1)

L. A. Shepp, Y. Vardi, “Maximum likelihood reconstruction for emission tomography,” IEEE Trans. Med. Imaging MI-1, 113–122 (1982).
[CrossRef]

J. Nucl. Med. (1)

H. H. Barrett, “Editorial: limited-angle tomography for the nineties,” J. Nucl. Med. 31, 1688–1692 (1990).
[PubMed]

J. Opt. Soc. Am. (1)

Opt. Eng. (1)

Y. Bétrémieux, T. A. Cook, D. M. Cotton, S. Chakrabarti, “SPINR: two-dimensional spectral imaging through tomographic reconstruction,” Opt. Eng. 32, 3133–3138 (1993).
[CrossRef]

Opt. Lett. (1)

Opt. Spectrosc. (USSR) (1)

F. V. Bulygin, G. N. Vishnyakov, G. G. Levin, D. V. Karpukhin, “Spectrotomography—a new method of obtaining spectrograms of 2-D objects,” Opt. Spectrosc. (USSR) 71, 561–563 (1991).

Remote Sensing Environ. (1)

G. Vane, A. F. H. Goetz, “Terrestrial imaging spectroscopy,” Remote Sensing Environ. 24, 1–29 (1988).
[CrossRef]

Other (11)

W. Blyleven, DALSA CCD Image Sensors Inc. Waterloo, Ontario, Canada N2V 2E9 (personal communication, 1994).

J. M. Mooney, “Spectral imaging via computed tomography,” in Proceedings of Infrared Information Symposia (IRIS) Specialty Group on Passive Sensors [Environmental Research Institute of Michigan (ERIM), Ann Arbor, Mich., 1994], Vol. 1, pp. 203–215.

J. Radon, “Über die Bestimmung von Funktionen durch ihre Integralwerte längs gewisser Mannigfaltigkeiten,” in Gesammelte Abhandlungen/Collected Works (Birkhäuser Verlag, Vienna, 1987), Vol. 2, pp. 11–26.

H. H. Barrett, W. Swindell, Radiological Imaging/The Theory of Image Formation, Detection, and Processing (Academic, New York, 1981), Vol. 2.

J. N. Aarsvold, “Multiple-pinhole transaxial tomography: a model and analysis,” Ph.D. dissertation (University of Arizona, Tucson, Ariz., 1993), Chap. 2.

B. R. Frieden, Probability, Statistical Optics, and Data Testing (Springer-Verlag, Berlin, 1991), Chap. 17.
[CrossRef]

D. L. Say, R. A. Hedler, L. L. Maninger, R. A. Momberger, J. D. Robbins, “Monochrome and color image-display devices,” in Television Engineering Handbook, K. B. Benson, ed. (McGraw-Hill, New York, 1992), Chap. 12.

H. H. Barrett, “Image reconstruction and the solution of inverse problems in medical imaging,” in The Formation, Handling, and Evaluation of Medical Images, A. Todd-Pokropek, M. A. Viergever, eds. (Springer-Verlag, Berlin, 1991), pp. 3–42.

J. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 4, pp. 69– 70.

M. R. Descour, “Non-scanning imaging spectrometry,” Ph.D. dissertation (University of Arizona, Tucson, Ariz., 1994), Chap. 2.

R. K. Rowe, “A system for three-dimensional SPECT without motion,” Ph.D. dissertation (University of Arizona, Tucson, Ariz., 1991), Chap. 5.

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Figures (14)

Fig. 1
Fig. 1

Object cube and data-acquisition modes characteristic of several conventional spectrometer types.

Fig. 2
Fig. 2

Nonscanning imaging spectrometer layout. The upper half of the diagram is a cutaway view of the system optics.

Fig. 3
Fig. 3

Irradiance distribution on the FPA.

Fig. 4
Fig. 4

Diffracted images of the field stop as parallel projections of a 3D object. The projection angle is defined as measured from the λ axis in this illustration. The projection marked 0 therefore corresponds to 0°, whereas the ±1 projections are taken at azimuth angles incremented by 60° and a projection angle of ~45°. See text for a discussion of the projection marked 90°. The center image plays a double role. In addition to being a valid projection it is also a direct image of the field stop forming a polychromatic image of the viewed scene.

Fig. 5
Fig. 5

Three-dimensional, frequency-space interpretation of Fig. 4. Only the upper half-space, l ≥ 0, is shown. Each plane corresponds to a unique projection and azimuth angle pair in Fig. 4.

Fig. 6
Fig. 6

Missing cones in 3D frequency space. For clarity only a half-space is shown. The half-angle of the missing cones is the complement of the maximum projection angle achievable with the imaging spectrometer.

Fig. 7
Fig. 7

Voxel representation. The full set of voxels fills the entire object cube.

Fig. 8
Fig. 8

Graphic representation of the produced gratings. The gratings’ efficiencies were measured at 632.8 nm and are denoted by plusses. The continuous curves and line represent the theoretical variations of efficiency with zero-to-peak phase delay for different orders (0th, 1st, and 2nd). All theoretical and measured diffraction efficiencies have been related to the ±1st diffraction-order efficiency.

Fig. 9
Fig. 9

System image of the U (green phosphor) and A (red phosphor) test target. Note that the dispersed images (i.e., object-cube projections) are collected only at a discrete set of polar coordinates.

Fig. 10
Fig. 10

Layout of the instrument in an operational configuration. The lower diagram provides a road map of the various components. The presence of the camera body allows us to install lenses of different focal lengths easily, changing field-stop image magnification.

Fig. 11
Fig. 11

Typical calibration image. The matrix is constructed from these kinds of data. Each image forms one column of the matrix. Certain sequences of diffraction orders cancel imperfectly and cause the double-dot images of the input voxel.

Fig. 12
Fig. 12

University of Arizona U and A phantoms and object-cube reconstruction results. Our test target consisted of two letters, each displayed on a different phosphor (top part of figure). The U was shown in green, and the A was shown in red.

Fig. 13
Fig. 13

Comparison of measured and reconstructed spectra. The broad gray curve represents the spectral emission of the monitor measured directly with a spectrometer. The thin, black curves represent the reconstructed estimates of the same spectral emission. Each of the two estimated spectra comes from a location on the target common to both the letter U and the letter A.

Fig. 14
Fig. 14

Background signal versus the spectral signature. The background signal (shown in gray) is associated with a nonemitting part of the target. The black curve indicates the spectrum found at a part of the target common to U and A.

Tables (1)

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Table 1 Imaging Spectrometer Parameters

Equations (4)

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g = H f + n ,
H = - i = 1 K τ i ln τ i ,
f ^ ( 0 ) = H T g .
f ^ n ( k + 1 ) = f ^ n ( k ) m = 1 M H m n m = 1 M H m n g m g ^ m ( k ) ,

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