Abstract

A new technique for hyperspectral imaging called spectrotomography collects all available photons and relies on computer tomography to reconstruct the three-dimensional data cube of an object. A rotational spectrotomographic (RST) imager is designed with a wide-aperture, objective-grating camera that is rotated in steps around its optical axis. The full range of spatial and spectral resolution is achieved by the use of a stepped-focal-length (zoom) lens to illuminate the grating. Two-dimensional projections of the object are analyzed with the use of both direct Fourier methods and filter-backprojection algorithms. The RST imager has applications to detection of optical emissions where large photon throughput is required.

© 1995 Optical Society of America

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References

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  1. P. S. Andersson, S. Montan, S. Svanberg, “Multispectral system for medical fluorescence imaging,” IEEE J. Quantum Electron. QE-23, 1798–1805 (1987).
    [CrossRef]
  2. S. Andersson-Engels, J. Johansson, S. Svanberg, “Medical diagnostic system based on simultaneous multispectral fluorescence imaging,” Appl. Opt. 33, 8022–8029 (1994).
    [CrossRef] [PubMed]
  3. S. Montan, S. Svanberg, “A system for industrial surface monitoring utilizing laser-induced fluorescence,” Appl. Phys. B 38, 241–247 (1985).
    [CrossRef]
  4. C. Elachi, Introduction to the Physics and Techniques of Remote Sensing (Wiley-Interscience, New York, 1987).
  5. E. R. Menzel, J. M. Duff, “Laser detection of latent fingerprints—treatment with fluorescers,” J. Forensic Sci. 24, 96–100 (1979).
    [PubMed]
  6. B. C. Wilson, S. L. Jacques, “Optical reflectance and transmittance of tissues: principles and applications,” IEEE J. Quantum Electron. 26, 2186–2199 (1990).
    [CrossRef]
  7. L. E. Blanchard, O. Weinstein, “Design challenges of the thematic mapper,” IEEE Trans. Geosci. Remote Sensing GE-18, 146–160 (1980).
    [CrossRef]
  8. M. H. Rees, Physics and Chemistry of the Upper Atmosphere (Cambridge U. Press, Cambridge, 1989).
    [CrossRef]
  9. P. A. Bernhardt, “Probing the magnetosphere using chemical releases from the Combined Release and Radiation Effects Satellite,” Phys. Fluids B 4, 2249–2256 (1992).
    [CrossRef]
  10. D. V. O’Connor, D. Phillips, Time Correlated Single Photon Counting (Academic, New York, 1984).
  11. P. A. Bernhardt, L. M. Duncan, C. A. Tepley, “Artificial airglow excited by high-power radio waves,” Science 242, 1022–1027 (1988).
    [CrossRef] [PubMed]
  12. T. Okamoto, I. Yamaguchi, “Simultaneous acquisition of spectral image information,” Opt. Lett. 16, 1277–1279 (1991).
    [CrossRef] [PubMed]
  13. T. Okamoto, A. Takahishi, I. Yamaguchi, “Simultaneous acquisition of spectral and spatial intensity distribution,” Appl. Spectrosc. 47, 1198–1202 (1993).
    [CrossRef]
  14. F. V. Bulygin, G. N. Vishnyakov, “Spectrotomography—a new method of obtaining spectrograms of two-dimensional objects,” in Analytical Methods for Optical Tomography, G. G. Levin, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1843, 315–322 (1991).
    [CrossRef]
  15. J. T. Mooney, “Spectral imaging via computed tomography,” in Proceedings of the IRIS Specialty Group on Passive Sensors (Environmental Research Institute of Michigan, Ann Arbor, Mich., 1994).
  16. T. Betremeiux, T. A. Cook, D. Cotton, S. Chakrabarti, “SPINR: high-resolution two-dimensional spectral imaging,” in Instrumentation for Magnetospheric Imagery II, S. Chakrabarti, ed., Proc. Soc. Photo-Opt. Instrum. Eng.2008, 114–119 (1993).
    [CrossRef]
  17. R. M. Lewitt, “Reconstruction algorithms: transform methods,” Proc. IEEE 71, 390–408 (1983).
    [CrossRef]
  18. Y. Censor, “Series-expansion reconstruction methods,” Proc. IEEE 71, 409–419 (1983).
    [CrossRef]
  19. G. T. Herman, “Algebraic reconstruction techniques can be made computationally efficient,” IEEE Trans. Med. Imaging 12, 600–609 (1993).
    [CrossRef]
  20. S. R. Deans, The Radon Transform and Some of Its Applications (Wiley, New York, 1983).
  21. S. W. Rowland, “Computer implementation of image reconstruction formulas,” in Image Reconstructions from Projections, G. T. Herman, ed. (Springer-Verlag, Berlin, 1979), pp. 9–78.
    [CrossRef]
  22. A. C. Kak, M. Slaney, Principles of Computerized Tomographic Imaging (IEEE, New York, 1988).
  23. R. M. Mersereau, A. V. Oppenheim, “Digital reconstruction of multidimensional signals from their projections,” Proc. IEEE 62, 1319–1338 (1974).
    [CrossRef]
  24. H. Stark, J. W. Woods, I. Paul, R. Hingorani, “Direct Fourier reconstruction in computer tomography,” IEEE Trans. Acoust. Speech Signal Process. ASSP-29, 237–245 (1981).
    [CrossRef]
  25. R. N. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 1965).
  26. R. N. Bracewell, “Strip integration in radio astronomy,” Aust. J. Phys. 9, 198–217 (1956).
    [CrossRef]
  27. D. Verhoeven, “Limited-data computed tomography algorithms for the physical sciences,” Appl. Opt. 32, 3736–3754 (1993).
    [CrossRef] [PubMed]
  28. J. S. Walker, Fast Fourier Transforms (CRC, Boca Raton, Fla., 1991).

1994 (1)

1993 (3)

1992 (1)

P. A. Bernhardt, “Probing the magnetosphere using chemical releases from the Combined Release and Radiation Effects Satellite,” Phys. Fluids B 4, 2249–2256 (1992).
[CrossRef]

1991 (1)

1990 (1)

B. C. Wilson, S. L. Jacques, “Optical reflectance and transmittance of tissues: principles and applications,” IEEE J. Quantum Electron. 26, 2186–2199 (1990).
[CrossRef]

1988 (1)

P. A. Bernhardt, L. M. Duncan, C. A. Tepley, “Artificial airglow excited by high-power radio waves,” Science 242, 1022–1027 (1988).
[CrossRef] [PubMed]

1987 (1)

P. S. Andersson, S. Montan, S. Svanberg, “Multispectral system for medical fluorescence imaging,” IEEE J. Quantum Electron. QE-23, 1798–1805 (1987).
[CrossRef]

1985 (1)

S. Montan, S. Svanberg, “A system for industrial surface monitoring utilizing laser-induced fluorescence,” Appl. Phys. B 38, 241–247 (1985).
[CrossRef]

1983 (2)

R. M. Lewitt, “Reconstruction algorithms: transform methods,” Proc. IEEE 71, 390–408 (1983).
[CrossRef]

Y. Censor, “Series-expansion reconstruction methods,” Proc. IEEE 71, 409–419 (1983).
[CrossRef]

1981 (1)

H. Stark, J. W. Woods, I. Paul, R. Hingorani, “Direct Fourier reconstruction in computer tomography,” IEEE Trans. Acoust. Speech Signal Process. ASSP-29, 237–245 (1981).
[CrossRef]

1980 (1)

L. E. Blanchard, O. Weinstein, “Design challenges of the thematic mapper,” IEEE Trans. Geosci. Remote Sensing GE-18, 146–160 (1980).
[CrossRef]

1979 (1)

E. R. Menzel, J. M. Duff, “Laser detection of latent fingerprints—treatment with fluorescers,” J. Forensic Sci. 24, 96–100 (1979).
[PubMed]

1974 (1)

R. M. Mersereau, A. V. Oppenheim, “Digital reconstruction of multidimensional signals from their projections,” Proc. IEEE 62, 1319–1338 (1974).
[CrossRef]

1956 (1)

R. N. Bracewell, “Strip integration in radio astronomy,” Aust. J. Phys. 9, 198–217 (1956).
[CrossRef]

Andersson, P. S.

P. S. Andersson, S. Montan, S. Svanberg, “Multispectral system for medical fluorescence imaging,” IEEE J. Quantum Electron. QE-23, 1798–1805 (1987).
[CrossRef]

Andersson-Engels, S.

Bernhardt, P. A.

P. A. Bernhardt, “Probing the magnetosphere using chemical releases from the Combined Release and Radiation Effects Satellite,” Phys. Fluids B 4, 2249–2256 (1992).
[CrossRef]

P. A. Bernhardt, L. M. Duncan, C. A. Tepley, “Artificial airglow excited by high-power radio waves,” Science 242, 1022–1027 (1988).
[CrossRef] [PubMed]

Betremeiux, T.

T. Betremeiux, T. A. Cook, D. Cotton, S. Chakrabarti, “SPINR: high-resolution two-dimensional spectral imaging,” in Instrumentation for Magnetospheric Imagery II, S. Chakrabarti, ed., Proc. Soc. Photo-Opt. Instrum. Eng.2008, 114–119 (1993).
[CrossRef]

Blanchard, L. E.

L. E. Blanchard, O. Weinstein, “Design challenges of the thematic mapper,” IEEE Trans. Geosci. Remote Sensing GE-18, 146–160 (1980).
[CrossRef]

Bracewell, R. N.

R. N. Bracewell, “Strip integration in radio astronomy,” Aust. J. Phys. 9, 198–217 (1956).
[CrossRef]

R. N. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 1965).

Bulygin, F. V.

F. V. Bulygin, G. N. Vishnyakov, “Spectrotomography—a new method of obtaining spectrograms of two-dimensional objects,” in Analytical Methods for Optical Tomography, G. G. Levin, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1843, 315–322 (1991).
[CrossRef]

Censor, Y.

Y. Censor, “Series-expansion reconstruction methods,” Proc. IEEE 71, 409–419 (1983).
[CrossRef]

Chakrabarti, S.

T. Betremeiux, T. A. Cook, D. Cotton, S. Chakrabarti, “SPINR: high-resolution two-dimensional spectral imaging,” in Instrumentation for Magnetospheric Imagery II, S. Chakrabarti, ed., Proc. Soc. Photo-Opt. Instrum. Eng.2008, 114–119 (1993).
[CrossRef]

Cook, T. A.

T. Betremeiux, T. A. Cook, D. Cotton, S. Chakrabarti, “SPINR: high-resolution two-dimensional spectral imaging,” in Instrumentation for Magnetospheric Imagery II, S. Chakrabarti, ed., Proc. Soc. Photo-Opt. Instrum. Eng.2008, 114–119 (1993).
[CrossRef]

Cotton, D.

T. Betremeiux, T. A. Cook, D. Cotton, S. Chakrabarti, “SPINR: high-resolution two-dimensional spectral imaging,” in Instrumentation for Magnetospheric Imagery II, S. Chakrabarti, ed., Proc. Soc. Photo-Opt. Instrum. Eng.2008, 114–119 (1993).
[CrossRef]

Deans, S. R.

S. R. Deans, The Radon Transform and Some of Its Applications (Wiley, New York, 1983).

Duff, J. M.

E. R. Menzel, J. M. Duff, “Laser detection of latent fingerprints—treatment with fluorescers,” J. Forensic Sci. 24, 96–100 (1979).
[PubMed]

Duncan, L. M.

P. A. Bernhardt, L. M. Duncan, C. A. Tepley, “Artificial airglow excited by high-power radio waves,” Science 242, 1022–1027 (1988).
[CrossRef] [PubMed]

Elachi, C.

C. Elachi, Introduction to the Physics and Techniques of Remote Sensing (Wiley-Interscience, New York, 1987).

Herman, G. T.

G. T. Herman, “Algebraic reconstruction techniques can be made computationally efficient,” IEEE Trans. Med. Imaging 12, 600–609 (1993).
[CrossRef]

Hingorani, R.

H. Stark, J. W. Woods, I. Paul, R. Hingorani, “Direct Fourier reconstruction in computer tomography,” IEEE Trans. Acoust. Speech Signal Process. ASSP-29, 237–245 (1981).
[CrossRef]

Jacques, S. L.

B. C. Wilson, S. L. Jacques, “Optical reflectance and transmittance of tissues: principles and applications,” IEEE J. Quantum Electron. 26, 2186–2199 (1990).
[CrossRef]

Johansson, J.

Kak, A. C.

A. C. Kak, M. Slaney, Principles of Computerized Tomographic Imaging (IEEE, New York, 1988).

Lewitt, R. M.

R. M. Lewitt, “Reconstruction algorithms: transform methods,” Proc. IEEE 71, 390–408 (1983).
[CrossRef]

Menzel, E. R.

E. R. Menzel, J. M. Duff, “Laser detection of latent fingerprints—treatment with fluorescers,” J. Forensic Sci. 24, 96–100 (1979).
[PubMed]

Mersereau, R. M.

R. M. Mersereau, A. V. Oppenheim, “Digital reconstruction of multidimensional signals from their projections,” Proc. IEEE 62, 1319–1338 (1974).
[CrossRef]

Montan, S.

P. S. Andersson, S. Montan, S. Svanberg, “Multispectral system for medical fluorescence imaging,” IEEE J. Quantum Electron. QE-23, 1798–1805 (1987).
[CrossRef]

S. Montan, S. Svanberg, “A system for industrial surface monitoring utilizing laser-induced fluorescence,” Appl. Phys. B 38, 241–247 (1985).
[CrossRef]

Mooney, J. T.

J. T. Mooney, “Spectral imaging via computed tomography,” in Proceedings of the IRIS Specialty Group on Passive Sensors (Environmental Research Institute of Michigan, Ann Arbor, Mich., 1994).

O’Connor, D. V.

D. V. O’Connor, D. Phillips, Time Correlated Single Photon Counting (Academic, New York, 1984).

Okamoto, T.

Oppenheim, A. V.

R. M. Mersereau, A. V. Oppenheim, “Digital reconstruction of multidimensional signals from their projections,” Proc. IEEE 62, 1319–1338 (1974).
[CrossRef]

Paul, I.

H. Stark, J. W. Woods, I. Paul, R. Hingorani, “Direct Fourier reconstruction in computer tomography,” IEEE Trans. Acoust. Speech Signal Process. ASSP-29, 237–245 (1981).
[CrossRef]

Phillips, D.

D. V. O’Connor, D. Phillips, Time Correlated Single Photon Counting (Academic, New York, 1984).

Rees, M. H.

M. H. Rees, Physics and Chemistry of the Upper Atmosphere (Cambridge U. Press, Cambridge, 1989).
[CrossRef]

Rowland, S. W.

S. W. Rowland, “Computer implementation of image reconstruction formulas,” in Image Reconstructions from Projections, G. T. Herman, ed. (Springer-Verlag, Berlin, 1979), pp. 9–78.
[CrossRef]

Slaney, M.

A. C. Kak, M. Slaney, Principles of Computerized Tomographic Imaging (IEEE, New York, 1988).

Stark, H.

H. Stark, J. W. Woods, I. Paul, R. Hingorani, “Direct Fourier reconstruction in computer tomography,” IEEE Trans. Acoust. Speech Signal Process. ASSP-29, 237–245 (1981).
[CrossRef]

Svanberg, S.

S. Andersson-Engels, J. Johansson, S. Svanberg, “Medical diagnostic system based on simultaneous multispectral fluorescence imaging,” Appl. Opt. 33, 8022–8029 (1994).
[CrossRef] [PubMed]

P. S. Andersson, S. Montan, S. Svanberg, “Multispectral system for medical fluorescence imaging,” IEEE J. Quantum Electron. QE-23, 1798–1805 (1987).
[CrossRef]

S. Montan, S. Svanberg, “A system for industrial surface monitoring utilizing laser-induced fluorescence,” Appl. Phys. B 38, 241–247 (1985).
[CrossRef]

Takahishi, A.

Tepley, C. A.

P. A. Bernhardt, L. M. Duncan, C. A. Tepley, “Artificial airglow excited by high-power radio waves,” Science 242, 1022–1027 (1988).
[CrossRef] [PubMed]

Verhoeven, D.

Vishnyakov, G. N.

F. V. Bulygin, G. N. Vishnyakov, “Spectrotomography—a new method of obtaining spectrograms of two-dimensional objects,” in Analytical Methods for Optical Tomography, G. G. Levin, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1843, 315–322 (1991).
[CrossRef]

Walker, J. S.

J. S. Walker, Fast Fourier Transforms (CRC, Boca Raton, Fla., 1991).

Weinstein, O.

L. E. Blanchard, O. Weinstein, “Design challenges of the thematic mapper,” IEEE Trans. Geosci. Remote Sensing GE-18, 146–160 (1980).
[CrossRef]

Wilson, B. C.

B. C. Wilson, S. L. Jacques, “Optical reflectance and transmittance of tissues: principles and applications,” IEEE J. Quantum Electron. 26, 2186–2199 (1990).
[CrossRef]

Woods, J. W.

H. Stark, J. W. Woods, I. Paul, R. Hingorani, “Direct Fourier reconstruction in computer tomography,” IEEE Trans. Acoust. Speech Signal Process. ASSP-29, 237–245 (1981).
[CrossRef]

Yamaguchi, I.

Appl. Opt. (2)

Appl. Phys. B (1)

S. Montan, S. Svanberg, “A system for industrial surface monitoring utilizing laser-induced fluorescence,” Appl. Phys. B 38, 241–247 (1985).
[CrossRef]

Appl. Spectrosc. (1)

Aust. J. Phys. (1)

R. N. Bracewell, “Strip integration in radio astronomy,” Aust. J. Phys. 9, 198–217 (1956).
[CrossRef]

IEEE J. Quantum Electron. (2)

P. S. Andersson, S. Montan, S. Svanberg, “Multispectral system for medical fluorescence imaging,” IEEE J. Quantum Electron. QE-23, 1798–1805 (1987).
[CrossRef]

B. C. Wilson, S. L. Jacques, “Optical reflectance and transmittance of tissues: principles and applications,” IEEE J. Quantum Electron. 26, 2186–2199 (1990).
[CrossRef]

IEEE Trans. Acoust. Speech Signal Process. (1)

H. Stark, J. W. Woods, I. Paul, R. Hingorani, “Direct Fourier reconstruction in computer tomography,” IEEE Trans. Acoust. Speech Signal Process. ASSP-29, 237–245 (1981).
[CrossRef]

IEEE Trans. Geosci. Remote Sensing (1)

L. E. Blanchard, O. Weinstein, “Design challenges of the thematic mapper,” IEEE Trans. Geosci. Remote Sensing GE-18, 146–160 (1980).
[CrossRef]

IEEE Trans. Med. Imaging (1)

G. T. Herman, “Algebraic reconstruction techniques can be made computationally efficient,” IEEE Trans. Med. Imaging 12, 600–609 (1993).
[CrossRef]

J. Forensic Sci. (1)

E. R. Menzel, J. M. Duff, “Laser detection of latent fingerprints—treatment with fluorescers,” J. Forensic Sci. 24, 96–100 (1979).
[PubMed]

Opt. Lett. (1)

Phys. Fluids B (1)

P. A. Bernhardt, “Probing the magnetosphere using chemical releases from the Combined Release and Radiation Effects Satellite,” Phys. Fluids B 4, 2249–2256 (1992).
[CrossRef]

Proc. IEEE (3)

R. M. Lewitt, “Reconstruction algorithms: transform methods,” Proc. IEEE 71, 390–408 (1983).
[CrossRef]

Y. Censor, “Series-expansion reconstruction methods,” Proc. IEEE 71, 409–419 (1983).
[CrossRef]

R. M. Mersereau, A. V. Oppenheim, “Digital reconstruction of multidimensional signals from their projections,” Proc. IEEE 62, 1319–1338 (1974).
[CrossRef]

Science (1)

P. A. Bernhardt, L. M. Duncan, C. A. Tepley, “Artificial airglow excited by high-power radio waves,” Science 242, 1022–1027 (1988).
[CrossRef] [PubMed]

Other (11)

F. V. Bulygin, G. N. Vishnyakov, “Spectrotomography—a new method of obtaining spectrograms of two-dimensional objects,” in Analytical Methods for Optical Tomography, G. G. Levin, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1843, 315–322 (1991).
[CrossRef]

J. T. Mooney, “Spectral imaging via computed tomography,” in Proceedings of the IRIS Specialty Group on Passive Sensors (Environmental Research Institute of Michigan, Ann Arbor, Mich., 1994).

T. Betremeiux, T. A. Cook, D. Cotton, S. Chakrabarti, “SPINR: high-resolution two-dimensional spectral imaging,” in Instrumentation for Magnetospheric Imagery II, S. Chakrabarti, ed., Proc. Soc. Photo-Opt. Instrum. Eng.2008, 114–119 (1993).
[CrossRef]

D. V. O’Connor, D. Phillips, Time Correlated Single Photon Counting (Academic, New York, 1984).

M. H. Rees, Physics and Chemistry of the Upper Atmosphere (Cambridge U. Press, Cambridge, 1989).
[CrossRef]

C. Elachi, Introduction to the Physics and Techniques of Remote Sensing (Wiley-Interscience, New York, 1987).

R. N. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 1965).

S. R. Deans, The Radon Transform and Some of Its Applications (Wiley, New York, 1983).

S. W. Rowland, “Computer implementation of image reconstruction formulas,” in Image Reconstructions from Projections, G. T. Herman, ed. (Springer-Verlag, Berlin, 1979), pp. 9–78.
[CrossRef]

A. C. Kak, M. Slaney, Principles of Computerized Tomographic Imaging (IEEE, New York, 1988).

J. S. Walker, Fast Fourier Transforms (CRC, Boca Raton, Fla., 1991).

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Figures (16)

Fig. 1
Fig. 1

Objective-grating camera with the chromatic dispersion (a) perpendicular to and (b) aligned with the plane containing axial and oblique rays. As the imaging system is rotated, the relationship between the spatial and the chromatic dispersion changes.

Fig. 2
Fig. 2

Geometry for the ST transform in three dimensions.

Fig. 3
Fig. 3

Effect of changing the focal length and the placement of the grating lens (lens 2) of the detector image. This zoom lens alters the spatial scale but not the chromatic displacement of the image.

Fig. 4
Fig. 4

RST transfer function for five values of tan Θ = 1/2k = f3/f2, where f2 and f3 are the instrument focal lengths in Figs. 1 and 3. The transfer function varies from zero in the black regions to unity in the white regions. The contour lines are separated by steps of 1/10. The five transfer functions (a)–(e) are combined to form a composite transfer function (f).

Fig. 5
Fig. 5

Transfer function for an imaging spectrometer with no attenuation (white) at low frequencies and complete attenuation (black) for spatial or spectral frequencies near unity. The frequencies are normalized by the size of the detector pixels or the width of the spectrograph slit.

Fig. 6
Fig. 6

Broad-spectrum, spatial image of the object to be reconstructed. The image is digitized on a 64 × 64 pixel array. The image is composed of a broadband point source and 18 monochromatic disks.

Fig. 7
Fig. 7

Images of the source object broken into 64 wavelength components. A point source (star) image is observed at each wavelength. Unit-intensity disks with random position and size are distributed through the wavelength spectral. The image indices identify wavelengths.

Fig. 8
Fig. 8

Images from an imaging spectrometer as the slit is stepped across the spectral object. The indices give the spatial location of each spectrographic image.

Fig. 9
Fig. 9

Output of the objective-grating camera for rotations from −π through 0 to π rad. Angle index 0 corresponds to the grating dispersion direction aligned with the horizontal dimension of the source. The simulated images are obtained with the instrument parameters tan Θ = f3/f2 = 1.

Fig. 10
Fig. 10

Sequence of images from the RST system in Fig. 3 as the focal length for grating lens 2 is increased (or zoomed) by factors of 2. The ratio of focal lengths is f3/f2 = 1/2k, where k is the index in the corner of each image. The spatial resolution decreases but the chromatic isolation increases as k is varied from 0 to 4. The horizontal line in the image for k = 0 is the spectral image of the star.

Fig. 11
Fig. 11

Computed tomographic reconstruction based on the single sequence of the rotated images in Fig. 9. Comparison with the source in Fig. 7 shows poor wavelength isolation of the monochromatic components.

Fig. 12
Fig. 12

Reconstructed object obtained by applying the direct Fourier algorithm to the RST images depicted in Fig. 9 and the range of instrument focal lengths used for Fig. 10. The spatial features at each wavelength are reconstructed. Some interference occurs between neighboring wavelength bins.

Fig. 13
Fig. 13

Comparison of the original (Fig. 7) and reconstructed (Fig. 12) features with wavelength index 15. The absolute intensity is less for the reconstructions than for the original object.

Fig. 14
Fig. 14

Reconstructed object obtained by applying the filter-backprojection algorithm to the RST images depicted in Figs. 9 and 10. The interference between wavelength bins is stronger than that obtained with the direct Fourier algorithm.

Fig. 15
Fig. 15

Intensity distribution of the unit-amplitude star and disk obtained with filter-backprojection techniques. A comparison with Fig. 13 shows that the recovered intensity is lower but the disk surface is flatter than that obtained with direct Fourier techniques.

Fig. 16
Fig. 16

Buildup of the recovered image as each new set of zoomed images is included in the reconstruction series. The indices k = 0 to 4 correspond to the transfer functions in Fig. 4 and the sampled detector images in Fig. 10.

Equations (71)

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

G ( δ s , δ λ ) = δ s δ λ ( x b - x a ) ( λ b - λ a ) ,
x 1 = f 1 α x ,             y 1 = f 1 α y ,
β x = x 1 / f 2 ,             β y = y 1 / f 2 ,
sin ψ λ = λ m ν ,
d ψ / d λ = m ν / cos ψ λ 0 .
γ x = β x + ( λ - λ 0 ) d ψ / d λ ,             γ y = β y .
x = γ x f 3 ,             y = γ y f 3 ,
x = x 1 ( f 3 / f 2 ) + z 1 ( λ ) ,             y = y 1 ( f 3 / f 2 ) ,
z 1 ( λ ) = m ν ( λ - λ 0 ) f 3 / cos ψ λ 0
x = ( x 1 cos Φ + y 1 sin Φ ) ( f 3 / f 2 ) + z 1 ,
y = ( y 1 cos Φ - x 1 sin Φ ) ( f 3 / f 2 ) .
tan Θ = f 3 / f 2
I ( x , y , Φ , Θ ) = - B ( x 1 , y 1 , z 1 ) δ [ x - ( x 1 cos Φ + y 1 sin Φ ) tan Θ - z 1 ] δ [ y - ( y 1 cos Φ - x 1 sin Φ ) tan Θ ] d x 1 d y 1 d z 1 ,
z a m ν ( λ a - λ 0 ) cos ψ λ 0 z 1 z b m ν ( λ b - λ 0 ) f 3 cos ψ λ 0 .
- r a tan Θ + z a x r a tan Θ + z b ,
- r a tan Θ y r a tan Θ .
χ ( x , y , Φ , Θ ) = - B ( x 1 , y 1 , z 1 ) δ ( x - x 1 tan Θ - z 1 cos Φ ) δ ( y - y 1 tan Θ - z 1 sin Φ ) d x 1 d y 1 d z 1 ,
F n [ A ( x ) ] ( f ) = - A ( x ) exp ( - j 2 π f · x ) d n x A ˜ ( f ) , A ( x ) = F n - 1 [ A ˜ ( f ) ] ( x ) = - A ˜ ( f ) exp ( j 2 π f · x ) d n x ,
n = 1 : x = ( x ) , f = ( f x ) , f · x = f x x , n = 2 : x = ( x , y ) , f = ( f x , f y ) , f · x = f x x + f y y , n = 3 : x = ( x , y , z ) , f = ( f x , f y , f z ) , f · x = f x x + f y y + f z z .
R 2 [ g ( x , y ) ] ( s , Θ ) = - g ( x , y ) δ ( s - x cos Θ - y sin Θ ) d x d y G ( s , Θ )
g ( x , y ) = 0 π F 1 - 1 [ f s F 1 [ G ( s , Θ ) ] ( f s , Θ ) ] × ( x cos Θ + y sin Θ ) d Θ .
F 1 [ G ( s , Θ ) ] ( f s , Θ ) = F 2 [ g ( x , y ) ] ( f s cos Θ , f s sin Θ )
R 3 [ g ( x , y , z ) ] ( p , Φ , Θ ) G ( p , Φ , Θ ) = - g ( x , y , z ) × δ ( p - x cos Φ sin Θ - y sin Φ sin Θ - z cos Θ ) d x d y d z ,
g ( x , y , z ) = 0 π 0 2 π F 1 - 1 [ F 1 [ G ( p , Φ , Θ ) ] ( f p , Φ , Θ ) f p 2 ] × ( x sin Θ cos Φ + y sin Θ sin Φ + z cos Θ , Φ , Θ ) sin Θ d Φ d Θ .
J y ( x , Φ = 0 , Θ ) = - C y ( x 1 , z 1 ) δ ( x - x 1 sin Θ - z 1 cos Θ ) d x 1 d z 1 ,
J ^ ( x , Φ , Θ ) - J y ( x , Φ , Θ ) d y = R 3 [ B ( x 1 , y 1 , z 1 ) ] ( x , Φ , Θ ) .
I k ( x , y , Φ ) = - B ( x 1 , y 1 , z 1 ) δ [ x - ( x 1 cos Φ + y 1 sin Φ ) tan Θ k - z 1 ] δ [ y - ( y 1 cos Φ - x 1 sin Φ ) tan Θ k ] d x 1 d y 1 d z 1 .
I k ( x , y , Φ ) = - B × [ ( x - z ) cos Φ - y sin Φ tan Θ k , ( x - z ) sin Φ + y cos Φ tan Θ k ] × d z tan 2 Θ k .
F 2 [ I k ( x , y , Φ ) ] ( f x , f y , Φ ) = F 3 [ B ( x 1 , y 1 , z 1 ) ] ( f x , f y , f z ) = B ˜ ( f x , f y , f z ) ,
f x = ( tan Θ k ) ( f x cos Φ - f y sin Φ ) ,
f y = ( tan Θ k ) ( f x sin Φ + f y cos Φ ) ,
f z = f x .
f x = f z ,
f y = ( f x 2 + f y 2 - f z 2 tan 2 Θ k ) 1 / 2 tan Θ k for f x 2 + f y 2 f z 2 tan 2 Θ k ,
tan Φ = f y f z - f x f y f x f z + f y f y .
B ^ k ( x 1 , y 1 , z 1 ) = F 3 - 1 [ B ˜ ( f x , f y , f z ) H ( f x 2 + f y 2 - f z 2 tan 2 Θ k ) ] ,
B ^ k ( x 1 , y 1 , z 1 ) = 0 2 π F 1 - 1 [ F 1 [ I k ( x , y , Φ ) ] ( x , f y , Φ ) × H ( f y ) f y ] ( α k , β k , Φ ) tan 2 Θ k d Φ ,
χ k ( x , y , Φ ) = - B ( x 1 , y 1 , z 1 ) δ ( x - x 1 tan Θ k - z 1 cos Φ ) δ ( y - y 1 tan Θ k - z 1 sin Φ ) d x 1 d y 1 d z 1 .
I k ( x , y , Φ ) = χ k ( x cos Φ - y sin Φ , x sin Φ + y cos Φ , Φ ) .
B ^ k ( x 1 , y 1 , z 1 ) = 0 2 π F 2 - 1 [ F 2 [ χ k ( x , y , Φ ) ] ( f x , f y , Φ ) × H ( f γ ) f γ ] ( η k , κ k , Φ ) tan 2 Θ k d Φ ,
B ( x 1 , y 1 , z 1 ) = exp [ - π ( x 1 2 + y 1 2 σ 2 + z 1 2 ξ 2 ) ] / σ 2 ξ ,
I k ( x , y , Φ ) = exp ( - π x 2 ξ 2 + σ 2 tan 2 Θ k - π y 2 σ 2 tan 2 Θ k ) σ tan Θ k ( ξ 2 + σ 2 tan 2 Θ k ) 1 / 2 .
B ^ k ( x 1 , y 1 , z 1 ) = 4 π 0 q 1 exp [ - π ( q 2 σ 2 + f 2 ξ 2 ) ] × J 0 ( 2 π q r ) q cos ( 2 π f z 1 ) d q d f ,
B ^ k ( 0 , 0 , z 1 ) = exp [ - π z 2 / ( ξ 2 + σ 2 tan 2 Θ k ) ] σ 2 ( ξ 2 + σ 2   tan 2 Θ k ) 1 / 2 .
I k s ( x , y , Φ ) = - I k ( x , y , Φ ) Π ( x - x Δ x ) × Π ( y - y Δ y ) d x d y .
I ˜ k s ( f x , f y , Φ ) = F 2 [ I k ( x , y , Φ ) ] ( f x , f y , Φ ) Δ x Δ y sinc ( Δ x f x ) × sinc ( Δ y f y ) ,
f x 1 / 2 Δ x ,             f y 1 / 2 Δ y .
f z 1 / 2 Δ x ,
( f x 2 + f y 2 - f z 2 tan 2 Θ k ) 1 / 2 tan Θ k / 2 Δ y .
I ˜ k s ( f x , f y , Φ ) = F 3 [ B ( x 1 , y 1 , z 1 ) ] ( f x , f y , f z ) H ( f x 2 + f y 2 - f z 2 tan 2 Θ k ) Δ x Δ y sinc ( f z Δ x ) × sinc [ ( f x 2 + f y 2 - f 2 2 tan 2 Θ k ) 1 / 2 Δ y tan Θ k ] ,
f z 2 > ( f x 2 + f y 2 ) / tan 2 Θ k .
f x 2 + f y 2 > [ f z 2 + 1 ( 2 Δ y ) 2 ] tan 2 Θ k ,             f z > 1 ( 2 Δ x ) 2 .
B ^ k ( x 1 , y 1 , z 1 ) = B ( x 1 , y 1 , z 1 ) * A k ( x 1 , y 1 , z 1 ) ,
C n + 1 ( x 1 , y 1 , z 1 ) = B ^ k + C n + * Λ k ,
I ˜ k s ( f x , f y , Φ ) = B ˜ ( f x , f y , f z ) G k ( f x , f y , f z ) ,
G k ( f x , f y , f z ) = H ( f r 2 - f z 2 / 2 2 k ) Δ x Δ y sinc ( f z Δ x ) × sinc ( f r 2 - f z 2 / 2 2 k Δ y 2 k ) ,
f x = ( f x cos Φ - f y sin Φ ) / 2 k , f y = ( f y sin Φ + f x cos Φ ) / 2 k ,             f z = f x ,
B ˜ R ( f x , f y , f z ) = k = 0 K I ˜ k s ( f x , f y , Φ ) W k ( f x , f y , f z ) ,
W k ( f x , f y , f z ) = 1 - H [ f r 2 - f z 2 / 2 2 ( k - 1 ) ]             for k > 0.
B R ( x 1 , y 1 , z 1 ) = F 3 - 1 [ B ˜ R ] = k = 0 K Δ B k s ( x 1 , y 1 , z 1 ) ,
Δ B k s = F 3 - 1 [ F 3 [ B ( x 1 , y 1 , z 1 ) ] ( f x , f y , f z ) G k ( f x , f y , f z ) × W k ( f x , f y , f z ) ] .
Δ B 0 s ( x 1 , y 1 , z 1 ) = 0 2 π F 1 - 1 [ F 1 [ I 0 s ( x , y , Φ ) ] ( x , f y , Φ ) × H ( f y ) f y ] ( α 0 , β 0 , Φ ) d Φ ,
H ( f r 2 - f z 2 / 2 2 k ) W ( f x , f y , f z ) = H ( f r 2 - f z / 2 2 k ) - H [ f r 2 - f z 2 / 2 2 ( k - 1 ) ] = H ( 3 f x 2 - f y 2 ) ,
Δ B k s = 0 2 π F 2 - 1 [ F 2 [ I k s ( x , y , Φ ) ] ( f x , f y , Φ ) H ( 3 f x 2 - f y 2 ) × H ( f y ) f y ] ( α k , β k , Φ ) / 2 2 k d Φ ,
G ^ ( f x , f y , f z ) = sinc ( f z Δ x ) sinc ( f y Δ y ) sinc ( f x δ s ) Δ x Δ y δ s ,
Γ ˜ 0 ( f x , f y ) H ( f y ) f y , Γ ˜ k ( f x , f y ) H ( 3 f x 2 - f y 2 ) H ( f y ) f y .
Δ B k s = 0 2 π [ I k s ( x , y , Φ ) * 2 Γ k ( x , y ) ] ( α k , β k , Φ ) / 2 2 k d Φ ,
Γ 0 ( x , y ) = - 1 / 2 1 / 2 H ( f y ) f y exp [ i 2 π ( f x x + f y y ) ] d f x d f y = sinc ( x ) [ ( 1 - i π y ) exp ( i π y ) - 1 ( 2 π y ) 2 ] ,
Γ k ( x , y ) = - 1 / 2 1 / 2 H ( 3 f x 2 - f y 2 ) H ( f y ) f y exp [ i 2 π ( f x x + f y y ) ] d f x d f y = 1 4 3 S ( π y , π x 3 ) ,
S ( a , b ) = exp ( i a c ) b ( b 2 - a 2 ) [ ( sin b ) ( i a + a 2 + b 2 b 2 - a 2 ) - ( cos b ) ( b - 2 i a b b 2 - a 2 ) ] - 2 i a b b ( b 2 - a 2 ) .
K = log 2 ( L x / Δ x ) - 1.

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