Abstract

We present a new algorithm for image restoration in limited-angle chromotomography. The algorithm is a generalization of the technique considered previously by the authors, based on a hybrid of a direct method of inversion and the iterative method of projections onto convex sets. The generalization is achieved by introducing a new object domain constraint. This constraint takes advantage of hyperspectral data redundancy and is realized by truncating the singular-value decomposition of the spatial–chromatic image matrix. As previously, the transform domain constraint is defined in terms of nonzero singular values of the system transfer function matrix. The new algorithm delivers high image fidelity, converges rapidly, and is easy to implement. Results of experiments on real data are included.

© 1999 Optical Society of America

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    [CrossRef]
  2. H. H. Barrett, “Limited-angle tomography for the nineties,” J. Nucl. Med. 31, 1689–1692 (1990).
  3. D. A. Hayner, W. K. Jenkins, “The missing cone problem in computer tomography,” in Advances in Computer Vision and Image Processing, T. S. Huang, ed. (JAI Press, London, 1984), Vol. 1, pp. 83–144.
  4. J. M. Mooney, V. E. Vickers, M. An, A. K. Brodzik, “A high-throughput hyperspectral infrared camera,” J. Opt. Soc. Am. A 14, 2951–2961 (1997).
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  9. R. W. Gerchberg, “Super-resolution through error energy reduction,” Opt. Acta 21, 709–720 (1974).
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    [CrossRef]
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  16. P. K. Sadasivan, D. N. Dutt, “SVD based technique for noise reduction in electroencephalographic signals,” Signal Process. 55, 179–189 (1996).
    [CrossRef]
  17. J. S. Goldstein, I. S. Reed, “Reduced-rank adaptive filtering,” IEEE Trans. Signal Process. 45, 492–496 (1997).
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  18. S. Heidari, C. L. Nikias, “Co-channel interference mitigation in the time-scale domain: the CIMTS algorithm,” IEEE Trans. Signal Process. 44, 2151–2162 (1996).
    [CrossRef]
  19. N. H. Endsley, “Spectral unmixing algorithms based on statistical models,” in Imaging Spectrometry, M. R. Descour, J. M. Mooney, D. L. Perry, L. Illing, eds., Proc. SPIE2480, 23–36 (1995).
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  20. A. A. Green, M. Berman, P. Switzer, M. D. Craig, “A transformation for ordering multispectral data in terms of image quality with implications for noise removal,” IEEE Trans. Geosci. Remote Sens. 26 (No. 1), 65–74 (1988).
    [CrossRef]
  21. J. B. Lee, A. S. Woodyatt, M. Berman, “Enhancement of high spectral resolution remote-sensing data by a noise-adjusted principal component transform,” IEEE Trans. Geosci. Remote Sens. 28 (No. 3), 295–304 (1990).
    [CrossRef]
  22. J. M. Mooney, “Spectral imaging via computed tomography,” in Proceedings of the 1994 Meeting of the Infrared Information Symposia Specialty Group on Passive Sensors (Defense Technical Information Center, Alexandria, Va., 1994), pp. 203–215.
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    [CrossRef]
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  26. A. Sano, “Optimally regularized inverse of singular value decomposition and application to signal extrapolation,” Signal Process. 30, 163–176 (1993).
    [CrossRef]
  27. L. M. Bregman, “The method of successive projections for finding a common point of convex sets,” Dokl. Akad. Nauk SSSR 162 (No. 3), 487–490 (1965).
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  29. D. C. Youla, H. Webb, “Image restoration by the method of convex projections: Part 1—theory,” IEEE Trans. Med. Imaging MI-1 (No. 2), 81–94 (1982).
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    [CrossRef]
  33. P. L. Combettes, “The foundations of set theoretic estimation,” Proc. IEEE 81 (No. 2), 182–208 (1993).
    [CrossRef]
  34. P. J. Ready, P. A. Wintz, “Information extraction, SNR improvement, and data compression in multispectral imagery,” IEEE Trans. Commun. COM-21, 1123–1130 (1973).
    [CrossRef]
  35. C. E. Shannon, “Coding theorems for discrete source with a fidelity criterion,” in Institute of Radio Engineers National Convention Record (Institute of Radio Engineers, New York, 1959), Part 4, pp. 142–163.
  36. L. L. Scharf, “The SVD and reduced-rank signal processing,” in SVD and Signal Processing II: Algorithms, Analysis and Applications, R. Vaccaro, ed. (Elsevier, Amsterdam, 1991), pp. 3–31.

1997 (4)

H. Hiriyannaiai, “X-ray computed tomography for medical imaging,” IEEE Signal Process. Mag. 14 (No. 2), 42–59 (1997).
[CrossRef]

K. Konstantinides, B. Natarajan, G. S. Yovanof, “Noise estimation and filtering using block-based singular value decomposition,” IEEE Trans. Image Process. 6 (No. 3), 479–483 (1997).
[CrossRef] [PubMed]

J. S. Goldstein, I. S. Reed, “Reduced-rank adaptive filtering,” IEEE Trans. Signal Process. 45, 492–496 (1997).
[CrossRef]

J. M. Mooney, V. E. Vickers, M. An, A. K. Brodzik, “A high-throughput hyperspectral infrared camera,” J. Opt. Soc. Am. A 14, 2951–2961 (1997).
[CrossRef]

1996 (2)

S. Heidari, C. L. Nikias, “Co-channel interference mitigation in the time-scale domain: the CIMTS algorithm,” IEEE Trans. Signal Process. 44, 2151–2162 (1996).
[CrossRef]

P. K. Sadasivan, D. N. Dutt, “SVD based technique for noise reduction in electroencephalographic signals,” Signal Process. 55, 179–189 (1996).
[CrossRef]

1993 (2)

A. Sano, “Optimally regularized inverse of singular value decomposition and application to signal extrapolation,” Signal Process. 30, 163–176 (1993).
[CrossRef]

P. L. Combettes, “The foundations of set theoretic estimation,” Proc. IEEE 81 (No. 2), 182–208 (1993).
[CrossRef]

1990 (2)

J. B. Lee, A. S. Woodyatt, M. Berman, “Enhancement of high spectral resolution remote-sensing data by a noise-adjusted principal component transform,” IEEE Trans. Geosci. Remote Sens. 28 (No. 3), 295–304 (1990).
[CrossRef]

H. H. Barrett, “Limited-angle tomography for the nineties,” J. Nucl. Med. 31, 1689–1692 (1990).

1988 (1)

A. A. Green, M. Berman, P. Switzer, M. D. Craig, “A transformation for ordering multispectral data in terms of image quality with implications for noise removal,” IEEE Trans. Geosci. Remote Sens. 26 (No. 1), 65–74 (1988).
[CrossRef]

1984 (1)

1983 (1)

1982 (2)

M. I. Sezan, H. Stark, “Image restoration by the method of convex projections: Part 2—applications and numerical results,” IEEE Trans. Med. Imaging MI-1 (No. 2), 95–101 (1982).
[CrossRef]

D. C. Youla, H. Webb, “Image restoration by the method of convex projections: Part 1—theory,” IEEE Trans. Med. Imaging MI-1 (No. 2), 81–94 (1982).
[CrossRef]

1981 (1)

1980 (1)

H. Knutsson, P. Edholm, G. Grandlund, C. Petersson, “Ectomography—a new radiographic reconstruction method—I. Theory and error estimates,” IEEE Trans. Biomed. Eng. BME-27, 640–648 (1980).
[CrossRef]

1976 (2)

H. C. Andrews, C. L. Patterson, “Singular value decomposition and digital image processing,” IEEE Trans. Acoust., Speech, Signal Process. ASSP-24, 26–53 (1976).
[CrossRef]

D. C. Solmon, “The x-ray transform,” J. Math. Anal. Appl. 56, 61–83 (1976).
[CrossRef]

1975 (1)

A. Papoulis, “A new algorithm in spectral analysis and band-limited extrapolation,” IEEE Trans. Circuits Syst. CAS-22, 735–742 (1975).
[CrossRef]

1974 (1)

R. W. Gerchberg, “Super-resolution through error energy reduction,” Opt. Acta 21, 709–720 (1974).
[CrossRef]

1973 (1)

P. J. Ready, P. A. Wintz, “Information extraction, SNR improvement, and data compression in multispectral imagery,” IEEE Trans. Commun. COM-21, 1123–1130 (1973).
[CrossRef]

1967 (1)

L. G. Gubin, B. T. Polyak, E. V. Raik, “The method of projections for finding a common point of convex sets,” USSR Comput. Math. Math. Phys. 7 (No. 6), 1–24 (1967).
[CrossRef]

1965 (1)

L. M. Bregman, “The method of successive projections for finding a common point of convex sets,” Dokl. Akad. Nauk SSSR 162 (No. 3), 487–490 (1965).

An, M.

J. M. Mooney, V. E. Vickers, M. An, A. K. Brodzik, “A high-throughput hyperspectral infrared camera,” J. Opt. Soc. Am. A 14, 2951–2961 (1997).
[CrossRef]

A. K. Brodzik, J. M. Mooney, M. An, “Image restoration by convex projections: application to image spectrometry,” in Imaging Spectrometry, M. R. Descour, J. M. Mooney, eds., Proc. SPIE2819, 231–242 (1996).
[CrossRef]

J. M. Mooney, A. K. Brodzik, M. An, “Principal component analysis in limited angle chromotomography,” in Imaging Spectrometry, M. R. Descour, S. S. Shen, eds., Proc. SPIE3118, 170–178 (1997).
[CrossRef]

Andrews, H. C.

H. C. Andrews, C. L. Patterson, “Singular value decomposition and digital image processing,” IEEE Trans. Acoust., Speech, Signal Process. ASSP-24, 26–53 (1976).
[CrossRef]

Barrett, H. H.

H. H. Barrett, “Limited-angle tomography for the nineties,” J. Nucl. Med. 31, 1689–1692 (1990).

Ben-Israel, A.

A. Ben-Israel, T. N. E. Greville, Generalized Inverses: Theory and Applications (Wiley, New York, 1980).

Berman, M.

J. B. Lee, A. S. Woodyatt, M. Berman, “Enhancement of high spectral resolution remote-sensing data by a noise-adjusted principal component transform,” IEEE Trans. Geosci. Remote Sens. 28 (No. 3), 295–304 (1990).
[CrossRef]

A. A. Green, M. Berman, P. Switzer, M. D. Craig, “A transformation for ordering multispectral data in terms of image quality with implications for noise removal,” IEEE Trans. Geosci. Remote Sens. 26 (No. 1), 65–74 (1988).
[CrossRef]

Bregman, L. M.

L. M. Bregman, “The method of successive projections for finding a common point of convex sets,” Dokl. Akad. Nauk SSSR 162 (No. 3), 487–490 (1965).

Brodzik, A. K.

J. M. Mooney, V. E. Vickers, M. An, A. K. Brodzik, “A high-throughput hyperspectral infrared camera,” J. Opt. Soc. Am. A 14, 2951–2961 (1997).
[CrossRef]

A. K. Brodzik, J. M. Mooney, M. An, “Image restoration by convex projections: application to image spectrometry,” in Imaging Spectrometry, M. R. Descour, J. M. Mooney, eds., Proc. SPIE2819, 231–242 (1996).
[CrossRef]

J. M. Mooney, A. K. Brodzik, M. An, “Principal component analysis in limited angle chromotomography,” in Imaging Spectrometry, M. R. Descour, S. S. Shen, eds., Proc. SPIE3118, 170–178 (1997).
[CrossRef]

Combettes, P. L.

P. L. Combettes, “The foundations of set theoretic estimation,” Proc. IEEE 81 (No. 2), 182–208 (1993).
[CrossRef]

Craig, M. D.

A. A. Green, M. Berman, P. Switzer, M. D. Craig, “A transformation for ordering multispectral data in terms of image quality with implications for noise removal,” IEEE Trans. Geosci. Remote Sens. 26 (No. 1), 65–74 (1988).
[CrossRef]

Dutt, D. N.

P. K. Sadasivan, D. N. Dutt, “SVD based technique for noise reduction in electroencephalographic signals,” Signal Process. 55, 179–189 (1996).
[CrossRef]

Edholm, P.

H. Knutsson, P. Edholm, G. Grandlund, C. Petersson, “Ectomography—a new radiographic reconstruction method—I. Theory and error estimates,” IEEE Trans. Biomed. Eng. BME-27, 640–648 (1980).
[CrossRef]

Endsley, N. H.

N. H. Endsley, “Spectral unmixing algorithms based on statistical models,” in Imaging Spectrometry, M. R. Descour, J. M. Mooney, D. L. Perry, L. Illing, eds., Proc. SPIE2480, 23–36 (1995).
[CrossRef]

Gerchberg, R. W.

R. W. Gerchberg, “Super-resolution through error energy reduction,” Opt. Acta 21, 709–720 (1974).
[CrossRef]

Goldstein, J. S.

J. S. Goldstein, I. S. Reed, “Reduced-rank adaptive filtering,” IEEE Trans. Signal Process. 45, 492–496 (1997).
[CrossRef]

Golub, G.

G. Golub, C. Van Loan, Matrix Computations (Johns Hopkins U. Press, Baltimore, Md., 1993).

Grandlund, G.

H. Knutsson, P. Edholm, G. Grandlund, C. Petersson, “Ectomography—a new radiographic reconstruction method—I. Theory and error estimates,” IEEE Trans. Biomed. Eng. BME-27, 640–648 (1980).
[CrossRef]

Green, A. A.

A. A. Green, M. Berman, P. Switzer, M. D. Craig, “A transformation for ordering multispectral data in terms of image quality with implications for noise removal,” IEEE Trans. Geosci. Remote Sens. 26 (No. 1), 65–74 (1988).
[CrossRef]

Greville, T. N. E.

A. Ben-Israel, T. N. E. Greville, Generalized Inverses: Theory and Applications (Wiley, New York, 1980).

Gubin, L. G.

L. G. Gubin, B. T. Polyak, E. V. Raik, “The method of projections for finding a common point of convex sets,” USSR Comput. Math. Math. Phys. 7 (No. 6), 1–24 (1967).
[CrossRef]

Hayner, D. A.

D. A. Hayner, W. K. Jenkins, “The missing cone problem in computer tomography,” in Advances in Computer Vision and Image Processing, T. S. Huang, ed. (JAI Press, London, 1984), Vol. 1, pp. 83–144.

Heidari, S.

S. Heidari, C. L. Nikias, “Co-channel interference mitigation in the time-scale domain: the CIMTS algorithm,” IEEE Trans. Signal Process. 44, 2151–2162 (1996).
[CrossRef]

Hiriyannaiai, H.

H. Hiriyannaiai, “X-ray computed tomography for medical imaging,” IEEE Signal Process. Mag. 14 (No. 2), 42–59 (1997).
[CrossRef]

Jenkins, W. K.

D. A. Hayner, W. K. Jenkins, “The missing cone problem in computer tomography,” in Advances in Computer Vision and Image Processing, T. S. Huang, ed. (JAI Press, London, 1984), Vol. 1, pp. 83–144.

Knutsson, H.

H. Knutsson, P. Edholm, G. Grandlund, C. Petersson, “Ectomography—a new radiographic reconstruction method—I. Theory and error estimates,” IEEE Trans. Biomed. Eng. BME-27, 640–648 (1980).
[CrossRef]

Konstantinides, K.

K. Konstantinides, B. Natarajan, G. S. Yovanof, “Noise estimation and filtering using block-based singular value decomposition,” IEEE Trans. Image Process. 6 (No. 3), 479–483 (1997).
[CrossRef] [PubMed]

Lee, J. B.

J. B. Lee, A. S. Woodyatt, M. Berman, “Enhancement of high spectral resolution remote-sensing data by a noise-adjusted principal component transform,” IEEE Trans. Geosci. Remote Sens. 28 (No. 3), 295–304 (1990).
[CrossRef]

Levi, A.

Mooney, J. M.

J. M. Mooney, V. E. Vickers, M. An, A. K. Brodzik, “A high-throughput hyperspectral infrared camera,” J. Opt. Soc. Am. A 14, 2951–2961 (1997).
[CrossRef]

A. K. Brodzik, J. M. Mooney, M. An, “Image restoration by convex projections: application to image spectrometry,” in Imaging Spectrometry, M. R. Descour, J. M. Mooney, eds., Proc. SPIE2819, 231–242 (1996).
[CrossRef]

J. M. Mooney, A. K. Brodzik, M. An, “Principal component analysis in limited angle chromotomography,” in Imaging Spectrometry, M. R. Descour, S. S. Shen, eds., Proc. SPIE3118, 170–178 (1997).
[CrossRef]

J. M. Mooney, “Spectral imaging via computed tomography,” in Proceedings of the 1994 Meeting of the Infrared Information Symposia Specialty Group on Passive Sensors (Defense Technical Information Center, Alexandria, Va., 1994), pp. 203–215.

Natarajan, B.

K. Konstantinides, B. Natarajan, G. S. Yovanof, “Noise estimation and filtering using block-based singular value decomposition,” IEEE Trans. Image Process. 6 (No. 3), 479–483 (1997).
[CrossRef] [PubMed]

Natterer, F.

F. Natterer, The Mathematics of Computerized Tomography (Wiley, Stuttgart, 1986).

Nikias, C. L.

S. Heidari, C. L. Nikias, “Co-channel interference mitigation in the time-scale domain: the CIMTS algorithm,” IEEE Trans. Signal Process. 44, 2151–2162 (1996).
[CrossRef]

Papoulis, A.

A. Papoulis, “A new algorithm in spectral analysis and band-limited extrapolation,” IEEE Trans. Circuits Syst. CAS-22, 735–742 (1975).
[CrossRef]

Patterson, C. L.

H. C. Andrews, C. L. Patterson, “Singular value decomposition and digital image processing,” IEEE Trans. Acoust., Speech, Signal Process. ASSP-24, 26–53 (1976).
[CrossRef]

Perez-Mendez, V.

Petersson, C.

H. Knutsson, P. Edholm, G. Grandlund, C. Petersson, “Ectomography—a new radiographic reconstruction method—I. Theory and error estimates,” IEEE Trans. Biomed. Eng. BME-27, 640–648 (1980).
[CrossRef]

Polyak, B. T.

L. G. Gubin, B. T. Polyak, E. V. Raik, “The method of projections for finding a common point of convex sets,” USSR Comput. Math. Math. Phys. 7 (No. 6), 1–24 (1967).
[CrossRef]

Raik, E. V.

L. G. Gubin, B. T. Polyak, E. V. Raik, “The method of projections for finding a common point of convex sets,” USSR Comput. Math. Math. Phys. 7 (No. 6), 1–24 (1967).
[CrossRef]

Ready, P. J.

P. J. Ready, P. A. Wintz, “Information extraction, SNR improvement, and data compression in multispectral imagery,” IEEE Trans. Commun. COM-21, 1123–1130 (1973).
[CrossRef]

Reed, I. S.

J. S. Goldstein, I. S. Reed, “Reduced-rank adaptive filtering,” IEEE Trans. Signal Process. 45, 492–496 (1997).
[CrossRef]

Sadasivan, P. K.

P. K. Sadasivan, D. N. Dutt, “SVD based technique for noise reduction in electroencephalographic signals,” Signal Process. 55, 179–189 (1996).
[CrossRef]

Sano, A.

A. Sano, “Optimally regularized inverse of singular value decomposition and application to signal extrapolation,” Signal Process. 30, 163–176 (1993).
[CrossRef]

Scharf, L. L.

L. L. Scharf, “The SVD and reduced-rank signal processing,” in SVD and Signal Processing II: Algorithms, Analysis and Applications, R. Vaccaro, ed. (Elsevier, Amsterdam, 1991), pp. 3–31.

Sezan, M. I.

M. I. Sezan, H. Stark, “Image restoration by the method of convex projections: Part 2—applications and numerical results,” IEEE Trans. Med. Imaging MI-1 (No. 2), 95–101 (1982).
[CrossRef]

Shannon, C. E.

C. E. Shannon, “Coding theorems for discrete source with a fidelity criterion,” in Institute of Radio Engineers National Convention Record (Institute of Radio Engineers, New York, 1959), Part 4, pp. 142–163.

Solmon, D. C.

D. C. Solmon, “The x-ray transform,” J. Math. Anal. Appl. 56, 61–83 (1976).
[CrossRef]

Stark, H.

Switzer, P.

A. A. Green, M. Berman, P. Switzer, M. D. Craig, “A transformation for ordering multispectral data in terms of image quality with implications for noise removal,” IEEE Trans. Geosci. Remote Sens. 26 (No. 1), 65–74 (1988).
[CrossRef]

Tam, K. C.

Van Loan, C.

G. Golub, C. Van Loan, Matrix Computations (Johns Hopkins U. Press, Baltimore, Md., 1993).

Vickers, V. E.

Webb, H.

D. C. Youla, H. Webb, “Image restoration by the method of convex projections: Part 1—theory,” IEEE Trans. Med. Imaging MI-1 (No. 2), 81–94 (1982).
[CrossRef]

Wintz, P. A.

P. J. Ready, P. A. Wintz, “Information extraction, SNR improvement, and data compression in multispectral imagery,” IEEE Trans. Commun. COM-21, 1123–1130 (1973).
[CrossRef]

Woodyatt, A. S.

J. B. Lee, A. S. Woodyatt, M. Berman, “Enhancement of high spectral resolution remote-sensing data by a noise-adjusted principal component transform,” IEEE Trans. Geosci. Remote Sens. 28 (No. 3), 295–304 (1990).
[CrossRef]

Youla, D. C.

D. C. Youla, H. Webb, “Image restoration by the method of convex projections: Part 1—theory,” IEEE Trans. Med. Imaging MI-1 (No. 2), 81–94 (1982).
[CrossRef]

Yovanof, G. S.

K. Konstantinides, B. Natarajan, G. S. Yovanof, “Noise estimation and filtering using block-based singular value decomposition,” IEEE Trans. Image Process. 6 (No. 3), 479–483 (1997).
[CrossRef] [PubMed]

Dokl. Akad. Nauk SSSR (1)

L. M. Bregman, “The method of successive projections for finding a common point of convex sets,” Dokl. Akad. Nauk SSSR 162 (No. 3), 487–490 (1965).

IEEE Signal Process. Mag. (1)

H. Hiriyannaiai, “X-ray computed tomography for medical imaging,” IEEE Signal Process. Mag. 14 (No. 2), 42–59 (1997).
[CrossRef]

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

H. C. Andrews, C. L. Patterson, “Singular value decomposition and digital image processing,” IEEE Trans. Acoust., Speech, Signal Process. ASSP-24, 26–53 (1976).
[CrossRef]

IEEE Trans. Biomed. Eng. (1)

H. Knutsson, P. Edholm, G. Grandlund, C. Petersson, “Ectomography—a new radiographic reconstruction method—I. Theory and error estimates,” IEEE Trans. Biomed. Eng. BME-27, 640–648 (1980).
[CrossRef]

IEEE Trans. Circuits Syst. (1)

A. Papoulis, “A new algorithm in spectral analysis and band-limited extrapolation,” IEEE Trans. Circuits Syst. CAS-22, 735–742 (1975).
[CrossRef]

IEEE Trans. Commun. (1)

P. J. Ready, P. A. Wintz, “Information extraction, SNR improvement, and data compression in multispectral imagery,” IEEE Trans. Commun. COM-21, 1123–1130 (1973).
[CrossRef]

IEEE Trans. Geosci. Remote Sens. (2)

A. A. Green, M. Berman, P. Switzer, M. D. Craig, “A transformation for ordering multispectral data in terms of image quality with implications for noise removal,” IEEE Trans. Geosci. Remote Sens. 26 (No. 1), 65–74 (1988).
[CrossRef]

J. B. Lee, A. S. Woodyatt, M. Berman, “Enhancement of high spectral resolution remote-sensing data by a noise-adjusted principal component transform,” IEEE Trans. Geosci. Remote Sens. 28 (No. 3), 295–304 (1990).
[CrossRef]

IEEE Trans. Image Process. (1)

K. Konstantinides, B. Natarajan, G. S. Yovanof, “Noise estimation and filtering using block-based singular value decomposition,” IEEE Trans. Image Process. 6 (No. 3), 479–483 (1997).
[CrossRef] [PubMed]

IEEE Trans. Med. Imaging (2)

M. I. Sezan, H. Stark, “Image restoration by the method of convex projections: Part 2—applications and numerical results,” IEEE Trans. Med. Imaging MI-1 (No. 2), 95–101 (1982).
[CrossRef]

D. C. Youla, H. Webb, “Image restoration by the method of convex projections: Part 1—theory,” IEEE Trans. Med. Imaging MI-1 (No. 2), 81–94 (1982).
[CrossRef]

IEEE Trans. Signal Process. (2)

J. S. Goldstein, I. S. Reed, “Reduced-rank adaptive filtering,” IEEE Trans. Signal Process. 45, 492–496 (1997).
[CrossRef]

S. Heidari, C. L. Nikias, “Co-channel interference mitigation in the time-scale domain: the CIMTS algorithm,” IEEE Trans. Signal Process. 44, 2151–2162 (1996).
[CrossRef]

J. Math. Anal. Appl. (1)

D. C. Solmon, “The x-ray transform,” J. Math. Anal. Appl. 56, 61–83 (1976).
[CrossRef]

J. Nucl. Med. (1)

H. H. Barrett, “Limited-angle tomography for the nineties,” J. Nucl. Med. 31, 1689–1692 (1990).

J. Opt. Soc. Am. (2)

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

Opt. Acta (1)

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Figures (11)

Fig. 1
Fig. 1

Schematic representation of the chromotomographic imager. The direct vision prism is shown spreading red, green, and blue light across the focal-plane array (FPA).

Fig. 2
Fig. 2

Chromotomographic projections of the hyperspectral data cube of Hanscom Air Force Base, Mass., registered by FPA for prism rotation angles ϕ=0, π/3, 2π/3, π, 4π/3, and 5π/3. The projections are ordered from middle left, clockwise. The center image corresponds to the sum of all monochromatic slices, as seen by the spectrometer.

Fig. 3
Fig. 3

Geometry of chromotomographic data collection and its relation to the x-ray transform.

Fig. 4
Fig. 4

Two-dimensional Fourier transform planes corresponding to the six chromotomographic projections shown in Fig. 2.

Fig. 5
Fig. 5

The set CA is convex, and the set CB is not convex. A convex set must contain every line segment with end points in the set.

Fig. 6
Fig. 6

POCS algorithm utilizing two convex sets CA and CB. A sequence of images f0, f1=PBPAf0, f2=PBPAf1,, fk=PBPAfk-1 converges to an element f in CACB.

Fig. 7
Fig. 7

Hyperspectral image of Moffett Field, Calif., taken with the AVIRIS instrument (courtesy of the Jet Propulsion Laboratory, Pasadena, Calif.).

Fig. 8
Fig. 8

Singular-value spectrum of Jasper Ridge, Calif.

Fig. 9
Fig. 9

First five eigenimages of Jasper Ridge: the original AVIRIS sequence (first column), the pseudoinverse reconstruction (second column), and the 20th iteration (third column).

Fig. 10
Fig. 10

First five chromatic singular vectors of Jasper Ridge: the original AVIRIS sequence (solid curves), the pseudoinverse reconstruction (dotted curves), and the 20th iteration (dashed curves).

Fig. 11
Fig. 11

First six eigenimages of Hanscom: the pseudoinverse (upper half) and the 20th iteration (lower half).

Equations (43)

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g(x¯, ϕ)=-+f(x¯-k(λ-λ0)p¯ϕ, λ)dλ,
g(ξ¯, ϕ)=-+ exp[-2πikp¯ϕ, ξ¯(λ-λ0)]f(ξ¯, λ)dλ,
g(ξ¯, ζϕ)=Fλ{f(ξ¯, λ-λ0)}.
gm(ξ¯)=n=0N-1 exp[-2πip¯m, ξ¯(n-n0)]fn(ξ¯),
g0(ξ¯)g1(ξ¯)...gM-1(ξ¯)=A(ξ¯)f0(ξ¯)f1(ξ¯)...fN-1(ξ¯),
Am,n(ξ¯)=exp[-2πip¯m, ξ¯(n-n0)].
g=Af.
A=UΣVH,
UHU=VVH=VHV=I,
Σ=diag(σ0, σ1,, σN-1),
A-1=VΣ-1UH,
Σ-1=diag(σ0-1, σ1-1,, σN-1-1).
f=A-1g.
Σ=ΣK=diag(σ0,, σK-1, 0,, 0),
A+=VΣ+UH,
Σ+=diag(σ0-1,, σK-1-1, 0,, 0).
f+=A+g.
g=Af+n.
A+g=A+Af+A+n=VΣ+(ΣVHf+UHn).
C0=r=1RCr.
f-Prf=minf-hoverallhCr,
fk+1=PRPR-1 P1fk.
limkfk, f=f, f.
fk+1=TRTR-1  T1fk,
f=f++fN.
fk+1=PAf+P¯APffk,
fk+1=f++P¯APffk
=PAPffk,
A+A=VΣK+ΣKVH=VIKVH,
fk+1=VIKVHf+(I-VIKVH)Pffk.
Σ+UHg=IKVHf,
y=IKx.
f+=Vy=VIKVHf
F=(f0-f¯0, f1-f¯1,, fN-1-f¯N-1)T,
fn=fn(x=x1+X1x2),
0x1<X1,0x2<X2,
F=UΣVT=n=0N-1σnu¯nv¯nT,
F=FL+FL,
FL=PfF.
fk+1=PAPffk=f0+P¯APffk.
RFF=FFT=UΛUT,
 compute f0=A+g,store VNK,}PAcompute the covariance matrix RF0F0,find U and Σ of F througheigendecomposition of RF0F0.}Pf
2pNK¯+2pNL,

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