Abstract

A method is proposed for superresolving multichannel data with application to video sequences. Based on a generalization of Papoulis’ sampling theorem, nonuniform samples of multiple channels are merged to generate high-resolution data. To overcome sampling ill posedness in the presence of noise, image frames are projected from standard orthonormal bases onto optimal Riesz bases defined by channel point spread functions (PSF’s). The method is therefore designed to perform under practical conditions of noise and other degradations. Unlike existing methods, where empirical models such as Gaussian, sinc, etc., are commonly used for characterizing channel PSF’s, the PSF’s are assumed unknown and possibly different and hence are blindly estimated from the observed data. The estimated PSF’s are then used to construct biorthogonal projection filters for the superresolution algorithm. This approach gives rise to a closed-form solution leading to a high-speed algorithm. The method has been tested and verified on PREDATOR video sequences (PREDATOR data are airborne video sequences of the Defense Advanced Research Projects Agency obtained by unmanned aircrafts).

© 1999 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. Y. Yacoob, L. Davis, R. Chellappa, Q. Zheng, “Visual surveillance and monitoring of human and vehicular activity,” in Proceedings of the Image Understanding Workshop (Omnipress, Madison, Wis., 1997), Vol. I, pp. 19–23.
  2. S. Srinivasan, R. Chellapa, “An integrated approach to image stabilization, mosaicking and super-resolution,” in Proceedings of the Image Understanding Workshop (Omnipress, Madison, Wis., 1997), Vol. I, pp. 247–253.
  3. J. L. Harris, “Diffraction and resolving power,” J. Opt. Soc. Am. 54, 931–936 (1964).
    [CrossRef]
  4. D. Slepian, H. O. Pollak, “Prolate spheroidal wave functions, Fourier analysis and uncertainty. I,” Bell Syst. Tech. J. 40, 43–63 (1961).
    [CrossRef]
  5. D. Slepian, “Prolate spheroidal wave functions, Fourier analysis and uncertainty. IV: Extension to many dimensions; generalized prolate spheroidal wave functions,” Bell Syst. Tech. J. 43, 3009–3057 (1964).
    [CrossRef]
  6. D. Slepian, “Prolate spheroidal wave functions, Fourier analysis and uncertainty. V: The discrete case,” Bell Syst. Tech. J. 57, 1371–1430 (1978).
    [CrossRef]
  7. R. W. Gerchberg, “Super-resolution through error energy reduction,” Opt. Acta 21, 709–720 (1974).
    [CrossRef]
  8. A. Papoulis, “A new algorithm in spectral analysis and bandlimited signal extrapolation,” IEEE Trans. Circuits Syst. CS-22, 735–742 (1975).
    [CrossRef]
  9. B. R. Hunt, P. Sementilli, “Description of Poisson imagery super resolution algorithm,” in Astronomical Data Analysis Software and Systems, C. Biemesderfer, D. Worral, J. Barnes, eds. (Astronomical Society of the Pacific, San Francisco, Calif., 1992), pp. 196–199.
  10. R. Tsai, T. Huang, “Multiframe image restoration and registration,” in Advances in Computer Vision and Image Processing (JAI, Greenwich, Conn., 1984), Vol. 1.
  11. S. P. Kim, N. K. Bose, H. M. Valenzuela, “Recursive reconstruction of high resolution image from noisy under-sampled multiframes,” IEEE Trans. Acoust., Speech, Signal Process. 38, 1013–1027 (1990).
    [CrossRef]
  12. S. P. Kim, W. Y. Su, “Recursive high-resolution reconstruction of blurred multiframe images,” IEEE Trans. Image Process. 2, 534–539 (1993).
    [CrossRef] [PubMed]
  13. M. Irani, S. Peleg, “Improving resolution by image registration,” Graphic. Models Image Process. 53, 231–239 (1991).
    [CrossRef]
  14. H. Ur, D. Gross, “Improved resolution from subpixel shifted pictures,” Comput. Vis. Graph. Image Process. 54, 181–186 (1991).
  15. K. Aizawa, T. Komatsu, T. Saito, “A scheme for acquiring very high resolution images using multiple cameras,” in Proceedings of the International Conference on Acoustics, Speech and Signal Processing (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1992), Vol. 3, pp. 289–292.
  16. A. M. Tekalp, M. K. Ozkan, M. I. Sezan, “High resolution image reconstruction from low resolution image sequences and space varying image restoration,” in Proceedings of the International Conference on Acoustics, Speech and Signal Processing (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1992), pp. 169–172.
  17. T. Numnonda, M. Andrews, R. Kakarala, “High resolution image reconstruction by simulated annealing,” in Proceedings of Image & Vision Computing (Industrial Research, Ltd., Aukland, N.Z., 1993), 213–220.
  18. M. Berthod, H. Shekarforoush, M. Werman, J. Zerubia, “Reconstruction of high resolution 3D visual information,” in Proceedings of Computer Vision and Pattern Recognition (IEEE Computer Society Press, Los Alamitos, Calif., 1994), pp. 654–657.
  19. H. Shekarforoush, M. Berthod, J. Zerubia, “3D super-resolution using generalized sampling expansion,” in Proceedings of the International Conference on Image Processing (IEEE Computer Society Press, Los Alamitos, Calif., 1995), pp. 300–303.
  20. H. Shekarforoush, M. Berthod, J. Zerubia, M. Werman, “Sub-pixel Bayesian estimation of albedo and height,” Int. J. Comput. Vis. 19, 289–300 (1996).
    [CrossRef]
  21. M. C. Chiang, T. E. Boult, “Local blur estimation and super-resolution,” in Proceedings of Computer Vision and Pattern Recognition (IEEE Computer Society Press, Los Alamitos, Calif., 1997), pp. 821–826.
  22. A. J. Patti, M. I. Sezan, A. M. Tekalp, “High resolution standards conversion of low resolution video,” in Proceedings of the International Conference on Acoustics, Speech and Signal Processing (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1995), pp. 2197–2200.
  23. A. J. Patti, M. I. Sezan, A. M. Tekalp, “Superresolution video reconstruction with arbitrary sampling lattices and nonzero aperture time,” IEEE Trans. Image Process. 6, 1064–1076 (1997).
    [CrossRef] [PubMed]
  24. M. Elad, A. Feuer, “Restoration of a single superresolution image from several blurred, noisy, and undersampled measured images,” IEEE Trans. Image Process. 6, 1646–1658 (1997).
    [CrossRef] [PubMed]
  25. C. A. Bernstein, E. V. Patrick, “Exact deconvolution or multiple operators: an overview plus performance characterizations for imaging sensors,” in Proceedings of Multidimensional Signal Processing (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1990), pp. 723–734.
  26. A. Lorette, H. Shekarforoush, J. Zerubia, “Super-resolution with adaptive regularization,” in Proceedings of the International Conference on Image Processing (IEEE Computer Society Press, Los Alamitos, Calif., 1997), pp. 169–172.
  27. A. Papoulis, “Generalized sampling expansion,” IEEE Trans. Circuits Syst. CS-24, 652–654 (1977).
    [CrossRef]
  28. R. E. A. C. Paley, N. Wiener, Fourier Transforms in the Complex Domain, Vol. 19 of Colloquium Publications (American Mathematical Society, New York, 1934).
  29. H. Shekarforoush, “Super-resolution in computer vision,” Ph.D. thesis (University of Nice, Institut National de la Recherche en Informatique et en Automatique, Sophia Antipolis, France, 1996).
  30. E. De-Castro, C. Morandi, “Registration of translated and rotated images using finite Fourier transforms,” IEEE Trans. Pattern. Anal. Mach. Intell. PAMI-9, 700–703 (1987).
    [CrossRef]
  31. H. Shekarforoush, M. Berthod, J. Zerubia, “Subpixel image registration by estimating the polyphase decomposition of cross power spectrum,” in Proceedings of Computer Vision and Pattern Recognition (IEEE Computer Society Press, Los Alamitos, Calif., 1996), pp. 532–537.
  32. G. B. Giannakis, R. W. Heath, “Blurs and perfect image restoration,” in Proceedings of the International Conference on Image Processing (IEEE Computer Society Press, Los Alamitos, Calif., 1997), Vol. 1, pp. 717–720.
  33. R. W. Heath, S. D. Halford, G. B. Giannakis, “Adaptive blind channel identification of FIR channels for Viterbi decoding,” in Proceedings of the 30th Asilomar Conference on Signals, Systems and Computers (IEEE Computer Society Press, Los Alamitos, Calif., 1996), Vol. 1, pp. 320–324.
  34. M. S. Scivier, M. A. Fiddy, “Phase ambiguities and the zeros of multidimensional band-limited functions,” J. Opt. Soc. Am. A 2, 693–697 (1985).
    [CrossRef]
  35. A. Papoulis, Signal Analysis (McGraw-Hill, New York, 1977).
  36. R. R. Coifman, M. V. Wickerhauser, “Entropy-based algorithms for best basis selection,” IEEE Trans. Inf. Theory 38, 713–718 (1992).
    [CrossRef]
  37. S. G. Mallat, Z. Zhang, “Matching pursuits with time-frequency dictionaries” IEEE Trans. Signal Process. 41, 3397–2415 (1993).
    [CrossRef]
  38. S. Chen, D. L. Donoho, “Basis pursuit,” in Proceedings of the 28th Asilomar Conference on Signals, Systems, and Computers (IEEE Computer Society Press, Los Alamitos, Calif., 1994), pp. 41–44.
  39. H. Krim, S. G. Mallat, D. L. Donoho, A. Willsky, “Best basis algorithm for signal enhancement,” in Proceedings of the International Conference on Acoustics, Speech and Signal Processing (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1995), pp. 1561–1564.
  40. R. M. Young, An Introduction to Nonharmonic Fourier Series (Academic, New York, 1980).
  41. F. Riesz, B. Sz.-Nagy, Functional Analysis (Dover, New York, 1990).
  42. A. Gray, T. M. Mathews, A Treatise on Bessel Functions and Their Applications to Physics (Dover, New York, 1966).

1997

A. J. Patti, M. I. Sezan, A. M. Tekalp, “Superresolution video reconstruction with arbitrary sampling lattices and nonzero aperture time,” IEEE Trans. Image Process. 6, 1064–1076 (1997).
[CrossRef] [PubMed]

M. Elad, A. Feuer, “Restoration of a single superresolution image from several blurred, noisy, and undersampled measured images,” IEEE Trans. Image Process. 6, 1646–1658 (1997).
[CrossRef] [PubMed]

1996

H. Shekarforoush, M. Berthod, J. Zerubia, M. Werman, “Sub-pixel Bayesian estimation of albedo and height,” Int. J. Comput. Vis. 19, 289–300 (1996).
[CrossRef]

1993

S. G. Mallat, Z. Zhang, “Matching pursuits with time-frequency dictionaries” IEEE Trans. Signal Process. 41, 3397–2415 (1993).
[CrossRef]

S. P. Kim, W. Y. Su, “Recursive high-resolution reconstruction of blurred multiframe images,” IEEE Trans. Image Process. 2, 534–539 (1993).
[CrossRef] [PubMed]

1992

R. R. Coifman, M. V. Wickerhauser, “Entropy-based algorithms for best basis selection,” IEEE Trans. Inf. Theory 38, 713–718 (1992).
[CrossRef]

1991

M. Irani, S. Peleg, “Improving resolution by image registration,” Graphic. Models Image Process. 53, 231–239 (1991).
[CrossRef]

H. Ur, D. Gross, “Improved resolution from subpixel shifted pictures,” Comput. Vis. Graph. Image Process. 54, 181–186 (1991).

1990

S. P. Kim, N. K. Bose, H. M. Valenzuela, “Recursive reconstruction of high resolution image from noisy under-sampled multiframes,” IEEE Trans. Acoust., Speech, Signal Process. 38, 1013–1027 (1990).
[CrossRef]

1987

E. De-Castro, C. Morandi, “Registration of translated and rotated images using finite Fourier transforms,” IEEE Trans. Pattern. Anal. Mach. Intell. PAMI-9, 700–703 (1987).
[CrossRef]

1985

1978

D. Slepian, “Prolate spheroidal wave functions, Fourier analysis and uncertainty. V: The discrete case,” Bell Syst. Tech. J. 57, 1371–1430 (1978).
[CrossRef]

1977

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

1975

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

1974

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

1964

D. Slepian, “Prolate spheroidal wave functions, Fourier analysis and uncertainty. IV: Extension to many dimensions; generalized prolate spheroidal wave functions,” Bell Syst. Tech. J. 43, 3009–3057 (1964).
[CrossRef]

J. L. Harris, “Diffraction and resolving power,” J. Opt. Soc. Am. 54, 931–936 (1964).
[CrossRef]

1961

D. Slepian, H. O. Pollak, “Prolate spheroidal wave functions, Fourier analysis and uncertainty. I,” Bell Syst. Tech. J. 40, 43–63 (1961).
[CrossRef]

Aizawa, K.

K. Aizawa, T. Komatsu, T. Saito, “A scheme for acquiring very high resolution images using multiple cameras,” in Proceedings of the International Conference on Acoustics, Speech and Signal Processing (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1992), Vol. 3, pp. 289–292.

Andrews, M.

T. Numnonda, M. Andrews, R. Kakarala, “High resolution image reconstruction by simulated annealing,” in Proceedings of Image & Vision Computing (Industrial Research, Ltd., Aukland, N.Z., 1993), 213–220.

Bernstein, C. A.

C. A. Bernstein, E. V. Patrick, “Exact deconvolution or multiple operators: an overview plus performance characterizations for imaging sensors,” in Proceedings of Multidimensional Signal Processing (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1990), pp. 723–734.

Berthod, M.

H. Shekarforoush, M. Berthod, J. Zerubia, M. Werman, “Sub-pixel Bayesian estimation of albedo and height,” Int. J. Comput. Vis. 19, 289–300 (1996).
[CrossRef]

H. Shekarforoush, M. Berthod, J. Zerubia, “Subpixel image registration by estimating the polyphase decomposition of cross power spectrum,” in Proceedings of Computer Vision and Pattern Recognition (IEEE Computer Society Press, Los Alamitos, Calif., 1996), pp. 532–537.

H. Shekarforoush, M. Berthod, J. Zerubia, “3D super-resolution using generalized sampling expansion,” in Proceedings of the International Conference on Image Processing (IEEE Computer Society Press, Los Alamitos, Calif., 1995), pp. 300–303.

M. Berthod, H. Shekarforoush, M. Werman, J. Zerubia, “Reconstruction of high resolution 3D visual information,” in Proceedings of Computer Vision and Pattern Recognition (IEEE Computer Society Press, Los Alamitos, Calif., 1994), pp. 654–657.

Bose, N. K.

S. P. Kim, N. K. Bose, H. M. Valenzuela, “Recursive reconstruction of high resolution image from noisy under-sampled multiframes,” IEEE Trans. Acoust., Speech, Signal Process. 38, 1013–1027 (1990).
[CrossRef]

Boult, T. E.

M. C. Chiang, T. E. Boult, “Local blur estimation and super-resolution,” in Proceedings of Computer Vision and Pattern Recognition (IEEE Computer Society Press, Los Alamitos, Calif., 1997), pp. 821–826.

Chellapa, R.

S. Srinivasan, R. Chellapa, “An integrated approach to image stabilization, mosaicking and super-resolution,” in Proceedings of the Image Understanding Workshop (Omnipress, Madison, Wis., 1997), Vol. I, pp. 247–253.

Chellappa, R.

Y. Yacoob, L. Davis, R. Chellappa, Q. Zheng, “Visual surveillance and monitoring of human and vehicular activity,” in Proceedings of the Image Understanding Workshop (Omnipress, Madison, Wis., 1997), Vol. I, pp. 19–23.

Chen, S.

S. Chen, D. L. Donoho, “Basis pursuit,” in Proceedings of the 28th Asilomar Conference on Signals, Systems, and Computers (IEEE Computer Society Press, Los Alamitos, Calif., 1994), pp. 41–44.

Chiang, M. C.

M. C. Chiang, T. E. Boult, “Local blur estimation and super-resolution,” in Proceedings of Computer Vision and Pattern Recognition (IEEE Computer Society Press, Los Alamitos, Calif., 1997), pp. 821–826.

Coifman, R. R.

R. R. Coifman, M. V. Wickerhauser, “Entropy-based algorithms for best basis selection,” IEEE Trans. Inf. Theory 38, 713–718 (1992).
[CrossRef]

Davis, L.

Y. Yacoob, L. Davis, R. Chellappa, Q. Zheng, “Visual surveillance and monitoring of human and vehicular activity,” in Proceedings of the Image Understanding Workshop (Omnipress, Madison, Wis., 1997), Vol. I, pp. 19–23.

De-Castro, E.

E. De-Castro, C. Morandi, “Registration of translated and rotated images using finite Fourier transforms,” IEEE Trans. Pattern. Anal. Mach. Intell. PAMI-9, 700–703 (1987).
[CrossRef]

Donoho, D. L.

H. Krim, S. G. Mallat, D. L. Donoho, A. Willsky, “Best basis algorithm for signal enhancement,” in Proceedings of the International Conference on Acoustics, Speech and Signal Processing (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1995), pp. 1561–1564.

S. Chen, D. L. Donoho, “Basis pursuit,” in Proceedings of the 28th Asilomar Conference on Signals, Systems, and Computers (IEEE Computer Society Press, Los Alamitos, Calif., 1994), pp. 41–44.

Elad, M.

M. Elad, A. Feuer, “Restoration of a single superresolution image from several blurred, noisy, and undersampled measured images,” IEEE Trans. Image Process. 6, 1646–1658 (1997).
[CrossRef] [PubMed]

Feuer, A.

M. Elad, A. Feuer, “Restoration of a single superresolution image from several blurred, noisy, and undersampled measured images,” IEEE Trans. Image Process. 6, 1646–1658 (1997).
[CrossRef] [PubMed]

Fiddy, M. A.

Gerchberg, R. W.

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

Giannakis, G. B.

G. B. Giannakis, R. W. Heath, “Blurs and perfect image restoration,” in Proceedings of the International Conference on Image Processing (IEEE Computer Society Press, Los Alamitos, Calif., 1997), Vol. 1, pp. 717–720.

R. W. Heath, S. D. Halford, G. B. Giannakis, “Adaptive blind channel identification of FIR channels for Viterbi decoding,” in Proceedings of the 30th Asilomar Conference on Signals, Systems and Computers (IEEE Computer Society Press, Los Alamitos, Calif., 1996), Vol. 1, pp. 320–324.

Gray, A.

A. Gray, T. M. Mathews, A Treatise on Bessel Functions and Their Applications to Physics (Dover, New York, 1966).

Gross, D.

H. Ur, D. Gross, “Improved resolution from subpixel shifted pictures,” Comput. Vis. Graph. Image Process. 54, 181–186 (1991).

Halford, S. D.

R. W. Heath, S. D. Halford, G. B. Giannakis, “Adaptive blind channel identification of FIR channels for Viterbi decoding,” in Proceedings of the 30th Asilomar Conference on Signals, Systems and Computers (IEEE Computer Society Press, Los Alamitos, Calif., 1996), Vol. 1, pp. 320–324.

Harris, J. L.

Heath, R. W.

R. W. Heath, S. D. Halford, G. B. Giannakis, “Adaptive blind channel identification of FIR channels for Viterbi decoding,” in Proceedings of the 30th Asilomar Conference on Signals, Systems and Computers (IEEE Computer Society Press, Los Alamitos, Calif., 1996), Vol. 1, pp. 320–324.

G. B. Giannakis, R. W. Heath, “Blurs and perfect image restoration,” in Proceedings of the International Conference on Image Processing (IEEE Computer Society Press, Los Alamitos, Calif., 1997), Vol. 1, pp. 717–720.

Huang, T.

R. Tsai, T. Huang, “Multiframe image restoration and registration,” in Advances in Computer Vision and Image Processing (JAI, Greenwich, Conn., 1984), Vol. 1.

Hunt, B. R.

B. R. Hunt, P. Sementilli, “Description of Poisson imagery super resolution algorithm,” in Astronomical Data Analysis Software and Systems, C. Biemesderfer, D. Worral, J. Barnes, eds. (Astronomical Society of the Pacific, San Francisco, Calif., 1992), pp. 196–199.

Irani, M.

M. Irani, S. Peleg, “Improving resolution by image registration,” Graphic. Models Image Process. 53, 231–239 (1991).
[CrossRef]

Kakarala, R.

T. Numnonda, M. Andrews, R. Kakarala, “High resolution image reconstruction by simulated annealing,” in Proceedings of Image & Vision Computing (Industrial Research, Ltd., Aukland, N.Z., 1993), 213–220.

Kim, S. P.

S. P. Kim, W. Y. Su, “Recursive high-resolution reconstruction of blurred multiframe images,” IEEE Trans. Image Process. 2, 534–539 (1993).
[CrossRef] [PubMed]

S. P. Kim, N. K. Bose, H. M. Valenzuela, “Recursive reconstruction of high resolution image from noisy under-sampled multiframes,” IEEE Trans. Acoust., Speech, Signal Process. 38, 1013–1027 (1990).
[CrossRef]

Komatsu, T.

K. Aizawa, T. Komatsu, T. Saito, “A scheme for acquiring very high resolution images using multiple cameras,” in Proceedings of the International Conference on Acoustics, Speech and Signal Processing (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1992), Vol. 3, pp. 289–292.

Krim, H.

H. Krim, S. G. Mallat, D. L. Donoho, A. Willsky, “Best basis algorithm for signal enhancement,” in Proceedings of the International Conference on Acoustics, Speech and Signal Processing (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1995), pp. 1561–1564.

Lorette, A.

A. Lorette, H. Shekarforoush, J. Zerubia, “Super-resolution with adaptive regularization,” in Proceedings of the International Conference on Image Processing (IEEE Computer Society Press, Los Alamitos, Calif., 1997), pp. 169–172.

Mallat, S. G.

S. G. Mallat, Z. Zhang, “Matching pursuits with time-frequency dictionaries” IEEE Trans. Signal Process. 41, 3397–2415 (1993).
[CrossRef]

H. Krim, S. G. Mallat, D. L. Donoho, A. Willsky, “Best basis algorithm for signal enhancement,” in Proceedings of the International Conference on Acoustics, Speech and Signal Processing (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1995), pp. 1561–1564.

Mathews, T. M.

A. Gray, T. M. Mathews, A Treatise on Bessel Functions and Their Applications to Physics (Dover, New York, 1966).

Morandi, C.

E. De-Castro, C. Morandi, “Registration of translated and rotated images using finite Fourier transforms,” IEEE Trans. Pattern. Anal. Mach. Intell. PAMI-9, 700–703 (1987).
[CrossRef]

Numnonda, T.

T. Numnonda, M. Andrews, R. Kakarala, “High resolution image reconstruction by simulated annealing,” in Proceedings of Image & Vision Computing (Industrial Research, Ltd., Aukland, N.Z., 1993), 213–220.

Ozkan, M. K.

A. M. Tekalp, M. K. Ozkan, M. I. Sezan, “High resolution image reconstruction from low resolution image sequences and space varying image restoration,” in Proceedings of the International Conference on Acoustics, Speech and Signal Processing (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1992), pp. 169–172.

Paley, R. E. A. C.

R. E. A. C. Paley, N. Wiener, Fourier Transforms in the Complex Domain, Vol. 19 of Colloquium Publications (American Mathematical Society, New York, 1934).

Papoulis, A.

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

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

A. Papoulis, Signal Analysis (McGraw-Hill, New York, 1977).

Patrick, E. V.

C. A. Bernstein, E. V. Patrick, “Exact deconvolution or multiple operators: an overview plus performance characterizations for imaging sensors,” in Proceedings of Multidimensional Signal Processing (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1990), pp. 723–734.

Patti, A. J.

A. J. Patti, M. I. Sezan, A. M. Tekalp, “Superresolution video reconstruction with arbitrary sampling lattices and nonzero aperture time,” IEEE Trans. Image Process. 6, 1064–1076 (1997).
[CrossRef] [PubMed]

A. J. Patti, M. I. Sezan, A. M. Tekalp, “High resolution standards conversion of low resolution video,” in Proceedings of the International Conference on Acoustics, Speech and Signal Processing (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1995), pp. 2197–2200.

Peleg, S.

M. Irani, S. Peleg, “Improving resolution by image registration,” Graphic. Models Image Process. 53, 231–239 (1991).
[CrossRef]

Pollak, H. O.

D. Slepian, H. O. Pollak, “Prolate spheroidal wave functions, Fourier analysis and uncertainty. I,” Bell Syst. Tech. J. 40, 43–63 (1961).
[CrossRef]

Riesz, F.

F. Riesz, B. Sz.-Nagy, Functional Analysis (Dover, New York, 1990).

Saito, T.

K. Aizawa, T. Komatsu, T. Saito, “A scheme for acquiring very high resolution images using multiple cameras,” in Proceedings of the International Conference on Acoustics, Speech and Signal Processing (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1992), Vol. 3, pp. 289–292.

Scivier, M. S.

Sementilli, P.

B. R. Hunt, P. Sementilli, “Description of Poisson imagery super resolution algorithm,” in Astronomical Data Analysis Software and Systems, C. Biemesderfer, D. Worral, J. Barnes, eds. (Astronomical Society of the Pacific, San Francisco, Calif., 1992), pp. 196–199.

Sezan, M. I.

A. J. Patti, M. I. Sezan, A. M. Tekalp, “Superresolution video reconstruction with arbitrary sampling lattices and nonzero aperture time,” IEEE Trans. Image Process. 6, 1064–1076 (1997).
[CrossRef] [PubMed]

A. M. Tekalp, M. K. Ozkan, M. I. Sezan, “High resolution image reconstruction from low resolution image sequences and space varying image restoration,” in Proceedings of the International Conference on Acoustics, Speech and Signal Processing (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1992), pp. 169–172.

A. J. Patti, M. I. Sezan, A. M. Tekalp, “High resolution standards conversion of low resolution video,” in Proceedings of the International Conference on Acoustics, Speech and Signal Processing (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1995), pp. 2197–2200.

Shekarforoush, H.

H. Shekarforoush, M. Berthod, J. Zerubia, M. Werman, “Sub-pixel Bayesian estimation of albedo and height,” Int. J. Comput. Vis. 19, 289–300 (1996).
[CrossRef]

H. Shekarforoush, M. Berthod, J. Zerubia, “Subpixel image registration by estimating the polyphase decomposition of cross power spectrum,” in Proceedings of Computer Vision and Pattern Recognition (IEEE Computer Society Press, Los Alamitos, Calif., 1996), pp. 532–537.

A. Lorette, H. Shekarforoush, J. Zerubia, “Super-resolution with adaptive regularization,” in Proceedings of the International Conference on Image Processing (IEEE Computer Society Press, Los Alamitos, Calif., 1997), pp. 169–172.

H. Shekarforoush, M. Berthod, J. Zerubia, “3D super-resolution using generalized sampling expansion,” in Proceedings of the International Conference on Image Processing (IEEE Computer Society Press, Los Alamitos, Calif., 1995), pp. 300–303.

H. Shekarforoush, “Super-resolution in computer vision,” Ph.D. thesis (University of Nice, Institut National de la Recherche en Informatique et en Automatique, Sophia Antipolis, France, 1996).

M. Berthod, H. Shekarforoush, M. Werman, J. Zerubia, “Reconstruction of high resolution 3D visual information,” in Proceedings of Computer Vision and Pattern Recognition (IEEE Computer Society Press, Los Alamitos, Calif., 1994), pp. 654–657.

Slepian, D.

D. Slepian, “Prolate spheroidal wave functions, Fourier analysis and uncertainty. V: The discrete case,” Bell Syst. Tech. J. 57, 1371–1430 (1978).
[CrossRef]

D. Slepian, “Prolate spheroidal wave functions, Fourier analysis and uncertainty. IV: Extension to many dimensions; generalized prolate spheroidal wave functions,” Bell Syst. Tech. J. 43, 3009–3057 (1964).
[CrossRef]

D. Slepian, H. O. Pollak, “Prolate spheroidal wave functions, Fourier analysis and uncertainty. I,” Bell Syst. Tech. J. 40, 43–63 (1961).
[CrossRef]

Srinivasan, S.

S. Srinivasan, R. Chellapa, “An integrated approach to image stabilization, mosaicking and super-resolution,” in Proceedings of the Image Understanding Workshop (Omnipress, Madison, Wis., 1997), Vol. I, pp. 247–253.

Su, W. Y.

S. P. Kim, W. Y. Su, “Recursive high-resolution reconstruction of blurred multiframe images,” IEEE Trans. Image Process. 2, 534–539 (1993).
[CrossRef] [PubMed]

Sz.-Nagy, B.

F. Riesz, B. Sz.-Nagy, Functional Analysis (Dover, New York, 1990).

Tekalp, A. M.

A. J. Patti, M. I. Sezan, A. M. Tekalp, “Superresolution video reconstruction with arbitrary sampling lattices and nonzero aperture time,” IEEE Trans. Image Process. 6, 1064–1076 (1997).
[CrossRef] [PubMed]

A. J. Patti, M. I. Sezan, A. M. Tekalp, “High resolution standards conversion of low resolution video,” in Proceedings of the International Conference on Acoustics, Speech and Signal Processing (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1995), pp. 2197–2200.

A. M. Tekalp, M. K. Ozkan, M. I. Sezan, “High resolution image reconstruction from low resolution image sequences and space varying image restoration,” in Proceedings of the International Conference on Acoustics, Speech and Signal Processing (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1992), pp. 169–172.

Tsai, R.

R. Tsai, T. Huang, “Multiframe image restoration and registration,” in Advances in Computer Vision and Image Processing (JAI, Greenwich, Conn., 1984), Vol. 1.

Ur, H.

H. Ur, D. Gross, “Improved resolution from subpixel shifted pictures,” Comput. Vis. Graph. Image Process. 54, 181–186 (1991).

Valenzuela, H. M.

S. P. Kim, N. K. Bose, H. M. Valenzuela, “Recursive reconstruction of high resolution image from noisy under-sampled multiframes,” IEEE Trans. Acoust., Speech, Signal Process. 38, 1013–1027 (1990).
[CrossRef]

Werman, M.

H. Shekarforoush, M. Berthod, J. Zerubia, M. Werman, “Sub-pixel Bayesian estimation of albedo and height,” Int. J. Comput. Vis. 19, 289–300 (1996).
[CrossRef]

M. Berthod, H. Shekarforoush, M. Werman, J. Zerubia, “Reconstruction of high resolution 3D visual information,” in Proceedings of Computer Vision and Pattern Recognition (IEEE Computer Society Press, Los Alamitos, Calif., 1994), pp. 654–657.

Wickerhauser, M. V.

R. R. Coifman, M. V. Wickerhauser, “Entropy-based algorithms for best basis selection,” IEEE Trans. Inf. Theory 38, 713–718 (1992).
[CrossRef]

Wiener, N.

R. E. A. C. Paley, N. Wiener, Fourier Transforms in the Complex Domain, Vol. 19 of Colloquium Publications (American Mathematical Society, New York, 1934).

Willsky, A.

H. Krim, S. G. Mallat, D. L. Donoho, A. Willsky, “Best basis algorithm for signal enhancement,” in Proceedings of the International Conference on Acoustics, Speech and Signal Processing (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1995), pp. 1561–1564.

Yacoob, Y.

Y. Yacoob, L. Davis, R. Chellappa, Q. Zheng, “Visual surveillance and monitoring of human and vehicular activity,” in Proceedings of the Image Understanding Workshop (Omnipress, Madison, Wis., 1997), Vol. I, pp. 19–23.

Young, R. M.

R. M. Young, An Introduction to Nonharmonic Fourier Series (Academic, New York, 1980).

Zerubia, J.

H. Shekarforoush, M. Berthod, J. Zerubia, M. Werman, “Sub-pixel Bayesian estimation of albedo and height,” Int. J. Comput. Vis. 19, 289–300 (1996).
[CrossRef]

H. Shekarforoush, M. Berthod, J. Zerubia, “Subpixel image registration by estimating the polyphase decomposition of cross power spectrum,” in Proceedings of Computer Vision and Pattern Recognition (IEEE Computer Society Press, Los Alamitos, Calif., 1996), pp. 532–537.

A. Lorette, H. Shekarforoush, J. Zerubia, “Super-resolution with adaptive regularization,” in Proceedings of the International Conference on Image Processing (IEEE Computer Society Press, Los Alamitos, Calif., 1997), pp. 169–172.

H. Shekarforoush, M. Berthod, J. Zerubia, “3D super-resolution using generalized sampling expansion,” in Proceedings of the International Conference on Image Processing (IEEE Computer Society Press, Los Alamitos, Calif., 1995), pp. 300–303.

M. Berthod, H. Shekarforoush, M. Werman, J. Zerubia, “Reconstruction of high resolution 3D visual information,” in Proceedings of Computer Vision and Pattern Recognition (IEEE Computer Society Press, Los Alamitos, Calif., 1994), pp. 654–657.

Zhang, Z.

S. G. Mallat, Z. Zhang, “Matching pursuits with time-frequency dictionaries” IEEE Trans. Signal Process. 41, 3397–2415 (1993).
[CrossRef]

Zheng, Q.

Y. Yacoob, L. Davis, R. Chellappa, Q. Zheng, “Visual surveillance and monitoring of human and vehicular activity,” in Proceedings of the Image Understanding Workshop (Omnipress, Madison, Wis., 1997), Vol. I, pp. 19–23.

Bell Syst. Tech. J.

D. Slepian, H. O. Pollak, “Prolate spheroidal wave functions, Fourier analysis and uncertainty. I,” Bell Syst. Tech. J. 40, 43–63 (1961).
[CrossRef]

D. Slepian, “Prolate spheroidal wave functions, Fourier analysis and uncertainty. IV: Extension to many dimensions; generalized prolate spheroidal wave functions,” Bell Syst. Tech. J. 43, 3009–3057 (1964).
[CrossRef]

D. Slepian, “Prolate spheroidal wave functions, Fourier analysis and uncertainty. V: The discrete case,” Bell Syst. Tech. J. 57, 1371–1430 (1978).
[CrossRef]

Comput. Vis. Graph. Image Process.

H. Ur, D. Gross, “Improved resolution from subpixel shifted pictures,” Comput. Vis. Graph. Image Process. 54, 181–186 (1991).

Graphic. Models Image Process.

M. Irani, S. Peleg, “Improving resolution by image registration,” Graphic. Models Image Process. 53, 231–239 (1991).
[CrossRef]

IEEE Trans. Acoust., Speech, Signal Process.

S. P. Kim, N. K. Bose, H. M. Valenzuela, “Recursive reconstruction of high resolution image from noisy under-sampled multiframes,” IEEE Trans. Acoust., Speech, Signal Process. 38, 1013–1027 (1990).
[CrossRef]

IEEE Trans. Circuits Syst.

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

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

IEEE Trans. Image Process.

S. P. Kim, W. Y. Su, “Recursive high-resolution reconstruction of blurred multiframe images,” IEEE Trans. Image Process. 2, 534–539 (1993).
[CrossRef] [PubMed]

A. J. Patti, M. I. Sezan, A. M. Tekalp, “Superresolution video reconstruction with arbitrary sampling lattices and nonzero aperture time,” IEEE Trans. Image Process. 6, 1064–1076 (1997).
[CrossRef] [PubMed]

M. Elad, A. Feuer, “Restoration of a single superresolution image from several blurred, noisy, and undersampled measured images,” IEEE Trans. Image Process. 6, 1646–1658 (1997).
[CrossRef] [PubMed]

IEEE Trans. Inf. Theory

R. R. Coifman, M. V. Wickerhauser, “Entropy-based algorithms for best basis selection,” IEEE Trans. Inf. Theory 38, 713–718 (1992).
[CrossRef]

IEEE Trans. Pattern. Anal. Mach. Intell.

E. De-Castro, C. Morandi, “Registration of translated and rotated images using finite Fourier transforms,” IEEE Trans. Pattern. Anal. Mach. Intell. PAMI-9, 700–703 (1987).
[CrossRef]

IEEE Trans. Signal Process.

S. G. Mallat, Z. Zhang, “Matching pursuits with time-frequency dictionaries” IEEE Trans. Signal Process. 41, 3397–2415 (1993).
[CrossRef]

Int. J. Comput. Vis.

H. Shekarforoush, M. Berthod, J. Zerubia, M. Werman, “Sub-pixel Bayesian estimation of albedo and height,” Int. J. Comput. Vis. 19, 289–300 (1996).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Acta

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

Other

B. R. Hunt, P. Sementilli, “Description of Poisson imagery super resolution algorithm,” in Astronomical Data Analysis Software and Systems, C. Biemesderfer, D. Worral, J. Barnes, eds. (Astronomical Society of the Pacific, San Francisco, Calif., 1992), pp. 196–199.

R. Tsai, T. Huang, “Multiframe image restoration and registration,” in Advances in Computer Vision and Image Processing (JAI, Greenwich, Conn., 1984), Vol. 1.

C. A. Bernstein, E. V. Patrick, “Exact deconvolution or multiple operators: an overview plus performance characterizations for imaging sensors,” in Proceedings of Multidimensional Signal Processing (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1990), pp. 723–734.

A. Lorette, H. Shekarforoush, J. Zerubia, “Super-resolution with adaptive regularization,” in Proceedings of the International Conference on Image Processing (IEEE Computer Society Press, Los Alamitos, Calif., 1997), pp. 169–172.

K. Aizawa, T. Komatsu, T. Saito, “A scheme for acquiring very high resolution images using multiple cameras,” in Proceedings of the International Conference on Acoustics, Speech and Signal Processing (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1992), Vol. 3, pp. 289–292.

A. M. Tekalp, M. K. Ozkan, M. I. Sezan, “High resolution image reconstruction from low resolution image sequences and space varying image restoration,” in Proceedings of the International Conference on Acoustics, Speech and Signal Processing (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1992), pp. 169–172.

T. Numnonda, M. Andrews, R. Kakarala, “High resolution image reconstruction by simulated annealing,” in Proceedings of Image & Vision Computing (Industrial Research, Ltd., Aukland, N.Z., 1993), 213–220.

M. Berthod, H. Shekarforoush, M. Werman, J. Zerubia, “Reconstruction of high resolution 3D visual information,” in Proceedings of Computer Vision and Pattern Recognition (IEEE Computer Society Press, Los Alamitos, Calif., 1994), pp. 654–657.

H. Shekarforoush, M. Berthod, J. Zerubia, “3D super-resolution using generalized sampling expansion,” in Proceedings of the International Conference on Image Processing (IEEE Computer Society Press, Los Alamitos, Calif., 1995), pp. 300–303.

M. C. Chiang, T. E. Boult, “Local blur estimation and super-resolution,” in Proceedings of Computer Vision and Pattern Recognition (IEEE Computer Society Press, Los Alamitos, Calif., 1997), pp. 821–826.

A. J. Patti, M. I. Sezan, A. M. Tekalp, “High resolution standards conversion of low resolution video,” in Proceedings of the International Conference on Acoustics, Speech and Signal Processing (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1995), pp. 2197–2200.

R. E. A. C. Paley, N. Wiener, Fourier Transforms in the Complex Domain, Vol. 19 of Colloquium Publications (American Mathematical Society, New York, 1934).

H. Shekarforoush, “Super-resolution in computer vision,” Ph.D. thesis (University of Nice, Institut National de la Recherche en Informatique et en Automatique, Sophia Antipolis, France, 1996).

H. Shekarforoush, M. Berthod, J. Zerubia, “Subpixel image registration by estimating the polyphase decomposition of cross power spectrum,” in Proceedings of Computer Vision and Pattern Recognition (IEEE Computer Society Press, Los Alamitos, Calif., 1996), pp. 532–537.

G. B. Giannakis, R. W. Heath, “Blurs and perfect image restoration,” in Proceedings of the International Conference on Image Processing (IEEE Computer Society Press, Los Alamitos, Calif., 1997), Vol. 1, pp. 717–720.

R. W. Heath, S. D. Halford, G. B. Giannakis, “Adaptive blind channel identification of FIR channels for Viterbi decoding,” in Proceedings of the 30th Asilomar Conference on Signals, Systems and Computers (IEEE Computer Society Press, Los Alamitos, Calif., 1996), Vol. 1, pp. 320–324.

A. Papoulis, Signal Analysis (McGraw-Hill, New York, 1977).

S. Chen, D. L. Donoho, “Basis pursuit,” in Proceedings of the 28th Asilomar Conference on Signals, Systems, and Computers (IEEE Computer Society Press, Los Alamitos, Calif., 1994), pp. 41–44.

H. Krim, S. G. Mallat, D. L. Donoho, A. Willsky, “Best basis algorithm for signal enhancement,” in Proceedings of the International Conference on Acoustics, Speech and Signal Processing (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1995), pp. 1561–1564.

R. M. Young, An Introduction to Nonharmonic Fourier Series (Academic, New York, 1980).

F. Riesz, B. Sz.-Nagy, Functional Analysis (Dover, New York, 1990).

A. Gray, T. M. Mathews, A Treatise on Bessel Functions and Their Applications to Physics (Dover, New York, 1966).

Y. Yacoob, L. Davis, R. Chellappa, Q. Zheng, “Visual surveillance and monitoring of human and vehicular activity,” in Proceedings of the Image Understanding Workshop (Omnipress, Madison, Wis., 1997), Vol. I, pp. 19–23.

S. Srinivasan, R. Chellapa, “An integrated approach to image stabilization, mosaicking and super-resolution,” in Proceedings of the Image Understanding Workshop (Omnipress, Madison, Wis., 1997), Vol. I, pp. 247–253.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (12)

Fig. 1
Fig. 1

Schematic view of the setup.

Fig. 2
Fig. 2

Multichannel interlaced sampling.

Fig. 3
Fig. 3

Typical spike patterns appearing in the magnitude of a cross-power spectrum corresponding to isolated zeros of one of the transfer functions.

Fig. 4
Fig. 4

(a) One of the test images in the sequence, (b) interpolation, (c) superresolved image.

Fig. 5
Fig. 5

(a) One of the test images in the sequence, (b) interpolation, (c) superresolved image.

Fig. 6
Fig. 6

(a) One of the test images in the sequence, (b) interpolation, (c) superresolved image.

Fig. 7
Fig. 7

Region extracted from the sequence in Fig. 4: (a) interpolated/sharpened, (b) superresolved.

Fig. 8
Fig. 8

Region extracted from the sequence in Fig. 4: (a) interpolated/sharpened, (b) superresolved.

Fig. 9
Fig. 9

Region extracted from the sequence in Fig. 4: (a) interpolated/sharpened, (b) superresolved.

Fig. 10
Fig. 10

Region extracted from the sequence in Fig. 5: (a) interpolated/sharpened, (b) superresolved.

Fig. 11
Fig. 11

Region extracted from the sequence in Fig. 5: (a) interpolated/sharpened, (b) superresolved.

Fig. 12
Fig. 12

Region extracted from the sequence in Fig. 6: (a) interpolated/sharpened, (b) superresolved.

Equations (49)

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

m=1M{Smk}kn=U(P(B)).
hˆm(u)φˆm(u)=1B.
f(t)=1Mm=1Mkncmkφm(t-Smk),
f(n)=1Mm=1Mkncmkφm(n-Smk).
{φˆm(u)exp(-iSmk, u)}m=1,,M,
hˆm(u)=kck exp(-iu, k),
hˆm(u)=Ak=0N[1-ak exp(-iu)]
=Ak=0N(1-akz-1),
limbk0 bkhˆk(u, v)=exp(-iωtk),
hˆm(u, v)=khˆmk(u, v),
hˆ(u, v)=kak exp(-iωtk),ak=bk-1.
hˆ(u, v)=Ak(1-ckz-1),
A=1k[1-exp(-iγk)],
gˆ1=hˆ1fˆ1+nˆ1,
gˆ2=hˆ2fˆ2+nˆ2,
gˆ1=hˆg1fˆ1,
gˆ2=hˆg2fˆ2,
|gˆ1gˆ2¯|=c|hˆg1hˆg2¯|,c=|fˆ|.
(1+φˆm)-1hˆm(1+hˆm)-1.
1-φˆm+O(φˆm2)hˆm(1+hˆm)-1
φˆmhˆm¯hˆmhˆm¯+hˆm¯,
{φˆm(u, v)gˆm(u, v)}m=1,,M.
fˆ(u, v)=1Mm=1Mx=0X-1y=0Y-1 exp[-i(xu+yv)]×gˆmxy(Xu, Yv),
gˆmxy(u, v)=φˆm(u, v)gˆm(u, v)×expix-xmXu+y-ymYv
{Smk}kn=U(P(Bm)),m=1,, M.
Φm={φm(t-Smk)}kn,m=1,, M.
Φˆm={φˆm(u)exp(-iSmk,u)}kn.
Pˆ(Bm)Pˆ(Bm), Oeˆmk=φˆm(u)eˆmk
=Φˆmkkn,
φˆm(u)eˆk|φˆm(u)eˆl=φˆm(u)φˆm(u)¯|eˆk¯eˆl=0k, lnandmm.
M(Φˆ)=m=1MM(Φˆm)Pˆ(B),
f(t)=1Mm=1Mkncmkφm(t-Smk),
cmk=f(t)|φm(t-Smk)=hm(t) * f(t)|hm(t) * φm(t-Smk)=gm(t)|δ(t-Smk)
=gm(Smk),
fk(z)=f(z)ifzsupp fk1card(K)f(zk)otherwise,
supp fk={zKn:|z-ζk|bk}.
f-kK1fkf-kKfk,
limcard(K)f-kKfk=0.
u-uk=ω cos φ,v-vk=ω sin φ, x=t cos θ,y=t sin θ,
hk(t)exp(-itρk)=0ωh˜k(ω)J0(tω)dω,
h˜k(ω)=0bkthk(t)exp(-itρk)J0(tω)dt,
0bkthk(t)exp(itρk)dt=0.
0bkt0ωh˜k(ω)J0(tω)dωdt=0,
0ωh˜k(ω)0bktJ0(tω)dtdω=0.
0atJ0(t)dt=tJ1(at),
0bktJ0(tω)dt=bkJ1(bkω)ω.
0bkh˜k(ω)J1(bkω)dω=0.
0Jn(bω)exp(±iωa)=i±(n+1)a2-b2(a-a2-b2)n
limbk0 bkh˜k(ω)=exp(-iωtk),

Metrics