Abstract

Several powerful iterative algorithms are being developed for the restoration and superresolution of diffraction-limited imagery data by use of diverse mathematical techniques. Notwithstanding the mathematical sophistication of the approaches used in their development and the potential for resolution enhancement possible with their implementation, the spectrum extrapolation that is central to superresolution comes in these algorithms only as a by-product and needs to be checked only after the completion of the processing steps to ensure that an expansion of the image bandwidth has indeed occurred. To overcome this limitation, a new approach of mathematically extrapolating the image spectrum and employing it to design constraint sets for implementing set-theoretic estimation procedures is described. Performance evaluation of a specific projection-onto-convex-sets algorithm by using this approach for the restoration and superresolution of degraded images is outlined. The primary goal of the method presented is to expand the power spectrum of the input image beyond the range of the sensor that captured the image.

© 2003 Optical Society of America

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  1. B. R. Hunt, “Super-resolution of images: algorithms, principles, and performance,” Int. J. Imaging Syst. Technol. 6, 297–304 (1995).
    [CrossRef]
  2. J. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1996).
  3. C. K. Rushforth, J. L. Harris, “Restoration, resolution, and noise,” J. Opt. Soc. Am. 58, 539–545 (1968).
    [CrossRef]
  4. D. L. Snyder, M. I. Miller, Random Point Processes in Time and Space, 2nd ed. (Springer-Verlag, New York, 1991).
  5. D. Slepian, H. O. Pollack, “Prolate spheroidal wave functions, Fourier analysis and uncertainty—I,” Bell Syst. Tech. J. 40, 43–62 (1961).
    [CrossRef]
  6. B. R. Frieden, “Band-unlimited reconstruction of optical objects and spectral sources,” J. Opt. Soc. Am. 57, 1013–1019 (1967).
    [CrossRef]
  7. B. R. Frieden, “Restoring with maximum likelihood and maximum entropy,” J. Opt. Soc. Am. 62, 51–55 (1972).
    [CrossRef]
  8. W. H. Richardson, “Bayesian-based iterative method of image restoration,” J. Opt. Soc. Am. 62, 55–60 (1972).
    [CrossRef]
  9. L. B. Lucy, “An iterative technique for the rectification of observed distributions,” Astron. J. 79, 745–759 (1974).
    [CrossRef]
  10. A. P. Dempster, N. M. Laird, D. B. Rubin, “Maximum likelihood from incomplete data via the EM algorithm,” J. R. Stat. Soc. Ser. B. Methodol. 39, 1–38 (1977).
  11. L. Shepp, Y. Vardi, “Maximum likelihood reconstruction for emission tomography,” IEEE Trans. Med. Imaging 1, 113–122 (1982).
    [CrossRef] [PubMed]
  12. B. R. Hunt, “Bayesian methods in digital image restoration,” IEEE Trans. Comput. C-26, 219–229 (1977).
    [CrossRef]
  13. S. Geman, D. Geman, “Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images,” IEEE Trans. Pattern Anal. Mach. Intell. PAMI-6, 721–741 (1984).
    [CrossRef]
  14. R. K. Pina, R. C. Puetter, “Bayesian image reconstruction: the pixon and optimal image modeling,” Publ. Astron. Soc. Pac. 105, 630–637 (1993).
    [CrossRef]
  15. H. H. Barrett, D. W. Wilson, B. M. W. Tsui, “Noise properties of the EM algorithm: 1. Theory,” Phys. Med. Biol. 39, 833–846 (1994).
    [CrossRef] [PubMed]
  16. H. Y. Pang, M. K. Sundareshan, S. Amphay, “Optimized maximum-likelihood algorithms for superresolution of passive millimeter-wave imagery,” in, Passive Millimeter-Wave Imaging TechnologyII, R. Smith , ed., Proc. SPIE3378, 148–160 (1998).
    [CrossRef]
  17. H. Y. Pang, M. K. Sundareshan, S. Amphay, “Superresolution of millimeter-wave images by iterative blind maximum-likelihood restoration,” in Passive Millimeter-Wave Imaging Technology, R. Smith, ed., Proc. SPIE3064, 227–238 (1997).
    [CrossRef]
  18. R. W. Gerchberg, “Super-resolution through error energy reduction,” Opt. Acta 21, 709–720 (1974).
    [CrossRef]
  19. A. Papoulis, “A new algorithm in spectral analysis and band-limited extrapolation,” IEEE Trans. Circuits Syst. CAS-22, 735–742 (1975).
    [CrossRef]
  20. L. G. Gubin, B. T. Polak, E. V. Raik, “The method of projection for finding the common point of convex sets,” USSR Comput. Math. Math. Phys. 7, 1–24 (1967).
    [CrossRef]
  21. A. Lent, H. Tuy, “An iterative method for extrapolation of band-limited functions,” J. Math. Anal. Appl. 83, 544–565 (1981).
    [CrossRef]
  22. D. C. Youla, “Generalized image restoration by the method of alternating orthogonal projections,” IEEE Trans. Circuits Syst. 25, 694–702 (1978).
    [CrossRef]
  23. D. C. Youla, H. Webb, “Image restoration by the method of convex projections: Part 1. Theory,” IEEE Trans. Med. Imaging MI-1, 81–94 (1982).
    [CrossRef]
  24. M. I. Sezan, H. Stark, “Image restoration by the method of convex projections: Part 2. Application and numerical results,” IEEE Trans. Med. Imaging MI-1, 95–101 (1982).
    [CrossRef]
  25. P. L. Combettes, “The foundations of set theoretic estimation,” Proc. IEEE 81, 182–208 (1993).
    [CrossRef]
  26. H. Stark, ed. Image Recovery: Theory and Application (Academic, San Diego, Calif., 1987).
  27. M. I. Sezan, “An overview of convex projections theory and its application to image recovery problems,” Ultramicroscopy 40, 55–67 (1992).
    [CrossRef]
  28. G. Crombez, “Image recovery by convex combinations of projections,” J. Math. Anal. Appl. 15, 413–419 (1991).
    [CrossRef]
  29. P. L. Combettes, H. Puh, “Iterations of parallel convex projections in Hilbert spaces,” Numer. Functi. Analy. Optim. 15, 225–243 (1994).
    [CrossRef]
  30. S. Bhattacharjee, M. K. Sundareshan, “Hybrid Bayesian and convex set projection algorithms for restoration and resolution enhancement of digital images,” in Applications of Digital Image Processing XXIII, A. G. Tescher, ed., Proc. SPIE4115, 12–22 (2000).
    [CrossRef]
  31. M. K. Sundareshan, S. Bhattacharjee, “Superresolution of passive millimeter-wave images using a combined maximum-likelihood optimization and projection-onto-convex-sets approach,” in Passive Millimeter-Wave Imaging Technology V, R. Smith, R. Appleby, eds., Proc. SPIE4373, 105–116 (2001).
    [CrossRef]
  32. D. Terzopoulos, “Regularization of the inverse visual problem involving discontinuities,” IEEE Trans. Pattern Anal. Mach. Intell. 1, 413–424 (1986).
    [CrossRef]
  33. E. L. Kosarev, “Shannon’s super-resolution limit for signal recovery,” Inverse Probl. 6, 55–76 (1990).
    [CrossRef]
  34. M. K. Sundareshan, P. Zegers, “Role of over-sampled data in superresolution processing and a progressive up-sampling scheme for optimized implementations of iterative restoration algorithms,” in Passive Millimeter-Wave Imaging Technology III, R. Smith, ed., Proc. SPIE3703, 155–166 (1999).
    [CrossRef]
  35. P. J. Sementilli, B. R. Hunt, M. S. Nadar, “An analysis of the limit to superresolution in incoherent imaging,” J. Opt. Soc. Am. A 10, 2265–2276 (1993).
    [CrossRef]
  36. J. Canny, “A computational approach to edge detection,” IEEE Trans. Pattern Anal. Mach. Intell. 8, 372–381 (1986).

1995

B. R. Hunt, “Super-resolution of images: algorithms, principles, and performance,” Int. J. Imaging Syst. Technol. 6, 297–304 (1995).
[CrossRef]

1994

P. L. Combettes, H. Puh, “Iterations of parallel convex projections in Hilbert spaces,” Numer. Functi. Analy. Optim. 15, 225–243 (1994).
[CrossRef]

H. H. Barrett, D. W. Wilson, B. M. W. Tsui, “Noise properties of the EM algorithm: 1. Theory,” Phys. Med. Biol. 39, 833–846 (1994).
[CrossRef] [PubMed]

1993

R. K. Pina, R. C. Puetter, “Bayesian image reconstruction: the pixon and optimal image modeling,” Publ. Astron. Soc. Pac. 105, 630–637 (1993).
[CrossRef]

P. L. Combettes, “The foundations of set theoretic estimation,” Proc. IEEE 81, 182–208 (1993).
[CrossRef]

P. J. Sementilli, B. R. Hunt, M. S. Nadar, “An analysis of the limit to superresolution in incoherent imaging,” J. Opt. Soc. Am. A 10, 2265–2276 (1993).
[CrossRef]

1992

M. I. Sezan, “An overview of convex projections theory and its application to image recovery problems,” Ultramicroscopy 40, 55–67 (1992).
[CrossRef]

1991

G. Crombez, “Image recovery by convex combinations of projections,” J. Math. Anal. Appl. 15, 413–419 (1991).
[CrossRef]

1990

E. L. Kosarev, “Shannon’s super-resolution limit for signal recovery,” Inverse Probl. 6, 55–76 (1990).
[CrossRef]

1986

D. Terzopoulos, “Regularization of the inverse visual problem involving discontinuities,” IEEE Trans. Pattern Anal. Mach. Intell. 1, 413–424 (1986).
[CrossRef]

J. Canny, “A computational approach to edge detection,” IEEE Trans. Pattern Anal. Mach. Intell. 8, 372–381 (1986).

1984

S. Geman, D. Geman, “Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images,” IEEE Trans. Pattern Anal. Mach. Intell. PAMI-6, 721–741 (1984).
[CrossRef]

1982

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

D. C. Youla, H. Webb, “Image restoration by the method of convex projections: Part 1. Theory,” IEEE Trans. Med. Imaging MI-1, 81–94 (1982).
[CrossRef]

M. I. Sezan, H. Stark, “Image restoration by the method of convex projections: Part 2. Application and numerical results,” IEEE Trans. Med. Imaging MI-1, 95–101 (1982).
[CrossRef]

1981

A. Lent, H. Tuy, “An iterative method for extrapolation of band-limited functions,” J. Math. Anal. Appl. 83, 544–565 (1981).
[CrossRef]

1978

D. C. Youla, “Generalized image restoration by the method of alternating orthogonal projections,” IEEE Trans. Circuits Syst. 25, 694–702 (1978).
[CrossRef]

1977

B. R. Hunt, “Bayesian methods in digital image restoration,” IEEE Trans. Comput. C-26, 219–229 (1977).
[CrossRef]

A. P. Dempster, N. M. Laird, D. B. Rubin, “Maximum likelihood from incomplete data via the EM algorithm,” J. R. Stat. Soc. Ser. B. Methodol. 39, 1–38 (1977).

1975

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

1974

L. B. Lucy, “An iterative technique for the rectification of observed distributions,” Astron. J. 79, 745–759 (1974).
[CrossRef]

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

1972

W. H. Richardson, “Bayesian-based iterative method of image restoration,” J. Opt. Soc. Am. 62, 55–60 (1972).
[CrossRef]

B. R. Frieden, “Restoring with maximum likelihood and maximum entropy,” J. Opt. Soc. Am. 62, 51–55 (1972).
[CrossRef]

1968

1967

B. R. Frieden, “Band-unlimited reconstruction of optical objects and spectral sources,” J. Opt. Soc. Am. 57, 1013–1019 (1967).
[CrossRef]

L. G. Gubin, B. T. Polak, E. V. Raik, “The method of projection for finding the common point of convex sets,” USSR Comput. Math. Math. Phys. 7, 1–24 (1967).
[CrossRef]

1961

D. Slepian, H. O. Pollack, “Prolate spheroidal wave functions, Fourier analysis and uncertainty—I,” Bell Syst. Tech. J. 40, 43–62 (1961).
[CrossRef]

Amphay, S.

H. Y. Pang, M. K. Sundareshan, S. Amphay, “Optimized maximum-likelihood algorithms for superresolution of passive millimeter-wave imagery,” in, Passive Millimeter-Wave Imaging TechnologyII, R. Smith , ed., Proc. SPIE3378, 148–160 (1998).
[CrossRef]

H. Y. Pang, M. K. Sundareshan, S. Amphay, “Superresolution of millimeter-wave images by iterative blind maximum-likelihood restoration,” in Passive Millimeter-Wave Imaging Technology, R. Smith, ed., Proc. SPIE3064, 227–238 (1997).
[CrossRef]

Barrett, H. H.

H. H. Barrett, D. W. Wilson, B. M. W. Tsui, “Noise properties of the EM algorithm: 1. Theory,” Phys. Med. Biol. 39, 833–846 (1994).
[CrossRef] [PubMed]

Bhattacharjee, S.

S. Bhattacharjee, M. K. Sundareshan, “Hybrid Bayesian and convex set projection algorithms for restoration and resolution enhancement of digital images,” in Applications of Digital Image Processing XXIII, A. G. Tescher, ed., Proc. SPIE4115, 12–22 (2000).
[CrossRef]

M. K. Sundareshan, S. Bhattacharjee, “Superresolution of passive millimeter-wave images using a combined maximum-likelihood optimization and projection-onto-convex-sets approach,” in Passive Millimeter-Wave Imaging Technology V, R. Smith, R. Appleby, eds., Proc. SPIE4373, 105–116 (2001).
[CrossRef]

Canny, J.

J. Canny, “A computational approach to edge detection,” IEEE Trans. Pattern Anal. Mach. Intell. 8, 372–381 (1986).

Combettes, P. L.

P. L. Combettes, H. Puh, “Iterations of parallel convex projections in Hilbert spaces,” Numer. Functi. Analy. Optim. 15, 225–243 (1994).
[CrossRef]

P. L. Combettes, “The foundations of set theoretic estimation,” Proc. IEEE 81, 182–208 (1993).
[CrossRef]

Crombez, G.

G. Crombez, “Image recovery by convex combinations of projections,” J. Math. Anal. Appl. 15, 413–419 (1991).
[CrossRef]

Dempster, A. P.

A. P. Dempster, N. M. Laird, D. B. Rubin, “Maximum likelihood from incomplete data via the EM algorithm,” J. R. Stat. Soc. Ser. B. Methodol. 39, 1–38 (1977).

Frieden, B. R.

B. R. Frieden, “Restoring with maximum likelihood and maximum entropy,” J. Opt. Soc. Am. 62, 51–55 (1972).
[CrossRef]

B. R. Frieden, “Band-unlimited reconstruction of optical objects and spectral sources,” J. Opt. Soc. Am. 57, 1013–1019 (1967).
[CrossRef]

Geman, D.

S. Geman, D. Geman, “Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images,” IEEE Trans. Pattern Anal. Mach. Intell. PAMI-6, 721–741 (1984).
[CrossRef]

Geman, S.

S. Geman, D. Geman, “Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images,” IEEE Trans. Pattern Anal. Mach. Intell. PAMI-6, 721–741 (1984).
[CrossRef]

Gerchberg, R. W.

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

Goodman, J.

J. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1996).

Gubin, L. G.

L. G. Gubin, B. T. Polak, E. V. Raik, “The method of projection for finding the common point of convex sets,” USSR Comput. Math. Math. Phys. 7, 1–24 (1967).
[CrossRef]

Harris, J. L.

Hunt, B. R.

B. R. Hunt, “Super-resolution of images: algorithms, principles, and performance,” Int. J. Imaging Syst. Technol. 6, 297–304 (1995).
[CrossRef]

P. J. Sementilli, B. R. Hunt, M. S. Nadar, “An analysis of the limit to superresolution in incoherent imaging,” J. Opt. Soc. Am. A 10, 2265–2276 (1993).
[CrossRef]

B. R. Hunt, “Bayesian methods in digital image restoration,” IEEE Trans. Comput. C-26, 219–229 (1977).
[CrossRef]

Kosarev, E. L.

E. L. Kosarev, “Shannon’s super-resolution limit for signal recovery,” Inverse Probl. 6, 55–76 (1990).
[CrossRef]

Laird, N. M.

A. P. Dempster, N. M. Laird, D. B. Rubin, “Maximum likelihood from incomplete data via the EM algorithm,” J. R. Stat. Soc. Ser. B. Methodol. 39, 1–38 (1977).

Lent, A.

A. Lent, H. Tuy, “An iterative method for extrapolation of band-limited functions,” J. Math. Anal. Appl. 83, 544–565 (1981).
[CrossRef]

Lucy, L. B.

L. B. Lucy, “An iterative technique for the rectification of observed distributions,” Astron. J. 79, 745–759 (1974).
[CrossRef]

Miller, M. I.

D. L. Snyder, M. I. Miller, Random Point Processes in Time and Space, 2nd ed. (Springer-Verlag, New York, 1991).

Nadar, M. S.

Pang, H. Y.

H. Y. Pang, M. K. Sundareshan, S. Amphay, “Optimized maximum-likelihood algorithms for superresolution of passive millimeter-wave imagery,” in, Passive Millimeter-Wave Imaging TechnologyII, R. Smith , ed., Proc. SPIE3378, 148–160 (1998).
[CrossRef]

H. Y. Pang, M. K. Sundareshan, S. Amphay, “Superresolution of millimeter-wave images by iterative blind maximum-likelihood restoration,” in Passive Millimeter-Wave Imaging Technology, R. Smith, ed., Proc. SPIE3064, 227–238 (1997).
[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]

Pina, R. K.

R. K. Pina, R. C. Puetter, “Bayesian image reconstruction: the pixon and optimal image modeling,” Publ. Astron. Soc. Pac. 105, 630–637 (1993).
[CrossRef]

Polak, B. T.

L. G. Gubin, B. T. Polak, E. V. Raik, “The method of projection for finding the common point of convex sets,” USSR Comput. Math. Math. Phys. 7, 1–24 (1967).
[CrossRef]

Pollack, H. O.

D. Slepian, H. O. Pollack, “Prolate spheroidal wave functions, Fourier analysis and uncertainty—I,” Bell Syst. Tech. J. 40, 43–62 (1961).
[CrossRef]

Puetter, R. C.

R. K. Pina, R. C. Puetter, “Bayesian image reconstruction: the pixon and optimal image modeling,” Publ. Astron. Soc. Pac. 105, 630–637 (1993).
[CrossRef]

Puh, H.

P. L. Combettes, H. Puh, “Iterations of parallel convex projections in Hilbert spaces,” Numer. Functi. Analy. Optim. 15, 225–243 (1994).
[CrossRef]

Raik, E. V.

L. G. Gubin, B. T. Polak, E. V. Raik, “The method of projection for finding the common point of convex sets,” USSR Comput. Math. Math. Phys. 7, 1–24 (1967).
[CrossRef]

Richardson, W. H.

Rubin, D. B.

A. P. Dempster, N. M. Laird, D. B. Rubin, “Maximum likelihood from incomplete data via the EM algorithm,” J. R. Stat. Soc. Ser. B. Methodol. 39, 1–38 (1977).

Rushforth, C. K.

Sementilli, P. J.

Sezan, M. I.

M. I. Sezan, “An overview of convex projections theory and its application to image recovery problems,” Ultramicroscopy 40, 55–67 (1992).
[CrossRef]

M. I. Sezan, H. Stark, “Image restoration by the method of convex projections: Part 2. Application and numerical results,” IEEE Trans. Med. Imaging MI-1, 95–101 (1982).
[CrossRef]

Shepp, L.

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

Slepian, D.

D. Slepian, H. O. Pollack, “Prolate spheroidal wave functions, Fourier analysis and uncertainty—I,” Bell Syst. Tech. J. 40, 43–62 (1961).
[CrossRef]

Snyder, D. L.

D. L. Snyder, M. I. Miller, Random Point Processes in Time and Space, 2nd ed. (Springer-Verlag, New York, 1991).

Stark, H.

M. I. Sezan, H. Stark, “Image restoration by the method of convex projections: Part 2. Application and numerical results,” IEEE Trans. Med. Imaging MI-1, 95–101 (1982).
[CrossRef]

Sundareshan, M. K.

M. K. Sundareshan, S. Bhattacharjee, “Superresolution of passive millimeter-wave images using a combined maximum-likelihood optimization and projection-onto-convex-sets approach,” in Passive Millimeter-Wave Imaging Technology V, R. Smith, R. Appleby, eds., Proc. SPIE4373, 105–116 (2001).
[CrossRef]

S. Bhattacharjee, M. K. Sundareshan, “Hybrid Bayesian and convex set projection algorithms for restoration and resolution enhancement of digital images,” in Applications of Digital Image Processing XXIII, A. G. Tescher, ed., Proc. SPIE4115, 12–22 (2000).
[CrossRef]

H. Y. Pang, M. K. Sundareshan, S. Amphay, “Optimized maximum-likelihood algorithms for superresolution of passive millimeter-wave imagery,” in, Passive Millimeter-Wave Imaging TechnologyII, R. Smith , ed., Proc. SPIE3378, 148–160 (1998).
[CrossRef]

M. K. Sundareshan, P. Zegers, “Role of over-sampled data in superresolution processing and a progressive up-sampling scheme for optimized implementations of iterative restoration algorithms,” in Passive Millimeter-Wave Imaging Technology III, R. Smith, ed., Proc. SPIE3703, 155–166 (1999).
[CrossRef]

H. Y. Pang, M. K. Sundareshan, S. Amphay, “Superresolution of millimeter-wave images by iterative blind maximum-likelihood restoration,” in Passive Millimeter-Wave Imaging Technology, R. Smith, ed., Proc. SPIE3064, 227–238 (1997).
[CrossRef]

Terzopoulos, D.

D. Terzopoulos, “Regularization of the inverse visual problem involving discontinuities,” IEEE Trans. Pattern Anal. Mach. Intell. 1, 413–424 (1986).
[CrossRef]

Tsui, B. M. W.

H. H. Barrett, D. W. Wilson, B. M. W. Tsui, “Noise properties of the EM algorithm: 1. Theory,” Phys. Med. Biol. 39, 833–846 (1994).
[CrossRef] [PubMed]

Tuy, H.

A. Lent, H. Tuy, “An iterative method for extrapolation of band-limited functions,” J. Math. Anal. Appl. 83, 544–565 (1981).
[CrossRef]

Vardi, Y.

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

Webb, H.

D. C. Youla, H. Webb, “Image restoration by the method of convex projections: Part 1. Theory,” IEEE Trans. Med. Imaging MI-1, 81–94 (1982).
[CrossRef]

Wilson, D. W.

H. H. Barrett, D. W. Wilson, B. M. W. Tsui, “Noise properties of the EM algorithm: 1. Theory,” Phys. Med. Biol. 39, 833–846 (1994).
[CrossRef] [PubMed]

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, 81–94 (1982).
[CrossRef]

D. C. Youla, “Generalized image restoration by the method of alternating orthogonal projections,” IEEE Trans. Circuits Syst. 25, 694–702 (1978).
[CrossRef]

Zegers, P.

M. K. Sundareshan, P. Zegers, “Role of over-sampled data in superresolution processing and a progressive up-sampling scheme for optimized implementations of iterative restoration algorithms,” in Passive Millimeter-Wave Imaging Technology III, R. Smith, ed., Proc. SPIE3703, 155–166 (1999).
[CrossRef]

Astron. J.

L. B. Lucy, “An iterative technique for the rectification of observed distributions,” Astron. J. 79, 745–759 (1974).
[CrossRef]

Bell Syst. Tech. J.

D. Slepian, H. O. Pollack, “Prolate spheroidal wave functions, Fourier analysis and uncertainty—I,” Bell Syst. Tech. J. 40, 43–62 (1961).
[CrossRef]

IEEE Trans. Circuits Syst.

D. C. Youla, “Generalized image restoration by the method of alternating orthogonal projections,” IEEE Trans. Circuits Syst. 25, 694–702 (1978).
[CrossRef]

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

IEEE Trans. Comput.

B. R. Hunt, “Bayesian methods in digital image restoration,” IEEE Trans. Comput. C-26, 219–229 (1977).
[CrossRef]

IEEE Trans. Med. Imaging

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

D. C. Youla, H. Webb, “Image restoration by the method of convex projections: Part 1. Theory,” IEEE Trans. Med. Imaging MI-1, 81–94 (1982).
[CrossRef]

M. I. Sezan, H. Stark, “Image restoration by the method of convex projections: Part 2. Application and numerical results,” IEEE Trans. Med. Imaging MI-1, 95–101 (1982).
[CrossRef]

IEEE Trans. Pattern Anal. Mach. Intell.

S. Geman, D. Geman, “Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images,” IEEE Trans. Pattern Anal. Mach. Intell. PAMI-6, 721–741 (1984).
[CrossRef]

J. Canny, “A computational approach to edge detection,” IEEE Trans. Pattern Anal. Mach. Intell. 8, 372–381 (1986).

D. Terzopoulos, “Regularization of the inverse visual problem involving discontinuities,” IEEE Trans. Pattern Anal. Mach. Intell. 1, 413–424 (1986).
[CrossRef]

Int. J. Imaging Syst. Technol.

B. R. Hunt, “Super-resolution of images: algorithms, principles, and performance,” Int. J. Imaging Syst. Technol. 6, 297–304 (1995).
[CrossRef]

Inverse Probl.

E. L. Kosarev, “Shannon’s super-resolution limit for signal recovery,” Inverse Probl. 6, 55–76 (1990).
[CrossRef]

J. Math. Anal. Appl.

G. Crombez, “Image recovery by convex combinations of projections,” J. Math. Anal. Appl. 15, 413–419 (1991).
[CrossRef]

A. Lent, H. Tuy, “An iterative method for extrapolation of band-limited functions,” J. Math. Anal. Appl. 83, 544–565 (1981).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. R. Stat. Soc. Ser. B. Methodol.

A. P. Dempster, N. M. Laird, D. B. Rubin, “Maximum likelihood from incomplete data via the EM algorithm,” J. R. Stat. Soc. Ser. B. Methodol. 39, 1–38 (1977).

Numer. Functi. Analy. Optim.

P. L. Combettes, H. Puh, “Iterations of parallel convex projections in Hilbert spaces,” Numer. Functi. Analy. Optim. 15, 225–243 (1994).
[CrossRef]

Opt. Acta

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

Phys. Med. Biol.

H. H. Barrett, D. W. Wilson, B. M. W. Tsui, “Noise properties of the EM algorithm: 1. Theory,” Phys. Med. Biol. 39, 833–846 (1994).
[CrossRef] [PubMed]

Proc. IEEE

P. L. Combettes, “The foundations of set theoretic estimation,” Proc. IEEE 81, 182–208 (1993).
[CrossRef]

Publ. Astron. Soc. Pac.

R. K. Pina, R. C. Puetter, “Bayesian image reconstruction: the pixon and optimal image modeling,” Publ. Astron. Soc. Pac. 105, 630–637 (1993).
[CrossRef]

Ultramicroscopy

M. I. Sezan, “An overview of convex projections theory and its application to image recovery problems,” Ultramicroscopy 40, 55–67 (1992).
[CrossRef]

USSR Comput. Math. Math. Phys.

L. G. Gubin, B. T. Polak, E. V. Raik, “The method of projection for finding the common point of convex sets,” USSR Comput. Math. Math. Phys. 7, 1–24 (1967).
[CrossRef]

Other

H. Stark, ed. Image Recovery: Theory and Application (Academic, San Diego, Calif., 1987).

H. Y. Pang, M. K. Sundareshan, S. Amphay, “Optimized maximum-likelihood algorithms for superresolution of passive millimeter-wave imagery,” in, Passive Millimeter-Wave Imaging TechnologyII, R. Smith , ed., Proc. SPIE3378, 148–160 (1998).
[CrossRef]

H. Y. Pang, M. K. Sundareshan, S. Amphay, “Superresolution of millimeter-wave images by iterative blind maximum-likelihood restoration,” in Passive Millimeter-Wave Imaging Technology, R. Smith, ed., Proc. SPIE3064, 227–238 (1997).
[CrossRef]

S. Bhattacharjee, M. K. Sundareshan, “Hybrid Bayesian and convex set projection algorithms for restoration and resolution enhancement of digital images,” in Applications of Digital Image Processing XXIII, A. G. Tescher, ed., Proc. SPIE4115, 12–22 (2000).
[CrossRef]

M. K. Sundareshan, S. Bhattacharjee, “Superresolution of passive millimeter-wave images using a combined maximum-likelihood optimization and projection-onto-convex-sets approach,” in Passive Millimeter-Wave Imaging Technology V, R. Smith, R. Appleby, eds., Proc. SPIE4373, 105–116 (2001).
[CrossRef]

M. K. Sundareshan, P. Zegers, “Role of over-sampled data in superresolution processing and a progressive up-sampling scheme for optimized implementations of iterative restoration algorithms,” in Passive Millimeter-Wave Imaging Technology III, R. Smith, ed., Proc. SPIE3703, 155–166 (1999).
[CrossRef]

J. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1996).

D. L. Snyder, M. I. Miller, Random Point Processes in Time and Space, 2nd ed. (Springer-Verlag, New York, 1991).

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Figures (6)

Fig. 1
Fig. 1

Spectrum extrapolation by using the Taylor series.

Fig. 2
Fig. 2

Spectrum symmetry.

Fig. 3
Fig. 3

Results of the superresolution experiment with POCS enforcement of the extrapolated spectrum constraint. (a) Ideal object, (b) blurred image, (c) restored image, (d) object spectrum, (e) blurred spectrum, (f) restored spectrum.

Fig. 4
Fig. 4

Convergence of the L2 norm of the restoration error in the frequency domain.

Fig. 5
Fig. 5

Blurring of the image used in experiment 2. (a) Original image, (b) blurred image, (c) spectrum of the original image, (d) spectrum of the blurred image.

Fig. 6
Fig. 6

Restoration results from experiment 2. (a) Restored image, (b) spectrum of the restored image, (c) convergence of the proximity measure.

Equations (38)

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g(y)=xXh(y, x)f(x)+noise,
fˆ=argmaxf p(g/f).
f^k+1(j)=f^k(j)g(j)f^k(j)h(j)h(j),
j=1, 2, 3,, N,
fˆ=argminfi=1MwiJi(fˆ, Si),
Ji(fˆ, Si)=fˆ-fp2,
f^n=PmPm-1Pm-2P1f^n-1,
Ti=(1-λ)I+λPi,
f^n=TmTm-1Tm-2T1f^n-1.
f^n+1=f^n+λniInwi,nTi(f^n)-f^n
rl=1ωclog2(1+SNR),
f(x+Δx, y+Δy)=f(x, y)+Δxfx(x, y)+Δyfy(x, y)+12![Δx2fxx(x, y)+2ΔxΔyfxy(x, y)+Δy2fyy(x, y)],
fxy=2f(x, y)xy,fx=f(x, y)x,fy=f(x, y)y,
fxx=2f(x, y)x2,fyy=2f(x, y)2y
f(x, y)xf(x, y)-f(x-hx, y)hx,
f(x, y)yf(x, y)-f(x, y-hy)hy,
2f(x, y)x2f(x, y)-2f(x-hx, y)+f(x-2hx, y)hx2,
2f(x, y)y2f(x, y)-2f(x, y-hy)+f(x, y-2hy)hy2,
2f(x, y)xyf(x, y)-f(x-hx, y)-f(x, y-hy)+f(x-hx, y-hy)hxhy,
f[i, j]i=f[i, j]-f[i-1, j]Δx,
f[i, j]j=f[i, j]-f[i, j-1]Δy,
2f[i, j]i2=f[i, j]-2f[i-1, j]+f[i-2, j]Δx2,
2f[i, j]j2=f[i, j]-2f[i, j-1]+f[i, j-2]Δy2,
2f[i, j]ij=f[i, j]-f[i, j-1]-f[i-1, j]+f[i-1, j-1]ΔxΔy.
F[i+1,j+1]=F[i, j]+ΔxF[i, j]i+ΔyF[i, j]j+12!Δx22F[i, j]i2+2ΔxΔy2F[i, j]ij+Δy22F[i, j]j2,
F[i, j]=F[N-i, M-j]* fori, j0;
F[i, j]=F[i, M-j]* fori=0, j0;
F[i, j]=F[N-i, j]* fori0, j=0,
s={fΞs : f(i, j)=0if(i, j)Ψ,f={fΞΔ : F(k, l)=Ge(k, l)if(k, l)Φ,
Psf=fp,
fp(i, j)=f(i, j)if(i, j)Ψ,fp(i, j)=0if(i, j)Ψ,
PΔf=fp,
Fp(k, l)=Ge(k, l)if(k, l)Φ,Fp(k, l)=F(k, l)if(k, l)Φ.
l2(Fideal, Fp)=(k,l)RFideal(k, l)-Fp(k, l)2,
s={fΞ:foreground ofimagefis bounded by ξ},
Pborder(f)=fp,
fp(i, j)=f(i, j) : (i, j)liesinsideξ0 : (i, j)liesoutsideξ.
Φnorm(n)=10 log10i=1MJi(f^n, Si)i=1MJi(f^0, Si),

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