Abstract

Six methods for the numerical calculation of zero-order Hankel transforms of oscillating functions were evaluated. One method based on Filon quadrature philosophy, two published projection-slice methods, and a third projection-slice method based on a new approach to computation of the Abel transform were implemented; alternative versions of two of the projection-slice methods were derived for more accurate approximations in the projection step. These six algorithms were tested with an oscillating sweep signal and with the calculation of a three-dimensional diffraction-limited point-spread function of a fluorescence microscope. We found that the Filon quadrature method is highly accurate but also computationally demanding. The projection-slice methods, in particular the new one that we derived, offer an excellent compromise between accuracy and computational efficiency.

© 2003 Optical Society of America

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  1. J. A. Conchello, Q. Yu, “Parametric blind deconvolution of fluorescence microscopy images: preliminary results,” in Three-Dimensional Microscopy: Image Acquisition and Processing III, C. J. Cogswell, G. Kino, T. Wilson, eds., Proc. SPIE2655, 164–174 (1996).
    [CrossRef]
  2. J. Markham, J. A. Conchello, “Parametric blind deconvolution of fluorescence microscopy images: further results,” in Three-Dimensional Microscopy: Image Acquisition and Processing V, C. J. Cogswell, J. A. Conchello, T. Wilson, eds., Proc. SPIE3261, 38–49 (1998).
    [CrossRef]
  3. J. Markham, J. A. Conchello, “Parametric blind deconvolution: a robust method for the simultaneous estimation of image and blur,” J. Opt. Soc. Am. A 16, 2377–2391 (1999).
    [CrossRef]
  4. T. J. Holmes, “Blind deconvolution of quantum-limited incoherent imagery: maximum-likelihood approach,” J. Opt. Soc. Am. A 9, 1052–1061 (1992).
    [CrossRef] [PubMed]
  5. R. G. Paxman, T. J. Schulz, J. R. Fienup, “Joint estimation of object and aberrations by using phase diversity,” J. Opt. Soc. Am. A 9, 1072–1085 (1992).
    [CrossRef]
  6. R. G. Lane, “Blind deconvolution of speckle images,” J. Opt. Soc. Am. A 9, 1508–1514 (1992).
    [CrossRef]
  7. T. J. Schulz, “Multiframe blind deconvolution of astronomical images,” J. Opt. Soc. Am. A 10, 1064–1073 (1993).
    [CrossRef]
  8. Y. Yang, N. P. Galastanos, H. Stark, “Projection-based blind deconvolution,” J. Opt. Soc. Am. A 11, 2401–2409 (1994).
    [CrossRef]
  9. V. Krishnamurthi, Y-H. Liu, S. Bhattacharyya, J. N. Turner, T. J. Holmes, “Blind deconvolution of fluorescence micrographs by maximum-likelihood estimation,” Appl. Opt. 34, 6633–6647 (1995).
    [CrossRef] [PubMed]
  10. E. Thiébaut, J.-M. Conan, “Strict a priori constraints for maximum-likelihood blind deconvolution,” J. Opt. Soc. Am. A 12, 485–492 (1995).
    [CrossRef]
  11. S. Bhattacharyya, D. H. Szarowski, J. N. Turner, N. O’Connor, T. J. Holmes, “ML-EM blind deconvolution algorithm: recent developments,” in Three-Dimensional Microscopy: Image Acquisition and Processing III, C. J. Cogswell, G. Kino, T. Wilson, eds., Proc. SPIE2655, 175–186 (1996).
    [CrossRef]
  12. M Born, E Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 6th ed. (Pergamon, New York, 1980).
  13. A. E. Siegman, “Quasi fast Hankel transform,” Opt. Lett. 1, 13–15 (1977).
    [CrossRef] [PubMed]
  14. M. J. Cree, P. J. Bones, “Algorithms to numerically evaluate the Hankel transform,” Comput. Math. Appl. 26, 1–12 (1993).
    [CrossRef]
  15. B. W. Suter, “Foundations of Hankel transform algorithms,” Q. Appl. Math. 49, 267–279 (1991).
  16. R. Barakat, E. Parshall, “Numerical evaluation of the zero-order Hankel transform using Filon quadrature philosophy,” Appl. Math. Lett. 9, 21–26 (1996).
    [CrossRef]
  17. R. Barakat, E. Parshall, B. H. Sandler, “Zero-order Hankel transformation algorithms based on Filon quadrature philosophy for diffraction optics and beam propagation,” J. Opt. Soc. Am. A 15, 652–659 (1998).
    [CrossRef]
  18. V. Magni, G. Cerullo, S. De Silvestri, “High-accuracy fast Hankel transform for optical beam propagation,” J. Opt. Soc. Am. A 9, 2031–2033 (1992).
    [CrossRef]
  19. A. Agnesi, G. C. Reali, G. Patrini, A. Tomaselli, “Numerical evaluation of the Hankel transform: remarks,” J. Opt. Soc. Am. A 10, 1872–1874 (1993).
    [CrossRef]
  20. Q. H. Liu, Z. Q. Zhang, “Nonuniform fast Hankel transform (NUFHT) algorithm,” Appl. Opt. 38, 6705–6708 (1999).
    [CrossRef]
  21. Q. H. Liu, N. Nguyen, “An accurate algorithm for nonuniform fast Fourier transforms (NUFFT’s),” IEEE Microwave Guid. Wave Lett. 8, 18–20 (1998).
    [CrossRef]
  22. Q. H. Liu, X. Y. Tang, “Iterative algorithm for nonuniform inverse fast Fourier transform (NU-IFFT),” Electron. Lett. 34, 1913–1914 (1998).
    [CrossRef]
  23. A. V. Oppenheim, G. V. Frisk, D. R. Martinez, “Computation of the Hankel transform using projections,” J. Acoust. Soc. Am. 68, 523–529 (1980).
    [CrossRef]
  24. A. V. Oppenheim, G. V. Frisk, D. R. Martinez, “An algorithm for the numerical evaluation of the Hankel transform,” Proc. IEEE 66, 264–265 (1978).
    [CrossRef]
  25. D. R. Mook, “An algorithm for the numerical calculation of Hankel and Abel transforms,” IEEE Trans. Acoust. Speech Signal Process. ASSP-31, 979–985 (1983).
    [CrossRef]
  26. E. W. Hansen, “Fast Hankel transform algorithm,” IEEE Trans. Acoust. Speech Signal Process. ASSP-33, 666–671 (1985).
    [CrossRef]
  27. E. W. Hansen, “Correction to ‘Fast Hankel transform algorithm’,” IEEE Trans. Acoust. Speech Signal Process. ASSP-34, 623–624 (1986).
    [CrossRef]
  28. E. W. Hansen, P-L. Law, “Recursive methods for computing the Abel transform and its inverse,” J. Opt. Soc. Am. A 2, 510–520 (1985).
    [CrossRef]
  29. B. W. Suter, “Fast nth-order Hankel transform algorithm,” IEEE Trans. Signal Process. 39, 532–536 (1991).
    [CrossRef]
  30. J. A. Ferrari, “Fast Hankel transform of order zero,” J. Opt. Soc. Am. A 12, 1812–1813 (1995).
    [CrossRef]
  31. J. A. Ferrari, D. Perciante, A. Dubra, “Fast Hankel transform of nth order,” J. Opt. Soc. Am. A 16, 2581–2582 (1999).
    [CrossRef]
  32. B. W. Suter, R. A. Hedges, “Understanding fast Hankel transforms,” J. Opt. Soc. Am. A 18, 717–720 (2001).
    [CrossRef]
  33. L. Knockaert, “Fast Hankel transform by fast sine and cosine transforms: the Mellin connection,” IEEE Trans. Signal Process. 48, 1695–1701 (2000).
    [CrossRef]
  34. R. N. Bracewell, The Fourier Transform and Its Applications, 3rd ed. (McGraw-Hill, Boston, 2000), pp. 351–356.
  35. The netlib routines are available at http://www.netlib.org .
  36. F. S. Gibson, F. Lanni, “Experimental test of an analytical model of aberration in an oil-immersion objective lens used in three-dimensional light microscopy,” J. Opt. Soc. Am. A 8, 1601–1613 (1991).
    [CrossRef]
  37. M Abramowitz, I. A. Stegun, eds., Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover, New York, 1972), Chap. 12.

2001 (1)

2000 (1)

L. Knockaert, “Fast Hankel transform by fast sine and cosine transforms: the Mellin connection,” IEEE Trans. Signal Process. 48, 1695–1701 (2000).
[CrossRef]

1999 (3)

1998 (3)

Q. H. Liu, N. Nguyen, “An accurate algorithm for nonuniform fast Fourier transforms (NUFFT’s),” IEEE Microwave Guid. Wave Lett. 8, 18–20 (1998).
[CrossRef]

Q. H. Liu, X. Y. Tang, “Iterative algorithm for nonuniform inverse fast Fourier transform (NU-IFFT),” Electron. Lett. 34, 1913–1914 (1998).
[CrossRef]

R. Barakat, E. Parshall, B. H. Sandler, “Zero-order Hankel transformation algorithms based on Filon quadrature philosophy for diffraction optics and beam propagation,” J. Opt. Soc. Am. A 15, 652–659 (1998).
[CrossRef]

1996 (1)

R. Barakat, E. Parshall, “Numerical evaluation of the zero-order Hankel transform using Filon quadrature philosophy,” Appl. Math. Lett. 9, 21–26 (1996).
[CrossRef]

1995 (3)

1994 (1)

1993 (3)

1992 (4)

1991 (3)

B. W. Suter, “Foundations of Hankel transform algorithms,” Q. Appl. Math. 49, 267–279 (1991).

B. W. Suter, “Fast nth-order Hankel transform algorithm,” IEEE Trans. Signal Process. 39, 532–536 (1991).
[CrossRef]

F. S. Gibson, F. Lanni, “Experimental test of an analytical model of aberration in an oil-immersion objective lens used in three-dimensional light microscopy,” J. Opt. Soc. Am. A 8, 1601–1613 (1991).
[CrossRef]

1986 (1)

E. W. Hansen, “Correction to ‘Fast Hankel transform algorithm’,” IEEE Trans. Acoust. Speech Signal Process. ASSP-34, 623–624 (1986).
[CrossRef]

1985 (2)

E. W. Hansen, P-L. Law, “Recursive methods for computing the Abel transform and its inverse,” J. Opt. Soc. Am. A 2, 510–520 (1985).
[CrossRef]

E. W. Hansen, “Fast Hankel transform algorithm,” IEEE Trans. Acoust. Speech Signal Process. ASSP-33, 666–671 (1985).
[CrossRef]

1983 (1)

D. R. Mook, “An algorithm for the numerical calculation of Hankel and Abel transforms,” IEEE Trans. Acoust. Speech Signal Process. ASSP-31, 979–985 (1983).
[CrossRef]

1980 (1)

A. V. Oppenheim, G. V. Frisk, D. R. Martinez, “Computation of the Hankel transform using projections,” J. Acoust. Soc. Am. 68, 523–529 (1980).
[CrossRef]

1978 (1)

A. V. Oppenheim, G. V. Frisk, D. R. Martinez, “An algorithm for the numerical evaluation of the Hankel transform,” Proc. IEEE 66, 264–265 (1978).
[CrossRef]

1977 (1)

Agnesi, A.

Barakat, R.

R. Barakat, E. Parshall, B. H. Sandler, “Zero-order Hankel transformation algorithms based on Filon quadrature philosophy for diffraction optics and beam propagation,” J. Opt. Soc. Am. A 15, 652–659 (1998).
[CrossRef]

R. Barakat, E. Parshall, “Numerical evaluation of the zero-order Hankel transform using Filon quadrature philosophy,” Appl. Math. Lett. 9, 21–26 (1996).
[CrossRef]

Bhattacharyya, S.

V. Krishnamurthi, Y-H. Liu, S. Bhattacharyya, J. N. Turner, T. J. Holmes, “Blind deconvolution of fluorescence micrographs by maximum-likelihood estimation,” Appl. Opt. 34, 6633–6647 (1995).
[CrossRef] [PubMed]

S. Bhattacharyya, D. H. Szarowski, J. N. Turner, N. O’Connor, T. J. Holmes, “ML-EM blind deconvolution algorithm: recent developments,” in Three-Dimensional Microscopy: Image Acquisition and Processing III, C. J. Cogswell, G. Kino, T. Wilson, eds., Proc. SPIE2655, 175–186 (1996).
[CrossRef]

Bones, P. J.

M. J. Cree, P. J. Bones, “Algorithms to numerically evaluate the Hankel transform,” Comput. Math. Appl. 26, 1–12 (1993).
[CrossRef]

Born, M

M Born, E Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 6th ed. (Pergamon, New York, 1980).

Bracewell, R. N.

R. N. Bracewell, The Fourier Transform and Its Applications, 3rd ed. (McGraw-Hill, Boston, 2000), pp. 351–356.

Cerullo, G.

Conan, J.-M.

Conchello, J. A.

J. Markham, J. A. Conchello, “Parametric blind deconvolution: a robust method for the simultaneous estimation of image and blur,” J. Opt. Soc. Am. A 16, 2377–2391 (1999).
[CrossRef]

J. A. Conchello, Q. Yu, “Parametric blind deconvolution of fluorescence microscopy images: preliminary results,” in Three-Dimensional Microscopy: Image Acquisition and Processing III, C. J. Cogswell, G. Kino, T. Wilson, eds., Proc. SPIE2655, 164–174 (1996).
[CrossRef]

J. Markham, J. A. Conchello, “Parametric blind deconvolution of fluorescence microscopy images: further results,” in Three-Dimensional Microscopy: Image Acquisition and Processing V, C. J. Cogswell, J. A. Conchello, T. Wilson, eds., Proc. SPIE3261, 38–49 (1998).
[CrossRef]

Cree, M. J.

M. J. Cree, P. J. Bones, “Algorithms to numerically evaluate the Hankel transform,” Comput. Math. Appl. 26, 1–12 (1993).
[CrossRef]

De Silvestri, S.

Dubra, A.

Ferrari, J. A.

Fienup, J. R.

Frisk, G. V.

A. V. Oppenheim, G. V. Frisk, D. R. Martinez, “Computation of the Hankel transform using projections,” J. Acoust. Soc. Am. 68, 523–529 (1980).
[CrossRef]

A. V. Oppenheim, G. V. Frisk, D. R. Martinez, “An algorithm for the numerical evaluation of the Hankel transform,” Proc. IEEE 66, 264–265 (1978).
[CrossRef]

Galastanos, N. P.

Gibson, F. S.

Hansen, E. W.

E. W. Hansen, “Correction to ‘Fast Hankel transform algorithm’,” IEEE Trans. Acoust. Speech Signal Process. ASSP-34, 623–624 (1986).
[CrossRef]

E. W. Hansen, “Fast Hankel transform algorithm,” IEEE Trans. Acoust. Speech Signal Process. ASSP-33, 666–671 (1985).
[CrossRef]

E. W. Hansen, P-L. Law, “Recursive methods for computing the Abel transform and its inverse,” J. Opt. Soc. Am. A 2, 510–520 (1985).
[CrossRef]

Hedges, R. A.

Holmes, T. J.

V. Krishnamurthi, Y-H. Liu, S. Bhattacharyya, J. N. Turner, T. J. Holmes, “Blind deconvolution of fluorescence micrographs by maximum-likelihood estimation,” Appl. Opt. 34, 6633–6647 (1995).
[CrossRef] [PubMed]

T. J. Holmes, “Blind deconvolution of quantum-limited incoherent imagery: maximum-likelihood approach,” J. Opt. Soc. Am. A 9, 1052–1061 (1992).
[CrossRef] [PubMed]

S. Bhattacharyya, D. H. Szarowski, J. N. Turner, N. O’Connor, T. J. Holmes, “ML-EM blind deconvolution algorithm: recent developments,” in Three-Dimensional Microscopy: Image Acquisition and Processing III, C. J. Cogswell, G. Kino, T. Wilson, eds., Proc. SPIE2655, 175–186 (1996).
[CrossRef]

Knockaert, L.

L. Knockaert, “Fast Hankel transform by fast sine and cosine transforms: the Mellin connection,” IEEE Trans. Signal Process. 48, 1695–1701 (2000).
[CrossRef]

Krishnamurthi, V.

Lane, R. G.

Lanni, F.

Law, P-L.

Liu, Q. H.

Q. H. Liu, Z. Q. Zhang, “Nonuniform fast Hankel transform (NUFHT) algorithm,” Appl. Opt. 38, 6705–6708 (1999).
[CrossRef]

Q. H. Liu, X. Y. Tang, “Iterative algorithm for nonuniform inverse fast Fourier transform (NU-IFFT),” Electron. Lett. 34, 1913–1914 (1998).
[CrossRef]

Q. H. Liu, N. Nguyen, “An accurate algorithm for nonuniform fast Fourier transforms (NUFFT’s),” IEEE Microwave Guid. Wave Lett. 8, 18–20 (1998).
[CrossRef]

Liu, Y-H.

Magni, V.

Markham, J.

J. Markham, J. A. Conchello, “Parametric blind deconvolution: a robust method for the simultaneous estimation of image and blur,” J. Opt. Soc. Am. A 16, 2377–2391 (1999).
[CrossRef]

J. Markham, J. A. Conchello, “Parametric blind deconvolution of fluorescence microscopy images: further results,” in Three-Dimensional Microscopy: Image Acquisition and Processing V, C. J. Cogswell, J. A. Conchello, T. Wilson, eds., Proc. SPIE3261, 38–49 (1998).
[CrossRef]

Martinez, D. R.

A. V. Oppenheim, G. V. Frisk, D. R. Martinez, “Computation of the Hankel transform using projections,” J. Acoust. Soc. Am. 68, 523–529 (1980).
[CrossRef]

A. V. Oppenheim, G. V. Frisk, D. R. Martinez, “An algorithm for the numerical evaluation of the Hankel transform,” Proc. IEEE 66, 264–265 (1978).
[CrossRef]

Mook, D. R.

D. R. Mook, “An algorithm for the numerical calculation of Hankel and Abel transforms,” IEEE Trans. Acoust. Speech Signal Process. ASSP-31, 979–985 (1983).
[CrossRef]

Nguyen, N.

Q. H. Liu, N. Nguyen, “An accurate algorithm for nonuniform fast Fourier transforms (NUFFT’s),” IEEE Microwave Guid. Wave Lett. 8, 18–20 (1998).
[CrossRef]

O’Connor, N.

S. Bhattacharyya, D. H. Szarowski, J. N. Turner, N. O’Connor, T. J. Holmes, “ML-EM blind deconvolution algorithm: recent developments,” in Three-Dimensional Microscopy: Image Acquisition and Processing III, C. J. Cogswell, G. Kino, T. Wilson, eds., Proc. SPIE2655, 175–186 (1996).
[CrossRef]

Oppenheim, A. V.

A. V. Oppenheim, G. V. Frisk, D. R. Martinez, “Computation of the Hankel transform using projections,” J. Acoust. Soc. Am. 68, 523–529 (1980).
[CrossRef]

A. V. Oppenheim, G. V. Frisk, D. R. Martinez, “An algorithm for the numerical evaluation of the Hankel transform,” Proc. IEEE 66, 264–265 (1978).
[CrossRef]

Parshall, E.

R. Barakat, E. Parshall, B. H. Sandler, “Zero-order Hankel transformation algorithms based on Filon quadrature philosophy for diffraction optics and beam propagation,” J. Opt. Soc. Am. A 15, 652–659 (1998).
[CrossRef]

R. Barakat, E. Parshall, “Numerical evaluation of the zero-order Hankel transform using Filon quadrature philosophy,” Appl. Math. Lett. 9, 21–26 (1996).
[CrossRef]

Patrini, G.

Paxman, R. G.

Perciante, D.

Reali, G. C.

Sandler, B. H.

Schulz, T. J.

Siegman, A. E.

Stark, H.

Suter, B. W.

B. W. Suter, R. A. Hedges, “Understanding fast Hankel transforms,” J. Opt. Soc. Am. A 18, 717–720 (2001).
[CrossRef]

B. W. Suter, “Fast nth-order Hankel transform algorithm,” IEEE Trans. Signal Process. 39, 532–536 (1991).
[CrossRef]

B. W. Suter, “Foundations of Hankel transform algorithms,” Q. Appl. Math. 49, 267–279 (1991).

Szarowski, D. H.

S. Bhattacharyya, D. H. Szarowski, J. N. Turner, N. O’Connor, T. J. Holmes, “ML-EM blind deconvolution algorithm: recent developments,” in Three-Dimensional Microscopy: Image Acquisition and Processing III, C. J. Cogswell, G. Kino, T. Wilson, eds., Proc. SPIE2655, 175–186 (1996).
[CrossRef]

Tang, X. Y.

Q. H. Liu, X. Y. Tang, “Iterative algorithm for nonuniform inverse fast Fourier transform (NU-IFFT),” Electron. Lett. 34, 1913–1914 (1998).
[CrossRef]

Thiébaut, E.

Tomaselli, A.

Turner, J. N.

V. Krishnamurthi, Y-H. Liu, S. Bhattacharyya, J. N. Turner, T. J. Holmes, “Blind deconvolution of fluorescence micrographs by maximum-likelihood estimation,” Appl. Opt. 34, 6633–6647 (1995).
[CrossRef] [PubMed]

S. Bhattacharyya, D. H. Szarowski, J. N. Turner, N. O’Connor, T. J. Holmes, “ML-EM blind deconvolution algorithm: recent developments,” in Three-Dimensional Microscopy: Image Acquisition and Processing III, C. J. Cogswell, G. Kino, T. Wilson, eds., Proc. SPIE2655, 175–186 (1996).
[CrossRef]

Wolf, E

M Born, E Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 6th ed. (Pergamon, New York, 1980).

Yang, Y.

Yu, Q.

J. A. Conchello, Q. Yu, “Parametric blind deconvolution of fluorescence microscopy images: preliminary results,” in Three-Dimensional Microscopy: Image Acquisition and Processing III, C. J. Cogswell, G. Kino, T. Wilson, eds., Proc. SPIE2655, 164–174 (1996).
[CrossRef]

Zhang, Z. Q.

Appl. Math. Lett. (1)

R. Barakat, E. Parshall, “Numerical evaluation of the zero-order Hankel transform using Filon quadrature philosophy,” Appl. Math. Lett. 9, 21–26 (1996).
[CrossRef]

Appl. Opt. (2)

Comput. Math. Appl. (1)

M. J. Cree, P. J. Bones, “Algorithms to numerically evaluate the Hankel transform,” Comput. Math. Appl. 26, 1–12 (1993).
[CrossRef]

Electron. Lett. (1)

Q. H. Liu, X. Y. Tang, “Iterative algorithm for nonuniform inverse fast Fourier transform (NU-IFFT),” Electron. Lett. 34, 1913–1914 (1998).
[CrossRef]

IEEE Microwave Guid. Wave Lett. (1)

Q. H. Liu, N. Nguyen, “An accurate algorithm for nonuniform fast Fourier transforms (NUFFT’s),” IEEE Microwave Guid. Wave Lett. 8, 18–20 (1998).
[CrossRef]

IEEE Trans. Acoust. Speech Signal Process. (3)

D. R. Mook, “An algorithm for the numerical calculation of Hankel and Abel transforms,” IEEE Trans. Acoust. Speech Signal Process. ASSP-31, 979–985 (1983).
[CrossRef]

E. W. Hansen, “Fast Hankel transform algorithm,” IEEE Trans. Acoust. Speech Signal Process. ASSP-33, 666–671 (1985).
[CrossRef]

E. W. Hansen, “Correction to ‘Fast Hankel transform algorithm’,” IEEE Trans. Acoust. Speech Signal Process. ASSP-34, 623–624 (1986).
[CrossRef]

IEEE Trans. Signal Process. (2)

B. W. Suter, “Fast nth-order Hankel transform algorithm,” IEEE Trans. Signal Process. 39, 532–536 (1991).
[CrossRef]

L. Knockaert, “Fast Hankel transform by fast sine and cosine transforms: the Mellin connection,” IEEE Trans. Signal Process. 48, 1695–1701 (2000).
[CrossRef]

J. Acoust. Soc. Am. (1)

A. V. Oppenheim, G. V. Frisk, D. R. Martinez, “Computation of the Hankel transform using projections,” J. Acoust. Soc. Am. 68, 523–529 (1980).
[CrossRef]

J. Opt. Soc. Am. A (15)

E. W. Hansen, P-L. Law, “Recursive methods for computing the Abel transform and its inverse,” J. Opt. Soc. Am. A 2, 510–520 (1985).
[CrossRef]

F. S. Gibson, F. Lanni, “Experimental test of an analytical model of aberration in an oil-immersion objective lens used in three-dimensional light microscopy,” J. Opt. Soc. Am. A 8, 1601–1613 (1991).
[CrossRef]

J. A. Ferrari, “Fast Hankel transform of order zero,” J. Opt. Soc. Am. A 12, 1812–1813 (1995).
[CrossRef]

J. A. Ferrari, D. Perciante, A. Dubra, “Fast Hankel transform of nth order,” J. Opt. Soc. Am. A 16, 2581–2582 (1999).
[CrossRef]

B. W. Suter, R. A. Hedges, “Understanding fast Hankel transforms,” J. Opt. Soc. Am. A 18, 717–720 (2001).
[CrossRef]

R. Barakat, E. Parshall, B. H. Sandler, “Zero-order Hankel transformation algorithms based on Filon quadrature philosophy for diffraction optics and beam propagation,” J. Opt. Soc. Am. A 15, 652–659 (1998).
[CrossRef]

V. Magni, G. Cerullo, S. De Silvestri, “High-accuracy fast Hankel transform for optical beam propagation,” J. Opt. Soc. Am. A 9, 2031–2033 (1992).
[CrossRef]

A. Agnesi, G. C. Reali, G. Patrini, A. Tomaselli, “Numerical evaluation of the Hankel transform: remarks,” J. Opt. Soc. Am. A 10, 1872–1874 (1993).
[CrossRef]

E. Thiébaut, J.-M. Conan, “Strict a priori constraints for maximum-likelihood blind deconvolution,” J. Opt. Soc. Am. A 12, 485–492 (1995).
[CrossRef]

J. Markham, J. A. Conchello, “Parametric blind deconvolution: a robust method for the simultaneous estimation of image and blur,” J. Opt. Soc. Am. A 16, 2377–2391 (1999).
[CrossRef]

T. J. Holmes, “Blind deconvolution of quantum-limited incoherent imagery: maximum-likelihood approach,” J. Opt. Soc. Am. A 9, 1052–1061 (1992).
[CrossRef] [PubMed]

R. G. Paxman, T. J. Schulz, J. R. Fienup, “Joint estimation of object and aberrations by using phase diversity,” J. Opt. Soc. Am. A 9, 1072–1085 (1992).
[CrossRef]

R. G. Lane, “Blind deconvolution of speckle images,” J. Opt. Soc. Am. A 9, 1508–1514 (1992).
[CrossRef]

T. J. Schulz, “Multiframe blind deconvolution of astronomical images,” J. Opt. Soc. Am. A 10, 1064–1073 (1993).
[CrossRef]

Y. Yang, N. P. Galastanos, H. Stark, “Projection-based blind deconvolution,” J. Opt. Soc. Am. A 11, 2401–2409 (1994).
[CrossRef]

Opt. Lett. (1)

Proc. IEEE (1)

A. V. Oppenheim, G. V. Frisk, D. R. Martinez, “An algorithm for the numerical evaluation of the Hankel transform,” Proc. IEEE 66, 264–265 (1978).
[CrossRef]

Q. Appl. Math. (1)

B. W. Suter, “Foundations of Hankel transform algorithms,” Q. Appl. Math. 49, 267–279 (1991).

Other (7)

J. A. Conchello, Q. Yu, “Parametric blind deconvolution of fluorescence microscopy images: preliminary results,” in Three-Dimensional Microscopy: Image Acquisition and Processing III, C. J. Cogswell, G. Kino, T. Wilson, eds., Proc. SPIE2655, 164–174 (1996).
[CrossRef]

J. Markham, J. A. Conchello, “Parametric blind deconvolution of fluorescence microscopy images: further results,” in Three-Dimensional Microscopy: Image Acquisition and Processing V, C. J. Cogswell, J. A. Conchello, T. Wilson, eds., Proc. SPIE3261, 38–49 (1998).
[CrossRef]

S. Bhattacharyya, D. H. Szarowski, J. N. Turner, N. O’Connor, T. J. Holmes, “ML-EM blind deconvolution algorithm: recent developments,” in Three-Dimensional Microscopy: Image Acquisition and Processing III, C. J. Cogswell, G. Kino, T. Wilson, eds., Proc. SPIE2655, 175–186 (1996).
[CrossRef]

M Born, E Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 6th ed. (Pergamon, New York, 1980).

M Abramowitz, I. A. Stegun, eds., Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover, New York, 1972), Chap. 12.

R. N. Bracewell, The Fourier Transform and Its Applications, 3rd ed. (McGraw-Hill, Boston, 2000), pp. 351–356.

The netlib routines are available at http://www.netlib.org .

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Figures (7)

Fig. 1
Fig. 1

Real part of the pupil function for out-of-focus 40 × / 1.00 - NA oil-immersion objective. The plot shown corresponds to an out-of-focus depth of 15.75 μm (plane 64), assuming a constant amplitude. The radial coordinate is normalized to a maximum of 1.

Fig. 2
Fig. 2

Sweep signal used as test function. The function is described by f ( x ) = sin π d b - a   x ( b - a ) d + a 2 - a 2 ,  where a = 5 ,   b = 40 , d = 1 , and 0 x 1 .

Fig. 3
Fig. 3

Zero-order Hankel transform of sweep signal computed by numerical Gauss–Kronrod quadrature.

Fig. 4
Fig. 4

PSF as function of r for in-focus plane ( z = 0 ) of test PSF computed by Gauss–Kronrod quadrature and AI-q method.

Fig. 5
Fig. 5

Magnitude of relative errors for in-focus plane ( z = 0 ) of test PSF computed by AI-q method.

Fig. 6
Fig. 6

Magnitude of relative errors in ( 0 ,   z ) point of each plane of test PSF as computed by Barakat’s method ( Δ ρ = 0.01 ) and AI-q method ( Δ ρ = 0.0049 ) .

Fig. 7
Fig. 7

Magnitude of relative errors in plane 64 of test PSF computed by Barakat’s method ( Δ ρ = 0.01 ) and AI-q method ( Δ ρ = 0.0049 ) .

Tables (4)

Tables Icon

Table 1 Errors in NHT for Sweep Signal

Tables Icon

Table 2 CPU Time for NHT of Sweep Signal a

Tables Icon

Table 3 Errors in PSF for N f = 2048 a

Tables Icon

Table 4 CPU Time for PSF Calculation a

Equations (42)

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H ( r ) = 2 π 0 B f ( ρ ) J 0 ( 2 π r ρ ) ρ d ρ ,
Φ ( μ ) = - f [ ( μ 2 + ν 2 ) 1 / 2 ] d ν .
k = 1 K h k   exp ( λ k t ) [ 1 - exp ( - 2 t ) ] - 1 / 2 , t 0 .
g ( y ) = y 2   f ( ρ ) d ρ [ 1 - ( y / ρ ) 2 ] 1 / 2 = y 2 ρ f ( ρ ) d ρ [ ρ 2 - y 2 ] 1 / 2 .
g ( y ) = [ 2   f ( ρ ) ρ 2 - y 2 ] | y - 2 y ρ 2 - y 2 d f ( ρ ) d ρ d ρ .
g ( y ) = 2   f ( Y ) Y 2 - y 2 - 2 y Y ρ 2 - y 2 d f ( ρ ) d ρ d ρ .
y n = n Δ , f n = f ( y n ) , n = 0 ,   1 , ,   N .
f ( ρ ) A n + B n ρ , y n ρ y n + 1 ,
d f ( ρ ) d ρ B n = f n + 1 - f n Δ .
g ( y k ) = 2   f N y N 2 - y k 2 - 2 n = k N - 1 B n y n y n + 1 ρ 2 - y k 2 d ρ
= 2   f N y N 2 - y k 2 + n = k N - 1 B n [ G ( k ,   n + 1 ) - G ( k ,   n ) ] ,
G ( k ,   n ) = y k 2   log ( y n + y n 2 - y k 2 ) - y n y n 2 - y k 2 .
G ( k ,   n ) = Δ 2 { k 2   log [ Δ ( n + n 2 - k 2 ) ] - n n 2 - k 2 } ,
f N y N 2 - y k 2 = Δ f N N 2 - k 2 .
f ( ρ ) A n + B n ρ + C n ρ 2 , y n ρ y n + 1 .
y n + 1 / 2 = ( n + 0.5 ) Δ , f n + 1 / 2 = f ( y n + 1 / 2 ) ,
n = 0 ,   1 , ,   N ,
B n = 2 Δ   [ - ( 2 n + 1.5 ) f n + ( 2 n + 1 ) f n + 1 / 2 - ( 2 n + 0.5 ) f n + 1 ] ,
C n = 2 Δ 2   ( f n - 2   f n + 1 / 2 + f n + 1 ) .
g ( y k ) = 2   f N y N 2 - y k 2 + n = k N - 1 { B n [ G ( k ,   n + 1 ) - G ( k ,   n ) ] + C n [ D ( k ,   n + 1 ) - D ( k ,   n ) ] } ,
D ( k ,   n ) = - 4 Δ 3 3   ( n 2 - k 2 ) 3 / 2 .
f ( x ) = sin π d b - a x ( b - a ) d + a 2 - a 2 ,
h ( r ,   z ) = 0 1 J 0 ( 2 π kr ρ ) P ( ρ ,   z ) ρ d ρ 2 ,
P ( ρ ,   z ) = A ( ρ ) exp [ jW ( ρ ,   z ) ]
$ 0 ( x ) = 0 x J 1 ( y ) y d y ,
$ 1 ( x 1 ,   x 2 ) = x 1 x 2 $ 0 ( y ) d y ,
$ 0 ( x ) = π 2   x [ J 1 ( x ) H 0 ( x ) - J 0 ( x ) H 1 ( x ) ] ,
$ 1 ( 0 ,   x ) = π 2   x 2 [ J 1 ( x ) H 0 ( x ) - J 0 ( x ) H 1 ( x ) ] + x 2 J 0 ( x ) - 2 xJ 1 ( x ) ,
$ 0 ( x ) = x 3 2 n = 0 ( - 1 4 ) n x 2 n ( n ! ) 2 ( n + 1 ) ( 2 n + 3 ) ,
$ 1 ( x 1 ,   x 2 ) = 1 2 n = 0 ( - 1 4 ) n ( x 2 2 n + 4 - x 1 2 n + 4 ) ( n ! ) 2 ( n + 1 ) ( 2 n + 3 ) ( 2 n + 4 )
x ( r ) = Φ ( r ,   r 0 ) x ( r 0 ) + 2 r 0 r Φ ( r ,   ξ ) B ˜ f ( ξ ) d ξ ,
x n - 1 = Φ ( n ) x n + B 0 ( n ) f n - 1 + B 1 ( n ) f n - 1 / 2 + B 2 ( n ) f n ,
g n = Δ Cx n ,
Φ ( n ) = diag n n - 1 λ 1   , ,   n n - 1 λ k ,
B 0 ( n ) = [ h 1 β 0 ( n ; λ 1 ) h k β 0 ( n ; λ k ) ] T ,
B 1 ( n ) = [ h 1 β 1 ( n ; λ 1 ) h k β 1 ( n ; λ k ) ] T ,
B 2 ( n ) = [ h 1 β 2 ( n ; λ 1 ) h k β 2 ( n ; λ k ) ] T ,
β 0 ( n ;   λ ) = D ( n ;   λ ) ( λ + 1 ) 2 + n ( 4 n + 3 λ + 1 ) + n ( λ - 4 n + 3 ) n n - 1 λ + 1 ,
β 1 ( n ;   λ ) = 4 D ( n ;   λ ) λ + 1 + n ( 1 - λ - 2 n ) + n ( 2 n - λ - 3 ) n n - 1 λ + 1 ,
β 2 ( n ;   λ ) = D ( n ;   λ ) ( n - 1 ) ( λ - 1 + 4 n ) + [ ( λ + 3 ) × ( 3 n - λ - 2 ) - 4 n 2 ] n n - 1 λ + 1 ,
D ( n ;   λ ) = - 2 ( n - 1 ) ( λ + 1 ) ( λ + 2 ) ( λ + 3 ) ,
f ( x ) = sin π d b - a   x ( b - a ) d + a 2 - a 2 ,

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