Abstract

It is widely believed, in the areas of optics, image analysis, and visual perception, that the Hilbert transform does not extend naturally and isotropically beyond one dimension. In some areas of image analysis, this belief has restricted the application of the analytic signal concept to multiple dimensions. We show that, contrary to this view, there is a natural, isotropic, and elegant extension. We develop a novel two-dimensional transform in terms of two multiplicative operators: a spiral phase spectral (Fourier) operator and an orientational phase spatial operator. Combining the two operators results in a meaningful two-dimensional quadrature (or Hilbert) transform. The new transform is applied to the problem of closed fringe pattern demodulation in two dimensions, resulting in a direct solution. The new transform has connections with the Riesz transform of classical harmonic analysis. We consider these connections, as well as others such as the propagation of optical phase singularities and the reconstruction of geomagnetic fields.

© 2001 Optical Society of America

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    [CrossRef]
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    [CrossRef]
  4. J. K. T. Eu, A. W. Lohmann, “Isotropic Hilbert spatial filtering,” Opt. Commun. 9, 257–262 (1973).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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  33. W. M. Moon, A. Ushah, V. Singh, B. Bruce, “Application of 2-D Hilbert transform in geophysical imaging with potential field data,” IEEE Trans. Geosci. Remote Sens. 26, 502–510 (1988).
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  38. P. Kube, P. Perona, “Scale-space properties of quadratic feature detectors,” IEEE Trans. Pattern Anal. Mach. Intell. 18, 987–999 (1996).
    [CrossRef]
  39. G. H. Granlund, H. Knutsson, Signal Processing for Computer Vision (Kluwer Academic, Dordrecht, The Netherlands, 1995).
  40. R. Muller, J. Marquard, “Die Hilberttransformation und ihre Verallgemeinerung in Optik und Bildeverarbeitung,” Optik (Stuttgart) 110, 99–109 (1999).
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    [CrossRef]
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    [CrossRef]
  51. I. V. Basistiy, V. Y. Bazhenov, M. S. Soskin, M. V. Vasnetsov, “Optics of light beams with screw dislocations,” Opt. Commun. 103, 422–428 (1993).
    [CrossRef]
  52. D. C. Ghiglia, M. D. Pritt, Two-Dimensional Phase Unwrapping (Wiley, New York, 1998).
  53. There are a number of ways to remove the offset. Low-pass filtering is the simplest but often not the best method. In situations with multiple phase-shifted interferograms, the difference between any two frames will have the offset nullified. Adaptive filtering methods can also provide more accurate offset removal. In practice, offset removal may be difficult. The difficulty exists even for 1-D signal demodulation using Hilbert techniques, as shown in detail by N. E. Huang, Z. Shen, S. Long, M. C. Wu et al., “The empirical mode decomposition and the Hilbert spectrum for nonlinear and nonstationary time series analysis,” Proc. R. Soc. London Ser. A 454, 903–995 (1998). We shall not discuss the difficulty further in this initial exposition, but it should be noted that failure to remove the offset signal correctly may introduce significant errors.
    [CrossRef]
  54. A. H. Nuttall, “On the quadrature approximation to the Hilbert transform of modulated signals,” Proc. IEEE 54, 1458–1459 (1966).
    [CrossRef]
  55. We shall refrain from calling this function the 2-D analytic signal at present because there are several conflicting definitions of analyticity in multiple dimensions. The alternative term “monogenic” does not seem appropriate because the word now has another widespread use in molecular genetics.
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    [CrossRef] [PubMed]
  57. J. G. Daugman, C. J. Downing, “Demodulation, predictive coding, and spatial vision,” J. Opt. Soc. Am. A 12, 641–660 (1995).
    [CrossRef]
  58. K. G. Larkin, B. F. Oreb, “Propagation of errors in different phase-shifting algorithms: a special property of the arctangent function,” in Interferometry: Techniques and Analysis, G. M. Brown, O. Y. Kwon, M. Kujawinska, G. T. Reid, eds., Proc. SPIE1755, 219–227 (1992).
    [CrossRef]
  59. D. Malacara, M. Servin, Z. Malacara, Interferogram Analysis for Optical Testing (Marcel Dekker, New York, 1998).
  60. Sharp-eyed readers may have noticed that the fringe pattern used in Figs. 4, 6, and 7actually contains some spiral discontinuities, which are manifested as ridge endings and bifurcations. The fringe pattern actually satisfies the local simplicity constraint everywhere except at the spiral center points. The robustness of the vortex transform to these discontinuities is significant but is not explored further in our initial exposition of the method.
  61. I. Amidror, “Fourier spectrum of radially periodic images,” J. Opt. Soc. Am. A 14, 816–826 (1997).
    [CrossRef]
  62. R. N. Bracewell, Two-Dimensional Imaging (Prentice-Hall, Englewood Cliffs, N.J., 1995).
  63. L. Cohen, P. Loughlin, D. Vakman, “On an ambiguity in the definition of the amplitude and phase of a signal,” Signal Process. 79, 301–307 (1999).
    [CrossRef]
  64. M. Felsberg, G. Sommer, “The multidimensional isotropic generalization of quadrature filters in geometric algebra,” in Algebraic Frames for the Perception–Action Cycle, AFPAC 2000 (Springer-Verlag, Heidelberg, 2000), pp. 175–185.
  65. S. Gull, A. Lasenby, C. Doran, “Imaginary numbers are not real—the geometric algebra of spacetime,” Found. Phys. 23, 1175–1201 (1993).
    [CrossRef]

2000 (2)

M. R. Arnison, C. J. Cogswell, N. I. Smith, P. W. Fekete, K. G. Larkin, “Using the Hilbert transform for 3D visualisation of differential interference contrast microscope images,” J. Microsc. 199, 79–84 (2000).
[CrossRef] [PubMed]

J. A. Davis, D. E. McNamara, D. Cottrel, J. Campos, “Image processing with the radial Hilbert transform: theory and experiments,” Opt. Lett. 25, 99–101 (2000).
[CrossRef]

1999 (2)

R. Muller, J. Marquard, “Die Hilberttransformation und ihre Verallgemeinerung in Optik und Bildeverarbeitung,” Optik (Stuttgart) 110, 99–109 (1999).

L. Cohen, P. Loughlin, D. Vakman, “On an ambiguity in the definition of the amplitude and phase of a signal,” Signal Process. 79, 301–307 (1999).
[CrossRef]

1998 (2)

There are a number of ways to remove the offset. Low-pass filtering is the simplest but often not the best method. In situations with multiple phase-shifted interferograms, the difference between any two frames will have the offset nullified. Adaptive filtering methods can also provide more accurate offset removal. In practice, offset removal may be difficult. The difficulty exists even for 1-D signal demodulation using Hilbert techniques, as shown in detail by N. E. Huang, Z. Shen, S. Long, M. C. Wu et al., “The empirical mode decomposition and the Hilbert spectrum for nonlinear and nonstationary time series analysis,” Proc. R. Soc. London Ser. A 454, 903–995 (1998). We shall not discuss the difficulty further in this initial exposition, but it should be noted that failure to remove the offset signal correctly may introduce significant errors.
[CrossRef]

A. Carbery, “Harmonic analysis of the Calderon–Zygmund school, 1970–1993,” Bull. London Math. Soc. 30, 11–23 (1998).
[CrossRef]

1997 (3)

1996 (4)

K. G. Larkin, “Efficient nonlinear algorithm for envelope detection in white light interferometry,” J. Opt. Soc. Am. A 13, 832–843 (1996).
[CrossRef]

A. E. Barnes, “Theory of 2-D complex seismic trace analysis,” Geophysics 61, 264–272 (1996).
[CrossRef]

P. Kube, P. Perona, “Scale-space properties of quadratic feature detectors,” IEEE Trans. Pattern Anal. Mach. Intell. 18, 987–999 (1996).
[CrossRef]

M. Craig, “Analytic signals for multivariate data,” Math. Geol. 28, 315–329 (1996).
[CrossRef]

1995 (1)

1994 (1)

1993 (2)

S. Gull, A. Lasenby, C. Doran, “Imaginary numbers are not real—the geometric algebra of spacetime,” Found. Phys. 23, 1175–1201 (1993).
[CrossRef]

I. V. Basistiy, V. Y. Bazhenov, M. S. Soskin, M. V. Vasnetsov, “Optics of light beams with screw dislocations,” Opt. Commun. 103, 422–428 (1993).
[CrossRef]

1990 (2)

Y. M. Zhu, F. Peyrin, R. Goutte, “The use of a two-dimensional Hilbert transform for Wigner analysis of 2-dimensional real signals,” Signal Process. 19, 205–220 (1990).
[CrossRef]

E. Peli, “Contrast in complex images,” J. Opt. Soc. Am. A 7, 2032–2040 (1990).
[CrossRef] [PubMed]

1989 (2)

E. Peli, “Hilbert transform pairs mechanisms,” Invest. Ophthalmol. Visual Sci. 30 (ARVO Suppl.), 110 (1989).

P. Coullet, L. Gil, F. Rocca, “Optical vortices,” Opt. Commun. 73, 403–408 (1989).
[CrossRef]

1988 (2)

W. M. Moon, A. Ushah, V. Singh, B. Bruce, “Application of 2-D Hilbert transform in geophysical imaging with potential field data,” IEEE Trans. Geosci. Remote Sens. 26, 502–510 (1988).
[CrossRef]

M. C. Morrone, D. C. Burr, “Feature detection in human vision: a phase-dependent energy model,” Proc. R. Soc. London Ser. B 235, 221–245 (1988).
[CrossRef]

1986 (3)

1985 (2)

1984 (1)

M. N. Nabighian, “Toward a three-dimensional automatic interpretation of potential field data via generalized Hilbert transform: fundamental relations,” Geophysics 49, 780–786 (1984).
[CrossRef]

1982 (1)

1974 (1)

J. F. Nye, M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. London, Ser. A 336, 165–190 (1974).
[CrossRef]

1973 (1)

J. K. T. Eu, A. W. Lohmann, “Isotropic Hilbert spatial filtering,” Opt. Commun. 9, 257–262 (1973).
[CrossRef]

1972 (1)

M. N. Nabighian, “The analytic signal of two-dimensional magnetic bodies with polygonal cross-section: its properties and use for automated anomaly interpretation,” Geophysics 37, 507–517 (1972).
[CrossRef]

1971 (1)

H. Stark, “An extension to the Hilbert transform product theorem,” Proc. IEEE 59, 1359–1360 (1971).
[CrossRef]

1967 (1)

S. Lowenthal, Y. Belvaux, “Observation of phase objects by optically processed Hilbert transform,” Appl. Phys. Lett. 11, 49–51 (1967).
[CrossRef]

1966 (1)

A. H. Nuttall, “On the quadrature approximation to the Hilbert transform of modulated signals,” Proc. IEEE 54, 1458–1459 (1966).
[CrossRef]

1957 (1)

A. Zygmund, “On singular integrals,” Rend. Mat. 16, 468–505 (1957).

1952 (1)

A. P. Calderon, A. Zygmund, “On the existence of certain singular integrals,” Acta Math. 88, 85–139 (1952).
[CrossRef]

1948 (1)

S. G. Mikhlin, “Singular integral equations,” Usp. Mat. Nauk 3, 29–112 (1948) (in Russian).

1947 (1)

D. Gabor, “Theory of communications,” J. Inst. Electr. Eng. 93, 429–457 (1947).

1936 (1)

G. Giraud, “Sur une classe generale d’equations a integrales principales,” C. R. Acad. Sci. 202, 2124–2126 (1936).

1932 (1)

D. G. Fulton, G. Y. Rainich, “Generalisations to higher dimensions of the Cauchy integral formula,” Am. J. Math. 54, 235–241 (1932).
[CrossRef]

1928 (1)

F. G. Tricomi, “Equazioni integrali contenenti il valor principale di un integrale doppio,” Math. Z. 27, 87–133 (1928).
[CrossRef]

1927 (1)

M. Riesz, “Sur les fonctions conjuguées,” Math. Z. 27, 218–244 (1927).
[CrossRef]

Amidror, I.

Andresen, K.

Arnison, M. R.

M. R. Arnison, C. J. Cogswell, N. I. Smith, P. W. Fekete, K. G. Larkin, “Using the Hilbert transform for 3D visualisation of differential interference contrast microscope images,” J. Microsc. 199, 79–84 (2000).
[CrossRef] [PubMed]

Bachor, H.-A.

Barnes, A. E.

A. E. Barnes, “Theory of 2-D complex seismic trace analysis,” Geophysics 61, 264–272 (1996).
[CrossRef]

Basistiy, I. V.

I. V. Basistiy, V. Y. Bazhenov, M. S. Soskin, M. V. Vasnetsov, “Optics of light beams with screw dislocations,” Opt. Commun. 103, 422–428 (1993).
[CrossRef]

Bazhenov, V. Y.

I. V. Basistiy, V. Y. Bazhenov, M. S. Soskin, M. V. Vasnetsov, “Optics of light beams with screw dislocations,” Opt. Commun. 103, 422–428 (1993).
[CrossRef]

Belvaux, Y.

S. Lowenthal, Y. Belvaux, “Observation of phase objects by optically processed Hilbert transform,” Appl. Phys. Lett. 11, 49–51 (1967).
[CrossRef]

Berry, M. V.

J. F. Nye, M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. London, Ser. A 336, 165–190 (1974).
[CrossRef]

Bone, D. J.

Bovik, A. C.

J. P. Havlicek, J. W. Havlicek, A. C. Bovik, “The analytic image,” in Proceedings of the IEEE International Conference on Image Processing (IEEE, New York, 1997), Vol. 2, pp. 446–449.

Bracewell, R. N.

R. N. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 1978).

R. N. Bracewell, Two-Dimensional Imaging (Prentice-Hall, Englewood Cliffs, N.J., 1995).

Bruce, B.

W. M. Moon, A. Ushah, V. Singh, B. Bruce, “Application of 2-D Hilbert transform in geophysical imaging with potential field data,” IEEE Trans. Geosci. Remote Sens. 26, 502–510 (1988).
[CrossRef]

Bülow, T.

T. Bülow, “Hypercomplex spectral signal representations for the processing and analysis of images,” Ph.D. dissertation (Christian Albrechts University, Kiel, Germany, 1999).

T. Bülow, G. Sommer, “A novel approach to the 2D analytic signal,” presented at the 8th International Conference on Computer Analysis of Images and Patterns, Ljubljana, Slovenia, September 1–3, 1999.

Burr, D. C.

M. C. Morrone, D. C. Burr, “Feature detection in human vision: a phase-dependent energy model,” Proc. R. Soc. London Ser. B 235, 221–245 (1988).
[CrossRef]

Calderon, A. P.

A. P. Calderon, A. Zygmund, “On the existence of certain singular integrals,” Acta Math. 88, 85–139 (1952).
[CrossRef]

Campos, J.

Carbery, A.

A. Carbery, “Harmonic analysis of the Calderon–Zygmund school, 1970–1993,” Bull. London Math. Soc. 30, 11–23 (1998).
[CrossRef]

Cogswell, C. J.

M. R. Arnison, C. J. Cogswell, N. I. Smith, P. W. Fekete, K. G. Larkin, “Using the Hilbert transform for 3D visualisation of differential interference contrast microscope images,” J. Microsc. 199, 79–84 (2000).
[CrossRef] [PubMed]

Cohen, L.

L. Cohen, P. Loughlin, D. Vakman, “On an ambiguity in the definition of the amplitude and phase of a signal,” Signal Process. 79, 301–307 (1999).
[CrossRef]

Condell, W. J.

Cottrel, D.

Coullet, P.

P. Coullet, L. Gil, F. Rocca, “Optical vortices,” Opt. Commun. 73, 403–408 (1989).
[CrossRef]

Craig, M.

M. Craig, “Analytic signals for multivariate data,” Math. Geol. 28, 315–329 (1996).
[CrossRef]

Daugman, J. G.

Davis, J. A.

Doran, C.

S. Gull, A. Lasenby, C. Doran, “Imaginary numbers are not real—the geometric algebra of spacetime,” Found. Phys. 23, 1175–1201 (1993).
[CrossRef]

Downing, C. J.

Eu, J. K. T.

J. K. T. Eu, A. W. Lohmann, “Isotropic Hilbert spatial filtering,” Opt. Commun. 9, 257–262 (1973).
[CrossRef]

Fekete, P. W.

M. R. Arnison, C. J. Cogswell, N. I. Smith, P. W. Fekete, K. G. Larkin, “Using the Hilbert transform for 3D visualisation of differential interference contrast microscope images,” J. Microsc. 199, 79–84 (2000).
[CrossRef] [PubMed]

Felsberg, M.

M. Felsberg, G. Sommer, “The multidimensional isotropic generalization of quadrature filters in geometric algebra,” in Algebraic Frames for the Perception–Action Cycle, AFPAC 2000 (Springer-Verlag, Heidelberg, 2000), pp. 175–185.

Fiddy, M. A.

M. A. Fiddy, “The role of analyticity in image recovery,” in Image Recovery: Theory and Application, H. Stark, ed. (Academic, Orlando, Fla., 1987).

Fienup, J. R.

Figueroa, J. E.

Fulton, D. G.

D. G. Fulton, G. Y. Rainich, “Generalisations to higher dimensions of the Cauchy integral formula,” Am. J. Math. 54, 235–241 (1932).
[CrossRef]

Gabor, D.

D. Gabor, “Theory of communications,” J. Inst. Electr. Eng. 93, 429–457 (1947).

Ghiglia, D. C.

D. C. Ghiglia, M. D. Pritt, Two-Dimensional Phase Unwrapping (Wiley, New York, 1998).

Gil, L.

P. Coullet, L. Gil, F. Rocca, “Optical vortices,” Opt. Commun. 73, 403–408 (1989).
[CrossRef]

Giraud, G.

G. Giraud, “Sur une classe generale d’equations a integrales principales,” C. R. Acad. Sci. 202, 2124–2126 (1936).

Goutte, R.

Y. M. Zhu, F. Peyrin, R. Goutte, “The use of a two-dimensional Hilbert transform for Wigner analysis of 2-dimensional real signals,” Signal Process. 19, 205–220 (1990).
[CrossRef]

Granlund, G. H.

G. H. Granlund, H. Knutsson, Signal Processing for Computer Vision (Kluwer Academic, Dordrecht, The Netherlands, 1995).

Gull, S.

S. Gull, A. Lasenby, C. Doran, “Imaginary numbers are not real—the geometric algebra of spacetime,” Found. Phys. 23, 1175–1201 (1993).
[CrossRef]

Hahn, S. L.

S. L. Hahn, Hilbert Transforms in Signal Processing (Artech House, Norwood, Mass., 1996).

Havlicek, J. P.

J. P. Havlicek, J. W. Havlicek, A. C. Bovik, “The analytic image,” in Proceedings of the IEEE International Conference on Image Processing (IEEE, New York, 1997), Vol. 2, pp. 446–449.

Havlicek, J. W.

J. P. Havlicek, J. W. Havlicek, A. C. Bovik, “The analytic image,” in Proceedings of the IEEE International Conference on Image Processing (IEEE, New York, 1997), Vol. 2, pp. 446–449.

Huang, N. E.

There are a number of ways to remove the offset. Low-pass filtering is the simplest but often not the best method. In situations with multiple phase-shifted interferograms, the difference between any two frames will have the offset nullified. Adaptive filtering methods can also provide more accurate offset removal. In practice, offset removal may be difficult. The difficulty exists even for 1-D signal demodulation using Hilbert techniques, as shown in detail by N. E. Huang, Z. Shen, S. Long, M. C. Wu et al., “The empirical mode decomposition and the Hilbert spectrum for nonlinear and nonstationary time series analysis,” Proc. R. Soc. London Ser. A 454, 903–995 (1998). We shall not discuss the difficulty further in this initial exposition, but it should be noted that failure to remove the offset signal correctly may introduce significant errors.
[CrossRef]

Ina, H.

Jara, E.

Knutsson, H.

G. H. Granlund, H. Knutsson, Signal Processing for Computer Vision (Kluwer Academic, Dordrecht, The Netherlands, 1995).

Kobayashi, S.

Kreis, T.

T. Kreis, Holographic Interferometry. Principles and Methods (Akademie, Berlin, 1996), Vol. 1.

Kube, P.

P. Kube, P. Perona, “Scale-space properties of quadratic feature detectors,” IEEE Trans. Pattern Anal. Mach. Intell. 18, 987–999 (1996).
[CrossRef]

Larkin, K. G.

M. R. Arnison, C. J. Cogswell, N. I. Smith, P. W. Fekete, K. G. Larkin, “Using the Hilbert transform for 3D visualisation of differential interference contrast microscope images,” J. Microsc. 199, 79–84 (2000).
[CrossRef] [PubMed]

K. G. Larkin, “Efficient nonlinear algorithm for envelope detection in white light interferometry,” J. Opt. Soc. Am. A 13, 832–843 (1996).
[CrossRef]

K. G. Larkin, B. F. Oreb, “Propagation of errors in different phase-shifting algorithms: a special property of the arctangent function,” in Interferometry: Techniques and Analysis, G. M. Brown, O. Y. Kwon, M. Kujawinska, G. T. Reid, eds., Proc. SPIE1755, 219–227 (1992).
[CrossRef]

Lasenby, A.

S. Gull, A. Lasenby, C. Doran, “Imaginary numbers are not real—the geometric algebra of spacetime,” Found. Phys. 23, 1175–1201 (1993).
[CrossRef]

Lohmann, A. W.

Long, S.

There are a number of ways to remove the offset. Low-pass filtering is the simplest but often not the best method. In situations with multiple phase-shifted interferograms, the difference between any two frames will have the offset nullified. Adaptive filtering methods can also provide more accurate offset removal. In practice, offset removal may be difficult. The difficulty exists even for 1-D signal demodulation using Hilbert techniques, as shown in detail by N. E. Huang, Z. Shen, S. Long, M. C. Wu et al., “The empirical mode decomposition and the Hilbert spectrum for nonlinear and nonstationary time series analysis,” Proc. R. Soc. London Ser. A 454, 903–995 (1998). We shall not discuss the difficulty further in this initial exposition, but it should be noted that failure to remove the offset signal correctly may introduce significant errors.
[CrossRef]

Loughlin, P.

L. Cohen, P. Loughlin, D. Vakman, “On an ambiguity in the definition of the amplitude and phase of a signal,” Signal Process. 79, 301–307 (1999).
[CrossRef]

Lowenthal, S.

S. Lowenthal, Y. Belvaux, “Observation of phase objects by optically processed Hilbert transform,” Appl. Phys. Lett. 11, 49–51 (1967).
[CrossRef]

Malacara, D.

D. Malacara, M. Servin, Z. Malacara, Interferogram Analysis for Optical Testing (Marcel Dekker, New York, 1998).

Malacara, Z.

D. Malacara, M. Servin, Z. Malacara, Interferogram Analysis for Optical Testing (Marcel Dekker, New York, 1998).

Marquard, J.

R. Muller, J. Marquard, “Die Hilberttransformation und ihre Verallgemeinerung in Optik und Bildeverarbeitung,” Optik (Stuttgart) 110, 99–109 (1999).

Marroquin, J. L.

McNamara, D. E.

Mikhlin, S. G.

S. G. Mikhlin, “Singular integral equations,” Usp. Mat. Nauk 3, 29–112 (1948) (in Russian).

S. G. Mikhlin, Multidimensional Singular Integrals and Integral Equations (Pergamon, Oxford, UK, 1965).

Moon, W. M.

W. M. Moon, A. Ushah, V. Singh, B. Bruce, “Application of 2-D Hilbert transform in geophysical imaging with potential field data,” IEEE Trans. Geosci. Remote Sens. 26, 502–510 (1988).
[CrossRef]

Morrone, M. C.

M. C. Morrone, D. C. Burr, “Feature detection in human vision: a phase-dependent energy model,” Proc. R. Soc. London Ser. B 235, 221–245 (1988).
[CrossRef]

Muller, R.

R. Muller, J. Marquard, “Die Hilberttransformation und ihre Verallgemeinerung in Optik und Bildeverarbeitung,” Optik (Stuttgart) 110, 99–109 (1999).

Nabighian, M. N.

M. N. Nabighian, “Toward a three-dimensional automatic interpretation of potential field data via generalized Hilbert transform: fundamental relations,” Geophysics 49, 780–786 (1984).
[CrossRef]

M. N. Nabighian, “The analytic signal of two-dimensional magnetic bodies with polygonal cross-section: its properties and use for automated anomaly interpretation,” Geophysics 37, 507–517 (1972).
[CrossRef]

Nugent, K. A.

Nuttall, A. H.

A. H. Nuttall, “On the quadrature approximation to the Hilbert transform of modulated signals,” Proc. IEEE 54, 1458–1459 (1966).
[CrossRef]

Nye, J. F.

J. F. Nye, M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. London, Ser. A 336, 165–190 (1974).
[CrossRef]

Ojeda-Castanada, J.

Oreb, B. F.

K. G. Larkin, B. F. Oreb, “Propagation of errors in different phase-shifting algorithms: a special property of the arctangent function,” in Interferometry: Techniques and Analysis, G. M. Brown, O. Y. Kwon, M. Kujawinska, G. T. Reid, eds., Proc. SPIE1755, 219–227 (1992).
[CrossRef]

Peli, E.

E. Peli, “Contrast in complex images,” J. Opt. Soc. Am. A 7, 2032–2040 (1990).
[CrossRef] [PubMed]

E. Peli, “Hilbert transform pairs mechanisms,” Invest. Ophthalmol. Visual Sci. 30 (ARVO Suppl.), 110 (1989).

Perona, P.

P. Kube, P. Perona, “Scale-space properties of quadratic feature detectors,” IEEE Trans. Pattern Anal. Mach. Intell. 18, 987–999 (1996).
[CrossRef]

Peyrin, F.

Y. M. Zhu, F. Peyrin, R. Goutte, “The use of a two-dimensional Hilbert transform for Wigner analysis of 2-dimensional real signals,” Signal Process. 19, 205–220 (1990).
[CrossRef]

Pritt, M. D.

D. C. Ghiglia, M. D. Pritt, Two-Dimensional Phase Unwrapping (Wiley, New York, 1998).

Pudney, C.

C. Pudney, M. Robbins, “Surface extraction from 3D images via local energy and ridge tracing,” in Digital Image Computing: Techniques and Analysis (Australian Pattern Recognition Society, Brisbane, Australia, 1995), pp. 240–245.

Rainich, G. Y.

D. G. Fulton, G. Y. Rainich, “Generalisations to higher dimensions of the Cauchy integral formula,” Am. J. Math. 54, 235–241 (1932).
[CrossRef]

Ramirez, J. G.

Riesz, M.

M. Riesz, “Sur les fonctions conjuguées,” Math. Z. 27, 218–244 (1927).
[CrossRef]

Robbins, M.

C. Pudney, M. Robbins, “Surface extraction from 3D images via local energy and ridge tracing,” in Digital Image Computing: Techniques and Analysis (Australian Pattern Recognition Society, Brisbane, Australia, 1995), pp. 240–245.

Rocca, F.

P. Coullet, L. Gil, F. Rocca, “Optical vortices,” Opt. Commun. 73, 403–408 (1989).
[CrossRef]

Sandeman, R. J.

Servin, M.

J. L. Marroquin, J. E. Figueroa, M. Servin, “Robust quadrature filters,” J. Opt. Soc. Am. A 14, 779–791 (1997).
[CrossRef]

D. Malacara, M. Servin, Z. Malacara, Interferogram Analysis for Optical Testing (Marcel Dekker, New York, 1998).

Shen, Z.

There are a number of ways to remove the offset. Low-pass filtering is the simplest but often not the best method. In situations with multiple phase-shifted interferograms, the difference between any two frames will have the offset nullified. Adaptive filtering methods can also provide more accurate offset removal. In practice, offset removal may be difficult. The difficulty exists even for 1-D signal demodulation using Hilbert techniques, as shown in detail by N. E. Huang, Z. Shen, S. Long, M. C. Wu et al., “The empirical mode decomposition and the Hilbert spectrum for nonlinear and nonstationary time series analysis,” Proc. R. Soc. London Ser. A 454, 903–995 (1998). We shall not discuss the difficulty further in this initial exposition, but it should be noted that failure to remove the offset signal correctly may introduce significant errors.
[CrossRef]

Singh, V.

W. M. Moon, A. Ushah, V. Singh, B. Bruce, “Application of 2-D Hilbert transform in geophysical imaging with potential field data,” IEEE Trans. Geosci. Remote Sens. 26, 502–510 (1988).
[CrossRef]

Smith, N. I.

M. R. Arnison, C. J. Cogswell, N. I. Smith, P. W. Fekete, K. G. Larkin, “Using the Hilbert transform for 3D visualisation of differential interference contrast microscope images,” J. Microsc. 199, 79–84 (2000).
[CrossRef] [PubMed]

Sommer, G.

M. Felsberg, G. Sommer, “The multidimensional isotropic generalization of quadrature filters in geometric algebra,” in Algebraic Frames for the Perception–Action Cycle, AFPAC 2000 (Springer-Verlag, Heidelberg, 2000), pp. 175–185.

T. Bülow, G. Sommer, “A novel approach to the 2D analytic signal,” presented at the 8th International Conference on Computer Analysis of Images and Patterns, Ljubljana, Slovenia, September 1–3, 1999.

Soskin, M. S.

I. V. Basistiy, V. Y. Bazhenov, M. S. Soskin, M. V. Vasnetsov, “Optics of light beams with screw dislocations,” Opt. Commun. 103, 422–428 (1993).
[CrossRef]

Stark, H.

H. Stark, “An extension to the Hilbert transform product theorem,” Proc. IEEE 59, 1359–1360 (1971).
[CrossRef]

Stein, E. M.

E. M. Stein, Singular Integrals and Differentiability Properties of Functions (Princeton U. Press, Princeton, N.J., 1970).

Takeda, M.

Tepichin, E.

Tricomi, F. G.

F. G. Tricomi, “Equazioni integrali contenenti il valor principale di un integrale doppio,” Math. Z. 27, 87–133 (1928).
[CrossRef]

Ushah, A.

W. M. Moon, A. Ushah, V. Singh, B. Bruce, “Application of 2-D Hilbert transform in geophysical imaging with potential field data,” IEEE Trans. Geosci. Remote Sens. 26, 502–510 (1988).
[CrossRef]

Vakman, D.

L. Cohen, P. Loughlin, D. Vakman, “On an ambiguity in the definition of the amplitude and phase of a signal,” Signal Process. 79, 301–307 (1999).
[CrossRef]

Vasnetsov, M. V.

I. V. Basistiy, V. Y. Bazhenov, M. S. Soskin, M. V. Vasnetsov, “Optics of light beams with screw dislocations,” Opt. Commun. 103, 422–428 (1993).
[CrossRef]

Wackerman, C. C.

Wu, M. C.

There are a number of ways to remove the offset. Low-pass filtering is the simplest but often not the best method. In situations with multiple phase-shifted interferograms, the difference between any two frames will have the offset nullified. Adaptive filtering methods can also provide more accurate offset removal. In practice, offset removal may be difficult. The difficulty exists even for 1-D signal demodulation using Hilbert techniques, as shown in detail by N. E. Huang, Z. Shen, S. Long, M. C. Wu et al., “The empirical mode decomposition and the Hilbert spectrum for nonlinear and nonstationary time series analysis,” Proc. R. Soc. London Ser. A 454, 903–995 (1998). We shall not discuss the difficulty further in this initial exposition, but it should be noted that failure to remove the offset signal correctly may introduce significant errors.
[CrossRef]

Yu, Q.

Zayed, A. I.

According to Ahmed Zayed [A. I. Zayed, Handbook of Generalized Function Transformations (CRC Press, Boca Raton, Fla., 1996)], the HT was so named by G. H. Hardy after David Hilbert (1862–1943), who was the first to observe the conjugate functions now known as a HT pair in D. Hilbert, Grundzüge einer allgemeinen Theorie der linearen Integralgleichungen (Tuebner, Leipzig, 1912).

Zhu, Y. M.

Y. M. Zhu, F. Peyrin, R. Goutte, “The use of a two-dimensional Hilbert transform for Wigner analysis of 2-dimensional real signals,” Signal Process. 19, 205–220 (1990).
[CrossRef]

Zygmund, A.

A. Zygmund, “On singular integrals,” Rend. Mat. 16, 468–505 (1957).

A. P. Calderon, A. Zygmund, “On the existence of certain singular integrals,” Acta Math. 88, 85–139 (1952).
[CrossRef]

Acta Math. (1)

A. P. Calderon, A. Zygmund, “On the existence of certain singular integrals,” Acta Math. 88, 85–139 (1952).
[CrossRef]

Am. J. Math. (1)

D. G. Fulton, G. Y. Rainich, “Generalisations to higher dimensions of the Cauchy integral formula,” Am. J. Math. 54, 235–241 (1932).
[CrossRef]

Appl. Opt. (5)

Appl. Phys. Lett. (1)

S. Lowenthal, Y. Belvaux, “Observation of phase objects by optically processed Hilbert transform,” Appl. Phys. Lett. 11, 49–51 (1967).
[CrossRef]

Bull. London Math. Soc. (1)

A. Carbery, “Harmonic analysis of the Calderon–Zygmund school, 1970–1993,” Bull. London Math. Soc. 30, 11–23 (1998).
[CrossRef]

C. R. Acad. Sci. (1)

G. Giraud, “Sur une classe generale d’equations a integrales principales,” C. R. Acad. Sci. 202, 2124–2126 (1936).

Found. Phys. (1)

S. Gull, A. Lasenby, C. Doran, “Imaginary numbers are not real—the geometric algebra of spacetime,” Found. Phys. 23, 1175–1201 (1993).
[CrossRef]

Geophysics (3)

A. E. Barnes, “Theory of 2-D complex seismic trace analysis,” Geophysics 61, 264–272 (1996).
[CrossRef]

M. N. Nabighian, “The analytic signal of two-dimensional magnetic bodies with polygonal cross-section: its properties and use for automated anomaly interpretation,” Geophysics 37, 507–517 (1972).
[CrossRef]

M. N. Nabighian, “Toward a three-dimensional automatic interpretation of potential field data via generalized Hilbert transform: fundamental relations,” Geophysics 49, 780–786 (1984).
[CrossRef]

IEEE Trans. Geosci. Remote Sens. (1)

W. M. Moon, A. Ushah, V. Singh, B. Bruce, “Application of 2-D Hilbert transform in geophysical imaging with potential field data,” IEEE Trans. Geosci. Remote Sens. 26, 502–510 (1988).
[CrossRef]

IEEE Trans. Pattern Anal. Mach. Intell. (1)

P. Kube, P. Perona, “Scale-space properties of quadratic feature detectors,” IEEE Trans. Pattern Anal. Mach. Intell. 18, 987–999 (1996).
[CrossRef]

Invest. Ophthalmol. Visual Sci. (1)

E. Peli, “Hilbert transform pairs mechanisms,” Invest. Ophthalmol. Visual Sci. 30 (ARVO Suppl.), 110 (1989).

J. Inst. Electr. Eng. (1)

D. Gabor, “Theory of communications,” J. Inst. Electr. Eng. 93, 429–457 (1947).

J. Microsc. (1)

M. R. Arnison, C. J. Cogswell, N. I. Smith, P. W. Fekete, K. G. Larkin, “Using the Hilbert transform for 3D visualisation of differential interference contrast microscope images,” J. Microsc. 199, 79–84 (2000).
[CrossRef] [PubMed]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (7)

Math. Geol. (1)

M. Craig, “Analytic signals for multivariate data,” Math. Geol. 28, 315–329 (1996).
[CrossRef]

Math. Z. (2)

M. Riesz, “Sur les fonctions conjuguées,” Math. Z. 27, 218–244 (1927).
[CrossRef]

F. G. Tricomi, “Equazioni integrali contenenti il valor principale di un integrale doppio,” Math. Z. 27, 87–133 (1928).
[CrossRef]

Opt. Commun. (3)

J. K. T. Eu, A. W. Lohmann, “Isotropic Hilbert spatial filtering,” Opt. Commun. 9, 257–262 (1973).
[CrossRef]

P. Coullet, L. Gil, F. Rocca, “Optical vortices,” Opt. Commun. 73, 403–408 (1989).
[CrossRef]

I. V. Basistiy, V. Y. Bazhenov, M. S. Soskin, M. V. Vasnetsov, “Optics of light beams with screw dislocations,” Opt. Commun. 103, 422–428 (1993).
[CrossRef]

Opt. Lett. (1)

Optik (Stuttgart) (1)

R. Muller, J. Marquard, “Die Hilberttransformation und ihre Verallgemeinerung in Optik und Bildeverarbeitung,” Optik (Stuttgart) 110, 99–109 (1999).

Proc. IEEE (2)

A. H. Nuttall, “On the quadrature approximation to the Hilbert transform of modulated signals,” Proc. IEEE 54, 1458–1459 (1966).
[CrossRef]

H. Stark, “An extension to the Hilbert transform product theorem,” Proc. IEEE 59, 1359–1360 (1971).
[CrossRef]

Proc. R. Soc. London Ser. A (1)

There are a number of ways to remove the offset. Low-pass filtering is the simplest but often not the best method. In situations with multiple phase-shifted interferograms, the difference between any two frames will have the offset nullified. Adaptive filtering methods can also provide more accurate offset removal. In practice, offset removal may be difficult. The difficulty exists even for 1-D signal demodulation using Hilbert techniques, as shown in detail by N. E. Huang, Z. Shen, S. Long, M. C. Wu et al., “The empirical mode decomposition and the Hilbert spectrum for nonlinear and nonstationary time series analysis,” Proc. R. Soc. London Ser. A 454, 903–995 (1998). We shall not discuss the difficulty further in this initial exposition, but it should be noted that failure to remove the offset signal correctly may introduce significant errors.
[CrossRef]

Proc. R. Soc. London Ser. B (1)

M. C. Morrone, D. C. Burr, “Feature detection in human vision: a phase-dependent energy model,” Proc. R. Soc. London Ser. B 235, 221–245 (1988).
[CrossRef]

Proc. R. Soc. London, Ser. A (1)

J. F. Nye, M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. London, Ser. A 336, 165–190 (1974).
[CrossRef]

Rend. Mat. (1)

A. Zygmund, “On singular integrals,” Rend. Mat. 16, 468–505 (1957).

Signal Process. (2)

Y. M. Zhu, F. Peyrin, R. Goutte, “The use of a two-dimensional Hilbert transform for Wigner analysis of 2-dimensional real signals,” Signal Process. 19, 205–220 (1990).
[CrossRef]

L. Cohen, P. Loughlin, D. Vakman, “On an ambiguity in the definition of the amplitude and phase of a signal,” Signal Process. 79, 301–307 (1999).
[CrossRef]

Usp. Mat. Nauk (1)

S. G. Mikhlin, “Singular integral equations,” Usp. Mat. Nauk 3, 29–112 (1948) (in Russian).

Other (21)

S. G. Mikhlin, Multidimensional Singular Integrals and Integral Equations (Pergamon, Oxford, UK, 1965).

C. Pudney, M. Robbins, “Surface extraction from 3D images via local energy and ridge tracing,” in Digital Image Computing: Techniques and Analysis (Australian Pattern Recognition Society, Brisbane, Australia, 1995), pp. 240–245.

There may have been some confusion about the priority of crucial results in the properties of multidimensional singular integrals. It seems that in 1948 Mikhlin24,25showed the L2boundedness of the 2-D RT, whereas in 1952 Calderon and Zygmund26,27proved the more general Lpboundedness of the n-dimensional RT.

E. M. Stein, Singular Integrals and Differentiability Properties of Functions (Princeton U. Press, Princeton, N.J., 1970).

Not surprisingly, a separable (i.e., orthant) definition of the multidimensional HT leads only to separable solutions.

S. L. Hahn, Hilbert Transforms in Signal Processing (Artech House, Norwood, Mass., 1996).

J. P. Havlicek, J. W. Havlicek, A. C. Bovik, “The analytic image,” in Proceedings of the IEEE International Conference on Image Processing (IEEE, New York, 1997), Vol. 2, pp. 446–449.

T. Bülow, G. Sommer, “A novel approach to the 2D analytic signal,” presented at the 8th International Conference on Computer Analysis of Images and Patterns, Ljubljana, Slovenia, September 1–3, 1999.

T. Bülow, “Hypercomplex spectral signal representations for the processing and analysis of images,” Ph.D. dissertation (Christian Albrechts University, Kiel, Germany, 1999).

M. A. Fiddy, “The role of analyticity in image recovery,” in Image Recovery: Theory and Application, H. Stark, ed. (Academic, Orlando, Fla., 1987).

According to Ahmed Zayed [A. I. Zayed, Handbook of Generalized Function Transformations (CRC Press, Boca Raton, Fla., 1996)], the HT was so named by G. H. Hardy after David Hilbert (1862–1943), who was the first to observe the conjugate functions now known as a HT pair in D. Hilbert, Grundzüge einer allgemeinen Theorie der linearen Integralgleichungen (Tuebner, Leipzig, 1912).

M. Felsberg, G. Sommer, “The multidimensional isotropic generalization of quadrature filters in geometric algebra,” in Algebraic Frames for the Perception–Action Cycle, AFPAC 2000 (Springer-Verlag, Heidelberg, 2000), pp. 175–185.

We shall refrain from calling this function the 2-D analytic signal at present because there are several conflicting definitions of analyticity in multiple dimensions. The alternative term “monogenic” does not seem appropriate because the word now has another widespread use in molecular genetics.

R. N. Bracewell, Two-Dimensional Imaging (Prentice-Hall, Englewood Cliffs, N.J., 1995).

K. G. Larkin, B. F. Oreb, “Propagation of errors in different phase-shifting algorithms: a special property of the arctangent function,” in Interferometry: Techniques and Analysis, G. M. Brown, O. Y. Kwon, M. Kujawinska, G. T. Reid, eds., Proc. SPIE1755, 219–227 (1992).
[CrossRef]

D. Malacara, M. Servin, Z. Malacara, Interferogram Analysis for Optical Testing (Marcel Dekker, New York, 1998).

Sharp-eyed readers may have noticed that the fringe pattern used in Figs. 4, 6, and 7actually contains some spiral discontinuities, which are manifested as ridge endings and bifurcations. The fringe pattern actually satisfies the local simplicity constraint everywhere except at the spiral center points. The robustness of the vortex transform to these discontinuities is significant but is not explored further in our initial exposition of the method.

T. Kreis, Holographic Interferometry. Principles and Methods (Akademie, Berlin, 1996), Vol. 1.

D. C. Ghiglia, M. D. Pritt, Two-Dimensional Phase Unwrapping (Wiley, New York, 1998).

G. H. Granlund, H. Knutsson, Signal Processing for Computer Vision (Kluwer Academic, Dordrecht, The Netherlands, 1995).

R. N. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 1978).

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Figures (7)

Fig. 1
Fig. 1

Half-plane signum function plotted as a function of frequency coordinates (u, v).

Fig. 2
Fig. 2

Spiral phase “signum” function exponent ϕ. The principal value of the complex exponent ϕ(u, v) is shown in the range ±π.

Fig. 3
Fig. 3

(a) Definition of local orientation angle β(x, y) for a locally simple fringe pattern. Each point in the fringe pattern has a well-defined orientation angle. (b) Spectral sidelobes related to the local fringe pattern. The lobes are located at a polar angle equal to the fringe normal angle.

Fig. 4
Fig. 4

Spiral phase algorithm for 2-D quadrature function estimation. A purely real fringe pattern is converted to a purely imaginary quadrature pattern by the spiral phase transform. The pattern shown could be an interferogram or a fingerprint pattern, for example. The operation consists of a frequency-domain spiral phase multiplication followed by a space-domain orientation phase multiplication. An additional multiplication can convert the output to real.

Fig. 5
Fig. 5

Comparison of 2-D Hilbert transform (HT) methods for a simple image. The complex images generated by three different methods are compared with the ideal. The vortex operator gives a result similar to the ideal magnitude and phase. Both the half-plane and quarter-plane Hilbert operators give highly anisotropic estimates of magnitude and phase.

Fig. 6
Fig. 6

Comparison of 2-D HT methods for an AM–FM image. The input image has both amplitude and frequency modulation with 10% uniform random noise added. The complex image generated by the vortex operator is visually (and numerically) close to the ideal. In contrast, the half-plane Hilbert result shows gross errors in both magnitude and phase estimates.

Fig. 7
Fig. 7

Comparison of demodulated magnitudes for vortex operator and half-plane Hilbert operator. The vortex operator derives a close estimate of the complex image magnitude, failing only at discontinuities in the phase. The half-plane Hilbert operator derives a highly oscillatory estimate of the magnitude with substantial errors in all regions.

Equations (39)

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

-sin(λx)=H{cos(λx)}forλ>0,
cos(λx)=H{sin(λx)}.
fˆ(x)=H{f(x)}
p(x)=f(x)-ifˆ(x)=|b(x)|exp[iψ(x)]
G(u)=-+g(x)exp(-2πiux)dx=F{g(x)},
i sign(u)G(u)=F{gˆ(x)}.
f(x, y)=a(x, y)+b(x, y)cos[ψ(x, y)].
S(u, v)=u+ivu2+v2=exp[iϕ(u, v)].
g(x, y)=f(x, y)-a(x, y)=b(x, y)cos[ψ(x, y)].
gˆ(x, y)=-b(x, y)sin[ψ(x, y)],
g-igˆ=b exp(iψ).
F -1{exp[iϕ(u, v)]F{b(x, y)cos[ψ(x, y)]}}
i exp[iβ(x, y)]b(x, y)sin[ψ(x, y)].
g1(x, y)=b0cos[2π(u0 x+v0 y)].
F {g1(x, y)}
=G1(u, v)=b02 [δ(u-u0, v-v0)+δ(u+u0, v+v0)].
exp(iϕ)G1(u, v)=exp(iβ0) b02 [δ(u-u0, v-v0)-δ(u+u0, v+v0)].
F -1{exp(iϕ)F{b0cos[2π(u0x+v0y)]}}
=i exp(iβ0)b0sin[2π(u0x+v0y)].
F -1{exp[iϕ(u, v)]F {g(x, y)}}=g(x, y) ** s(x, y),
s(x, y)=i(x+iy)2π(x2+y2)3/2=i exp(iθ)2πr2.
x=r cos(θ),y=r sin(θ).
F -1-i uq=x2πr3,F -1-i vq=y2πr3,
q2=u2+v2.
F -1{i sign(u)}=1πx.
g-igˆ=g-exp(-iβe)F -1{exp(iϕ)F {g}}.
V {g}=-i exp(-iβ)F -1{exp(iϕ)F {g}},
V{g}=-i exp[-i(β+)]F -1{exp(iϕ)F {g}}
=b(x, y)sin[iψ(x, y)]exp(-i).
V{g}b(x, y)sin[iψ(x, y)]1-i-22.
tan[ψ(x, y)]=R[V {g(x, y)}]g(x, y)=1-22tan[ψ(x, y)].
δψ(x, y)-24sin[2ψ(x, y)].
b2b2cos2(ψ)+b21-222sin2(ψ),
δb-22 b sin2(ψ)=-24 b[1-cos(2ψ)].
|g|2+|gˆ|2=|g|2+|F -1{exp(iϕ)F {g}}|2=|b|2δb0.
V {b(r)cos(λr)}b(r)sin(λr),β=θ,0<λ.
V {J0(λr)}J1(λr),0<λ.
V -1{J1(λr)}=J0(λr),
V -1{g}=iF -1{exp(-iϕ)F{exp(iβ)g}}.

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