Natural demodulation of two-dimensional fringe patterns. I. General background of the spiral phase quadrature transform
JOSA A, Vol. 18, Issue 8, pp. 1862-1870 (2001)
http://dx.doi.org/10.1364/JOSAA.18.001862
Acrobat PDF (1172 KB)
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
OCIS Codes
(100.2650) Image processing : Fringe analysis
(100.2960) Image processing : Image analysis
(100.5070) Image processing : Phase retrieval
(100.5090) Image processing : Phase-only filters
(120.3180) Instrumentation, measurement, and metrology : Interferometry
(120.5050) Instrumentation, measurement, and metrology : Phase measurement
Citation
Kieran G. Larkin, Donald J. Bone, and Michael A. Oldfield, "Natural demodulation of two-dimensional fringe patterns. I. General background of the spiral phase quadrature transform," J. Opt. Soc. Am. A 18, 1862-1870 (2001)
http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-18-8-1862
Sort: Year | Journal | Reset
References
- D. Gabor, “Theory of communications,” J. Inst. Electr. Eng. 93, 429–457 (1947).
- S. Lowenthal and Y. Belvaux, “Observation of phase objects by optically processed Hilbert transform,” Appl. Phys. Lett. 11, 49–51 (1967).
- H. Stark, “An extension to the Hilbert transform product theorem,” Proc. IEEE 59, 1359–1360 (1971).
- J. K. T. Eu and A. W. Lohmann, “Isotropic Hilbert spatial filtering,” Opt. Commun. 9, 257–262 (1973).
- J. Ojeda-Castanada and E. Jara, “Isotropic Hilbert transform by anisotropic spatial filtering,” Appl. Opt. 25, 4035–4038 (1986).
- E. Peli, “Hilbert transform pairs mechanisms,” Invest. Ophthalmol. Visual Sci. 30 (ARVO Suppl.), 110 (1989).
- A. W. Lohmann, E. Tepichin, and J. G. Ramirez, “Optical implementation of the fractional Hilbert transform for two-dimensional objects,” Appl. Opt. 36, 6620–6626 (1997).
- Y. M. Zhu, F. Peyrin, and R. Goutte, “The use of a two-dimensional Hilbert transform for Wigner analysis of 2-dimensional real signals,” Signal Process. 19, 205–220 (1990).
- S. L. Hahn, Hilbert Transforms in Signal Processing (Artech House, Norwood, Mass., 1996).
- J. P. Havlicek, J. W. Havlicek, and 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 and 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. Craig, “Analytic signals for multivariate data,” Math. Geol. 28, 315–329 (1996).
- M. A. Fiddy, “The role of analyticity in image recovery,” in Image Recovery: Theory and Application, H. Stark, ed. (Academic, Orlando, Fla., 1987).
- J. R. Fienup and C. C. Wackerman, “Phase retrieval stagnation problems and solution,” J. Opt. Soc. Am. A 3, 1897–1907 (1986).
- Not surprisingly, a separable (i.e., orthant) definition of the multidimensional HT leads only to separable solutions.
- D. G. Fulton and G. Y. Rainich, “Generalisations to higher dimensions of the Cauchy integral formula,” Am. J. Math. 54, 235–241 (1932).
- M. R. Arnison, C. J. Cogswell, N. I. Smith, P. W. Fekete, and K. G. Larkin, “Using the Hilbert transform for 3D visualisation of differential interference contrast microscope images,” J. Microsc. 199, 79–84 (2000).
- K. G. Larkin, “Efficient nonlinear algorithm for envelope detection in white light interferometry,” J. Opt. Soc. Am. A 13, 832–843 (1996).
- 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. Riesz, “Sur les fonctions conjuguées,” Math. Z. 27, 218–244 (1927).
- F. G. Tricomi, “Equazioni integrali contenenti il valor principale di un integrale doppio,” Math. Z. 27, 87–133 (1928).
- G. Giraud, “Sur une classe generale d’equations a integrales principales,” C. R. Acad. Sci. 202, 2124–2126 (1936).
- 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).
- A. P. Calderon and A. Zygmund, “On the existence of certain singular integrals,” Acta Math. 88, 85–139 (1952).
- A. Zygmund, “On singular integrals,” Rend. Mat. 16, 468–505 (1957).
- 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 25 showed the L ^{2} boundedness of the 2-D RT, whereas in 1952 Calderon and Zygmund26 27 proved the more general L ^{p} boundedness of the n-dimensional RT.
- E. M. Stein, Singular Integrals and Differentiability Properties of Functions (Princeton U. Press, Princeton, N.J., 1970).
- A. Carbery, “Harmonic analysis of the Calderon–Zygmund school, 1970–1993,” Bull. London Math. Soc. 30, 11–23 (1998).
- 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).
- M. N. Nabighian, “Toward a three-dimensional automatic interpretation of potential field data via generalized Hilbert transform: fundamental relations,” Geophysics 49, 780–786 (1984).
- W. M. Moon, A. Ushah, V. Singh, and B. Bruce, “Application of 2-D Hilbert transform in geophysical imaging with potential field data,” IEEE Trans. Geosci. Remote Sens. 26, 502–510 (1988).
- A. E. Barnes, “Theory of 2-D complex seismic trace analysis,” Geophysics 61, 264–272 (1996).
- C. Pudney and 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.
- M. C. Morrone and D. C. Burr, “Feature detection in human vision: a phase-dependent energy model,” Proc. R. Soc. London Ser. B 235, 221–245 (1988).
- E. Peli, “Contrast in complex images,” J. Opt. Soc. Am. A 7, 2032–2040 (1990).
- P. Kube and P. Perona, “Scale-space properties of quadratic feature detectors,” IEEE Trans. Pattern Anal. Mach. Intell. 18, 987–999 (1996).
- G. H. Granlund and H. Knutsson, Signal Processing for Computer Vision (Kluwer Academic, Dordrecht, The Netherlands, 1995).
- R. Muller and J. Marquard, “Die Hilberttransformation und ihre Verallgemeinerung in Optik und Bildeverarbeitung,” Optik (Stuttgart) 110, 99–109 (1999).
- J. A. Davis, D. E. McNamara, D. Cottrel, and J. Campos, “Image processing with the radial Hilbert transform: theory and experiments,” Opt. Lett. 25, 99–101 (2000).
- R. N. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 1978).
- M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. 72, 156–160 (1982).
- K. A. Nugent, “Interferogram analysis using an accurate fully automatic algorithm,” Appl. Opt. 24, 3101–3105 (1985).
- D. J. Bone, H.-A. Bachor, and R. J. Sandeman, “Fringe-pattern analysis using a 2-D Fourier transform,” Appl. Opt. 25, 1653–1660 (1986).
- T. Kreis, Holographic Interferometry. Principles and Methods (Akademie, Berlin, 1996), Vol. 1.
- J. L. Marroquin, J. E. Figueroa, and M. Servin, “Robust quadrature filters,” J. Opt. Soc. Am. A 14, 779–791 (1997).
- J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. London, Ser. A 336, 165–190 (1974).
- W. J. Condell, “Fraunhofer diffraction from a circular annular aperture with helical phase factor,” J. Opt. Soc. Am. A 2, 206–208 (1985).
- P. Coullet, L. Gil, and F. Rocca, “Optical vortices,” Opt. Commun. 73, 403–408 (1989).
- I. V. Basistiy, V. Y. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, “Optics of light beams with screw dislocations,” Opt. Commun. 103, 422–428 (1993).
- D. C. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping (Wiley, New York, 1998).
- 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.
- A. H. Nuttall, “On the quadrature approximation to the Hilbert transform of modulated signals,” Proc. IEEE 54, 1458–1459 (1966).
- 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.
- Q. Yu and K. Andresen, “Fringe-orientation maps and fringe skeleton extraction by the two-dimensional derivative-sign binary-fringe method,” Appl. Opt. 33, 6873–6878 (1994).
- J. G. Daugman and C. J. Downing, “Demodulation, predictive coding, and spatial vision,” J. Opt. Soc. Am. A 12, 641–660 (1995).
- K. G. Larkin and 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, and G. T. Reid, eds., Proc. SPIE 1755, 219–227 (1992).
- D. Malacara, M. Servin, and 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 7 actually 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.
- I. Amidror, “Fourier spectrum of radially periodic images,” J. Opt. Soc. Am. A 14, 816–826 (1997).
- R. N. Bracewell, Two-Dimensional Imaging (Prentice-Hall, Englewood Cliffs, N.J., 1995).
- L. Cohen, P. Loughlin, and D. Vakman, “On an ambiguity in the definition of the amplitude and phase of a signal,” Signal Process. 79, 301–307 (1999).
- M. Felsberg and 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.
- S. Gull, A. Lasenby, and C. Doran, “Imaginary numbers are not real—the geometric algebra of spacetime,” Found. Phys. 23, 1175–1201 (1993).
Cited By |
Alert me when this paper is cited |
OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.
« Previous Article | Next Article »
OSA is a member of CrossRef.