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Applied Optics

Applied Optics


  • Vol. 41, Iss. 11 — Apr. 8, 2002
  • pp: 2129–2148

Least-mean-squares phase unwrapping by use of an incomplete set of residue branch cuts

Igor V. Lyuboshenko, Henri Maı̂tre, and Alain Maruani  »View Author Affiliations

Applied Optics, Vol. 41, Issue 11, pp. 2129-2148 (2002)

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A new technique for the least-mean-squares (LMS) phase-unwrapping method is developed that incorporates the concept of branch cuts between phase singularities (residues), which are usually associated with the path-following gradient integration technique. These branch cuts are introduced by decomposition of the least-mean-squares unwrapped phase into two separate components. The first results from the transverse part of the wrapped phase gradient, which is induced by residues of the original phase, and the second component is due to a potential component, independent of the residues. This decomposition allows the reconstruction of phase patterns with a high level of accuracy and consistency with the initial (wrapped) phase, even when only partial knowledge of the placement of branch cuts between residues is available. We show how the residue-induced phase, ignored by conventional LMS phase estimators, is reconstructed for a given boundary-value problem. The method is illustrated with interferometric quality-control measurements of optical fiber-connector terminations and also with synthetic aperture radar interferometry. These experiments demonstrate the high accuracy of the method in practical situations in which only a limited number of branch cuts are available.

© 2002 Optical Society of America

OCIS Codes
(100.2650) Image processing : Fringe analysis
(100.3010) Image processing : Image reconstruction techniques
(100.5070) Image processing : Phase retrieval
(120.0280) Instrumentation, measurement, and metrology : Remote sensing and sensors
(120.3180) Instrumentation, measurement, and metrology : Interferometry

Original Manuscript: February 9, 2001
Revised Manuscript: November 29, 2001
Published: April 10, 2002

Igor V. Lyuboshenko, Henri Maı̂tre, and Alain Maruani, "Least-mean-squares phase unwrapping by use of an incomplete set of residue branch cuts," Appl. Opt. 41, 2129-2148 (2002)

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  1. D. L. Fried, “Branch point problem in adaptive optics,” J. Opt. Soc. Am. A 15, 2759–2768 (1998). [CrossRef]
  2. V. A. Banakh, A. V. Falits, “Phase unwrapping from measured phase differences for optical wave propagating through the turbulent atmosphere,” Atmos. Oceanic Opt. 14, 383–390 (2001).
  3. W. W. Arrasmith, “Branch-point-tolerant least-squares phase reconstructor,” J. Opt. Soc. Am. A 16, 1864–1872 (1999). [CrossRef]
  4. V. Aksenov, V. Banakh, O. Tikhomirova, “Potential and vortex features of optical speckle fields and visualization of wave-front singularities,” Appl. Opt. 37, 4536–4540 (1998). [CrossRef]
  5. J. M. Huntley, J. R. Buckland, “Characterization of sources of 2π phase discontinuity in speckle interferograms,” J. Opt. Soc. Am. A 12, 1990–1996 (1995). [CrossRef]
  6. M. S. Scivier, M. A. Fiddy, “Phase ambiguities and the zeros of multidimensional band-limited functions,” J. Opt. Soc. Am. A 2, 693–697 (1985). [CrossRef]
  7. H. A. Zebker, Y. Lu, “Phase unwrapping algorithms for radar interferometry: residue-cut, least-squares, and synthesis algorithms,” J. Opt. Soc. Am. A 15, 586–598 (1998). [CrossRef]
  8. M. Constantini, “A novel phase unwrapping method based on network programming,” IEEE Trans. Geosci. Remote Sens. 36, 813–821 (1998). [CrossRef]
  9. W. Xu, I. Cumming, “A region-growing algorithm for InSAR phase unwrapping,” IEEE Trans. Geosci. Remote Sens. 37, 124–134 (1999). [CrossRef]
  10. R. M. Goldstein, H. A. Zebker, C. L. Werner, “Satellite radar interferometry: two-dimensional phase unwrapping,” Radio Sci. 23, 713–720 (1988). [CrossRef]
  11. B. Gutmann, H. Weber, “Phase unwrapping with the branch-cut method: role of phase-field direction,” Appl. Opt. 39, 4802–4816 (2000). [CrossRef]
  12. M. D. Pritt, J. S. Shipman, “Least-squares two-dimensional phase unwrapping using FFT’s,” IEEE Trans. Geosci. Remote Sens. 32, 706–708 (1994). [CrossRef]
  13. D. C. Ghiglia, L. A. Romero, “Direct phase estimation from phase differences using fast elliptic partial differential equation solvers,” Opt. Lett. 14, 1107–1109 (1989). [CrossRef] [PubMed]
  14. D. C. Ghiglia, L. A. Romero, “Robust two-dimensional weighted and unweighted phase unwrapping that uses fast transforms and iterative methods,” J. Opt. Soc. Am. A 11, 107–117 (1994). [CrossRef]
  15. P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953).
  16. G. Fornaro, G. Franceschetti, R. Lanari, “Interferometric phase unwrapping using Green’s formulation,” IEEE Trans. Geosci. Remote Sens. 34, 720–727 (1996). [CrossRef]
  17. D. C. Ghiglia, L. A. Romero, “Minimum Lp-norm two-dimensional phase unwrapping,” J. Opt. Soc. Am. A 13, 1999–2013 (1996). [CrossRef]
  18. H. Takajo, T. Takahashi, “Noniterative methods for obtaining the exact solution for the normal equation in least-squares phase estimation from the phase difference,” J. Opt. Soc. Am. A 5, 1818–1827 (1988). [CrossRef]
  19. D. C. Ghiglia, M. C. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms, and Software (Wiley, New York, 1998).
  20. B. D. Bobrov, “Screw dislocations of laser speckle fields in interferograms with a circular line structure,” Soviet J. Quantum Electron. 21, 802–806 (1991). [CrossRef]
  21. I. V. Basistiy, V. Yu, “Optics of light beams with screw dislocations,” Opt. Commun. 103, 422–428 (1993). [CrossRef]
  22. J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975).
  23. I. Lyuboshenko, H. Maı̂tre, “Phase unwrapping for interferometric synthetic aperture radar by use of Helmholtz equation eigenfunctions and the first Green’s identity,” J. Opt. Soc. Am. A 16, 378–395 (1999). [CrossRef]
  24. I. Lyuboshenko, “Unwrapping circular interferograms,” Appl. Opt. 39, 4817–4825 (2000). [CrossRef]
  25. G. Fornaro, G. Franceschetti, R. Lanari, E. Sansosti, “Robust phase-unwrapping techniques: a comparison,” J. Opt. Soc. Am. A 13, 2355–2366 (1996). [CrossRef]
  26. E. G. Abramochkin, V. G. Volostnikov, “Relationship between two-dimensional intensity and phase in a Fresnel diffraction zone,” Opt. Commun. 74, 144–148 (1989). [CrossRef]
  27. N. B. Baranova, B. Ya, “Dislocations of the wave-front surface and zeros of the amplitude,” Sov. Phys. JETP 53, 925–929 (1981).
  28. G. Nico, “Noise-residue filtering of interferometric phase images,” J. Opt. Soc. Am. A 17, 1962–1974 (2000). [CrossRef]
  29. W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C: The Art of Scientific Computing (Cambridge U. Press, Cambridge, England, 1988).
  30. A. P. Prudnikov, Y. A. Brychkov, I. O. Marichev, Integrals and Series (Gordon and Breach, New York, 1986).
  31. I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series and Products (Academic, New York, 1980).

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