Abstract

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

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References

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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).
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    [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).

2001

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).

2000

1999

1998

1996

1995

1994

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]

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]

1993

I. V. Basistiy, V. Yu, “Optics of light beams with screw dislocations,” Opt. Commun. 103, 422–428 (1993).
[CrossRef]

1991

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]

1989

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]

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]

1988

1985

1981

N. B. Baranova, B. Ya, “Dislocations of the wave-front surface and zeros of the amplitude,” Sov. Phys. JETP 53, 925–929 (1981).

Abramochkin, E. G.

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]

Aksenov, V.

Arrasmith, W. W.

Banakh, V.

Banakh, V. A.

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).

Baranova, N. B.

N. B. Baranova, B. Ya, “Dislocations of the wave-front surface and zeros of the amplitude,” Sov. Phys. JETP 53, 925–929 (1981).

Basistiy, I. V.

I. V. Basistiy, V. Yu, “Optics of light beams with screw dislocations,” Opt. Commun. 103, 422–428 (1993).
[CrossRef]

Bobrov, B. D.

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]

Brychkov, Y. A.

A. P. Prudnikov, Y. A. Brychkov, I. O. Marichev, Integrals and Series (Gordon and Breach, New York, 1986).

Buckland, J. R.

Constantini, M.

M. Constantini, “A novel phase unwrapping method based on network programming,” IEEE Trans. Geosci. Remote Sens. 36, 813–821 (1998).
[CrossRef]

Cumming, I.

W. Xu, I. Cumming, “A region-growing algorithm for InSAR phase unwrapping,” IEEE Trans. Geosci. Remote Sens. 37, 124–134 (1999).
[CrossRef]

Falits, A. V.

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).

Feshbach, H.

P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953).

Fiddy, M. A.

Flannery, B. P.

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).

Fornaro, G.

G. Fornaro, G. Franceschetti, R. Lanari, E. Sansosti, “Robust phase-unwrapping techniques: a comparison,” J. Opt. Soc. Am. A 13, 2355–2366 (1996).
[CrossRef]

G. Fornaro, G. Franceschetti, R. Lanari, “Interferometric phase unwrapping using Green’s formulation,” IEEE Trans. Geosci. Remote Sens. 34, 720–727 (1996).
[CrossRef]

Franceschetti, G.

G. Fornaro, G. Franceschetti, R. Lanari, “Interferometric phase unwrapping using Green’s formulation,” IEEE Trans. Geosci. Remote Sens. 34, 720–727 (1996).
[CrossRef]

G. Fornaro, G. Franceschetti, R. Lanari, E. Sansosti, “Robust phase-unwrapping techniques: a comparison,” J. Opt. Soc. Am. A 13, 2355–2366 (1996).
[CrossRef]

Fried, D. L.

Ghiglia, D. C.

Goldstein, R. M.

R. M. Goldstein, H. A. Zebker, C. L. Werner, “Satellite radar interferometry: two-dimensional phase unwrapping,” Radio Sci. 23, 713–720 (1988).
[CrossRef]

Gradshteyn, I. S.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series and Products (Academic, New York, 1980).

Gutmann, B.

Huntley, J. M.

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975).

Lanari, R.

G. Fornaro, G. Franceschetti, R. Lanari, E. Sansosti, “Robust phase-unwrapping techniques: a comparison,” J. Opt. Soc. Am. A 13, 2355–2366 (1996).
[CrossRef]

G. Fornaro, G. Franceschetti, R. Lanari, “Interferometric phase unwrapping using Green’s formulation,” IEEE Trans. Geosci. Remote Sens. 34, 720–727 (1996).
[CrossRef]

Lu, Y.

Lyuboshenko, I.

Mai^tre, H.

Marichev, I. O.

A. P. Prudnikov, Y. A. Brychkov, I. O. Marichev, Integrals and Series (Gordon and Breach, New York, 1986).

Morse, P. M.

P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953).

Nico, G.

Press, W. H.

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).

Pritt, M. C.

D. C. Ghiglia, M. C. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms, and Software (Wiley, New York, 1998).

Pritt, M. D.

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]

Prudnikov, A. P.

A. P. Prudnikov, Y. A. Brychkov, I. O. Marichev, Integrals and Series (Gordon and Breach, New York, 1986).

Romero, L. A.

Ryzhik, I. M.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series and Products (Academic, New York, 1980).

Sansosti, E.

Scivier, M. S.

Shipman, J. S.

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]

Takahashi, T.

Takajo, H.

Teukolsky, S. A.

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).

Tikhomirova, O.

Vetterling, W. T.

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).

Volostnikov, V. G.

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]

Weber, H.

Werner, C. L.

R. M. Goldstein, H. A. Zebker, C. L. Werner, “Satellite radar interferometry: two-dimensional phase unwrapping,” Radio Sci. 23, 713–720 (1988).
[CrossRef]

Xu, W.

W. Xu, I. Cumming, “A region-growing algorithm for InSAR phase unwrapping,” IEEE Trans. Geosci. Remote Sens. 37, 124–134 (1999).
[CrossRef]

Ya, B.

N. B. Baranova, B. Ya, “Dislocations of the wave-front surface and zeros of the amplitude,” Sov. Phys. JETP 53, 925–929 (1981).

Yu, V.

I. V. Basistiy, V. Yu, “Optics of light beams with screw dislocations,” Opt. Commun. 103, 422–428 (1993).
[CrossRef]

Zebker, H. A.

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]

R. M. Goldstein, H. A. Zebker, C. L. Werner, “Satellite radar interferometry: two-dimensional phase unwrapping,” Radio Sci. 23, 713–720 (1988).
[CrossRef]

Appl. Opt.

Atmos. Oceanic Opt.

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).

IEEE Trans. Geosci. Remote Sens.

M. Constantini, “A novel phase unwrapping method based on network programming,” IEEE Trans. Geosci. Remote Sens. 36, 813–821 (1998).
[CrossRef]

W. Xu, I. Cumming, “A region-growing algorithm for InSAR phase unwrapping,” IEEE Trans. Geosci. Remote Sens. 37, 124–134 (1999).
[CrossRef]

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]

G. Fornaro, G. Franceschetti, R. Lanari, “Interferometric phase unwrapping using Green’s formulation,” IEEE Trans. Geosci. Remote Sens. 34, 720–727 (1996).
[CrossRef]

J. Opt. Soc. Am. A

D. C. Ghiglia, L. A. Romero, “Minimum Lp-norm two-dimensional phase unwrapping,” J. Opt. Soc. Am. A 13, 1999–2013 (1996).
[CrossRef]

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]

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]

W. W. Arrasmith, “Branch-point-tolerant least-squares phase reconstructor,” J. Opt. Soc. Am. A 16, 1864–1872 (1999).
[CrossRef]

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]

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]

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]

G. Fornaro, G. Franceschetti, R. Lanari, E. Sansosti, “Robust phase-unwrapping techniques: a comparison,” J. Opt. Soc. Am. A 13, 2355–2366 (1996).
[CrossRef]

D. L. Fried, “Branch point problem in adaptive optics,” J. Opt. Soc. Am. A 15, 2759–2768 (1998).
[CrossRef]

G. Nico, “Noise-residue filtering of interferometric phase images,” J. Opt. Soc. Am. A 17, 1962–1974 (2000).
[CrossRef]

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]

Opt. Commun.

I. V. Basistiy, V. Yu, “Optics of light beams with screw dislocations,” Opt. Commun. 103, 422–428 (1993).
[CrossRef]

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]

Opt. Lett.

Radio Sci.

R. M. Goldstein, H. A. Zebker, C. L. Werner, “Satellite radar interferometry: two-dimensional phase unwrapping,” Radio Sci. 23, 713–720 (1988).
[CrossRef]

Sov. Phys. JETP

N. B. Baranova, B. Ya, “Dislocations of the wave-front surface and zeros of the amplitude,” Sov. Phys. JETP 53, 925–929 (1981).

Soviet J. Quantum Electron.

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]

Other

J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975).

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).

A. P. Prudnikov, Y. A. Brychkov, I. O. Marichev, Integrals and Series (Gordon and Breach, New York, 1986).

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series and Products (Academic, New York, 1980).

P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953).

D. C. Ghiglia, M. C. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms, and Software (Wiley, New York, 1998).

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Figures (14)

Fig. 1
Fig. 1

(a) Optical interferogram corresponding to an optical-fiber-waveguide-connector termination. (b) Phase unwrapped from (a) by use of branch cuts and the path-following gradient integration. Size of images, 513 × 513 pixels.

Fig. 2
Fig. 2

Interferometric phase support region S (gray area) and its boundary C, vectors and ŷ in Eq. (1), an example of the integration contour C 0 used in Eq. (2) to determine the residue sign, an example of the unit outward vector S used in Eq. (7), and an example of the vector r i pointing at the position of the ith residue.

Fig. 3
Fig. 3

(a) Squared norm and (b) the maximum value of χ̂(r) as a function of the residue position in an interferogram. Both surfaces reach maximum values for residues situated near the boundary. Residues close to the center of an interferogram produce vanishing values of χ̂(r).

Fig. 4
Fig. 4

Example of (a) a harmonic component and (b) a hidden phase for a pair of residues. Plus and negative signs denote positions of the positive and negative residues, respectively.

Fig. 5
Fig. 5

Sample configuration of the two pairs of closely spaced unmatched residues: A (residues at r i and r j ) and B (residues at r k and at r l ). Residues at r m and r n forming the pair C are spaced a few pixels apart from each other.

Fig. 6
Fig. 6

Residue map associated with the phase in Fig. 1. Bright pixels denote positions of positive residues, whereas dark pixels indicate positions of negative ones. The residue map is divided into four sectors with varying number of enclosed residues. Regions 1, 2, 3, and 4 contain 32%, 27%, 23%, and 18% of the total of 141 residue pairs, respectively. The image size is 512 × 512 pixels.

Fig. 7
Fig. 7

Phase unwrapping by use of the harmonic component and the hidden phase computed from all residues in Fig. 6: (a) a harmonic component computed with Eq. (19); (b) a hidden phase computed with Eq. (24) and residue branch cuts determined by employing the criteria of proximity of residues and the direction of the phase-gradient field11; (c) unwrapped phase computed according to Eq. (27); (d) modulo 2π unwrapping error between the interferometric phase in Fig. 1(a) and the unwrapped phase in (c). The error is due to limits imposed by numerical accuracy and is less than 0.01 of the cycle. The size of images is 512 × 512 pixels.

Fig. 8
Fig. 8

Phase unwrapping by use of branch cuts between residues only in region 1 in Fig. 6: (a) a harmonic component; (b) a hidden phase; (c) an unwrapped phase in which the harmonic component in (a) and the hidden phase in (b) are combined according to Eq. (27) with the phase unwrapped by the LMS method [Eq. (11)]; (d) modulo 2π phase-unwrapping error between the phase in (c) and the interferogram in Fig. 1(a). Size of images, 512 × 512 pixels.

Fig. 9
Fig. 9

(a–c) Phase unwrapping by use of branch cuts between residues only in regions 1 and 2 in Fig. 6: (a) a harmonic component; (b) a hidden phase; (c) a modulo 2π phase-unwrapping error. (d–f) Phase unwrapping by use of branch cuts between residues only in regions 1, 2, and 3 in Fig. 6: (d) a harmonic component; (e) a hidden phase; (f) a modulo 2π phase-unwrapping error. Size of images, 512 × 512 pixels.

Fig. 10
Fig. 10

(a) Unwrapping error and (b) unwrapping error decrease as a function of the number of matched residue pairs in Fig. 6. In (b) the corresponding increment in pairs formed from residues spaced more than one pixel apart (with the distance between residues >2 of a pixel’s width) is also plotted (squares). Triangles denote unwrapping errors computed from 3 × 3-pixel window median-filtered data and filled circles denote those from 5 × 5-pixel window median-filtered data.

Fig. 11
Fig. 11

(a) Interferogram of the Bern area (the image size is 129 × 129 pixels) and (b) the associated residue map (the image size is 128 × 128 pixels). There are 195 positive and 198 negative residues. Locations of the negative residues are depicted as dark pixels, and locations of the positive ones are represented as bright pixels.

Fig. 12
Fig. 12

(a) Harmonic component and (b) a hidden phase computed from 92% of the total amount of residues in the residue map in Fig. 11(b) by use of Eqs. (19) and (24), respectively. Size of images, 128 × 128 pixels.

Fig. 13
Fig. 13

Phase unwrapped by use of the developed method [Eqs. (11), (19), (24), and (27)] after matching 92% of the total amount of residues in Fig. 11(b). The unwrapped phase is superimposed with the corresponding amplitude values. Image size, 128 × 128 pixels.

Fig. 14
Fig. 14

Images of a modulo 2π unwrapping error corresponding to the interferogram in Fig. 11(a). (a) A modulo 2π unwrapping error between the phase in Fig. 13(a) unwrapped by the proposed method [Eqs. (11), (19), (24), and (27)]. Remaining shears in the unwrapping error image are due to unmatched residues. (b) A modulo 2π unwrapping error between the phase unwrapped by the LMS method [Eq. (11)]. Shears in the unwrapping error image are between all residues with positions indicated in Fig. 11(b), as no residues are matched when the LMS phase unwrapping is used. Image sizes, 128 × 128 pixels.

Equations (93)

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

Fr=limΔ0θr+Δxˆ-θrmod 2πΔxˆ+θr+Δyˆ-θrmod 2πΔyˆ=mod 2π θr,
C0Fr · dc=±2π,
×Fr=××Ar=2π iRnˆiriδr-ri,
Ar=S×Frgr, rdr,
Ar=S2π iRnˆiriδr-ri ×gr, rdr=2π iRnˆirigri, r.
×Ar=2π iRgri, r×nˆiri+gri, r×nˆiri =-2π iRnˆiri×gri, r,
ϕr=SFr · gr, rdr-C ϕrnˆS · gr, rdc,
nˆS · gr, r=0.
2gr, r=-δr, r.
gr, r=m,k=0,vmk20 vmk-2FmkrFmkr,
ϕr=SFr · gr, rdr,
r2ϕr=r · Fr.
r2ϕr=SFr · rr2gr, rdr=-SFr · rδr, rdr=r · Fr,
 fxnδ x, x/xndx=-1nnfx/ xn
ϕ-F2=min,
χˆr=-2π iRSg r, r · nˆiri×g r, ridr.
χˆ r=-2π iRSnˆiri · g r, ri×g r, rdr=-iR ωiriSgr, ri×gr, rzdr,
χˆ r=-iR ωiriS-gyr, rigxr, r+gxr, rigyr, rdr.
χˆ x, y=-iR ωiriTr-Wyi, xiWx, y+Wxi, yiWy, x,
Wm,kv, u=2a-1 cosηkvξm2+ηk2sinhkπ-1×ηk sinhkπsinξmu+ξmcoshηku-kπ- -1m coshηku, m, k=0,, M-1.
χˆr=Sχ r · gr, rdr,
Gr=-iR ωirigr, ri.
2χˆ= · χ= · -Gyr, GxrT=0.
χr=iR ωirik=1M-1 ηk-2Qkx, xiFky, yi-Qky, yiFkx, xi+a-1Hx-xiy-Hy-yix,
Fku, v=- sinhkπ-1× Hu-vsinhkπ-ηkucoshηkv-Hv-usinhηkucoshkπ-ηkv.
C×Ar · dc=Cχr · dc=2π P-N.
Φr=ϕ r-χˆ r+χ r,
FUr=mod 2π Φr,
E=FU-F2F2.
F2=ψ2-4π iR Cigri, rψr · dc+4π2i,jRnˆiri · nˆjrjgri, rj,
FU-F2=4π2i,jR-Unˆiri · nˆjrjg ri, rj.
FU-F2=4π2g ri, ri+grj, rj+grk, rk+grl, rl+8π2-gri, rj-gri, rl+gri, rk-grj, rk+grj, rl-grk, rl.
FU-F28π2grA, rA+g rB, rB+8π2-g rA, rA-grA, rB+g rA, rB-grA, rB+g rA, rB-grB, rB=0.
FU-F2=4π2grm, rm+grn, rn+8π2grA, rm-grA, rn-grA, rm+grA, rn+grB, rm-grB, rn-grB, rm+grB, rn-grm, rn =4π2grm, rm+grn, rn-8π2grm, rn.
C0Fr · dc θxi, yj+1-θ xi, yjmod 2π+ θxi+1, yj+1-θ xi, yj+1mod 2π- θxi+1, yj+1-θ xi+1, yjmod 2π- θxi+1, yj-θ xi, yjmod 2π.
Hn,l= m=0M-1k=0M-1 hm,k expj2πmn+kl/M.
hm,k=1M2n=0M-1l=0M-1 Hn,l exp-j2πmn+kl/M.
ϕ xn, yl=N Φn,l+Φn,-l,
Φ˜m,k=mXm,k+kYm,kπm2+k2, Xm,k=X˜m,k+X˜m,-k, Ym,k=Y˜m,k-Y˜m,-k,
αmk2=Amk2a2π2=2π-2if m=0 or k=04π-2otherwise,
gr, r=k=0, m2+k20 αmk2 cosηkycosηky×m=0cosξmxcosξmxm2+k2.
gr, r=2π-2m=1cosξmxcosξmxm2+2π-2k=1cosηkycosηkyk2+4π-2k=1cosηkycosηky×m=1cosξmxcosξmxm2+k2.
m=1cosmum2+k2= π2kcoshkπ-usinhkπ- 12k2,0u2π,
m=1cosξmxcosξmxm2+k2=π2k fkx, x-12k2,k0,
fkx, x= 1sinhkπcoshkπ-ηkxcoshηkxif x>xcoshkπ-ηkxcoshηkxif x<x.
g r, r=2π-1k=1 k-1 cosηkycosηkyfkx, x+2π-2m=1 m-2 cosξmxcosξmx.
m=1 m-2 cosmucosmv=1123u2+3v-π2-π2if 0uv3v2+3u-π2-π2if v<uπ.
gr, r=2π-1k=1 k-1 cosηkycosηkyfkx, x+6a-13x2+3x-a2-a2if xx3x2+3x-a2-a2if x<x.
gr, r=2π-1m=1 m-1 cosξmxcosξmxfmy, y+6a-13y2+3y-a2-a2if yy3y2+3y-a2-a2if y<y.
gxr, r =-2a-1m=1sinξmxcosξmxfmy, y,
gyr, r =-2a-1k=1sinηkycosηkyfkx, x.
χˆ r=4a-2m=1M-1cosξmxi ωiri×k=1M-1 sk, m, ysm, k, xicosηkyi-4a-2k=1M-1cosηkyi ωiri×m=1M-1 sm, k, xsk, m, yicosξmxi,
sm, k, u=0adu sinξmufku, u.
s-m, k, u=0udu sinξmuƒku, u,
s+m, k, u=uadu sinξmufku, u,
sm, k, u=s-m, k, u+s+m, k, u,
du coshαu+βsinγu=αα2+γ2sinhαu+βsinγu-γα2+γ2coshαu+βcosγu.
s-m, k, u= coshηku-kπsinhkπ0udu coshηkusinξmu.
s-m, k, u=coshηku-kπξm2+ηk2sinhkπηk sinhηkusinξmu-ξm coshηkucosξmu|0u= coshηku-kπξm2+ηk2sinhkπηk sinhηkusinξmu-ξm coshηkucosξmu+ξm.
s+m, k, u= coshηkusinhkπuadu coshηku-kπsinξmu.
s+m, k, u=coshηkuξm2+ηk2sinhkπηk sinhηku-kπsinξmu-ξm coshηku-kπcosξmu|ua =coshηkuξm2+ηk2sinhkπ-ηk sinhηku-kπsinξmu+ξm coshηku-kπcosξmu-ξm-1m,
sinhηka-kπ=0, coshηka-kπ=1, cosξma=cosmπ=-1m.
sm, k, u=s-m, k, u+s+m, k, u
=ξm2+ηk2sinhkπ-1× ηk sinξmusinhηkucoshηku-kπ-coshηkusinhηku-kπ+ξm cosξmucoshηkucoshηku-kπ-coshηkucoshηku-kπ+ξmcoshηku-kπ- -1m coshηku.
sinhu-v=sinhucoshv-coshusinhv,
sm, k, u= ξm2+ηk2sinhkπ-1ηk sinhkπ×sinξmu+ξmcoshηku-kπ- -1m coshηku.
Wm,kv, u=2a-1 cosηkvs m, k, u,m, k=0,, M-1.
gxr, r=2π-1k=1 k-1 cosηkycosηkyFkx, x+a-2xif xxx-aif x<x,
gyr, r=2π-1m=1 m-1 cosξmxcosξmxFmy, y+a-2yif yyy-aif y<y,
Fku, v= fku, vu= sinhkπ-1uHu-vcoshkπ-ηkucoshηkv+Hv-ucoshkπ-ηkvcoshηku=- sinhkπ-1Hu-vsinhkπ-ηkucoshηkv-Hv-ucoshkπ-ηkvsinhηku.
χxr=iR ωirigyr, ri,
χyr=-iR ωirigxr, ri.
χr=iR ωiri0x gyu, y; ridu-0y gyx, v; ridv.
0x gyu, y; ridu =2π-1m=1 ξm-22a-1 sinξmxcosξmxiFmy, yi+a-2xyif yyixy-aif yi<y,
0y gxx, v; ridv=2π-1k=1 ηk-22a-1 sinηkycosηkyiFkx, xi+a-2xyif xxiyx-aif xi<x.
χr=iR ωirik=1M-1 ηk-2Qkx, xiFky, yi-Qky, yiFkx, xi-a-10if xxi and yyix-yif x>xi and y>yi-yif x>xi and yyixif xxi and y>yi.
Hn=m=0M-1 hm expj2πmn/M.
Hn=m=0M-1 hm cos2πmn/M.
Δx=Δy=aM/2,
χˆ1xn, yl=m=0M-1 hmylcos2πmn/M,
hmyl=i ωirik=0M-1 bkm, xi, ylcos2πkpi/M,
bkm, xi, yl=4a-2s k, m, ylsm, k, xi.
Hn=k=0M-1 hk sin2πkn/M.
χ1r=2a-1iR ωirik=1M-1 ηk-2Fky, yi×sinηkxcosηkxi.
χ1xn, yl=k=1M-12a-1ηk-2iR ωiriFkyl, yi×cosηkxisin2πkn/M.
Fr=ψ r-2π iRnˆiri×gri, r.
F2=ψ2-4π iRSψ r · nˆiri×g ri, rdr+4π2i,jRSnˆiri×gri, r · nˆjrj×grj, rdr.
Sψr · nˆiri×gri, rdr= Snˆiri · gri, r×ψrdr.
gri, r×ψ r=×gri, rψr.
Snˆiri · gri, r×ψrdr=Cigri, rψr · dc,
Snˆiri×gri, r · nˆjrj×grj, rdr= nˆiri · nˆjrjSgri, r · grj, rdr-Snˆiri · grj, rnˆjrj · gri, rdr.
nˆiri · nˆjrjSgri, r · grj, rdr= nˆiri · nˆjrjgri, rj.
FUr=ψrCompletely reconstructed potential component-2π iURnˆiri×gri, rPartially reconstructed rotational component.

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