Abstract

We describe a novel algorithm for two-dimensional phase unwrapping. The technique combines the principles of agglomerative clustering and use of heuristics to construct a discontinuous quality-guided path. Unlike other quality-guided algorithms, which establish the path at the start of the unwrapping process, our technique constructs the path as the unwrapping process evolves. This makes the technique less prone to error propagation, although it presents higher execution times than other existing algorithms. The algorithm reacts satisfactorily to random noise and breaks in the phase distribution. A variation of the algorithm is also presented that considerably reduces the execution time without affecting the results significantly.

© 2005 Optical Society of America

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  1. D. C. Ghiglia, M. D. Pritt, Two-dimensional Phase Unwrapping: Theory, Algorithm, and Software (Wiley, New York, 1998).
  2. R. W. Schafer, A. V. Oppenheim, Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975), pp. 507–511.
  3. D. Robinson, “Phase unwrapping methods,” in Interferogram Analysis: Digital Fringe Pattern Measurement Techniques, W. R. Robinson, T. Reid, eds. (Institute of Physics, Philadelphia, Pa., 1993), pp. 194–229.
  4. H. A. Vrooman, A. A. M. Maas, “Image processing algorithms for the analysis of phase-shifted speckle interference patterns,” Appl. Opt. 30, 1636–1641 (1991).
    [Crossref] [PubMed]
  5. J. M. Huntley, “Noise-immune phase unwrapping algorithm,” Appl. Opt. 28, 3268–3270 (1989).
    [Crossref] [PubMed]
  6. R. M. Goldstein, H. A. Zebker, C. L. Werner, “Two-dimensional phase unwrapping,” Radio Sci. 23, 713–720 (1988).
    [Crossref]
  7. K. M. Hung, T. Yamada, “Phase unwrapping by regions using least-squares approach,” Opt. Eng. 37, 2965–2970 (1998).
    [Crossref]
  8. O. Y. Kwon, D. M. Shough, R. A. Williams, “Stroboscopic phase-shifting interferometry,” Opt. Lett. 12, 855–857 (1987).
    [Crossref] [PubMed]
  9. J. Schörner, A. Ettemeyer, U. Neupert, H. Rottenkolber, C. Winter, P. Obermeier, “New approaches in interpreting holographic images,” Opt. Lasers Eng. 14, 283–291 (1991).
    [Crossref]
  10. J. A. Quiroga, E. Bernabeu, “Phase-unwrapping algorithm for noisy phase-map processing,” Appl. Opt. 33, 6725–6731 (1994).
    [Crossref] [PubMed]
  11. J. A. Quiroga, A. González-Cano, E. Bernabeu, “Phase-unwrapping algorithm based on an adaptive criterion,” Appl. Opt. 34, 2560–2563 (1995).
    [Crossref] [PubMed]
  12. M. Arevalillo Herráez, D. R. Burton, M. J. Lalor, M. A. Gdeisat, “Fast two-dimensional phase unwrapping algorithm based on sorting by reliability following a noncontinuous path,” Appl. Opt. 41, 7437–7443 (2002).
    [Crossref]
  13. D. C. Ghiglia, G. A. Mastin, L. A. Romero, “Cellular-automata method for phase unwrapping,” J. Opt. Soc. Am. A 4, 267–280 (1987).
    [Crossref]
  14. D. P. Towers, T. R. Judge, P. J. Bryanston-Cross, “A quasi heterodyne holographic technique and automatic algorithms for phase unwrapping,” in Fringe Pattern Analysis, G. T. Reid, ed., Proc. SPIE1163, 95–119 (1989).
    [Crossref]
  15. M. Arevalillo Herráez, D. R. Burton, M. J. Lalor, D. B. Clegg, “Robust, simple and fast algorithm for phase unwrapping,” Appl. Opt. 35, 5847–5852 (1996).
    [Crossref]
  16. A. Baldi, “Two-dimensional phase unwrapping by quad-tree decomposition,” Appl. Opt. 40, 1187–1194 (2001).
    [Crossref]
  17. J. J. Gierloff, “Phase unwrapping by regions,” in Current Developments in Optical Engineering II, R. E. Fischer, W. J. Smith, eds., Proc. SPIE818, 2–9 (1987).
  18. M. Arevalillo Herráez, M. A. Gdeisat, D. R. Burton, M. J. Lalor, “Robust, fast, and effective two-dimensional automatic phase unwrapping algorithm based on image decomposition,” Appl. Opt. 41, 7445–7455 (2002).
    [Crossref]
  19. M. Servin, R. Rodriguez-Vera, A. J. Moore, “A robust cellular processor for phase unwrapping,” J. Mod. Opt. 41, 119–127 (1994).
    [Crossref]
  20. 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]
  21. J. M. Huntley, “New methods for unwrapping noisy phase maps,” in Interferometry ’94: New Techniques and Analysis in Optical Measurements, M. Kujawinska, K. Przysztof, eds., Proc. SPIE, 2340, 110–122 (1994).
  22. J. M. Huntley, H. O. Saldner, “Temporal phase-unwrapping algorithm for automated interferogram analysis,” Appl. Opt. 32, 3047–3052 (1993).
    [Crossref] [PubMed]
  23. A. K. Jain, M. N. Murty, P. J. Flynn, “Data clustering: a review,” ACM Comput. Surveys 31, 264–323 (1999).
    [Crossref]
  24. A. K. Jain, P. J. Flynn, Three-Dimensional Object Recognition Systems (Elsevier Science, New York, 1993).
  25. T. Kurita, “An efficient agglomerative clustering algorithm using a heap,” Pattern Recogn. 24, 205–209 (1991).
    [Crossref]
  26. J. W. J. Williams, “Heapsort,” Commun. ACM 7, 347–348 (1964).
  27. P. H. A. Sneath, R. R. Sokal, Numerical Taxonomy (Freeman, London, 1973).
  28. B. King, “Step-wise clustering procedures,” J. Am. Stat. Assoc. 69, 86–101 (1967).
    [Crossref]

2002 (2)

2001 (1)

1999 (1)

A. K. Jain, M. N. Murty, P. J. Flynn, “Data clustering: a review,” ACM Comput. Surveys 31, 264–323 (1999).
[Crossref]

1998 (1)

K. M. Hung, T. Yamada, “Phase unwrapping by regions using least-squares approach,” Opt. Eng. 37, 2965–2970 (1998).
[Crossref]

1996 (1)

1995 (1)

1994 (3)

1993 (1)

1991 (3)

T. Kurita, “An efficient agglomerative clustering algorithm using a heap,” Pattern Recogn. 24, 205–209 (1991).
[Crossref]

J. Schörner, A. Ettemeyer, U. Neupert, H. Rottenkolber, C. Winter, P. Obermeier, “New approaches in interpreting holographic images,” Opt. Lasers Eng. 14, 283–291 (1991).
[Crossref]

H. A. Vrooman, A. A. M. Maas, “Image processing algorithms for the analysis of phase-shifted speckle interference patterns,” Appl. Opt. 30, 1636–1641 (1991).
[Crossref] [PubMed]

1989 (1)

1988 (1)

R. M. Goldstein, H. A. Zebker, C. L. Werner, “Two-dimensional phase unwrapping,” Radio Sci. 23, 713–720 (1988).
[Crossref]

1987 (2)

1967 (1)

B. King, “Step-wise clustering procedures,” J. Am. Stat. Assoc. 69, 86–101 (1967).
[Crossref]

1964 (1)

J. W. J. Williams, “Heapsort,” Commun. ACM 7, 347–348 (1964).

Arevalillo Herráez, M.

Baldi, A.

Bernabeu, E.

Bryanston-Cross, P. J.

D. P. Towers, T. R. Judge, P. J. Bryanston-Cross, “A quasi heterodyne holographic technique and automatic algorithms for phase unwrapping,” in Fringe Pattern Analysis, G. T. Reid, ed., Proc. SPIE1163, 95–119 (1989).
[Crossref]

Burton, D. R.

Clegg, D. B.

Ettemeyer, A.

J. Schörner, A. Ettemeyer, U. Neupert, H. Rottenkolber, C. Winter, P. Obermeier, “New approaches in interpreting holographic images,” Opt. Lasers Eng. 14, 283–291 (1991).
[Crossref]

Flynn, P. J.

A. K. Jain, M. N. Murty, P. J. Flynn, “Data clustering: a review,” ACM Comput. Surveys 31, 264–323 (1999).
[Crossref]

A. K. Jain, P. J. Flynn, Three-Dimensional Object Recognition Systems (Elsevier Science, New York, 1993).

Gdeisat, M. A.

Ghiglia, D. C.

Gierloff, J. J.

J. J. Gierloff, “Phase unwrapping by regions,” in Current Developments in Optical Engineering II, R. E. Fischer, W. J. Smith, eds., Proc. SPIE818, 2–9 (1987).

Goldstein, R. M.

R. M. Goldstein, H. A. Zebker, C. L. Werner, “Two-dimensional phase unwrapping,” Radio Sci. 23, 713–720 (1988).
[Crossref]

González-Cano, A.

Hung, K. M.

K. M. Hung, T. Yamada, “Phase unwrapping by regions using least-squares approach,” Opt. Eng. 37, 2965–2970 (1998).
[Crossref]

Huntley, J. M.

J. M. Huntley, H. O. Saldner, “Temporal phase-unwrapping algorithm for automated interferogram analysis,” Appl. Opt. 32, 3047–3052 (1993).
[Crossref] [PubMed]

J. M. Huntley, “Noise-immune phase unwrapping algorithm,” Appl. Opt. 28, 3268–3270 (1989).
[Crossref] [PubMed]

J. M. Huntley, “New methods for unwrapping noisy phase maps,” in Interferometry ’94: New Techniques and Analysis in Optical Measurements, M. Kujawinska, K. Przysztof, eds., Proc. SPIE, 2340, 110–122 (1994).

Jain, A. K.

A. K. Jain, M. N. Murty, P. J. Flynn, “Data clustering: a review,” ACM Comput. Surveys 31, 264–323 (1999).
[Crossref]

A. K. Jain, P. J. Flynn, Three-Dimensional Object Recognition Systems (Elsevier Science, New York, 1993).

Judge, T. R.

D. P. Towers, T. R. Judge, P. J. Bryanston-Cross, “A quasi heterodyne holographic technique and automatic algorithms for phase unwrapping,” in Fringe Pattern Analysis, G. T. Reid, ed., Proc. SPIE1163, 95–119 (1989).
[Crossref]

King, B.

B. King, “Step-wise clustering procedures,” J. Am. Stat. Assoc. 69, 86–101 (1967).
[Crossref]

Kurita, T.

T. Kurita, “An efficient agglomerative clustering algorithm using a heap,” Pattern Recogn. 24, 205–209 (1991).
[Crossref]

Kwon, O. Y.

Lalor, M. J.

Maas, A. A. M.

Mastin, G. A.

Moore, A. J.

M. Servin, R. Rodriguez-Vera, A. J. Moore, “A robust cellular processor for phase unwrapping,” J. Mod. Opt. 41, 119–127 (1994).
[Crossref]

Murty, M. N.

A. K. Jain, M. N. Murty, P. J. Flynn, “Data clustering: a review,” ACM Comput. Surveys 31, 264–323 (1999).
[Crossref]

Neupert, U.

J. Schörner, A. Ettemeyer, U. Neupert, H. Rottenkolber, C. Winter, P. Obermeier, “New approaches in interpreting holographic images,” Opt. Lasers Eng. 14, 283–291 (1991).
[Crossref]

Obermeier, P.

J. Schörner, A. Ettemeyer, U. Neupert, H. Rottenkolber, C. Winter, P. Obermeier, “New approaches in interpreting holographic images,” Opt. Lasers Eng. 14, 283–291 (1991).
[Crossref]

Oppenheim, A. V.

R. W. Schafer, A. V. Oppenheim, Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975), pp. 507–511.

Pritt, M. D.

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

Quiroga, J. A.

Robinson, D.

D. Robinson, “Phase unwrapping methods,” in Interferogram Analysis: Digital Fringe Pattern Measurement Techniques, W. R. Robinson, T. Reid, eds. (Institute of Physics, Philadelphia, Pa., 1993), pp. 194–229.

Rodriguez-Vera, R.

M. Servin, R. Rodriguez-Vera, A. J. Moore, “A robust cellular processor for phase unwrapping,” J. Mod. Opt. 41, 119–127 (1994).
[Crossref]

Romero, L. A.

Rottenkolber, H.

J. Schörner, A. Ettemeyer, U. Neupert, H. Rottenkolber, C. Winter, P. Obermeier, “New approaches in interpreting holographic images,” Opt. Lasers Eng. 14, 283–291 (1991).
[Crossref]

Saldner, H. O.

Schafer, R. W.

R. W. Schafer, A. V. Oppenheim, Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975), pp. 507–511.

Schörner, J.

J. Schörner, A. Ettemeyer, U. Neupert, H. Rottenkolber, C. Winter, P. Obermeier, “New approaches in interpreting holographic images,” Opt. Lasers Eng. 14, 283–291 (1991).
[Crossref]

Servin, M.

M. Servin, R. Rodriguez-Vera, A. J. Moore, “A robust cellular processor for phase unwrapping,” J. Mod. Opt. 41, 119–127 (1994).
[Crossref]

Shough, D. M.

Sneath, P. H. A.

P. H. A. Sneath, R. R. Sokal, Numerical Taxonomy (Freeman, London, 1973).

Sokal, R. R.

P. H. A. Sneath, R. R. Sokal, Numerical Taxonomy (Freeman, London, 1973).

Towers, D. P.

D. P. Towers, T. R. Judge, P. J. Bryanston-Cross, “A quasi heterodyne holographic technique and automatic algorithms for phase unwrapping,” in Fringe Pattern Analysis, G. T. Reid, ed., Proc. SPIE1163, 95–119 (1989).
[Crossref]

Vrooman, H. A.

Werner, C. L.

R. M. Goldstein, H. A. Zebker, C. L. Werner, “Two-dimensional phase unwrapping,” Radio Sci. 23, 713–720 (1988).
[Crossref]

Williams, J. W. J.

J. W. J. Williams, “Heapsort,” Commun. ACM 7, 347–348 (1964).

Williams, R. A.

Winter, C.

J. Schörner, A. Ettemeyer, U. Neupert, H. Rottenkolber, C. Winter, P. Obermeier, “New approaches in interpreting holographic images,” Opt. Lasers Eng. 14, 283–291 (1991).
[Crossref]

Yamada, T.

K. M. Hung, T. Yamada, “Phase unwrapping by regions using least-squares approach,” Opt. Eng. 37, 2965–2970 (1998).
[Crossref]

Zebker, H. A.

R. M. Goldstein, H. A. Zebker, C. L. Werner, “Two-dimensional phase unwrapping,” Radio Sci. 23, 713–720 (1988).
[Crossref]

ACM Comput. Surveys (1)

A. K. Jain, M. N. Murty, P. J. Flynn, “Data clustering: a review,” ACM Comput. Surveys 31, 264–323 (1999).
[Crossref]

Appl. Opt. (9)

J. M. Huntley, H. O. Saldner, “Temporal phase-unwrapping algorithm for automated interferogram analysis,” Appl. Opt. 32, 3047–3052 (1993).
[Crossref] [PubMed]

H. A. Vrooman, A. A. M. Maas, “Image processing algorithms for the analysis of phase-shifted speckle interference patterns,” Appl. Opt. 30, 1636–1641 (1991).
[Crossref] [PubMed]

J. M. Huntley, “Noise-immune phase unwrapping algorithm,” Appl. Opt. 28, 3268–3270 (1989).
[Crossref] [PubMed]

J. A. Quiroga, E. Bernabeu, “Phase-unwrapping algorithm for noisy phase-map processing,” Appl. Opt. 33, 6725–6731 (1994).
[Crossref] [PubMed]

J. A. Quiroga, A. González-Cano, E. Bernabeu, “Phase-unwrapping algorithm based on an adaptive criterion,” Appl. Opt. 34, 2560–2563 (1995).
[Crossref] [PubMed]

M. Arevalillo Herráez, D. R. Burton, M. J. Lalor, M. A. Gdeisat, “Fast two-dimensional phase unwrapping algorithm based on sorting by reliability following a noncontinuous path,” Appl. Opt. 41, 7437–7443 (2002).
[Crossref]

M. Arevalillo Herráez, D. R. Burton, M. J. Lalor, D. B. Clegg, “Robust, simple and fast algorithm for phase unwrapping,” Appl. Opt. 35, 5847–5852 (1996).
[Crossref]

A. Baldi, “Two-dimensional phase unwrapping by quad-tree decomposition,” Appl. Opt. 40, 1187–1194 (2001).
[Crossref]

M. Arevalillo Herráez, M. A. Gdeisat, D. R. Burton, M. J. Lalor, “Robust, fast, and effective two-dimensional automatic phase unwrapping algorithm based on image decomposition,” Appl. Opt. 41, 7445–7455 (2002).
[Crossref]

Commun. ACM (1)

J. W. J. Williams, “Heapsort,” Commun. ACM 7, 347–348 (1964).

J. Am. Stat. Assoc. (1)

B. King, “Step-wise clustering procedures,” J. Am. Stat. Assoc. 69, 86–101 (1967).
[Crossref]

J. Mod. Opt. (1)

M. Servin, R. Rodriguez-Vera, A. J. Moore, “A robust cellular processor for phase unwrapping,” J. Mod. Opt. 41, 119–127 (1994).
[Crossref]

J. Opt. Soc. Am. A (2)

Opt. Eng. (1)

K. M. Hung, T. Yamada, “Phase unwrapping by regions using least-squares approach,” Opt. Eng. 37, 2965–2970 (1998).
[Crossref]

Opt. Lasers Eng. (1)

J. Schörner, A. Ettemeyer, U. Neupert, H. Rottenkolber, C. Winter, P. Obermeier, “New approaches in interpreting holographic images,” Opt. Lasers Eng. 14, 283–291 (1991).
[Crossref]

Opt. Lett. (1)

Pattern Recogn. (1)

T. Kurita, “An efficient agglomerative clustering algorithm using a heap,” Pattern Recogn. 24, 205–209 (1991).
[Crossref]

Radio Sci. (1)

R. M. Goldstein, H. A. Zebker, C. L. Werner, “Two-dimensional phase unwrapping,” Radio Sci. 23, 713–720 (1988).
[Crossref]

Other (8)

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

R. W. Schafer, A. V. Oppenheim, Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975), pp. 507–511.

D. Robinson, “Phase unwrapping methods,” in Interferogram Analysis: Digital Fringe Pattern Measurement Techniques, W. R. Robinson, T. Reid, eds. (Institute of Physics, Philadelphia, Pa., 1993), pp. 194–229.

D. P. Towers, T. R. Judge, P. J. Bryanston-Cross, “A quasi heterodyne holographic technique and automatic algorithms for phase unwrapping,” in Fringe Pattern Analysis, G. T. Reid, ed., Proc. SPIE1163, 95–119 (1989).
[Crossref]

J. J. Gierloff, “Phase unwrapping by regions,” in Current Developments in Optical Engineering II, R. E. Fischer, W. J. Smith, eds., Proc. SPIE818, 2–9 (1987).

J. M. Huntley, “New methods for unwrapping noisy phase maps,” in Interferometry ’94: New Techniques and Analysis in Optical Measurements, M. Kujawinska, K. Przysztof, eds., Proc. SPIE, 2340, 110–122 (1994).

P. H. A. Sneath, R. R. Sokal, Numerical Taxonomy (Freeman, London, 1973).

A. K. Jain, P. J. Flynn, Three-Dimensional Object Recognition Systems (Elsevier Science, New York, 1993).

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Figures (13)

Fig. 1
Fig. 1

Concept of an edge.

Fig. 2
Fig. 2

Conceptual representation of data structures: (a) groups, (b) frontiers between group (each arrow represents a frontier), (c) binary heap storing the frontiers.

Fig. 3
Fig. 3

Grouping of clusters 4 and 5.

Fig. 4
Fig. 4

Update of frontiers graph.

Fig. 5
Fig. 5

Update of the binary heap structure.

Fig. 6
Fig. 6

Concept of a frontier.

Fig. 7
Fig. 7

Single-linkage clustering.

Fig. 8
Fig. 8

Calculation of the second differences for a pixel.

Fig. 9
Fig. 9

Changes in quality values of the frontiers after a grouping operation.

Fig. 10
Fig. 10

Unwrapping of a simple simulated wrapped distribution. (a) Original phase map, (b) resulting unwrapped phase.

Fig. 11
Fig. 11

(a) Unwrapping of the phase distribution, (b) result of unwrapping the phase distribution, (c) three-dimensional view of the result, (d) the groups isolated by the algorithm.

Fig. 12
Fig. 12

Unwrapping of a wrapped phase distribution obtained from a real fringe pattern. The region crossing the image is a product of the shadow. (a) Wrapped phase distribution, (b) unwrapped phase distribution after carrier removal, (c) a three-dimensional view of the result, (d) groups isolated by the algorithm.

Fig. 13
Fig. 13

Unwrapping of a wrapped phase distributed obtained from the measurement of a human face. (a) Wrapped phase distribution, (b) unwrapped phase distribution, (c) a three-dimensional view of the result, (d) groups isolated by the algorithm.

Equations (3)

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ψ 1 ( p i , j ) = ϒ ( p i - 1 , j - 1 - p i , j ) - ϒ ( p i , j - p i + 1 , j + 1 ) , ψ 2 ( p i , j ) = ϒ ( p i - 1 , j - p i , j ) - ϒ ( p i , j - p i + 1 , j ) , ψ 3 ( p i , j ) = ϒ ( p i , j - p i - 1 , j + 1 ) - ϒ ( p i + 1 , j - 1 - p i , j ) , ψ 4 ( p i , j ) = ϒ ( p i , j - 1 - p i , j ) - ϒ ( p i , j - p i , j + 1 ) .
p q ( p i j ) = ψ 1 2 + ψ 2 2 + ψ 3 2 + ψ 4 2 .
e q ( p ,     q ) = [ p q ( p ) + p q ( q ) ] 2 .

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