Abstract

Phase and amplitude information obtained from phase-shifting interferometry may be combined, to be a complex-valued phasor for every pixel of the image. Phasor image processing is presented as a simple yet effective concept for filtering, visualization, masking, and unwrapping interferometric phase maps. The results from two electronic speckle pattern interferometry data sets illustrate the new method.

© 1996 Optical Society of America

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References

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  1. For an overview see K. Creath, “Temporal phase measurement methods,” in Interferogram Analysis, D. W. Robinson, G. T. Reid, eds. (Institute of Physics Publishing, Bristol, U.K., 1993), pp. 94–140.
  2. For an overview see W. Jüptner, W. Osten, eds., Fringe ’93, Proceedings of the Second International Workshop on Automatic Processing of Fringe Patterns (Akademie Verlag, Berlin, 1993).
  3. D. W. Robinson, “Phase unwrapping methods,” in Interferogram Analysis, D. W. Robinson, G. T. Reid, eds. (Institute of Physics Publishing, Bristol, U.K., 1993), pp. 194–229.
  4. R. Jones, C. Wykes, Holographic and Speckle Interferometry (Cambridge U. Press, Cambridge, U.K., 1989).
  5. K. Creath, “Phase-shifting speckle interferometry,” Appl. Opt. 24, 3053–3058 (1985).
  6. J. M. Huntley, “Noise-immune phase unwrapping algorithm,” Appl. Opt. 28, 3268–3270 (1989).
  7. K. Creath, “Phase-measurement interferometry techniques,” Prog. Opt. 26, 349–369 (1988).
  8. H. A. Vrooman, A. M. Maas, “Image processing algorithms for the analysis of phase-shifted speckle interference patterns,” Appl. Opt. 30, 1636–1641 (1991).
  9. J. A. Quiroga, E. Bernabeu, “Phase-unwrapping algorithm for noisy phase-map processing,” Appl. Opt. 33, 6725–6731 (1994).
  10. R. Dändliker, R. Thalmann, “Heterodyne and quasi-heterodyne holographic interferometry,” Opt. Eng. 24, 824–831 (1985).
  11. X. Y. Su, A. M. Zarubin, G. von Bally, “Modulation analysis of phase-shifted holographic interferograms,” Opt. Commun. 105, 379–387 (1994).
  12. B. Ströbel, “Fiber-optic modular speckle interferometer,” in Proceedings of LASER ’95 Congress on Optical Measuring Techniques, W. Waidelich, ed., (Springer-Verlag, Berlin, 1996).
  13. By courtesy of T. Floureux, Institut für Konstruktion und Bauweisen, ETH Zürich, Switzerland.

1994 (2)

X. Y. Su, A. M. Zarubin, G. von Bally, “Modulation analysis of phase-shifted holographic interferograms,” Opt. Commun. 105, 379–387 (1994).

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

1991 (1)

1989 (1)

1988 (1)

K. Creath, “Phase-measurement interferometry techniques,” Prog. Opt. 26, 349–369 (1988).

1985 (2)

R. Dändliker, R. Thalmann, “Heterodyne and quasi-heterodyne holographic interferometry,” Opt. Eng. 24, 824–831 (1985).

K. Creath, “Phase-shifting speckle interferometry,” Appl. Opt. 24, 3053–3058 (1985).

Bernabeu, E.

Creath, K.

K. Creath, “Phase-measurement interferometry techniques,” Prog. Opt. 26, 349–369 (1988).

K. Creath, “Phase-shifting speckle interferometry,” Appl. Opt. 24, 3053–3058 (1985).

For an overview see K. Creath, “Temporal phase measurement methods,” in Interferogram Analysis, D. W. Robinson, G. T. Reid, eds. (Institute of Physics Publishing, Bristol, U.K., 1993), pp. 94–140.

Dändliker, R.

R. Dändliker, R. Thalmann, “Heterodyne and quasi-heterodyne holographic interferometry,” Opt. Eng. 24, 824–831 (1985).

Floureux, T.

By courtesy of T. Floureux, Institut für Konstruktion und Bauweisen, ETH Zürich, Switzerland.

Huntley, J. M.

Jones, R.

R. Jones, C. Wykes, Holographic and Speckle Interferometry (Cambridge U. Press, Cambridge, U.K., 1989).

Maas, A. M.

Quiroga, J. A.

Robinson, D. W.

D. W. Robinson, “Phase unwrapping methods,” in Interferogram Analysis, D. W. Robinson, G. T. Reid, eds. (Institute of Physics Publishing, Bristol, U.K., 1993), pp. 194–229.

Ströbel, B.

B. Ströbel, “Fiber-optic modular speckle interferometer,” in Proceedings of LASER ’95 Congress on Optical Measuring Techniques, W. Waidelich, ed., (Springer-Verlag, Berlin, 1996).

Su, X. Y.

X. Y. Su, A. M. Zarubin, G. von Bally, “Modulation analysis of phase-shifted holographic interferograms,” Opt. Commun. 105, 379–387 (1994).

Thalmann, R.

R. Dändliker, R. Thalmann, “Heterodyne and quasi-heterodyne holographic interferometry,” Opt. Eng. 24, 824–831 (1985).

von Bally, G.

X. Y. Su, A. M. Zarubin, G. von Bally, “Modulation analysis of phase-shifted holographic interferograms,” Opt. Commun. 105, 379–387 (1994).

Vrooman, H. A.

Wykes, C.

R. Jones, C. Wykes, Holographic and Speckle Interferometry (Cambridge U. Press, Cambridge, U.K., 1989).

Zarubin, A. M.

X. Y. Su, A. M. Zarubin, G. von Bally, “Modulation analysis of phase-shifted holographic interferograms,” Opt. Commun. 105, 379–387 (1994).

Appl. Opt. (4)

Opt. Commun. (1)

X. Y. Su, A. M. Zarubin, G. von Bally, “Modulation analysis of phase-shifted holographic interferograms,” Opt. Commun. 105, 379–387 (1994).

Opt. Eng. (1)

R. Dändliker, R. Thalmann, “Heterodyne and quasi-heterodyne holographic interferometry,” Opt. Eng. 24, 824–831 (1985).

Prog. Opt. (1)

K. Creath, “Phase-measurement interferometry techniques,” Prog. Opt. 26, 349–369 (1988).

Other (6)

B. Ströbel, “Fiber-optic modular speckle interferometer,” in Proceedings of LASER ’95 Congress on Optical Measuring Techniques, W. Waidelich, ed., (Springer-Verlag, Berlin, 1996).

By courtesy of T. Floureux, Institut für Konstruktion und Bauweisen, ETH Zürich, Switzerland.

For an overview see K. Creath, “Temporal phase measurement methods,” in Interferogram Analysis, D. W. Robinson, G. T. Reid, eds. (Institute of Physics Publishing, Bristol, U.K., 1993), pp. 94–140.

For an overview see W. Jüptner, W. Osten, eds., Fringe ’93, Proceedings of the Second International Workshop on Automatic Processing of Fringe Patterns (Akademie Verlag, Berlin, 1993).

D. W. Robinson, “Phase unwrapping methods,” in Interferogram Analysis, D. W. Robinson, G. T. Reid, eds. (Institute of Physics Publishing, Bristol, U.K., 1993), pp. 194–229.

R. Jones, C. Wykes, Holographic and Speckle Interferometry (Cambridge U. Press, Cambridge, U.K., 1989).

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Figures (13)

Fig. 1
Fig. 1

Phase of a complex number determined from its real and imaginary parts. In the presence of noise (indicated by a point cloud) the phase value is more reliable if the amplitude is high.

Fig. 2
Fig. 2

Effect of filtering a phasor image in the complex plane for a simple model case. Phasors (sites of the arrowheads) of three pixels that are to be averaged, ○. Two of these pixels are assumed to have low amplitudes and therefore less reliable phase values. Consequently the phase ϕav of the average, •, is dominated by the pixel with the large amplitude.

Fig. 3
Fig. 3

Application of the sine/cosine transformation means assuming the same amplitude of 1 for all pixels. As a result, ϕav is now incorrectly dominated by the phases of the two low-amplitude pixels.

Fig. 4
Fig. 4

White-light image of sample 1. The sites of some features of the PSPI data set are marked: 1, laser-light reflection causing camera saturation; 2, phase discontinuity; 3, air inclusion causing dense fringes; 4, poor illumination by laser light; 5, hole.

Fig. 5
Fig. 5

Unfiltered phase map of sample 1.

Fig. 6
Fig. 6

Unfiltered amplitude map of sample 1.

Fig. 7
Fig. 7

Filtered phase map of sample 1.

Fig. 8
Fig. 8

Filtered amplitude map of sample 1. Note the amplitude decrease at the site of the dense fringes and the phase discontinuities.

Fig. 9
Fig. 9

Unwrapped phase map of sample 1 after dilation. An unwrapping error is indicated by the arrow.

Fig. 10
Fig. 10

Unwrapping path record of sample 1. Light areas are unwrapped early, dark areas late; white denotes the masked areas that are not unwrapped. Note that the dark speckles and the low-amplitude regions are processed later.

Fig. 11
Fig. 11

Filtered phase map of sample 2. Note the dense fringes in the center that are potential error sources.

Fig. 12
Fig. 12

Unwrapped phase map of sample 2.

Fig. 13
Fig. 13

Unwrapping path record of sample 2. The central region is unwrapped late owing to its reduced amplitude.

Equations (13)

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I 1 ( x , y ) = I 0 ( x , y ) + A ( x , y ) cos [ ϕ ( x , y ) ] + N 1 ( x , y , t ) , I 2 ( x , y ) = I 0 ( x , y ) + A ( x , y ) cos [ ϕ ( x , y ) + π / 2 ] + N 2 ( x , y , t ) , I 3 ( x , y ) = I 0 ( x , y ) + A ( x , y ) cos [ ϕ ( x , y ) + π ] + N 3 ( x , y , t ) , I 4 ( x , y ) = I 0 ( x , y ) + A ( x , y ) cos [ ϕ ( x , y ) + 3 π / 2 ] + N 4 ( x , y , t ) ,
ϕ ( x , y ) = arctan [ I 4 ( x , y ) I 2 ( x , y ) I 1 ( x , y ) I 3 ( x , y ) ] .
A = [ ( I 4 I 2 2 ) 2 + ( I 1 I 3 2 ) 2 ] 1 / 2 .
ϕ = arctan ( Im z / Re z ) ,
A = [ ( Re z ) 2 + ( Im z ) 2 ] 1 / 2 .
ϕ = ϕ load ϕ unl ( + 2 π if ϕ unl > ϕ load ) ,
A = ( A load + A unl ) / 2 arithmetic average ,
A = ( A load / A unl ) 1 / 2 geometric average ,
A = [ ( A load 1 + A unl 1 ) / 2 ] 1 average of inverses ,
A = [ ( A load 2 + A unl 2 ) / 2 ] 1 / 2 average of squared inverses ,
A = min ( A load, A unl ) minimum .
Re z = A cos ϕ , Im z = A sin ϕ ,
ϕ unwr = ϕ + 2 π N .

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