Abstract

We explore an approach to the unwrapping of two-dimensional phase functions using a robust extrapolation–projection algorithm. Phase unwrapping is essential for imaging systems that construct the image from phase information. Unlike some existing methods where unwrapping is performed locally on a pixel-by-pixel basis, this work approaches the unwrapping problem from a global point of view. The unwrapping is done iteratively by a modification of the Gerchberg–Papoulis extrapolation algorithm, and the solution is refined by projecting onto the available global data at each iteration. Robustness of the algorithm is demonstrated through its performance in a noisy environment, and in comparison with a least-squares algorithm well-known in the literature.

© 2006 Optical Society of America

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  1. R. M. Goldstein, H. A. Zebker, and C. L. Werner, "Satellite radar interferometry: Two-dimensional phase unwrap-ping," Radio Sci. 23, 713-720 (1988).
    [CrossRef]
  2. W. Schwartzkopf, T. E. Milner, J. Ghosh, B. L. Evans, and A. C. Bovik, "Two-dimensional phase unwrapping using neural networks," in Proceedings of the IEEE Southwest Symposium on Image Analysis and Interpretation, (IEEE, New York, 2000), pp. 274-277.
    [CrossRef]
  3. T. J. Flynn, "Two-dimensional phase unwrapping with minimum weighted discontinuity," J. Opt. Soc. Am. A 14, 2692-2701 (1997).
    [CrossRef]
  4. T. J. Flynn, "Phase unwrapping using discontinuity optimization," in Proceedings of the IEEE International Geoscience and Remote Sensing Symposium (IGARSS), (IEEE, New York, 1998), Vol. 1, pp. 80-82.
  5. C. R. Guarino, "Weighted two-dimensional phase unwrapping," in Proceedings of the IEEE International Geoscience and Remote Sensing Symposium (IGARSS), (IEEE, New York, 1995), Vol. 1, pp. 193-195.
  6. L. Ying, B. J. Frey, R. Koetter, and D. C. Munson, Jr., "Analysis of an iterative dynamic programming approach to 2-D phase unwrapping," in Proceedings of the IEEE International Conference on Image Processing (ICIP), (IEEE, New York, 2002), Vol. 1, pp. 469-471.
  7. B. J. Frey, R. Koetter, N. Petrovic, and D. C. Munson, Jr., "Unwrapping phase images by propagating probabilities across graphs," in Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP), (IEEE, New York, 2001), Vol. 3, pp. 1845-1848.
  8. G. Nico, G. Palubinskas, and M. Datcu, "Bayesian approaches to phase unwrapping: theoretical study," IEEE Trans. Signal Process. 48, 2545-2556 (2000).
    [CrossRef]
  9. M. D. Pritt and J. S. Shipman, "Least-squares two-dimensional phase unwrapping using FFT's," IEEE Trans. Geosci. Remote Sens. 32, 706-708 (1994).
    [CrossRef]
  10. M. Constantini, "A novel phase unwrapping method based on network programming," IEEE Trans. Geosci. Remote Sens. 36, 813-821 (1998).
    [CrossRef]
  11. C. W. Chen and H. A. Zebker, "Two-dimensional phase unwrapping with use of statistical models for cost functions in nonlinear optimization," J. Opt. Soc. Am. A 18, 338-351 (2001).
    [CrossRef]
  12. D. C. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms, and Software (Wiley-Interscience, 1998).
  13. H. A. Zebker and 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]
  14. Y. Yang and H. Stark, Vector Space Projections: A Numerical Approach to Signal and Image Processing, Neural Nets, and Optics (Wiley-Interscience, 1998).
    [PubMed]
  15. A. Levi and H. Stark, "Image restoration by the method of generalized projections with application to restoration from magnitude," J. Opt. Soc. Am. A 1, 932-943 (1984).
    [CrossRef]
  16. A. Papoulis, "A new algorithm in spectral analysis and band-limited extrapolation," IEEE Trans. Circuits Syst. CAS-22, 735-742 (1975).
    [CrossRef]
  17. R. C. Gonzalez and R. E. Woods, Digital Image Processing (Addison-Wesley, 1992).
  18. J. Canny, "A computational approach to edge detection," IEEE Trans. Pattern Anal. Mach. Intell. 8, 679-698 (1986).
    [CrossRef] [PubMed]
  19. I. Cumming and W. Xu, "A region-growing algorithm for InSAR phase unwrapping," IEEE Trans. Geosci. Remote Sens. 37, 124-134 (1999).
    [CrossRef]
  20. J. S. Lee, K. W. Hoppel, S. A. Mango, and A. R. Miller, "Intensity and phase statistics of multilook polarimetric and interferometric SAR imagery," IEEE Trans. Geosci. Remote Sens. 32, 1017-1028 (1994).
    [CrossRef]
  21. R. Bamler, N. Adam, G. W. Davidson, and D. Just, "Noise-induced slope distortion in 2-D phase unwrapping linear estimators with application to SAR interferometry," IEEE Trans. Geosci. Remote Sens. 36, 913-921 (1998).
    [CrossRef]

2001 (1)

2000 (1)

G. Nico, G. Palubinskas, and M. Datcu, "Bayesian approaches to phase unwrapping: theoretical study," IEEE Trans. Signal Process. 48, 2545-2556 (2000).
[CrossRef]

1999 (1)

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

1998 (3)

R. Bamler, N. Adam, G. W. Davidson, and D. Just, "Noise-induced slope distortion in 2-D phase unwrapping linear estimators with application to SAR interferometry," IEEE Trans. Geosci. Remote Sens. 36, 913-921 (1998).
[CrossRef]

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

H. A. Zebker and 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]

1997 (1)

1994 (2)

M. D. Pritt and J. S. Shipman, "Least-squares two-dimensional phase unwrapping using FFT's," IEEE Trans. Geosci. Remote Sens. 32, 706-708 (1994).
[CrossRef]

J. S. Lee, K. W. Hoppel, S. A. Mango, and A. R. Miller, "Intensity and phase statistics of multilook polarimetric and interferometric SAR imagery," IEEE Trans. Geosci. Remote Sens. 32, 1017-1028 (1994).
[CrossRef]

1988 (1)

R. M. Goldstein, H. A. Zebker, and C. L. Werner, "Satellite radar interferometry: Two-dimensional phase unwrap-ping," Radio Sci. 23, 713-720 (1988).
[CrossRef]

1986 (1)

J. Canny, "A computational approach to edge detection," IEEE Trans. Pattern Anal. Mach. Intell. 8, 679-698 (1986).
[CrossRef] [PubMed]

1984 (1)

1975 (1)

A. Papoulis, "A new algorithm in spectral analysis and band-limited extrapolation," IEEE Trans. Circuits Syst. CAS-22, 735-742 (1975).
[CrossRef]

Adam, N.

R. Bamler, N. Adam, G. W. Davidson, and D. Just, "Noise-induced slope distortion in 2-D phase unwrapping linear estimators with application to SAR interferometry," IEEE Trans. Geosci. Remote Sens. 36, 913-921 (1998).
[CrossRef]

Bamler, R.

R. Bamler, N. Adam, G. W. Davidson, and D. Just, "Noise-induced slope distortion in 2-D phase unwrapping linear estimators with application to SAR interferometry," IEEE Trans. Geosci. Remote Sens. 36, 913-921 (1998).
[CrossRef]

Bovik, A. C.

W. Schwartzkopf, T. E. Milner, J. Ghosh, B. L. Evans, and A. C. Bovik, "Two-dimensional phase unwrapping using neural networks," in Proceedings of the IEEE Southwest Symposium on Image Analysis and Interpretation, (IEEE, New York, 2000), pp. 274-277.
[CrossRef]

Canny, J.

J. Canny, "A computational approach to edge detection," IEEE Trans. Pattern Anal. Mach. Intell. 8, 679-698 (1986).
[CrossRef] [PubMed]

Chen, C. W.

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.

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

Datcu, M.

G. Nico, G. Palubinskas, and M. Datcu, "Bayesian approaches to phase unwrapping: theoretical study," IEEE Trans. Signal Process. 48, 2545-2556 (2000).
[CrossRef]

Davidson, G. W.

R. Bamler, N. Adam, G. W. Davidson, and D. Just, "Noise-induced slope distortion in 2-D phase unwrapping linear estimators with application to SAR interferometry," IEEE Trans. Geosci. Remote Sens. 36, 913-921 (1998).
[CrossRef]

Evans, B. L.

W. Schwartzkopf, T. E. Milner, J. Ghosh, B. L. Evans, and A. C. Bovik, "Two-dimensional phase unwrapping using neural networks," in Proceedings of the IEEE Southwest Symposium on Image Analysis and Interpretation, (IEEE, New York, 2000), pp. 274-277.
[CrossRef]

Flynn, T. J.

T. J. Flynn, "Two-dimensional phase unwrapping with minimum weighted discontinuity," J. Opt. Soc. Am. A 14, 2692-2701 (1997).
[CrossRef]

T. J. Flynn, "Phase unwrapping using discontinuity optimization," in Proceedings of the IEEE International Geoscience and Remote Sensing Symposium (IGARSS), (IEEE, New York, 1998), Vol. 1, pp. 80-82.

Frey, B. J.

B. J. Frey, R. Koetter, N. Petrovic, and D. C. Munson, Jr., "Unwrapping phase images by propagating probabilities across graphs," in Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP), (IEEE, New York, 2001), Vol. 3, pp. 1845-1848.

L. Ying, B. J. Frey, R. Koetter, and D. C. Munson, Jr., "Analysis of an iterative dynamic programming approach to 2-D phase unwrapping," in Proceedings of the IEEE International Conference on Image Processing (ICIP), (IEEE, New York, 2002), Vol. 1, pp. 469-471.

Ghiglia, D. C.

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

Ghosh, J.

W. Schwartzkopf, T. E. Milner, J. Ghosh, B. L. Evans, and A. C. Bovik, "Two-dimensional phase unwrapping using neural networks," in Proceedings of the IEEE Southwest Symposium on Image Analysis and Interpretation, (IEEE, New York, 2000), pp. 274-277.
[CrossRef]

Goldstein, R. M.

R. M. Goldstein, H. A. Zebker, and C. L. Werner, "Satellite radar interferometry: Two-dimensional phase unwrap-ping," Radio Sci. 23, 713-720 (1988).
[CrossRef]

Gonzalez, R. C.

R. C. Gonzalez and R. E. Woods, Digital Image Processing (Addison-Wesley, 1992).

Guarino, C. R.

C. R. Guarino, "Weighted two-dimensional phase unwrapping," in Proceedings of the IEEE International Geoscience and Remote Sensing Symposium (IGARSS), (IEEE, New York, 1995), Vol. 1, pp. 193-195.

Hoppel, K. W.

J. S. Lee, K. W. Hoppel, S. A. Mango, and A. R. Miller, "Intensity and phase statistics of multilook polarimetric and interferometric SAR imagery," IEEE Trans. Geosci. Remote Sens. 32, 1017-1028 (1994).
[CrossRef]

Just, D.

R. Bamler, N. Adam, G. W. Davidson, and D. Just, "Noise-induced slope distortion in 2-D phase unwrapping linear estimators with application to SAR interferometry," IEEE Trans. Geosci. Remote Sens. 36, 913-921 (1998).
[CrossRef]

Koetter, R.

L. Ying, B. J. Frey, R. Koetter, and D. C. Munson, Jr., "Analysis of an iterative dynamic programming approach to 2-D phase unwrapping," in Proceedings of the IEEE International Conference on Image Processing (ICIP), (IEEE, New York, 2002), Vol. 1, pp. 469-471.

B. J. Frey, R. Koetter, N. Petrovic, and D. C. Munson, Jr., "Unwrapping phase images by propagating probabilities across graphs," in Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP), (IEEE, New York, 2001), Vol. 3, pp. 1845-1848.

Lee, J. S.

J. S. Lee, K. W. Hoppel, S. A. Mango, and A. R. Miller, "Intensity and phase statistics of multilook polarimetric and interferometric SAR imagery," IEEE Trans. Geosci. Remote Sens. 32, 1017-1028 (1994).
[CrossRef]

Levi, A.

Lu, Y.

Mango, S. A.

J. S. Lee, K. W. Hoppel, S. A. Mango, and A. R. Miller, "Intensity and phase statistics of multilook polarimetric and interferometric SAR imagery," IEEE Trans. Geosci. Remote Sens. 32, 1017-1028 (1994).
[CrossRef]

Miller, A. R.

J. S. Lee, K. W. Hoppel, S. A. Mango, and A. R. Miller, "Intensity and phase statistics of multilook polarimetric and interferometric SAR imagery," IEEE Trans. Geosci. Remote Sens. 32, 1017-1028 (1994).
[CrossRef]

Milner, T. E.

W. Schwartzkopf, T. E. Milner, J. Ghosh, B. L. Evans, and A. C. Bovik, "Two-dimensional phase unwrapping using neural networks," in Proceedings of the IEEE Southwest Symposium on Image Analysis and Interpretation, (IEEE, New York, 2000), pp. 274-277.
[CrossRef]

Munson, D. C.

L. Ying, B. J. Frey, R. Koetter, and D. C. Munson, Jr., "Analysis of an iterative dynamic programming approach to 2-D phase unwrapping," in Proceedings of the IEEE International Conference on Image Processing (ICIP), (IEEE, New York, 2002), Vol. 1, pp. 469-471.

B. J. Frey, R. Koetter, N. Petrovic, and D. C. Munson, Jr., "Unwrapping phase images by propagating probabilities across graphs," in Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP), (IEEE, New York, 2001), Vol. 3, pp. 1845-1848.

Nico, G.

G. Nico, G. Palubinskas, and M. Datcu, "Bayesian approaches to phase unwrapping: theoretical study," IEEE Trans. Signal Process. 48, 2545-2556 (2000).
[CrossRef]

Palubinskas, G.

G. Nico, G. Palubinskas, and M. Datcu, "Bayesian approaches to phase unwrapping: theoretical study," IEEE Trans. Signal Process. 48, 2545-2556 (2000).
[CrossRef]

Papoulis, A.

A. Papoulis, "A new algorithm in spectral analysis and band-limited extrapolation," IEEE Trans. Circuits Syst. CAS-22, 735-742 (1975).
[CrossRef]

Petrovic, N.

B. J. Frey, R. Koetter, N. Petrovic, and D. C. Munson, Jr., "Unwrapping phase images by propagating probabilities across graphs," in Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP), (IEEE, New York, 2001), Vol. 3, pp. 1845-1848.

Pritt, M. D.

M. D. Pritt and J. S. Shipman, "Least-squares two-dimensional phase unwrapping using FFT's," IEEE Trans. Geosci. Remote Sens. 32, 706-708 (1994).
[CrossRef]

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

Schwartzkopf, W.

W. Schwartzkopf, T. E. Milner, J. Ghosh, B. L. Evans, and A. C. Bovik, "Two-dimensional phase unwrapping using neural networks," in Proceedings of the IEEE Southwest Symposium on Image Analysis and Interpretation, (IEEE, New York, 2000), pp. 274-277.
[CrossRef]

Shipman, J. S.

M. D. Pritt and J. S. Shipman, "Least-squares two-dimensional phase unwrapping using FFT's," IEEE Trans. Geosci. Remote Sens. 32, 706-708 (1994).
[CrossRef]

Stark, H.

A. Levi and H. Stark, "Image restoration by the method of generalized projections with application to restoration from magnitude," J. Opt. Soc. Am. A 1, 932-943 (1984).
[CrossRef]

Y. Yang and H. Stark, Vector Space Projections: A Numerical Approach to Signal and Image Processing, Neural Nets, and Optics (Wiley-Interscience, 1998).
[PubMed]

Werner, C. L.

R. M. Goldstein, H. A. Zebker, and C. L. Werner, "Satellite radar interferometry: Two-dimensional phase unwrap-ping," Radio Sci. 23, 713-720 (1988).
[CrossRef]

Woods, R. E.

R. C. Gonzalez and R. E. Woods, Digital Image Processing (Addison-Wesley, 1992).

Xu, W.

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

Yang, Y.

Y. Yang and H. Stark, Vector Space Projections: A Numerical Approach to Signal and Image Processing, Neural Nets, and Optics (Wiley-Interscience, 1998).
[PubMed]

Ying, L.

L. Ying, B. J. Frey, R. Koetter, and D. C. Munson, Jr., "Analysis of an iterative dynamic programming approach to 2-D phase unwrapping," in Proceedings of the IEEE International Conference on Image Processing (ICIP), (IEEE, New York, 2002), Vol. 1, pp. 469-471.

Zebker, H. A.

IEEE Trans. Circuits Syst. (1)

A. Papoulis, "A new algorithm in spectral analysis and band-limited extrapolation," IEEE Trans. Circuits Syst. CAS-22, 735-742 (1975).
[CrossRef]

IEEE Trans. Geosci. Remote Sens. (5)

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

J. S. Lee, K. W. Hoppel, S. A. Mango, and A. R. Miller, "Intensity and phase statistics of multilook polarimetric and interferometric SAR imagery," IEEE Trans. Geosci. Remote Sens. 32, 1017-1028 (1994).
[CrossRef]

R. Bamler, N. Adam, G. W. Davidson, and D. Just, "Noise-induced slope distortion in 2-D phase unwrapping linear estimators with application to SAR interferometry," IEEE Trans. Geosci. Remote Sens. 36, 913-921 (1998).
[CrossRef]

M. D. Pritt and J. S. Shipman, "Least-squares two-dimensional phase unwrapping using FFT's," IEEE Trans. Geosci. Remote Sens. 32, 706-708 (1994).
[CrossRef]

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

IEEE Trans. Pattern Anal. Mach. Intell. (1)

J. Canny, "A computational approach to edge detection," IEEE Trans. Pattern Anal. Mach. Intell. 8, 679-698 (1986).
[CrossRef] [PubMed]

IEEE Trans. Signal Process. (1)

G. Nico, G. Palubinskas, and M. Datcu, "Bayesian approaches to phase unwrapping: theoretical study," IEEE Trans. Signal Process. 48, 2545-2556 (2000).
[CrossRef]

J. Opt. Soc. Am. A (4)

Radio Sci. (1)

R. M. Goldstein, H. A. Zebker, and C. L. Werner, "Satellite radar interferometry: Two-dimensional phase unwrap-ping," Radio Sci. 23, 713-720 (1988).
[CrossRef]

Other (8)

W. Schwartzkopf, T. E. Milner, J. Ghosh, B. L. Evans, and A. C. Bovik, "Two-dimensional phase unwrapping using neural networks," in Proceedings of the IEEE Southwest Symposium on Image Analysis and Interpretation, (IEEE, New York, 2000), pp. 274-277.
[CrossRef]

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

T. J. Flynn, "Phase unwrapping using discontinuity optimization," in Proceedings of the IEEE International Geoscience and Remote Sensing Symposium (IGARSS), (IEEE, New York, 1998), Vol. 1, pp. 80-82.

C. R. Guarino, "Weighted two-dimensional phase unwrapping," in Proceedings of the IEEE International Geoscience and Remote Sensing Symposium (IGARSS), (IEEE, New York, 1995), Vol. 1, pp. 193-195.

L. Ying, B. J. Frey, R. Koetter, and D. C. Munson, Jr., "Analysis of an iterative dynamic programming approach to 2-D phase unwrapping," in Proceedings of the IEEE International Conference on Image Processing (ICIP), (IEEE, New York, 2002), Vol. 1, pp. 469-471.

B. J. Frey, R. Koetter, N. Petrovic, and D. C. Munson, Jr., "Unwrapping phase images by propagating probabilities across graphs," in Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP), (IEEE, New York, 2001), Vol. 3, pp. 1845-1848.

R. C. Gonzalez and R. E. Woods, Digital Image Processing (Addison-Wesley, 1992).

Y. Yang and H. Stark, Vector Space Projections: A Numerical Approach to Signal and Image Processing, Neural Nets, and Optics (Wiley-Interscience, 1998).
[PubMed]

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Figures (16)

Fig. 1
Fig. 1

Original phantom phase function φ [ m , n ] .

Fig. 2
Fig. 2

Wrapped phase function ψ [ m , n ] from Fig. 1.

Fig. 3
Fig. 3

Segmented image with fat boundaries.

Fig. 4
Fig. 4

Largest smooth region for ψ [ m , n ] in Fig. 3 as detected with region growing.

Fig. 5
Fig. 5

Solution φ ̂ [ m , n ] as generated by the modified GP algorithm.

Fig. 6
Fig. 6

Solution φ ̂ [ m , n ] generated by the extrapolation–projection algorithm.

Fig. 7
Fig. 7

Original phantom phase function φ [ m , n ] .

Fig. 8
Fig. 8

Wrapped phase function ψ [ m , n ] .

Fig. 9
Fig. 9

Solution φ ̂ [ m , n ] generated by the extrapolation–projection algorithm.

Fig. 10
Fig. 10

(a) Noisy wrapped phase with P N = 0.365 . (b) Unwrapped φ ̂ [ m , n ] from (a) using the extrapolation–projection algorithm.

Fig. 11
Fig. 11

(a) Noisy wrapped phase with P N = 0.653 + 1.6 dB . (b) Unwrapped φ ̂ [ m , n ] from (a) using the extrapolation–projection algorithm.

Fig. 12
Fig. 12

(a) Noisy wrapped phase with P N = 0.653 + 3.4 dB . (b) Unwrapped φ ̂ [ m , n ] from (a) using the extrapolation–projection algorithm.

Fig. 13
Fig. 13

Solution φ ̂ [ m , n ] with Pritt and Shipman’s algorithm[9] at (a) P N = 0.653 + 1.6 dB , (b) after projecting on C 1 ( δ 2 ) .

Fig. 14
Fig. 14

Error phase for reconstruction of φ ̂ [ m , n ] with P N = 0.653 + 1.6 dB : (a) VSP, (b) Pritt and Shipman.[9]

Fig. 15
Fig. 15

Real wrapped SAR data.

Fig. 16
Fig. 16

Unwrapped real SAR phase with (a) Pritt and Shipman’s algorithm,[9] (b) the extrapolation–projection algorithm.

Equations (18)

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

x 2 = ( i = 1 M N x i 2 ) 1 2 .
P C x = x * = arg min y C x y ,
x n + 1 = P m P m 1 P 1 x n ,
C 1 ( δ 2 ) = { φ R M N : φ Φ ( ω 1 , ω 2 ) , B C Φ ( ω 1 , ω 2 ) 2 d ω 1 d ω 2 δ 2 } ;
C 2 = { φ R M N : [ φ [ m , n ] ] MODULO 2 π = ψ [ m , n ] , 1 m , n M , N } .
F T 2 ( P 1 θ ) = { Θ ( ω 1 , ω 2 ) , if ( ω 1 , ω 2 ) B Θ ( ω 1 , ω 2 ) , if ( ω 1 , ω 2 ) B C and B C Θ ( ω 1 , ω 2 ) 2 d ω 1 d ω 2 δ 2 Θ ( ω 1 , ω 2 ) B C Θ ( ω 1 , ω 2 ) 2 d ω 1 d ω 2 δ , if ( ω 1 , ω 2 ) B C and B C Θ ( ω 1 , ω 2 ) 2 d ω 1 d ω 2 > δ 2 } ,
P 2 θ = ψ [ m , n ] + 2 π [ θ [ m , n ] ψ [ m , n ] 2 π ] .
C 3 = { φ R M N : φ [ m , n ] = f [ m , n ] , ( m , n ) R } ,
P 3 θ = { θ [ m , n ] , if ( m , n ) R f [ m , n ] , if ( m , n ) R }
ψ [ m , n ] = { ( ψ [ m + 1 , n ] ψ [ m , n ] ) 2 + ( ψ [ m , n + 1 ] ψ [ m , n ] ) 2 } 1 2 .
φ K + 1 = P 2 ( P 1 P 3 { R φ K } )
φ END = P 1 P 3 φ END 1 ,
φ [ n ] = k = 0 K 1 Φ [ k ] exp ( j 2 π N k n ) + k = K N 1 Φ [ k ] exp ( j 2 π N k n ) ,
φ [ n ] φ [ n + 1 ] = ( 2 j ) { k = 0 K 1 Φ [ k ] exp [ j 2 π k ( n + 1 2 N ) ] sin π k N + k = K N Φ [ k ] exp [ j 2 π k ( n + 1 2 N ) ] sin π k N } .
φ [ n ] φ [ n + 1 ] 2 { k = 0 K 1 Φ [ k ] exp [ j 2 π k ( n + 1 2 N ) ] sin π k N + k = K N Φ [ k ] exp [ j 2 π k ( n + 1 2 N ) ] sin π k N } 2 [ k = 0 K 1 π k N Φ [ k ] + k = K N 1 Φ [ k ] 2 ] 2 π Φ MAX N k = 0 K 1 k + 2 δ ,
φ [ n ] φ [ n + 1 ] π K 2 N Φ MAX + 2 δ .
φ [ m + 1 ] φ * [ m + 1 ] φ [ m + 1 ] φ [ m ] + φ [ m ] φ * [ m + 1 ] 2 D .
φ [ m ] φ [ m + 1 ] π K 2 N Φ MAX + 2 δ = D .

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