Abstract

We present a performance evaluation of eight two-dimensional phase unwrapping methods with respect to correct phase unwrapping and execution times. The evaluated methods are block least squares (BLS), adaptive integration (AI), quality guided path following (QUAL), mask cut (MCUT), multigrid (MGRID), preconditioned conjugate gradient (PCG), Flynn’s (FLYNN), and Liang’s (LIANG). This set included integration- (path following), least-squares-, L 1-, and model-based methods. The methods were tested on several synthetic images, on two magnetic resonance images, and on two interferometry images. The synthetic images were designed to demonstrate different aspects of the phase unwrapping problem. To test the noise robustness of the methods, independent noise was added to the synthetic images to yield different signal-to-noise ratios. Each experiment was performed 50 times with different noise realizations to test the stability of the methods. The results of the experiments showed that the congruent minimum L 1 norm FLYNN method was best overall and the most noise robust of the methods, but it was also one of the slowest methods. The integration-based QUAL method was the only method that correctly unwrapped the two interferometry images. The least-squares-based methods (MGRID, PCG) gave worse results on average than did the integration- (or path following) based methods (BLS, AI, QUAL, MCUT) and were also slower. The model-based LIANG method was sensitive to noise and resulted in large errors for the magnetic resonance images and the interferometry images. In conclusion, for a particular application there is a trade-off between the quality of the unwrapping and the execution time when we attempt to select the most appropriate method.

© 1999 Optical Society of America

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  36. R. C. Gonzalez, P. Wintz, Digital Image Processing (Addison-Wesley, Reading, Mass., 1987).
  37. O. D. Trier, T. Taxt, “Evaluation of binarization methods for document images,” IEEE Trans. Pattern Anal. Mach. Intell. 17, 312–315 (1995).
    [CrossRef]

1999

J. Strand, T. Taxt, A. K. Jain, “Two-dimensional phase unwrapping using a block least-squares method,” IEEE Trans. Image Process. 8, 375–386 (1999).
[CrossRef]

1997

1996

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

M. D. Pritt, “Phase unwrapping by means of multigrid techniques for interferometric SAR,” IEEE Trans. Geosci. Remote Sens. 34, 728–738 (1996).
[CrossRef]

Z.-P. Liang, “A model-based method for phase unwrapping,” IEEE Trans. Med. Imaging 15, 893–897 (1996).
[CrossRef] [PubMed]

1995

H. Gudbjartsson, S. Patz, “The Rician distribution of noisy MRI data,” Magn. Reson. Med. 34, 910–914 (1995).
[CrossRef] [PubMed]

O. D. Trier, T. Taxt, “Evaluation of binarization methods for document images,” IEEE Trans. Pattern Anal. Mach. Intell. 17, 312–315 (1995).
[CrossRef]

S. M.-H. Song, S. Napel, N. J. Pelc, G. H. Glover, “Phase unwrapping of MR phase images using Poisson equation,” IEEE Trans. Image Process. 4, 667–676 (1995).
[CrossRef]

1994

1992

M. Hedley, D. Rosenfeld, “A new two-dimensional phase unwrapping algorithm for MRI images,” Magn. Reson. Med. 24, 177–181 (1992).
[CrossRef] [PubMed]

1991

1988

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

1985

H. Hjalmarson, L. Romeo, D. Ghiglia, E. Jones, C. Norris, “Extraction of photon density-of-states from optical spectra,” Phys. Rev. B 32, 4300–4303 (1985).
[CrossRef]

1982

1979

B. R. Hunt, “Matrix formulation of the reconstruction of phase values from phase differences,” J. Opt. Soc. Am. A 69, 393–399 (1979).
[CrossRef]

1977

1976

Adam, N.

D. Just, N. Adam, M. Schwabisch, R. Bamler, “Comparison of phase unwrapping algorithms for SAR interferograms,” in Proceedings of the International Geoscience and Remote Sensing Symposium (Institute of Electrical and Electronics Engineers, New York, 1995), pp. 767–769.

Ai, C.

Y. Xu, C. Ai, “Simple and effective phase unwrapping technique,” in Interferometry VI: Techniques and Analysis, O. Y. Kwon, G. M. Brown, M. Kujawinska, eds., Proc. SPIE2003, 254–263 (1993).
[CrossRef]

Ames, W. F.

W. F. Ames, Numerical Methods for Partial Differential Equations (Academic, San Diego, Calif., 1992).

Bamler, R.

D. Just, N. Adam, M. Schwabisch, R. Bamler, “Comparison of phase unwrapping algorithms for SAR interferograms,” in Proceedings of the International Geoscience and Remote Sensing Symposium (Institute of Electrical and Electronics Engineers, New York, 1995), pp. 767–769.

Beheim, G.

C. R. Mercer, G. Beheim, “Fiber-optic projected-fringe digital interferometry,” in (NASA, Washington, D.C., 1990).

Berthod, M.

S. Dupont, D. Labrousse, M. Berthod, “SAR interferometry: an application of simulated annealing to phase unwrapping,” in Proceedings of the 9th Scandinavian Conference on Image Analysis (IAPR, Surrey, UK, 1995), pp. 697–705.

Bone, D. J.

Bydder, G. M.

I. R. Young, G. M. Bydder, “Phase Imaging,” in Magnetic Resonance Imaging, D. D. Stark, W. G. Bradley, eds. (Mosby Year Book, St. Louis, Mo., 1992), Chap. 9.

Dupont, S.

S. Dupont, D. Labrousse, M. Berthod, “SAR interferometry: an application of simulated annealing to phase unwrapping,” in Proceedings of the 9th Scandinavian Conference on Image Analysis (IAPR, Surrey, UK, 1995), pp. 697–705.

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C, Cambridge University, Cambridge, UK, 1992).

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, “Consistent 2-D Phase unwrapping guided by a quality map,” in Proceedings of the International Geoscience and Remote Sensing Symposium (Institute of Electrical and Electronics Engineers, New York, 1996), pp. 2057–2059.

Fornaro, G.

G. Fornaro, G. Franceschetti, R. Lanari, E. Sansosti, M. Tesauro, “Global and local phase-unwrapping techniques: a comparison,” J. Opt. Soc. Am. A 14, 2702–2708 (1997).
[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]

G. Fornaro, G. Franceschetti, R. Lanari, E. Sansosti, “A theoretical analysis on the robust phase unwrapping algorithms for SAR interferometry,” in Proceedings of the International Geoscience and Remote Sensing Symposium (Institute of Electrical and Electronics Engineers, New York, 1996), pp. 2047–2049.

Franceschetti, G.

G. Fornaro, G. Franceschetti, R. Lanari, E. Sansosti, M. Tesauro, “Global and local phase-unwrapping techniques: a comparison,” J. Opt. Soc. Am. A 14, 2702–2708 (1997).
[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]

G. Fornaro, G. Franceschetti, R. Lanari, E. Sansosti, “A theoretical analysis on the robust phase unwrapping algorithms for SAR interferometry,” in Proceedings of the International Geoscience and Remote Sensing Symposium (Institute of Electrical and Electronics Engineers, New York, 1996), pp. 2047–2049.

Ghiglia, D.

H. Hjalmarson, L. Romeo, D. Ghiglia, E. Jones, C. Norris, “Extraction of photon density-of-states from optical spectra,” Phys. Rev. B 32, 4300–4303 (1985).
[CrossRef]

Ghiglia, D. C.

Glover, G. H.

S. M.-H. Song, S. Napel, N. J. Pelc, G. H. Glover, “Phase unwrapping of MR phase images using Poisson equation,” IEEE Trans. Image Process. 4, 667–676 (1995).
[CrossRef]

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]

Golub, G. H.

G. H. Golub, C. F. Van Loan, Matrix Computations (John Hopkins U. Press, Baltimore, Md., 1989).

Gonzalez, R. C.

R. C. Gonzalez, P. Wintz, Digital Image Processing (Addison-Wesley, Reading, Mass., 1987).

Gudbjartsson, H.

H. Gudbjartsson, S. Patz, “The Rician distribution of noisy MRI data,” Magn. Reson. Med. 34, 910–914 (1995).
[CrossRef] [PubMed]

Hedley, M.

M. Hedley, D. Rosenfeld, “A new two-dimensional phase unwrapping algorithm for MRI images,” Magn. Reson. Med. 24, 177–181 (1992).
[CrossRef] [PubMed]

Hjalmarson, H.

H. Hjalmarson, L. Romeo, D. Ghiglia, E. Jones, C. Norris, “Extraction of photon density-of-states from optical spectra,” Phys. Rev. B 32, 4300–4303 (1985).
[CrossRef]

Holmes, J. F.

Hunt, B. R.

B. R. Hunt, “Matrix formulation of the reconstruction of phase values from phase differences,” J. Opt. Soc. Am. A 69, 393–399 (1979).
[CrossRef]

Itoh, K.

Jain, A. K.

J. Strand, T. Taxt, A. K. Jain, “Two-dimensional phase unwrapping using a block least-squares method,” IEEE Trans. Image Process. 8, 375–386 (1999).
[CrossRef]

Jones, E.

H. Hjalmarson, L. Romeo, D. Ghiglia, E. Jones, C. Norris, “Extraction of photon density-of-states from optical spectra,” Phys. Rev. B 32, 4300–4303 (1985).
[CrossRef]

Just, D.

D. Just, N. Adam, M. Schwabisch, R. Bamler, “Comparison of phase unwrapping algorithms for SAR interferograms,” in Proceedings of the International Geoscience and Remote Sensing Symposium (Institute of Electrical and Electronics Engineers, New York, 1995), pp. 767–769.

Kaufmann, G. H.

R. Rodriguez-Vera, G. H. Kaufmann, A. J. Moore, “Unwrapping of noisy phase maps: a comparison of two methods,” in Laser Interferometry III: Techniques and Analysis, M. Kujawinska, R. J. Pryputniewicz, M. Takeda, eds., Proc. SPIE2860, 61–71 (1996).

Kerr, J. R.

Labrousse, D.

S. Dupont, D. Labrousse, M. Berthod, “SAR interferometry: an application of simulated annealing to phase unwrapping,” in Proceedings of the 9th Scandinavian Conference on Image Analysis (IAPR, Surrey, UK, 1995), pp. 697–705.

Lanari, R.

G. Fornaro, G. Franceschetti, R. Lanari, E. Sansosti, M. Tesauro, “Global and local phase-unwrapping techniques: a comparison,” J. Opt. Soc. Am. A 14, 2702–2708 (1997).
[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]

G. Fornaro, G. Franceschetti, R. Lanari, E. Sansosti, “A theoretical analysis on the robust phase unwrapping algorithms for SAR interferometry,” in Proceedings of the International Geoscience and Remote Sensing Symposium (Institute of Electrical and Electronics Engineers, New York, 1996), pp. 2047–2049.

Lee, M. H.

Liang, Z.-P.

Z.-P. Liang, “A model-based method for phase unwrapping,” IEEE Trans. Med. Imaging 15, 893–897 (1996).
[CrossRef] [PubMed]

Mercer, C. R.

C. R. Mercer, G. Beheim, “Fiber-optic projected-fringe digital interferometry,” in (NASA, Washington, D.C., 1990).

Moore, A. J.

R. Rodriguez-Vera, G. H. Kaufmann, A. J. Moore, “Unwrapping of noisy phase maps: a comparison of two methods,” in Laser Interferometry III: Techniques and Analysis, M. Kujawinska, R. J. Pryputniewicz, M. Takeda, eds., Proc. SPIE2860, 61–71 (1996).

Napel, S.

S. M.-H. Song, S. Napel, N. J. Pelc, G. H. Glover, “Phase unwrapping of MR phase images using Poisson equation,” IEEE Trans. Image Process. 4, 667–676 (1995).
[CrossRef]

Norris, C.

H. Hjalmarson, L. Romeo, D. Ghiglia, E. Jones, C. Norris, “Extraction of photon density-of-states from optical spectra,” Phys. Rev. B 32, 4300–4303 (1985).
[CrossRef]

Oppenheim, A. V.

A. V. Oppenheim, R. W. Schafer, Discrete-Time Signal Processing (Prentice-Hall, Englewood Cliffs., N.J., 1989).

Patz, S.

H. Gudbjartsson, S. Patz, “The Rician distribution of noisy MRI data,” Magn. Reson. Med. 34, 910–914 (1995).
[CrossRef] [PubMed]

Pelc, N. J.

S. M.-H. Song, S. Napel, N. J. Pelc, G. H. Glover, “Phase unwrapping of MR phase images using Poisson equation,” IEEE Trans. Image Process. 4, 667–676 (1995).
[CrossRef]

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C, Cambridge University, Cambridge, UK, 1992).

Pritt, M. D.

M. D. Pritt, “Phase unwrapping by means of multigrid techniques for interferometric SAR,” IEEE Trans. Geosci. Remote Sens. 34, 728–738 (1996).
[CrossRef]

M. D. Pritt, “Congruence in least-squares phase unwrapping,” in Proceedings of the International Geoscience and Remote Sensing Symposium (Institute of Electrical and Electronics Engineers, New York, 1997), Vol. 2, pp. 875–877.

M. D. Pritt, “Comparison of path-following and least-squares phase unwrapping algorithms,” in Proceedings of the International Geoscience and Remote Sensing Symposium (Institute of Electrical and Electronics Engineers, New York, 1997). Vol. 2, pp. 872–874.

D. C. Ghiglia, M. D. Pritt, Two-Dimensional Phase Unwrapping (Wiley, New York, 1998).

Rodriguez-Vera, R.

R. Rodriguez-Vera, G. H. Kaufmann, A. J. Moore, “Unwrapping of noisy phase maps: a comparison of two methods,” in Laser Interferometry III: Techniques and Analysis, M. Kujawinska, R. J. Pryputniewicz, M. Takeda, eds., Proc. SPIE2860, 61–71 (1996).

Romeo, L.

H. Hjalmarson, L. Romeo, D. Ghiglia, E. Jones, C. Norris, “Extraction of photon density-of-states from optical spectra,” Phys. Rev. B 32, 4300–4303 (1985).
[CrossRef]

Romero, L. A.

Rosenfeld, D.

M. Hedley, D. Rosenfeld, “A new two-dimensional phase unwrapping algorithm for MRI images,” Magn. Reson. Med. 24, 177–181 (1992).
[CrossRef] [PubMed]

Sansosti, E.

G. Fornaro, G. Franceschetti, R. Lanari, E. Sansosti, M. Tesauro, “Global and local phase-unwrapping techniques: a comparison,” J. Opt. Soc. Am. A 14, 2702–2708 (1997).
[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]

G. Fornaro, G. Franceschetti, R. Lanari, E. Sansosti, “A theoretical analysis on the robust phase unwrapping algorithms for SAR interferometry,” in Proceedings of the International Geoscience and Remote Sensing Symposium (Institute of Electrical and Electronics Engineers, New York, 1996), pp. 2047–2049.

Schafer, R. W.

A. V. Oppenheim, R. W. Schafer, Discrete-Time Signal Processing (Prentice-Hall, Englewood Cliffs., N.J., 1989).

Schwabisch, M.

D. Just, N. Adam, M. Schwabisch, R. Bamler, “Comparison of phase unwrapping algorithms for SAR interferograms,” in Proceedings of the International Geoscience and Remote Sensing Symposium (Institute of Electrical and Electronics Engineers, New York, 1995), pp. 767–769.

Song, S. M.-H.

S. M.-H. Song, S. Napel, N. J. Pelc, G. H. Glover, “Phase unwrapping of MR phase images using Poisson equation,” IEEE Trans. Image Process. 4, 667–676 (1995).
[CrossRef]

Strand, J.

J. Strand, T. Taxt, A. K. Jain, “Two-dimensional phase unwrapping using a block least-squares method,” IEEE Trans. Image Process. 8, 375–386 (1999).
[CrossRef]

J. Strand, T. Taxt are preparing a manuscript to be called “Two-dimensional phase unwrapping using robust derivative estimation and adaptive integration.”

Taxt, T.

J. Strand, T. Taxt, A. K. Jain, “Two-dimensional phase unwrapping using a block least-squares method,” IEEE Trans. Image Process. 8, 375–386 (1999).
[CrossRef]

O. D. Trier, T. Taxt, “Evaluation of binarization methods for document images,” IEEE Trans. Pattern Anal. Mach. Intell. 17, 312–315 (1995).
[CrossRef]

J. Strand, T. Taxt are preparing a manuscript to be called “Two-dimensional phase unwrapping using robust derivative estimation and adaptive integration.”

Tesauro, M.

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C, Cambridge University, Cambridge, UK, 1992).

Thomas, J. B.

J. B. Thomas, An Introduction to Statistical Communication Theory (Wiley, New York, 1969).

Trier, O. D.

O. D. Trier, T. Taxt, “Evaluation of binarization methods for document images,” IEEE Trans. Pattern Anal. Mach. Intell. 17, 312–315 (1995).
[CrossRef]

Van Loan, C. F.

G. H. Golub, C. F. Van Loan, Matrix Computations (John Hopkins U. Press, Baltimore, Md., 1989).

Vetterling, W. T.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C, Cambridge University, Cambridge, UK, 1992).

Wang, J. Y.

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]

Wintz, P.

R. C. Gonzalez, P. Wintz, Digital Image Processing (Addison-Wesley, Reading, Mass., 1987).

Xu, Y.

Y. Xu, C. Ai, “Simple and effective phase unwrapping technique,” in Interferometry VI: Techniques and Analysis, O. Y. Kwon, G. M. Brown, M. Kujawinska, eds., Proc. SPIE2003, 254–263 (1993).
[CrossRef]

Young, I. R.

I. R. Young, G. M. Bydder, “Phase Imaging,” in Magnetic Resonance Imaging, D. D. Stark, W. G. Bradley, eds. (Mosby Year Book, St. Louis, Mo., 1992), Chap. 9.

Zebker, H. A.

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.

IEEE Trans. Geosci. Remote Sens.

M. D. Pritt, “Phase unwrapping by means of multigrid techniques for interferometric SAR,” IEEE Trans. Geosci. Remote Sens. 34, 728–738 (1996).
[CrossRef]

IEEE Trans. Image Process.

J. Strand, T. Taxt, A. K. Jain, “Two-dimensional phase unwrapping using a block least-squares method,” IEEE Trans. Image Process. 8, 375–386 (1999).
[CrossRef]

S. M.-H. Song, S. Napel, N. J. Pelc, G. H. Glover, “Phase unwrapping of MR phase images using Poisson equation,” IEEE Trans. Image Process. 4, 667–676 (1995).
[CrossRef]

IEEE Trans. Med. Imaging

Z.-P. Liang, “A model-based method for phase unwrapping,” IEEE Trans. Med. Imaging 15, 893–897 (1996).
[CrossRef] [PubMed]

IEEE Trans. Pattern Anal. Mach. Intell.

O. D. Trier, T. Taxt, “Evaluation of binarization methods for document images,” IEEE Trans. Pattern Anal. Mach. Intell. 17, 312–315 (1995).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Magn. Reson. Med.

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Figures (10)

Fig. 1
Fig. 1

Phase noise density functions for different SNR levels.

Fig. 2
Fig. 2

COSINE test image. (a) True phase values, (b) wrapped phase values.

Fig. 3
Fig. 3

DISC test image. (a) True phase values, (b) wrapped phase values.

Fig. 4
Fig. 4

Masks for the DISC image. (a) Mask used as input to the phase unwrapping methods, (b) mask used for evaluation.

Fig. 5
Fig. 5

MR images. (a) Phase image MR1, (b) amplitude corresponding to (a), (c) phase image MR2, (d) amplitude corresponding to (b).

Fig. 6
Fig. 6

Interferometry images. (a) Image IF1, (b) wrapped gradient of (a), (c) image IF2, (d) fringe contrast image corresponding to (c).

Fig. 7
Fig. 7

Image MR1. Discontinuities in the unwrapped results are marked in black for methods (a) BLS, (b) AI, (c) QUAL, (d) MCUT, (e) MGRID, (f) PCG, (g) FLYNN, and (h) LIANG.

Fig. 8
Fig. 8

Image MR2. Discontinuities in the unwrapped results are marked in black for methods (a) BLS, (b) AI, (c) QUAL, (d) MCUT, (e) MGRID, (f) PCG, (g) FLYNN, and (h) LIANG.

Fig. 9
Fig. 9

Image IF1. Discontinuities in the unwrapped results are marked in black for methods (a) BLS, (b) AI, (c) QUAL, (d) MCUT, (e) MGRID, (f) PCG, (g) FLYNN, and (h) LIANG.

Fig. 10
Fig. 10

Image IF2. Discontinuities in the unwrapped results are marked in black for methods (a) BLS, (b) AI, (c) QUAL, (d) MCUT, (e) MGRID, (f) PCG, (g) FLYNN, and (h) LIANG.

Tables (6)

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Table 1 Linear Plane Test

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Table 2 COSINE Image Test

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Table 3 DISC Image, Test 1

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Table 4 DISC Image, Test 2

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Table 5 Number of Discontinuities in Unwrapped Real Images

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Table 6 Execution Timesa, b

Equations (3)

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

Pwnx, y=arctanInx, y/Rnx, y,
σ2=1MNx=0M-1y=0N-1Punx, y-Panx, y-μ2,
μ=1MNx=0M-1y=0N-1 Punx, y-Panx, y.

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