Abstract

For the wrapped phase map with regional abnormal fringes, a new phase unwrapping algorithm that combines the image-inpainting theory and the quality-guided phase unwrapping algorithm is proposed. First, by applying a threshold to the modulation map, the valid region (i.e., the interference region) is divided into the doubtful region (called the target region during the inpainting period) and the reasonable one (the source region). The wrapped phase of the doubtful region is thought to be unreliable, and the data are abandoned temporarily. Using the region-filling image-inpainting method, the blank target region is filled with new data, while nothing is changed in the source region. A new wrapped phase map is generated, and then it is unwrapped with the quality-guided phase unwrapping algorithm. Finally, a postprocessing operation is proposed for the final result. Experimental results have shown that the performance of the proposed algorithm is effective.

© 2012 Optical Society of America

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References

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  1. D. C. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithm, and Software (Wiley, 1998).
  2. H. A. Aebischer and S. Waldner, “A simple and effective method for filtering speckle-interferometric phase fringe patterns,” Opt. Commun. 162, 205–210 (1999).
    [CrossRef]
  3. K. Qian, W. Gao, and H. Wang, “Windowed Fourier-filtered and quality-guided phase-unwrapping algorithm,” Appl. Opt. 47, 5420–5428 (2008).
    [CrossRef]
  4. T. J. Flynn, “Consistent 2-D phase unwrapping guided by a quality map,” in Proceedings of IEEE Conference on Geoscience and Remote Sensing Symposium (IEEE, 1996), pp. 2057–2059.
  5. T. J. Flynn, “Two-dimensional phase unwrapping with minimum weighted discontinuity,” J. Opt. Soc. Am. A 14, 2692–2701 (1997).
    [CrossRef]
  6. W. Xu and I. Cumming, “A region-growing algorithm for InSAR phase unwrapping,” IEEE Trans. Geosci. Remote Sensing 37, 124–134 (1999).
    [CrossRef]
  7. M. A. Herráez, D. R. Burton, M. J. Lalor, and M. A. Gdeisat, “Fast two-dimensional phase-unwrapping algorithm based on sorting by reliability following a noncontinuous path,” Appl. Opt. 41, 7437–7444 (2002).
    [CrossRef]
  8. J. Bioucas-Dias and G. Valadão, “Phase unwrapping via graph cuts,” IEEE Trans. Image Process. 16, 698–709 (2007).
    [CrossRef]
  9. L. Meng, S. Fang, P. Yang, and L. Wang, “Quality-guided phase unwrapping algorithm based on reliability evaluation,” Appl. Opt. 50, 1925–1932 (2011).
    [CrossRef]
  10. M. Servin, F. J. Cuevas, D. Malacara, J. L. Marroguin, and R. Rodriguez-Vera, “Phase unwrapping through demodulation by use of the regularized phase-tracking technique,” Appl. Opt. 38, 1934–1941 (1999).
    [CrossRef]
  11. H. Y. Yun, C. K. Hong, and S. W. Chang, “Least-square phase estimation with multiple parameters in phase-shifting electronic speckle pattern interferometry,” J. Opt. Soc. Am. A 20, 240–247 (2003).
    [CrossRef]
  12. K. Qian, S. H. Soon, and A. Asundi, “A simple phase unwrapping approach based on filtering by windowed Fourier transform,” Opt. Laser Technol. 37, 458–462 (2005).
    [CrossRef]
  13. J. Bioucas-Dias, V. Katkovnik, J. Astola, and K. Egiazarian, “Absolute phase estimation: adaptive local denoising and global unwrapping,” Appl. Opt. 47, 5358–5369 (2008).
    [CrossRef]
  14. S. Fang, A. Kubo, H. Fujio, K. Oyama, Y. Saitoh, and M. Suzuki, “Digital phase data processing method for laser interferometry measurement of gear tooth flank,” in Proceedings of VDI International Conference on Gears, Vol. 1230 (VDI-Verlag, 1996), pp. 1111–1123.
  15. S. Fang, L. Wang, M. Komori, and A. Kubo, “Simulation method for interference fringe patterns in measuring gear tooth flanks by laser interferometry,” Appl. Opt. 49, 6409–6415 (2010).
    [CrossRef]
  16. S. Fang, L. Wang, M. Komori, and A. Kubo, “Design of laser interferometric system for measurement of gear tooth flank,” Optik 122, 1301–1304 (2011).
    [CrossRef]
  17. S. Fang, L. Wang, P. Yang, L. Meng, M. Komori, and A. Kubo, “Improvement of the oblique-incidence optical interferometric system to measure tooth flanks of involute helical gears,” J. Opt. Soc. Am. A 28, 590–595 (2011).
    [CrossRef]
  18. X. Su and W. Chen, “Reliability-guided phase unwrapping algorithm: a review,” Opt. Lasers Eng. 42, 245–261 (2004).
    [CrossRef]
  19. S. Esedoglu and J. Shen, “Digital inpainting based on the Mumford–Shah–Euler image model,” Eur. J. Appl. Math. 13, 353–370 (2002).
    [CrossRef]
  20. T. F. Chan and J. Shen, “Non-texture inpainting by curvature-driven diffusions,” J. Visual Commun. Image Rep. 4, 436–449 (2001).
    [CrossRef]
  21. T. F. Chan, S. H. Kang, and J. H. Shen, “Euler’s elastica and curvature based inpainting,” SIAM J. Appl. Math. 63, 564–592 (2002).
  22. A. Efros and W. T. Freeman, “Image quilting for texture synthesis and transfer,” in Proceedings of the 28th Annual Conference on Computer Graphics and Interactive Techniques (ASSOC Computing Machinery, 2001), pp. 341–346.
  23. I. Drori, D. Cohen-Or, and H. Yeshurun, “Fragment based image completion,” ACM Trans. Graph. 22, 303–312 (2003).
    [CrossRef]
  24. A. Criminisi, P. Perez, and K. Toyama, “Object removal by exemplar-based inpainting,” in Proceedings of IEEE Conference on Computer Vision and Pattern Recognition (IEEE, 2003), pp. 1–8.
  25. A. Criminisi, P. Perez, and K. Toyama, “Region filling and object removal by exemplar-based image inpainting,” IEEE Trans. Image Process. 13, 1200–1212 (2004).
    [CrossRef]

2011 (3)

2010 (1)

2008 (2)

2007 (1)

J. Bioucas-Dias and G. Valadão, “Phase unwrapping via graph cuts,” IEEE Trans. Image Process. 16, 698–709 (2007).
[CrossRef]

2005 (1)

K. Qian, S. H. Soon, and A. Asundi, “A simple phase unwrapping approach based on filtering by windowed Fourier transform,” Opt. Laser Technol. 37, 458–462 (2005).
[CrossRef]

2004 (2)

X. Su and W. Chen, “Reliability-guided phase unwrapping algorithm: a review,” Opt. Lasers Eng. 42, 245–261 (2004).
[CrossRef]

A. Criminisi, P. Perez, and K. Toyama, “Region filling and object removal by exemplar-based image inpainting,” IEEE Trans. Image Process. 13, 1200–1212 (2004).
[CrossRef]

2003 (2)

2002 (3)

S. Esedoglu and J. Shen, “Digital inpainting based on the Mumford–Shah–Euler image model,” Eur. J. Appl. Math. 13, 353–370 (2002).
[CrossRef]

M. A. Herráez, D. R. Burton, M. J. Lalor, and M. A. Gdeisat, “Fast two-dimensional phase-unwrapping algorithm based on sorting by reliability following a noncontinuous path,” Appl. Opt. 41, 7437–7444 (2002).
[CrossRef]

T. F. Chan, S. H. Kang, and J. H. Shen, “Euler’s elastica and curvature based inpainting,” SIAM J. Appl. Math. 63, 564–592 (2002).

2001 (1)

T. F. Chan and J. Shen, “Non-texture inpainting by curvature-driven diffusions,” J. Visual Commun. Image Rep. 4, 436–449 (2001).
[CrossRef]

1999 (3)

M. Servin, F. J. Cuevas, D. Malacara, J. L. Marroguin, and R. Rodriguez-Vera, “Phase unwrapping through demodulation by use of the regularized phase-tracking technique,” Appl. Opt. 38, 1934–1941 (1999).
[CrossRef]

H. A. Aebischer and S. Waldner, “A simple and effective method for filtering speckle-interferometric phase fringe patterns,” Opt. Commun. 162, 205–210 (1999).
[CrossRef]

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

1997 (1)

Aebischer, H. A.

H. A. Aebischer and S. Waldner, “A simple and effective method for filtering speckle-interferometric phase fringe patterns,” Opt. Commun. 162, 205–210 (1999).
[CrossRef]

Astola, J.

Asundi, A.

K. Qian, S. H. Soon, and A. Asundi, “A simple phase unwrapping approach based on filtering by windowed Fourier transform,” Opt. Laser Technol. 37, 458–462 (2005).
[CrossRef]

Bioucas-Dias, J.

Burton, D. R.

Chan, T. F.

T. F. Chan, S. H. Kang, and J. H. Shen, “Euler’s elastica and curvature based inpainting,” SIAM J. Appl. Math. 63, 564–592 (2002).

T. F. Chan and J. Shen, “Non-texture inpainting by curvature-driven diffusions,” J. Visual Commun. Image Rep. 4, 436–449 (2001).
[CrossRef]

Chang, S. W.

Chen, W.

X. Su and W. Chen, “Reliability-guided phase unwrapping algorithm: a review,” Opt. Lasers Eng. 42, 245–261 (2004).
[CrossRef]

Cohen-Or, D.

I. Drori, D. Cohen-Or, and H. Yeshurun, “Fragment based image completion,” ACM Trans. Graph. 22, 303–312 (2003).
[CrossRef]

Criminisi, A.

A. Criminisi, P. Perez, and K. Toyama, “Region filling and object removal by exemplar-based image inpainting,” IEEE Trans. Image Process. 13, 1200–1212 (2004).
[CrossRef]

A. Criminisi, P. Perez, and K. Toyama, “Object removal by exemplar-based inpainting,” in Proceedings of IEEE Conference on Computer Vision and Pattern Recognition (IEEE, 2003), pp. 1–8.

Cuevas, F. J.

Cumming, I.

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

Drori, I.

I. Drori, D. Cohen-Or, and H. Yeshurun, “Fragment based image completion,” ACM Trans. Graph. 22, 303–312 (2003).
[CrossRef]

Efros, A.

A. Efros and W. T. Freeman, “Image quilting for texture synthesis and transfer,” in Proceedings of the 28th Annual Conference on Computer Graphics and Interactive Techniques (ASSOC Computing Machinery, 2001), pp. 341–346.

Egiazarian, K.

Esedoglu, S.

S. Esedoglu and J. Shen, “Digital inpainting based on the Mumford–Shah–Euler image model,” Eur. J. Appl. Math. 13, 353–370 (2002).
[CrossRef]

Fang, S.

S. Fang, L. Wang, M. Komori, and A. Kubo, “Design of laser interferometric system for measurement of gear tooth flank,” Optik 122, 1301–1304 (2011).
[CrossRef]

S. Fang, L. Wang, P. Yang, L. Meng, M. Komori, and A. Kubo, “Improvement of the oblique-incidence optical interferometric system to measure tooth flanks of involute helical gears,” J. Opt. Soc. Am. A 28, 590–595 (2011).
[CrossRef]

L. Meng, S. Fang, P. Yang, and L. Wang, “Quality-guided phase unwrapping algorithm based on reliability evaluation,” Appl. Opt. 50, 1925–1932 (2011).
[CrossRef]

S. Fang, L. Wang, M. Komori, and A. Kubo, “Simulation method for interference fringe patterns in measuring gear tooth flanks by laser interferometry,” Appl. Opt. 49, 6409–6415 (2010).
[CrossRef]

S. Fang, A. Kubo, H. Fujio, K. Oyama, Y. Saitoh, and M. Suzuki, “Digital phase data processing method for laser interferometry measurement of gear tooth flank,” in Proceedings of VDI International Conference on Gears, Vol. 1230 (VDI-Verlag, 1996), pp. 1111–1123.

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 IEEE Conference on Geoscience and Remote Sensing Symposium (IEEE, 1996), pp. 2057–2059.

Freeman, W. T.

A. Efros and W. T. Freeman, “Image quilting for texture synthesis and transfer,” in Proceedings of the 28th Annual Conference on Computer Graphics and Interactive Techniques (ASSOC Computing Machinery, 2001), pp. 341–346.

Fujio, H.

S. Fang, A. Kubo, H. Fujio, K. Oyama, Y. Saitoh, and M. Suzuki, “Digital phase data processing method for laser interferometry measurement of gear tooth flank,” in Proceedings of VDI International Conference on Gears, Vol. 1230 (VDI-Verlag, 1996), pp. 1111–1123.

Gao, W.

Gdeisat, M. A.

Ghiglia, D. C.

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

Herráez, M. A.

Hong, C. K.

Kang, S. H.

T. F. Chan, S. H. Kang, and J. H. Shen, “Euler’s elastica and curvature based inpainting,” SIAM J. Appl. Math. 63, 564–592 (2002).

Katkovnik, V.

Komori, M.

Kubo, A.

S. Fang, L. Wang, M. Komori, and A. Kubo, “Design of laser interferometric system for measurement of gear tooth flank,” Optik 122, 1301–1304 (2011).
[CrossRef]

S. Fang, L. Wang, P. Yang, L. Meng, M. Komori, and A. Kubo, “Improvement of the oblique-incidence optical interferometric system to measure tooth flanks of involute helical gears,” J. Opt. Soc. Am. A 28, 590–595 (2011).
[CrossRef]

S. Fang, L. Wang, M. Komori, and A. Kubo, “Simulation method for interference fringe patterns in measuring gear tooth flanks by laser interferometry,” Appl. Opt. 49, 6409–6415 (2010).
[CrossRef]

S. Fang, A. Kubo, H. Fujio, K. Oyama, Y. Saitoh, and M. Suzuki, “Digital phase data processing method for laser interferometry measurement of gear tooth flank,” in Proceedings of VDI International Conference on Gears, Vol. 1230 (VDI-Verlag, 1996), pp. 1111–1123.

Lalor, M. J.

Malacara, D.

Marroguin, J. L.

Meng, L.

Oyama, K.

S. Fang, A. Kubo, H. Fujio, K. Oyama, Y. Saitoh, and M. Suzuki, “Digital phase data processing method for laser interferometry measurement of gear tooth flank,” in Proceedings of VDI International Conference on Gears, Vol. 1230 (VDI-Verlag, 1996), pp. 1111–1123.

Perez, P.

A. Criminisi, P. Perez, and K. Toyama, “Region filling and object removal by exemplar-based image inpainting,” IEEE Trans. Image Process. 13, 1200–1212 (2004).
[CrossRef]

A. Criminisi, P. Perez, and K. Toyama, “Object removal by exemplar-based inpainting,” in Proceedings of IEEE Conference on Computer Vision and Pattern Recognition (IEEE, 2003), pp. 1–8.

Pritt, M. D.

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

Qian, K.

K. Qian, W. Gao, and H. Wang, “Windowed Fourier-filtered and quality-guided phase-unwrapping algorithm,” Appl. Opt. 47, 5420–5428 (2008).
[CrossRef]

K. Qian, S. H. Soon, and A. Asundi, “A simple phase unwrapping approach based on filtering by windowed Fourier transform,” Opt. Laser Technol. 37, 458–462 (2005).
[CrossRef]

Rodriguez-Vera, R.

Saitoh, Y.

S. Fang, A. Kubo, H. Fujio, K. Oyama, Y. Saitoh, and M. Suzuki, “Digital phase data processing method for laser interferometry measurement of gear tooth flank,” in Proceedings of VDI International Conference on Gears, Vol. 1230 (VDI-Verlag, 1996), pp. 1111–1123.

Servin, M.

Shen, J.

S. Esedoglu and J. Shen, “Digital inpainting based on the Mumford–Shah–Euler image model,” Eur. J. Appl. Math. 13, 353–370 (2002).
[CrossRef]

T. F. Chan and J. Shen, “Non-texture inpainting by curvature-driven diffusions,” J. Visual Commun. Image Rep. 4, 436–449 (2001).
[CrossRef]

Shen, J. H.

T. F. Chan, S. H. Kang, and J. H. Shen, “Euler’s elastica and curvature based inpainting,” SIAM J. Appl. Math. 63, 564–592 (2002).

Soon, S. H.

K. Qian, S. H. Soon, and A. Asundi, “A simple phase unwrapping approach based on filtering by windowed Fourier transform,” Opt. Laser Technol. 37, 458–462 (2005).
[CrossRef]

Su, X.

X. Su and W. Chen, “Reliability-guided phase unwrapping algorithm: a review,” Opt. Lasers Eng. 42, 245–261 (2004).
[CrossRef]

Suzuki, M.

S. Fang, A. Kubo, H. Fujio, K. Oyama, Y. Saitoh, and M. Suzuki, “Digital phase data processing method for laser interferometry measurement of gear tooth flank,” in Proceedings of VDI International Conference on Gears, Vol. 1230 (VDI-Verlag, 1996), pp. 1111–1123.

Toyama, K.

A. Criminisi, P. Perez, and K. Toyama, “Region filling and object removal by exemplar-based image inpainting,” IEEE Trans. Image Process. 13, 1200–1212 (2004).
[CrossRef]

A. Criminisi, P. Perez, and K. Toyama, “Object removal by exemplar-based inpainting,” in Proceedings of IEEE Conference on Computer Vision and Pattern Recognition (IEEE, 2003), pp. 1–8.

Valadão, G.

J. Bioucas-Dias and G. Valadão, “Phase unwrapping via graph cuts,” IEEE Trans. Image Process. 16, 698–709 (2007).
[CrossRef]

Waldner, S.

H. A. Aebischer and S. Waldner, “A simple and effective method for filtering speckle-interferometric phase fringe patterns,” Opt. Commun. 162, 205–210 (1999).
[CrossRef]

Wang, H.

Wang, L.

Xu, W.

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

Yang, P.

Yeshurun, H.

I. Drori, D. Cohen-Or, and H. Yeshurun, “Fragment based image completion,” ACM Trans. Graph. 22, 303–312 (2003).
[CrossRef]

Yun, H. Y.

ACM Trans. Graph. (1)

I. Drori, D. Cohen-Or, and H. Yeshurun, “Fragment based image completion,” ACM Trans. Graph. 22, 303–312 (2003).
[CrossRef]

Appl. Opt. (6)

Eur. J. Appl. Math. (1)

S. Esedoglu and J. Shen, “Digital inpainting based on the Mumford–Shah–Euler image model,” Eur. J. Appl. Math. 13, 353–370 (2002).
[CrossRef]

IEEE Trans. Geosci. Remote Sensing (1)

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

IEEE Trans. Image Process. (2)

J. Bioucas-Dias and G. Valadão, “Phase unwrapping via graph cuts,” IEEE Trans. Image Process. 16, 698–709 (2007).
[CrossRef]

A. Criminisi, P. Perez, and K. Toyama, “Region filling and object removal by exemplar-based image inpainting,” IEEE Trans. Image Process. 13, 1200–1212 (2004).
[CrossRef]

J. Opt. Soc. Am. A (3)

J. Visual Commun. Image Rep. (1)

T. F. Chan and J. Shen, “Non-texture inpainting by curvature-driven diffusions,” J. Visual Commun. Image Rep. 4, 436–449 (2001).
[CrossRef]

Opt. Commun. (1)

H. A. Aebischer and S. Waldner, “A simple and effective method for filtering speckle-interferometric phase fringe patterns,” Opt. Commun. 162, 205–210 (1999).
[CrossRef]

Opt. Laser Technol. (1)

K. Qian, S. H. Soon, and A. Asundi, “A simple phase unwrapping approach based on filtering by windowed Fourier transform,” Opt. Laser Technol. 37, 458–462 (2005).
[CrossRef]

Opt. Lasers Eng. (1)

X. Su and W. Chen, “Reliability-guided phase unwrapping algorithm: a review,” Opt. Lasers Eng. 42, 245–261 (2004).
[CrossRef]

Optik (1)

S. Fang, L. Wang, M. Komori, and A. Kubo, “Design of laser interferometric system for measurement of gear tooth flank,” Optik 122, 1301–1304 (2011).
[CrossRef]

SIAM J. Appl. Math. (1)

T. F. Chan, S. H. Kang, and J. H. Shen, “Euler’s elastica and curvature based inpainting,” SIAM J. Appl. Math. 63, 564–592 (2002).

Other (5)

A. Efros and W. T. Freeman, “Image quilting for texture synthesis and transfer,” in Proceedings of the 28th Annual Conference on Computer Graphics and Interactive Techniques (ASSOC Computing Machinery, 2001), pp. 341–346.

A. Criminisi, P. Perez, and K. Toyama, “Object removal by exemplar-based inpainting,” in Proceedings of IEEE Conference on Computer Vision and Pattern Recognition (IEEE, 2003), pp. 1–8.

S. Fang, A. Kubo, H. Fujio, K. Oyama, Y. Saitoh, and M. Suzuki, “Digital phase data processing method for laser interferometry measurement of gear tooth flank,” in Proceedings of VDI International Conference on Gears, Vol. 1230 (VDI-Verlag, 1996), pp. 1111–1123.

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

T. J. Flynn, “Consistent 2-D phase unwrapping guided by a quality map,” in Proceedings of IEEE Conference on Geoscience and Remote Sensing Symposium (IEEE, 1996), pp. 2057–2059.

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Figures (7)

Fig. 1.
Fig. 1.

Problem of the filtering method: (a) original wrapped phase map (105×69 pixels), (b) after filtering (the window size σ=3), (c) unwrapped result of the original wrapped phase, and (d) unwrapped result of the filtered wrapped phase map.

Fig. 2.
Fig. 2.

Structure propagation of RF algorithm: (a) original image with the target region Ω, the contour δΩ, and the source region Φ; (b) patch Ψp centered at point p (pδΩ); (c) most likely candidate Ψq; (d) partial filling of Ψp. Only the target pixel in the patch Ψp is filled with the value of the corresponding pixel in Ψq.

Fig. 3.
Fig. 3.

(a) Modulation map, (b) source region (white pixels) and target region (dark pixels), (c) restored wrapped phase map, (d) unwrapped result of the restored phase data using the quality-guided algorithm, (e) after the congruence operation, and (f) after the continuity operation.

Fig. 4.
Fig. 4.

Comparison between Fig. 1(c) and Fig. 3(e): (a) error map, (b) diagram generated with the 40th row phase data.

Fig. 5.
Fig. 5.

Wrapped phase maps: (a) original wrapped phase map (250×200 pixels), (b) after inpainting.

Fig. 6.
Fig. 6.

Unwrapped results: (a) unwrapped result of the original wrapped phase map, (b) unwrapped result of the restored wrapped phase map, (c) after the congruence operation, and (d) after the continuity operation.

Fig. 7.
Fig. 7.

Comparison between Figs. 6(a) and 6(c): (a) error map, (b) diagram generated with the 75th row phase data.

Equations (8)

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

M(x,y)=2N[(k=0N1Ik(x,y)cos(2πk/N))2+(k=0N1Ik(x,y)sin(2πk/N))2]1/2,
Tn=αTo,
P(p)=C(p).D(p),
C(p)=sΨpΦC(s)|Ψp|,
D(p)=|Ip.np|a,
Ψq^=argminΨqΦdist(Ψp^,Ψq),
C(t)=C(p^)tΨp^Ω.
ϕu(x,y)=φ(x,y)+int(ϕr(x,y)φ(x,y)2π)2π,

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