Abstract

A nonorthogonal model for 2D signals with rotational components is presented, which enables estimation of phase values from observations of its local gradients. In this research, the rotational components are caused by the presence of branch points, which indicates phase wrapping. Using the proposed model, the phase is estimated using standard least-squares or recently proposed wavelet techniques by processing a linear combination of the wrapped observed gradients and the curl generated by phase wrapping.

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References

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  1. J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. A 336, 165–190 (1974).
    [CrossRef]
  2. D. L. Fried and J. L. Vaughn, “Branch cuts in the phase function,” Appl. Opt. 31, 2865–2882 (1992).
    [CrossRef]
  3. D. L. Fried, “Branch point problem in adaptive optics,” J. Opt. Soc. Am. A 15, 2759–2768 (1998).
    [CrossRef]
  4. G. Tyler, “Reconstruction and assessment of the least-squares and slope discrepancy components of the phase,” J. Opt. Soc. Am. A 17, 1828–1839 (2000).
    [CrossRef]
  5. D. L. Fried, “Adaptive optics wave function reconstruction and phase unwrapping when branch points are present,” Opt. Commun. 200, 43–72 (2001).
    [CrossRef]
  6. R. Goldstein, H. A. Zebker, and C. L. Werner, “Satellite radar interferometry: two-dimensional phase unwrapping,” Radio Sci. 23, 713–720 (1988).
    [CrossRef]
  7. D. C. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms, and Software (Wiley, 1998).
  8. T. M. Venema and J. D. Schmidt, “Optical phase unwrapping in the presence of branch points,” Opt. Express 16, 6985–6998 (2008).
    [CrossRef]
  9. C. Pellizzari, “Phase unwrapping in the presence of strong turbulence,” in IEEE Aerospace Conference (IEEE, 2010).
  10. C. Pellizzari and J. D. Schmidt, “Phase unwrapping in the presence of strong turbulence,” IEEE Aerospace Conference (2010), pp. 1–10.
  11. J. D. Barchers, D. L. Fried, and D. J. Link, “Evaluation of the performance of Hartmann sensors in strong scintillation,” Appl. Opt. 41, 1012–1021 (2002).
    [CrossRef]
  12. J. D. Barchers, D. L. Fried, D. J. Link, G. A. Tyler, W. Moretti, T. J. Brennan, and R. Q. Fugate, “The performance of wavefront sensors in strong scintillation,” Proc. SPIE 4839, 217–227 (2003).
    [CrossRef]
  13. E.-O. Le Bigot and W. J. Wild, “Theory of branch-point detection and its implementation,” J. Opt. Soc. Am. A 16, 1724–1729 (1999).
    [CrossRef]
  14. K. Murphy, R. Mackey, and C. Dainty, “Branch point detection and correction using the branch point potential method,” Proc. SPIE 6951, 695105 (2008).
    [CrossRef]
  15. V. E. Zetterlind and E. P. Magee, “Performance of various branch point tolerant phase reconstructors with finite time delays and measurement noise,” Proc. SPIE 4632, 85–94 (2002).
    [CrossRef]
  16. D. J. Sanchez and D. W. Oesch, “The aggregate behavior of branch points: the creation and evolution of branch points,” Proc. SPIE 7466, 746605 (2009).
    [CrossRef]
  17. D. W. Oesch, D. J. Sanchez, C. M. Tewksbury-Christle, and P. R. Kelly, “The aggregate behavior of branch points: altitude and strength of atmospheric turbulence layers,” Proc. SPIE 7816, 781605 (2010).
    [CrossRef]
  18. D. J. Sanchez, D. W. Oesch, and P. R. Kelly, “The aggregate behavior of branch points: theoretical calculation of branch point velocity,” Proc. SPIE 8380, 83800P (2012).
  19. D. W. Oesch, D. J. Sanchez, and C. M. Tewksbury-Christle, “Aggregate behavior of branch points–persistent pairs,” Opt. Express 20, 1046–1059 (2012).
    [CrossRef]
  20. D. W. Oesch, D. J. Sanchez, and C. L. Matson, “The aggregate behavior of branch points-measuring the number and velocity of atmospheric turbulence layers,” Opt. Express 18, 22377–22392 (2010).
    [CrossRef]
  21. D. J. Sanchez and D. W. Oesch, “Localization of angular momentum in optical waves propagating through turbulence,” Opt. Express 19, 25388–25396 (2011).
    [CrossRef]
  22. D. J. Sanchez and D. W. Oesch, “Orbital angular momentum in optical waves propagating through distributed turbulence,” Opt. Express 19, 24596–24608 (2011).
    [CrossRef]
  23. F. Roddier, Adaptive Optics in Astronomy (Cambridge University, 1999).
  24. R. C. Olsen, Remote Sensing from Air and Space (SPIE, 2007).
  25. D. L. Fried, “Least-square fitting a wave-front distortion estimate to an array of phase-difference measurements,” J. Opt. Soc. Am. 67, 370–375 (1977).
    [CrossRef]
  26. R. H. Hudgin, “Wave-front reconstruction for compensated imaging,” J. Opt. Soc. Am. 67, 375–378 (1977).
    [CrossRef]
  27. L. A. Poyneer, D. T. Gavel, and J. M. Brase, “Fast wave-front reconstruction in large adaptive optics systems with use of the Fourier transform,” J. Opt. Soc. Am. A 19, 2100–2111 (2002).
    [CrossRef]
  28. T. W. Axtell and R. Cristi, “Generalized orthogonal wavelet phase reconstruction,” J. Opt. Soc. Am. A 30, 859–870 (2013).
    [CrossRef]
  29. T. J. Brennan and P. H. Roberts, AOTools: the Adaptive Optics Toolbox for Use with MATLAB, Optical Sciences Company User’s Guide 1.4a (Optical Sciences Company, 2010).
  30. T. J. Brennan, P. H. Roberts, and D. C. Mann, WaveProp: a Wave Optics Simulation System for Use with MATLAB, Optical Sciences Company User’s Guide 1.3 (Optical Sciences Company, 2010).

2013 (1)

2012 (2)

D. J. Sanchez, D. W. Oesch, and P. R. Kelly, “The aggregate behavior of branch points: theoretical calculation of branch point velocity,” Proc. SPIE 8380, 83800P (2012).

D. W. Oesch, D. J. Sanchez, and C. M. Tewksbury-Christle, “Aggregate behavior of branch points–persistent pairs,” Opt. Express 20, 1046–1059 (2012).
[CrossRef]

2011 (2)

2010 (2)

D. W. Oesch, D. J. Sanchez, and C. L. Matson, “The aggregate behavior of branch points-measuring the number and velocity of atmospheric turbulence layers,” Opt. Express 18, 22377–22392 (2010).
[CrossRef]

D. W. Oesch, D. J. Sanchez, C. M. Tewksbury-Christle, and P. R. Kelly, “The aggregate behavior of branch points: altitude and strength of atmospheric turbulence layers,” Proc. SPIE 7816, 781605 (2010).
[CrossRef]

2009 (1)

D. J. Sanchez and D. W. Oesch, “The aggregate behavior of branch points: the creation and evolution of branch points,” Proc. SPIE 7466, 746605 (2009).
[CrossRef]

2008 (2)

K. Murphy, R. Mackey, and C. Dainty, “Branch point detection and correction using the branch point potential method,” Proc. SPIE 6951, 695105 (2008).
[CrossRef]

T. M. Venema and J. D. Schmidt, “Optical phase unwrapping in the presence of branch points,” Opt. Express 16, 6985–6998 (2008).
[CrossRef]

2003 (1)

J. D. Barchers, D. L. Fried, D. J. Link, G. A. Tyler, W. Moretti, T. J. Brennan, and R. Q. Fugate, “The performance of wavefront sensors in strong scintillation,” Proc. SPIE 4839, 217–227 (2003).
[CrossRef]

2002 (3)

2001 (1)

D. L. Fried, “Adaptive optics wave function reconstruction and phase unwrapping when branch points are present,” Opt. Commun. 200, 43–72 (2001).
[CrossRef]

2000 (1)

1999 (1)

1998 (1)

1992 (1)

1988 (1)

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

1977 (2)

1974 (1)

J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. A 336, 165–190 (1974).
[CrossRef]

Axtell, T. W.

Barchers, J. D.

J. D. Barchers, D. L. Fried, D. J. Link, G. A. Tyler, W. Moretti, T. J. Brennan, and R. Q. Fugate, “The performance of wavefront sensors in strong scintillation,” Proc. SPIE 4839, 217–227 (2003).
[CrossRef]

J. D. Barchers, D. L. Fried, and D. J. Link, “Evaluation of the performance of Hartmann sensors in strong scintillation,” Appl. Opt. 41, 1012–1021 (2002).
[CrossRef]

Berry, M. V.

J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. A 336, 165–190 (1974).
[CrossRef]

Brase, J. M.

Brennan, T. J.

J. D. Barchers, D. L. Fried, D. J. Link, G. A. Tyler, W. Moretti, T. J. Brennan, and R. Q. Fugate, “The performance of wavefront sensors in strong scintillation,” Proc. SPIE 4839, 217–227 (2003).
[CrossRef]

T. J. Brennan, P. H. Roberts, and D. C. Mann, WaveProp: a Wave Optics Simulation System for Use with MATLAB, Optical Sciences Company User’s Guide 1.3 (Optical Sciences Company, 2010).

T. J. Brennan and P. H. Roberts, AOTools: the Adaptive Optics Toolbox for Use with MATLAB, Optical Sciences Company User’s Guide 1.4a (Optical Sciences Company, 2010).

Cristi, R.

Dainty, C.

K. Murphy, R. Mackey, and C. Dainty, “Branch point detection and correction using the branch point potential method,” Proc. SPIE 6951, 695105 (2008).
[CrossRef]

Fried, D. L.

Fugate, R. Q.

J. D. Barchers, D. L. Fried, D. J. Link, G. A. Tyler, W. Moretti, T. J. Brennan, and R. Q. Fugate, “The performance of wavefront sensors in strong scintillation,” Proc. SPIE 4839, 217–227 (2003).
[CrossRef]

Gavel, D. T.

Ghiglia, D. C.

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

Goldstein, R.

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

Hudgin, R. H.

Kelly, P. R.

D. J. Sanchez, D. W. Oesch, and P. R. Kelly, “The aggregate behavior of branch points: theoretical calculation of branch point velocity,” Proc. SPIE 8380, 83800P (2012).

D. W. Oesch, D. J. Sanchez, C. M. Tewksbury-Christle, and P. R. Kelly, “The aggregate behavior of branch points: altitude and strength of atmospheric turbulence layers,” Proc. SPIE 7816, 781605 (2010).
[CrossRef]

Le Bigot, E.-O.

Link, D. J.

J. D. Barchers, D. L. Fried, D. J. Link, G. A. Tyler, W. Moretti, T. J. Brennan, and R. Q. Fugate, “The performance of wavefront sensors in strong scintillation,” Proc. SPIE 4839, 217–227 (2003).
[CrossRef]

J. D. Barchers, D. L. Fried, and D. J. Link, “Evaluation of the performance of Hartmann sensors in strong scintillation,” Appl. Opt. 41, 1012–1021 (2002).
[CrossRef]

Mackey, R.

K. Murphy, R. Mackey, and C. Dainty, “Branch point detection and correction using the branch point potential method,” Proc. SPIE 6951, 695105 (2008).
[CrossRef]

Magee, E. P.

V. E. Zetterlind and E. P. Magee, “Performance of various branch point tolerant phase reconstructors with finite time delays and measurement noise,” Proc. SPIE 4632, 85–94 (2002).
[CrossRef]

Mann, D. C.

T. J. Brennan, P. H. Roberts, and D. C. Mann, WaveProp: a Wave Optics Simulation System for Use with MATLAB, Optical Sciences Company User’s Guide 1.3 (Optical Sciences Company, 2010).

Matson, C. L.

Moretti, W.

J. D. Barchers, D. L. Fried, D. J. Link, G. A. Tyler, W. Moretti, T. J. Brennan, and R. Q. Fugate, “The performance of wavefront sensors in strong scintillation,” Proc. SPIE 4839, 217–227 (2003).
[CrossRef]

Murphy, K.

K. Murphy, R. Mackey, and C. Dainty, “Branch point detection and correction using the branch point potential method,” Proc. SPIE 6951, 695105 (2008).
[CrossRef]

Nye, J. F.

J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. A 336, 165–190 (1974).
[CrossRef]

Oesch, D. W.

D. J. Sanchez, D. W. Oesch, and P. R. Kelly, “The aggregate behavior of branch points: theoretical calculation of branch point velocity,” Proc. SPIE 8380, 83800P (2012).

D. W. Oesch, D. J. Sanchez, and C. M. Tewksbury-Christle, “Aggregate behavior of branch points–persistent pairs,” Opt. Express 20, 1046–1059 (2012).
[CrossRef]

D. J. Sanchez and D. W. Oesch, “Localization of angular momentum in optical waves propagating through turbulence,” Opt. Express 19, 25388–25396 (2011).
[CrossRef]

D. J. Sanchez and D. W. Oesch, “Orbital angular momentum in optical waves propagating through distributed turbulence,” Opt. Express 19, 24596–24608 (2011).
[CrossRef]

D. W. Oesch, D. J. Sanchez, and C. L. Matson, “The aggregate behavior of branch points-measuring the number and velocity of atmospheric turbulence layers,” Opt. Express 18, 22377–22392 (2010).
[CrossRef]

D. W. Oesch, D. J. Sanchez, C. M. Tewksbury-Christle, and P. R. Kelly, “The aggregate behavior of branch points: altitude and strength of atmospheric turbulence layers,” Proc. SPIE 7816, 781605 (2010).
[CrossRef]

D. J. Sanchez and D. W. Oesch, “The aggregate behavior of branch points: the creation and evolution of branch points,” Proc. SPIE 7466, 746605 (2009).
[CrossRef]

Olsen, R. C.

R. C. Olsen, Remote Sensing from Air and Space (SPIE, 2007).

Pellizzari, C.

C. Pellizzari, “Phase unwrapping in the presence of strong turbulence,” in IEEE Aerospace Conference (IEEE, 2010).

C. Pellizzari and J. D. Schmidt, “Phase unwrapping in the presence of strong turbulence,” IEEE Aerospace Conference (2010), pp. 1–10.

Poyneer, L. A.

Pritt, M. D.

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

Roberts, P. H.

T. J. Brennan and P. H. Roberts, AOTools: the Adaptive Optics Toolbox for Use with MATLAB, Optical Sciences Company User’s Guide 1.4a (Optical Sciences Company, 2010).

T. J. Brennan, P. H. Roberts, and D. C. Mann, WaveProp: a Wave Optics Simulation System for Use with MATLAB, Optical Sciences Company User’s Guide 1.3 (Optical Sciences Company, 2010).

Roddier, F.

F. Roddier, Adaptive Optics in Astronomy (Cambridge University, 1999).

Sanchez, D. J.

D. J. Sanchez, D. W. Oesch, and P. R. Kelly, “The aggregate behavior of branch points: theoretical calculation of branch point velocity,” Proc. SPIE 8380, 83800P (2012).

D. W. Oesch, D. J. Sanchez, and C. M. Tewksbury-Christle, “Aggregate behavior of branch points–persistent pairs,” Opt. Express 20, 1046–1059 (2012).
[CrossRef]

D. J. Sanchez and D. W. Oesch, “Orbital angular momentum in optical waves propagating through distributed turbulence,” Opt. Express 19, 24596–24608 (2011).
[CrossRef]

D. J. Sanchez and D. W. Oesch, “Localization of angular momentum in optical waves propagating through turbulence,” Opt. Express 19, 25388–25396 (2011).
[CrossRef]

D. W. Oesch, D. J. Sanchez, and C. L. Matson, “The aggregate behavior of branch points-measuring the number and velocity of atmospheric turbulence layers,” Opt. Express 18, 22377–22392 (2010).
[CrossRef]

D. W. Oesch, D. J. Sanchez, C. M. Tewksbury-Christle, and P. R. Kelly, “The aggregate behavior of branch points: altitude and strength of atmospheric turbulence layers,” Proc. SPIE 7816, 781605 (2010).
[CrossRef]

D. J. Sanchez and D. W. Oesch, “The aggregate behavior of branch points: the creation and evolution of branch points,” Proc. SPIE 7466, 746605 (2009).
[CrossRef]

Schmidt, J. D.

T. M. Venema and J. D. Schmidt, “Optical phase unwrapping in the presence of branch points,” Opt. Express 16, 6985–6998 (2008).
[CrossRef]

C. Pellizzari and J. D. Schmidt, “Phase unwrapping in the presence of strong turbulence,” IEEE Aerospace Conference (2010), pp. 1–10.

Tewksbury-Christle, C. M.

D. W. Oesch, D. J. Sanchez, and C. M. Tewksbury-Christle, “Aggregate behavior of branch points–persistent pairs,” Opt. Express 20, 1046–1059 (2012).
[CrossRef]

D. W. Oesch, D. J. Sanchez, C. M. Tewksbury-Christle, and P. R. Kelly, “The aggregate behavior of branch points: altitude and strength of atmospheric turbulence layers,” Proc. SPIE 7816, 781605 (2010).
[CrossRef]

Tyler, G.

Tyler, G. A.

J. D. Barchers, D. L. Fried, D. J. Link, G. A. Tyler, W. Moretti, T. J. Brennan, and R. Q. Fugate, “The performance of wavefront sensors in strong scintillation,” Proc. SPIE 4839, 217–227 (2003).
[CrossRef]

Vaughn, J. L.

Venema, T. M.

Werner, C. L.

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

Wild, W. J.

Zebker, H. A.

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

Zetterlind, V. E.

V. E. Zetterlind and E. P. Magee, “Performance of various branch point tolerant phase reconstructors with finite time delays and measurement noise,” Proc. SPIE 4632, 85–94 (2002).
[CrossRef]

Appl. Opt. (2)

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (5)

Opt. Commun. (1)

D. L. Fried, “Adaptive optics wave function reconstruction and phase unwrapping when branch points are present,” Opt. Commun. 200, 43–72 (2001).
[CrossRef]

Opt. Express (5)

Proc. R. Soc. A (1)

J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. A 336, 165–190 (1974).
[CrossRef]

Proc. SPIE (6)

K. Murphy, R. Mackey, and C. Dainty, “Branch point detection and correction using the branch point potential method,” Proc. SPIE 6951, 695105 (2008).
[CrossRef]

V. E. Zetterlind and E. P. Magee, “Performance of various branch point tolerant phase reconstructors with finite time delays and measurement noise,” Proc. SPIE 4632, 85–94 (2002).
[CrossRef]

D. J. Sanchez and D. W. Oesch, “The aggregate behavior of branch points: the creation and evolution of branch points,” Proc. SPIE 7466, 746605 (2009).
[CrossRef]

D. W. Oesch, D. J. Sanchez, C. M. Tewksbury-Christle, and P. R. Kelly, “The aggregate behavior of branch points: altitude and strength of atmospheric turbulence layers,” Proc. SPIE 7816, 781605 (2010).
[CrossRef]

D. J. Sanchez, D. W. Oesch, and P. R. Kelly, “The aggregate behavior of branch points: theoretical calculation of branch point velocity,” Proc. SPIE 8380, 83800P (2012).

J. D. Barchers, D. L. Fried, D. J. Link, G. A. Tyler, W. Moretti, T. J. Brennan, and R. Q. Fugate, “The performance of wavefront sensors in strong scintillation,” Proc. SPIE 4839, 217–227 (2003).
[CrossRef]

Radio Sci. (1)

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

Other (7)

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

C. Pellizzari, “Phase unwrapping in the presence of strong turbulence,” in IEEE Aerospace Conference (IEEE, 2010).

C. Pellizzari and J. D. Schmidt, “Phase unwrapping in the presence of strong turbulence,” IEEE Aerospace Conference (2010), pp. 1–10.

F. Roddier, Adaptive Optics in Astronomy (Cambridge University, 1999).

R. C. Olsen, Remote Sensing from Air and Space (SPIE, 2007).

T. J. Brennan and P. H. Roberts, AOTools: the Adaptive Optics Toolbox for Use with MATLAB, Optical Sciences Company User’s Guide 1.4a (Optical Sciences Company, 2010).

T. J. Brennan, P. H. Roberts, and D. C. Mann, WaveProp: a Wave Optics Simulation System for Use with MATLAB, Optical Sciences Company User’s Guide 1.3 (Optical Sciences Company, 2010).

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Figures (18)

Fig. 1.
Fig. 1.

Block diagram form (a) the traditional Fried gradient and (b) the wrapped Fried gradient.

Fig. 2.
Fig. 2.

Function u¯[n] is used to create c¯[n] from c[n].

Fig. 3.
Fig. 3.

Block diagram comparison of the traditional least-squares approach to the new form that is capable of handling branch points. When the curl is equal to zero, the output is exactly the same for both forms.

Fig. 4.
Fig. 4.

Original ϕ[n] phase data for example 1.

Fig. 5.
Fig. 5.

Reconstructed Wϕ^[n] phase data for example 1.

Fig. 6.
Fig. 6.

Wrapped gradient ψ1[n] data for example 1. The large discontinuity seen in Fig. 4 is not apparent in the wrapped measured gradient.

Fig. 7.
Fig. 7.

Wrapped gradient ψ2[n] data for example 1. The large discontinuity seen in Fig. 4 is not apparent in the wrapped measured gradient. The correction term proposed in this paper will be added to ψ2[n] to create the large discontinuity.

Fig. 8.
Fig. 8.

Original ϕ[n] plotted for example 2.

Fig. 9.
Fig. 9.

Estimated phase Wϕ^[n] plotted for example 2.

Fig. 10.
Fig. 10.

Estimated phase Wϕ^[n] with 40 dB SNR for example 2. Pixels with values close to π or π may wrap due to the noise.

Fig. 11.
Fig. 11.

High turbulence wavefront phase ϕ[n] created using WaveProp for example 3.

Fig. 12.
Fig. 12.

Estimated wavefront Wϕ^[n] reconstructed for example 3.

Fig. 13.
Fig. 13.

Estimated wavefront Wϕ^[n] reconstructed with 40 dB SNR.

Fig. 14.
Fig. 14.

Detected branch points for example 3 with no noise. Positive branch points are indicated with a red plus while negative branch points are indicated with a blue dot. Because the Fried geometry averages neighboring values, the locations on this plot are quadrupled compared to Fig. 15.

Fig. 15.
Fig. 15.

Known branch points for example 3 with no noise. Locations were determined by WaveProp. Positive branch points are indicated with a red plus while negative branch points are indicated with a blue dot.

Fig. 16.
Fig. 16.

Spiral dataset ϕ[n] from [7]. This dataset is known to be difficult to process correctly.

Fig. 17.
Fig. 17.

Estimated phase Wϕ^[n] reconstructed for the spiral dataset.

Fig. 18.
Fig. 18.

Estimated phase Wϕ^[n] reconstructed for spiral dataset with 40 dB SNR.

Equations (84)

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

F(κx,κy)=FT{f(x,y)}=++f(x,y)ej2π(κxx+κyy)dxdy
f[n1,n2]=f(n1Δx,n2Δy),
F(z1,z2)=Z{f[n1,n2]}=n1=+n2=+f[n1,n2]z1n1z2n2.
Z{f[n1+L1,n2+L2]}=z1L1z2L2Z{f[n1,n2]}
f[n1+L1,n2+L2]=z1L1z2L2f[n1,n2].
F(ω)=n1=+n2=+f[n1,n2]ej(ω1n1+ω2n2).
ω1=2πκxΔx,ω2=2πκyΔy,
πωi<π
[n1,n2]Z,n1,n2,
ψ=ϕ+×v,
ψ(x,y)=[ψx(x,y)ψy(x,y)],
[ψx(x,y)ψy(x,y)]=[xyyx][ϕ(x,y)v(x,y)].
[Ψx(κ)Ψy(κ)]=[j2πκxj2πκyj2πκyj2πκx][Φ(κ)V(κ)],
((j2πκx)2+(j2πκy)2)[Φ(κ)V(κ)]=[j2πκxj2πκyj2πκyj2πκx][Ψx(κ)Ψy(κ)].
ψ(x,y)=ϕ(x,y).
vec(ψ[:,:])=Γvec(ϕ[:,:]),
vec(ϕ^[:,:])=(ΓTΓ)1ΓTvec(ψ[:,:]).
[ψx(x,y)ψy(x,y)]=[x0yy][ϕ0(x,y)c¯(x,y)],
[ψx(x,y)ψy(x,y)]=[xxy0][ϕ1(x,y)c¯(x,y)].
e1=[j2πκxj2πκy],e2=[0j2πκy]ore2=[j2πκx0],
c(x,y)=yψx(x,y)xψy(x,y),
c(x,y)xyc¯(x,y),v(x,y)xyv¯(x,y)
c¯(x,y)0y0xc(λ1,λ2)dλ1dλ2+w(x,y).
ϕ0(x,y)=ϕ(x,y)+y2v¯(x,y)+wy(y),ϕ1(x,y)=ϕ(x,y)x2v¯(x,y)+wx(x)
(x2+y2)v(x,y)=c(x,y).
(x2+y2)v¯(x,y)=c¯(x,y)+w(x,y),
xyw(x,y)=0.
w(x,y)=wx(x)wy(y).
[ψxψy]=[xy]ϕ+[xy2v¯yx2(v¯)].
[ψxψy]=[xy](ϕ+y2v¯+wy)[0y]c¯.
ϕ1(x,y)=ϕ0(x,y)c¯(x,y).
[ψx(x,y)ψy(x,y)]+[0yc¯(x,y)]=[xy]ϕ0(x,y)
c¯(x,y)=c(x,y)**u(x)u(y),xc¯(x,y)=c(x,y)*u(y),yc¯(x,y)=c(x,y)*u(x),
θ(x,y)=phase(x+jy)
ψx(x,y)=yx2+y2Ψx(κx,κy)=jκyκx2+κy2,ψy(x,y)=xx2+y2Ψy(κx,κy)=jκxκx2+κy2.
[xy]θ(x,y)=[ψx(x,y)ψy(x,y)]+[02πδ(y)u(x)].
Φ(κ)=0,V(κ)=12π(κx2+κy2),C(κ)=2π.
xy2v¯(x,y)=yv(x,y)=IFT{jκyκx2+κy2}.
y2v¯(x,y)=θ(x,y)+wy(y)
ϕ0(x,y)=θ(x,y)+wy(y).
[ψx(x,y)ψy(x,y)]+[02πu(x)δ(y)]=[xy]ϕ0(x,y).
θ(x,y)=ϕ0(x,y)+C,
ψ1[n1,n2]=12(ϕ0[n1+1,n2+1]+ϕ0[n1,n2+1])12(ϕ0[n1+1,n2]+ϕ0[n1,n2]),ψ2[n1,n2]=12(ϕ0[n1+1,n2+1]+ϕ0[n1+1,n2])12(ϕ0[n1,n2+1]+ϕ0[n1,n2]).
ψ1[n1,n2]=12(z1+1)(z21)ϕ0[n1,n2],ψ2[n1,n2]=12(z11)(z2+1)ϕ0[n1,n2]
1(z1,z2)12(z1+1)(z21),2(z1,z2)12(z11)(z2+1).
1(ω)=2ejω1+ω22cos(ω12)sin(ω22),2(ω)=2ejω1+ω22sin(ω12)cos(ω22).
x[n]=0x[n]=C0+C1(1)n1+n2
θ[n]=W(α[n])α[n]+2π[n]
ψ[n1,n2]=[W1W2]ϕ0[n1,n2].
Wϕ0[n]=ϕ0[n]+π[n]
[ψ1[n1,n2]ψ2[n1,n2]]=[1221][ϕ[n1,n2]v[n1,n2]].
[Ψ1(ω)Ψ2(ω)]=ejω1+ω22[c1s2s1c2s1c2c1s2][Φ(ω)V(ω)]
c[n1,n2]2ψ1[n1,n2]1ψ2[n1,n2],
[ψ1[n1,n2]ψ2[n1,n2]]=[1022][ϕ0[n1,n2]c¯[n1,n2]]
c[n1,n2]12c¯[n1,n2].
c[n1,n2]=c[n1,n2]**δ[n1]δ[n2]
1(z)2(z)=14(z11)(z1+1)(z21)(z2+1)=14(z121)(z221).
u¯[n](1+(1)n)u[n2],
δ[n1]δ[n2]=1(z)2(z)u¯[n1]u¯[n2],
c¯[n1,n2]=c[n1,n2]**u¯[n1]u¯[n2].
[ψ1[n]ψ2[n]]=[W1W2]ϕ0[n].
c[n]=π[n],
[ψ1[n]ψ2[n]]+[0c[n]**q[n]]=[12]ϕ^[n],
q[n1,n2]12(z11)(z2+1)u¯[n1,n2].
ϕ0[n]=ϕ^[n]+C0+C1(1)n1+n2+2π[n],
c[n]**q[n]=2π[n],
[12](ϕ0[n]ϕ^[n])=π[1[n]2[n]],
[n]=[n]**δ[n1]δ[n2]
δ[n1]δ[n2]=1(2(1)n11u[n11]u[n21])=2(2(1)n21u[n11]u[n21]),
[n]=21[n][n]=22[n].
(ϕ0[n]ϕ^[n]+2π[n])=0,
ϕ0[n]=W(ϕ^[n]+C0)
c[n1,n2]=2ψ1[n1,n2]1ψ2[n1,n2].
2c¯[n1,n2]=c[n1,n2]**q[n1,n2].
ψ2,new[n1,n2]=ψ2[n1,n2]+2c¯[n1,n2].
[ψ1[n1,n2]ψ2,new[n1,n2]]=[12]ϕ^[n1,n2].
[ψx,noisy[n1,n2]ψy,noisy[n1,n2]]=[ψx[n1,n2]ψy[n1,n2]]+[αnx[n1,n2]αny[n1,n2]],
ϕ(x,y)=phase(s)
ϕ(x,y)=phase((sb1)(sb2)(sa))
xyw(x,y)=0
w(x,y)=a(x)+b(y).
g(x,y)=xw(x,y).
g(x,y)=g(x,0),
w(x,y)=w(0,y)+0xg(λ,0)dλ,

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