Abstract

We develop an algorithm for the minimum Lp-norm solution to the two-dimensional phase unwrapping problem. Rather than its being a mathematically intractable problem, we show that the governing equations are equivalent to those that describe weighted least-squares phase unwrapping. The only exception is that the weights are data dependent. In addition, we show that the minimum Lp-norm solution is obtained by embedding the transform-based methods for unweighted and weighted least squares within a simple iterative structure. The data-dependent weights are generated within the algorithm and need not be supplied explicitly by the user. Interesting and useful solutions to many phase unwrapping problems can be obtained when p< 2. Specifically, the minimum L0-norm solution requires the solution phase gradients to equal the input data phase gradients in as many places as possible. This concept provides an interesting link to branch-cut unwrapping methods, where none existed previously.

© 1996 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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  40. The wrapped phase data sets shown in Fig. 11 and below in Fig. 13 are courtesy of Gang Zhu. They come from the ESTEEM clinical MRI scanner manufactured by Elscint MR Inc., Fort Collins. Colo.
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  42. M. D. Pritt, J. S. Shipman, “Least-squares two-dimensional phase unwrapping using FFTs,” IEEE Trans. Geosci. Remote Sens. 32, 706–708 (1994).
    [CrossRef]

1996

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

1995

1994

J. Szumowski, W. R. Coshow, F. Li, S. F. Quinn, “Phase unwrapping in the three-point Dixon method for fat suppression MR imaging,” Radiology, 192, 555–561 (1994).
[PubMed]

D. C. Ghiglia, L. A. Romero, “Robust two-dimensional weighted and unweighted phase unwrapping that uses fast transforms and iterative methods,” J. Opt. Soc. Am. A 11, 107–117 (1994).
[CrossRef]

M. D. Pritt, J. S. Shipman, “Least-squares two-dimensional phase unwrapping using FFTs,” IEEE Trans. Geosci. Remote Sens. 32, 706–708 (1994).
[CrossRef]

1993

A. L. Gray, P. J. Farris-Manning, “Repeat-pass interferometry with airborne synthetic aperture radar,” IEEE Trans. Geosci. Remote Sens. 31, 180–191 (1993).
[CrossRef]

D. Massonnet, M. Rossi, C. Carmona, F. Adragna, G. Peltzer, K. Feigl, T. Rabaute, “The displacement field of the Landers earthquake mapped by radar interferometry,” Nature (London) 364, 138–142 (1993).
[CrossRef]

1992

N. H. Ching, D. Rosenfeld, M. Braun, “Two-dimensional phase unwrapping using a minimum spanning tree algorithm,” IEEE Trans. Image Process. 1, 355–365 (1992).
[CrossRef] [PubMed]

1991

D. J. Bone, “Fourier fringe analysis: the two-dimensional phase unwrapping problem,” Appl. Opt. 30, 3627–3632 (1991).
[CrossRef] [PubMed]

G. H. Glover, E. Schneider, “Three-point Dixon technique for true water/fat decomposition with B0inhomogeneity correction,” Magn. Reson. Med. 18, 371–383 (1991).
[CrossRef] [PubMed]

1990

F. K. Li, R. M. Goldstein, “Studies of multibaseline spaceborne interferometric synthetic aperture radars,” IEEE Trans. Geosci. Remote Sens. 28, 88–97 (1990).
[CrossRef]

1989

1988

1987

1986

H. A. Zebker, R. M. Goldstein, “Topographic mapping from interferometric synthetic aperture radar observations,” J. Geophys. Res. 91, 4993–4999 (1986).
[CrossRef]

1982

1979

1978

1977

1974

L. C. Graham, “Synthetic interferometer radar for topographic mapping,” Proc. IEEE 62, 763–768 (1974).
[CrossRef]

1970

B. L. Busbee, G. H. Gollub, C. W. Nielson, “On direct methods for solving Poisson’s equations,” SIAM J. Numer. Anal. 7, 627–656 (1970).
[CrossRef]

Adam, N.

D. Just, N. Adam, M. Schwäbisch, R. Bamler, “Comparison of phase unwrapping algorithms for SAR interferograms,” presented at the International Geoscience and Remote Sensing Symposium (IGARSS ’95), Firenze, Italy, July 10–14, 1995.

Adragna, F.

D. Massonnet, M. Rossi, C. Carmona, F. Adragna, G. Peltzer, K. Feigl, T. Rabaute, “The displacement field of the Landers earthquake mapped by radar interferometry,” Nature (London) 364, 138–142 (1993).
[CrossRef]

Arfken, G. B.

G. B. Arfken, H. J. Weber, Mathematical Methods for Physicists, 4th ed. (Academic, San Diego, 1995), Chap. 17.

Bamler, R.

D. Just, N. Adam, M. Schwäbisch, R. Bamler, “Comparison of phase unwrapping algorithms for SAR interferograms,” presented at the International Geoscience and Remote Sensing Symposium (IGARSS ’95), Firenze, Italy, July 10–14, 1995.

Bernabeu, E.

Bone, D. J.

Braun, M.

N. H. Ching, D. Rosenfeld, M. Braun, “Two-dimensional phase unwrapping using a minimum spanning tree algorithm,” IEEE Trans. Image Process. 1, 355–365 (1992).
[CrossRef] [PubMed]

Buckland, J. R.

Busbee, B. L.

B. L. Busbee, G. H. Gollub, C. W. Nielson, “On direct methods for solving Poisson’s equations,” SIAM J. Numer. Anal. 7, 627–656 (1970).
[CrossRef]

Carmona, C.

D. Massonnet, M. Rossi, C. Carmona, F. Adragna, G. Peltzer, K. Feigl, T. Rabaute, “The displacement field of the Landers earthquake mapped by radar interferometry,” Nature (London) 364, 138–142 (1993).
[CrossRef]

Ching, N. H.

N. H. Ching, D. Rosenfeld, M. Braun, “Two-dimensional phase unwrapping using a minimum spanning tree algorithm,” IEEE Trans. Image Process. 1, 355–365 (1992).
[CrossRef] [PubMed]

Coshow, W. R.

J. Szumowski, W. R. Coshow, F. Li, S. F. Quinn, “Phase unwrapping in the three-point Dixon method for fat suppression MR imaging,” Radiology, 192, 555–561 (1994).
[PubMed]

Cusack, R.

Eichel, P. H.

P. H. Eichel, D. C. Ghiglia, C. V. Jakowatz, P. A. Thompson, D. E. Wahl, “Spotlight SAR interferometry for terrain elevation mapping and interferometric change detection,” Rep. SAND93-2072J (Sandia National Laboratories, Albuquerque, N.M., 1993).

Ewing, G. M.

G. M. Ewing, Calculus of Variations with Applications (Dover, New York, 1985).

Farris-Manning, P. J.

A. L. Gray, P. J. Farris-Manning, “Repeat-pass interferometry with airborne synthetic aperture radar,” IEEE Trans. Geosci. Remote Sens. 31, 180–191 (1993).
[CrossRef]

Feigl, K.

D. Massonnet, M. Rossi, C. Carmona, F. Adragna, G. Peltzer, K. Feigl, T. Rabaute, “The displacement field of the Landers earthquake mapped by radar interferometry,” Nature (London) 364, 138–142 (1993).
[CrossRef]

Flannery, B. P.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes: The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, Cambridge, 1992), Chap. 18.

Fried, D. L.

Gåsvik, K. J.

K. J. Gåsvik, Optical Metrology, 2nd ed. (Wiley, Chichester, UK, 1995).

Ghiglia, D. C.

Glover, G. H.

S. M. 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]

G. H. Glover, E. Schneider, “Three-point Dixon technique for true water/fat decomposition with B0inhomogeneity correction,” Magn. Reson. Med. 18, 371–383 (1991).
[CrossRef] [PubMed]

Goldrein, H. T.

Goldstein, R. M.

F. K. Li, R. M. Goldstein, “Studies of multibaseline spaceborne interferometric synthetic aperture radars,” IEEE Trans. Geosci. Remote Sens. 28, 88–97 (1990).
[CrossRef]

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

H. A. Zebker, R. M. Goldstein, “Topographic mapping from interferometric synthetic aperture radar observations,” J. Geophys. Res. 91, 4993–4999 (1986).
[CrossRef]

Gollub, G. H.

B. L. Busbee, G. H. Gollub, C. W. Nielson, “On direct methods for solving Poisson’s equations,” SIAM J. Numer. Anal. 7, 627–656 (1970).
[CrossRef]

González-Cano, A.

Graham, L. C.

L. C. Graham, “Synthetic interferometer radar for topographic mapping,” Proc. IEEE 62, 763–768 (1974).
[CrossRef]

Gray, A. L.

A. L. Gray, P. J. Farris-Manning, “Repeat-pass interferometry with airborne synthetic aperture radar,” IEEE Trans. Geosci. Remote Sens. 31, 180–191 (1993).
[CrossRef]

Hudgin, R. H.

Hunt, B. R.

Huntley, J. M.

Itoh, K.

Jakowatz, C. V.

P. H. Eichel, D. C. Ghiglia, C. V. Jakowatz, P. A. Thompson, D. E. Wahl, “Spotlight SAR interferometry for terrain elevation mapping and interferometric change detection,” Rep. SAND93-2072J (Sandia National Laboratories, Albuquerque, N.M., 1993).

Just, D.

D. Just, Deutsche Forschungsanstalt für Luft-und Raumfahrt (DLR), Wessling, Germany, July1995 (personal communication).

D. Just, N. Adam, M. Schwäbisch, R. Bamler, “Comparison of phase unwrapping algorithms for SAR interferograms,” presented at the International Geoscience and Remote Sensing Symposium (IGARSS ’95), Firenze, Italy, July 10–14, 1995.

Li, F.

J. Szumowski, W. R. Coshow, F. Li, S. F. Quinn, “Phase unwrapping in the three-point Dixon method for fat suppression MR imaging,” Radiology, 192, 555–561 (1994).
[PubMed]

Li, F. K.

F. K. Li, R. M. Goldstein, “Studies of multibaseline spaceborne interferometric synthetic aperture radars,” IEEE Trans. Geosci. Remote Sens. 28, 88–97 (1990).
[CrossRef]

Lim, J. S.

J. S. Lim, “The discrete cosine transform,” in Two-Dimensional Signal and Image Processing (Prentice-Hall, Englewood Cliffs, N.J., 1990), pp. 148–157.

Marroquin, J. L.

Massonnet, D.

D. Massonnet, M. Rossi, C. Carmona, F. Adragna, G. Peltzer, K. Feigl, T. Rabaute, “The displacement field of the Landers earthquake mapped by radar interferometry,” Nature (London) 364, 138–142 (1993).
[CrossRef]

Mastin, G. A.

Napel, S.

S. M. 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]

Nielson, C. W.

B. L. Busbee, G. H. Gollub, C. W. Nielson, “On direct methods for solving Poisson’s equations,” SIAM J. Numer. Anal. 7, 627–656 (1970).
[CrossRef]

Noll, R. J.

Pelc, N. J.

S. M. 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]

Peltzer, G.

D. Massonnet, M. Rossi, C. Carmona, F. Adragna, G. Peltzer, K. Feigl, T. Rabaute, “The displacement field of the Landers earthquake mapped by radar interferometry,” Nature (London) 364, 138–142 (1993).
[CrossRef]

Press, W. H.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes: The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, Cambridge, 1992), Chap. 18.

Pritt, M. D.

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

M. D. Pritt, J. S. Shipman, “Least-squares two-dimensional phase unwrapping using FFTs,” IEEE Trans. Geosci. Remote Sens. 32, 706–708 (1994).
[CrossRef]

M. D. Pritt, “Multigrid phase unwrapping for interferometric SAR,” in Proceedings of the IEEE International Geoscience Remote Sensing Symposium (IEEE, Au:city, 1995), Vol. 1, pp. 562–564.

M. D. Pritt, Loral Federal Systems, Gaithersburg, Md., September1995 (personal communication).

Quinn, S. F.

J. Szumowski, W. R. Coshow, F. Li, S. F. Quinn, “Phase unwrapping in the three-point Dixon method for fat suppression MR imaging,” Radiology, 192, 555–561 (1994).
[PubMed]

Quiroga, J. A.

Rabaute, T.

D. Massonnet, M. Rossi, C. Carmona, F. Adragna, G. Peltzer, K. Feigl, T. Rabaute, “The displacement field of the Landers earthquake mapped by radar interferometry,” Nature (London) 364, 138–142 (1993).
[CrossRef]

Reid, G. T.

D. W. Robinson, G. T. Reid, Interferogram Analysis: Digital Fringe Pattern Measurement Techniques (Institute of Physics, Bristol, UK, 1993).

Rivera, M.

Robinson, D. W.

D. W. Robinson, G. T. Reid, Interferogram Analysis: Digital Fringe Pattern Measurement Techniques (Institute of Physics, Bristol, UK, 1993).

Romero, L. A.

Rosenfeld, D.

N. H. Ching, D. Rosenfeld, M. Braun, “Two-dimensional phase unwrapping using a minimum spanning tree algorithm,” IEEE Trans. Image Process. 1, 355–365 (1992).
[CrossRef] [PubMed]

Rossi, M.

D. Massonnet, M. Rossi, C. Carmona, F. Adragna, G. Peltzer, K. Feigl, T. Rabaute, “The displacement field of the Landers earthquake mapped by radar interferometry,” Nature (London) 364, 138–142 (1993).
[CrossRef]

Schneider, E.

G. H. Glover, E. Schneider, “Three-point Dixon technique for true water/fat decomposition with B0inhomogeneity correction,” Magn. Reson. Med. 18, 371–383 (1991).
[CrossRef] [PubMed]

Schwäbisch, M.

D. Just, N. Adam, M. Schwäbisch, R. Bamler, “Comparison of phase unwrapping algorithms for SAR interferograms,” presented at the International Geoscience and Remote Sensing Symposium (IGARSS ’95), Firenze, Italy, July 10–14, 1995.

Shipman, J. S.

M. D. Pritt, J. S. Shipman, “Least-squares two-dimensional phase unwrapping using FFTs,” IEEE Trans. Geosci. Remote Sens. 32, 706–708 (1994).
[CrossRef]

Song, S. M.

S. M. 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]

Szumowski, J.

J. Szumowski, W. R. Coshow, F. Li, S. F. Quinn, “Phase unwrapping in the three-point Dixon method for fat suppression MR imaging,” Radiology, 192, 555–561 (1994).
[PubMed]

Takahashi, T.

Takajo, H.

Teukolsky, S. A.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes: The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, Cambridge, 1992), Chap. 18.

Thompson, P. A.

P. H. Eichel, D. C. Ghiglia, C. V. Jakowatz, P. A. Thompson, D. E. Wahl, “Spotlight SAR interferometry for terrain elevation mapping and interferometric change detection,” Rep. SAND93-2072J (Sandia National Laboratories, Albuquerque, N.M., 1993).

Turner, S. R. E.

Vetterling, W. T.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes: The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, Cambridge, 1992), Chap. 18.

Wahl, D. E.

P. H. Eichel, D. C. Ghiglia, C. V. Jakowatz, P. A. Thompson, D. E. Wahl, “Spotlight SAR interferometry for terrain elevation mapping and interferometric change detection,” Rep. SAND93-2072J (Sandia National Laboratories, Albuquerque, N.M., 1993).

Weber, H. J.

G. B. Arfken, H. J. Weber, Mathematical Methods for Physicists, 4th ed. (Academic, San Diego, 1995), Chap. 17.

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]

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]

H. A. Zebker, R. M. Goldstein, “Topographic mapping from interferometric synthetic aperture radar observations,” J. Geophys. Res. 91, 4993–4999 (1986).
[CrossRef]

Appl. Opt.

IEEE Trans. Geosci. Remote Sens.

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

F. K. Li, R. M. Goldstein, “Studies of multibaseline spaceborne interferometric synthetic aperture radars,” IEEE Trans. Geosci. Remote Sens. 28, 88–97 (1990).
[CrossRef]

A. L. Gray, P. J. Farris-Manning, “Repeat-pass interferometry with airborne synthetic aperture radar,” IEEE Trans. Geosci. Remote Sens. 31, 180–191 (1993).
[CrossRef]

M. D. Pritt, J. S. Shipman, “Least-squares two-dimensional phase unwrapping using FFTs,” IEEE Trans. Geosci. Remote Sens. 32, 706–708 (1994).
[CrossRef]

IEEE Trans. Image Process.

N. H. Ching, D. Rosenfeld, M. Braun, “Two-dimensional phase unwrapping using a minimum spanning tree algorithm,” IEEE Trans. Image Process. 1, 355–365 (1992).
[CrossRef] [PubMed]

S. M. 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]

J. Geophys. Res.

H. A. Zebker, R. M. Goldstein, “Topographic mapping from interferometric synthetic aperture radar observations,” J. Geophys. Res. 91, 4993–4999 (1986).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Magn. Reson. Med.

G. H. Glover, E. Schneider, “Three-point Dixon technique for true water/fat decomposition with B0inhomogeneity correction,” Magn. Reson. Med. 18, 371–383 (1991).
[CrossRef] [PubMed]

Nature (London)

D. Massonnet, M. Rossi, C. Carmona, F. Adragna, G. Peltzer, K. Feigl, T. Rabaute, “The displacement field of the Landers earthquake mapped by radar interferometry,” Nature (London) 364, 138–142 (1993).
[CrossRef]

Opt. Lett.

Proc. IEEE

L. C. Graham, “Synthetic interferometer radar for topographic mapping,” Proc. IEEE 62, 763–768 (1974).
[CrossRef]

Radio Sci.

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

Radiology

J. Szumowski, W. R. Coshow, F. Li, S. F. Quinn, “Phase unwrapping in the three-point Dixon method for fat suppression MR imaging,” Radiology, 192, 555–561 (1994).
[PubMed]

SIAM J. Numer. Anal.

B. L. Busbee, G. H. Gollub, C. W. Nielson, “On direct methods for solving Poisson’s equations,” SIAM J. Numer. Anal. 7, 627–656 (1970).
[CrossRef]

Other

P. H. Eichel, D. C. Ghiglia, C. V. Jakowatz, P. A. Thompson, D. E. Wahl, “Spotlight SAR interferometry for terrain elevation mapping and interferometric change detection,” Rep. SAND93-2072J (Sandia National Laboratories, Albuquerque, N.M., 1993).

K. J. Gåsvik, Optical Metrology, 2nd ed. (Wiley, Chichester, UK, 1995).

D. W. Robinson, G. T. Reid, Interferogram Analysis: Digital Fringe Pattern Measurement Techniques (Institute of Physics, Bristol, UK, 1993).

M. D. Pritt, “Multigrid phase unwrapping for interferometric SAR,” in Proceedings of the IEEE International Geoscience Remote Sensing Symposium (IEEE, Au:city, 1995), Vol. 1, pp. 562–564.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes: The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, Cambridge, 1992), Chap. 18.

D. Just, Deutsche Forschungsanstalt für Luft-und Raumfahrt (DLR), Wessling, Germany, July1995 (personal communication).

D. Just, N. Adam, M. Schwäbisch, R. Bamler, “Comparison of phase unwrapping algorithms for SAR interferograms,” presented at the International Geoscience and Remote Sensing Symposium (IGARSS ’95), Firenze, Italy, July 10–14, 1995.

M. D. Pritt, Loral Federal Systems, Gaithersburg, Md., September1995 (personal communication).

G. B. Arfken, H. J. Weber, Mathematical Methods for Physicists, 4th ed. (Academic, San Diego, 1995), Chap. 17.

G. M. Ewing, Calculus of Variations with Applications (Dover, New York, 1985).

The wrapped phase data sets shown in Fig. 11 and below in Fig. 13 are courtesy of Gang Zhu. They come from the ESTEEM clinical MRI scanner manufactured by Elscint MR Inc., Fort Collins. Colo.

J. S. Lim, “The discrete cosine transform,” in Two-Dimensional Signal and Image Processing (Prentice-Hall, Englewood Cliffs, N.J., 1990), pp. 148–157.

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Figures (17)

Fig. 1
Fig. 1

(a) Wrapped values of a 128 × 128 pixel 2-D phase surface with shear (scaled for display). (b) The five negative residues are located at the black dots. No other residues are present.

Fig. 2
Fig. 2

Progress of Algorithm LP-NORM on the phase-shear problem. In this and all other examples in this paper we set p = 0. (a) Data-dependent x-gradient weights, U0(i, j), from the initial guess, ϕ(i, j) = 0; (b) data-dependent y-gradient weights, V0(i, j), from the initial guess, ϕ(i, j) = 0; rewrapped solution after iteration l = 1; (d) U1(i, j); (e) V1(i, j); (f) rewrapped solution after iteration l = 2.

Fig. 3
Fig. 3

Continued progress of Algorithm LP-NORM on the phase-shear problem: (a) U2(i, j), (b) V2(i, j), (c) rewrapped solution after iteration l = 3, (d) U3(i, j), (e) V3(i, j), (f) rewrapped solution after iteration l = 4.

Fig. 4
Fig. 4

Convergence on the phase-shear problem. (a) U4(i, j). (b) V4(i, j). (c) Rewrapped solution after iteration l = 5. This is the result obtained (at convergence) after correcting the solution with the unwrapped residual. It agrees perfectly with the original input, as it should. (d) The x gradient weights, U5(i, j), are all unity at convergence because there is no disagreement between the x gradients of the solution and the x gradients of the input data. (e) The y-gradient weights, V5(i, j), differ from unity at only 101 locations. Based on the locations of the residues, this is the minimum number of locations at which the gradients of the solution can differ from the gradients of the original data. For this example, a minimum L0-norm solution was obtained. (f) Unwrapped minimum L0-norm solution after iteration l = 5.

Fig. 5
Fig. 5

Rewrapped unweighted least-squares solution to the phase-shear problem.

Fig. 6
Fig. 6

Rotated and translated version of the wrapped phase-shear data.

Fig. 7
Fig. 7

Progress of Algorithm LP-NORM on the rotated and translated phase-shear problem: (a) data-dependent x-gradient weights, U0(i, j), from the initial guess, ϕ(i, j) = 0; (b) data-dependent y-gradient weights, V0(i, j), from the initial guess, ϕ(i, j) = 0; (c) rewrapped solution after iteration l = 1; (d) U1(i, j); (e) V1(i, j); (f) rewrapped solution after iteration l = 2.

Fig. 8
Fig. 8

Further progress and final convergence on the rotated phase-shear problem: (a) U4(i, j). (b) V4(i, j). (c) Rewrapped solution after iteration l = 5. (d) x-gradient weights, U17(i, j), after convergence to a local minimum. (e) y-gradient weights, V17(i, j), after convergence to a local minimum. (f) Unwrapped result after iteration l = 17. Remember that the rewrapped solution at this point agrees exactly with the original wrapped input.

Fig. 9
Fig. 9

Another phase surface with shear. The perceived horizontal shear line is within 13 pixels of the top border. This data set is simply a vertically shifted version of the data shown in Fig. 1(a). The corresponding residues [as in Fig. 1(b)] shift accordingly.

Fig. 10
Fig. 10

Converged solution for the vertically shifted phase-shear problem: (a) U9(i, j). (b) V9(i, j). (c) Minimum L0-norm solution after l = 9 iterations. This is one of two possible mathematically optimal solutions. Neither mathematically optimal solution maintains the phase shear solely along the horizontal shear line. There are only 64 locations in which the solution gradients disagree with the input data gradients. In order to maintain the intuitively correct horizontal shear, more than 64 low-valued weights would be required. This example shows that the mathematically optimal solution may not be what we desire in practice.

Fig. 11
Fig. 11

Magnetic resonance example: (a) Wrapped phase of MR knee image for use in the water/fat separation problem. These data must be successfully unwrapped to allow correct water/fat separation. (b) Residue map containing 5620 positive and 5620 negative residues. Most of the residues come from the noise region outside the boundary of the knee.

Fig. 12
Fig. 12

Unwrapped result of MR knee data: (a) Unwrapped phase after convergence at l = 11 iterations. This unwrapped phase produced the correct water/fat separation. (b) x-gradient weights at convergence, U11(i, j). (c) y-gradient weights at convergence, V11(i, j).

Fig. 13
Fig. 13

Another MR example: (a) Wrapped phase of MR head image. (b) Residue map containing 963 positive and 963 negative residues. (c) Unwrapped phase at convergence (l = 8). (d) The rewrapped phase is identical to the input wrapped phase as expected. However, the unwrapped result did not produce the correct water/fat separation in all regions of the MR image.

Fig. 14
Fig. 14

Sequence of data-dependent weights for the first four iterations: (a) initial x-gradient weights (l = 0), (b) initial y-gradient weights (l = 0), (c) x-gradient weights (l = 1), (d) y-gradient weights (l = 1), (e) x-gradient weights (l = 2), (f) y-gradient weights (l = 2), (g) x-gradient weights (l = 3), (h) y-gradient weights (l = 3).

Fig. 15
Fig. 15

Data-dependent weights at convergence (l = 8): (a) x-gradient weights, (b) y-gradient weights.

Fig. 16
Fig. 16

A SAR interferometry example: (a) Wrapped phase (512 × 512) representing a scaled and wrapped version of actual terrain elevation. This image also represents the rewrapped solution of Algorithm LP-NORM at convergence. (b) Unwrapped result (with planar trend removed to allow easier visualization). The algorithm converged after l = 10 iterations. (c) x-gradient weights at convergence. (d) y-gradient weights at convergence.

Fig. 17
Fig. 17

(Rewrapped) unweighted least-squares solution for the SAR interferometry example. This least-squares solution has underestimated the magnitude of the phase gradients everywhere. The correct (rewrapped) solution for this SAR problem comes from Algorithm LP-NORM and is shown in Fig. 16(a).

Equations (55)

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ψ i , j = ϕ i , j + 2 π k i , j ,             k i , j an integer array , - π < ψ i , j π , i = 0 , , M - 1 , j = 0 , , N - 1.
f i , j = { W { ψ i + 1 , j - ψ i , j } i = 0 , , M - 2 , j = 0 , , N - 1 , 0 otherwise
g i , j = { W { ψ i , j + 1 - ψ i , j } i = 0 , , M - 1 , j = 0 , , N - 2. 0 otherwise
r i , j = g i , j + f i , j + 1 - g i + 1 , j + f i , j , i = 0 , , M - 1 ,             j = 0 , , N - 1.
ɛ 2 = i = 0 M - 2 j = 0 N - 1 ( ϕ i + 1 , j - ϕ i , j - f i , j ) 2 + i = 0 M - 1 j = 0 N - 2 ( ϕ i , j + 1 - ϕ i , j - g i , j ) 2
J = ɛ p = i = 0 M - 2 j = 0 N - 1 ϕ i + 1 , j - ϕ i , j - f i , j p + i = 0 M - 1 j = 0 N - 2 ϕ i , j + 1 - ϕ i , j - g i , j p
J = f ( ϕ x , ϕ y , x , y ) d x d y ,
f = ϕ x - ψ x p + ϕ y - ψ y p
x ( f ϕ x ) + y ( f ϕ y ) = 0.
z ( z p ) = z [ ( z 2 ) p / 2 ] = p z z p - 2
f ϕ x = p ( ϕ x - ψ x ) ϕ x - ψ x p - 2 , f ϕ y = p ( ϕ y - ψ y ) ϕ y - ψ y p - 2 .
{ x [ ( ϕ x - ψ x ) ϕ x - ψ x p - 2 ] + y [ ( ϕ y - ψ y ) ϕ y - ψ y p - 2 ] } = 0.
U ( x , y ) = ϕ x - ψ x p - 2 ,             V ( x , y ) = ϕ y - ψ y p - 2
{ x [ U ( x , y ) ( ϕ x - ψ x ) ] + y [ V ( x , y ) ( ϕ y - ψ y ) ] } = 0.
ϕ x x + ϕ y y = ψ x x + ψ y y ,
2 ϕ = ρ
δ J = p ( δ 1 + δ 2 ) .
δ 1 = i = 0 M - 2 j = 0 N - 1 ( ϕ i + 1 , j - ϕ i , j - f i , j ) ϕ i + 1 , j - ϕ i , j - f i , j p - 2 ( δ ϕ i + 1 , j - δ ϕ i , j ) ,
δ 2 = i = 0 M - 1 j = 0 N - 2 ( ϕ i , j + 1 - ϕ i , j - g i , j ) ϕ i , j + 1 - ϕ i , j - g i , j p - 2 ( δ ϕ i , j + 1 - δ ϕ i , j ) .
a i , j = ( ϕ i + 1 , j - ϕ i , j - f i , j ) ϕ i + 1 , j - ϕ i , j - f i , j p - 2 , 0 i M - 2 ,             0 j N - 1.
a i , j = 0 ,             i = - 1 ,             i = M - 1 ,             0 j N - 1.
b i , j = ( ϕ i , j + 1 - ϕ i , j - g i , j ) ϕ i , j + 1 - ϕ i , j - g i , j p - 2 , 0 i M - 1 ,             0 j N - 2 ,
b i , j = 0 ,             j = - 1 ,             j = N - 1 ,             0 i M - 1.
δ 1 = i = 0 M - 2 j = 0 N - 1 a i , j ( δ ϕ i + 1 , j - δ ϕ i , j ) ,
δ 2 = i = 0 M - 1 j = 0 N - 2 b i , j ( δ ϕ i , j + 1 - δ ϕ i , j ) .
δ 1 = - i = 0 M - 1 j = 0 N - 1 ( a i , j - a i - 1 , j ) δ ϕ i , j ,
δ 2 = - i = 0 M - 1 j = 0 N - 1 ( b i , j - b i , j - 1 ) δ ϕ i , j .
δ J = - p i = 0 M - 1 j = 0 N - 1 ( b i , j - b i , j - 1 + a i , j - a i - 1 , j ) δ ϕ i , j .
b i , j - b i , j - 1 + a i , j - a i - 1 , j = 0 , 0 i M - 1 ,     0 j N - 1.
ϕ - 1 , j - ϕ 0 , j = ψ - 1 , j - ψ 0 , j ,             0 j N - 1 , ϕ M , j - ϕ M - 1 , j = ψ M , j - ψ M - 1 , j ,             0 j N - 1 , ϕ i , - 1 - ϕ i , 0 = ψ i , - 1 - ψ i , 0 ,             0 i M - 1 , ϕ i , N - ϕ i , N - 1 = ψ i , N - ψ i , N - 1 ,             0 i M - 1.
( ϕ i + 1 , j - ϕ i , j - f i , j ) U ( i , j ) + ( ϕ i , j + 1 - ϕ i , j - g i , j ) × V ( i , j ) - ( ϕ i , j - ϕ i - 1 , j - f i - 1 , j ) U ( i - 1 , j ) - ( ϕ i , j - ϕ i , j - 1 - g i , j - 1 ) V ( i , j - 1 ) = 0 ,
U ( i , j ) = { ϕ i + 1 , j - ϕ i , j - f i , j p - 2 i = 0 , , M - 2 j = 0 , , N - 1 , 0 otherwise
V ( i , j ) = { ϕ i , j + 1 - ϕ i , j - g i , j p - 2 i = 0 , , M - 1 j = 0 , , N = 2. 0 otherwise
( ϕ i + 1 , j - 2 ϕ i , j + ϕ i - 1 , j ) + ( ϕ i , j + 1 - 2 ϕ i , j + ϕ i , j - 1 ) = ρ i , j ,
ρ i , j = f i , j - f i - 1 , j + g i , j - g i , j - 1 .
( ϕ i + 1 , j - ϕ i , j ) U ( i , j ) + ( ϕ i , j + 1 - ϕ i , j ) V ( i , j ) - ( ϕ i , j - ϕ i - 1 , j ) U ( i - 1 , j ) - ( ϕ i , j - ϕ i , j - 1 ) V ( i , j - 1 ) = c ( i , j ) ,
c ( i , j ) = f i , j U ( i , j ) - f i - 1 , j U ( i - 1 , j ) + g i , j V ( i , j ) - g i , j - 1 V ( i , j - 1 ) ,
U ( i , j ) = { ϕ i + 1 , j - ϕ i , j - f i , j p - 2 i = 0 , , M - 2 , j = 0 , , N - 1 0 otherwise ,
V ( i , j ) = { ϕ i , j + 1 - ϕ i , j - g i , j p - 2 i = 0 , , M - 1 , j = 0 , , N - 2 0 otherwise . .
Q ϕ = c .
U ( i , j ) = ɛ 0 ϕ i + 1 , j - ϕ i , j - f i , j 2 - p + ɛ 0 ,
V ( i , j ) = ɛ 0 ϕ i , j + 1 - ϕ i , j - g i , j 2 - p + ɛ 0 .
R ( i , j ) = W { ψ ( i , j ) - ϕ l ( i , j ) } .
ϕ ( i , j ) = ϕ l ( i , j ) + W - 1 { R ( i , j ) } ,
s = [ W + F exp ( j Φ ) ] exp [ j ( θ 0 + φ ) ]
s 0 = ( W + F ) exp ( j θ 0 ) ,
s 1 = ( W - F ) exp [ j ( θ 0 + φ ) ] .
s ¯ 0 s 1 s 0 s ¯ 1 = exp ( j 2 φ ) .
I 0 = 0.5 [ s 0 + s 1 exp ( - j φ ^ ) ] ,
I 1 = 0.5 [ s 0 - s 1 exp ( - j φ ^ ) ] .
P ϕ = ρ ,
A T W T WA ϕ = c .
Q ϕ = c ,
( ϕ i + 1 , j - ϕ i , j ) U ˜ ( i , j ) + ( ϕ i , j + 1 - ϕ i , j ) V ˜ ( i , j ) - ( ϕ i , j - ϕ i - 1 , j ) U ˜ ( i - 1 , j ) - ( ϕ i , j - ϕ i , j - 1 ) V ˜ ( i , j - 1 ) = c ( i , j ) ,
c ( i , j ) = f i , j U ˜ ( i , j ) - f i - 1 , j U ˜ ( i - 1 , j ) + g i , j V ˜ ( i , j ) - g i , j - 1 V ˜ ( i , j - 1 )

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