Abstract

A fully self-contained discrete framework with discrete equivalents of Stokes’s, Gauss’s, and Green’s theorems is presented. The formulation is analogous to that of continuous operators, but totally discrete in nature, and the exact relationships derived are shown to hold provided that a set of predefined rules is followed in building discrete contours and domains. The method allows for an analytical rigor that is not guaranteed if one translates the classical continuous formulations onto a discretized approximated framework. We clarify several issues related to the use of discrete operators, which may play a crucial role in specific applications such as the two-dimensional phase-unwrapping problem, chosen as our main application example, and we show that reconstruction on irregular domains and/or in the presence of undersampling and noise is better formulated in the discrete framework than in the continuous domain.

© 2002 Optical Society of America

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References

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  1. A. V. Oppheneim, R. W. Schafer, Discrete Time Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1989).
  2. S. Marano, F. Palmieri, G. Franceschetti, “Integral-differential relationships reformulated for image processing applications,” manuscript in preparation.
  3. J. M. Tribolet, “A new phase unwrapping algorithm,” IEEE Trans. Acoust., Speech, Signal Process. ASP-25, 170–177 (1977).
    [CrossRef]
  4. R. M. Goldstein, H. A. Zebker, C. L. Werner, “Satellite radar interferometry: two-dimensional phase unwrapping,” Radio Sci. 23, 713–720 (1988).
    [CrossRef]
  5. G. Fornaro, G. Franceschetti, R. Lanari, “Interferometric SAR phase unwrapping using Green’s formulation,” IEEE Trans. Geosci. Remote Sens. 34, 720–727 (1996).
    [CrossRef]
  6. M. D. Pritt, J. S. Shipman, “Least-squares two dimensional phase unwrapping using FFT’s,” IEEE Trans. Geosci. Remote Sens. 32, 706–708 (1994).
    [CrossRef]
  7. J. L. Marroquin, M. Rivera, “Quadratic regularization functionals for phase unwrapping,” J. Opt. Soc. Am. A 12, 2393–2400 (1995).
    [CrossRef]
  8. B. R. Hunt, “Matrix formulation of the reconstruction of phase value from phase differences,” J. Opt. Soc. Am. 69, 393–399 (1979).
    [CrossRef]
  9. D. C. Ghiglia, L. A. Romero, “Robust two-dimensional weighted and unweighted phase unwrapping that uses fast transforms and iterative method,” J. Opt. Soc. Am. A 11, 107–117 (1994).
    [CrossRef]
  10. G. Fornaro, G. Franceschetti, R. Lanari, E. Sansosti, “A theoretical analysis of the robust phase unwrapping algorithms for SAR interferometry,” in Proceedings of the IEEE International Symposium on Geoscience and Remote Sensing (Institute of Electrical and Electronics Engineers, New York, 1996), pp. 2047–2049.
  11. S. Moon-Ho. Song, S. Napel, N. J. Pelc, G. H. Glover, “Phase unwrapping of MR images using Poisson equation,” IEEE Trans. Image Process. 4, 667–676 (1995).
    [CrossRef]
  12. G. Fornaro, G. Franceschetti, R. Lanari, E. Sansosti, “Robust phase-unwrapping techniques: a comparison,” J. Opt. Soc. Am. A 13, 2355–2366 (1996).
    [CrossRef]
  13. 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]
  14. G. Fornaro, G. Franceschetti, R. Lanari, D. Rossi, M. Tesauro, “Interferometric SAR phase unwrapping using the finite element method,” IEE Proc. Radar Sonar Navig. 144 (No. 4), 1–9 (1997).
    [CrossRef]
  15. J. Strand, T. Taxt, A. K. Jain, “Two-dimensional phase unwrapping using a block least-square method,” IEEE Trans. Image Process. 8, 375–386 (1999).
    [CrossRef]
  16. D. C. Ghiglia, M. D. Pritt, Two-Dimensional Phase Unwrapping (Wiley, New York, 1998).
  17. G. Franceschetti, R. Lanari, Synthetic Aperture Radar Processing (CRC Press, Boca Raton, Fla., 1999).
  18. R. A. Horn, C. R. Johnson, Matrix Analysis (Cambridge U. Press, Cambridge, UK, 1990).
  19. W. H. Press, S. A. Teukolsky, W. A. Vetterling, B. P. Flannery, Numerical Recipes in C: The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, Cambridge, UK, 1993).
  20. S. S. Haykin, Adaptive Filter Theory, 3rd ed. (Prentice-Hall, Englewood Cliffs, N.J., 1995).

1999 (1)

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

1997 (2)

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, D. Rossi, M. Tesauro, “Interferometric SAR phase unwrapping using the finite element method,” IEE Proc. Radar Sonar Navig. 144 (No. 4), 1–9 (1997).
[CrossRef]

1996 (2)

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, “Interferometric SAR phase unwrapping using Green’s formulation,” IEEE Trans. Geosci. Remote Sens. 34, 720–727 (1996).
[CrossRef]

1995 (2)

J. L. Marroquin, M. Rivera, “Quadratic regularization functionals for phase unwrapping,” J. Opt. Soc. Am. A 12, 2393–2400 (1995).
[CrossRef]

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

1994 (2)

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

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

1988 (1)

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

1979 (1)

1977 (1)

J. M. Tribolet, “A new phase unwrapping algorithm,” IEEE Trans. Acoust., Speech, Signal Process. ASP-25, 170–177 (1977).
[CrossRef]

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. A. Vetterling, B. P. Flannery, Numerical Recipes in C: The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, Cambridge, UK, 1993).

Fornaro, G.

G. Fornaro, G. Franceschetti, R. Lanari, D. Rossi, M. Tesauro, “Interferometric SAR phase unwrapping using the finite element method,” IEE Proc. Radar Sonar Navig. 144 (No. 4), 1–9 (1997).
[CrossRef]

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, “Interferometric SAR phase unwrapping using Green’s formulation,” IEEE Trans. Geosci. Remote Sens. 34, 720–727 (1996).
[CrossRef]

G. Fornaro, G. Franceschetti, R. Lanari, E. Sansosti, “A theoretical analysis of the robust phase unwrapping algorithms for SAR interferometry,” in Proceedings of the IEEE International Symposium on Geoscience and Remote Sensing (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, D. Rossi, M. Tesauro, “Interferometric SAR phase unwrapping using the finite element method,” IEE Proc. Radar Sonar Navig. 144 (No. 4), 1–9 (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, “Interferometric SAR phase unwrapping using Green’s formulation,” IEEE Trans. Geosci. Remote Sens. 34, 720–727 (1996).
[CrossRef]

S. Marano, F. Palmieri, G. Franceschetti, “Integral-differential relationships reformulated for image processing applications,” manuscript in preparation.

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

G. Franceschetti, R. Lanari, Synthetic Aperture Radar Processing (CRC Press, Boca Raton, Fla., 1999).

Ghiglia, D. C.

Glover, G. H.

S. Moon-Ho. Song, S. Napel, N. J. Pelc, G. H. Glover, “Phase unwrapping of MR 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]

Haykin, S. S.

S. S. Haykin, Adaptive Filter Theory, 3rd ed. (Prentice-Hall, Englewood Cliffs, N.J., 1995).

Horn, R. A.

R. A. Horn, C. R. Johnson, Matrix Analysis (Cambridge U. Press, Cambridge, UK, 1990).

Hunt, B. R.

Jain, A. K.

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

Johnson, C. R.

R. A. Horn, C. R. Johnson, Matrix Analysis (Cambridge U. Press, Cambridge, UK, 1990).

Lanari, R.

G. Fornaro, G. Franceschetti, R. Lanari, D. Rossi, M. Tesauro, “Interferometric SAR phase unwrapping using the finite element method,” IEE Proc. Radar Sonar Navig. 144 (No. 4), 1–9 (1997).
[CrossRef]

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, “Interferometric SAR phase unwrapping using Green’s formulation,” IEEE Trans. Geosci. Remote Sens. 34, 720–727 (1996).
[CrossRef]

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

G. Franceschetti, R. Lanari, Synthetic Aperture Radar Processing (CRC Press, Boca Raton, Fla., 1999).

Marano, S.

S. Marano, F. Palmieri, G. Franceschetti, “Integral-differential relationships reformulated for image processing applications,” manuscript in preparation.

Marroquin, J. L.

Napel, S.

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

Oppheneim, A. V.

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

Palmieri, F.

S. Marano, F. Palmieri, G. Franceschetti, “Integral-differential relationships reformulated for image processing applications,” manuscript in preparation.

Pelc, N. J.

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

Press, W. H.

W. H. Press, S. A. Teukolsky, W. A. Vetterling, B. P. Flannery, Numerical Recipes in C: The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, Cambridge, UK, 1993).

Pritt, M. D.

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

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

Rivera, M.

Romero, L. A.

Rossi, D.

G. Fornaro, G. Franceschetti, R. Lanari, D. Rossi, M. Tesauro, “Interferometric SAR phase unwrapping using the finite element method,” IEE Proc. Radar Sonar Navig. 144 (No. 4), 1–9 (1997).
[CrossRef]

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 of the robust phase unwrapping algorithms for SAR interferometry,” in Proceedings of the IEEE International Symposium on Geoscience and Remote Sensing (Institute of Electrical and Electronics Engineers, New York, 1996), pp. 2047–2049.

Schafer, R. W.

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

Shipman, J. S.

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

Song, S. Moon-Ho.

S. Moon-Ho. Song, S. Napel, N. J. Pelc, G. H. Glover, “Phase unwrapping of MR 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-square method,” IEEE Trans. Image Process. 8, 375–386 (1999).
[CrossRef]

Taxt, T.

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

Tesauro, M.

G. Fornaro, G. Franceschetti, R. Lanari, D. Rossi, M. Tesauro, “Interferometric SAR phase unwrapping using the finite element method,” IEE Proc. Radar Sonar Navig. 144 (No. 4), 1–9 (1997).
[CrossRef]

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]

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. A. Vetterling, B. P. Flannery, Numerical Recipes in C: The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, Cambridge, UK, 1993).

Tribolet, J. M.

J. M. Tribolet, “A new phase unwrapping algorithm,” IEEE Trans. Acoust., Speech, Signal Process. ASP-25, 170–177 (1977).
[CrossRef]

Vetterling, W. A.

W. H. Press, S. A. Teukolsky, W. A. Vetterling, B. P. Flannery, Numerical Recipes in C: The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, Cambridge, UK, 1993).

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]

IEE Proc. Radar Sonar Navig. (1)

G. Fornaro, G. Franceschetti, R. Lanari, D. Rossi, M. Tesauro, “Interferometric SAR phase unwrapping using the finite element method,” IEE Proc. Radar Sonar Navig. 144 (No. 4), 1–9 (1997).
[CrossRef]

IEEE Trans. Acoust., Speech, Signal Process. (1)

J. M. Tribolet, “A new phase unwrapping algorithm,” IEEE Trans. Acoust., Speech, Signal Process. ASP-25, 170–177 (1977).
[CrossRef]

IEEE Trans. Geosci. Remote Sens. (2)

G. Fornaro, G. Franceschetti, R. Lanari, “Interferometric SAR phase unwrapping using Green’s formulation,” IEEE Trans. Geosci. Remote Sens. 34, 720–727 (1996).
[CrossRef]

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

IEEE Trans. Image Process. (2)

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

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

J. Opt. Soc. Am. (1)

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

Radio Sci. (1)

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

Other (8)

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

S. Marano, F. Palmieri, G. Franceschetti, “Integral-differential relationships reformulated for image processing applications,” manuscript in preparation.

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

G. Franceschetti, R. Lanari, Synthetic Aperture Radar Processing (CRC Press, Boca Raton, Fla., 1999).

R. A. Horn, C. R. Johnson, Matrix Analysis (Cambridge U. Press, Cambridge, UK, 1990).

W. H. Press, S. A. Teukolsky, W. A. Vetterling, B. P. Flannery, Numerical Recipes in C: The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, Cambridge, UK, 1993).

S. S. Haykin, Adaptive Filter Theory, 3rd ed. (Prentice-Hall, Englewood Cliffs, N.J., 1995).

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

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Figures (16)

Fig. 1
Fig. 1

Coordinate system and discrete lattice.

Fig. 2
Fig. 2

Horizontal and vertical integration paths. The boxes indicate points that must be included in the summation, and the arrows indicate the direction of integration.

Fig. 3
Fig. 3

Examples of integration paths. Note that in the last example the path is closed, i.e., (a, b)=(c, d), and we have chosen to run it counterclockwise.

Fig. 4
Fig. 4

Contour C (marked with arrows to denote the direction along which the line integral is evaluated) and domain D (shaded boxes) for Stokes’s theorem on a rectangular region.

Fig. 5
Fig. 5

Contour C (marked with boxes or dashed boxes, in both cases containing arrows) and the associated doubly connected domain D (shaded boxes).

Fig. 6
Fig. 6

Example of how the doubly connected rectangular domain of Fig. 5 is decomposed into rectangular subsets of pixels. For each subset, relationship (5) is applicable, and the appropriate definition of D and C, as illustrated in Fig. 5, immediately follows.

Fig. 7
Fig. 7

Examples of application of the discrete version of Stokes’s theorem. The contour C is made up of the pixels marked with an arrow. The domain D consists of the shaded boxed pixels. Note how some points of C belong also to D. Points marked with a dot are shown simply for reference, but they are never involved in the computation.

Fig. 8
Fig. 8

Two examples of construction of the contour C (on the right), starting from a domain D (on the left). The shaded boxed pixels constitute the given domain, and the arrows denote pixels belonging to C. In the first example, DC. In the second example, the pixel (p, q) at the center of the cross ∈ D but ∉ C.

Fig. 9
Fig. 9

Another example of construction of the contour C (bottom), starting from a quite arbitrary domain D (top).

Fig. 10
Fig. 10

Illustration of the contour C, the enclosed domain D (two of those already shown in Fig. 7), and the extended contour Ce, made up of points marked with the symbol ×. For application of Green’s identity, all these points are involved. Note that some pixels are marked with ⊕ and ⊖ according to the sign with which they enter the definition of normal derivative [Eq. (9)], computed at the pixel belonging to C.

Fig. 11
Fig. 11

Some examples of test functions. In (a) the rectangular domain of Fig. 4 is considered with the assumption of a=1, b=1, c=12, and d=22. The test function is computed for the central pixel (p, q)=(7, 12) with just a minimum-norm constraint and with no boundary conditions. In (b) the test function is computed for the doubly connected domain of Fig. 5, where a=b=1, c=12, d=22, m=5, n=9, r=8, and s=14. The GF g refers to pixel (p, q)=(5, 9), and the constraint on Eq. (12) is again a minimum norm with no boundary conditions. In (c) and (d), the same domains as those of (a) and (b), respectively, are considered, but the test function is computed with the requirement of having identically null (relevant component of the) normal derivative on the contour C, except for the nine pixels that constitute the lower-right corner of the domains, i.e., pixels (8, 22), (9, 22), (10, 22), (11, 22), (12, 22), (12, 18), (12, 19), (12, 20), and (12, 21). Since the system is still underdetermined, we have taken the minimum-norm solution. Note how in the more constrained case the numerical values tend to spread over a wider range.

Fig. 12
Fig. 12

Domain D (shaded boxes) and contour C (depicted as boxes containing arrows indicating the direction according to which the contour is spanned) for the application of Green’s method to a square domain. The pixels marked with the symbol × are those involved in the definition of the test function g but not belonging to DC. On each pixel is shown the value of a phase sequence, with values given in radians. The chosen phase pattern represents a pyramid with a dent at ϕ(5, 2): If one substitutes ϕ(5,2)=3.25 rad for ϕ(5, 2)=1 rad, a perfect pyramidal shape is obtained.

Fig. 13
Fig. 13

Example of the wrapped process applied to the 2D phase sequence of Fig. 12. The PhU problem is the retrieval of the original sequence from its (-π, π) determination. All values are given in radians.

Fig. 14
Fig. 14

Numerical example of phase retrieval procedure by means of the proposed unwrapping technique, applied to the phase pattern of Fig. 12, with which the present values should be compared. Note how the error, because of undersampling, has spread itself over the whole domain D.

Fig. 15
Fig. 15

First possibility for choosing the new domain. We cut a certain portion of the initial domain and eliminate the pixels that are known to give undersampling. We do not develop further the method for this domain, as better choices are possible.

Fig. 16
Fig. 16

New definition of the domain D that excludes all the undersampled pixels that give rise to errors in the reconstruction of the pyramid-with-dent pattern of Fig. 12. Being free from undersampling,the retrieval procedure is able to recover exactly the true phase from the knowledge of ϕ over the first row, the first column, the last row, and the last column and from the knowledge of ϕw over the remaining pixels.

Tables (1)

Tables Icon

Table 1 Summary of Definitions of Discrete Operators

Equations (58)

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

Dxϕ(p, q)ϕ(p, q)-ϕ(p-1, q),
Dyϕ(p, q)ϕ(p, q)-ϕ(p, q-1).
(a,b)(c,d)f(p, q)cˆ=ϕ(c, d)-ϕ(a, b) if f(p, q)=ϕ(p, q).
Cf(p, q)cˆ=q=b+1dfy(a, q)+p=a+1cfx(p, d)-q=b+1dfy(c, q)-p=a+1cfx(p, b)=p=a+1c[fx(p, d)-fx(p, b)]-q=b+1d[fy(c, q)-fy(a, q)]=p=a+1cq=b+1d[Dyfx(p, q)-Dxfy(p, q)]
Cf(p, q)cˆ=p=a+1cq=b+1d×f(p, q)(-zˆ)=D×f(p, q)(-zˆ);
Cf(p, q)cˆ=q=b+1dfy(a, q)+p=a+1cfx(p, d)-q=b+1dfy(c, q)-p=a+1cfx(p, b)-q=n+1sfy(m, q)+p=m+1rfx(p, n)+q=n+1sfy(r, q)-p=m+1rfx(p, s)
=p=a+1cq=b+1d[Dyfx(p, q)-Dxfy(p, q)]-p=m+1rq=n+1s[Dyfx(p, q)-Dxfy(p, q)]
=p=a+1cq=b+1d×f(p, q)(-zˆ)-p=m+1rq=n+1s×f(p, q)(-zˆ)=D×f(p, q)(-zˆ).
Cf(p, q)nˆ=-q=b+1dfx(a, q)+p=a+1cfy(p, d)+q=b+1dfx(c, q)-p=a+1cfy(p, b),
Cf(p, q)nˆ=p=a+1cq=b+1d[Dxfx(p, q)+Dyfy(p, q)]=Df(p, q).
Dnˆϕ(p, q)[Dyϕ(p, q+1)xˆ-Dxϕ(p+1, q)yˆ]cˆ,
(p, q)C.
 2ϕ(p, q)Dx[Dxϕ(p+1, q)]
+Dy[Dyϕ(p, q+1)]
=ϕ(p+1, q)+ϕ(p, q+1)-4ϕ(p, q)+ϕ(p-1, q)+ϕ(p, q-1).
Cϕ(p, q)Dnˆu(p, q)=Dϕ(p, q)u(p, q)+Dϕ(p, q)2u(p, q),
2g(p, q; p, q)=δ(p-p, q-q),
g(p+1, q;p, q)+g(p, q+1;p, q)-4g(p, q;p, q)+g(p-1, q;p, q)+g(p, q-1;p, q)=δ(p-p, q-q),
ϕ(p, q)=Cϕ(p, q)Dnˆg(p, q;p, q)-Dϕ(p, q)g(p, q;p, q).
U1[Dϕw(p)]
Dϕw(p)+2πifDϕw(p)-πDϕw(p)if-π<Dϕw(p)πDϕw(p)-2πifDϕw(p)>π.
fsx(p, q)=fx(p, q)-sx(p, q)2π,
fsy(p, q)=fy(p, q)-sy(p, q)2π,
sx(p, q):[2sx(p, q)-1]π
<fx(p, q)[2sx(p, q)+1]π,
sy(p, q):[2sy(p, q)-1]π
<fy(p, q)[2sy(p, q)+1]π.
ϕr(p, q)=(pi, qi)(p,q)fs(p,q)cˆ=(pi,qi)(p,q)f(p,q)cˆ-2π(pi,qi)(p,q)s(p,q)cˆ=ϕ(p, q)-ϕ(pi,qi)-2π(pi,qi)(p,q)s(p,q)cˆ,
ϕr(p,q)=Cϕ(p,q)Dnˆg(p,q;p,q)-Dfs(p,q)g(p,q;p,q)
=ϕ(p,q)+2πDs(p,q)g(p,q;p,q),
×fszˆ=-2π,(p,q)=(5, 3)2π,(p,q)=(6, 3)0,elsewhere.
r(p,q)=2πDs(p,q)g(p,q;p,q)=2πsy(5, 3)Dyg(5, 3;p,q)=2πDyg(5, 3;p,q),
Cϕ(p,q)Dnˆu(p,q)
=-q=b+1dϕ(a,q)Dxu(a+1, q)+p=a+1cϕ(p, d)Dyu(p, d+1)+q=b+1dϕ(c,q)Dxu(c+1, q)-p=a+1cϕ(p, b)Dyu(p, b+1)
=q=b+1d[ϕ(c,q)Dxu(c+1, q)
-ϕ(a,q)Dxu(a+1, q)]
+p=a+1c[ϕ(p, d)Dyu(p, d+1)
-ϕ(p, b)Dyu(p, b+1)]
=p=a+1cq=b+1dDx[ϕ(p,q)Dxu(p+1, q)]
+p=a+1cq=b+1dDy[ϕ(p,q)Dyu(p,q+1)]=D[ϕ(p,q)Dxu(p+1, q)-ϕ(p-1, q)Dxu(p,q)+ϕ(p,q)Dyu(p,q+1)-ϕ(p,q-1)Dyu(p,q)]=Dϕ(p,q)[u(p+1, q)+u(p,q+1)-2u(p,q)]+D{ϕ(p,q-1)[u(p,q-1)-u(p,q)]+ϕ(p-1, q)[u(p-1, q)-u(p,q)]}.
Cϕ(p,q)Dnˆu(p,q)=Dϕ(p,q)2u(p,q)+Dϕ(p,q)u(p,q),
Au=c,
ck=1,k=h1(p,q)0,elsewhere,
(p,q)=h1-1(k),
A[k, h2(p+1, q)]=1,
A[k, h2(p,q+1)]=1,
A[k, h2(p,q)]=-4,
A[k, h2(p-1, q)]=1,
A[k, h2(p,q-1)]=1,
k=1, 2 ,, ND.
(p,q, d)=h3-1(j),
Ae[ND+j, h2(p,q+1)]=1,
Ae[ND+j, h2(p,q)]=-1
ifd=horizontal,
Ae[ND+j, h2(p+1, q)]=1
Ae[ND+j, h2(p,q)]=-1
ifd=vertical,
ce=[cT0000NP]T,

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