Abstract

A new algorithm for filtering noise in phase maps that contain 2π discontinuities is presented. The algorithm is based on a thermal model that uses the heat equation to perform low-pass filtering. A similar approach is used in image processing for filtering noise, but the edges are generally distorted because of their inherent high frequencies. A solution that consists in redefining the spatial derivatives is proposed here. Simulation results are presented and discussed.

© 2001 Optical Society of America

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References

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  1. E. Frins, W. Dultz, J. A. Ferrari, “Polarization-shifting method for step interferometry,” Pure Appl. Opt. 7, 53–60 (1998).
    [Crossref]
  2. M. Takeda, H. Ina, S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. 72, 156–160 (1982).
    [Crossref]
  3. 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]
  4. J. L. Marroquin, M. Tapia, R. Rodriguez-Vera, M. Servin, “Parallel algorithms for phase unwrapping based on Markov random fields models,” J. Opt. Soc. Am. A 12, 2578–2585 (1995).
    [Crossref]
  5. H. A. Zebker, Y.-P. Lu, “Phase unwrapping algorithms for radar interferometry: residue-cut, least-squares, and synthesis algorithms,” J. Opt. Soc. Am. A 15, 586–598 (1998).
    [Crossref]
  6. G. Páez, M. Strojnik, “Fringe analysis and phase reconstruction from modulated intensity patterns,” Opt. Lett. 22, 1669–1671 (1997).
    [Crossref]
  7. G. Fornaro, G. Franceschetti, R. Lanari, “Interferometric SAR phase unwrapping using Green’s formulation,” IEEE Trans. Geosci. Remote Sens. 34, 720–727 (1996).
    [Crossref]
  8. A. Collaro, G. Franceschetti, F. Palmieri, M. S. Ferreiro, “Phase unwrapping by means of genetic algorithms,” J. Opt. Soc. Am. A 15, 407–418 (1997).
    [Crossref]
  9. L. Guerriero, G. Nico, G. Pasquariello, S. Stramaglia, “New regularization scheme for phase unwrapping,” Appl. Opt. 37, 3053–3058 (1998).
    [Crossref]
  10. R. Seara, A. A. Gonçalves, P. B. Uliana, “Filtering algorithm for noise reduction in phase-map images with 2π phase jumps,” Appl. Opt. 37, 2046–1050 (1998).
    [Crossref]
  11. B. M. Ter, Haar Romeny, ed., Geometry-Driven Diffusion in Computer Vision (Kluwer Scientific, Dordrecht, The Netherlands, 1994).
  12. F. P. Incropera, D. P. De Witt, Fundamentals of Heat and Mass Transfer (Wiley, New York, 1990), Chap. 5.
  13. W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C (Cambridge U. Press, Cambridge, UK, 1996), Chap. 19.

1998 (4)

1997 (2)

1996 (1)

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 (1)

1994 (1)

1982 (1)

Collaro, A.

De Witt, D. P.

F. P. Incropera, D. P. De Witt, Fundamentals of Heat and Mass Transfer (Wiley, New York, 1990), Chap. 5.

Dultz, W.

E. Frins, W. Dultz, J. A. Ferrari, “Polarization-shifting method for step interferometry,” Pure Appl. Opt. 7, 53–60 (1998).
[Crossref]

Ferrari, J. A.

E. Frins, W. Dultz, J. A. Ferrari, “Polarization-shifting method for step interferometry,” Pure Appl. Opt. 7, 53–60 (1998).
[Crossref]

Ferreiro, M. S.

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C (Cambridge U. Press, Cambridge, UK, 1996), Chap. 19.

Fornaro, G.

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

Franceschetti, G.

A. Collaro, G. Franceschetti, F. Palmieri, M. S. Ferreiro, “Phase unwrapping by means of genetic algorithms,” J. Opt. Soc. Am. A 15, 407–418 (1997).
[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]

Frins, E.

E. Frins, W. Dultz, J. A. Ferrari, “Polarization-shifting method for step interferometry,” Pure Appl. Opt. 7, 53–60 (1998).
[Crossref]

Ghiglia, D. C.

Gonçalves, A. A.

Guerriero, L.

Ina, H.

Incropera, F. P.

F. P. Incropera, D. P. De Witt, Fundamentals of Heat and Mass Transfer (Wiley, New York, 1990), Chap. 5.

Kobayashi, S.

Lanari, R.

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

Lu, Y.-P.

Marroquin, J. L.

Nico, G.

Páez, G.

Palmieri, F.

Pasquariello, G.

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C (Cambridge U. Press, Cambridge, UK, 1996), Chap. 19.

Rodriguez-Vera, R.

Romero, L. A.

Seara, R.

Servin, M.

Stramaglia, S.

Strojnik, M.

Takeda, M.

Tapia, M.

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C (Cambridge U. Press, Cambridge, UK, 1996), Chap. 19.

Uliana, P. B.

Vetterling, W. T.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C (Cambridge U. Press, Cambridge, UK, 1996), Chap. 19.

Zebker, H. A.

Appl. Opt. (2)

IEEE Trans. Geosci. Remote Sens. (1)

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

J. Opt. Soc. Am. (1)

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

Opt. Lett. (1)

Pure Appl. Opt. (1)

E. Frins, W. Dultz, J. A. Ferrari, “Polarization-shifting method for step interferometry,” Pure Appl. Opt. 7, 53–60 (1998).
[Crossref]

Other (3)

B. M. Ter, Haar Romeny, ed., Geometry-Driven Diffusion in Computer Vision (Kluwer Scientific, Dordrecht, The Netherlands, 1994).

F. P. Incropera, D. P. De Witt, Fundamentals of Heat and Mass Transfer (Wiley, New York, 1990), Chap. 5.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C (Cambridge U. Press, Cambridge, UK, 1996), Chap. 19.

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Figures (2)

Fig. 1
Fig. 1

Modulo 2π operator m(ξ) defined in Eq. (4).

Fig. 2
Fig. 2

(a) Original simulated phase map. (b) Noisy phase map obtained by addition of a normal distributed random variable. (c) Result of the filtering process after the first iteration. (d) Result after the fourth iteration.

Equations (12)

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

ϕx, y=Φx, y+2πkx, y,
k2T-Tt=0,
-kωx2+ωy2T˜-T˜t=0,
Hωx, ωy=exp-ktωx2+ωy2.
mξ=ξ-2π Roundξ/2π,
dπfxdx=limdx0mfx+dx-fxdx.
πΦx, ydx=limdx0mΦx+dx, y-Φx, ydx=limdx0ϕx+dx, y-ϕx, ydx=ϕx, ydx.
π2T=π2Tx2+π2Ty2,
kπ2T-Tt=0.
Tt1ΔtTi,jn-Ti,jn-1,π2Tx21Δx2 mmTi+1,jn-1-Ti,jn-1-mTi,jn-1-Ti-1,jn-1,π2Ty21Δy2 mmTi,j+1n-1-Ti,jn-1-mTi,jn-1-Ti,j-1n-1.
Tn=FTn-1,
NPn=1NijTi,jn-Ti,j2,

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