Abstract

We present a generic regularized formulation, based on robust half-quadratic regularization, for unwrapping noisy and discontinuous wrapped phase maps. Two cases are presented: the convex case and the nonconvex case. The unwrapped phase with the convex formulation is unique and robust to noise; however, the convex function solution deteriorates as a result of real discontinuities in phase maps. Therefore we also present a nonconvex formulation that, with a parameter continuation strategy, shows superior performance.

© 2004 Optical Society of America

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References

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  1. J. L. Marroquin and M. Rivera, J. Opt. Soc. Am. A 12, 2393 (1995).
    [CrossRef]
  2. M. J. Black and A. Rangarajan, Int. J. Comput. Vision 19, 57 (1996).
    [CrossRef]
  3. P. Charbonnier, L. Blanc-Féraud, G. Aubert, and M. Barlaud, IEEE Trans. Image Process. 6, 298 (1997).
    [CrossRef]
  4. M. Rivera and J. L. Marroquin, Image Vision Comput. 21, 345 (2003).
    [CrossRef]
  5. M. Rivera and J. L. Marroquin, Comput. Vision Image Understand. 88, 76 (2002).
    [CrossRef]
  6. D. C. Ghiglia and L. A. Romero, J. Opt. Soc. Am. A 13, 1999 (1996).
    [CrossRef]

2003

M. Rivera and J. L. Marroquin, Image Vision Comput. 21, 345 (2003).
[CrossRef]

2002

M. Rivera and J. L. Marroquin, Comput. Vision Image Understand. 88, 76 (2002).
[CrossRef]

1997

P. Charbonnier, L. Blanc-Féraud, G. Aubert, and M. Barlaud, IEEE Trans. Image Process. 6, 298 (1997).
[CrossRef]

1996

D. C. Ghiglia and L. A. Romero, J. Opt. Soc. Am. A 13, 1999 (1996).
[CrossRef]

M. J. Black and A. Rangarajan, Int. J. Comput. Vision 19, 57 (1996).
[CrossRef]

1995

Aubert, G.

P. Charbonnier, L. Blanc-Féraud, G. Aubert, and M. Barlaud, IEEE Trans. Image Process. 6, 298 (1997).
[CrossRef]

Barlaud, M.

P. Charbonnier, L. Blanc-Féraud, G. Aubert, and M. Barlaud, IEEE Trans. Image Process. 6, 298 (1997).
[CrossRef]

Black, M. J.

M. J. Black and A. Rangarajan, Int. J. Comput. Vision 19, 57 (1996).
[CrossRef]

Blanc-Féraud, L.

P. Charbonnier, L. Blanc-Féraud, G. Aubert, and M. Barlaud, IEEE Trans. Image Process. 6, 298 (1997).
[CrossRef]

Charbonnier, P.

P. Charbonnier, L. Blanc-Féraud, G. Aubert, and M. Barlaud, IEEE Trans. Image Process. 6, 298 (1997).
[CrossRef]

Ghiglia, D. C.

Marroquin, J. L.

M. Rivera and J. L. Marroquin, Image Vision Comput. 21, 345 (2003).
[CrossRef]

M. Rivera and J. L. Marroquin, Comput. Vision Image Understand. 88, 76 (2002).
[CrossRef]

J. L. Marroquin and M. Rivera, J. Opt. Soc. Am. A 12, 2393 (1995).
[CrossRef]

Rangarajan, A.

M. J. Black and A. Rangarajan, Int. J. Comput. Vision 19, 57 (1996).
[CrossRef]

Rivera, M.

M. Rivera and J. L. Marroquin, Image Vision Comput. 21, 345 (2003).
[CrossRef]

M. Rivera and J. L. Marroquin, Comput. Vision Image Understand. 88, 76 (2002).
[CrossRef]

J. L. Marroquin and M. Rivera, J. Opt. Soc. Am. A 12, 2393 (1995).
[CrossRef]

Romero, L. A.

Comput. Vision Image Understand.

M. Rivera and J. L. Marroquin, Comput. Vision Image Understand. 88, 76 (2002).
[CrossRef]

IEEE Trans. Image Process.

P. Charbonnier, L. Blanc-Féraud, G. Aubert, and M. Barlaud, IEEE Trans. Image Process. 6, 298 (1997).
[CrossRef]

Image Vision Comput.

M. Rivera and J. L. Marroquin, Image Vision Comput. 21, 345 (2003).
[CrossRef]

Int. J. Comput. Vision

M. J. Black and A. Rangarajan, Int. J. Comput. Vision 19, 57 (1996).
[CrossRef]

J. Opt. Soc. Am. A

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Figures (3)

Fig. 1
Fig. 1

Test wrapped phases: (a) synthetic, (b) deformation of a steel plate (ESPI), (c) fracture in a steel plate (ESPI).

Fig. 2
Fig. 2

Unwrapped phase (first column), rewrapping of the unwrapped phase (second column), and the l field (third column) with (a) quadratic potential1 and (b) the Ghiglia–Romero L0Norm6 {in our formulation it is equivalent to the Herbert–Leahy nonconvex half-quadratic potential,3 i.e., Ψlrs=μlrs2-lnlrs2-1, where μ is the positive parameter that controls the outlier detection and with λ=0}, and the proposed half-quadratic cost function with (c) convex, (d) nonconvex with four neighbors, and (e) nonconvex with eight neighbors.

Fig. 3
Fig. 3

Results computed with the proposed method, in the same order as Fig. 2. (a), (b) Convex half-quadratic cost function; (c), (d) nonconvex cost function.

Equations (9)

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Δfˆrs-WΔgrs=0
Uf=r,sαrs2+λβrs2,
Uhqf,l=r,sρhqf,l;r,s=r,slrs2αrs2+λβrs2+Ψlrs,
sNr2lrs2αrsαrsfr+λβrsβrsfr=0
ρˆαrs def¯¯ 0αrs2wx2xdx,
Uˆhqf,l=r,sρˆαrs,
Uˆhqfr=sNrρˆαrsαrsfr=sNr2wαrs2αrsαrsfr,
lrs2=wαrs2=ρˆαrs2αrs=1αrs2ϵϵ/αrs2otherwise,
lrs2=wαrs2+λβrs2=μμ+αrs2+λβrs22,

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