Abstract

A discrete Fourier transform (DFT) based algorithm for solving a quadratic cost functional is proposed; this regularized functional allows one to obtain a consistent gradient field from an inconsistent one. The calculated consistent gradient may then be integrated by use of simple methods. The technique is presented in the context of the phase-unwrapping problem; however, it may be applied to other problems, such as shapes from shading (a robot-vision technique) when inconsistent gradient fields with irregular domains are obtained. The regularized functional introduced here has advantages over existing techniques; in particular, it is able to manage complex irregular domains and to interpolate over regions with invalid data without any smoothness assumptions over the rest of the lattice, so that the estimation error is reduced. Furthermore, there are no free parameters to adjust. The DFT is used to compute a preconditioner because there is highly efficient hardware to perform the calculations and also because it may be computed by optical means.

© 1997 Optical Society of America

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References

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  1. M. V. Mantravadi, “Lateral shearing interferometers,” in Optical Shop Testing, 2nd ed., D. Malacara, ed. (Wiley, New York, 1992), pp. 123–206.
  2. K. Itoh, “Analysis of the phase unwrapping algorithm,” Appl. Opt. 21, 2470 (1982).
    [CrossRef] [PubMed]
  3. J. M. Tribolet, “A new phase unwrapping algorithm,” IEEE Trans. Acoust. Speech Signal Process. ASSP-25, 170–177 (1977).
    [CrossRef]
  4. D. L. Fried, “Least-squares fitting of a wave-front distortion estimate to an array of phase-difference measurements,” J. Opt. Soc. Am. 67, 370–375 (1977).
    [CrossRef]
  5. R. H. Hudgin, “Wave-front reconstruction for compensated imaging,” J. Opt. Soc. Am. 67, 375–378 (1977).
    [CrossRef]
  6. H. Takajo, T. Takahashi, “Least-squares phase estimation from the phase difference,” J. Opt. Soc. Am. A 5, 416–425 (1988).
    [CrossRef]
  7. J. L. Marroquin, M. Rivera, “Quadratic regularization functional for phase unwrapping,” J. Opt. Soc. Am. A 12, 2393–2400 (1995).
    [CrossRef]
  8. K. P. Horn, Robot Vision (MIT Press, Cambridge, Mass., 1986), Chap. 11.
  9. 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]
  10. R. Fletcher, Practical Methods of Optimization, 2nd ed. (Wiley, New York, 1987).
  11. R. Courant, “Variational methods for the solution of problems of equilibrium and vibrations,” Bull. Am. Math. Soc. 49, 1–23 (1943).
    [CrossRef]
  12. 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]
  13. G. H. Golub, C. F. van Loan, Matrix Computations (Johns Hopkins U. Press, Baltimore, Md., 1989), Chap. 10.
  14. R. Jones, C. Wykes, Holographic and Speckle Interferometry (Cambridge U. Press, Cambridge, 1989).
    [CrossRef]
  15. D. C. Ghiglia, G. A. Masting, L. A. Romero, “Cellular-automata method for phase unwrapping,” J. Opt. Soc. Am. A 4, 267–280 (1987).
    [CrossRef]
  16. H. Takajo, T. Takahashi, “Noniterative method for obtaining the exact solution for the normal equation in least phase estimation from the phase difference,” J. Opt. Soc. Am. A 5, 1818–1827 (1988).
    [CrossRef]

1995 (1)

1994 (1)

1988 (2)

1987 (1)

1982 (2)

1977 (3)

1943 (1)

R. Courant, “Variational methods for the solution of problems of equilibrium and vibrations,” Bull. Am. Math. Soc. 49, 1–23 (1943).
[CrossRef]

Courant, R.

R. Courant, “Variational methods for the solution of problems of equilibrium and vibrations,” Bull. Am. Math. Soc. 49, 1–23 (1943).
[CrossRef]

Fletcher, R.

R. Fletcher, Practical Methods of Optimization, 2nd ed. (Wiley, New York, 1987).

Fried, D. L.

Ghiglia, D. C.

Golub, G. H.

G. H. Golub, C. F. van Loan, Matrix Computations (Johns Hopkins U. Press, Baltimore, Md., 1989), Chap. 10.

Horn, K. P.

K. P. Horn, Robot Vision (MIT Press, Cambridge, Mass., 1986), Chap. 11.

Hudgin, R. H.

Ina, H.

Itoh, K.

Jones, R.

R. Jones, C. Wykes, Holographic and Speckle Interferometry (Cambridge U. Press, Cambridge, 1989).
[CrossRef]

Kobayashi, S.

Mantravadi, M. V.

M. V. Mantravadi, “Lateral shearing interferometers,” in Optical Shop Testing, 2nd ed., D. Malacara, ed. (Wiley, New York, 1992), pp. 123–206.

Marroquin, J. L.

Masting, G. A.

Rivera, M.

Romero, L. A.

Takahashi, T.

Takajo, H.

Takeda, M.

Tribolet, J. M.

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

van Loan, C. F.

G. H. Golub, C. F. van Loan, Matrix Computations (Johns Hopkins U. Press, Baltimore, Md., 1989), Chap. 10.

Wykes, C.

R. Jones, C. Wykes, Holographic and Speckle Interferometry (Cambridge U. Press, Cambridge, 1989).
[CrossRef]

Appl. Opt. (1)

Bull. Am. Math. Soc. (1)

R. Courant, “Variational methods for the solution of problems of equilibrium and vibrations,” Bull. Am. Math. Soc. 49, 1–23 (1943).
[CrossRef]

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

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

J. Opt. Soc. Am. (3)

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

Other (5)

R. Fletcher, Practical Methods of Optimization, 2nd ed. (Wiley, New York, 1987).

M. V. Mantravadi, “Lateral shearing interferometers,” in Optical Shop Testing, 2nd ed., D. Malacara, ed. (Wiley, New York, 1992), pp. 123–206.

K. P. Horn, Robot Vision (MIT Press, Cambridge, Mass., 1986), Chap. 11.

G. H. Golub, C. F. van Loan, Matrix Computations (Johns Hopkins U. Press, Baltimore, Md., 1989), Chap. 10.

R. Jones, C. Wykes, Holographic and Speckle Interferometry (Cambridge U. Press, Cambridge, 1989).
[CrossRef]

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Figures (5)

Fig. 1
Fig. 1

(a) Sintetic test phase. (b) Wrapped phase. Sites in white (c) with valid information and (d) where the unwrapped phase will be computed.

Fig. 2
Fig. 2

Unwrapped phase fields of the wrapped phase showed in Fig. 1(b) after 13 iterations with (a) the PCGMR with λ = 1.0 and (b) the PCG-CC with λ1 = 10,000 (ν = λ1-1).

Fig. 3
Fig. 3

Map errors of solutions computed with (a) PCGMR with λ = 1.0, (b) PCGMR with λ = 0.01, (c) PCG-CC with λ1 = 10,000 (ν = λ1-1).

Fig. 4
Fig. 4

Mean-square error (MSE) versus iteration number for the PCGMR algorithm with λ = 1.0 (open circles), λ = 0.1 (stars), and λ = 0.01 (plus signs) and for the PCG-CC with λ1 = 10,000 (dashes).

Fig. 5
Fig. 5

(a) ESPI phase map. (b) Unwrapped phase after 20 iterations with the PCG-CC algorithm (λ1 = 10,000).

Equations (63)

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gx, y=ϕx, y+nx, y+2πkx, y,
Uϕˆx, y=x,ygxx, y-ΔxB ϕˆx, y2+gyx, y-ΔyBϕˆ x, y2,
gxx, y, gyx, y=WΔxBgx, y, WΔyBgx, y
Wϕx, y=ϕx, y+2πkx, y,
fxx, yΔxBfx, y=fx, y-fx-1, y,
fyx, yΔyBfx, y=fx, y-fx, y-1.
U1ϕˆx, y=x,ygxx, y-ϕˆx, y+ϕˆx-1, y2+x,ygyx, y-ϕˆx, y+ϕˆx, y-12,
Δlfx, yπ, lx, y.
2ϕˆx, y=ρx, y,
ρx, ygxx+1, y-gxx, y+gyx, y+1-gyx, y
2ϕˆx, ySx, y-4ϕˆx, y=ρx, ySx, y,
Sx,y=1if x+1, y,x, y,x-1, y,x, y+1,x, y-1S,0otherwise
U0ϕˆx, ϕˆy=x,yS1gxx, y-ϕˆxx, y2+x,yS2gyx, y-ϕˆyx, y2,
Ccϕˆx, ϕˆyΔyBϕˆxx, y-ΔxBϕˆyx, y=0, x, yL,
S1=x, y such that an observation gxx, y exists, S2=x, y such that an observation gyx, y exists,
Ucϕˆx, ϕˆy=U0ϕˆx, ϕˆy+λ1x,yLCcϕˆx, ϕˆy2,
U2ϕˆx, ϕˆx=x,yS1gxx, y-ϕˆxx, y2+x,yS2gyx, y-ϕˆyx, y2+λ1x,yLΔyBϕˆxx, y-ΔxBϕˆyx, y2+νx,yS1ϕˆxx, y2+νx,yS2ϕˆyx, y2.
U2ϕˆx, ϕˆyˆϕx=0, U2ϕˆx, ϕˆyˆϕy=0.
ϕˆxx, ySxx, y1-ν+ν-λ1ΔyFΔyBϕˆxx, y-ΔxBϕˆyx, y=gxx, ySxx, y, ϕˆyx, ySyx, y1-ν+ν+λ1ΔxFΔyBϕˆxx, y-ΔxBϕˆyx, y=gyx, ySyx, y,
Sxx, y=1 if x, yS1, 0 otherwise, Syx, y=1 if x, yS2, 0 otherwise,
ΔxBfx, y=fx, y-fx-1, y1xM-1f0, y-fM-1, yx=0.
Gxu, v=1+λ14 sin2πvNΦˆxu, v+λ1expj2πvN-1×1-exp-j2πuM Φˆyu, v, Gyu, v=λ1expj2πuM-11-exp-j2πvN ×Φˆxu, v+1+λ14 sin2πuMΦˆyu, v,
egxx, y, gyx, yΔyBgxx, y-ΔxBgyx, y,
ϕˆxx, y=gxx, y+ΔyFIDFTIHu, v×DFTegxx, y, gyx, y, ϕˆyx, y=gyx, y-ΔxFIDFTIHu, v×DFTegxx, y, gyx, y,
IHu, v=λ11+λ14 sin2πuM+4 sin2πvN.
ϕˆxx, y=gxx, y, ϕˆyx, y=gyx, y.
ΔxBWf0, xΔxBf0, x.
Sxx, y=1 if x, yS1,x00 otherwise,Syx, y=1 if x, yS2,y00 otherwise.
min fx=12xTAx-xTb,
Ax-b=0.
xi+1=xi+αisi,
fxi+αisiαi=0.
Ãx-b˜=0,
Hzi=ri.
Z=1H0R,
zxx, y=rxx, y+ΔyFIDFTIHu, v×DFTerxx, y, ryx, y, zyx, y=ryx, y-ΔxFIDFTIHu, v×DFTerxx, y, ryx, y,
ϕx, y=13 sinπx2M-1sinπy-10N-1+stepx, y*Sx, y, stepx, y=intx/10/3y < N/20otherwise.
gx, yWϕx, y=tan-1tanϕx, y.
gxx, y=gxx, ySxx, y, gyx, y=gyx, ySyx, y.
λ2P2xϕˆx2=λ2x,yRxΔxFΔxBϕˆxx, y2, λ2P2yϕˆy2=λ2x,yRyΔyFΔyBϕˆyx, y2,
Rx=x, yR: x+1, y, x-1, y, ×x-2, yR, Ry=x, yR: x, y+1, x, y-1, ×x, y-1R,
0×M-1, 0yN-1
0uM-1, 0vN-1
ΔxFfx, y=fx+1, y-fx, y0xM-2f0, y-fM-1, yx=M-1
ΔyFfx, y=fx, y+1-fx, y0yN-2fx, 0-fx, N-1y=N-1
ΔxBfx, y=fx, y-fx-1, y1xM-1f0, y-fM-1, yx=0
ΔyBfx, y=fx, y-fx, y-11yN-1fx, 0-fx, N-1y=0
Wf=tan-1tanf:-π, π.
incofxx, y, fyx, y=ΔyBfxx, y-ΔxBfyx, y
IHu, v=λ11+λ14 sin2πuM+4 sin2πvN.
Sx0, y=0,Syx, 0=0,rxx, y=gxx, y,ryx, y=gyx, y,ϕˆxx, y=0,ϕˆyx, y=0,t=0.
cx, y=IDFT{IHu, vDFTincorxx, y, ryx, y.
zxx, y=rxx, y+ΔyFcx, y, zyx, y=ryx, y-ΔxFcx, y.
W=zxx, yrxx, y+zyx, yryx, y.
pxx, y=zxx, y+βpxx, y, pyx, y=zyx, y+βpyx, y.
uxx, y=-λ1 ΔyFincopxx, y, pyx, y,
uxx, y=pxx, ySxx, y1-ν+v-λ1ΔyFincopxx, y, pyx, y.
uyx, y=λ1ΔxFincopxx, y, pyx, y,
uyx, y=pyx, ySyx, y1-ν+ν+λ1 ΔxFincopxx, y, pyx, y.
α=Wpxx, yuxx, y+pyx, yuyx, y.
ϕˆxx, y=ϕˆxx, y+αpxx, y, ϕˆyx, y=ϕˆyx, y+αpyx, y, rxx, y=rxx, y-αuxx, y, ryx, y=ryx, y-αuyx, y.
ϕˆx0, 0=0,
ϕˆx, y=i=0x ϕˆxi, 0+j=1y ϕˆyx, jRx, y.

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