Abstract

In interferometric tomography, two-dimensional projection extraction, i.e., wave-front retrieval from interferograms, is a critical step for a successful three-dimensional reconstruction. We analyze the difficulty of locating the first-order frequency spectrum—a difficulty inherent in interferogram phase unwrapping by means of the traditional Fourier transform. The problem becomes more serious when we deal with interferograms obtained in real experiments, which usually lack exactly parallel and equally spaced references. To overcome this difficulty, we propose the method of localized matching filtration with Gabor filters and apply it, for the first time to our knowledge, to interferogram phase demodulation. A multi-channel Gabor spatial filter set is constructed to yield the optimally matched local spatial frequency. The optimization is aimed at maximizing the L2(R2) norm of the output of the filters. At the same time, noise and opaque objects are removed. The method, applied to both simulated and real interferograms, proves to be more effective and more flexible than the conventional Fourier transformation method, especially in cases in which references are not equally spaced and fringes are seriously deformed.

© 1999 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
  3. D. Gabor, “Theory of communication,” J. Inst. Electr. Eng. 93, 429–457 (1946).
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    [CrossRef] [PubMed]
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    [CrossRef]

1990 (1)

A. C. Bovik, M. Clark, W. S. Geisler, “Multichannel texture analysis using localized spatial filters,” IEEE Trans. Pattern. Anal. Mach. Intell. 12, 55–73 (1990).
[CrossRef]

1986 (1)

1985 (1)

1982 (1)

1946 (1)

D. Gabor, “Theory of communication,” J. Inst. Electr. Eng. 93, 429–457 (1946).

Bovik, A. C.

A. C. Bovik, M. Clark, W. S. Geisler, “Multichannel texture analysis using localized spatial filters,” IEEE Trans. Pattern. Anal. Mach. Intell. 12, 55–73 (1990).
[CrossRef]

Clark, M.

A. C. Bovik, M. Clark, W. S. Geisler, “Multichannel texture analysis using localized spatial filters,” IEEE Trans. Pattern. Anal. Mach. Intell. 12, 55–73 (1990).
[CrossRef]

Daugman, J. G.

Gabor, D.

D. Gabor, “Theory of communication,” J. Inst. Electr. Eng. 93, 429–457 (1946).

Geisler, W. S.

A. C. Bovik, M. Clark, W. S. Geisler, “Multichannel texture analysis using localized spatial filters,” IEEE Trans. Pattern. Anal. Mach. Intell. 12, 55–73 (1990).
[CrossRef]

Ina, H.

Kobayashi, S.

Kreis, Thomas

Takeda, M.

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Figures (7)

Fig. 1
Fig. 1

Real part of the Gabor function. The imaginary part is only a π/2 phase lag.

Fig. 2
Fig. 2

Block diagram of Gabor multichannel matching filtering.

Fig. 3
Fig. 3

(a) Simulated interferogram with parallel and equally spaced references (64×64). (b) Simulated interferogram with a gradually increasing reference frequency. (c) Fourier frequency spectrum of the interferogram shown in (b).

Fig. 4
Fig. 4

(a) Wrapped phase map demodulated by application of the Fourier transform method to Fig. 3(a). (b) Retrieved wave front. It is clear that the result cannot represent the trend toward an increase along the x direction.

Fig. 5
Fig. 5

Configuration of a Gabor multichannel matching filter set in the 2D frequency domain.

Fig. 6
Fig. 6

(a) Modulation frequencies in the x direction (U). (b) Modulation frequencies in the y direction (V). (c) Wrapped phase map demodulated by application of the Gabor matching filtration method to Fig. 3(a). (d) Wave front retrieved from (c). It is clear that the result does represent the trend toward an increase in the x direction. Its appearance is not smooth enough because of an insufficient number of filters.

Fig. 7
Fig. 7

(a) Interferogram of a two-peak temperature field (256×256). (b) Wrapped phase map demodulated by application of the Fourier transform method to (a). (c) Wrapped phase map demodulated by application of the Gabor transform method to (a). (d) Wave front retrieved from (c).

Equations (25)

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f(x, y)=a(x, y)+b(x, y)cos[2πFx+ϵ(x, y)],
Gmn[f(x, y)]=hmn(x, y)f(x, y),
hmn(x, y)=g(x, y)exp[2πj(Umx+Vny)],
g(x, y)=12πλσ2exp-(x/λ)2+y22σ2.
(hmnf )(x, y)=Σhmn(x-s, y-t)f(s, t)dsdt,
G(F, 0)[f(x, y)]=kr(x, y)+jki(x, y).
f(x, y)=a(x, y)+2[c(x, y)cos(2πFx)+s(x, y)sin(2πFx)],
[c2(x, y)+s2(x, y)]1/2=12b(x, y),
-tan-1[s(x, y)/c(x, y)]=ϵ(x, y).
kr(x, y)=hr(x, y)f(x, y)=hr(x, y)a(x, y)+[g(x, y)c(x, y)]cos(2πFx)+[g(x, y)s(x, y)]sin(2πFx),
ki(x, y)=hi(x, y)f(x, y)=hi(x, y)a(x, y)+[g(x, y)c(x, y)]sin(2πFx)-[g(x, y)s(x, y)]cos(2πFx),
m(x, y)=[kr2(x, y)+ki2(x, y)]1/2{[g(x, y)c(x, y)]2+[g(x, y)s(x, y)]2}1/2.
ψ(x, y)=tan-1ki(x, y)kr(x, y)=2πFx-tan-1g(x, y)s(x, y)g(x, y)c(x, y).
ϵ(x, y)=-tan-1s(x, y)c(x, y)-tan-1g(x, y)s(x, y)g(x, y)c(x, y)=tan-1ki(x, y)kr(x, y)-2πFx.
f(x, y)=a(x, y)+m=1Mn=1Nfmn(x, y)Zmn(x, y)+f(x, y),
fmn(x, y)=2{cmn(x, y)cos[2π(Umxm+Vnxn)]+smn(x, y)sin[2π(Umxm+Vnyn)]},
Zmn(x, y)=1(x, y)Smn0(x, y)SmnisthesetofindicatorsofSmn.
|hmn(x, y)fkl(x, y)||hmn(x, y)fmn(x, y)|,
kmorln.
F(x, y)=(Um, Vn)if
(m, n)=argmax1iM1jN[mij(x, y)].
m(x, y)=[krmn2(x, y)+kimn2(x, y)]1/2{[g(x, y)cmn(x, y)]2+[g(x, y)smn(x, y)]2}1/2,
ϵ(x, y)tan-1kimn(x, y)krmn(x, y)-2πUmx.
F(x, y)=(Um, Vn)if
(m, n)=argmax1iM1jNgxγ, yγmij(x, y),γ>1.

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