Abstract

This paper presents a new method to obtain a wrapped phase distribution from a single interferogram with a spatial carrier modulation. The Fourier transform of the interferogram has three peaks: one is a dc peak around the origin in the Fourier domain, and the other two are carrier peaks that have information of phase modulation by an object placed in the interferometer. Since the wrapped phase can be evaluated by one of the two carrier peaks, the dc peak and the adjoint peak that is the other peak of two carrier peaks should be removed by filters. The proposed filtering process consists of two stages: dc peak filtering and adjoint peak filtering. A spectrum shift filter based on symmetrical characteristics of the spectrum is applied in both stages as a basic filter that can remove most of the undesired spectrum. An additional two filters are applied to remove the remaining spectrum. The new method can automatically isolate the carrier peak, even when the boundary of peaks is not very clear. Numerical evaluations of simulation data and experimental data demonstrate that the proposed method can successfully isolate the carrier peak.

© 2014 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  23. L.-C. Chen, H.-W. Ho, and X.-L. Nguyen, “Fourier transform profilometry (FTP) using an innovative band-pass filter for accurate 3-D surface reconstruction,” Opt. Lasers Eng. 48, 182–190 (2010).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  34. I. Gurov and M. Volynsky, “Interference fringe analysis based on recurrence computational algorithms,” Opt. Lasers Eng. 50, 514–521 (2012).
    [CrossRef]
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    [CrossRef]

2013 (2)

2012 (6)

2011 (2)

2010 (1)

L.-C. Chen, H.-W. Ho, and X.-L. Nguyen, “Fourier transform profilometry (FTP) using an innovative band-pass filter for accurate 3-D surface reconstruction,” Opt. Lasers Eng. 48, 182–190 (2010).
[CrossRef]

2008 (1)

2007 (2)

Q. Kemao, “Two-dimensional windowed Fourier transform for fringe pattern analysis: principles, applications and implementations,” Opt. Lasers Eng. 45, 304–317 (2007).
[CrossRef]

J. A. Ferrari and E. M. Frins, “Multiple phase-shifted interferograms obtained from a single interferogram with linear carrier,” Opt. Commun. 271, 59–64 (2007).
[CrossRef]

2006 (2)

2004 (1)

2001 (2)

2000 (1)

J. Yañez-Mendiola, M. Servín, and D. Malacara-Hernández, “Iterative method to obtain the wrapped phase in an interferogram with a linear carrier,” Opt. Commun. 178, 291–296 (2000).
[CrossRef]

1999 (1)

M. A. Herráez, D. Burton, and M. Lalor, “Accelerating fast Fourier transform and filtering operations in Fourier fringe analysis for accurate measurement of three-dimensional surfaces,” Opt. Lasers Eng. 31, 135–145 (1999).
[CrossRef]

1997 (2)

M. Servin, J. L. Marroquin, and F. J. Cuevas, “Demodulation of a single interferogram by use of a two-dimensional regularized phase-tracking technique,” Appl. Opt. 36, 4540–4548 (1997).
[CrossRef]

Y. Arai, S. Yokozeki, K. Shiraki, and T. Yamada, “High precision two-dimensional spatial fringe analysis method,” J. Mod. Opt. 44, 739–751 (1997).
[CrossRef]

1995 (1)

J.-F. Lin and X. Su, “Two-dimensional Fourier transform profilometry for the automatic measurement of three-dimensional object shapes,” Opt. Eng. 34, 3297–3302 (1995).
[CrossRef]

1993 (1)

1991 (1)

M. Kujawinska and J. Wójciak, “High accuracy Fourier transform fringe pattern analysis,” Opt. Lasers Eng. 14, 325–339 (1991).
[CrossRef]

1990 (1)

M. Takeda, “Spatial-carrier fringe-pattern analysis and its applications to precision interferometry and profilometry: an overview,” Ind. Metrol. 1, 79–99 (1990).

1987 (1)

1986 (2)

1983 (1)

1982 (1)

1974 (1)

R. W. Gerchberg, “Super-resolution through error energy reduction,” Opt. Acta 21, 709–720 (1974).
[CrossRef]

Arai, Y.

Y. Arai, S. Yokozeki, K. Shiraki, and T. Yamada, “High precision two-dimensional spatial fringe analysis method,” J. Mod. Opt. 44, 739–751 (1997).
[CrossRef]

Bachor, H.-A.

Bauer, T.

Bone, D. J.

Burton, D.

M. A. Herráez, D. Burton, and M. Lalor, “Accelerating fast Fourier transform and filtering operations in Fourier fringe analysis for accurate measurement of three-dimensional surfaces,” Opt. Lasers Eng. 31, 135–145 (1999).
[CrossRef]

Chen, L.-C.

L.-C. Chen, H.-W. Ho, and X.-L. Nguyen, “Fourier transform profilometry (FTP) using an innovative band-pass filter for accurate 3-D surface reconstruction,” Opt. Lasers Eng. 48, 182–190 (2010).
[CrossRef]

Creath, K.

K. Creath and G. Goldstein, “Dynamic quantitative phase imaging for biological objects using a pixelated phase mask,” Biomed. Opt. Express 3, 2866–2880 (2012).
[CrossRef]

K. Creath, “Temporal phase measurement methods,” in Interferogram Analysis: Digital Fringe Pattern Measurement Techniques, D. W. Robinson and G. T. Reid, eds. (IOP, 1993), pp. 94–140.

Cuevas, F. J.

Desse, J.-M.

Estrada, J. C.

Ferrari, J. A.

J. A. Ferrari and E. M. Frins, “Multiple phase-shifted interferograms obtained from a single interferogram with linear carrier,” Opt. Commun. 271, 59–64 (2007).
[CrossRef]

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, “Optimal (Wiener) filtering with the FFT,” in Numerical Recipes: the Art of Scientific Computing, 3rd ed. (Cambridge University, 2007), Chap. 13.3, pp. 649–652.

Frins, E. M.

J. A. Ferrari and E. M. Frins, “Multiple phase-shifted interferograms obtained from a single interferogram with linear carrier,” Opt. Commun. 271, 59–64 (2007).
[CrossRef]

Ge, Z.

Gerchberg, R. W.

R. W. Gerchberg, “Super-resolution through error energy reduction,” Opt. Acta 21, 709–720 (1974).
[CrossRef]

Goldstein, G.

Gurov, I.

I. Gurov and M. Volynsky, “Interference fringe analysis based on recurrence computational algorithms,” Opt. Lasers Eng. 50, 514–521 (2012).
[CrossRef]

Heil, J.

Heppner, J.

Herráez, M. A.

M. A. Herráez, D. Burton, and M. Lalor, “Accelerating fast Fourier transform and filtering operations in Fourier fringe analysis for accurate measurement of three-dimensional surfaces,” Opt. Lasers Eng. 31, 135–145 (1999).
[CrossRef]

Ho, H.-W.

L.-C. Chen, H.-W. Ho, and X.-L. Nguyen, “Fourier transform profilometry (FTP) using an innovative band-pass filter for accurate 3-D surface reconstruction,” Opt. Lasers Eng. 48, 182–190 (2010).
[CrossRef]

Ina, H.

Kemao, Q.

Q. Kemao, “Two-dimensional windowed Fourier transform for fringe pattern analysis: principles, applications and implementations,” Opt. Lasers Eng. 45, 304–317 (2007).
[CrossRef]

Kobayashi, F.

Kobayashi, S.

Kostianovski, S.

Kreis, T.

Kujawinska, M.

M. Kujawinska and J. Wójciak, “High accuracy Fourier transform fringe pattern analysis,” Opt. Lasers Eng. 14, 325–339 (1991).
[CrossRef]

Lalor, M.

M. A. Herráez, D. Burton, and M. Lalor, “Accelerating fast Fourier transform and filtering operations in Fourier fringe analysis for accurate measurement of three-dimensional surfaces,” Opt. Lasers Eng. 31, 135–145 (1999).
[CrossRef]

Lin, J.-F.

J.-F. Lin and X. Su, “Two-dimensional Fourier transform profilometry for the automatic measurement of three-dimensional object shapes,” Opt. Eng. 34, 3297–3302 (1995).
[CrossRef]

Ling, T.

Lipson, S. G.

Macy, W. W.

Malacara-Hernández, D.

J. Yañez-Mendiola, M. Servín, and D. Malacara-Hernández, “Iterative method to obtain the wrapped phase in an interferogram with a linear carrier,” Opt. Commun. 178, 291–296 (2000).
[CrossRef]

Marroquin, J. L.

Marroquín, J. L.

Martínez-García, A.

D. I. Serrano-García, N.-I. Toto-Arellano, A. Martínez-García, and G. R. Zurita, “Radial slope measurement of dynamic transparent samples,” J. Opt. 14, 045706 (2012).
[CrossRef]

Massig, J. H.

Matsuda, S.

Nguyen, X.-L.

L.-C. Chen, H.-W. Ho, and X.-L. Nguyen, “Fourier transform profilometry (FTP) using an innovative band-pass filter for accurate 3-D surface reconstruction,” Opt. Lasers Eng. 48, 182–190 (2010).
[CrossRef]

Nishiyama, S.

S. Tomioka and S. Nishiyama, “Phase unwrapping for noisy phase map using localized compensator,” Appl. Opt. 51, 4984–4994 (2012).
[CrossRef]

S. Tomioka and S. Nishiyama, “Nondestructive three-dimensional measurement of gas temperature distribution by phase tomography,” Proc. SPIE 8296, 829617 (2011).
[CrossRef]

Picart, P.

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, “Optimal (Wiener) filtering with the FFT,” in Numerical Recipes: the Art of Scientific Computing, 3rd ed. (Cambridge University, 2007), Chap. 13.3, pp. 649–652.

Quiroga, J. A.

Rajshekhar, G.

Rastogi, P.

Ribak, E. N.

Roddier, C.

Roddier, F.

Sandeman, R. J.

Serrano-García, D. I.

D. I. Serrano-García, N.-I. Toto-Arellano, A. Martínez-García, and G. R. Zurita, “Radial slope measurement of dynamic transparent samples,” J. Opt. 14, 045706 (2012).
[CrossRef]

Servin, M.

Servín, M.

J. C. Estrada, M. Servín, J. A. Quiroga, and J. L. Marroquín, “Path independent demodulation method for single image interferograms with closed fringes within the function space c2,” Opt. Express 14, 9687–9698 (2006).
[CrossRef]

J. Yañez-Mendiola, M. Servín, and D. Malacara-Hernández, “Iterative method to obtain the wrapped phase in an interferogram with a linear carrier,” Opt. Commun. 178, 291–296 (2000).
[CrossRef]

Shiraki, K.

Y. Arai, S. Yokozeki, K. Shiraki, and T. Yamada, “High precision two-dimensional spatial fringe analysis method,” J. Mod. Opt. 44, 739–751 (1997).
[CrossRef]

Su, X.

J.-F. Lin and X. Su, “Two-dimensional Fourier transform profilometry for the automatic measurement of three-dimensional object shapes,” Opt. Eng. 34, 3297–3302 (1995).
[CrossRef]

Sure, T.

Takeda, M.

Tankam, P.

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, “Optimal (Wiener) filtering with the FFT,” in Numerical Recipes: the Art of Scientific Computing, 3rd ed. (Cambridge University, 2007), Chap. 13.3, pp. 649–652.

Tian, C.

Tomioka, S.

S. Tomioka and S. Nishiyama, “Phase unwrapping for noisy phase map using localized compensator,” Appl. Opt. 51, 4984–4994 (2012).
[CrossRef]

S. Tomioka and S. Nishiyama, “Nondestructive three-dimensional measurement of gas temperature distribution by phase tomography,” Proc. SPIE 8296, 829617 (2011).
[CrossRef]

Toto-Arellano, N.-I.

D. I. Serrano-García, N.-I. Toto-Arellano, A. Martínez-García, and G. R. Zurita, “Radial slope measurement of dynamic transparent samples,” J. Opt. 14, 045706 (2012).
[CrossRef]

Vetterling, W. T.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, “Optimal (Wiener) filtering with the FFT,” in Numerical Recipes: the Art of Scientific Computing, 3rd ed. (Cambridge University, 2007), Chap. 13.3, pp. 649–652.

Volynsky, M.

I. Gurov and M. Volynsky, “Interference fringe analysis based on recurrence computational algorithms,” Opt. Lasers Eng. 50, 514–521 (2012).
[CrossRef]

Wei, T.

Wesner, J.

Wójciak, J.

M. Kujawinska and J. Wójciak, “High accuracy Fourier transform fringe pattern analysis,” Opt. Lasers Eng. 14, 325–339 (1991).
[CrossRef]

Wyant, J. C.

Yamada, T.

Y. Arai, S. Yokozeki, K. Shiraki, and T. Yamada, “High precision two-dimensional spatial fringe analysis method,” J. Mod. Opt. 44, 739–751 (1997).
[CrossRef]

Yañez-Mendiola, J.

J. Yañez-Mendiola, M. Servín, and D. Malacara-Hernández, “Iterative method to obtain the wrapped phase in an interferogram with a linear carrier,” Opt. Commun. 178, 291–296 (2000).
[CrossRef]

Yang, Y.

Yokozeki, S.

Y. Arai, S. Yokozeki, K. Shiraki, and T. Yamada, “High precision two-dimensional spatial fringe analysis method,” J. Mod. Opt. 44, 739–751 (1997).
[CrossRef]

Zhuo, Y.

Zurita, G. R.

D. I. Serrano-García, N.-I. Toto-Arellano, A. Martínez-García, and G. R. Zurita, “Radial slope measurement of dynamic transparent samples,” J. Opt. 14, 045706 (2012).
[CrossRef]

Appl. Opt. (12)

W. W. Macy, “Two-dimensional fringe-pattern analysis,” Appl. Opt. 22, 3898–3901 (1983).
[CrossRef]

D. J. Bone, H.-A. Bachor, and R. J. Sandeman, “Fringe-pattern analysis using a 2-D Fourier transform,” Appl. Opt. 25, 1653–1660 (1986).
[CrossRef]

S. Kostianovski, S. G. Lipson, and E. N. Ribak, “Interference microscopy and Fourier fringe analysis applied to measuring the spatial refractive-index distribution,” Appl. Opt. 32, 4744–4750 (1993).
[CrossRef]

M. Servin, J. L. Marroquin, and F. J. Cuevas, “Demodulation of a single interferogram by use of a two-dimensional regularized phase-tracking technique,” Appl. Opt. 36, 4540–4548 (1997).
[CrossRef]

C. Roddier and F. Roddier, “Interferogram analysis using Fourier transform techniques,” Appl. Opt. 26, 1668–1673 (1987).
[CrossRef]

Z. Ge, F. Kobayashi, S. Matsuda, and M. Takeda, “Coordinate-transform technique for closed-fringe analysis by the Fourier-transform method,” Appl. Opt. 40, 1649–1657 (2001).
[CrossRef]

J. H. Massig and J. Heppner, “Fringe-pattern analysis with high accuracy by use of the Fourier-transform method: theory and experimental tests,” Appl. Opt. 40, 2081–2088 (2001).
[CrossRef]

J. Heil, T. Bauer, T. Sure, and J. Wesner, “Iterative full-bandwidth wavefront reconstruction from a set of low-tilt Fizeau interferograms for high-numerical-aperture surface characterization,” Appl. Opt. 45, 4270–4283 (2006).
[CrossRef]

S. Tomioka and S. Nishiyama, “Phase unwrapping for noisy phase map using localized compensator,” Appl. Opt. 51, 4984–4994 (2012).
[CrossRef]

G. Rajshekhar and P. Rastogi, “Multiple signal classification technique for phase estimation from a fringe pattern,” Appl. Opt. 51, 5869–5875 (2012).
[CrossRef]

J. C. Wyant, “Computerized interferometric surface measurements,” Appl. Opt. 52, 1–8 (2013).
[CrossRef]

M. Takeda, “Fourier fringe analysis and its application to metrology of extreme physical phenomena: a review,” Appl. Opt. 52, 20–29 (2013).
[CrossRef]

Biomed. Opt. Express (1)

Ind. Metrol. (1)

M. Takeda, “Spatial-carrier fringe-pattern analysis and its applications to precision interferometry and profilometry: an overview,” Ind. Metrol. 1, 79–99 (1990).

J. Mod. Opt. (1)

Y. Arai, S. Yokozeki, K. Shiraki, and T. Yamada, “High precision two-dimensional spatial fringe analysis method,” J. Mod. Opt. 44, 739–751 (1997).
[CrossRef]

J. Opt. (1)

D. I. Serrano-García, N.-I. Toto-Arellano, A. Martínez-García, and G. R. Zurita, “Radial slope measurement of dynamic transparent samples,” J. Opt. 14, 045706 (2012).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (2)

Opt. Acta (1)

R. W. Gerchberg, “Super-resolution through error energy reduction,” Opt. Acta 21, 709–720 (1974).
[CrossRef]

Opt. Commun. (2)

J. Yañez-Mendiola, M. Servín, and D. Malacara-Hernández, “Iterative method to obtain the wrapped phase in an interferogram with a linear carrier,” Opt. Commun. 178, 291–296 (2000).
[CrossRef]

J. A. Ferrari and E. M. Frins, “Multiple phase-shifted interferograms obtained from a single interferogram with linear carrier,” Opt. Commun. 271, 59–64 (2007).
[CrossRef]

Opt. Eng. (1)

J.-F. Lin and X. Su, “Two-dimensional Fourier transform profilometry for the automatic measurement of three-dimensional object shapes,” Opt. Eng. 34, 3297–3302 (1995).
[CrossRef]

Opt. Express (2)

Opt. Lasers Eng. (5)

M. Kujawinska and J. Wójciak, “High accuracy Fourier transform fringe pattern analysis,” Opt. Lasers Eng. 14, 325–339 (1991).
[CrossRef]

L.-C. Chen, H.-W. Ho, and X.-L. Nguyen, “Fourier transform profilometry (FTP) using an innovative band-pass filter for accurate 3-D surface reconstruction,” Opt. Lasers Eng. 48, 182–190 (2010).
[CrossRef]

M. A. Herráez, D. Burton, and M. Lalor, “Accelerating fast Fourier transform and filtering operations in Fourier fringe analysis for accurate measurement of three-dimensional surfaces,” Opt. Lasers Eng. 31, 135–145 (1999).
[CrossRef]

Q. Kemao, “Two-dimensional windowed Fourier transform for fringe pattern analysis: principles, applications and implementations,” Opt. Lasers Eng. 45, 304–317 (2007).
[CrossRef]

I. Gurov and M. Volynsky, “Interference fringe analysis based on recurrence computational algorithms,” Opt. Lasers Eng. 50, 514–521 (2012).
[CrossRef]

Opt. Lett. (2)

Proc. SPIE (1)

S. Tomioka and S. Nishiyama, “Nondestructive three-dimensional measurement of gas temperature distribution by phase tomography,” Proc. SPIE 8296, 829617 (2011).
[CrossRef]

Other (3)

D. Malacara, ed., “Interferometric optical profilers,” in Optical Shop Testing, 3rd ed. (Wiley, 2007), Chap. 15.4, pp. 695–702.

K. Creath, “Temporal phase measurement methods,” in Interferogram Analysis: Digital Fringe Pattern Measurement Techniques, D. W. Robinson and G. T. Reid, eds. (IOP, 1993), pp. 94–140.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, “Optimal (Wiener) filtering with the FFT,” in Numerical Recipes: the Art of Scientific Computing, 3rd ed. (Cambridge University, 2007), Chap. 13.3, pp. 649–652.

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Figures (7)

Fig. 1.
Fig. 1.

Scheme of dc peak isolation by spectrum shift technique.

Fig. 2.
Fig. 2.

Scheme of two-dimensional dc peak isolation: (a) P(k), (b) D1(k) by spectrum shifting, (c) Pos{D1(k)}, and (d) |A(k)|2 by a single connected support region filter. The number of small circles represents the spectrum density. The filled small circles and open ones express positive and negative spectrum, respectively. The enclosed circles represent dc peaks. Triplets of squares represent asymmetrical carrier peaks in which the carrier frequency is positioned at the center of each triplet.

Fig. 3.
Fig. 3.

Scheme of two-dimensional carrier peak isolation: (a) Ptwin(k), (b) DCs(k) by spectrum shifting, (c) PCs(k), and (d) |Cs(k)|2 by a symmetrical point comparison filter. The triplet of squares in kx>0 and that in kx<0 represent a carrier peak spectrum and an adjoint one, respectively. The carrier frequency, ±s, is positioned at the center of each triplet, and each spectrum is asymmetric with respect to the carrier frequency.

Fig. 4.
Fig. 4.

Result of peak isolation and phase unwrapping for simulation data with a hash-sign-shaped spectrum. The input data for filtering are shown in (a)–(d); the images in (e)–(i) show the sequence of the filtering process; (j) and (k) show the wrapped phase with carrier modulation and that without modulation, respectively; and (l) is the unwrapped phase. The images shown in (m)–(o) are the results when using a half-plane filter, and those in (p)–(r) are the ones when using a circular bandpass filter. In the spectra, the displayed range for each axis is reduced to 50% of the range determined by the sampling theorem where the origin is the center of the image, and the brightness is converted to enhance the small spectra by an arc-tangent function. The filtering weight to obtain (e) is γ=1.

Fig. 5.
Fig. 5.

Result of peak isolation and phase unwrapping for simulation data with a partially closed interferogram. The assumed true continuous phase, ϕTrue(r), and its wrapped phase, ϕwTrue, are shown at the top. The bottom tables show the simulated interferogram as the input of filters, the results of the spectrum shift filter, and the results of the half-plane filter. Each row shows the dependency of the normalized carrier frequency, k^bg. The input image for each k^bg consists of three images: ibg(r) (the background fringe), i(r) (the interferogram with noise), and |I(k)| [the spectrum of i(r)]. The results for both filters consist of three images: |Cs(k)| (the filtrated spectrum), ϕw(r) (the wrapped phase), and ϕ(r) (the unwrapped phase). The scales of the brightness of each figure are shown at the top. In the images of the spectra, the displayed range for each axis is reduced to 25% of the range determined by the sampling theorem. The filtering weight in the spectrum shift filter is not considered, i.e., γ=1.

Fig. 6.
Fig. 6.

Error of unwrapped phase for partially closed fringe simulation. The horizontal axis shows the normalized carrier frequency, and the vertical axis shows the root mean square of the error between the unwrapped phase and the true phase, i.e., (|ϕ(r)ϕ(r)|2r)1/2.

Fig. 7.
Fig. 7.

Phase extraction for an experimental fringe pattern. The fringe pattern is obtained by using an interferometer with an object that is flame of a horizontally placed gas burner. (a) Schematic view of image configuration. (b) Observed fringe pattern, i(r), with 512×512 pixels; the circle shows the laser beam profile with 50 mm diameter, and the bottom figure is a magnified image for the analyses with 256×256 pixels, whose region is determined by the box in the top image. (c) Spectrum, |I(k)|, of (b). (d) Background fringe pattern without object. (e) Filtered results by the spectrum shift filter; |Cs(k)| is the filtered spectrum, ϕw(r) is the wrapped phase, and ϕ(r) is the unwrapped phase. (f) Filtered results by the half-plane filter. In the spectral images, the horizontal and vertical ranges are reduced to 25% of the range defined by the sampling theorem. The filtering weight in the spectrum shift filter is not considered, i.e., γ=1.

Equations (66)

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i(r)=a(r)+b(r)cosϕs(r),
ϕs(r)=ϕ(r)+s·r,
c^(r)=12b(r)ejϕ(r),
i(r)=a(r)+c^(r)ejs·r+c^*(r)ejs·r,
F(k)F{f^(r)}=f^(r)ejk·rdr
I(k)=A(k)+C(ks)+C*(ks),
F1{C(ks)}=c^(r)ejs·r=12b(r)ej(ϕ+s·r),
ϕws(r)arg(F1{C(ks)})=W{ϕ(r)+s·r},
ϕw(r)W{ϕws(r)s·r},
ϕ(r)=U{ϕw(r)},
|s|>max{|ϕ(r)|}.
P(k)|I(k)|2PA(k)+PC(ks)+PC(ks),
P(k2s)PA(k2s)+PC(k3s)+PC(k+s),
P(k+2s)PA(k+2s)+PC(k+s)+PC(k3s).
D1(k)P(k)(P(k2s)+P(k+2s)).
PA(k)=PA(k),
PC(ks)=PC(k±s).
D1(k)PA(k)PA(k2s)PA(k+2s)PC(k3s)PC(k3s).
Pos{D1(k)}PA(k),
Pos{f}max{f,0},
P(k±2s)=|F{i(r)e2js·r}|2.
F(k)=F*(k),
PF(k)=PF(k).
c^(r)=cr(r)+jci(r)(c^C,cr,ciR).
Cr(k)F{cr(r)},Ci(k)F{ci(r)},
C(k)=Cr(k)+jCi(k),
C*(k)=Cr*(k)jCi*(k),
C(k)=Cr*(k)+jCi*(k).
PC(k)PC(k).
Cr(k)=|Cr(k)|ejΘr(k),Ci(k)=|Ci(k)|ejΘi(k).
PC(k)=PCe(k)+PCo(k),
PC(k)=PCe(k)PCo(k),
PCe(k)|Cr(k)|2+|Ci(k)|20,
PCo(k)2|Cr(k)||Ci(k)|sin(Θr(k)Θi(k)).
PCe(k)=PCe(k),PCo(k)=PCo(k).
PC(ks)=PCe(ks)+PCo(ks),
PC(k±s)=PCe(ks)PCo(ks).
PCe(k±s)=PCe(ks),
PCo(k±s)=PCo(ks).
D1(k)PA(k)PA(k2s)PA(k+2s)PC(k3s)PC(k3s)+2PCo(ks)+2PCo(ks).
Pos{D1(k)}PA(k)+2Pos{PCo(ks)}+2Pos{PCo(ks)}.
Dγ(k)=P(k)γ[P(k2s)+P(k+2s)].
A(k)=Pos{Dγ(k)}P(k)I(k),
S˜(k)=Ψ(k)O(k),
Ψ(k)=|S(k)|2|S(k)|2+|N(k)|2Pos{|O(k)|2|N(k)|2}|O(k)|2Ψ{|O(k)|2,|N(k)|2}.
Ib(j)=|A(k)|kLb(j),
σb(j)=|A(k)|2kLb(j)(Ib(j))2,
Lb(j+1)={k|0<|A(k)|<Ib(j)+3σb(j)}.
A(k)={Ψ{|A(k)|2,Ib2}A(k),kL0,0,kL0.
Ptwin(k)=P(k)PA(k),
Itwin(k)=Ψ{P(k),PA(k)}I(k),
Ptwin(k)PC(ks)+PC(ks),
Ptwin(k+2s)PC(k+s)+PC(k3s).
DCs(k)Ptwin(k)Ptwin(k+2s)PC(ks)+2PCo(ks)PC(k3s).
PCs(k)=Pos{DCs(k)}PC(ks)+2Pos{PCo(ks)}.
Cs(k)=Pos{DCs(k)}Ptwin(k)Itwin(k).
Δs(k)Pos{DCs(k)}Pos{DCs(k)}.
Lc={k|Δs(k)>0},
La={k|Δs(k)<0},
Cs(k)={Cs(k),kLc,0,kLc.
k^bgsmax{|ϕTrue|}=e2σϕ1ϕ1s.
GI(r;rc,σ)e((xxc)2+(yyc)2)/σ2,
GE(r;rc,σ,θ)e(X2(r;rc,θ)/σX2+Y2(r;rc,θ)/σY2),
i(r)=|fref(r)+fobj(r)|2+N[n¯,σn],fref(r)=fmaxrefGI(r;rfc,σf),fobj(r)=fmaxobjGI(r;rfc,σf)ej(s·r+ϕ(r)),ϕ(r)=n=1NϕϕnGE(r;rϕn,σϕn,θϕn),
(fmaxref,fmaxobj)=(1,0.9),(n¯,σn)=(0,0),(rfc,σf)=((128,128),200),(Nsx,Nsy)=(10.4,10.4),ϕ(12)=±4π(rϕ1c,σϕ1,θϕ1)=((128,108),(240,10),π6),(rϕ2c,σϕ2,θϕ2)=((108,128),(10,240),0).
(fmaxref,fmaxobj)=(1,1),(n¯,σn)=(0,0.3),(rfc,σf)=((128,128),128),ϕ1=10π(rϕ1c,σϕ1)=((128,128),(100,100)),

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