Abstract

This paper presents a method of significantly improving the previously proposed simple, flexible, and accurate phase retrieval algorithm for the random phase-shifting interferometry named HEFS [ K. Yatabe, J. Opt. Soc. Am. A 34, 87 (2017)]. Although its effectiveness and performance were confirmed by numerical experiments in the original paper, it is found that the algorithm may not work properly if observed fringe images contains untrusted (extremely noisy) pixels. In this paper, a method of avoiding such untrusted pixels within the estimation processes of HEFS is proposed for the practical use of the algorithm. In addition to the proposal, an experiment of measuring a sound field in air was conducted to show the performance for real data, where the proposed improvement is critical for that situation. MATLAB codes (which can be downloaded from http://goo.gl/upcsFe) are provided within the paper to aid understanding the main concept of the proposed methods.

© 2017 Optical Society of America

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2018 (1)

Y. Koyano, K. Yatabe, and Y. Oikawa, “Infinite-dimensional SVD for revealing microphone array’s characteristics,” Appl. Acoust. 129, 116–125 (2018).
[Crossref]

2017 (4)

N. Chitanont, K. Yatabe, K. Ishikawa, and Y. Oikawa, “Spatio-temporal filter bank for visualizing audible sound field by Schlieren method,” Appl. Acoust. 115, 109–120 (2017).
[Crossref]

K. Yatabe, K. Ishikawa, and Y. Oikawa, “Acousto-optic back-projection: Physical-model-based sound field reconstruction from optical projections,” J. Sound Vib. 394, 171–184 (2017).
[Crossref]

K. Ishikawa, K. Yatabe, Y. Ikeda, Y. Oikawa, T. Onuma, H. Niwa, and M. Yoshii, “Interferometric imaging of acoustical phenomena using high-speed polarization camera and 4-step parallel phase-shifting technique,” Proc. SPIE 10328, 103280I (2017).

K. Yatabe, K. Ishikawa, and Y. Oikawa, “Simple, flexible, and accurate phase retrieval method for generalized phase-shifting interferometry,” J. Opt. Soc. Am. A 34, 87–96 (2017).
[Crossref]

2016 (11)

K. Yatabe, K. Ishikawa, and Y. Oikawa, “Improving principal component analysis based phase extraction method for phase-shifting interferometry by integrating spatial information,” Opt. Express 24, 22881–22891 (2016).
[Crossref] [PubMed]

F. Liu, Y. Wu, F. Wu, and W. Song, “Generalized phase shifting interferometry based on Lissajous calibration technology,” Opt. Lasers Eng. 83, 106–115 (2016).
[Crossref]

F. Liu, J. Wang, Y. Wu, F. Wu, M. Trusiak, K. Patorski, Y. Wan, Q. Chen, and X. Hou, “Simultaneous extraction of phase and phase shift from two interferograms using Lissajous figure and ellipse fitting technology with Hilbert–Huang prefiltering,” J. Opt. 18, 105604 (2016).
[Crossref]

A. V. Fantin, D. P. Willemann, M. E. Benedet, and A. G. Albertazzi, “Robust method to improve the quality of shearographic phase maps obtained in harsh environments,” Appl. Opt. 55, 1318–1323 (2016).
[Crossref] [PubMed]

J. Deng, D. Wu, K. Wang, and J. Vargas, “Precise phase retrieval under harsh conditions by constructing new connected interferograms,” Sci. Rep. 6, 24416 (2016).
[Crossref] [PubMed]

Y. Xu, Y. Wang, Y. Ji, H. Han, and W. Jin, “Three-frame generalized phase-shifting interferometry by a Euclidean matrix norm algorithm,” Opt. Lasers Eng. 84, 89–95 (2016).
[Crossref]

X. Xu, X. Lu, J. Tian, J. Shou, D. Zheng, and L. Zhong, “Random phase-shifting interferometry based on independent component analysis,” Opt. Commun. 370, 75–80 (2016).
[Crossref]

K. Ishikawa, K. Yatabe, Y. Ikeda, Y. Oikawa, T. Onuma, H. Niwa, and M. Yoshii, “Optical sensing of sound fields: Non-contact, quantitative, and single-shot imaging of sound using high-speed polarization camera,” Proc. Meet. Acoust. (POMA) 29, 030005 (2016).
[Crossref]

K. Yatabe, K. Ishikawa, and Y. Oikawa, “Compensation of fringe distortion for phase-shifting three-dimensional shape measurement by inverse map estimation,” Appl. Opt. 55, 6017–6024 (2016).
[Crossref] [PubMed]

K. Yatabe and Y. Oikawa, “Convex optimization based windowed Fourier filtering with multiple windows for wrapped phase denoising,” Appl. Opt.,  55, 4632–4641 (2016).
[Crossref] [PubMed]

K. Ishikawa, K. Yatabe, N. Chitanont, Y. Ikeda, Y. Oikawa, T. Onuma, H. Niwa, and M. Yoshii, “High-speed imaging of sound using parallel phase-shifting interferometry,” Opt. Express 24, 12922–12932 (2016).
[Crossref] [PubMed]

2015 (5)

2014 (3)

A. Albertazzi, A. V. Fantin, D. P. Willemann, and M. E. Benedet, “Phase maps retrieval from sequences of phase shifted images with unknown phase steps using generalized N-dimensional Lissajous figures — principles and applications,” Int. J. Optomechatronics 8, 340–356 (2014).
[Crossref]

J. Deng, L. Zhong, H. Wang, H. Wang, W. Zhang, F. Zhang, S. Ma, and X. Lu, “1-norm character of phase shifting interferograms and its application in phase shift extraction,” Opt. Commun. 316, 156–160 (2014).
[Crossref]

T. Onuma and Y. Otani, “A development of two-dimensional birefringence distribution measurement system with a sampling rate of 1.3 MHz,” Opt. Commun. 315, 69–73 (2014).
[Crossref]

2013 (5)

J. Deng, H. Wang, D. Zhang, L. Zhong, J. Fan, and X. Lu, “Phase shift extraction algorithm based on Euclidean matrix norm,” Opt. Lett. 38, 1506–1508 (2013).
[Crossref] [PubMed]

H. Guo and Z. Zhang, “Phase shift estimation from variances of fringe pattern differences,” Appl. Opt. 52, 6572–6578 (2013).
[Crossref] [PubMed]

R. Juarez-Salazar, C. Robledo-Sánchez, C. Meneses-Fabian, F. Guerrero-Sánchez, and L. A. Aguilar, “Generalized phase-shifting interferometry by parameter estimation with the least squares method,” Opt. Lasers Eng. 51, 626–632 (2013).
[Crossref]

J. Vargas, C. Sorzano, J. Estrada, and J. Carazo, “Generalization of the principal component analysis algorithm for interferometry,” Opt. Commun. 286, 130–134 (2013).
[Crossref]

J. Vargas and C. Sorzano, “Quadrature component analysis for interferometry,” Opt. Lasers Eng. 51, 637–641 (2013).
[Crossref]

2011 (5)

2009 (1)

2008 (1)

2007 (1)

2005 (2)

2004 (3)

Z. Wang and B. Han, “Advanced iterative algorithm for phase extraction of randomly phase-shifted interferograms,” Opt. Lett. 29, 1671–1673 (2004).
[Crossref] [PubMed]

Y. Awatsuji, M. Sasada, and T. Kubota, “Parallel quasi-phase-shifting digital holography,” Appl. Phys. Lett. 85, 1069–1071 (2004).
[Crossref]

J. E. Millerd, N. J. Brock, J. B. Hayes, M. B. North-Morris, M. Novak, and J. C. Wyant, “Pixelated phase-mask dynamic interferometer,” Proc. SPIE 5531, 304–314 (2004).
[Crossref]

1995 (1)

I.-B. Kong and S.-W. Kim, “General algorithm of phase-shifting interferometry by iterative least-squares fitting,” Opt. Eng. 34, 183–188 (1995).
[Crossref]

1994 (1)

C. T. Farrell and M. A. Player, “Phase-step insensitive algorithms for phase-shifting interferometry,” Meas. Sci. Technol. 5, 648 (1994).
[Crossref]

1992 (1)

C. T. Farrell and M. A. Player, “Phase step measurement and variable step algorithms in phase-shifting interferometry,” Meas. Sci. Technol. 3, 953–958 (1992).
[Crossref]

1991 (2)

G. Lai and T. Yatagai, “Generalized phase-shifting interferometry,” J. Opt. Soc. Am. A 8, 822–827 (1991).
[Crossref]

K. Okada, A. Sato, and J. Tsujiuchi, “Simultaneous calculation of phase distribution and scanning phase shift in phase shifting interferometry,” Opt. Commun. 84, 118–124 (1991).
[Crossref]

1988 (1)

Aguilar, L. A.

R. Juarez-Salazar, C. Robledo-Sánchez, C. Meneses-Fabian, F. Guerrero-Sánchez, and L. A. Aguilar, “Generalized phase-shifting interferometry by parameter estimation with the least squares method,” Opt. Lasers Eng. 51, 626–632 (2013).
[Crossref]

Albertazzi, A.

A. Albertazzi, A. V. Fantin, D. P. Willemann, and M. E. Benedet, “Phase maps retrieval from sequences of phase shifted images with unknown phase steps using generalized N-dimensional Lissajous figures — principles and applications,” Int. J. Optomechatronics 8, 340–356 (2014).
[Crossref]

Albertazzi, A. G.

Awatsuji, Y.

Y. Awatsuji, M. Sasada, and T. Kubota, “Parallel quasi-phase-shifting digital holography,” Appl. Phys. Lett. 85, 1069–1071 (2004).
[Crossref]

Belenguer, T.

Benedet, M. E.

A. V. Fantin, D. P. Willemann, M. E. Benedet, and A. G. Albertazzi, “Robust method to improve the quality of shearographic phase maps obtained in harsh environments,” Appl. Opt. 55, 1318–1323 (2016).
[Crossref] [PubMed]

A. Albertazzi, A. V. Fantin, D. P. Willemann, and M. E. Benedet, “Phase maps retrieval from sequences of phase shifted images with unknown phase steps using generalized N-dimensional Lissajous figures — principles and applications,” Int. J. Optomechatronics 8, 340–356 (2014).
[Crossref]

Brock, N.

Brock, N. J.

J. E. Millerd, N. J. Brock, J. B. Hayes, M. B. North-Morris, M. Novak, and J. C. Wyant, “Pixelated phase-mask dynamic interferometer,” Proc. SPIE 5531, 304–314 (2004).
[Crossref]

Cai, L. Z.

Carazo, J.

J. Vargas, C. Sorzano, J. Estrada, and J. Carazo, “Generalization of the principal component analysis algorithm for interferometry,” Opt. Commun. 286, 130–134 (2013).
[Crossref]

Chai, L.

Chen, M.

Chen, Q.

F. Liu, J. Wang, Y. Wu, F. Wu, M. Trusiak, K. Patorski, Y. Wan, Q. Chen, and X. Hou, “Simultaneous extraction of phase and phase shift from two interferograms using Lissajous figure and ellipse fitting technology with Hilbert–Huang prefiltering,” J. Opt. 18, 105604 (2016).
[Crossref]

Chen, W.

Cheng, X. C.

Chitanont, N.

N. Chitanont, K. Yatabe, K. Ishikawa, and Y. Oikawa, “Spatio-temporal filter bank for visualizing audible sound field by Schlieren method,” Appl. Acoust. 115, 109–120 (2017).
[Crossref]

K. Ishikawa, K. Yatabe, N. Chitanont, Y. Ikeda, Y. Oikawa, T. Onuma, H. Niwa, and M. Yoshii, “High-speed imaging of sound using parallel phase-shifting interferometry,” Opt. Express 24, 12922–12932 (2016).
[Crossref] [PubMed]

Deng, J.

J. Deng, D. Wu, K. Wang, and J. Vargas, “Precise phase retrieval under harsh conditions by constructing new connected interferograms,” Sci. Rep. 6, 24416 (2016).
[Crossref] [PubMed]

J. Deng, K. Wang, D. Wu, X. Lv, C. Li, J. Hao, J. Qin, and W. Chen, “Advanced principal component analysis method for phase reconstruction,” Opt. Express 23, 12222–12231 (2015).
[Crossref] [PubMed]

J. Deng, L. Zhong, H. Wang, H. Wang, W. Zhang, F. Zhang, S. Ma, and X. Lu, “1-norm character of phase shifting interferograms and its application in phase shift extraction,” Opt. Commun. 316, 156–160 (2014).
[Crossref]

J. Deng, H. Wang, D. Zhang, L. Zhong, J. Fan, and X. Lu, “Phase shift extraction algorithm based on Euclidean matrix norm,” Opt. Lett. 38, 1506–1508 (2013).
[Crossref] [PubMed]

Dong, G. Y.

Estrada, J.

J. Vargas, C. Sorzano, J. Estrada, and J. Carazo, “Generalization of the principal component analysis algorithm for interferometry,” Opt. Commun. 286, 130–134 (2013).
[Crossref]

Fan, J.

Fantin, A. V.

A. V. Fantin, D. P. Willemann, M. E. Benedet, and A. G. Albertazzi, “Robust method to improve the quality of shearographic phase maps obtained in harsh environments,” Appl. Opt. 55, 1318–1323 (2016).
[Crossref] [PubMed]

A. Albertazzi, A. V. Fantin, D. P. Willemann, and M. E. Benedet, “Phase maps retrieval from sequences of phase shifted images with unknown phase steps using generalized N-dimensional Lissajous figures — principles and applications,” Int. J. Optomechatronics 8, 340–356 (2014).
[Crossref]

Farrell, C. T.

C. T. Farrell and M. A. Player, “Phase-step insensitive algorithms for phase-shifting interferometry,” Meas. Sci. Technol. 5, 648 (1994).
[Crossref]

C. T. Farrell and M. A. Player, “Phase step measurement and variable step algorithms in phase-shifting interferometry,” Meas. Sci. Technol. 3, 953–958 (1992).
[Crossref]

Gao, P.

Geist, E.

Guerrero-Sánchez, F.

R. Juarez-Salazar, C. Robledo-Sánchez, C. Meneses-Fabian, F. Guerrero-Sánchez, and L. A. Aguilar, “Generalized phase-shifting interferometry by parameter estimation with the least squares method,” Opt. Lasers Eng. 51, 626–632 (2013).
[Crossref]

Guo, H.

Han, B.

Han, H.

Y. Xu, Y. Wang, Y. Ji, H. Han, and W. Jin, “Three-frame generalized phase-shifting interferometry by a Euclidean matrix norm algorithm,” Opt. Lasers Eng. 84, 89–95 (2016).
[Crossref]

Hao, J.

Harder, I.

Hayes, J.

Hayes, J. B.

J. E. Millerd, N. J. Brock, J. B. Hayes, M. B. North-Morris, M. Novak, and J. C. Wyant, “Pixelated phase-mask dynamic interferometer,” Proc. SPIE 5531, 304–314 (2004).
[Crossref]

He, J.

Q. Liu, Y. Wang, J. He, and F. Ji, “Phase shift extraction and wavefront retrieval from interferograms with background and contrast fluctuations,” J. Opt. 17, 025704 (2015).
[Crossref]

Hou, X.

F. Liu, J. Wang, Y. Wu, F. Wu, M. Trusiak, K. Patorski, Y. Wan, Q. Chen, and X. Hou, “Simultaneous extraction of phase and phase shift from two interferograms using Lissajous figure and ellipse fitting technology with Hilbert–Huang prefiltering,” J. Opt. 18, 105604 (2016).
[Crossref]

Ikeda, Y.

K. Ishikawa, K. Yatabe, Y. Ikeda, Y. Oikawa, T. Onuma, H. Niwa, and M. Yoshii, “Interferometric imaging of acoustical phenomena using high-speed polarization camera and 4-step parallel phase-shifting technique,” Proc. SPIE 10328, 103280I (2017).

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T. Yatagai, B. J. Jackin, A. Ono, K. Kiyohara, M. Noguchi, M. Yoshii, M. Kiyohara, H. Niwa, K. Ikuo, and T. Onuma, “Instantaneous phase-shifting fizeau interferometry with high-speed pixelated phase-mask camera,” Proc. SPIE 9660, 966018 (2015).
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K. Yatabe, K. Ishikawa, and Y. Oikawa, “Acousto-optic back-projection: Physical-model-based sound field reconstruction from optical projections,” J. Sound Vib. 394, 171–184 (2017).
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K. Yatabe, K. Ishikawa, and Y. Oikawa, “Simple, flexible, and accurate phase retrieval method for generalized phase-shifting interferometry,” J. Opt. Soc. Am. A 34, 87–96 (2017).
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K. Yatabe, K. Ishikawa, and Y. Oikawa, “Compensation of fringe distortion for phase-shifting three-dimensional shape measurement by inverse map estimation,” Appl. Opt. 55, 6017–6024 (2016).
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Y. Koyano, K. Yatabe, and Y. Oikawa, “Infinite-dimensional SVD for revealing microphone array’s characteristics,” Appl. Acoust. 129, 116–125 (2018).
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K. Yatabe, K. Ishikawa, and Y. Oikawa, “Improving principal component analysis based phase extraction method for phase-shifting interferometry by integrating spatial information,” Opt. Express 24, 22881–22891 (2016).
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K. Ishikawa, K. Yatabe, Y. Ikeda, Y. Oikawa, T. Onuma, H. Niwa, and M. Yoshii, “Optical sensing of sound fields: Non-contact, quantitative, and single-shot imaging of sound using high-speed polarization camera,” Proc. Meet. Acoust. (POMA) 29, 030005 (2016).
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T. Yatagai, B. J. Jackin, A. Ono, K. Kiyohara, M. Noguchi, M. Yoshii, M. Kiyohara, H. Niwa, K. Ikuo, and T. Onuma, “Instantaneous phase-shifting fizeau interferometry with high-speed pixelated phase-mask camera,” Proc. SPIE 9660, 966018 (2015).
[Crossref]

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Sci. Rep. (1)

J. Deng, D. Wu, K. Wang, and J. Vargas, “Precise phase retrieval under harsh conditions by constructing new connected interferograms,” Sci. Rep. 6, 24416 (2016).
[Crossref] [PubMed]

Other (3)

K. Kanatani, Y. Sugaya, and Y. Kanazawa, Ellipse Fitting for Computer Vision: Implementation and Applications, Synthesis Lectures on Computer Vision (Morgan & Claypool Publishers, 2016).

Y. Oikawa, K. Yatabe, K. Ishikawa, and Y. Ikeda, “Optical sound field measurement and imaging using laser and high-speed camera,” Proc. 45th Int. Congr. Noise Control Eng. (INTER-NOISE 2016) pp. 258–266 (2016).

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[Crossref]

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Figures (11)

Fig. 1
Fig. 1 MATLAB implementation of the improved HEFS. Only a subset of the data matrix D, which is named as Dsub, is used for the parameter estimation. In addition to HEFS, a code for constructing the extended data matrix is included at the bottom right, where the input variable X is an Nv × Nh × M three-dimensional array which is obtained by concatenating M fringe images, whose size is Nv × Nh, along the third dimension, and the shifting position is an L × 2 matrix consisting of the indices relative to the center pixel (for example, the shifting position corresponding to Eq. (11) is obtained by vertically concatenating [0, 0], [1, 0], [−1, 0], [0, 1], [0, −1], [1, 1], [1, −1], [−1, 1], and [−1, −1]).
Fig. 2
Fig. 2 MATLAB implementation of the proposed SNR-based pixel selection method. Input variable D is the data matrix, and threshold is a real number between 0 and 1.
Fig. 3
Fig. 3 Illustration of the experimental setup. A polarized interferometer is combined with a polarized high-speed camera for realizing the high-speed PPSI. A loudspeaker was placed between the optical flats which reflect the object and reference lights.
Fig. 4
Fig. 4 The loudspeaker used in the experiment (110 × 180 × 130 mm). The red dotted circle roughly indicate the observable area apparent in the measured results.
Fig. 5
Fig. 5 Illustration of recovering procedure of the measured sound field. Four fringe images were observed simultaneously by PPSI. Phase retrieval algorithms were applied to each time instance for obtaining time-sequential phase maps, and the time-directional filtering was utilized to eliminate time-invariant components unrelated to sound as in [42,43].
Fig. 6
Fig. 6 Visualized sound field obtained by PPSI and the traditional four-step algorithm. Eight consecutive time instances of the results were shown, where time evolves from left to right (from top left to bottom right).
Fig. 7
Fig. 7 Visualized sound field obtained by PPSI and the ordinary HEFS [32]. The untrusted pixels due to the shadow and unobservable area caused huge error on the phase maps.
Fig. 8
Fig. 8 Selected pixels used for the improved HEFS. The left figure shows sum of the four fringe images m I [ m ] which approximates the background illumination, and the red rectangle is the manually selected region. The center figure shows estimated SNR normalized by its maximum. The right figure shows the pixels chosen by the proposed SNR-based selection method with ϒ = 0.1 [see Eq. (30)], where the yellow pixels are the selected ones.
Fig. 9
Fig. 9 Visualized sound field obtained by PPSI and the improved HEFS. The selected pixels used for the parameter estimations in Section 3 were chosen manually (Fig. 8 left).
Fig. 10
Fig. 10 Visualized sound field obtained by PPSI and the improved HEFS. The selected pixels used for the parameter estimations in Section 3 were chosen automatically by the SNR-based pixel selection method proposed in Section 4 with ϒ = 0.1 (Fig. 8 right).
Fig. 11
Fig. 11 One-dimensional plot of the obtained results by each method. Each line was extracted vertically from each figure in Figs. 6, 9 and 10, where the horizontal position is the center.

Equations (31)

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I i , j [ m ] = B i , j + A i , j cos ( ϕ i , j + δ [ m ] ) ,
d i , j = [ I i , j [ 1 ] , I i , j [ 2 ] , , I i , j [ M ] ] T ,
I i , j [ m ] = B i , j + A i , j [ cos ( ϕ i , j ) cos ( δ [ m ] ) sin ( ϕ i , j ) sin ( δ [ m ] ) ] ,
d i , j = B i , j o + C i , j c + S i , j s ,
c = [ cos ( δ [ 1 ] ) , cos ( δ [ 2 ] ) , , cos ( δ [ M ] ) ] T ,
s = [ sin ( δ [ 1 ] ) , sin ( δ [ 2 ] ) , , sin ( δ [ M ] ) ] T .
ϕ i , j = Arg [ C i , j ι S i , j ] = Arg [ A i , j { cos ( ϕ i , j ) + ι sin ( ϕ i , j ) } ] ,
D = [ d 1 , 1 , d 2 , 1 , d N v , 1 , d 1 , 2 , d 2 , 2 , d N v , N h ] ,
D = U Σ V T ,
d i , j ( 2 ) = [ d i , j T , d i + 1 , j T ] T ,
d i , j ( 9 ) = [ d i , j T , d i + 1 , j T , d i 1 , j T , d i , j + 1 T , d i , j 1 T , d i + 1 , j + 1 T , d i + 1 , j 1 T , d i 1 , j + 1 T , d i 1 , j 1 T ] T .
α 1 x 2 + 2 α 2 x y + α 3 y 2 + 2 β ( α 4 x + α 5 y ) + β 2 α 6 = 0 ,
ϕ = Arg [ α 1 + α 3 + κ ( y ˜ + τ x ˜ ) + ι α 1 + α 3 κ ( x ˜ τ y ˜ ) ] ,
x ˜ = x β ( α 3 α 4 α 2 α 5 ) / ( α 2 2 α 1 α 3 ) ,
y ˜ = y β ( α 1 α 5 α 2 α 4 ) / ( α 2 2 α 1 α 3 ) ,
κ = 4 α 2 2 + ( α 1 α 3 ) 2 , τ = ( α 1 α 3 + κ ) / ( 2 α 2 ) .
α T χ = 0 ,
X α = λ α ,
X = 1 N n = 1 N χ n χ n T ,
W = [ 6 x ¯ 2 6 x y ¯ x ¯ 2 + y ¯ 2 6 β x ¯ 2 β y ¯ β 2 6 x y ¯ 4 ( x ¯ 2 + y ¯ 2 ) 6 x y ¯ 4 β y ¯ 4 β x ¯ 0 x ¯ 2 + y ¯ 2 6 x y ¯ 6 y ¯ 2 2 β x ¯ 6 β y ¯ β 2 6 β x ¯ 4 β y ¯ 2 β x ¯ 4 β 2 0 0 2 β y ¯ 4 β x ¯ 6 β y ¯ 0 4 β 2 0 β 2 0 β 2 0 0 0 ] ,
X α = λ W α ,
x ¯ 2 = 1 N n = 1 N x n 2 , y ¯ 2 = 1 N n = 1 N y n 2 , x y ¯ = 1 N n = 1 N x n y n .
W α = λ X α ,
P = Σ ˜ 1 U ˜ T ,
PD = Σ ˜ 1 U ˜ T U Σ V T = V ˜ T ,
D D T = U Σ V T V Σ U T = U Σ 2 U T .
P = Σ 1 U T ,
P d i , j ( L ) 0
SNR i , j = P d i , j ( L ) 2 2 P d i , j ( L ) 2 2 ,
SNR i , j > ϒ max i , j { SNR i , j } .
ϕ = Arg [ ( m = 0 M 1 I [ m ] cos ( 2 π m / M ) ι ( m = 0 M 1 I [ m ] sin ( 2 π m / M ) ) ) ] = Arg [ m = 0 M 1 I [ m ] e 2 π ι m M ] .

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