Abstract

Fourier single-pixel imaging is one of the main single-pixel imaging techniques. To improve the imaging efficiency, some of the recent method typically select the low-frequency and discard the high-frequency information to reduce the number of acquired samples. However, sampling only a small amount of low-frequency components will lead to the loss of object details and will reduce the imaging resolution. At the same time, the ringing effect of the restored image due to frequency truncation is significant. In this paper, a new sparse Fourier single-pixel imaging method is proposed that reduces the number of samples explorations while maintaining increased image quality. The proposed method makes a special use of the characteristics of the Fourier spectrum distribution based on which the power of image information decreases gradually from low to high frequencies in the Fourier space. A variable density random sampling matrix is employed to achieve random sampling with Fourier single-pixel imaging technology, followed by the processing of the sparse Fourier spectra using compressive sensing algorithms to recover the high-quality information of the object. The new algorithm can effectively improve the quality of object restoration comparing with the existing Fourier single-pixel imaging methods which only acquire the low-frequency parts. Additionally, considering that the resolution of the system is diffraction limited, super-resolution imaging can also be achieved. Experimental results demonstrate the mainly correctness but also effectiveness of the proposed method.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2018 (3)

2017 (3)

2016 (4)

R. I. Stantchev, B. Q. Sun, S. M. Hornett, P. A. Hobson, G. M. Gibson, M. J. Padgett, and E. Hendry, “Noninvasive, near-field terahertz imaging of hidden objects using a single-pixel detector,” Sci. Adv. 2(6), e1600190 (2016).
[Crossref]

M. J. Sun, M. P. Edgar, G. M. Gibson, B. Sun, N. Radwell, R. Lamb, and M. J. Padgett, “Single-pixel three-dimensional imaging with time-based depth resolution,” Nat. Commun. 7(1), 12010 (2016).
[Crossref]

N. Huynh, E. Zhang, M. Betcke, S. Arridge, P. Beard, and B. Cox, “Single-pixel optical camera for video rate ultrasonic imaging,” Optica 3(1), 26–29 (2016).
[Crossref]

L. Bian, J. Suo, X. Hu, F. Chen, and Q. Dai, “Efficient single pixel imaging in Fourier space,” J. Opt. 18(8), 085704 (2016).
[Crossref]

2015 (6)

2014 (1)

2013 (2)

L. Llull, X. Liao, X. Yuan, J. Yang, D. Kittle, L. Carlin, G. Sapiro, and D. J. Brady, “Coded aperture compressive temporal imaging,” Opt. Express 21(9), 10526–10545 (2013).
[Crossref]

B. Sun, M. P. Edgar, R. Bowman, L. E. Vittert, S. Welsh, A. Bowman, and M. J. Padgett, “3D computational imaging with single-pixel detectors,” Science 340(6134), 844–847 (2013).
[Crossref]

2008 (2)

M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, T. Sun, K. F. Kelly, and R. G. Baraniuk, “Single-pixel imaging via compressive sampling,” IEEE Signal Process. Mag. 25(2), 83–91 (2008).
[Crossref]

E. J. Candes and M. B. Wakin, “An introduction to compressive sampling,” IEEE Signal Process. Mag. 25(2), 21–30 (2008).
[Crossref]

Arce, G. R.

Arguello, H.

Arridge, S.

Aspden, R. S.

Baraniuk, R. G.

M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, T. Sun, K. F. Kelly, and R. G. Baraniuk, “Single-pixel imaging via compressive sampling,” IEEE Signal Process. Mag. 25(2), 83–91 (2008).
[Crossref]

Beard, P.

Betcke, M.

Bian, L.

L. Bian, J. Suo, X. Hu, F. Chen, and Q. Dai, “Efficient single pixel imaging in Fourier space,” J. Opt. 18(8), 085704 (2016).
[Crossref]

Bowman, A.

B. Sun, M. P. Edgar, R. Bowman, L. E. Vittert, S. Welsh, A. Bowman, and M. J. Padgett, “3D computational imaging with single-pixel detectors,” Science 340(6134), 844–847 (2013).
[Crossref]

Bowman, R.

B. Sun, M. P. Edgar, R. Bowman, L. E. Vittert, S. Welsh, A. Bowman, and M. J. Padgett, “3D computational imaging with single-pixel detectors,” Science 340(6134), 844–847 (2013).
[Crossref]

Bowman, R. W.

M. P. Edgar, G. M. Gibson, R. W. Bowman, B. Sun, N. Radwell, K. J. Mitchell, S. S. Welsh, and M. J. Padgett, “Simultaneous real-time visible and infrared video with single-pixel detectors,” Sci. Rep. 5(1), 10669 (2015).
[Crossref]

Boyd, R. W.

Brady, D. J.

Buller, G. S.

Candes, E. J.

E. J. Candes and M. B. Wakin, “An introduction to compressive sampling,” IEEE Signal Process. Mag. 25(2), 21–30 (2008).
[Crossref]

Carlin, L.

Chen, F.

L. Bian, J. Suo, X. Hu, F. Chen, and Q. Dai, “Efficient single pixel imaging in Fourier space,” J. Opt. 18(8), 085704 (2016).
[Crossref]

Cossairt, O.

Cox, B.

Dai, Q.

L. Bian, J. Suo, X. Hu, F. Chen, and Q. Dai, “Efficient single pixel imaging in Fourier space,” J. Opt. 18(8), 085704 (2016).
[Crossref]

Dai, Q. H.

Davenport, M. A.

M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, T. Sun, K. F. Kelly, and R. G. Baraniuk, “Single-pixel imaging via compressive sampling,” IEEE Signal Process. Mag. 25(2), 83–91 (2008).
[Crossref]

Duarte, M. F.

M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, T. Sun, K. F. Kelly, and R. G. Baraniuk, “Single-pixel imaging via compressive sampling,” IEEE Signal Process. Mag. 25(2), 83–91 (2008).
[Crossref]

Edgar, M. P.

G. M. Gibson, B. Q. Sun, M. P. Edgar, D. B. Phillips, N. Hempler, G. T. Maker, G. P. A. Malcolm, and M. J. Padgett, “Real-time imaging of methane gas leaks using a single-pixel camera,” Opt. Express 25(4), 2998–3005 (2017).
[Crossref]

M. J. Sun, M. P. Edgar, G. M. Gibson, B. Sun, N. Radwell, R. Lamb, and M. J. Padgett, “Single-pixel three-dimensional imaging with time-based depth resolution,” Nat. Commun. 7(1), 12010 (2016).
[Crossref]

M. P. Edgar, G. M. Gibson, R. W. Bowman, B. Sun, N. Radwell, K. J. Mitchell, S. S. Welsh, and M. J. Padgett, “Simultaneous real-time visible and infrared video with single-pixel detectors,” Sci. Rep. 5(1), 10669 (2015).
[Crossref]

B. Sun, M. P. Edgar, R. Bowman, L. E. Vittert, S. Welsh, A. Bowman, and M. J. Padgett, “3D computational imaging with single-pixel detectors,” Science 340(6134), 844–847 (2013).
[Crossref]

Galvis, L.

Gemmell, N. R.

Gibson, G. M.

G. M. Gibson, B. Q. Sun, M. P. Edgar, D. B. Phillips, N. Hempler, G. T. Maker, G. P. A. Malcolm, and M. J. Padgett, “Real-time imaging of methane gas leaks using a single-pixel camera,” Opt. Express 25(4), 2998–3005 (2017).
[Crossref]

M. J. Sun, M. P. Edgar, G. M. Gibson, B. Sun, N. Radwell, R. Lamb, and M. J. Padgett, “Single-pixel three-dimensional imaging with time-based depth resolution,” Nat. Commun. 7(1), 12010 (2016).
[Crossref]

R. I. Stantchev, B. Q. Sun, S. M. Hornett, P. A. Hobson, G. M. Gibson, M. J. Padgett, and E. Hendry, “Noninvasive, near-field terahertz imaging of hidden objects using a single-pixel detector,” Sci. Adv. 2(6), e1600190 (2016).
[Crossref]

M. P. Edgar, G. M. Gibson, R. W. Bowman, B. Sun, N. Radwell, K. J. Mitchell, S. S. Welsh, and M. J. Padgett, “Simultaneous real-time visible and infrared video with single-pixel detectors,” Sci. Rep. 5(1), 10669 (2015).
[Crossref]

Gu, J. W.

Y. Hitomi, J. W. Gu, M. Gupta, T. Mitsunaga, and S. K. Nayar, “Video from a single coded exposure photograph using a learned over-complete dictionary,” in Proceedings of 2011 IEEE International Conference on Computer Vision (ICCV), (2011), pp. 287–294.

Gupta, M.

Y. Hitomi, J. W. Gu, M. Gupta, T. Mitsunaga, and S. K. Nayar, “Video from a single coded exposure photograph using a learned over-complete dictionary,” in Proceedings of 2011 IEEE International Conference on Computer Vision (ICCV), (2011), pp. 287–294.

Hadfield, R. H.

Hempler, N.

Hendry, E.

R. I. Stantchev, B. Q. Sun, S. M. Hornett, P. A. Hobson, G. M. Gibson, M. J. Padgett, and E. Hendry, “Noninvasive, near-field terahertz imaging of hidden objects using a single-pixel detector,” Sci. Adv. 2(6), e1600190 (2016).
[Crossref]

Hitomi, Y.

Y. Hitomi, J. W. Gu, M. Gupta, T. Mitsunaga, and S. K. Nayar, “Video from a single coded exposure photograph using a learned over-complete dictionary,” in Proceedings of 2011 IEEE International Conference on Computer Vision (ICCV), (2011), pp. 287–294.

Hobson, P. A.

R. I. Stantchev, B. Q. Sun, S. M. Hornett, P. A. Hobson, G. M. Gibson, M. J. Padgett, and E. Hendry, “Noninvasive, near-field terahertz imaging of hidden objects using a single-pixel detector,” Sci. Adv. 2(6), e1600190 (2016).
[Crossref]

Hornett, S. M.

R. I. Stantchev, B. Q. Sun, S. M. Hornett, P. A. Hobson, G. M. Gibson, M. J. Padgett, and E. Hendry, “Noninvasive, near-field terahertz imaging of hidden objects using a single-pixel detector,” Sci. Adv. 2(6), e1600190 (2016).
[Crossref]

Hu, S.

Hu, X.

L. Bian, J. Suo, X. Hu, F. Chen, and Q. Dai, “Efficient single pixel imaging in Fourier space,” J. Opt. 18(8), 085704 (2016).
[Crossref]

Huang, J.

Huynh, N.

Katsaggelos, A. K.

Kelly, K. F.

M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, T. Sun, K. F. Kelly, and R. G. Baraniuk, “Single-pixel imaging via compressive sampling,” IEEE Signal Process. Mag. 25(2), 83–91 (2008).
[Crossref]

Kirkwood, R. A.

Kittle, D.

Koller, R.

Lamb, R.

M. J. Sun, M. P. Edgar, G. M. Gibson, B. Sun, N. Radwell, R. Lamb, and M. J. Padgett, “Single-pixel three-dimensional imaging with time-based depth resolution,” Nat. Commun. 7(1), 12010 (2016).
[Crossref]

Laska, J. N.

M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, T. Sun, K. F. Kelly, and R. G. Baraniuk, “Single-pixel imaging via compressive sampling,” IEEE Signal Process. Mag. 25(2), 83–91 (2008).
[Crossref]

Lau, D.

Liao, X.

Lin, X.

Liu, S.

Liu, Y. B.

Llull, L.

Ma, X.

Z. Zhang, X. Ma, and J. Zhong, “Single-pixel imaging by means of Fourier spectrum acquisition,” Nat. Commun. 6(1), 6225 (2015).
[Crossref]

Maker, G. T.

Malcolm, G. P. A.

Matsuda, N.

Mertens, L.

Mitchell, K. J.

M. P. Edgar, G. M. Gibson, R. W. Bowman, B. Sun, N. Radwell, K. J. Mitchell, S. S. Welsh, and M. J. Padgett, “Simultaneous real-time visible and infrared video with single-pixel detectors,” Sci. Rep. 5(1), 10669 (2015).
[Crossref]

Mitsunaga, T.

Y. Hitomi, J. W. Gu, M. Gupta, T. Mitsunaga, and S. K. Nayar, “Video from a single coded exposure photograph using a learned over-complete dictionary,” in Proceedings of 2011 IEEE International Conference on Computer Vision (ICCV), (2011), pp. 287–294.

Morris, P. A.

Nayar, S. K.

Y. Hitomi, J. W. Gu, M. Gupta, T. Mitsunaga, and S. K. Nayar, “Video from a single coded exposure photograph using a learned over-complete dictionary,” in Proceedings of 2011 IEEE International Conference on Computer Vision (ICCV), (2011), pp. 287–294.

Niederberger, T.

Olivieri, L.

L. Olivieri, J. S. Totero Gongora, A. Pasquazi, and M. Peccianti, “Time-Resolved Nonlinear Ghost Imaging,” ACS Photonics 5(8), 3379–3388 (2018).
[Crossref]

Padgett, M. J.

G. M. Gibson, B. Q. Sun, M. P. Edgar, D. B. Phillips, N. Hempler, G. T. Maker, G. P. A. Malcolm, and M. J. Padgett, “Real-time imaging of methane gas leaks using a single-pixel camera,” Opt. Express 25(4), 2998–3005 (2017).
[Crossref]

M. J. Sun, M. P. Edgar, G. M. Gibson, B. Sun, N. Radwell, R. Lamb, and M. J. Padgett, “Single-pixel three-dimensional imaging with time-based depth resolution,” Nat. Commun. 7(1), 12010 (2016).
[Crossref]

R. I. Stantchev, B. Q. Sun, S. M. Hornett, P. A. Hobson, G. M. Gibson, M. J. Padgett, and E. Hendry, “Noninvasive, near-field terahertz imaging of hidden objects using a single-pixel detector,” Sci. Adv. 2(6), e1600190 (2016).
[Crossref]

M. P. Edgar, G. M. Gibson, R. W. Bowman, B. Sun, N. Radwell, K. J. Mitchell, S. S. Welsh, and M. J. Padgett, “Simultaneous real-time visible and infrared video with single-pixel detectors,” Sci. Rep. 5(1), 10669 (2015).
[Crossref]

R. S. Aspden, N. R. Gemmell, P. A. Morris, D. S. Tasca, L. Mertens, M. G. Tanner, R. A. Kirkwood, A. Ruggeri, A. Tosi, R. W. Boyd, G. S. Buller, R. H. Hadfield, and M. J. Padgett, “Photon-sparse microscopy: visible light imaging using infrared illumination,” Optica 2(12), 1049–1052 (2015).
[Crossref]

B. Sun, M. P. Edgar, R. Bowman, L. E. Vittert, S. Welsh, A. Bowman, and M. J. Padgett, “3D computational imaging with single-pixel detectors,” Science 340(6134), 844–847 (2013).
[Crossref]

Pasquazi, A.

L. Olivieri, J. S. Totero Gongora, A. Pasquazi, and M. Peccianti, “Time-Resolved Nonlinear Ghost Imaging,” ACS Photonics 5(8), 3379–3388 (2018).
[Crossref]

Peccianti, M.

L. Olivieri, J. S. Totero Gongora, A. Pasquazi, and M. Peccianti, “Time-Resolved Nonlinear Ghost Imaging,” ACS Photonics 5(8), 3379–3388 (2018).
[Crossref]

Peng, J.

Phillips, D. B.

Radwell, N.

M. J. Sun, M. P. Edgar, G. M. Gibson, B. Sun, N. Radwell, R. Lamb, and M. J. Padgett, “Single-pixel three-dimensional imaging with time-based depth resolution,” Nat. Commun. 7(1), 12010 (2016).
[Crossref]

M. P. Edgar, G. M. Gibson, R. W. Bowman, B. Sun, N. Radwell, K. J. Mitchell, S. S. Welsh, and M. J. Padgett, “Simultaneous real-time visible and infrared video with single-pixel detectors,” Sci. Rep. 5(1), 10669 (2015).
[Crossref]

Rueda, H.

Ruggeri, A.

Sapiro, G.

Schmid, L.

Schuster, G.

Shi, D.

Spinoula, L.

Stantchev, R. I.

R. I. Stantchev, B. Q. Sun, S. M. Hornett, P. A. Hobson, G. M. Gibson, M. J. Padgett, and E. Hendry, “Noninvasive, near-field terahertz imaging of hidden objects using a single-pixel detector,” Sci. Adv. 2(6), e1600190 (2016).
[Crossref]

Sun, B.

M. J. Sun, M. P. Edgar, G. M. Gibson, B. Sun, N. Radwell, R. Lamb, and M. J. Padgett, “Single-pixel three-dimensional imaging with time-based depth resolution,” Nat. Commun. 7(1), 12010 (2016).
[Crossref]

M. P. Edgar, G. M. Gibson, R. W. Bowman, B. Sun, N. Radwell, K. J. Mitchell, S. S. Welsh, and M. J. Padgett, “Simultaneous real-time visible and infrared video with single-pixel detectors,” Sci. Rep. 5(1), 10669 (2015).
[Crossref]

B. Sun, M. P. Edgar, R. Bowman, L. E. Vittert, S. Welsh, A. Bowman, and M. J. Padgett, “3D computational imaging with single-pixel detectors,” Science 340(6134), 844–847 (2013).
[Crossref]

Sun, B. Q.

G. M. Gibson, B. Q. Sun, M. P. Edgar, D. B. Phillips, N. Hempler, G. T. Maker, G. P. A. Malcolm, and M. J. Padgett, “Real-time imaging of methane gas leaks using a single-pixel camera,” Opt. Express 25(4), 2998–3005 (2017).
[Crossref]

R. I. Stantchev, B. Q. Sun, S. M. Hornett, P. A. Hobson, G. M. Gibson, M. J. Padgett, and E. Hendry, “Noninvasive, near-field terahertz imaging of hidden objects using a single-pixel detector,” Sci. Adv. 2(6), e1600190 (2016).
[Crossref]

Sun, M. J.

M. J. Sun, M. P. Edgar, G. M. Gibson, B. Sun, N. Radwell, R. Lamb, and M. J. Padgett, “Single-pixel three-dimensional imaging with time-based depth resolution,” Nat. Commun. 7(1), 12010 (2016).
[Crossref]

Sun, T.

M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, T. Sun, K. F. Kelly, and R. G. Baraniuk, “Single-pixel imaging via compressive sampling,” IEEE Signal Process. Mag. 25(2), 83–91 (2008).
[Crossref]

Suo, J.

L. Bian, J. Suo, X. Hu, F. Chen, and Q. Dai, “Efficient single pixel imaging in Fourier space,” J. Opt. 18(8), 085704 (2016).
[Crossref]

Takhar, D.

M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, T. Sun, K. F. Kelly, and R. G. Baraniuk, “Single-pixel imaging via compressive sampling,” IEEE Signal Process. Mag. 25(2), 83–91 (2008).
[Crossref]

Tanner, M. G.

Tasca, D. S.

Tosi, A.

Totero Gongora, J. S.

L. Olivieri, J. S. Totero Gongora, A. Pasquazi, and M. Peccianti, “Time-Resolved Nonlinear Ghost Imaging,” ACS Photonics 5(8), 3379–3388 (2018).
[Crossref]

Vittert, L. E.

B. Sun, M. P. Edgar, R. Bowman, L. E. Vittert, S. Welsh, A. Bowman, and M. J. Padgett, “3D computational imaging with single-pixel detectors,” Science 340(6134), 844–847 (2013).
[Crossref]

Wakin, M. B.

E. J. Candes and M. B. Wakin, “An introduction to compressive sampling,” IEEE Signal Process. Mag. 25(2), 21–30 (2008).
[Crossref]

Wang, X.

Z. Zhang, X. Wang, G. Zheng, and J. Zhong, “Fast Fourier single-pixel imaging via binary illumination,” Sci. Rep. 7(1), 12029 (2017).
[Crossref]

Z. Zhang, X. Wang, G. Zheng, and J. Zhong, “Hadamard single-pixel imaging versus Fourier single-pixel imaging,” Opt. Express 25(16), 19619–19639 (2017).
[Crossref]

Wang, Y.

Welsh, S.

B. Sun, M. P. Edgar, R. Bowman, L. E. Vittert, S. Welsh, A. Bowman, and M. J. Padgett, “3D computational imaging with single-pixel detectors,” Science 340(6134), 844–847 (2013).
[Crossref]

Welsh, S. S.

M. P. Edgar, G. M. Gibson, R. W. Bowman, B. Sun, N. Radwell, K. J. Mitchell, S. S. Welsh, and M. J. Padgett, “Simultaneous real-time visible and infrared video with single-pixel detectors,” Sci. Rep. 5(1), 10669 (2015).
[Crossref]

Wetzstein, G.

Yang, J.

Yao, M.

Yuan, K.

Yuan, X.

Zhang, E.

Zhang, Z.

Z. Zhang, S. Liu, J. Peng, M. Yao, G. Zheng, and J. Zhong, “Simultaneous spatial, spectrum, and 3D compressive imaging via efficient Fourier single-pixel measurements,” Optica 5(3), 315–319 (2018).
[Crossref]

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[Crossref]

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Figures (9)

Fig. 1.
Fig. 1. Low-frequency sampling: (A) and (B) represent the original image and its spectrum, (C) and (D) represent the low-frequency sampling spectrum and the restored image.
Fig. 2.
Fig. 2. Sampling schemes. (Left) Equal probability random sampling scheme, (middle) variable-density random sampling scheme, and (right) circular sampling scheme.
Fig. 3.
Fig. 3. Restoration results of the binary fringe. Restoration results following low-resolution sampling exhibit edge expansion and details loss. Noted is the presence of noise in the zero-filled restoration results, while the intensity of noise for uniform sampling is larger than that of variable density sampling. CS restoration results yield high-quality outcomes, and a large number of image details can be presented. At the same time, the quality of the CS restoration results with variable density sampling is better than those of other methods.
Fig. 4.
Fig. 4. Grayscale image restoration results. Restoration results following low-resolution sampling exhibit edge expansion and details loss. There is considerable noise in the zero-filled restored results, and the noise of uniform sampling has a larger intensity than that associated with variable density sampling. CS restoration results are of high quality, and a large number of image details can be observed. At the same time, the quality of the CS restoration results with variable density sampling is better than those of other methods.
Fig. 5.
Fig. 5. Restoration results based on the diffraction limit. The label “Low-resolution” represents the image and spectral information subject to the diffraction limit. The label “CS restored image” represents the spatial and spectral information that is restored by CS. The label “Original” represents the original image and its spectral information.
Fig. 6.
Fig. 6. Experimental setup.
Fig. 7.
Fig. 7. Restoration outcomes. Low-resolution sampling recovers lost details. Extensive noise is observed in the zero-filled restoration results, and the noise of uniform sampling is larger than that of variable density sampling. The quality of the CS restoration results obtained from variable density sampling is higher than those of other methods. Considering a sampling rate of 10% as an example, the reconstructed results of CS with variable density sampling can identify the non-recognizable digits in other results. The red box is the selected image enlargement area.
Fig. 8.
Fig. 8. Magnified views of experimental results.
Fig. 9.
Fig. 9. Restoration outcomes based on the diffraction limit. The “low-resolution” label represents the spatial and spectral information of the image subject to the diffraction limit. The “CS restored” label represents the spatial and spectral information of the image restored by CS. The label “Original” represents the spatial and spectral information of the original image. The third line enlarges the corresponding local areas of the three groups of images.

Tables (3)

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Table 1. RMSE of restoration results of binary stripe resolution plate

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Table 2. RMSE of cameraman image restoration

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Table 3. RMSE of experimental restoration results

Equations (8)

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I θ ( f x , f y ) = x,y B θ ( x , y ; f x , f y ) t ( x , y ) ,
B θ ( x , y ; f x , f y ) = a + b cos ( 2 π f x x / M + 2 π f y y / N + θ ) ,
I 0 ( f x , f y ) = x,y [ a + b cos ( 2 π f x x / M + 2 π f y y / N + 0 ) ] t ( x , y ) I π /2 ( f x , f y ) = x,y [ a + b cos ( 2 π f x x / M + 2 π f y y / N + π /2 ) ] t ( x , y ) I π ( f x , f y ) = x,y [ a + b cos ( 2 π f x x / M + 2 π f y y / N + π ) ] t ( x , y ) I 3 π /2 ( f x , f y ) = x,y [ a + b cos ( 2 π f x x / M + 2 π f y y / N + 3 π /2 ) ] t ( x , y ) ,
T ( f x , f y ) = 1 2 b [ I π ( f x , f y ) I 0 ( f x , f y ) ] + j [ I 3 π / 2 ( f x , f y ) I π /2 ( f x , f y ) ] ,
arg min   | | F t T | | 2 2 + λ 1 | | t | | 1 + λ 2 T V ( t ) ,
B s = [ B π ( x , y ; f x , f y ) B 0 ( x , y ; f x , f y ) ]   + j [ B 3 π / 2 ( x , y ; f x , f y ) B π /2 ( x , y ; f x , f y ) ] .
ρ ( r ) = { 1 r R ( 1 r ) p r > R
R M S E = x , y = 1 M , N [ t r ( x , y ) t o ( x , y ) ] 2 / [ t r ( x , y ) t o ( x , y ) ] 2 ( M × N ) ( M × N )

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