Abstract

The single-pixel imaging technique, which is significantly different from conventional multi-pixel imaging, utilizes the signal recorded by a single-pixel detector and a stream of structured illumination patterns to reconstruct an image. We design and experimentally demonstrate a real-time single-pixel foreground imaging system with fewer samples and without a priori sensing of the background by performing incremental principal component analysis on online compressed sampling data. A fast ℓ1 compressed sensing algorithm is adopted to realize real-time foreground imaging of 10 frames per second with an image size of 127 × 127 pixels and a compression ratio of 3%. When applied to a surveillance system that requires long-distance video transmission, this scheme can greatly reduce the compression ratio and allow the system to work with smaller communication bandwidths.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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2019 (6)

R. Chen, Y. Tong, J. Yang, and M. Wu, “Video foreground detection algorithm based on fast principal component pursuit and motion saliency,” Comput. Intel. Neurosc. 2019, 1–11 (2019).
[Crossref]

M. P. Edgar, G. M. Gibson, and M. J. Padgett, “Principles and prospects for single-pixel imaging,” Nat. Photonics 13(1), 13–20 (2019).
[Crossref]

R. Liu, S. Zhao, P. Zhang, H. Gao, and F. Li, “Complex wavefront reconstruction with single-pixel detector,” Appl. Phys. Lett. 114(16), 161901 (2019).
[Crossref]

S. Zhao, R. Liu, P. Zhang, H. Gao, and F. Li, “Fourier single-pixel reconstruction of a complex amplitude optical field,” Opt. Lett. 44(13), 3278–3281 (2019).
[Crossref]

D. Shi, K. Yin, J. Huang, K. Yuan, W. Zhu, C. Xie, D. Liu, and Y. Wang, “Fast tracking of moving objects using single-pixel imaging,” Opt. Commun. 440, 155–162 (2019).
[Crossref]

S. Sun, H. Lin, Y. Xu, J. Gu, and W. Liu, “Tracking and imaging of moving objects with temporal intensity difference correlation,” Opt. Express 27(20), 27851–27861 (2019).
[Crossref]

2018 (4)

C. F. Higham, R. Murray-Smith, M. J. Padgett, and M. P. Edgar, “Deep learning for real-time single-pixel video,” Sci. Rep. 8(1), 2369 (2018).
[Crossref]

S. Ota, R. Horisaki, Y. Kawamura, M. Ugawa, I. Sato, K. Hashimoto, R. Kamesawa, K. Setoyama, S. Yamaguchi, K. Fujiu, and H. N. Kayo Waki, “Ghost cytometry,” Science 360(6394), 1246–1251 (2018).
[Crossref]

K. Shibuya, H. Araki, and T. Iwata, “Photon-counting-based diffraction phase microscopy combined with single-pixel imaging,” Jpn. J. Appl. Phys. 57(4), 042501 (2018).
[Crossref]

A.-X. Zhang, Y.-H. He, L.-A. Wu, L.-M. Chen, and B.-B. Wang, “Tabletop x-ray ghost imaging with ultra-low radiation,” Optica 5(4), 374–377 (2018).
[Crossref]

2017 (3)

2016 (3)

F. Soldevila, P. Clemente, E. Tajahuerce, N. Uribe-Patarroyo, P. Andrés, and J. Lancis, “Computational imaging with a balanced detector,” Sci. Rep. 6(1), 29181 (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]

W. Cao, Y. Wang, J. Sun, D. Meng, C. Yang, A. Cichocki, and Z. Xu, “Total variation regularized tensor rpca for background subtraction from compressive measurements,” IEEE Trans. on Image Process. 25(9), 4075–4090 (2016).
[Crossref]

2015 (4)

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]

S. M. Khamoushi, Y. Nosrati, and S. H. Tavassoli, “Sinusoidal ghost imaging,” Opt. Lett. 40(15), 3452–3455 (2015).
[Crossref]

W.-K. Yu, X.-R. Yao, X.-F. Liu, L.-Z. Li, and G.-J. Zhai, “Compressive moving target tracking with thermal light based on complementary sampling,” Appl. Opt. 54(13), 4249–4254 (2015).
[Crossref]

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

2014 (2)

N. Radwell, K. J. Mitchell, G. M. Gibson, M. P. Edgar, R. Bowman, and M. J. Padgett, “Single-pixel infrared and visible microscope,” Optica 1(5), 285–289 (2014).
[Crossref]

A. Sobral and A. Vacavant, “A comprehensive review of background subtraction algorithms evaluated with synthetic and real videos,” Comput Vis Image Underst. 122, 4–21 (2014).
[Crossref]

2013 (2)

D. Shrekenhamer, C. M. Watts, and W. J. Padilla, “Terahertz single pixel imaging with an optically controlled dynamic spatial light modulator,” Opt. Express 21(10), 12507–12518 (2013).
[Crossref]

O. S. Magana-Loaiza, G. A. Howland, M. Malik, J. C. Howell, and R. W. Boyd, “Compressive object tracking using entangled photons,” Appl. Phys. Lett. 102(23), 231104 (2013).
[Crossref]

2011 (1)

T. Bouwmans, “Recent advanced statistical background modeling for foreground detection-a systematic survey,” CSENG 4(3), 147–176 (2011).
[Crossref]

2010 (2)

Y. Benezeth, P.-M. Jodoin, B. Emile, H. Laurent, and C. Rosenberger, “Comparative study of background subtraction algorithms,” J. Electron. Imaging 19(3), 033003 (2010).
[Crossref]

J. Yang, Y. Zhang, and W. Yin, “A fast alternating direction method for tvl1-l2 signal reconstruction from partial fourier data,” IEEE J. Sel. Top. Signal Process. 4(2), 288–297 (2010).
[Crossref]

2009 (1)

O. Katz, Y. Bromberg, and Y. Silberberg, “Compressive ghost imaging,” Appl. Phys. Lett. 95(13), 131110 (2009).
[Crossref]

2006 (1)

E. Candes, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52(2), 489–509 (2006).
[Crossref]

2005 (1)

S.-C. S. Cheung and C. Kamath, “Robust background subtraction with foreground validation for urban traffic video,” EURASIP J. Adv. Signal Process. 2005(14), 726261 (2005).
[Crossref]

2003 (1)

J. Weng, Y. Zhang, and W.-S. Hwang, “Candid covariance-free incremental principal component analysis,” IEEE Trans. Pattern Anal. Machine Intell. 25(8), 1034–1040 (2003).
[Crossref]

Andrés, P.

F. Soldevila, P. Clemente, E. Tajahuerce, N. Uribe-Patarroyo, P. Andrés, and J. Lancis, “Computational imaging with a balanced detector,” Sci. Rep. 6(1), 29181 (2016).
[Crossref]

Araki, H.

K. Shibuya, H. Araki, and T. Iwata, “Photon-counting-based diffraction phase microscopy combined with single-pixel imaging,” Jpn. J. Appl. Phys. 57(4), 042501 (2018).
[Crossref]

Baraniuk, R. G.

M. B. Wakin, J. N. Laska, M. F. Duarte, D. Baron, S. Sarvotham, D. Takhar, K. F. Kelly, and R. G. Baraniuk, “An architecture for compressive imaging,” in 2006 International Conference on Image Processing, (IEEE, 2006), pp. 1273–1276.

Baron, D.

M. B. Wakin, J. N. Laska, M. F. Duarte, D. Baron, S. Sarvotham, D. Takhar, K. F. Kelly, and R. G. Baraniuk, “An architecture for compressive imaging,” in 2006 International Conference on Image Processing, (IEEE, 2006), pp. 1273–1276.

Beleznai, C.

C. Beleznai, B. Fruhstuck, and H. Bischof, “Multiple object tracking using local pca,” in 18th International Conference on Pattern Recognition (ICPR’06), vol. 3 (IEEE, 2006), pp. 79–82.

Benezeth, Y.

Y. Benezeth, P.-M. Jodoin, B. Emile, H. Laurent, and C. Rosenberger, “Comparative study of background subtraction algorithms,” J. Electron. Imaging 19(3), 033003 (2010).
[Crossref]

Bischof, H.

C. Beleznai, B. Fruhstuck, and H. Bischof, “Multiple object tracking using local pca,” in 18th International Conference on Pattern Recognition (ICPR’06), vol. 3 (IEEE, 2006), pp. 79–82.

Bouwmans, T.

T. Bouwmans, “Recent advanced statistical background modeling for foreground detection-a systematic survey,” CSENG 4(3), 147–176 (2011).
[Crossref]

C. Guyon, T. Bouwmans, and E.-H. Zahzah, “Foreground detection by robust pca solved via a linearized alternating direction method,” in International Conference Image Analysis and Recognition, (Springer, 2012), pp. 115–122.

Bowman, R.

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.

O. S. Magana-Loaiza, G. A. Howland, M. Malik, J. C. Howell, and R. W. Boyd, “Compressive object tracking using entangled photons,” Appl. Phys. Lett. 102(23), 231104 (2013).
[Crossref]

Bromberg, Y.

O. Katz, Y. Bromberg, and Y. Silberberg, “Compressive ghost imaging,” Appl. Phys. Lett. 95(13), 131110 (2009).
[Crossref]

Brutzer, S.

S. Brutzer, B. Höferlin, and G. Heidemann, “Evaluation of background subtraction techniques for video surveillance,” in CVPR 2011, (IEEE, 2011), pp. 1937–1944.

Candes, E.

E. Candes, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52(2), 489–509 (2006).
[Crossref]

Cao, W.

W. Cao, Y. Wang, J. Sun, D. Meng, C. Yang, A. Cichocki, and Z. Xu, “Total variation regularized tensor rpca for background subtraction from compressive measurements,” IEEE Trans. on Image Process. 25(9), 4075–4090 (2016).
[Crossref]

Chartrand, R.

R. Chartrand, “Nonconvex compressive sensing and reconstruction of gradient-sparse images: random vs. tomographic fourier sampling,” in 2008 15th IEEE International Conference on Image Processing, (IEEE, 2008), pp. 2624–2627.

Chen, L.-M.

Chen, R.

R. Chen, Y. Tong, J. Yang, and M. Wu, “Video foreground detection algorithm based on fast principal component pursuit and motion saliency,” Comput. Intel. Neurosc. 2019, 1–11 (2019).
[Crossref]

Cheung, S.-C. S.

S.-C. S. Cheung and C. Kamath, “Robust background subtraction with foreground validation for urban traffic video,” EURASIP J. Adv. Signal Process. 2005(14), 726261 (2005).
[Crossref]

Cichocki, A.

W. Cao, Y. Wang, J. Sun, D. Meng, C. Yang, A. Cichocki, and Z. Xu, “Total variation regularized tensor rpca for background subtraction from compressive measurements,” IEEE Trans. on Image Process. 25(9), 4075–4090 (2016).
[Crossref]

Clemente, P.

F. Soldevila, P. Clemente, E. Tajahuerce, N. Uribe-Patarroyo, P. Andrés, and J. Lancis, “Computational imaging with a balanced detector,” Sci. Rep. 6(1), 29181 (2016).
[Crossref]

Duarte, M. F.

M. B. Wakin, J. N. Laska, M. F. Duarte, D. Baron, S. Sarvotham, D. Takhar, K. F. Kelly, and R. G. Baraniuk, “An architecture for compressive imaging,” in 2006 International Conference on Image Processing, (IEEE, 2006), pp. 1273–1276.

Edgar, M. P.

M. P. Edgar, G. M. Gibson, and M. J. Padgett, “Principles and prospects for single-pixel imaging,” Nat. Photonics 13(1), 13–20 (2019).
[Crossref]

C. F. Higham, R. Murray-Smith, M. J. Padgett, and M. P. Edgar, “Deep learning for real-time single-pixel video,” Sci. Rep. 8(1), 2369 (2018).
[Crossref]

G. M. Gibson, B. Sun, M. P. Edgar, D. B. Phillips, N. Hempler, G. T. Maker, G. P. 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]

N. Radwell, K. J. Mitchell, G. M. Gibson, M. P. Edgar, R. Bowman, and M. J. Padgett, “Single-pixel infrared and visible microscope,” Optica 1(5), 285–289 (2014).
[Crossref]

Emile, B.

Y. Benezeth, P.-M. Jodoin, B. Emile, H. Laurent, and C. Rosenberger, “Comparative study of background subtraction algorithms,” J. Electron. Imaging 19(3), 033003 (2010).
[Crossref]

Fruhstuck, B.

C. Beleznai, B. Fruhstuck, and H. Bischof, “Multiple object tracking using local pca,” in 18th International Conference on Pattern Recognition (ICPR’06), vol. 3 (IEEE, 2006), pp. 79–82.

Fujiu, K.

S. Ota, R. Horisaki, Y. Kawamura, M. Ugawa, I. Sato, K. Hashimoto, R. Kamesawa, K. Setoyama, S. Yamaguchi, K. Fujiu, and H. N. Kayo Waki, “Ghost cytometry,” Science 360(6394), 1246–1251 (2018).
[Crossref]

Gao, H.

S. Zhao, R. Liu, P. Zhang, H. Gao, and F. Li, “Fourier single-pixel reconstruction of a complex amplitude optical field,” Opt. Lett. 44(13), 3278–3281 (2019).
[Crossref]

R. Liu, S. Zhao, P. Zhang, H. Gao, and F. Li, “Complex wavefront reconstruction with single-pixel detector,” Appl. Phys. Lett. 114(16), 161901 (2019).
[Crossref]

Gibson, G. M.

M. P. Edgar, G. M. Gibson, and M. J. Padgett, “Principles and prospects for single-pixel imaging,” Nat. Photonics 13(1), 13–20 (2019).
[Crossref]

G. M. Gibson, B. Sun, M. P. Edgar, D. B. Phillips, N. Hempler, G. T. Maker, G. P. 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]

N. Radwell, K. J. Mitchell, G. M. Gibson, M. P. Edgar, R. Bowman, and M. J. Padgett, “Single-pixel infrared and visible microscope,” Optica 1(5), 285–289 (2014).
[Crossref]

Gu, J.

Guyon, C.

C. Guyon, T. Bouwmans, and E.-H. Zahzah, “Foreground detection by robust pca solved via a linearized alternating direction method,” in International Conference Image Analysis and Recognition, (Springer, 2012), pp. 115–122.

Hashimoto, K.

S. Ota, R. Horisaki, Y. Kawamura, M. Ugawa, I. Sato, K. Hashimoto, R. Kamesawa, K. Setoyama, S. Yamaguchi, K. Fujiu, and H. N. Kayo Waki, “Ghost cytometry,” Science 360(6394), 1246–1251 (2018).
[Crossref]

He, Y.-H.

Heidemann, G.

S. Brutzer, B. Höferlin, and G. Heidemann, “Evaluation of background subtraction techniques for video surveillance,” in CVPR 2011, (IEEE, 2011), pp. 1937–1944.

Hempler, N.

Higham, C. F.

C. F. Higham, R. Murray-Smith, M. J. Padgett, and M. P. Edgar, “Deep learning for real-time single-pixel video,” Sci. Rep. 8(1), 2369 (2018).
[Crossref]

Höferlin, B.

S. Brutzer, B. Höferlin, and G. Heidemann, “Evaluation of background subtraction techniques for video surveillance,” in CVPR 2011, (IEEE, 2011), pp. 1937–1944.

Horisaki, R.

S. Ota, R. Horisaki, Y. Kawamura, M. Ugawa, I. Sato, K. Hashimoto, R. Kamesawa, K. Setoyama, S. Yamaguchi, K. Fujiu, and H. N. Kayo Waki, “Ghost cytometry,” Science 360(6394), 1246–1251 (2018).
[Crossref]

Howell, J. C.

O. S. Magana-Loaiza, G. A. Howland, M. Malik, J. C. Howell, and R. W. Boyd, “Compressive object tracking using entangled photons,” Appl. Phys. Lett. 102(23), 231104 (2013).
[Crossref]

Howland, G. A.

O. S. Magana-Loaiza, G. A. Howland, M. Malik, J. C. Howell, and R. W. Boyd, “Compressive object tracking using entangled photons,” Appl. Phys. Lett. 102(23), 231104 (2013).
[Crossref]

Huang, J.

D. Shi, K. Yin, J. Huang, K. Yuan, W. Zhu, C. Xie, D. Liu, and Y. Wang, “Fast tracking of moving objects using single-pixel imaging,” Opt. Commun. 440, 155–162 (2019).
[Crossref]

Hwang, W.-S.

J. Weng, Y. Zhang, and W.-S. Hwang, “Candid covariance-free incremental principal component analysis,” IEEE Trans. Pattern Anal. Machine Intell. 25(8), 1034–1040 (2003).
[Crossref]

Iwata, T.

K. Shibuya, H. Araki, and T. Iwata, “Photon-counting-based diffraction phase microscopy combined with single-pixel imaging,” Jpn. J. Appl. Phys. 57(4), 042501 (2018).
[Crossref]

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Appl. Opt. (1)

Appl. Phys. Lett. (3)

O. S. Magana-Loaiza, G. A. Howland, M. Malik, J. C. Howell, and R. W. Boyd, “Compressive object tracking using entangled photons,” Appl. Phys. Lett. 102(23), 231104 (2013).
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Comput Vis Image Underst. (1)

A. Sobral and A. Vacavant, “A comprehensive review of background subtraction algorithms evaluated with synthetic and real videos,” Comput Vis Image Underst. 122, 4–21 (2014).
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Comput. Intel. Neurosc. (1)

R. Chen, Y. Tong, J. Yang, and M. Wu, “Video foreground detection algorithm based on fast principal component pursuit and motion saliency,” Comput. Intel. Neurosc. 2019, 1–11 (2019).
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EURASIP J. Adv. Signal Process. (1)

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IEEE J. Sel. Top. Signal Process. (1)

J. Yang, Y. Zhang, and W. Yin, “A fast alternating direction method for tvl1-l2 signal reconstruction from partial fourier data,” IEEE J. Sel. Top. Signal Process. 4(2), 288–297 (2010).
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IEEE Trans. Inf. Theory (1)

E. Candes, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52(2), 489–509 (2006).
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IEEE Trans. on Image Process. (1)

W. Cao, Y. Wang, J. Sun, D. Meng, C. Yang, A. Cichocki, and Z. Xu, “Total variation regularized tensor rpca for background subtraction from compressive measurements,” IEEE Trans. on Image Process. 25(9), 4075–4090 (2016).
[Crossref]

IEEE Trans. Pattern Anal. Machine Intell. (1)

J. Weng, Y. Zhang, and W.-S. Hwang, “Candid covariance-free incremental principal component analysis,” IEEE Trans. Pattern Anal. Machine Intell. 25(8), 1034–1040 (2003).
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J. Electron. Imaging (1)

Y. Benezeth, P.-M. Jodoin, B. Emile, H. Laurent, and C. Rosenberger, “Comparative study of background subtraction algorithms,” J. Electron. Imaging 19(3), 033003 (2010).
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Jpn. J. Appl. Phys. (1)

K. Shibuya, H. Araki, and T. Iwata, “Photon-counting-based diffraction phase microscopy combined with single-pixel imaging,” Jpn. J. Appl. Phys. 57(4), 042501 (2018).
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Nat. Commun. (2)

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

M. P. Edgar, G. M. Gibson, and M. J. Padgett, “Principles and prospects for single-pixel imaging,” Nat. Photonics 13(1), 13–20 (2019).
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Opt. Commun. (1)

D. Shi, K. Yin, J. Huang, K. Yuan, W. Zhu, C. Xie, D. Liu, and Y. Wang, “Fast tracking of moving objects using single-pixel imaging,” Opt. Commun. 440, 155–162 (2019).
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Opt. Express (4)

Opt. Lett. (2)

Optica (2)

Sci. Rep. (4)

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Supplementary Material (10)

NameDescription
» Visualization 1       Visualization 1
» Visualization 2       Visualization 2
» Visualization 3       Visualization 3
» Visualization 4       Visualization 4
» Visualization 5       Visualization 5
» Visualization 6       Visualization 6
» Visualization 7       Visualization 7
» Visualization 8       Visualization 8
» Visualization 9       Visualization 9
» Visualization 10       Visualization 10

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Figures (7)

Fig. 1.
Fig. 1. Schematic of foreground surveillance system based on single-pixel imaging: (a) the scene, (b) the measurement system, and (c) the terminal of the system.
Fig. 2.
Fig. 2. Simulated results from two sets of real surveillance videos. (a1)–(a5) and (d1)–(d5) are 10 image frames obtained from raw surveillance videos, (b1)–(b5) and (e1)–(e5) are the corresponding foreground ground-truth. (c1)–(c5) and (f1)–(f5) are the reconstructed foreground using the proposed method with a compression ratio of $4\%$. (g) and (h) are the curves of the mean MSE versus the compression ratio for the two sets of videos.
Fig. 3.
Fig. 3. Experimental setup. L1, L2: lens, Scene: the moving targets and static background, DMD: digital micromirror device, PDA: amplified photodetector, PC: personal computer, ADC: analogue-to-digital converter, M: mirror, LED: white LED light source.
Fig. 4.
Fig. 4. Experimental scenes and sampling matrices in Fourier domain, (a) and (d) are two real scenes recorded by a color camera, in which the moving targets have different sizes. (b) is the sampling matrix corresponding to scene (a), it contains $10$ radial lines in the Fourier domain. (c) is the experimental sampling matrix of the scene when considering the symmetry of Fourier spectrum of the real function, and it contains $621$ Fourier components. (e) is the sampling matrix corresponding to scene (d), it contains $8$ radial lines in Fourier domain. (f) is the experimental sampling matrix of scene (d), and it contains $501$ Fourier components.
Fig. 5.
Fig. 5. Sample frames of foreground extraction. (a)–(e) are the results of inverse Fourier transform after partial Fourier spectral sampling of a moving target, which is equivalent to ${\Phi }^T b_n$ in Eq. (3). (f)–(j) are obtained by removing the largest principal component in (a)–(e), respectively, and they are the results of ${\Phi }^T b_n-{\Phi }^T b^g$ in Eq. (3). $b_{n}^{f}$ in Eq. (4) is obtained by performing measurement operator $\Phi$ on (f)–(j), and the sample frames (k)–(o) from the raw video data (Visualization 1) are reconstructed by solving the optimization problem Eq. (4).
Fig. 6.
Fig. 6. Foreground extraction with different scenes. (a)–(d) are four frames taken from the raw video data (Visualization 2, Visualization 3, Visualization 4, and Visualization 5), respectively, which are real-time imaging results of four different toy soldiers moving from left to right in the scene shown in Fig. 3(a). (e)–(h) are four frames taken from the raw video data (Visualization 6, Visualization 7, Visualization 8, and Visualization 9), respectively, which are real-time imaging results of four different cards moving from right to left in the scene shown in Fig. 3(d). The insets in the upper left corner of each result are the experimental targets.
Fig. 7.
Fig. 7. Sample frames from the raw video data (Visualization 10) when the background of the scene changes drastically. (a) The scene captured by a color camera. (b) Part of the target enters the area with the cardboard background (changing background area). (c) Target fully enters the changing background area. (d) Part of the target leaves the changing background area.

Equations (4)

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min x Ψ x 1 , subject to Φ x = b ,
M C K log ( N / K ) ,
b n f = Φ ( Φ T b n Φ T b g ) .
min x n f x n f 1 + 1 2 ρ Φ x n f b n f 2 2 ,

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