Abstract

Compressed sensing is a theory which can reconstruct an image almost perfectly with only a few measurements by finding its sparsest representation. However, the computation time consumed for large images may be a few hours or more. In this work, we both theoretically and experimentally demonstrate a method that combines the advantages of both adaptive computational ghost imaging and compressed sensing, which we call adaptive compressive ghost imaging, whereby both the reconstruction time and measurements required for any image size can be significantly reduced. The technique can be used to improve the performance of all computational ghost imaging protocols, especially when measuring ultra-weak or noisy signals, and can be extended to imaging applications at any wavelength.

© 2014 Optical Society of America

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2013 (4)

2012 (3)

A. Averbuch, S. Dekel, S. Deutsch, “Adaptive compressed image sensing using dictionaries,” SIAM J. Imaging Sci. 5(1), 57–89 (2012).
[CrossRef]

W. L. Gong, S. S. Han, “Experimental investigation of the quality of lensless super-resolution ghost imaging via sparsity constraints,” Phys. Lett. A 376(17), 1519–1522 (2012).
[CrossRef]

N. B. Karahanoglu, H. Erdogan, “A* orthogonal matching pursuit: best-first search for compressed sensing signal recovery,” Digit. Sig. Process. 22(4), 555–568 (2012).
[CrossRef]

2010 (1)

J. Yang, Y. Zhang, W. Yin, “A fast alternating direction method for TVL1-L2 signal reconstruction from partial Fourier data,” IEEE J. Sel. Top. Signal Processing 4(2), 288–297 (2010).
[CrossRef]

2009 (3)

Y. Bromberg, O. Katz, Y. Silberberg, “Ghost imaging with a single detector,” Phys. Rev. A 79(5), 053840 (2009).
[CrossRef]

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

P. Sen, S. Darabi, “Compressive dual photography,” Computer Graphics Forum 28(2), 609–618 (2009).
[CrossRef]

2008 (5)

M. Fornasier, H. Rauhut, “Iterative thresholding algorithms,” Appl. Comput. Harmon. Anal. 25(2), 187–208 (2008).
[CrossRef]

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

W. L. Chan, K. Charan, D. Takhar, K. F. Kelly, R. G. Baraniuk, D. M. Mittleman, “A single-pixel terahertz imaging system based on compressed sensing,” Appl. Phys. Lett. 93(12), 121105 (2008).
[CrossRef]

B. I. Erkmen, J. H. Shapiro, “Unified theory of ghost imaging with Gaussian-state light,” Phys. Rev. A 77(4), 043809 (2008).
[CrossRef]

J. H. Shapiro, “Computational ghost imaging,” Phys. Rev. A 78, 061802 (2008).
[CrossRef]

2006 (2)

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

D. Donoho, “Compressed sensing,” IEEE Trans. Inf. Theory 52(4), 1289–1306 (2006).
[CrossRef]

2005 (1)

2004 (1)

A. Gatti, E. Brambilla, M. Bache, L. A. Lugiato, “Ghost imaging with thermal light: comparing entanglement and classical correlation,” Phys. Rev. Lett. 93, 093602 (2004).
[CrossRef] [PubMed]

2000 (1)

S. G. Chang, B. Yu, M. Vetterli, “Adaptive wavelet thresholding for image denoising and compression,” IEEE Trans. Image Process. 9(9), 1532–1546 (2000).
[CrossRef]

1998 (1)

S. S. Chen, D. L. Donoho, M. A. Saunders, “Atomic decomposition by basis pursuit,” SIAM J. Sci. Comput. 20(1), 33–61 (1998).
[CrossRef]

1996 (1)

A. Said, W. Pearlman, “A new, fast, and efficient image codec based on set partitioning in hierarchical trees,” IEEE Trans. Circ. Syst. Video Technol. 6(3), 243–250 (1996).
[CrossRef]

1995 (1)

D. V. Strekalov, A. V. Sergienko, D. N. Klyshko, Y. H. Shih, “Observation of two-photon “ghost” interference and diffraction,” Phys. Rev. Lett. 74, 3600–3603 (1995).
[CrossRef] [PubMed]

1993 (1)

J. Shapiro, “Embedded image coding using zerotrees of wavelet coefficients,” IEEE Trans. Signal Proces. 41(12), 3445–3462 (1993).
[CrossRef]

1989 (1)

S. Mallat, “A theory for multiresolution signal decomposition: the wavelet representation,” IEEE Trans. Pattern Anal. 11(7), 674–693 (1989).
[CrossRef]

Aßmann, M.

M. Aßmann, M. Bayer, “Compressive adaptive computational ghost imaging,” Sci. Rep. 3, 1545 (2013).

Averbuch, A.

A. Averbuch, S. Dekel, S. Deutsch, “Adaptive compressed image sensing using dictionaries,” SIAM J. Imaging Sci. 5(1), 57–89 (2012).
[CrossRef]

Bache, M.

A. Gatti, E. Brambilla, M. Bache, L. A. Lugiato, “Ghost imaging with thermal light: comparing entanglement and classical correlation,” Phys. Rev. Lett. 93, 093602 (2004).
[CrossRef] [PubMed]

Baraniuk, R. G.

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

W. L. Chan, K. Charan, D. Takhar, K. F. Kelly, R. G. Baraniuk, D. M. Mittleman, “A single-pixel terahertz imaging system based on compressed sensing,” Appl. Phys. Lett. 93(12), 121105 (2008).
[CrossRef]

Bayer, M.

M. Aßmann, M. Bayer, “Compressive adaptive computational ghost imaging,” Sci. Rep. 3, 1545 (2013).

Berinde, R.

R. Berinde, P. Indyk, “Sequential sparse matching pursuit,” in Proc. 47th Annu. Allerton Conf. Commun. Control Comput., (2009), 36–43.

Bobin, J.

V. Studer, J. Bobin, M. Chahid, H. Moussavi, E. J. Candès, M. Dahan, “Compressive fluorescence microscopy for biological and hyperspectral imaging,” in Proceedings of the National Academy of Sciences, (2012), 109(26), E1679–E1687.
[CrossRef]

Brambilla, E.

A. Gatti, E. Brambilla, M. Bache, L. A. Lugiato, “Ghost imaging with thermal light: comparing entanglement and classical correlation,” Phys. Rev. Lett. 93, 093602 (2004).
[CrossRef] [PubMed]

Bromberg, Y.

Y. Bromberg, O. Katz, Y. Silberberg, “Ghost imaging with a single detector,” Phys. Rev. A 79(5), 053840 (2009).
[CrossRef]

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

Candès, E. J.

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

E. J. Candès, “Compressive sampling,” in Proc. Int. Cong. Math, (European Mathematical Society, Madrid, Spain, 2006), 3, pp. 1433–1452.

V. Studer, J. Bobin, M. Chahid, H. Moussavi, E. J. Candès, M. Dahan, “Compressive fluorescence microscopy for biological and hyperspectral imaging,” in Proceedings of the National Academy of Sciences, (2012), 109(26), E1679–E1687.
[CrossRef]

Castro, R.

J. Haupt, R. Nowak, R. Castro, “Adaptive sensing for sparse signal recovery,” in Proceedings of the 2009 IEEE Digital Signal Processing Workshop and 5th IEEE Signal Processing Education Workshop, (Marco Island, FL, Jan., 2009), 702–707.

Chahid, M.

V. Studer, J. Bobin, M. Chahid, H. Moussavi, E. J. Candès, M. Dahan, “Compressive fluorescence microscopy for biological and hyperspectral imaging,” in Proceedings of the National Academy of Sciences, (2012), 109(26), E1679–E1687.
[CrossRef]

Chan, W. L.

W. L. Chan, K. Charan, D. Takhar, K. F. Kelly, R. G. Baraniuk, D. M. Mittleman, “A single-pixel terahertz imaging system based on compressed sensing,” Appl. Phys. Lett. 93(12), 121105 (2008).
[CrossRef]

Chang, S. G.

S. G. Chang, B. Yu, M. Vetterli, “Adaptive wavelet thresholding for image denoising and compression,” IEEE Trans. Image Process. 9(9), 1532–1546 (2000).
[CrossRef]

Charan, K.

W. L. Chan, K. Charan, D. Takhar, K. F. Kelly, R. G. Baraniuk, D. M. Mittleman, “A single-pixel terahertz imaging system based on compressed sensing,” Appl. Phys. Lett. 93(12), 121105 (2008).
[CrossRef]

Chen, S. S.

S. S. Chen, D. L. Donoho, M. A. Saunders, “Atomic decomposition by basis pursuit,” SIAM J. Sci. Comput. 20(1), 33–61 (1998).
[CrossRef]

Chen, X. H.

Dahan, M.

V. Studer, J. Bobin, M. Chahid, H. Moussavi, E. J. Candès, M. Dahan, “Compressive fluorescence microscopy for biological and hyperspectral imaging,” in Proceedings of the National Academy of Sciences, (2012), 109(26), E1679–E1687.
[CrossRef]

Darabi, S.

P. Sen, S. Darabi, “Compressive dual photography,” Computer Graphics Forum 28(2), 609–618 (2009).
[CrossRef]

Davenport, M. A.

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

Dekel, S.

A. Averbuch, S. Dekel, S. Deutsch, “Adaptive compressed image sensing using dictionaries,” SIAM J. Imaging Sci. 5(1), 57–89 (2012).
[CrossRef]

Deutsch, S.

A. Averbuch, S. Dekel, S. Deutsch, “Adaptive compressed image sensing using dictionaries,” SIAM J. Imaging Sci. 5(1), 57–89 (2012).
[CrossRef]

Donoho, D.

D. Donoho, “Compressed sensing,” IEEE Trans. Inf. Theory 52(4), 1289–1306 (2006).
[CrossRef]

Donoho, D. L.

S. S. Chen, D. L. Donoho, M. A. Saunders, “Atomic decomposition by basis pursuit,” SIAM J. Sci. Comput. 20(1), 33–61 (1998).
[CrossRef]

Duarte, M. F.

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

Erdogan, H.

N. B. Karahanoglu, H. Erdogan, “A* orthogonal matching pursuit: best-first search for compressed sensing signal recovery,” Digit. Sig. Process. 22(4), 555–568 (2012).
[CrossRef]

Erkmen, B. I.

B. I. Erkmen, J. H. Shapiro, “Unified theory of ghost imaging with Gaussian-state light,” Phys. Rev. A 77(4), 043809 (2008).
[CrossRef]

Fan, H.

M. F. Li, Y. R. Zhang, X. F. Liu, X. R. Yao, K. H. Luo, H. Fan, L. A. Wu, “A double-threshold technique for fast time-correspondence imaging,” Appl. Phys. Lett. 103, 211119 (2013).
[CrossRef]

Fornasier, M.

M. Fornasier, H. Rauhut, “Iterative thresholding algorithms,” Appl. Comput. Harmon. Anal. 25(2), 187–208 (2008).
[CrossRef]

Gatti, A.

A. Gatti, E. Brambilla, M. Bache, L. A. Lugiato, “Ghost imaging with thermal light: comparing entanglement and classical correlation,” Phys. Rev. Lett. 93, 093602 (2004).
[CrossRef] [PubMed]

Gong, W. L.

W. L. Gong, S. S. Han, “Experimental investigation of the quality of lensless super-resolution ghost imaging via sparsity constraints,” Phys. Lett. A 376(17), 1519–1522 (2012).
[CrossRef]

Han, S. S.

W. L. Gong, S. S. Han, “Experimental investigation of the quality of lensless super-resolution ghost imaging via sparsity constraints,” Phys. Lett. A 376(17), 1519–1522 (2012).
[CrossRef]

Haupt, J.

J. Haupt, R. Nowak, R. Castro, “Adaptive sensing for sparse signal recovery,” in Proceedings of the 2009 IEEE Digital Signal Processing Workshop and 5th IEEE Signal Processing Education Workshop, (Marco Island, FL, Jan., 2009), 702–707.

Indyk, P.

R. Berinde, P. Indyk, “Sequential sparse matching pursuit,” in Proc. 47th Annu. Allerton Conf. Commun. Control Comput., (2009), 36–43.

Karahanoglu, N. B.

N. B. Karahanoglu, H. Erdogan, “A* orthogonal matching pursuit: best-first search for compressed sensing signal recovery,” Digit. Sig. Process. 22(4), 555–568 (2012).
[CrossRef]

Katz, O.

Y. Bromberg, O. Katz, Y. Silberberg, “Ghost imaging with a single detector,” Phys. Rev. A 79(5), 053840 (2009).
[CrossRef]

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

Kelly, K. F.

W. L. Chan, K. Charan, D. Takhar, K. F. Kelly, R. G. Baraniuk, D. M. Mittleman, “A single-pixel terahertz imaging system based on compressed sensing,” Appl. Phys. Lett. 93(12), 121105 (2008).
[CrossRef]

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

Klyshko, D. N.

D. V. Strekalov, A. V. Sergienko, D. N. Klyshko, Y. H. Shih, “Observation of two-photon “ghost” interference and diffraction,” Phys. Rev. Lett. 74, 3600–3603 (1995).
[CrossRef] [PubMed]

Laska, J. N.

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

Li, C. B.

C. B. Li, “An efficient algorithm for total variation regularization with applications to the single pixel camera and compressive sensing,” Master Thesis, Rice University, (2010).

Li, M. F.

M. F. Li, Y. R. Zhang, X. F. Liu, X. R. Yao, K. H. Luo, H. Fan, L. A. Wu, “A double-threshold technique for fast time-correspondence imaging,” Appl. Phys. Lett. 103, 211119 (2013).
[CrossRef]

Li, S.

Liu, X. F.

W. K. Yu, S. Li, X. R. Yao, X. F. Liu, L. A. Wu, G. J. Zhai, “Protocol based on compressed sensing for high-speed authentication and cryptographic key distribution over a multiparty optical network,” Appl. Opt. 52(33), 7882–7888 (2013).
[CrossRef]

M. F. Li, Y. R. Zhang, X. F. Liu, X. R. Yao, K. H. Luo, H. Fan, L. A. Wu, “A double-threshold technique for fast time-correspondence imaging,” Appl. Phys. Lett. 103, 211119 (2013).
[CrossRef]

Lugiato, L. A.

A. Gatti, E. Brambilla, M. Bache, L. A. Lugiato, “Ghost imaging with thermal light: comparing entanglement and classical correlation,” Phys. Rev. Lett. 93, 093602 (2004).
[CrossRef] [PubMed]

Luo, K. H.

M. F. Li, Y. R. Zhang, X. F. Liu, X. R. Yao, K. H. Luo, H. Fan, L. A. Wu, “A double-threshold technique for fast time-correspondence imaging,” Appl. Phys. Lett. 103, 211119 (2013).
[CrossRef]

Mallat, S.

S. Mallat, “A theory for multiresolution signal decomposition: the wavelet representation,” IEEE Trans. Pattern Anal. 11(7), 674–693 (1989).
[CrossRef]

S. Mallat, A wavelet tour of signal processing, the sparse way (Elsevier, 2009), pp. 340–346.

Mittleman, D. M.

W. L. Chan, K. Charan, D. Takhar, K. F. Kelly, R. G. Baraniuk, D. M. Mittleman, “A single-pixel terahertz imaging system based on compressed sensing,” Appl. Phys. Lett. 93(12), 121105 (2008).
[CrossRef]

Moussavi, H.

V. Studer, J. Bobin, M. Chahid, H. Moussavi, E. J. Candès, M. Dahan, “Compressive fluorescence microscopy for biological and hyperspectral imaging,” in Proceedings of the National Academy of Sciences, (2012), 109(26), E1679–E1687.
[CrossRef]

Nowak, R.

J. Haupt, R. Nowak, R. Castro, “Adaptive sensing for sparse signal recovery,” in Proceedings of the 2009 IEEE Digital Signal Processing Workshop and 5th IEEE Signal Processing Education Workshop, (Marco Island, FL, Jan., 2009), 702–707.

Pearlman, W.

A. Said, W. Pearlman, “A new, fast, and efficient image codec based on set partitioning in hierarchical trees,” IEEE Trans. Circ. Syst. Video Technol. 6(3), 243–250 (1996).
[CrossRef]

Rauhut, H.

M. Fornasier, H. Rauhut, “Iterative thresholding algorithms,” Appl. Comput. Harmon. Anal. 25(2), 187–208 (2008).
[CrossRef]

Romberg, J.

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

Said, A.

A. Said, W. Pearlman, “A new, fast, and efficient image codec based on set partitioning in hierarchical trees,” IEEE Trans. Circ. Syst. Video Technol. 6(3), 243–250 (1996).
[CrossRef]

Saunders, M. A.

S. S. Chen, D. L. Donoho, M. A. Saunders, “Atomic decomposition by basis pursuit,” SIAM J. Sci. Comput. 20(1), 33–61 (1998).
[CrossRef]

Sen, P.

P. Sen, S. Darabi, “Compressive dual photography,” Computer Graphics Forum 28(2), 609–618 (2009).
[CrossRef]

Sergienko, A. V.

D. V. Strekalov, A. V. Sergienko, D. N. Klyshko, Y. H. Shih, “Observation of two-photon “ghost” interference and diffraction,” Phys. Rev. Lett. 74, 3600–3603 (1995).
[CrossRef] [PubMed]

Shapiro, J.

J. Shapiro, “Embedded image coding using zerotrees of wavelet coefficients,” IEEE Trans. Signal Proces. 41(12), 3445–3462 (1993).
[CrossRef]

Shapiro, J. H.

B. I. Erkmen, J. H. Shapiro, “Unified theory of ghost imaging with Gaussian-state light,” Phys. Rev. A 77(4), 043809 (2008).
[CrossRef]

J. H. Shapiro, “Computational ghost imaging,” Phys. Rev. A 78, 061802 (2008).
[CrossRef]

Shih, Y. H.

D. V. Strekalov, A. V. Sergienko, D. N. Klyshko, Y. H. Shih, “Observation of two-photon “ghost” interference and diffraction,” Phys. Rev. Lett. 74, 3600–3603 (1995).
[CrossRef] [PubMed]

Silberberg, Y.

Y. Bromberg, O. Katz, Y. Silberberg, “Ghost imaging with a single detector,” Phys. Rev. A 79(5), 053840 (2009).
[CrossRef]

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

Strekalov, D. V.

D. V. Strekalov, A. V. Sergienko, D. N. Klyshko, Y. H. Shih, “Observation of two-photon “ghost” interference and diffraction,” Phys. Rev. Lett. 74, 3600–3603 (1995).
[CrossRef] [PubMed]

Studer, V.

V. Studer, J. Bobin, M. Chahid, H. Moussavi, E. J. Candès, M. Dahan, “Compressive fluorescence microscopy for biological and hyperspectral imaging,” in Proceedings of the National Academy of Sciences, (2012), 109(26), E1679–E1687.
[CrossRef]

Sun, T.

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

Takhar, D.

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

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Figures (10)

Fig. 1
Fig. 1

(a) A 256 × 256 pixels image of living cells, taken from the photo gallery of Matlab. (b) Two-level wavelet transform of image (a). (c) The three-level wavelet subtree.

Fig. 2
Fig. 2

Flowchart of adaptive compressive ghost imaging.

Fig. 3
Fig. 3

The simulation procedure for a gray-scale map. All figures are automatically gray-scale compensated. (a) 64 × 64 pixels shrunken image of Fig. 1(a). (b) Regions of large coefficients in Fig. 3(a) are adaptively searched by a one-step wavelet transform, and these areas are scanned with higher resolution in the next 128 × 128 image. (c) As in Step (b) but for Fig. 3(b) to produce the next 256 × 256 image. (d)–(f) are the reconstructed images corresponding, respectively, to 64 × 64, 128 × 128 and 256 × 256 pixels, using speckles of size 4 × 4, 2 × 2 and 1 × 1 pixels in our algorithm. (g) Wavelet transform of the final image; large wavelet coefficients are shown in white, small ones in black. (h) The result (g) converted back to a real space image using the inverse wavelet transform. For this example the total number of measurements needed is roughly 24.2% of the number of pixels of Fig. 1(a).

Fig. 4
Fig. 4

Experimental setup of adaptive compressive ghost imaging.

Fig. 5
Fig. 5

(a) The original target is a black-and-white 1951 USAF resolution test chart. (b) The experimental reconstruction image consisting of 256 × 256 pixels which is automatically gray-scale compensated.

Fig. 6
Fig. 6

Noise-added simulation results: (a) The original image also comes from the photo gallery of Matlab. Top row: images retrieved by Aßmann’s method after adding white noise obeying a normal distribution of (b) N(0, 102), (c) N(0, 202), (d) N(0, 302), with PSNRs of 27.470, 23.516 and 20.068 dB, respectively. Bottom row: (f)–(h) are the corresponding images reconstructed by our method, with PSNRs of 27.643, 24.344 and 20.322 dB, respectively. (e) and (i) are enlarged sections of (d) and (h), respectively.

Fig. 7
Fig. 7

The PSNRs of CCGI and ACGI vs. (a) the standard deviation of Gaussian white noise; (b) the mean Poisson noise; (c) the total acquisition rate.

Fig. 8
Fig. 8

Comparison between ACGI and other standard CS techniques which reconstruct the full 256 × 256 pixel image. (a)–(d): Images recovered by ACGI, RecPF, TVAL3 and SSMP, with a PSNR of 30.882, 35.663, 28.270 and 21.781 dB, and core computation times used for reconstruction of 3.024, 11.900, 25.217 and 702.618 s, respectively.

Fig. 9
Fig. 9

Simulation results of time-resolved 256 × 256 pixel images: (a)–(d) Original time-resolved images; (e)–(h) Corresponding reconstructed images with a PSNR of 25.730, 26.268, 24.228 and 26.243 dB, respectively.

Fig. 10
Fig. 10

Simulation procedure for color images. (a) A 256 × 256 pixel photo taken by ourselves. (b)–(d) correspond, respectively, to the red, green and blue components of (a). (e)–(g) are the corresponding ACGI recovered images, with a PSNR of 32.812, 32.633 and 33.185 dB, respectively. (h) is the color image retrieved by synthesizing the three components (e)–(g).

Equations (4)

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y = Φ x + e , x = Ψ x ,
min x x 1 subject to y Φ Ψ x 2 2 < ε ,
PSNR = 10 log 255 2 MSE
MSE = 1 s t i , j = 1 s , t [ T o ( i , j ) T ˜ ( i , j ) ] 2 ,

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