Abstract

We present a pseudo-inverse ghost imaging (PGI) technique which can dramatically enhance the spatial transverse resolution of pseudo-thermal ghost imaging (GI). In comparison with conventional GI, PGI can break the limitation on the imaging resolution imposed by the speckle’s transverse size on the object plane and also enables the reconstruction of an N-pixel image from much less than N measurements. This feature also allows high-resolution imaging of gray-scale objects. Experimental and numerical data assessing the performance of the technique are presented.

© 2015 Chinese Laser Press

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References

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2015 (2)

2014 (2)

C. Zhang, S. Guo, J. Cao, J. Guan, and F. Gao, “Object reconstitution using pseudo-inverse for ghost imaging,” Opt. Express 22, 30063–30073 (2014).
[Crossref]

R. Barankov and J. Mertz, “High-throughput imaging of self-luminous objects through a single optical fibre,” Nat. Commun. 5, 5581 (2014).
[Crossref]

2013 (1)

2012 (5)

J. Du, W. Gong, and S. Han, “The influence of sparsity property of images on ghost imaging with thermal light,” Opt. Lett. 37, 1067–1069 (2012).
[Crossref]

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

J. H. Shapiro and R. W. Boyd, “The physics of ghost imaging,” Quantum Inf. Process. 11, 949–993 (2012).
[Crossref]

J. Bertolotti, E. G. van Putten, C. Blum, A. Lagendijk, W. L. Vos, and A. P. Mosk, “Non-invasive imaging through opaque scattering layers,” Nature 491, 232–234 (2012).
[Crossref]

C. Zhao, W. Gong, M. Chen, E. Li, H. Wang, W. Xu, and S. Han, “Ghost imaging lidar via sparsity constraints,” Appl. Phys. Lett. 101, 141123 (2012).
[Crossref]

2011 (3)

2010 (2)

W. Gong and S. Han, “A method to improve the visibility of ghost images obtained by thermal light,” Phys. Lett. A. 374, 1005–1008 (2010).
[Crossref]

P. Clemente, V. Durán, V. Torres-Company, E. Tajahuerce, and J. Lancis, “Optical encryption based on computational ghost imaging,” Opt. Lett. 35, 2391–2393 (2010).
[Crossref]

2009 (2)

W. Gong, P. Zhang, X. Shen, and S. Han, “Ghost ‘pinhole’ imaging in Fraunhofer region,” Appl. Phys. Lett. 95, 071110 (2009).
[Crossref]

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

2005 (4)

D. Zhang, Y.-H. Zhai, L.-A. Wu, and X.-H. Chen, “Correlated two-photon imaging with true thermal light,” Opt. Lett. 30, 2354–2356 (2005).
[Crossref]

D. Z. Cao, J. Xiong, and K. Wang, “Geometrical optics in correlated imaging systems,” Phys. Rev. A 71, 013801 (2005).
[Crossref]

F. Ferri, D. Magatti, A. Gatti, M. Bache, E. Brambilla, and L. A. Lugiato, “High-resolution ghost image and ghost diffraction experiments with thermal light,” Phys. Rev. Lett. 94, 183602 (2005).
[Crossref]

M. D. Angelo and Y. H. Shih, “Quantum imaging,” Laser Phys. Lett. 2, 567–596 (2005).
[Crossref]

2004 (2)

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

R. S. Bennink, S. J. Bentley, R. W. Boyd, and J. C. Howell, “Quantum and classical coincidence imaging,” Phys. Rev. Lett. 92, 033601 (2004).
[Crossref]

Angelo, M. D.

M. D. Angelo and Y. H. Shih, “Quantum imaging,” Laser Phys. Lett. 2, 567–596 (2005).
[Crossref]

Bache, M.

F. Ferri, D. Magatti, A. Gatti, M. Bache, E. Brambilla, and L. A. Lugiato, “High-resolution ghost image and ghost diffraction experiments with thermal light,” Phys. Rev. Lett. 94, 183602 (2005).
[Crossref]

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

Barankov, R.

R. Barankov and J. Mertz, “High-throughput imaging of self-luminous objects through a single optical fibre,” Nat. Commun. 5, 5581 (2014).
[Crossref]

Bennink, R. S.

R. S. Bennink, S. J. Bentley, R. W. Boyd, and J. C. Howell, “Quantum and classical coincidence imaging,” Phys. Rev. Lett. 92, 033601 (2004).
[Crossref]

Bentley, S. J.

R. S. Bennink, S. J. Bentley, R. W. Boyd, and J. C. Howell, “Quantum and classical coincidence imaging,” Phys. Rev. Lett. 92, 033601 (2004).
[Crossref]

Bertolotti, J.

J. Bertolotti, E. G. van Putten, C. Blum, A. Lagendijk, W. L. Vos, and A. P. Mosk, “Non-invasive imaging through opaque scattering layers,” Nature 491, 232–234 (2012).
[Crossref]

Blum, C.

J. Bertolotti, E. G. van Putten, C. Blum, A. Lagendijk, W. L. Vos, and A. P. Mosk, “Non-invasive imaging through opaque scattering layers,” Nature 491, 232–234 (2012).
[Crossref]

Boyd, R. W.

J. H. Shapiro and R. W. Boyd, “The physics of ghost imaging,” Quantum Inf. Process. 11, 949–993 (2012).
[Crossref]

P. Zerom, K. W. C. Chan, J. C. Howell, and R. W. Boyd, “Entangled-photon compressive ghost imaging,” Phys. Rev. A 84, 061804 (2011).
[Crossref]

R. S. Bennink, S. J. Bentley, R. W. Boyd, and J. C. Howell, “Quantum and classical coincidence imaging,” Phys. Rev. Lett. 92, 033601 (2004).
[Crossref]

Brambilla, E.

F. Ferri, D. Magatti, A. Gatti, M. Bache, E. Brambilla, and L. A. Lugiato, “High-resolution ghost image and ghost diffraction experiments with thermal light,” Phys. Rev. Lett. 94, 183602 (2005).
[Crossref]

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

Bromberg, Y.

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

Cao, D. Z.

D. Z. Cao, J. Xiong, and K. Wang, “Geometrical optics in correlated imaging systems,” Phys. Rev. A 71, 013801 (2005).
[Crossref]

Cao, J.

Chan, K. W. C.

P. Zerom, K. W. C. Chan, J. C. Howell, and R. W. Boyd, “Entangled-photon compressive ghost imaging,” Phys. Rev. A 84, 061804 (2011).
[Crossref]

Chen, M.

C. Zhao, W. Gong, M. Chen, E. Li, H. Wang, W. Xu, and S. Han, “Ghost imaging lidar via sparsity constraints,” Appl. Phys. Lett. 101, 141123 (2012).
[Crossref]

W. Gong, C. Zhao, J. Jiao, E. Li, M. Chen, H. Wang, W. Xu, and S. Han, “Three-dimensional ghost imaging ladar,” arXiv: 1301.5767 (2013).

Chen, X.-H.

Chountasis, S.

S. Chountasis, V. N. Katsikis, and D. Pappas, “Digital image reconstruction in the spectral domain utilizing the Moore-Penrose inverse,” Math. Prob. Eng.2010, 750352 (2010).

Clemente, P.

Du, J.

Durán, V.

Ferri, F.

F. Ferri, D. Magatti, A. Gatti, M. Bache, E. Brambilla, and L. A. Lugiato, “High-resolution ghost image and ghost diffraction experiments with thermal light,” Phys. Rev. Lett. 94, 183602 (2005).
[Crossref]

Fink, M.

Fu, L.

Gao, F.

Gatti, A.

F. Ferri, D. Magatti, A. Gatti, M. Bache, E. Brambilla, and L. A. Lugiato, “High-resolution ghost image and ghost diffraction experiments with thermal light,” Phys. Rev. Lett. 94, 183602 (2005).
[Crossref]

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

Gigan, S.

Gong, W.

W. Gong and S. Han, “High-resolution far-field ghost imaging via sparsity constraint,” Sci. Rep. 5, 9280 (2015).
[Crossref]

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

C. Zhao, W. Gong, M. Chen, E. Li, H. Wang, W. Xu, and S. Han, “Ghost imaging lidar via sparsity constraints,” Appl. Phys. Lett. 101, 141123 (2012).
[Crossref]

J. Du, W. Gong, and S. Han, “The influence of sparsity property of images on ghost imaging with thermal light,” Opt. Lett. 37, 1067–1069 (2012).
[Crossref]

W. Gong and S. Han, “Correlated imaging in scattering media,” Opt. Lett. 36, 394–396 (2011).
[Crossref]

W. Gong and S. Han, “A method to improve the visibility of ghost images obtained by thermal light,” Phys. Lett. A. 374, 1005–1008 (2010).
[Crossref]

W. Gong, P. Zhang, X. Shen, and S. Han, “Ghost ‘pinhole’ imaging in Fraunhofer region,” Appl. Phys. Lett. 95, 071110 (2009).
[Crossref]

W. Gong, C. Zhao, J. Jiao, E. Li, M. Chen, H. Wang, W. Xu, and S. Han, “Three-dimensional ghost imaging ladar,” arXiv: 1301.5767 (2013).

Guan, J.

Guo, Q.

Guo, S.

Han, S.

W. Gong and S. Han, “High-resolution far-field ghost imaging via sparsity constraint,” Sci. Rep. 5, 9280 (2015).
[Crossref]

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

C. Zhao, W. Gong, M. Chen, E. Li, H. Wang, W. Xu, and S. Han, “Ghost imaging lidar via sparsity constraints,” Appl. Phys. Lett. 101, 141123 (2012).
[Crossref]

J. Du, W. Gong, and S. Han, “The influence of sparsity property of images on ghost imaging with thermal light,” Opt. Lett. 37, 1067–1069 (2012).
[Crossref]

W. Gong and S. Han, “Correlated imaging in scattering media,” Opt. Lett. 36, 394–396 (2011).
[Crossref]

W. Gong and S. Han, “A method to improve the visibility of ghost images obtained by thermal light,” Phys. Lett. A. 374, 1005–1008 (2010).
[Crossref]

W. Gong, P. Zhang, X. Shen, and S. Han, “Ghost ‘pinhole’ imaging in Fraunhofer region,” Appl. Phys. Lett. 95, 071110 (2009).
[Crossref]

W. Gong, C. Zhao, J. Jiao, E. Li, M. Chen, H. Wang, W. Xu, and S. Han, “Three-dimensional ghost imaging ladar,” arXiv: 1301.5767 (2013).

Howell, J. C.

P. Zerom, K. W. C. Chan, J. C. Howell, and R. W. Boyd, “Entangled-photon compressive ghost imaging,” Phys. Rev. A 84, 061804 (2011).
[Crossref]

R. S. Bennink, S. J. Bentley, R. W. Boyd, and J. C. Howell, “Quantum and classical coincidence imaging,” Phys. Rev. Lett. 92, 033601 (2004).
[Crossref]

Jiao, J.

W. Gong, C. Zhao, J. Jiao, E. Li, M. Chen, H. Wang, W. Xu, and S. Han, “Three-dimensional ghost imaging ladar,” arXiv: 1301.5767 (2013).

Katsikis, V. N.

S. Chountasis, V. N. Katsikis, and D. Pappas, “Digital image reconstruction in the spectral domain utilizing the Moore-Penrose inverse,” Math. Prob. Eng.2010, 750352 (2010).

Katz, O.

Kolenderska, S. M.

Lagendijk, A.

J. Bertolotti, E. G. van Putten, C. Blum, A. Lagendijk, W. L. Vos, and A. P. Mosk, “Non-invasive imaging through opaque scattering layers,” Nature 491, 232–234 (2012).
[Crossref]

Lancis, J.

Li, E.

C. Zhao, W. Gong, M. Chen, E. Li, H. Wang, W. Xu, and S. Han, “Ghost imaging lidar via sparsity constraints,” Appl. Phys. Lett. 101, 141123 (2012).
[Crossref]

W. Gong, C. Zhao, J. Jiao, E. Li, M. Chen, H. Wang, W. Xu, and S. Han, “Three-dimensional ghost imaging ladar,” arXiv: 1301.5767 (2013).

Li, S.

Lugiato, L. A.

F. Ferri, D. Magatti, A. Gatti, M. Bache, E. Brambilla, and L. A. Lugiato, “High-resolution ghost image and ghost diffraction experiments with thermal light,” Phys. Rev. Lett. 94, 183602 (2005).
[Crossref]

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

Magatti, D.

F. Ferri, D. Magatti, A. Gatti, M. Bache, E. Brambilla, and L. A. Lugiato, “High-resolution ghost image and ghost diffraction experiments with thermal light,” Phys. Rev. Lett. 94, 183602 (2005).
[Crossref]

Mertz, J.

R. Barankov and J. Mertz, “High-throughput imaging of self-luminous objects through a single optical fibre,” Nat. Commun. 5, 5581 (2014).
[Crossref]

Mosk, A. P.

J. Bertolotti, E. G. van Putten, C. Blum, A. Lagendijk, W. L. Vos, and A. P. Mosk, “Non-invasive imaging through opaque scattering layers,” Nature 491, 232–234 (2012).
[Crossref]

Pappas, D.

S. Chountasis, V. N. Katsikis, and D. Pappas, “Digital image reconstruction in the spectral domain utilizing the Moore-Penrose inverse,” Math. Prob. Eng.2010, 750352 (2010).

Shapiro, J. H.

J. H. Shapiro and R. W. Boyd, “The physics of ghost imaging,” Quantum Inf. Process. 11, 949–993 (2012).
[Crossref]

Shen, X.

W. Gong, P. Zhang, X. Shen, and S. Han, “Ghost ‘pinhole’ imaging in Fraunhofer region,” Appl. Phys. Lett. 95, 071110 (2009).
[Crossref]

Shih, Y. H.

M. D. Angelo and Y. H. Shih, “Quantum imaging,” Laser Phys. Lett. 2, 567–596 (2005).
[Crossref]

Silberberg, Y.

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

Tajahuerce, E.

Tian, N.

Torres-Company, V.

van Putten, E. G.

J. Bertolotti, E. G. van Putten, C. Blum, A. Lagendijk, W. L. Vos, and A. P. Mosk, “Non-invasive imaging through opaque scattering layers,” Nature 491, 232–234 (2012).
[Crossref]

Vos, W. L.

J. Bertolotti, E. G. van Putten, C. Blum, A. Lagendijk, W. L. Vos, and A. P. Mosk, “Non-invasive imaging through opaque scattering layers,” Nature 491, 232–234 (2012).
[Crossref]

Wang, A.

Wang, H.

C. Zhao, W. Gong, M. Chen, E. Li, H. Wang, W. Xu, and S. Han, “Ghost imaging lidar via sparsity constraints,” Appl. Phys. Lett. 101, 141123 (2012).
[Crossref]

W. Gong, C. Zhao, J. Jiao, E. Li, M. Chen, H. Wang, W. Xu, and S. Han, “Three-dimensional ghost imaging ladar,” arXiv: 1301.5767 (2013).

Wang, K.

D. Z. Cao, J. Xiong, and K. Wang, “Geometrical optics in correlated imaging systems,” Phys. Rev. A 71, 013801 (2005).
[Crossref]

Wu, L.

Wu, L.-A.

Xiong, J.

D. Z. Cao, J. Xiong, and K. Wang, “Geometrical optics in correlated imaging systems,” Phys. Rev. A 71, 013801 (2005).
[Crossref]

Xu, D.

Xu, W.

C. Zhao, W. Gong, M. Chen, E. Li, H. Wang, W. Xu, and S. Han, “Ghost imaging lidar via sparsity constraints,” Appl. Phys. Lett. 101, 141123 (2012).
[Crossref]

W. Gong, C. Zhao, J. Jiao, E. Li, M. Chen, H. Wang, W. Xu, and S. Han, “Three-dimensional ghost imaging ladar,” arXiv: 1301.5767 (2013).

Yao, X.

Yu, W.

Zerom, P.

P. Zerom, K. W. C. Chan, J. C. Howell, and R. W. Boyd, “Entangled-photon compressive ghost imaging,” Phys. Rev. A 84, 061804 (2011).
[Crossref]

Zhai, G.

Zhai, Y.-H.

Zhang, C.

Zhang, D.

Zhang, P.

W. Gong, P. Zhang, X. Shen, and S. Han, “Ghost ‘pinhole’ imaging in Fraunhofer region,” Appl. Phys. Lett. 95, 071110 (2009).
[Crossref]

Zhao, C.

C. Zhao, W. Gong, M. Chen, E. Li, H. Wang, W. Xu, and S. Han, “Ghost imaging lidar via sparsity constraints,” Appl. Phys. Lett. 101, 141123 (2012).
[Crossref]

W. Gong, C. Zhao, J. Jiao, E. Li, M. Chen, H. Wang, W. Xu, and S. Han, “Three-dimensional ghost imaging ladar,” arXiv: 1301.5767 (2013).

Appl. Phys. Lett. (3)

W. Gong, P. Zhang, X. Shen, and S. Han, “Ghost ‘pinhole’ imaging in Fraunhofer region,” Appl. Phys. Lett. 95, 071110 (2009).
[Crossref]

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

C. Zhao, W. Gong, M. Chen, E. Li, H. Wang, W. Xu, and S. Han, “Ghost imaging lidar via sparsity constraints,” Appl. Phys. Lett. 101, 141123 (2012).
[Crossref]

Laser Phys. Lett. (1)

M. D. Angelo and Y. H. Shih, “Quantum imaging,” Laser Phys. Lett. 2, 567–596 (2005).
[Crossref]

Nat. Commun. (1)

R. Barankov and J. Mertz, “High-throughput imaging of self-luminous objects through a single optical fibre,” Nat. Commun. 5, 5581 (2014).
[Crossref]

Nature (1)

J. Bertolotti, E. G. van Putten, C. Blum, A. Lagendijk, W. L. Vos, and A. P. Mosk, “Non-invasive imaging through opaque scattering layers,” Nature 491, 232–234 (2012).
[Crossref]

Opt. Express (1)

Opt. Lett. (7)

Phys. Lett. A (1)

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

Phys. Lett. A. (1)

W. Gong and S. Han, “A method to improve the visibility of ghost images obtained by thermal light,” Phys. Lett. A. 374, 1005–1008 (2010).
[Crossref]

Phys. Rev. A (2)

D. Z. Cao, J. Xiong, and K. Wang, “Geometrical optics in correlated imaging systems,” Phys. Rev. A 71, 013801 (2005).
[Crossref]

P. Zerom, K. W. C. Chan, J. C. Howell, and R. W. Boyd, “Entangled-photon compressive ghost imaging,” Phys. Rev. A 84, 061804 (2011).
[Crossref]

Phys. Rev. Lett. (3)

F. Ferri, D. Magatti, A. Gatti, M. Bache, E. Brambilla, and L. A. Lugiato, “High-resolution ghost image and ghost diffraction experiments with thermal light,” Phys. Rev. Lett. 94, 183602 (2005).
[Crossref]

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

R. S. Bennink, S. J. Bentley, R. W. Boyd, and J. C. Howell, “Quantum and classical coincidence imaging,” Phys. Rev. Lett. 92, 033601 (2004).
[Crossref]

Quantum Inf. Process. (1)

J. H. Shapiro and R. W. Boyd, “The physics of ghost imaging,” Quantum Inf. Process. 11, 949–993 (2012).
[Crossref]

Sci. Rep. (1)

W. Gong and S. Han, “High-resolution far-field ghost imaging via sparsity constraint,” Sci. Rep. 5, 9280 (2015).
[Crossref]

Other (2)

S. Chountasis, V. N. Katsikis, and D. Pappas, “Digital image reconstruction in the spectral domain utilizing the Moore-Penrose inverse,” Math. Prob. Eng.2010, 750352 (2010).

W. Gong, C. Zhao, J. Jiao, E. Li, M. Chen, H. Wang, W. Xu, and S. Han, “Three-dimensional ghost imaging ladar,” arXiv: 1301.5767 (2013).

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Figures (4)

Fig. 1.
Fig. 1. Experimental schematic of PGI with pseudo-thermal light. The blue dashed line and the red solid line in the bottom right of the schematic correspond to the GI reconstruction method ( O GI = Ψ T ( B I B ) ) and PGI reconstruction method ( O PGI = Ψ ( B I B ) ), respectively.
Fig. 2.
Fig. 2. Results of experimental demonstration of high-resolution PGI. (a)  Ψ T Ψ ; (b) GI reconstruction results using K = 10,000 measurements. Left column of (c)–(g), Ψ Ψ in different measurements K ; middle column of (c)–(g), five slits reconstructed by PGI method in different measurements K ; right column of (c)–(g), transmission aperture (“zhong” ring) recovered by PGI method in different measurements K . (c)  K = 1000 (the compression ratio η = K N pix = 0.1 , namely 10% Nyquist limit); (d)  K = 3000 ( η = 0.3 , namely 30% Nyquist limit); (e)  K = 5000 ( η = 0.5 , namely 50% Nyquist limit); (f)  K = 8000 ( η = 0.8 , namely 80% Nyquist limit); (g)  K = 10,000 ( η = 1.0 , namely 100% Nyquist limit).
Fig. 3.
Fig. 3. Numerical experimental demonstration of high-resolution PGI for gray-scale objects. (a) Original object; (b) GI reconstruction result using K = 10,000 measurements; (c)–(g) reconstruction results obtained by PGI method with compression ratio η = 0.1 , 0.3, 0.5, 0.8, and 1.0, respectively.
Fig. 4.
Fig. 4. Performance between PSNR and compression ratio η based on the results obtained in Fig. 2. Solid curve with red squares shows PGI reconstruction results of five slits and dashed curve with green circles corresponds to PGI reconstruction results of the transmission aperture (“zhong” ring).

Equations (7)

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O GI ( x , y ) = 1 K s = 1 K I r s ( x , y ) ( B s B s ) = 1 K s = 1 K ( I r s ( x , y ) I r s ( x , y ) ) ( B s B s ) .
O GI = 1 K ( Φ I Φ ) T ( B I B ) .
Φ = [ I r 1 ( 1,1 ) I r 1 ( 1,2 ) I r 1 ( m , n ) I r 2 ( 1,1 ) I r 2 ( 1,2 ) I r 2 ( m , n ) I r s ( 1,1 ) I r s ( 1,2 ) I r s ( m , n ) I r K ( 1,1 ) I r K ( 1,2 ) I r K ( m , n ) ] .
O GI = α K ( Φ I Φ ) T ( Φ I Φ ) O = α K Ψ T Ψ O .
O PGI = 1 K Ψ ( B I B ) = α K Ψ Ψ O .
PSNR = 10 × log 10 [ ( 2 p 1 ) 2 MSE ] .
MSE = 1 N pix i , j [ O PGI ( x i , y j ) O ( x i , y j ) ] 2 .

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