Abstract

We present an experimental demonstration of edge detection based on ghost imaging (GI) in the gradient domain. Through modification of a random light field, gradient GI (GGI) can directly give the edge of an object without needing the original image. As edges of real objects are usually sparser than the original objects, the signal-to-noise ratio (SNR) of the edge detection result will be dramatically enhanced, especially for large-area, high-transmittance objects. In this study, we experimentally perform one- and two-dimensional edge detection with a double-slit based on GI and GGI. The use of GGI improves the SNR significantly in both cases. Gray-scale objects are also studied by the use of simulation. The special advantages of GI will make the edge detection based on GGI be valuable in real applications.

© 2015 Optical Society of America

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References

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    [Crossref]
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2015 (2)

M. Kmiec and A. Glowacz, “Object detection in security applications using dominant edge directions,” Pattern Recognit. Lett. 52, 72–79 (2015).
[Crossref]

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

2014 (1)

2012 (3)

2011 (2)

2010 (3)

M. N. O’Sullivan, K. W. C. Chan, and R. W. Boyd, “Comparison of the signal-to-noise characteristics of quantum versus thermal ghost imaging,” Phys. Rev. A 82, 053803 (2010).
[Crossref]

X. F. Li, S. Q. Zhang, X. Pan, P. Dale, and R. Cropp, “Straight road edge detection from high-resolution remote sensing images based on the ridgelet transform with the revised parallel-beam Radon transform,” Int. J. Remote Sensing 31, 5041–5059 (2010).
[Crossref]

F. Ferri, D. Magatti, L. A. Lugiato, and A. Gatti, “Differential ghost imaging,” Phys. Rev. Lett. 104, 253603 (2010).
[Crossref] [PubMed]

2009 (3)

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

B. I. Erkmen and J. H. Shapiro, “Signal-to-noise ratio of Gaussian-state ghost imaging,” Phys. Rev. A 79, 023833 (2009).
[Crossref]

S. Yi, D. Labate, G. R. Easley, and H. Krim, “A shearlet approach to edge analysis and detection,” IEEE Trans. Image Process. 18, 929–941 (2009).
[Crossref] [PubMed]

2008 (1)

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

2006 (2)

G. Scarcelli, V. Berardi, and Y. H. Shih, “Phase-conjugate mirror via two-photon thermal light imaging,” Appl. Phys. Lett. 88, 061106 (2006).
[Crossref]

A. Gatti, M. Bache, D. Magatti, E. Brambilla, F. Ferri, and L. A. Lugiato, “Coherent imaging with pseudo-thermal incoherent light,” J. Mod. Opt. 53, 739–760 (2006).
[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] [PubMed]

A. Valencia, G. Scarcelli, M. D. Angelo, and Y. H. Shih, “Two-photon imaging with thermal light,” Phys. Rev. Lett. 94, 063601 (2005).
[Crossref] [PubMed]

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

P. Bao, L. Zhang, and X. L. Wu, “Canny edge detection enhancement by scale multiplication,” IEEE Trans. PAMI 27, 1485 (2005).
[Crossref]

2002 (1)

L. Zhang and P. Bao, “Edge detection by scale multiplication in wavelet domain,” Pattern Recognit. Lett. 23, 1771–1784 (2002).
[Crossref]

2001 (1)

O. K. Kwon, D. G. Sim, and R. H. Park, “Robust Hausdorff distance matching algorithms using pyramidal structures,” Pattern Recognit. 34, 2005–2013 (2001).
[Crossref]

1986 (1)

J. Canny, “A computational approach to edge detection,” IEEE Trans. PAMI 8, 679–698 (1986).
[Crossref]

Angelo, M. D.

A. Valencia, G. Scarcelli, M. D. Angelo, and Y. H. Shih, “Two-photon imaging with thermal light,” Phys. Rev. Lett. 94, 063601 (2005).
[Crossref] [PubMed]

Bache, M.

A. Gatti, M. Bache, D. Magatti, E. Brambilla, F. Ferri, and L. A. Lugiato, “Coherent imaging with pseudo-thermal incoherent light,” J. Mod. Opt. 53, 739–760 (2006).
[Crossref]

Bao, P.

P. Bao, L. Zhang, and X. L. Wu, “Canny edge detection enhancement by scale multiplication,” IEEE Trans. PAMI 27, 1485 (2005).
[Crossref]

L. Zhang and P. Bao, “Edge detection by scale multiplication in wavelet domain,” Pattern Recognit. Lett. 23, 1771–1784 (2002).
[Crossref]

Berardi, V.

G. Scarcelli, V. Berardi, and Y. H. Shih, “Phase-conjugate mirror via two-photon thermal light imaging,” Appl. Phys. Lett. 88, 061106 (2006).
[Crossref]

Boyd, R. W.

M. N. O’Sullivan, K. W. C. Chan, and R. W. Boyd, “Comparison of the signal-to-noise characteristics of quantum versus thermal ghost imaging,” Phys. Rev. A 82, 053803 (2010).
[Crossref]

Brambilla, E.

A. Gatti, M. Bache, D. Magatti, E. Brambilla, F. Ferri, and L. A. Lugiato, “Coherent imaging with pseudo-thermal incoherent light,” J. Mod. Opt. 53, 739–760 (2006).
[Crossref]

Bromberg, Y.

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

Canny, J.

J. Canny, “A computational approach to edge detection,” IEEE Trans. PAMI 8, 679–698 (1986).
[Crossref]

Cao, D. Z.

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

Chan, K. W. C.

M. N. O’Sullivan, K. W. C. Chan, and R. W. Boyd, “Comparison of the signal-to-noise characteristics of quantum versus thermal ghost imaging,” Phys. Rev. A 82, 053803 (2010).
[Crossref]

Chen, M. L.

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

Chen, X. H.

Cropp, R.

X. F. Li, S. Q. Zhang, X. Pan, P. Dale, and R. Cropp, “Straight road edge detection from high-resolution remote sensing images based on the ridgelet transform with the revised parallel-beam Radon transform,” Int. J. Remote Sensing 31, 5041–5059 (2010).
[Crossref]

Dale, P.

X. F. Li, S. Q. Zhang, X. Pan, P. Dale, and R. Cropp, “Straight road edge detection from high-resolution remote sensing images based on the ridgelet transform with the revised parallel-beam Radon transform,” Int. J. Remote Sensing 31, 5041–5059 (2010).
[Crossref]

Easley, G. R.

S. Yi, D. Labate, G. R. Easley, and H. Krim, “A shearlet approach to edge analysis and detection,” IEEE Trans. Image Process. 18, 929–941 (2009).
[Crossref] [PubMed]

Edgar, M P.

Erkmen, B. I.

B. I. Erkmen and J. H. Shapiro, “Signal-to-noise ratio of Gaussian-state ghost imaging,” Phys. Rev. A 79, 023833 (2009).
[Crossref]

Ferri, F.

F. Ferri, D. Magatti, L. A. Lugiato, and A. Gatti, “Differential ghost imaging,” Phys. Rev. Lett. 104, 253603 (2010).
[Crossref] [PubMed]

A. Gatti, M. Bache, D. Magatti, E. Brambilla, F. Ferri, and L. A. Lugiato, “Coherent imaging with pseudo-thermal incoherent light,” J. Mod. Opt. 53, 739–760 (2006).
[Crossref]

Fu, L.

Gatti, A.

F. Ferri, D. Magatti, L. A. Lugiato, and A. Gatti, “Differential ghost imaging,” Phys. Rev. Lett. 104, 253603 (2010).
[Crossref] [PubMed]

A. Gatti, M. Bache, D. Magatti, E. Brambilla, F. Ferri, and L. A. Lugiato, “Coherent imaging with pseudo-thermal incoherent light,” J. Mod. Opt. 53, 739–760 (2006).
[Crossref]

Glowacz, A.

M. Kmiec and A. Glowacz, “Object detection in security applications using dominant edge directions,” Pattern Recognit. Lett. 52, 72–79 (2015).
[Crossref]

Gong, W. L.

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

Guo, Q. C.

Han, S. S.

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

Hardy, N. D.

N. D. Hardy and J. H. Shapiro, “Reflective ghost imaging through turbulence,” Phys. Rev. A 84, 063824 (2011).
[Crossref]

Katz, O.

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

Kmiec, M.

M. Kmiec and A. Glowacz, “Object detection in security applications using dominant edge directions,” Pattern Recognit. Lett. 52, 72–79 (2015).
[Crossref]

Krim, H.

S. Yi, D. Labate, G. R. Easley, and H. Krim, “A shearlet approach to edge analysis and detection,” IEEE Trans. Image Process. 18, 929–941 (2009).
[Crossref] [PubMed]

Kwon, O. K.

O. K. Kwon, D. G. Sim, and R. H. Park, “Robust Hausdorff distance matching algorithms using pyramidal structures,” Pattern Recognit. 34, 2005–2013 (2001).
[Crossref]

Labate, D.

S. Yi, D. Labate, G. R. Easley, and H. Krim, “A shearlet approach to edge analysis and detection,” IEEE Trans. Image Process. 18, 929–941 (2009).
[Crossref] [PubMed]

Li, E. R.

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

Li, L. Z.

Li, X. F.

X. F. Li, S. Q. Zhang, X. Pan, P. Dale, and R. Cropp, “Straight road edge detection from high-resolution remote sensing images based on the ridgelet transform with the revised parallel-beam Radon transform,” Int. J. Remote Sensing 31, 5041–5059 (2010).
[Crossref]

Liu, X. F.

Lugiato, L. A.

F. Ferri, D. Magatti, L. A. Lugiato, and A. Gatti, “Differential ghost imaging,” Phys. Rev. Lett. 104, 253603 (2010).
[Crossref] [PubMed]

A. Gatti, M. Bache, D. Magatti, E. Brambilla, F. Ferri, and L. A. Lugiato, “Coherent imaging with pseudo-thermal incoherent light,” J. Mod. Opt. 53, 739–760 (2006).
[Crossref]

Ma, X.

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

Magatti, D.

F. Ferri, D. Magatti, L. A. Lugiato, and A. Gatti, “Differential ghost imaging,” Phys. Rev. Lett. 104, 253603 (2010).
[Crossref] [PubMed]

A. Gatti, M. Bache, D. Magatti, E. Brambilla, F. Ferri, and L. A. Lugiato, “Coherent imaging with pseudo-thermal incoherent light,” J. Mod. Opt. 53, 739–760 (2006).
[Crossref]

O’Sullivan, M. N.

M. N. O’Sullivan, K. W. C. Chan, and R. W. Boyd, “Comparison of the signal-to-noise characteristics of quantum versus thermal ghost imaging,” Phys. Rev. A 82, 053803 (2010).
[Crossref]

Padgett, M. J.

Pan, X.

X. F. Li, S. Q. Zhang, X. Pan, P. Dale, and R. Cropp, “Straight road edge detection from high-resolution remote sensing images based on the ridgelet transform with the revised parallel-beam Radon transform,” Int. J. Remote Sensing 31, 5041–5059 (2010).
[Crossref]

Park, R. H.

O. K. Kwon, D. G. Sim, and R. H. Park, “Robust Hausdorff distance matching algorithms using pyramidal structures,” Pattern Recognit. 34, 2005–2013 (2001).
[Crossref]

Scarcelli, G.

G. Scarcelli, V. Berardi, and Y. H. Shih, “Phase-conjugate mirror via two-photon thermal light imaging,” Appl. Phys. Lett. 88, 061106 (2006).
[Crossref]

A. Valencia, G. Scarcelli, M. D. Angelo, and Y. H. Shih, “Two-photon imaging with thermal light,” Phys. Rev. Lett. 94, 063601 (2005).
[Crossref] [PubMed]

Shapiro, J. H.

B. Sun, S. S. Welsh, M P. Edgar, J. H. Shapiro, and M. J. Padgett, “Normalized ghost imaging,” Opt. Express 20, 16892–16901 (2012).
[Crossref]

N. D. Hardy and J. H. Shapiro, “Reflective ghost imaging through turbulence,” Phys. Rev. A 84, 063824 (2011).
[Crossref]

B. I. Erkmen and J. H. Shapiro, “Signal-to-noise ratio of Gaussian-state ghost imaging,” Phys. Rev. A 79, 023833 (2009).
[Crossref]

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

Shih, Y. H.

G. Scarcelli, V. Berardi, and Y. H. Shih, “Phase-conjugate mirror via two-photon thermal light imaging,” Appl. Phys. Lett. 88, 061106 (2006).
[Crossref]

A. Valencia, G. Scarcelli, M. D. Angelo, and Y. H. Shih, “Two-photon imaging with thermal light,” Phys. Rev. Lett. 94, 063601 (2005).
[Crossref] [PubMed]

Silberberg, Y.

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

Sim, D. G.

O. K. Kwon, D. G. Sim, and R. H. Park, “Robust Hausdorff distance matching algorithms using pyramidal structures,” Pattern Recognit. 34, 2005–2013 (2001).
[Crossref]

Sun, B.

Tian, N.

Valencia, A.

A. Valencia, G. Scarcelli, M. D. Angelo, and Y. H. Shih, “Two-photon imaging with thermal light,” Phys. Rev. Lett. 94, 063601 (2005).
[Crossref] [PubMed]

Wang, A. L.

Wang, H.

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

Wang, K. G.

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

Welsh, S. S.

Wu, L. A.

Wu, X. L.

P. Bao, L. Zhang, and X. L. Wu, “Canny edge detection enhancement by scale multiplication,” IEEE Trans. PAMI 27, 1485 (2005).
[Crossref]

Xiong, J.

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

Xu, D. L.

Xu, W. D.

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

Yao, X. R.

Yi, S.

S. Yi, D. Labate, G. R. Easley, and H. Krim, “A shearlet approach to edge analysis and detection,” IEEE Trans. Image Process. 18, 929–941 (2009).
[Crossref] [PubMed]

Yu, W. K.

Zhai, G. J.

Zhai, Y. H.

Zhang, D.

Zhang, L.

P. Bao, L. Zhang, and X. L. Wu, “Canny edge detection enhancement by scale multiplication,” IEEE Trans. PAMI 27, 1485 (2005).
[Crossref]

L. Zhang and P. Bao, “Edge detection by scale multiplication in wavelet domain,” Pattern Recognit. Lett. 23, 1771–1784 (2002).
[Crossref]

Zhang, S. Q.

X. F. Li, S. Q. Zhang, X. Pan, P. Dale, and R. Cropp, “Straight road edge detection from high-resolution remote sensing images based on the ridgelet transform with the revised parallel-beam Radon transform,” Int. J. Remote Sensing 31, 5041–5059 (2010).
[Crossref]

Zhang, Z. B.

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

Zhao, C. Q.

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

Zhong, J. G.

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

Appl. Phys. Lett. (2)

G. Scarcelli, V. Berardi, and Y. H. Shih, “Phase-conjugate mirror via two-photon thermal light imaging,” Appl. Phys. Lett. 88, 061106 (2006).
[Crossref]

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

IEEE Trans. Image Process. (1)

S. Yi, D. Labate, G. R. Easley, and H. Krim, “A shearlet approach to edge analysis and detection,” IEEE Trans. Image Process. 18, 929–941 (2009).
[Crossref] [PubMed]

IEEE Trans. PAMI (2)

J. Canny, “A computational approach to edge detection,” IEEE Trans. PAMI 8, 679–698 (1986).
[Crossref]

P. Bao, L. Zhang, and X. L. Wu, “Canny edge detection enhancement by scale multiplication,” IEEE Trans. PAMI 27, 1485 (2005).
[Crossref]

Int. J. Remote Sensing (1)

X. F. Li, S. Q. Zhang, X. Pan, P. Dale, and R. Cropp, “Straight road edge detection from high-resolution remote sensing images based on the ridgelet transform with the revised parallel-beam Radon transform,” Int. J. Remote Sensing 31, 5041–5059 (2010).
[Crossref]

J. Mod. Opt. (1)

A. Gatti, M. Bache, D. Magatti, E. Brambilla, F. Ferri, and L. A. Lugiato, “Coherent imaging with pseudo-thermal incoherent light,” J. Mod. Opt. 53, 739–760 (2006).
[Crossref]

J. Opt. Soc. Am. A (1)

Nat. Commun. (1)

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

Opt. Express (2)

Opt. Lett. (2)

Pattern Recognit. (1)

O. K. Kwon, D. G. Sim, and R. H. Park, “Robust Hausdorff distance matching algorithms using pyramidal structures,” Pattern Recognit. 34, 2005–2013 (2001).
[Crossref]

Pattern Recognit. Lett. (2)

M. Kmiec and A. Glowacz, “Object detection in security applications using dominant edge directions,” Pattern Recognit. Lett. 52, 72–79 (2015).
[Crossref]

L. Zhang and P. Bao, “Edge detection by scale multiplication in wavelet domain,” Pattern Recognit. Lett. 23, 1771–1784 (2002).
[Crossref]

Phys. Rev. A (6)

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

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

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

B. I. Erkmen and J. H. Shapiro, “Signal-to-noise ratio of Gaussian-state ghost imaging,” Phys. Rev. A 79, 023833 (2009).
[Crossref]

M. N. O’Sullivan, K. W. C. Chan, and R. W. Boyd, “Comparison of the signal-to-noise characteristics of quantum versus thermal ghost imaging,” Phys. Rev. A 82, 053803 (2010).
[Crossref]

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[Crossref]

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[Crossref] [PubMed]

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Figures (6)

Fig. 1
Fig. 1 Schematic diagram of GI.
Fig. 2
Fig. 2 Experimental setup for edge detection based on GGI. BS, beamsplitter. DMD, digital micro-mirror device. CCD, charge coupled device. The source is a halogen lamp, and Lens1 is used for collimation.
Fig. 3
Fig. 3 Experimental results of one-dimensional edge detection based on GGI. (a) Original object image. (b) GI result in spatial domain. (c), (d) are the results of edge detection based on GI and GGI, respectively. The figures are normalized to a range of −1∼1.
Fig. 4
Fig. 4 Experimental results of two-dimensional edge detection based on GGI. (a) Original object image. (b) GI result in spatial domain. (c), (d) are the results of edge detection based on GI and GGI, respectively. In (b), (c) and (d), the object edges are shown as absolute values. The size of one pixel is 35.6 μm × 35.6 μm.
Fig. 5
Fig. 5 SNR of edge detections based on GI and GGI. (a) Original object. (b) Object edge. (c) SNR1, SNR2 are the SNRs of edge detections based on GI and GGI, respectively, for different object sizes. (d) SNR enhancement versus object size. Data points are simulation results and dashed curves are fitted according to the theory.
Fig. 6
Fig. 6 Edge detection with gray-scale object. (a) Object. (b) Traditional GI result. (c, d) Edge detection based on GI with 45° and −45° gradient vectors, respectively. (e, f) Edge detection based on GGI with 45° and −45° gradient vector. Gradient vectors are marked in the figures.

Equations (13)

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y k = x i A k ( x i ) T ( x i ) ,
T ( x ) = g ( 2 ) ( A ( x ) , y ) = A ( x ) y A ( x ) y ,
SNR = N S coh S spa ,
T ( x i ) = T ( x i ) T ( x i + 1 ) = g ( 2 ) ( A ( x i ) , y ) g ( 2 ) ( A ( x i + 1 ) , y ) .
SNR = N S coh 2 S spa ,
B ( x i ) = A ( x i 1 ) .
y a = x i A ( x i ) T ( x i ) y b = x i B ( x i ) T ( x i ) .
y a y b = x i ( A ( x i ) T ( x i ) A ( x i 1 ) T ( x i ) ) = x i A ( x i ) ( T ( x i ) T ( x i + 1 ) ) = x i A ( x i ) T ( x i ) .
T ( x ) = A ( x ) Δ y A ( x ) Δ y .
η = S spa S gra ,
SNR = mean ( I edge ) mean ( I back ) ( var ( I back ) ) 0.5 ,
T ( x i , y j ) = T ( x i , y j ) T ( x i + m , y j + n ) , with φ = arctan ( n m ) ,
B ( x i , y j ) = A ( x i m , y j n ) .

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