Abstract

Free space optical communications utilizing orbital angular momentum beams have recently emerged as a new technique for communications with potential for increased channel capacity. Turbulence due to changes in the index of refraction emanating from temperature, humidity, and air flow patterns, however, add nonlinear effects to the received patterns, thus making the demultiplexing task more difficult. Deep learning techniques have been previously been applied to solve the demultiplexing problem as an image classification task. Here we make use of a newly developed theory suggesting a link between image turbulence and photon transport through the continuity equation to describe a method that utilizes a “shallow” learning method instead. The decoding technique is tested and compared against previous approaches using deep convolutional neural networks. Results show that the new method can obtain similar classification accuracies (bit error ratio) at a small fraction (1/90) of the computational cost, thus enabling higher bit rates.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

Full Article  |  PDF Article
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2017 (3)

J. Li, M. Zhang, and D. Wang, “Adaptive demodulator using machine learning for orbital angular momentum shift keying,” IEEE Photon. Technol. Lett. 25, 1455–1458 (2017).
[Crossref]

S. Kolouri, S. R. Park, M. Thorpe, D. Slepcev, and G. K. Rohde, “Optimal mass transport: Signal processing and machine-learning applications,” IEEE Signal Process. Mag 34, 43–59 (2017).
[Crossref]

T. Doster and A. T. Watnik, “Machine learning approach to OAM beam demultiplexing via convolutional neural networks,” Appl. Opt. 56, 3386–3396 (2017).
[Crossref] [PubMed]

2016 (2)

T. Doster and A. T. Watnik, “Laguerre-Gauss and Bessel-Gauss beams propagation through turbulence: analysis of channel efficiency,” Appl. Opt. 55, 10239–10246 (2016).
[Crossref]

S. Kolouri, S. R. Park, and G. K. Rohde, “The radon cumulative distribution transform and its application to image classification,” IEEE Trans. Image Process. 25, 920–934 (2016).
[Crossref]

2015 (1)

A. E. Willer, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7, 66–106 (2015).
[Crossref]

2014 (3)

2013 (3)

D. J. Richardson, J. M. Fini, and L. E. Nelson, “Space-division multiplexing in optical fibres,” Nat. Photonics 7, 354–362 (2013).
[Crossref]

M. Mirhosseini, M. Malik, Z. Shi, and R. W. Boyd, “Efficient separation of the orbital angular momentum eigenstates of light,” Nat. Commun. 4, 2781 (2013).
[Crossref] [PubMed]

M. P. Lavery, F. C. Speirits, S. M. Barnett, and M. J. Padgett, “Detection of a spinning object using light’s orbital angular momentum,” Science 341, 537–540 (2013).
[Crossref] [PubMed]

2012 (1)

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and et al., “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6, 488–496 (2012).
[Crossref]

2011 (3)

G. Xie, F. Wang, A. Dang, and H. Guo, “A novel polarization-multiplexing system for free-space optical links,” IEEE Photon. Technol. Lett. 23, 1484–1486 (2011).
[Crossref]

M. P. Lavery, G. C. Berkhout, J. Courtial, and M. J. Padgett, “Measurement of the light orbital angular momentum spectrum using an optical geometric transformation,” J. Opt-UK 13, 064006 (2011).
[Crossref]

P. J. Winzer and G. J. Foschini, “MIMO capacities and outage probabilities in spatially multiplexed optical transport systems,” Opt. Express 19, 16680–16696 (2011).
[Crossref] [PubMed]

2010 (1)

2008 (1)

2005 (2)

2004 (1)

2002 (1)

J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, “Measuring the orbital angular momentum of a single photon,” Phys. Rev. Lett. 88, 257901 (2002).
[Crossref] [PubMed]

2000 (1)

J.-D. Benamou and Y. Brenier, “A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem,” Numerische Mathematik 84, 375–393 (2000).
[Crossref]

1997 (1)

M. Soskin, V. Gorshkov, M. Vasnetsov, J. Malos, and N. Heckenberg, “Topological charge and angular momentum of light beams carrying optical vortices,” Phys. Rev. A 56, 4064 (1997).
[Crossref]

1992 (2)

R. Lane, A. Glindemann, and J. Dainty, “Simulation of a kolmogorov phase screen,” Wave. Random Media 2, 209–224 (1992).
[Crossref]

L. C. Andrews, “An analytical model for the refractive index power spectrum and its application to optical scintillations in the atmosphere,” J. of Mod. Opt. 39, 1849–1853 (1992).
[Crossref]

1936 (1)

R. A. Fisher, “The use of multiple measurements in taxonomic problems,” Ann. Hum. Genet. 7, 179–188 (1936).

1927 (1)

E. Madelung, “Quantum theory in hydrodynamic form,” Zeit. f. Phys. 40, 322–326 (1927).
[Crossref]

Ahmed, N.

A. E. Willer, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7, 66–106 (2015).
[Crossref]

H. Huang, G. Xie, Y. Yan, N. Ahmed, Y. Ren, Y. Yue, D. Rogawski, M. J. Willner, B. I. Erkmen, K. M. Birnbaum, S. J. Dolinar, M. P. J. Lavery, M. J. Padgett, and A. E. Willner, “100 Tbit/s free-space data link enabled by three-dimensional multiplexing of orbital angular momentum, polarization, and wavelength,” Opt. Lett. 39, 197–200 (2014).
[Crossref] [PubMed]

Y. Ren, G. Xie, H. Huang, C. Bao, Y. Yan, N. Ahmed, M. P. J. Lavery, B. I. Erkmen, S. Dolinar, M. Tur, M. A. Neifeld, M. J. Padgett, R. W. Boyd, J. H. Shapiro, and A. E. Willner, “Adaptive optics compensation of multiple orbital angular momentum beams propagating through emulated atmospheric turbulence,” Opt. Lett. 39, 2845–2848 (2014).
[Crossref] [PubMed]

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and et al., “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6, 488–496 (2012).
[Crossref]

Aksyuk, V. A.

Andrews, L. C.

L. C. Andrews, “An analytical model for the refractive index power spectrum and its application to optical scintillations in the atmosphere,” J. of Mod. Opt. 39, 1849–1853 (1992).
[Crossref]

Anguita, J. A.

Arabaci, M.

Ashrafi, N.

A. E. Willer, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7, 66–106 (2015).
[Crossref]

Ashrafi, S.

A. E. Willer, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7, 66–106 (2015).
[Crossref]

Ba, J.

J. Ba and R. Caruana, “Do deep nets really need to be deep?” in “Advances in Neural Information Processing Systems 27,” Z. Ghahramani, M. Welling, C. Cortes, N. D. Lawrence, and K. Q. Weinberger, eds. (Curran Associates, Inc., 2014), pp. 2654–2662.

D. Kingma and J. Ba, “Adam: A method for stochastic optimization,” arXiv preprint arXiv:1412.6980 (2014).

Banerjee, A.

Bao, C.

A. E. Willer, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7, 66–106 (2015).
[Crossref]

Y. Ren, G. Xie, H. Huang, C. Bao, Y. Yan, N. Ahmed, M. P. J. Lavery, B. I. Erkmen, S. Dolinar, M. Tur, M. A. Neifeld, M. J. Padgett, R. W. Boyd, J. H. Shapiro, and A. E. Willner, “Adaptive optics compensation of multiple orbital angular momentum beams propagating through emulated atmospheric turbulence,” Opt. Lett. 39, 2845–2848 (2014).
[Crossref] [PubMed]

Barnett, S. M.

M. P. Lavery, F. C. Speirits, S. M. Barnett, and M. J. Padgett, “Detection of a spinning object using light’s orbital angular momentum,” Science 341, 537–540 (2013).
[Crossref] [PubMed]

G. Gibson, J. Courtial, M. J. Padgett, M. Vasnetsov, V. Pas’ko, S. M. Barnett, and S. Franke-Arnold, “Free-space information transfer using light beams carrying orbital angular momentum free-space information transfer using light beams carrying orbital angular momentum,” Opt. Express 12, 5448–5456 (2004).
[Crossref] [PubMed]

J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, “Measuring the orbital angular momentum of a single photon,” Phys. Rev. Lett. 88, 257901 (2002).
[Crossref] [PubMed]

Basavanhally, N. R.

Benamou, J.-D.

J.-D. Benamou and Y. Brenier, “A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem,” Numerische Mathematik 84, 375–393 (2000).
[Crossref]

Bengio, Y.

X. Glorot and Y. Bengio, “Understanding the difficulty of training deep feedforward neural networks,” in “Proceedings of the Thirteenth International Conference on Artificial Intelligence and Statistics,” (2010), pp. 249–256.

Berkhout, G. C.

M. P. Lavery, G. C. Berkhout, J. Courtial, and M. J. Padgett, “Measurement of the light orbital angular momentum spectrum using an optical geometric transformation,” J. Opt-UK 13, 064006 (2011).
[Crossref]

Birnbaum, K. M.

Bolle, C. A.

Boyd, R. W.

Brenier, Y.

J.-D. Benamou and Y. Brenier, “A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem,” Numerische Mathematik 84, 375–393 (2000).
[Crossref]

Cao, Y.

A. E. Willer, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7, 66–106 (2015).
[Crossref]

Caruana, R.

J. Ba and R. Caruana, “Do deep nets really need to be deep?” in “Advances in Neural Information Processing Systems 27,” Z. Ghahramani, M. Welling, C. Cortes, N. D. Lawrence, and K. Q. Weinberger, eds. (Curran Associates, Inc., 2014), pp. 2654–2662.

Cattell, L.

J. M. Nichols, A. T. Watnik, T. Doster, S. Park, A. Kanaev, L. Cattell, and G. K. Rohde, “An optimal transport model for imaging in atmospheric turbulence,” arXiv preprint arXiv:1705.01050 (2017).

Clarke, F.

Courtial, J.

M. P. Lavery, G. C. Berkhout, J. Courtial, and M. J. Padgett, “Measurement of the light orbital angular momentum spectrum using an optical geometric transformation,” J. Opt-UK 13, 064006 (2011).
[Crossref]

G. Gibson, J. Courtial, M. J. Padgett, M. Vasnetsov, V. Pas’ko, S. M. Barnett, and S. Franke-Arnold, “Free-space information transfer using light beams carrying orbital angular momentum free-space information transfer using light beams carrying orbital angular momentum,” Opt. Express 12, 5448–5456 (2004).
[Crossref] [PubMed]

J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, “Measuring the orbital angular momentum of a single photon,” Phys. Rev. Lett. 88, 257901 (2002).
[Crossref] [PubMed]

Dainty, J.

R. Lane, A. Glindemann, and J. Dainty, “Simulation of a kolmogorov phase screen,” Wave. Random Media 2, 209–224 (1992).
[Crossref]

Danaci, O.

E. Knutson, S. Lohani, O. Danaci, S. D. Huver, and R. T. Glasser, “Deep learning as a tool to distinguish between high orbital angular momentum optical modes,” in “Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series,”, vol. 9970 (2016), vol. 9970, p. 997013.

Dang, A.

G. Xie, F. Wang, A. Dang, and H. Guo, “A novel polarization-multiplexing system for free-space optical links,” IEEE Photon. Technol. Lett. 23, 1484–1486 (2011).
[Crossref]

Djordjevic, I. B.

Dolinar, S.

Dolinar, S. J.

Doster, T.

Erkmen, B. I.

Fazal, I. M.

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and et al., “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6, 488–496 (2012).
[Crossref]

Fickler, R.

M. Krenn, R. Fickler, M. Fink, J. Handsteiner, M. Malik, T. Scheidl, R. Ursin, and A. Zeilinger, “Communication with spatially modulated light through turbulent air across Vienna,” New J. Phys. 16, 113028 (2014).
[Crossref]

Fini, J. M.

D. J. Richardson, J. M. Fini, and L. E. Nelson, “Space-division multiplexing in optical fibres,” Nat. Photonics 7, 354–362 (2013).
[Crossref]

Fink, M.

M. Krenn, R. Fickler, M. Fink, J. Handsteiner, M. Malik, T. Scheidl, R. Ursin, and A. Zeilinger, “Communication with spatially modulated light through turbulent air across Vienna,” New J. Phys. 16, 113028 (2014).
[Crossref]

Fisher, R. A.

R. A. Fisher, “The use of multiple measurements in taxonomic problems,” Ann. Hum. Genet. 7, 179–188 (1936).

Foschini, G. J.

Franke-Arnold, S.

Gibson, G.

Glasser, R. T.

E. Knutson, S. Lohani, O. Danaci, S. D. Huver, and R. T. Glasser, “Deep learning as a tool to distinguish between high orbital angular momentum optical modes,” in “Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series,”, vol. 9970 (2016), vol. 9970, p. 997013.

Glindemann, A.

R. Lane, A. Glindemann, and J. Dainty, “Simulation of a kolmogorov phase screen,” Wave. Random Media 2, 209–224 (1992).
[Crossref]

Glorot, X.

X. Glorot and Y. Bengio, “Understanding the difficulty of training deep feedforward neural networks,” in “Proceedings of the Thirteenth International Conference on Artificial Intelligence and Statistics,” (2010), pp. 249–256.

Gorshkov, V.

M. Soskin, V. Gorshkov, M. Vasnetsov, J. Malos, and N. Heckenberg, “Topological charge and angular momentum of light beams carrying optical vortices,” Phys. Rev. A 56, 4064 (1997).
[Crossref]

Greywall, D. S.

Guo, H.

G. Xie, F. Wang, A. Dang, and H. Guo, “A novel polarization-multiplexing system for free-space optical links,” IEEE Photon. Technol. Lett. 23, 1484–1486 (2011).
[Crossref]

Handsteiner, J.

M. Krenn, R. Fickler, M. Fink, J. Handsteiner, M. Malik, T. Scheidl, R. Ursin, and A. Zeilinger, “Communication with spatially modulated light through turbulent air across Vienna,” New J. Phys. 16, 113028 (2014).
[Crossref]

Heckenberg, N.

M. Soskin, V. Gorshkov, M. Vasnetsov, J. Malos, and N. Heckenberg, “Topological charge and angular momentum of light beams carrying optical vortices,” Phys. Rev. A 56, 4064 (1997).
[Crossref]

Hinton, G. E.

A. Krizhevsky, I. Sutskever, and G. E. Hinton, “Imagenet classification with deep convolutional neural networks,” in “Advances in neural information processing systems,” (NIPS2012), pp. 1097–1105.

Huang, H.

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Rohde, G. K.

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Shi, Z.

M. Mirhosseini, M. Malik, Z. Shi, and R. W. Boyd, “Efficient separation of the orbital angular momentum eigenstates of light,” Nat. Commun. 4, 2781 (2013).
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M. Krenn, R. Fickler, M. Fink, J. Handsteiner, M. Malik, T. Scheidl, R. Ursin, and A. Zeilinger, “Communication with spatially modulated light through turbulent air across Vienna,” New J. Phys. 16, 113028 (2014).
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J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and et al., “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6, 488–496 (2012).
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Willer, A. E.

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IEEE Signal Process. Mag (1)

S. Kolouri, S. R. Park, M. Thorpe, D. Slepcev, and G. K. Rohde, “Optimal mass transport: Signal processing and machine-learning applications,” IEEE Signal Process. Mag 34, 43–59 (2017).
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J. Opt. Netw. (1)

Nat. Commun. (1)

M. Mirhosseini, M. Malik, Z. Shi, and R. W. Boyd, “Efficient separation of the orbital angular momentum eigenstates of light,” Nat. Commun. 4, 2781 (2013).
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Nat. Photonics (2)

D. J. Richardson, J. M. Fini, and L. E. Nelson, “Space-division multiplexing in optical fibres,” Nat. Photonics 7, 354–362 (2013).
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New J. Phys. (1)

M. Krenn, R. Fickler, M. Fink, J. Handsteiner, M. Malik, T. Scheidl, R. Ursin, and A. Zeilinger, “Communication with spatially modulated light through turbulent air across Vienna,” New J. Phys. 16, 113028 (2014).
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Phys. Rev. Lett. (1)

J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, “Measuring the orbital angular momentum of a single photon,” Phys. Rev. Lett. 88, 257901 (2002).
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Science (1)

M. P. Lavery, F. C. Speirits, S. M. Barnett, and M. J. Padgett, “Detection of a spinning object using light’s orbital angular momentum,” Science 341, 537–540 (2013).
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Figures (9)

Fig. 1
Fig. 1 Analytical vortex modes (top row), actual vortex modes generated with no turbulence present (middle row) and observed vortex modes after passage through a phase screen designed to mimic turbulence quantified by C n 2 = 1 e 14 (bottom row).
Fig. 2
Fig. 2 The influence of turbulence on an image from a ray optics perspective. Changes in the refractive index will cause the ray to fluctuate in direction, eventually reaching the receiver at a perturbed distance u ( x , Z ) from the true location. The optical path can be reasonably approximated as a straight line connecting the true position on ρ 0 ( x ) to the observed position on the image ρ 1 ( x ). Thus, for fixed distance to the receiver Z, minimizing the optical path length is equivalent to minimizing u ( x , z ). The transport-based approach to mapping ρ 0 ( x ) to ρ 1 ( x ) creates a model with precisely this objective function hence would seem a reasonable approach to modeling the influence of turbulence.
Fig. 3
Fig. 3 Pictorial description of data distribution in image space (a) and R-CDT space (b). 3 classes of data (OAM beams with different mode number, color-coded as pink, blue, and sepia) form more complex decision boundary in the image space then they would in the R-CDT space.
Fig. 4
Fig. 4 Depiction of linear separation properties of the cumulative distribution transform (CDT). Under certain conditions (see text), the CDT can render signal classes linearly separable in transform domain. It could thus be possible to build a decoder for the received pattern our of a linear classifier in transform domain.
Fig. 5
Fig. 5 Multiplexed OAM beam patterns of mode set 1
Fig. 6
Fig. 6 Multiplexed OAM beam patterns of mode set 1 in the R-CDT space. The R-CDT is computed with respect to the pattern ‘00001’, and therefore the corresponding R-CDT image for ’00001’ is shown as an empty image. The R-CDT represents the transport map between the pattern ‘00001’ and the pattern ρm. More precisely, each column of ρm represents a transport map between the projection of ‘00001’ and ρm at an angle θ.
Fig. 7
Fig. 7 1-Layer shallow CNN architecture used to decode the Radon-CDT data
Fig. 8
Fig. 8 Examples of BGB under different turbulence levels, mode set1
Fig. 9
Fig. 9 Downsampled image (top) and corresponding Radon-CDT (bottom) for downsampling factors {1, 4, 8, 16} (left to right).

Tables (10)

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Table 1 Mode sets used in the experiment

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Table 2 Linear classification accuracy (%) for testing set

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Table 3 1-Layer CNN performance for testing set, R-CDT

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Table 4 Alexnet performance in the image space

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Table 5 1-Layer CNN performance in the image space

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Table 6 Computational complexity for input image of size 151×151

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Table 7 Classification accuracy (%) for testing set for 1-Layer CNN, D/r0 = 15

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Table 8 Dimensions of the downsampled images and their corresponding Radon-CDTs.

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Table 9 Linear classification accuracy (%) for down-sampled training set (TR) and testing set (TS), D/r0 = 15.

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Table 10 Dense neural network classification accuracy (%) for down-sampled testing set, D/r0 = 15.

Equations (19)

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2 U ( x , z ) + k 2 U ( x , z ) = 0 ,
U B ( m ) ( r , θ , z ) = C b J m ( k r r ) exp ( i k z z ) exp ( i m θ ) ,
U B G ( m ) ( r , θ , z ) = C b g w 0 w ( z ) J m ( k r r 1 + i z / z r ) exp [ i ( k k r 2 2 k ) z ζ ( z ) + 1 w 2 ( z ) ] × exp [ i k 2 R ( z ) ( r 2 + k r 2 z r k 2 ) ] exp ( i m θ ) ,
r θ U B G ( m 1 ) U B G ( m 2 ) * d r d θ = 0 .
ρ ( x , z ) z + X ( ρ ( x , z ) v ( x , z ) ) = 0
v ( x , z ) z + ( v ( x , z ) X ) v ( x , z ) = 2 X η ( x , z ) .
A Z 2 0 Z ρ ( x , z ) | v ( x , z ) | 2 d z d x .
inf f X | f ( x ) x | 2 ρ 0 ( x ) d x
P m { ρ ˜ m | ρ ˜ m ( x ) = D f ( x ) ρ m ( f ( x ) ) , f F }
ρ ˜ ^ ( x ) = ( h ( x ) x ) I 0 ( x ) .
h ( x ) ρ ˜ ( τ ) d τ = x I 0 ( τ ) d τ
( ρ ˜ ) ( t , θ ) = ρ ˜ ( x ) δ ( t x 1 cos ( θ ) x 2 sin ( θ ) ) d x 1 d x 2
{ ( ρ ˜ ) ^ ( t , θ ) = ( f ( t , θ ) t ) ( I 0 ) ( t , θ ) f ( t , θ ) ( I ) ( τ , θ ) d τ = t ( I 0 ) ( τ , θ ) d τ , θ [ 0 , π ]
y ( L ) = a ( L ) a ( L 1 ) a ( 1 ) ( ρ ˜ ) .
y ( L ) = e y ( L 1 ) k = 1 32 e y k ( L 1 ) , [ 1 , 32 ]
L = log p ( | ρ ˜ ) = log ( e y ( L 1 ) k = 1 32 e y k ( L 1 ) )
t e s t * = arg max = 1 , 32 e y t e s t , ( L 1 ) k = 1 32 e y t e s t , k ( L 1 ) .
Acc = n = 1 N 1 ( ^ n = n ) N × 100 ,
BER = n = 1 N m = 1 5 1 ( p ^ n m = p n m ) n = 1 N m = 1 5 p ^ n m .

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