Abstract

This paper, written for interdisciplinary audience, presents computational image reconstruction implementable by quantum optics. The input-triggered selection of a high-resolution image among many stored ones, and its reconstruction if the input is occluded or noisy, has been successfully simulated. The original algorithm, based on the Hopfield associative neural net, was transformed in order to enable its quantum-wave implementation based on holography. The main limitations of the classical Hopfield net are much reduced with the simulated new quantum-optical implementation.

© 2004 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
  6. D. Ventura, T. Martinez, “Quantum associative memory,” Info. Sci. 124, 273–296 (2000).
    [CrossRef]
  7. J. Howell, J. Yeazell, D. Ventura, “Optically simulating a quantum associative memory,” Phys. Rev. A 62, 042303 (2000).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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2003

M. Andrecut, M. K. Ali, “Quantum associative memory,” Int. J. Mod. Phys. B 17, 2447–2472 (2003).
[CrossRef]

D. Goswami, “Optical pulse shaping approaches to coherent control,” Phys. Reports 376, 385–481 (2003).
[CrossRef]

2001

B. Travaglione, G. Milburn, “Generation of eigenstates using the phase estimation algorithm,” Phys. Rev. A 63, 032301 (2001).
[CrossRef]

G. D’Ariano, P. Lo Presti, “Quantum tomography for measuring experimentally the matrix elements of an arbitrary quantum operation,” Phys. Rev. Lett. 86, 4195–4198 (2001).
[CrossRef] [PubMed]

C. Trugenberger, “Probabilistic quantum memories,” Phys. Rev. Lett. 87, 067901 (2001).
[CrossRef] [PubMed]

C. Trugenberger, “Phase transitions in quantum pattern recognition,” Phys. Rev. Lett. 89, 277903 (2001).
[CrossRef]

R. Spencer, “Bipolar spectral associative memories,” IEEE Transac. Neural Netw. 12, 463–474 (2001).
[CrossRef]

A. V. Pavlov, “On algebraic foundations of Fourier holography,” Opt. Spectrosc. 90, 452–457 (2001).
[CrossRef]

2000

M. Peruš, “Neural networks as a basis for quantum associative networks,” Neural Netw. World 10, 1001–1013 (2000).

D. Ventura, T. Martinez, “Quantum associative memory,” Info. Sci. 124, 273–296 (2000).
[CrossRef]

J. Howell, J. Yeazell, D. Ventura, “Optically simulating a quantum associative memory,” Phys. Rev. A 62, 042303 (2000).
[CrossRef]

M. Peruš, S. K. Dey, “Quantum systems can implement content-addressable associative memory,” Appl. Math. Lett. 13, 31–36 (2000).
[CrossRef]

1999

T. C. Weinacht, J. Ahn, P. H. Baucksbaum, “Controlling the shape of a quantum wavefunction,” Nature 397, 233–235 (1999).
[CrossRef]

1998

C. Leichtle, W. Schleich, I. Averbukh, M. Shapiro, “Quantum state holography,” Phys. Rev. Lett. 80, 1418–1421 (1998).
[CrossRef]

1993

D. Smithey, M. Beck, M. Raymer, A. Faridani, “Measurement of the Wigner distribution and the density matrix of a light mode using optical homodyne tomography,” Phys. Rev. Lett. 70, 1244–1247 (1993).
[CrossRef] [PubMed]

1990

1989

1985

C. Bennett, R. Landauer, “The fundamental physical limits of computation,” Sci. Am. 253, 48–56 (1985).
[CrossRef]

Ahn, J.

T. C. Weinacht, J. Ahn, P. H. Baucksbaum, “Controlling the shape of a quantum wavefunction,” Nature 397, 233–235 (1999).
[CrossRef]

Ali, M. K.

M. Andrecut, M. K. Ali, “Quantum associative memory,” Int. J. Mod. Phys. B 17, 2447–2472 (2003).
[CrossRef]

Andrecut, M.

M. Andrecut, M. K. Ali, “Quantum associative memory,” Int. J. Mod. Phys. B 17, 2447–2472 (2003).
[CrossRef]

Averbukh, I.

C. Leichtle, W. Schleich, I. Averbukh, M. Shapiro, “Quantum state holography,” Phys. Rev. Lett. 80, 1418–1421 (1998).
[CrossRef]

Baucksbaum, P. H.

T. C. Weinacht, J. Ahn, P. H. Baucksbaum, “Controlling the shape of a quantum wavefunction,” Nature 397, 233–235 (1999).
[CrossRef]

Beck, M.

D. Smithey, M. Beck, M. Raymer, A. Faridani, “Measurement of the Wigner distribution and the density matrix of a light mode using optical homodyne tomography,” Phys. Rev. Lett. 70, 1244–1247 (1993).
[CrossRef] [PubMed]

Bennett, C.

C. Bennett, R. Landauer, “The fundamental physical limits of computation,” Sci. Am. 253, 48–56 (1985).
[CrossRef]

Bischof, H.

M. Peruš, H. Bischof, “Quantum-wave pattern recognition,” in Proceedings of the 7th Joint Conference on Information Sciences, K. Chen, ed. (JCIS/Association for Intelligent Machinery, Durham, N.C., 2003), pp. 1536–1539.

Caulfield, H. J.

D’Ariano, G.

G. D’Ariano, P. Lo Presti, “Quantum tomography for measuring experimentally the matrix elements of an arbitrary quantum operation,” Phys. Rev. Lett. 86, 4195–4198 (2001).
[CrossRef] [PubMed]

Dey, S. K.

M. Peruš, S. K. Dey, “Quantum systems can implement content-addressable associative memory,” Appl. Math. Lett. 13, 31–36 (2000).
[CrossRef]

Faridani, A.

D. Smithey, M. Beck, M. Raymer, A. Faridani, “Measurement of the Wigner distribution and the density matrix of a light mode using optical homodyne tomography,” Phys. Rev. Lett. 70, 1244–1247 (1993).
[CrossRef] [PubMed]

Goswami, D.

D. Goswami, “Optical pulse shaping approaches to coherent control,” Phys. Reports 376, 385–481 (2003).
[CrossRef]

Granik, A.

A. Granik, H. J. Caulfield, “Quantum holography,” in Holography, Vol. IS 8 of SPIE Institute for Advanced Optical Technologies Series (SPIE Press, Bellingham, Wash., 1990), pp. 33–38.

Greguss, P.

Howell, J.

J. Howell, J. Yeazell, D. Ventura, “Optically simulating a quantum associative memory,” Phys. Rev. A 62, 042303 (2000).
[CrossRef]

Landauer, R.

C. Bennett, R. Landauer, “The fundamental physical limits of computation,” Sci. Am. 253, 48–56 (1985).
[CrossRef]

Leichtle, C.

C. Leichtle, W. Schleich, I. Averbukh, M. Shapiro, “Quantum state holography,” Phys. Rev. Lett. 80, 1418–1421 (1998).
[CrossRef]

Lo Presti, P.

G. D’Ariano, P. Lo Presti, “Quantum tomography for measuring experimentally the matrix elements of an arbitrary quantum operation,” Phys. Rev. Lett. 86, 4195–4198 (2001).
[CrossRef] [PubMed]

Ludman, J.

Martinez, T.

D. Ventura, T. Martinez, “Quantum associative memory,” Info. Sci. 124, 273–296 (2000).
[CrossRef]

Milburn, G.

B. Travaglione, G. Milburn, “Generation of eigenstates using the phase estimation algorithm,” Phys. Rev. A 63, 032301 (2001).
[CrossRef]

Pavlov, A. V.

A. V. Pavlov, “On algebraic foundations of Fourier holography,” Opt. Spectrosc. 90, 452–457 (2001).
[CrossRef]

Peruš, M.

M. Peruš, S. K. Dey, “Quantum systems can implement content-addressable associative memory,” Appl. Math. Lett. 13, 31–36 (2000).
[CrossRef]

M. Peruš, “Neural networks as a basis for quantum associative networks,” Neural Netw. World 10, 1001–1013 (2000).

M. Peruš, H. Bischof, “Quantum-wave pattern recognition,” in Proceedings of the 7th Joint Conference on Information Sciences, K. Chen, ed. (JCIS/Association for Intelligent Machinery, Durham, N.C., 2003), pp. 1536–1539.

Raymer, M.

D. Smithey, M. Beck, M. Raymer, A. Faridani, “Measurement of the Wigner distribution and the density matrix of a light mode using optical homodyne tomography,” Phys. Rev. Lett. 70, 1244–1247 (1993).
[CrossRef] [PubMed]

Rigatos, G.

G. Rigatos, S. Tzafestas, “Fuzzy learning compatible with quantum mechanics postulates,” in Proceedings of the 7th Joint Conference on Information Sciences, K. Chen, ed. (JCIS/Association for Intelligent Machinery, Durham, N.C., 2003), pp. 1532–1535, sec. 2.

Schleich, W.

C. Leichtle, W. Schleich, I. Averbukh, M. Shapiro, “Quantum state holography,” Phys. Rev. Lett. 80, 1418–1421 (1998).
[CrossRef]

W. Schleich, Quantum Optics in Phase Space (Wiley-VCH, Berlin, 2001).
[CrossRef]

Shamir, J.

Shapiro, M.

C. Leichtle, W. Schleich, I. Averbukh, M. Shapiro, “Quantum state holography,” Phys. Rev. Lett. 80, 1418–1421 (1998).
[CrossRef]

Smithey, D.

D. Smithey, M. Beck, M. Raymer, A. Faridani, “Measurement of the Wigner distribution and the density matrix of a light mode using optical homodyne tomography,” Phys. Rev. Lett. 70, 1244–1247 (1993).
[CrossRef] [PubMed]

Spencer, R.

R. Spencer, “Bipolar spectral associative memories,” IEEE Transac. Neural Netw. 12, 463–474 (2001).
[CrossRef]

Sutherland, J.

J. Sutherland, “Holographic model of memory, learning and expression,” Int. J. Neural Sys. 1, 256–267 (1990).
[CrossRef]

Travaglione, B.

B. Travaglione, G. Milburn, “Generation of eigenstates using the phase estimation algorithm,” Phys. Rev. A 63, 032301 (2001).
[CrossRef]

Trugenberger, C.

C. Trugenberger, “Probabilistic quantum memories,” Phys. Rev. Lett. 87, 067901 (2001).
[CrossRef] [PubMed]

C. Trugenberger, “Phase transitions in quantum pattern recognition,” Phys. Rev. Lett. 89, 277903 (2001).
[CrossRef]

Tzafestas, S.

G. Rigatos, S. Tzafestas, “Fuzzy learning compatible with quantum mechanics postulates,” in Proceedings of the 7th Joint Conference on Information Sciences, K. Chen, ed. (JCIS/Association for Intelligent Machinery, Durham, N.C., 2003), pp. 1532–1535, sec. 2.

Ventura, D.

D. Ventura, T. Martinez, “Quantum associative memory,” Info. Sci. 124, 273–296 (2000).
[CrossRef]

J. Howell, J. Yeazell, D. Ventura, “Optically simulating a quantum associative memory,” Phys. Rev. A 62, 042303 (2000).
[CrossRef]

Weinacht, T. C.

T. C. Weinacht, J. Ahn, P. H. Baucksbaum, “Controlling the shape of a quantum wavefunction,” Nature 397, 233–235 (1999).
[CrossRef]

Yeazell, J.

J. Howell, J. Yeazell, D. Ventura, “Optically simulating a quantum associative memory,” Phys. Rev. A 62, 042303 (2000).
[CrossRef]

Appl. Math. Lett.

M. Peruš, S. K. Dey, “Quantum systems can implement content-addressable associative memory,” Appl. Math. Lett. 13, 31–36 (2000).
[CrossRef]

Appl. Opt.

IEEE Transac. Neural Netw.

R. Spencer, “Bipolar spectral associative memories,” IEEE Transac. Neural Netw. 12, 463–474 (2001).
[CrossRef]

Info. Sci.

D. Ventura, T. Martinez, “Quantum associative memory,” Info. Sci. 124, 273–296 (2000).
[CrossRef]

Int. J. Mod. Phys. B

M. Andrecut, M. K. Ali, “Quantum associative memory,” Int. J. Mod. Phys. B 17, 2447–2472 (2003).
[CrossRef]

Int. J. Neural Sys.

J. Sutherland, “Holographic model of memory, learning and expression,” Int. J. Neural Sys. 1, 256–267 (1990).
[CrossRef]

J. Opt. Soc. Am. A

Nature

T. C. Weinacht, J. Ahn, P. H. Baucksbaum, “Controlling the shape of a quantum wavefunction,” Nature 397, 233–235 (1999).
[CrossRef]

Neural Netw. World

M. Peruš, “Neural networks as a basis for quantum associative networks,” Neural Netw. World 10, 1001–1013 (2000).

Opt. Lett.

Opt. Spectrosc.

A. V. Pavlov, “On algebraic foundations of Fourier holography,” Opt. Spectrosc. 90, 452–457 (2001).
[CrossRef]

Phys. Reports

D. Goswami, “Optical pulse shaping approaches to coherent control,” Phys. Reports 376, 385–481 (2003).
[CrossRef]

Phys. Rev. A

B. Travaglione, G. Milburn, “Generation of eigenstates using the phase estimation algorithm,” Phys. Rev. A 63, 032301 (2001).
[CrossRef]

J. Howell, J. Yeazell, D. Ventura, “Optically simulating a quantum associative memory,” Phys. Rev. A 62, 042303 (2000).
[CrossRef]

Phys. Rev. Lett.

C. Trugenberger, “Probabilistic quantum memories,” Phys. Rev. Lett. 87, 067901 (2001).
[CrossRef] [PubMed]

C. Trugenberger, “Phase transitions in quantum pattern recognition,” Phys. Rev. Lett. 89, 277903 (2001).
[CrossRef]

C. Leichtle, W. Schleich, I. Averbukh, M. Shapiro, “Quantum state holography,” Phys. Rev. Lett. 80, 1418–1421 (1998).
[CrossRef]

D. Smithey, M. Beck, M. Raymer, A. Faridani, “Measurement of the Wigner distribution and the density matrix of a light mode using optical homodyne tomography,” Phys. Rev. Lett. 70, 1244–1247 (1993).
[CrossRef] [PubMed]

G. D’Ariano, P. Lo Presti, “Quantum tomography for measuring experimentally the matrix elements of an arbitrary quantum operation,” Phys. Rev. Lett. 86, 4195–4198 (2001).
[CrossRef] [PubMed]

Sci. Am.

C. Bennett, R. Landauer, “The fundamental physical limits of computation,” Sci. Am. 253, 48–56 (1985).
[CrossRef]

Other

G. Rigatos, S. Tzafestas, “Fuzzy learning compatible with quantum mechanics postulates,” in Proceedings of the 7th Joint Conference on Information Sciences, K. Chen, ed. (JCIS/Association for Intelligent Machinery, Durham, N.C., 2003), pp. 1532–1535, sec. 2.

M. Peruš, H. Bischof, “Quantum-wave pattern recognition,” in Proceedings of the 7th Joint Conference on Information Sciences, K. Chen, ed. (JCIS/Association for Intelligent Machinery, Durham, N.C., 2003), pp. 1536–1539.

H. Bjelkhagen, H. J. Caulfield, eds., Selected Papers on the Fundamental Techniques in Holography (SPIE Press, Bellingham, Wash., 2001).

W. Schleich, Quantum Optics in Phase Space (Wiley-VCH, Berlin, 2001).
[CrossRef]

A. Granik, H. J. Caulfield, “Quantum holography,” in Holography, Vol. IS 8 of SPIE Institute for Advanced Optical Technologies Series (SPIE Press, Bellingham, Wash., 1990), pp. 33–38.

D. Bouwmeester, A. Ekert, A. Zeilinger, eds., The Physics of Quantum Information (Springer, Berlin, 2000).
[CrossRef]

F. T. S. Yu, S. Jutamulia, eds., Optical Pattern Recognition (Cambridge U. Press, Cambridge, UK, 1998).

R. Schützhold, “Pattern recognition on a quantum computer,” http://xxx.lanl.gov/pdf/quant-ph/0208063 (3Dec2002).

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Figures (3)

Fig. 1
Fig. 1

Plots of peak signal-to-noise ratio of reconstructed image from “query-image” versus number of simultaneously-stored images of (a) Chinese pictograms and (b) fingerprints, where (a) query is a Chinese pictogram with salt-and-pepper noise, and (b) query is an occluded fingerprint.

Fig. 2
Fig. 2

(a), Original image; (b), original image (a) with 80% salt-and-pepper noise; and (c)–(g), image restored from memory of 10 different simultaneously stored fingerprints after presentation of the “query-image,” which is (c) whole original image (a), (d) 25%-occluded image (a), (e) 50%-occluded (a), (f) 75%-occluded (a), and (g) noisy image (b).

Fig. 3
Fig. 3

Reconstruction from 30 simultaneously stored images (10 different Chinese pictograms and 10 different fingerprints as on Fig. 2(a) and 10 different face poses, such as in (a) and (b). “Query” (c) triggers reconstruction (d). (e) and (f), reconstructions from 25%- and 50%-occluded “query”-pictogram.

Equations (7)

Equations on this page are rendered with MathJax. Learn more.

Ghj=k=1P ψhkψjk*,
Ψhoutput=j=1N GhjΨjinput=j=1Nk=1P ψhkψjk*Ψjinput=k=1Pj=1Nψjk*Ψjinputψhk  ψhk0
|Ψoutput=G|Ψinput=k|ψkψk||Ψinput=k ψk|Ψinput|ψk  ψk0.
ψkr, t=Akr, texpiφkr, t=Ak expipkr-Ekt.
Ghj=k=1Pexpiφhkexp-iφjk=k=1Pexpiφhk-φjk;
expiφhoutput=j=1Nk=1Pexpiφhkexp-iφjkexpiφjinput=k=1Pj=1Nexp-iφjkexpiφjinputexpiφhk expiφhk0.
PSNR=20 log10255RMSE,RMSE=1Nj=1Nvjoriginal-vjreconstructed21/2.

Metrics