Abstract

A new optical transformation that combines geometrical coordinate transformations with the conventional optical Fourier transform is described. The resultant transformations are invariant to both scale and rotational changes in the input object or function. Extensions of these operations to optical pattern recognition and initial experimental demonstrations are also presented.

© 1976 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. O. Bryngdahl, J. Opt. Soc. Am. 64, 1092 (1974).
  2. A. Sawchuck, in Proc. Elec. Opt. Sys. Des. Conf. (Anaheim, Calif.
  3. S. Lee, Opt. Eng. 13, 196 (1974).
  4. D. Casasent, D. Psaltis, Opt. Eng. (1976) to appear.
  5. D. Casasent, D. Psaltis, Opt. Commun. to appear.
  6. W. R. Callen, J. E. Weaver, in Proc. Elec. Opt. Sys. Des. Conf. (Anaheim, Calif., 1975).
  7. D. Casasent, Proc. IEEE (Nov.1976).
  8. D. Casasent, “Materials and Devices for Optical Computing,” in Optical Information Processing, G. W. Stroke et al. Eds. (Plenum, New York, 1976).
  9. D. Casasent, “Recyclable Input Devices and Spatial Filter Materials,” in Laser Applications, M. Ross, Ed. (Academic, New York, 1976).
  10. A. Vander Lugt, IEEE Trans. Inf. Theory IT-10, 139 (1964).

1976 (1)

D. Casasent, Proc. IEEE (Nov.1976).

1974 (2)

O. Bryngdahl, J. Opt. Soc. Am. 64, 1092 (1974).

S. Lee, Opt. Eng. 13, 196 (1974).

1964 (1)

A. Vander Lugt, IEEE Trans. Inf. Theory IT-10, 139 (1964).

Bryngdahl, O.

Callen, W. R.

W. R. Callen, J. E. Weaver, in Proc. Elec. Opt. Sys. Des. Conf. (Anaheim, Calif., 1975).

Casasent, D.

D. Casasent, Proc. IEEE (Nov.1976).

D. Casasent, “Materials and Devices for Optical Computing,” in Optical Information Processing, G. W. Stroke et al. Eds. (Plenum, New York, 1976).

D. Casasent, “Recyclable Input Devices and Spatial Filter Materials,” in Laser Applications, M. Ross, Ed. (Academic, New York, 1976).

D. Casasent, D. Psaltis, Opt. Commun. to appear.

D. Casasent, D. Psaltis, Opt. Eng. (1976) to appear.

Lee, S.

S. Lee, Opt. Eng. 13, 196 (1974).

Psaltis, D.

D. Casasent, D. Psaltis, Opt. Eng. (1976) to appear.

D. Casasent, D. Psaltis, Opt. Commun. to appear.

Sawchuck, A.

A. Sawchuck, in Proc. Elec. Opt. Sys. Des. Conf. (Anaheim, Calif.

Vander Lugt, A.

A. Vander Lugt, IEEE Trans. Inf. Theory IT-10, 139 (1964).

Weaver, J. E.

W. R. Callen, J. E. Weaver, in Proc. Elec. Opt. Sys. Des. Conf. (Anaheim, Calif., 1975).

IEEE Trans. Inf. Theory (1)

A. Vander Lugt, IEEE Trans. Inf. Theory IT-10, 139 (1964).

J. Opt. Soc. Am. (1)

Opt. Eng. (1)

S. Lee, Opt. Eng. 13, 196 (1974).

Proc. IEEE (1)

D. Casasent, Proc. IEEE (Nov.1976).

Other (6)

D. Casasent, “Materials and Devices for Optical Computing,” in Optical Information Processing, G. W. Stroke et al. Eds. (Plenum, New York, 1976).

D. Casasent, “Recyclable Input Devices and Spatial Filter Materials,” in Laser Applications, M. Ross, Ed. (Academic, New York, 1976).

D. Casasent, D. Psaltis, Opt. Eng. (1976) to appear.

D. Casasent, D. Psaltis, Opt. Commun. to appear.

W. R. Callen, J. E. Weaver, in Proc. Elec. Opt. Sys. Des. Conf. (Anaheim, Calif., 1975).

A. Sawchuck, in Proc. Elec. Opt. Sys. Des. Conf. (Anaheim, Calif.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (10)

Fig. 1
Fig. 1

SNR of the correlation peak vs the percent scale change between the input and holographic spatial filter functions.

Fig. 2
Fig. 2

SNR of the correlation peak vs the rotational angle θ0 between the input and holographic spatial filter functions.

Fig. 3
Fig. 3

Effects of a rotation by θ0 in the input function F(ωx,ωy) on the transformed function F(r,θ): (a) input function; (b) polar coordinate transform of (a); (c) rotated input function; (d) polar coordinate transform of (c).

Fig. 4
Fig. 4

Block diagram of the positional, rotational, and scale invariant PRSI transformation system. FT denotes Fourier transform.

Fig. 5
Fig. 5

Block diagram of the real-time implementation of a PRSI transformation using a real-time optically or electron beam addressed spatial light modulator.

Fig. 6
Fig. 6

PRSI correlator schematic diagram.

Fig. 7
Fig. 7

(a) Input function F(ωx,ωy); (b) polar transform of (a) with logarithmic scaling in r to form F(expρ,θ); (c) Fourier transform of (b) or M(ωρ,ωθ).

Fig. 8
Fig. 8

(a) Rotated and scaled input function F′(ωx,ωy); (b) polar transform of (a) with logarithmic scaling in r to form F′(expρ,θ); (c) Fourier transform of (b) or M′(ωρ,ωθ).

Fig. 9
Fig. 9

(a) Autocorrelation of F(ωx,ωy); (b) cross correlation of F(ωx,ωy) and F′(ωx,ωy).

Fig. 10
Fig. 10

(a) Cross section of autocorrelation peak in Fig. 9(a); (b) cross section of cross-correlation peaks in Fig. 9(b).

Equations (10)

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

M ( ω ρ , θ ) = 0 F ( r , θ ) r - j ω - 1 d r ,
M ( ω ρ , θ ) = a - j ω M ( ω ρ , θ ) ,
M ( ω ρ , θ ) = - F ( exp ρ , θ ) exp ( - j ω ρ ) d ρ ,
F ( r , θ ) = F 1 ( r , θ ) + F 2 ( r , θ ) ,
G ( r , ω θ ) = G 1 ( r , ω θ ) + G 2 ( r , ω θ ) ,
F ( r , θ ) = F 1 ( r , θ - θ 0 ) + F 2 ( r , θ + 2 π - θ 0 ) ,
G ( r , ω θ ) = G 1 ( r , ω θ ) exp ( - j ω θ θ 0 ) + G 2 ( r , ω θ ) exp [ j ω θ ( 2 π - θ 0 ) ] .
M ( ω ρ , ω θ ) = M 1 ( ω ρ , ω θ ) + M 2 ( ω ρ , ω θ ) ,
M ( ω ρ , ω θ ) = M 1 ( ω ρ , ω θ ) exp [ - j ( ω ρ ln a + ω θ θ 0 ) ] + M 2 ( ω ρ , ω θ ) exp { - j [ ω ρ ln a - ω θ ( 2 π - θ 0 ) ] .
M * M = M * M 1 exp [ - j ( ω ρ ln a + ω θ θ 0 ) ] + M * M 2 exp { - j [ ω ρ ln a - ω θ ( 2 π - θ 0 ) ] .

Metrics