Abstract

Fourier- and Hartley-related transforms are realized in a family of interferometers. The implementation of these interferometers as image correlators is investigated theoretically and experimentally with both coherent and spatially incoherent illumination. Several correlators that can be used for pattern recognition are studied and demonstrated experimentally as special cases.

© 1992 Optical Society of America

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References

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  1. L. Mertz, Transformations in Optics (Wiley, New York, 1965).
  2. K. V. Konjaev, “Interference method of two-dimensional Fourier transform with spatially incoherent illumination,” Phys. Lett. A 24, 490–491 (1967).
    [CrossRef]
  3. S. Wang, N. George, “Fresnel zone transforms in spatially incoherent illumination,” Appl. Opt. 24, 84–850 (1985).
    [CrossRef]
  4. E. Ribak, C. Roddier, F. Roddier, J. B. Breckinridge, “Signal-to-noise limitations in white light holography,” Appl. Opt. 27, 1183–1186 (1988).
    [CrossRef] [PubMed]
  5. J. B. Breckinridge, E. Ribak, C. Roddier, F. Roddier, C. Habecker, “Real time optical correlation using white light Fourier transforms,” final report of the technical work (Jet Propulsion Laboratory, California Institute of Technology, Pasadena, Calif., 1988).
  6. R. N. Bracewell, H. Bartelt, A. W. Lohmann, N. Streibl, “Optical synthesis of the Hartley transform,” Appl. Opt. 24, 1401–1402 (1985).
    [CrossRef] [PubMed]
  7. J. Villasenor, R. N. Bracewell, “Optical phase obtained by analogue Hartley transformation,” Nature (London) 330, 735–737 (1987).
    [CrossRef]
  8. T. Nomura, K. Itoh, Y. Ichioka, “Hartley transformation for hybrid pattern matching,” Appl. Opt. 29, 4345–4350 (1990).
    [CrossRef] [PubMed]
  9. C. S. Weaver, J. W. Goodman, “A technique for optically convolving two functions,” Appl. Opt. 5, 1248–1249 (1966).
    [CrossRef] [PubMed]
  10. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1980).
  11. R. N. Bracewell, “Discrete Hartley transform,” J. Opt. Soc. Am. 73, 1832–1835 (1983).
    [CrossRef]
  12. Y. Fainman, E. Lenz, J. Shamir, “Contouring by phase conjugation,” Appl. Opt. 20, 158–163 (1981).
    [CrossRef] [PubMed]
  13. M. Segev, A. Yariv, “Optical interferometry between image-bearing beams and their redirected phase conjugates,” Opt. Lett. 17, 145–147 (1992).
    [CrossRef] [PubMed]
  14. M. Cronin-Golomb, B. Fischer, J. O. White, A. Yariv, “Passive (self pump) phase conjugate mirror: theoretical and experimental investigation,” Appl. Phys. Lett. 41, 689–691 (1982).
    [CrossRef]
  15. O. I. Potaturkin, “Incoherent diffraction correlator with a holographic filter,” Appl. Opt. 18, 4203–4205 (1979).
    [CrossRef] [PubMed]
  16. D. Mendlovic, E. Marom, N. Konforti, “Complex reference functions in joint transform correlator,” Opt. Lett. 15, 1224–1226 (1990).
    [CrossRef] [PubMed]
  17. U. Mahlab, J. Rosen, J. Shamir, “Iterative generation of complex reference functions in joint transform correlator,” Opt. Lett. 16, 330–332 (1991).
    [CrossRef] [PubMed]
  18. J. Rosen, T. Kotzer, J. Shamir, “Optical implementation of phase extraction pattern recognition,” Opt. Commun. 83, 10–14 (1991).
    [CrossRef]
  19. J. L. Horner, P. D. Gianino, “Phase-only matched filtering,” Appl. Opt. 23, 812–816 (1984).
    [CrossRef] [PubMed]

1992 (1)

1991 (2)

U. Mahlab, J. Rosen, J. Shamir, “Iterative generation of complex reference functions in joint transform correlator,” Opt. Lett. 16, 330–332 (1991).
[CrossRef] [PubMed]

J. Rosen, T. Kotzer, J. Shamir, “Optical implementation of phase extraction pattern recognition,” Opt. Commun. 83, 10–14 (1991).
[CrossRef]

1990 (2)

1988 (1)

1987 (1)

J. Villasenor, R. N. Bracewell, “Optical phase obtained by analogue Hartley transformation,” Nature (London) 330, 735–737 (1987).
[CrossRef]

1985 (2)

R. N. Bracewell, H. Bartelt, A. W. Lohmann, N. Streibl, “Optical synthesis of the Hartley transform,” Appl. Opt. 24, 1401–1402 (1985).
[CrossRef] [PubMed]

S. Wang, N. George, “Fresnel zone transforms in spatially incoherent illumination,” Appl. Opt. 24, 84–850 (1985).
[CrossRef]

1984 (1)

1983 (1)

1982 (1)

M. Cronin-Golomb, B. Fischer, J. O. White, A. Yariv, “Passive (self pump) phase conjugate mirror: theoretical and experimental investigation,” Appl. Phys. Lett. 41, 689–691 (1982).
[CrossRef]

1981 (1)

1979 (1)

1967 (1)

K. V. Konjaev, “Interference method of two-dimensional Fourier transform with spatially incoherent illumination,” Phys. Lett. A 24, 490–491 (1967).
[CrossRef]

1966 (1)

Bartelt, H.

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1980).

Bracewell, R. N.

Breckinridge, J. B.

E. Ribak, C. Roddier, F. Roddier, J. B. Breckinridge, “Signal-to-noise limitations in white light holography,” Appl. Opt. 27, 1183–1186 (1988).
[CrossRef] [PubMed]

J. B. Breckinridge, E. Ribak, C. Roddier, F. Roddier, C. Habecker, “Real time optical correlation using white light Fourier transforms,” final report of the technical work (Jet Propulsion Laboratory, California Institute of Technology, Pasadena, Calif., 1988).

Cronin-Golomb, M.

M. Cronin-Golomb, B. Fischer, J. O. White, A. Yariv, “Passive (self pump) phase conjugate mirror: theoretical and experimental investigation,” Appl. Phys. Lett. 41, 689–691 (1982).
[CrossRef]

Fainman, Y.

Fischer, B.

M. Cronin-Golomb, B. Fischer, J. O. White, A. Yariv, “Passive (self pump) phase conjugate mirror: theoretical and experimental investigation,” Appl. Phys. Lett. 41, 689–691 (1982).
[CrossRef]

George, N.

S. Wang, N. George, “Fresnel zone transforms in spatially incoherent illumination,” Appl. Opt. 24, 84–850 (1985).
[CrossRef]

Gianino, P. D.

Goodman, J. W.

Habecker, C.

J. B. Breckinridge, E. Ribak, C. Roddier, F. Roddier, C. Habecker, “Real time optical correlation using white light Fourier transforms,” final report of the technical work (Jet Propulsion Laboratory, California Institute of Technology, Pasadena, Calif., 1988).

Horner, J. L.

Ichioka, Y.

Itoh, K.

Konforti, N.

Konjaev, K. V.

K. V. Konjaev, “Interference method of two-dimensional Fourier transform with spatially incoherent illumination,” Phys. Lett. A 24, 490–491 (1967).
[CrossRef]

Kotzer, T.

J. Rosen, T. Kotzer, J. Shamir, “Optical implementation of phase extraction pattern recognition,” Opt. Commun. 83, 10–14 (1991).
[CrossRef]

Lenz, E.

Lohmann, A. W.

Mahlab, U.

Marom, E.

Mendlovic, D.

Mertz, L.

L. Mertz, Transformations in Optics (Wiley, New York, 1965).

Nomura, T.

Potaturkin, O. I.

Ribak, E.

E. Ribak, C. Roddier, F. Roddier, J. B. Breckinridge, “Signal-to-noise limitations in white light holography,” Appl. Opt. 27, 1183–1186 (1988).
[CrossRef] [PubMed]

J. B. Breckinridge, E. Ribak, C. Roddier, F. Roddier, C. Habecker, “Real time optical correlation using white light Fourier transforms,” final report of the technical work (Jet Propulsion Laboratory, California Institute of Technology, Pasadena, Calif., 1988).

Roddier, C.

E. Ribak, C. Roddier, F. Roddier, J. B. Breckinridge, “Signal-to-noise limitations in white light holography,” Appl. Opt. 27, 1183–1186 (1988).
[CrossRef] [PubMed]

J. B. Breckinridge, E. Ribak, C. Roddier, F. Roddier, C. Habecker, “Real time optical correlation using white light Fourier transforms,” final report of the technical work (Jet Propulsion Laboratory, California Institute of Technology, Pasadena, Calif., 1988).

Roddier, F.

E. Ribak, C. Roddier, F. Roddier, J. B. Breckinridge, “Signal-to-noise limitations in white light holography,” Appl. Opt. 27, 1183–1186 (1988).
[CrossRef] [PubMed]

J. B. Breckinridge, E. Ribak, C. Roddier, F. Roddier, C. Habecker, “Real time optical correlation using white light Fourier transforms,” final report of the technical work (Jet Propulsion Laboratory, California Institute of Technology, Pasadena, Calif., 1988).

Rosen, J.

U. Mahlab, J. Rosen, J. Shamir, “Iterative generation of complex reference functions in joint transform correlator,” Opt. Lett. 16, 330–332 (1991).
[CrossRef] [PubMed]

J. Rosen, T. Kotzer, J. Shamir, “Optical implementation of phase extraction pattern recognition,” Opt. Commun. 83, 10–14 (1991).
[CrossRef]

Segev, M.

Shamir, J.

Streibl, N.

Villasenor, J.

J. Villasenor, R. N. Bracewell, “Optical phase obtained by analogue Hartley transformation,” Nature (London) 330, 735–737 (1987).
[CrossRef]

Wang, S.

S. Wang, N. George, “Fresnel zone transforms in spatially incoherent illumination,” Appl. Opt. 24, 84–850 (1985).
[CrossRef]

Weaver, C. S.

White, J. O.

M. Cronin-Golomb, B. Fischer, J. O. White, A. Yariv, “Passive (self pump) phase conjugate mirror: theoretical and experimental investigation,” Appl. Phys. Lett. 41, 689–691 (1982).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1980).

Yariv, A.

M. Segev, A. Yariv, “Optical interferometry between image-bearing beams and their redirected phase conjugates,” Opt. Lett. 17, 145–147 (1992).
[CrossRef] [PubMed]

M. Cronin-Golomb, B. Fischer, J. O. White, A. Yariv, “Passive (self pump) phase conjugate mirror: theoretical and experimental investigation,” Appl. Phys. Lett. 41, 689–691 (1982).
[CrossRef]

Appl. Opt. (8)

Appl. Phys. Lett. (1)

M. Cronin-Golomb, B. Fischer, J. O. White, A. Yariv, “Passive (self pump) phase conjugate mirror: theoretical and experimental investigation,” Appl. Phys. Lett. 41, 689–691 (1982).
[CrossRef]

J. Opt. Soc. Am. (1)

Nature (London) (1)

J. Villasenor, R. N. Bracewell, “Optical phase obtained by analogue Hartley transformation,” Nature (London) 330, 735–737 (1987).
[CrossRef]

Opt. Commun. (1)

J. Rosen, T. Kotzer, J. Shamir, “Optical implementation of phase extraction pattern recognition,” Opt. Commun. 83, 10–14 (1991).
[CrossRef]

Opt. Lett. (3)

Phys. Lett. A (1)

K. V. Konjaev, “Interference method of two-dimensional Fourier transform with spatially incoherent illumination,” Phys. Lett. A 24, 490–491 (1967).
[CrossRef]

Other (3)

L. Mertz, Transformations in Optics (Wiley, New York, 1965).

J. B. Breckinridge, E. Ribak, C. Roddier, F. Roddier, C. Habecker, “Real time optical correlation using white light Fourier transforms,” final report of the technical work (Jet Propulsion Laboratory, California Institute of Technology, Pasadena, Calif., 1988).

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1980).

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Figures (8)

Fig. 1
Fig. 1

Schematic illustration of the generalized model of the FI.

Fig. 2
Fig. 2

Shearing interferometer, as assigned to perform the RHT.

Fig. 3
Fig. 3

Experimental results for the modified HT obtained with the interferometer of Fig. 2.

Fig. 4
Fig. 4

Incoherent-recording JTC: L1–L3, lenses; P1–P4, planes; M’s, mirrors; S, light source; CCD, charge-coupled device; PC, personal computer; BS, beam splitter.

Fig. 5
Fig. 5

(a) Input mask to the incoherent-recording JTC. The two upper letters are the tested objects, and the lower letter is the reference. (b) Intensity distribution as recorded in plane P2 of Fig. 4 when the input was the mask of (a). (c) Pattern of (b) after the bias term was subtracted. (d) Transmissivity of the SLM in Fig. 4 [the distribution shown in (c), squared]. (e) Correlation results with a cross section from right to left through the upper peak.

Fig. 6
Fig. 6

Reconstruction of the grating shown in Fig. 5(b) (after binarization) obtained by coherent optical FT.

Fig. 7
Fig. 7

(a) Input mask to the incoherent-recording JTC for the second experiment. (b) Intensity distribution as recorded in plane P2 of Fig. 4 (without the dc term) with the input mask of (a). (c) Reconstruction of the grating shown in (b) (after binarization) obtained by coherent optical FT. (d) Correlation results with a cross section from right to left through the lower peak.

Fig. 8
Fig. 8

(a) Input mask to the incoherent correlator. (b) Binary hologram used as the phase-only filter of the letter T. (c) Intensity distribution as recorded in plane P2 of Fig. 4 with the input mask of (a). (d) Correlation results of the incoherent correlator with a cross section from right to left through the upper peak.

Equations (34)

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I ( u ) = 1 T - T / 2 T / 2 | R e 1 ( r , t ) g 1 ( r ) C 1 ( u ) exp ( j k L u · r ) + e 2 ( r , t ) g 2 ( r ) C 2 ( u ) exp ( j k L u · r ) d r | 2 d t ,
I ( u ) = C 1 ( u ) 2 R R [ I 1 ( r ) I 1 ( r ) ] 1 / 2 g 1 ( r ) g 1 * ( r ) γ 11 ( r , r ) × exp [ j k L u · ( r - r ) ] d r d r + C 2 ( u ) 2 R R [ I 2 ( r ) I 2 ( r ) ] 1 / 2 g 2 ( r ) g 2 * ( r ) γ 22 ( r , r ) × exp [ j k L u · ( r - r ) ] d r d r + C 1 ( u ) C 2 * ( u ) R R [ I 1 ( r ) I 2 ( r ) ] 1 / 2 g 1 ( r ) g 2 * ( r ) γ 12 ( r , r ) × exp [ j k L u · ( r - r ) ] d r d r + C 1 * ( u ) C 2 ( u ) R R [ I 1 ( r ) I 2 ( r ) ] 1 / 2 g 1 * ( r ) g 2 ( r ) γ 21 ( r , r ) × exp [ j k L u · ( r - r ) ] d r d r ,
γ i j ( r , r ) e i ( r , t ) , e j * ( r , t ) [ I i ( r ) I j ( r ) ] 1 / 2 ,
e i ( r , t ) , e j * ( r , t ) 1 T - T / 2 T / 2 e i ( r , t ) e j * ( r , t ) d t , I i ( r ) 1 T - T / 2 T / 2 e i ( r , t ) 2 d t .
γ ( r , r ) = exp ( j ψ ) I s ( ξ ) exp [ - j ( 2 π / λ L s ) ξ · ( r - r ) ] d ξ I s ( ξ ) d ξ ,
ψ = π λ L s ( r 2 - r 2 ) .
γ 11 ( r , r ) = γ ( r - r ) , γ 22 ( r , r ) = γ [ Φ ( r - r ) ] , γ 12 ( r , r ) = γ ( r - Φ r ) , γ 21 ( r , r ) = γ ( Φ r - r ) ,
I ( u ) = C 1 ( u ) 2 R R g 1 ( r ) g 1 * ( r ) γ ( r - r ) × exp [ j k L u · ( r - r ) ] d r d r + C 2 ( u ) 2 R R g 2 ( r ) g 2 * ( r ) γ [ Φ ( r - r ) ] × exp [ j k L u · ( r - r ) ] d r d r + W ( u ) R R g 1 ( r ) g 2 * ( r ) γ ( r - Φ r ) × exp [ j k L u · ( r - r ) ] d r d r + W * ( u ) R R g 1 * ( r ) g 2 ( r ) γ ( Φ r - r ) × exp [ j k L u · ( r - r ) ] d r d r ,
E H ( f r ) exp ( j α 2 ) G ( f r ) + exp ( - j α 2 ) G ( - f r ) ,
I H ( f r ) 2 R g ( r ) 2 d r + exp ( j α ) G ˜ ( 2 f r ) + exp ( - j α ) G ˜ ( - 2 f r ) ,
I H ( f r ) R g ( r ) 2 d r + R A 2 d r + A * exp ( j α ) G ( 2 f r ) + A exp ( - j α ) G * ( 2 f r ) .
H α ( f r ) = 2 g ( r ) cos [ 2 π f r · r + Φ ( r ) + α / 2 ] d r ,
H α ( f r ) = 1 2 exp ( j α 2 ) g ( r ) exp ( j 2 π f r · r ) d r + 1 2 exp ( - j α 2 ) g * ( r ) exp ( - j 2 π f r · r ) d r = 1 2 exp ( j α 2 ) [ G ( f r ) + exp ( - j α ) G * ( f r ) ] .
g ( r ) = exp [ j ( α / 2 ) ] 2 [ H ˜ α ( f r ) + H ˜ α + π ( f r ) ] × exp ( - j 2 π f r · r ) d f r .
c ( r ˜ ) = F { C 1 ( u ) 2 } * γ ( r ˜ ) R g 1 ( r ) g 1 * ( r - r ˜ ) d r + F { C 2 ( u ) 2 } * γ ( Φ r ˜ ) R g 2 ( r ) g 2 * ( r - r ˜ ) d r + F { W ( u ) } * R g 1 ( r ) g 2 * ( r - r ˜ ) γ ( r - Φ r + Φ r ˜ ) d r + F { W * ( u ) } * R g 2 ( r ) g 1 * ( r - r ˜ ) γ ( Φ r - r + r ˜ ) d r ,
s ( r ˜ ) = R g 1 ( r ) g 2 * ( r - r ˜ ) γ ( 2 r - r ˜ ) d r .
Q [ d ] exp ( j π λ d u 2 ) ,
I Q ( f r ) = 2 I α + Q [ 4 f 1 f 2 2 ] G ˜ ( f r ) + Q [ - 4 f 1 f 2 2 ] G ˜ * ( f r ) ,
I α = R g ( r ) 2 d r .
T Q ( f r ) = Q [ 8 f 1 f 2 2 ] [ G ˜ ( f r ) ] 2 + Q [ - 8 f 1 f 2 2 ] [ G ˜ * ( f r ) ] 2 + 2 G ˜ ( f r ) G ˜ * ( f r ) .
c ( r ˜ ) = | g ( f 2 2 f 3 r ˜ ) | 2 | g ( f 2 2 f 3 r ˜ ) | 2 ,
L [ r ] = exp ( j π λ r · u ) ,
I L ( f r ) = 2 I α + L [ 4 r d f 2 ] G ˜ ( f r ) + L [ - 4 r d f 2 ] G ˜ * ( f r ) .
T L ( f r ) = L [ 8 r d f 2 ] [ G ˜ ( f r ) ] 2 + L [ - 8 r d f 2 ] [ G ˜ * ( f r ) ] 2 + 2 G ˜ ( f r ) G ˜ * ( f r ) .
g ( r ) 2 = f ( r - r e ) + r ( r + r e ) ,
c ( r ¯ ) = f ( r ¯ ) f ( r ¯ ) + r ( r ¯ ) r ( r ¯ ) + f ( r ¯ - 2 r e ) r ( r ¯ ) + r ( r ¯ + 2 r e ) f ( r ¯ ) ,
R ( f r ) = A δ ˜ ( f r ) + R ˜ ( f r - f 0 ) + R ˜ * ( - f r - f 0 ) ,
I Q ( f r ) = rect ( f r W R ˜ ) [ 2 I α + Q [ 4 f 1 f 2 2 ] G ˜ ( f r + f 0 ) + Q [ - 4 f 1 f 2 2 ] G ˜ * ( f r + f 0 ) ] = 2 I α + Q [ 4 f 1 f 2 2 ] U ( f r ) + Q [ - 4 f 1 f 2 2 ] U * ( f r ) ,
U ( f r ) = R ˜ ( f r ) exp ( j 2 π f r · r e ) + F ( f r ) exp ( - j 2 π f r · r e ) .
c ( r ¯ ) = h ( r ¯ ) [ 2 δ ( r ¯ ) + g ( r ¯ ) 2 δ ( r ¯ - d g ) + g ( - r ¯ ) 2 δ ( r ¯ + d g ) ] ,
I h ( f r ) = 2 I α + L [ 4 d f f 2 ] F ˜ * ( 2 f r ) + L [ - 4 d f f 2 ] F ˜ ( 2 f r ) ,
H ( f r ) = { 1 if I h ( f r ) 2 I α - 1 otherwise .
H ( f r ) = n = - , odd n = 2 n π ( - 1 ) ( n - 1 ) / 2 exp { j n [ 2 π f r · d f + φ F ( f r ) ] } ,
s ( r ¯ ) = F { exp [ j ϕ F ( f r ) ] } * g ( r ¯ ) ( 2 ) .

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