Abstract

A computerized optical system has been constructed for the design and generation of space-variant holographic filters. Special considerations of the computerized design of the space-variant filters and the operation and performances of the system are addressed and discussed. The system was experimentally employed to generate holographic optical filters for the optical implementations of Hough transform, coordinate transformations, etc. Experimental performances of the generated space-variant filters are evaluated and discussed.

© 1988 Optical Society of America

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References

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  1. J. W. Goodman, “Linear Space-Variant Optical Data Processing,” in Optical Data Processing Fundamentals, S. H. Lee, Ed. (Springer-Verlag, New York, 1981), pp. 235–259.
  2. W. T. Rhodes, “Space-Variant Optical Systems and Processing,” in Applications of Optical Fourier Transforms, H. Stark, Ed. (Academic, New York, 1982), Chap. 8.
    [CrossRef]
  3. J. F. Walkup, “Space-Variant Coherent Optical Processing,” Opt. Eng. 19, 339 (1980).
    [CrossRef]
  4. O. Bryngdahl, “Optical Map Transformations,” Opt. Commun. 10, 164 (1974).
    [CrossRef]
  5. O. Bryngdahl, “Geometrical Transformations in Optics,” J. Opt. Soc. Am. 64, 1092 (1974).
    [CrossRef]
  6. L. M. Deen, J. F. Walkup, M. O. Hagler, “Representations of Space-Variant Optical Systems Using Volume Holograms,” Appl. Opt. 14, 2438 (1975).
    [CrossRef] [PubMed]
  7. T. F. Krile, R. J. Marks, J. F. Walkup, M. O. Hagler, “Holographic Representations of Space-Variant Systems Using Phase Coded Reference Beams,” Appl. Opt. 16, 3131 (1977).
    [CrossRef] [PubMed]
  8. H. Bartelt, F. Sauer, “Space-Variant Filtering with Holographic Multifacet Elements,” Opt. Commun. 53, 296 (1985).
    [CrossRef]
  9. T. Georgekutty, H. K. Liu, “A Simplified Dichromated Gelatin Hologram Recording Process,” Appl. Opt. 26, 372 (1987).
    [CrossRef] [PubMed]
  10. R. O. Duda, P. E. Hart, “Use of the Hough Transform to Detect Lines and Curves in Pictures,” Commun. ACM 15, 11 (1972).
    [CrossRef]
  11. P. Ambs, S. H. Lee, Q. Tian, Y. Fainman, “Optical Implementation of the Hough Transform by a Matrix of Holograms,” Appl. Opt. 25, 4039 (1986).
    [CrossRef] [PubMed]
  12. D. Casasent, R. Krishnapuram, “Detection of Target Trajectories Using the Hough Transform,” Appl. Opt. 26, 247 (1987).
    [CrossRef] [PubMed]
  13. D. Casasent, R. Krishnapuram, “Curved Object Location by Hough Transformations and Inversions,” Pattern Recognition 20, 181 (1987).
    [CrossRef]
  14. H. Arsenault, Y. Hsu, Y. Yang, “Incoherent Method for Rotation-Invariant Recognition,” Appl. Opt. 21, 610 (1982).
    [CrossRef] [PubMed]
  15. H. Arsenault, Y. Hsu, “Rotation-Invariant Discrimination Between Almost Similar Objects,” Appl. Opt. 22, 130 (1983).
    [CrossRef] [PubMed]
  16. D. Casasent, D. Psaltis, “Scale Invariant Optical Correlation Using Mellin Transforms,” Opt. Commun. 17, 59 (1976).
    [CrossRef]
  17. D. Casasent, M. Kraus, “Polar Camera for Space-Variant Pattern Recognition,” Appl. Opt. 17, 1559 (1978).
    [CrossRef] [PubMed]
  18. H. Bartelt, S. K. Case, “Coordinate Transformation via Multifacet Holographic Optical Elements,” Opt. Eng. 22, 497 (1983).
    [CrossRef]
  19. R. Sandstrom, S. H. Lee, “Production of Optical Coordinate Transform Filters by a Computer Controlled Scanning Interferometric Pattern System,” Proc. Soc. Photo-Opt. Instrum. Eng. 437, 64 (1983).
  20. Y. Saito, S. Komatsu, H. Ohzu, “Scale and Rotation Invariant Real Time Optical Correlator Using Computer Generated Hologram,” Opt. Commun. 47, 8 (1987).
    [CrossRef]

1987 (4)

D. Casasent, R. Krishnapuram, “Curved Object Location by Hough Transformations and Inversions,” Pattern Recognition 20, 181 (1987).
[CrossRef]

Y. Saito, S. Komatsu, H. Ohzu, “Scale and Rotation Invariant Real Time Optical Correlator Using Computer Generated Hologram,” Opt. Commun. 47, 8 (1987).
[CrossRef]

D. Casasent, R. Krishnapuram, “Detection of Target Trajectories Using the Hough Transform,” Appl. Opt. 26, 247 (1987).
[CrossRef] [PubMed]

T. Georgekutty, H. K. Liu, “A Simplified Dichromated Gelatin Hologram Recording Process,” Appl. Opt. 26, 372 (1987).
[CrossRef] [PubMed]

1986 (1)

1985 (1)

H. Bartelt, F. Sauer, “Space-Variant Filtering with Holographic Multifacet Elements,” Opt. Commun. 53, 296 (1985).
[CrossRef]

1983 (3)

H. Bartelt, S. K. Case, “Coordinate Transformation via Multifacet Holographic Optical Elements,” Opt. Eng. 22, 497 (1983).
[CrossRef]

R. Sandstrom, S. H. Lee, “Production of Optical Coordinate Transform Filters by a Computer Controlled Scanning Interferometric Pattern System,” Proc. Soc. Photo-Opt. Instrum. Eng. 437, 64 (1983).

H. Arsenault, Y. Hsu, “Rotation-Invariant Discrimination Between Almost Similar Objects,” Appl. Opt. 22, 130 (1983).
[CrossRef] [PubMed]

1982 (1)

1980 (1)

J. F. Walkup, “Space-Variant Coherent Optical Processing,” Opt. Eng. 19, 339 (1980).
[CrossRef]

1978 (1)

1977 (1)

1976 (1)

D. Casasent, D. Psaltis, “Scale Invariant Optical Correlation Using Mellin Transforms,” Opt. Commun. 17, 59 (1976).
[CrossRef]

1975 (1)

1974 (2)

O. Bryngdahl, “Geometrical Transformations in Optics,” J. Opt. Soc. Am. 64, 1092 (1974).
[CrossRef]

O. Bryngdahl, “Optical Map Transformations,” Opt. Commun. 10, 164 (1974).
[CrossRef]

1972 (1)

R. O. Duda, P. E. Hart, “Use of the Hough Transform to Detect Lines and Curves in Pictures,” Commun. ACM 15, 11 (1972).
[CrossRef]

Ambs, P.

Arsenault, H.

Bartelt, H.

H. Bartelt, F. Sauer, “Space-Variant Filtering with Holographic Multifacet Elements,” Opt. Commun. 53, 296 (1985).
[CrossRef]

H. Bartelt, S. K. Case, “Coordinate Transformation via Multifacet Holographic Optical Elements,” Opt. Eng. 22, 497 (1983).
[CrossRef]

Bryngdahl, O.

O. Bryngdahl, “Geometrical Transformations in Optics,” J. Opt. Soc. Am. 64, 1092 (1974).
[CrossRef]

O. Bryngdahl, “Optical Map Transformations,” Opt. Commun. 10, 164 (1974).
[CrossRef]

Casasent, D.

D. Casasent, R. Krishnapuram, “Curved Object Location by Hough Transformations and Inversions,” Pattern Recognition 20, 181 (1987).
[CrossRef]

D. Casasent, R. Krishnapuram, “Detection of Target Trajectories Using the Hough Transform,” Appl. Opt. 26, 247 (1987).
[CrossRef] [PubMed]

D. Casasent, M. Kraus, “Polar Camera for Space-Variant Pattern Recognition,” Appl. Opt. 17, 1559 (1978).
[CrossRef] [PubMed]

D. Casasent, D. Psaltis, “Scale Invariant Optical Correlation Using Mellin Transforms,” Opt. Commun. 17, 59 (1976).
[CrossRef]

Case, S. K.

H. Bartelt, S. K. Case, “Coordinate Transformation via Multifacet Holographic Optical Elements,” Opt. Eng. 22, 497 (1983).
[CrossRef]

Deen, L. M.

Duda, R. O.

R. O. Duda, P. E. Hart, “Use of the Hough Transform to Detect Lines and Curves in Pictures,” Commun. ACM 15, 11 (1972).
[CrossRef]

Fainman, Y.

Georgekutty, T.

Goodman, J. W.

J. W. Goodman, “Linear Space-Variant Optical Data Processing,” in Optical Data Processing Fundamentals, S. H. Lee, Ed. (Springer-Verlag, New York, 1981), pp. 235–259.

Hagler, M. O.

Hart, P. E.

R. O. Duda, P. E. Hart, “Use of the Hough Transform to Detect Lines and Curves in Pictures,” Commun. ACM 15, 11 (1972).
[CrossRef]

Hsu, Y.

Komatsu, S.

Y. Saito, S. Komatsu, H. Ohzu, “Scale and Rotation Invariant Real Time Optical Correlator Using Computer Generated Hologram,” Opt. Commun. 47, 8 (1987).
[CrossRef]

Kraus, M.

Krile, T. F.

Krishnapuram, R.

D. Casasent, R. Krishnapuram, “Curved Object Location by Hough Transformations and Inversions,” Pattern Recognition 20, 181 (1987).
[CrossRef]

D. Casasent, R. Krishnapuram, “Detection of Target Trajectories Using the Hough Transform,” Appl. Opt. 26, 247 (1987).
[CrossRef] [PubMed]

Lee, S. H.

P. Ambs, S. H. Lee, Q. Tian, Y. Fainman, “Optical Implementation of the Hough Transform by a Matrix of Holograms,” Appl. Opt. 25, 4039 (1986).
[CrossRef] [PubMed]

R. Sandstrom, S. H. Lee, “Production of Optical Coordinate Transform Filters by a Computer Controlled Scanning Interferometric Pattern System,” Proc. Soc. Photo-Opt. Instrum. Eng. 437, 64 (1983).

Liu, H. K.

Marks, R. J.

Ohzu, H.

Y. Saito, S. Komatsu, H. Ohzu, “Scale and Rotation Invariant Real Time Optical Correlator Using Computer Generated Hologram,” Opt. Commun. 47, 8 (1987).
[CrossRef]

Psaltis, D.

D. Casasent, D. Psaltis, “Scale Invariant Optical Correlation Using Mellin Transforms,” Opt. Commun. 17, 59 (1976).
[CrossRef]

Rhodes, W. T.

W. T. Rhodes, “Space-Variant Optical Systems and Processing,” in Applications of Optical Fourier Transforms, H. Stark, Ed. (Academic, New York, 1982), Chap. 8.
[CrossRef]

Saito, Y.

Y. Saito, S. Komatsu, H. Ohzu, “Scale and Rotation Invariant Real Time Optical Correlator Using Computer Generated Hologram,” Opt. Commun. 47, 8 (1987).
[CrossRef]

Sandstrom, R.

R. Sandstrom, S. H. Lee, “Production of Optical Coordinate Transform Filters by a Computer Controlled Scanning Interferometric Pattern System,” Proc. Soc. Photo-Opt. Instrum. Eng. 437, 64 (1983).

Sauer, F.

H. Bartelt, F. Sauer, “Space-Variant Filtering with Holographic Multifacet Elements,” Opt. Commun. 53, 296 (1985).
[CrossRef]

Tian, Q.

Walkup, J. F.

Yang, Y.

Appl. Opt. (8)

Commun. ACM (1)

R. O. Duda, P. E. Hart, “Use of the Hough Transform to Detect Lines and Curves in Pictures,” Commun. ACM 15, 11 (1972).
[CrossRef]

J. Opt. Soc. Am. (1)

Opt. Commun. (4)

D. Casasent, D. Psaltis, “Scale Invariant Optical Correlation Using Mellin Transforms,” Opt. Commun. 17, 59 (1976).
[CrossRef]

O. Bryngdahl, “Optical Map Transformations,” Opt. Commun. 10, 164 (1974).
[CrossRef]

H. Bartelt, F. Sauer, “Space-Variant Filtering with Holographic Multifacet Elements,” Opt. Commun. 53, 296 (1985).
[CrossRef]

Y. Saito, S. Komatsu, H. Ohzu, “Scale and Rotation Invariant Real Time Optical Correlator Using Computer Generated Hologram,” Opt. Commun. 47, 8 (1987).
[CrossRef]

Opt. Eng. (2)

J. F. Walkup, “Space-Variant Coherent Optical Processing,” Opt. Eng. 19, 339 (1980).
[CrossRef]

H. Bartelt, S. K. Case, “Coordinate Transformation via Multifacet Holographic Optical Elements,” Opt. Eng. 22, 497 (1983).
[CrossRef]

Pattern Recognition (1)

D. Casasent, R. Krishnapuram, “Curved Object Location by Hough Transformations and Inversions,” Pattern Recognition 20, 181 (1987).
[CrossRef]

Proc. Soc. Photo-Opt. Instrum. Eng. (1)

R. Sandstrom, S. H. Lee, “Production of Optical Coordinate Transform Filters by a Computer Controlled Scanning Interferometric Pattern System,” Proc. Soc. Photo-Opt. Instrum. Eng. 437, 64 (1983).

Other (2)

J. W. Goodman, “Linear Space-Variant Optical Data Processing,” in Optical Data Processing Fundamentals, S. H. Lee, Ed. (Springer-Verlag, New York, 1981), pp. 235–259.

W. T. Rhodes, “Space-Variant Optical Systems and Processing,” in Applications of Optical Fourier Transforms, H. Stark, Ed. (Academic, New York, 1982), Chap. 8.
[CrossRef]

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Figures (10)

Fig. 1
Fig. 1

Schematic of the hybrid system for producing computer-generated space-variant holographic filters. A matrix of 256 by 256 holograms of different PSF is recorded with this system. The following focal length f in millimeters and f/No. lenses are employed f1 = 100,f2 = 20,f3 = 360,f4 = 80 of f/1.6,f5 = 80 of f/1.6,f6 = 80 of f/1.6, f7 = 20, f8 = 120, f9 = 120.

Fig. 2
Fig. 2

Recording system; on the right it is easy to distinguish the CRT/LCLV and on the left the translation stage supporting the holographic plate.

Fig. 3
Fig. 3

Photomicrograph of a matrix of 256 × 256 holograms.

Fig. 4
Fig. 4

Real-time space-variant optical processor (top view). The input pattern is displayed on the CRT/LCLV and imaged on the space-variant holographic filter. A camera is placed in the reconstruction plane of the holograms.

Fig. 5
Fig. 5

Principle of the Hough transform of a straight line in normal parametrization: (a) input image plane; (b) parameter domain plane.

Fig. 6
Fig. 6

Detection of the parameters of a straight line defined in normal parametrization by ρ = 0, θ = 45°(y = x). Parameter domain of the HT: (a) computer simulation of the HT; (b) output of the optical HT processor; (c) optical output after threshold, one point remains at θ = 45°, ρ = 0.

Fig. 7
Fig. 7

Detection of the parameters of a circle with the HT: (a) input image of a circle of radius r = 40 centered at (x0,y0) = (−10,20); (b) computer calculation of the parameter domain. The parameters of the circle are determined from the coordinate of the points A,B,C, and D as explained in the text. For A and B with coordinates (0,30) and (0,−50), respecively, the extracted parameters are r = 40 and x0 = −10. For C and D with coordinates (90°,60) and (90°,−20) the extracted parameters are r = 40 and y0 = 20. (c) Output of the optical processor from which the parameters r = 40, x0 = −10, and y0 = 20 can be extracted.

Fig. 8
Fig. 8

Detection of the parameters of an ellipse with the HT: (a) input image of an ellipse centered at (x0,y0) = (0,0) and of axis a = 30 and b = 50; (b) computer calculation of the parameter domain. The parameters of the ellipse are determined from the coordinate of the points A,B,C, and D. For A and B with coordinates (0,+30) and (0,−30), respectively, the extracted parameters are a = 30 and x0 = 0. For C and D with coordinates (90,50), and (90,−50) the extracted parameters are b = 50 and y0 = 0. (c) Output of the optical processor for the same input ellipse from which the following parameters a = 30, b = 50, x0 = 0, and y0 can also be extracted.

Fig. 9
Fig. 9

The lnrθ coordinate transform of a rotation variant pattern: (a) original input image of an off-center circle; (b) optical coordinate transform lnrθ of the input pattern; (c) optical coordinate transform of the input pattern rotated by an angle of 45° around the origin of the coordinate system; (d) optical coordinate transform of the input pattern rotated by an angle of 90° around the origin of the coordinate system.

Fig. 10
Fig. 10

The lnrθ coordinate transform of another scale-variant pattern: (a) input image of an airplane with original scale 1; (b) optical coordinate transform of the image in (a); (c) input image of (a) scaled by a factor of α = 0.7; (d) optical coordinate transform of the image in (c). The transformed pattern of (b) is shifted upward by an amount of lnα.

Tables (1)

Tables Icon

Table I Maximum Number of Holograms Nmax in one Direction in a 2-D Plane which can be Recorded According to Eq. (10), where f/No. is the f/No. of the FT Lens and Ih is the Size of the Matrix of Holograms

Equations (19)

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F ( x , y ) = - + f ( ζ , η ) h ( x , y , ζ , η ) d ζ d η ,
D 2 l .
d = ( 2 λ f ) / Δ ,
a d .
D = f / f / No . = ( a Δ ) / ( 2 λ f / No . ) .
a 2 ( 2 ) λ f / No . l / Δ .
N = l / Δ .
a f / No .2 ( 2 ) λ N .
l h = a N ,
N max { l h / [ f / No .2 ( 2 ) λ ] } 1 / 2 .
ρ = x i cos θ + y i sin θ ,
F ( θ , ρ ) = - + f ( x , y ) δ ( ρ - x cos θ - y sin θ ) d x d y ,
x = x 0 + r cos α , y = y 0 + r sin α ,
F ( θ , ρ ) = 0 2 π δ [ ρ - ( r cos α + x 0 ) cos θ - ( r sin α + y 0 ) sin θ ] d α .
F ( 0 , ρ ) = 0 2 π δ [ ρ - ( r cos α + x 0 ) ] d α ,
F ( 90 ° , ρ ) = 0 2 π δ [ ρ - ( r cos α + y 0 ) ] d α .
F ( 0 , ρ ) = 0 2 π δ [ p - ( a cos α + x 0 ) ] d α ;
F ( 90 ° , ρ ) = 0 2 π δ [ p - ( b sin α + y 0 ) ] d α .
r 2 = x 2 + y 2             or             ln r 2 = ln ( x 2 + y 2 ) θ = arctan ( x / y ) .

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