Abstract

Optical techniques are introduced for performing general types of map transformations. Geometrical image modifications like coordinate transformations and local translation, inversion, reflection, stretching, and rotation require highly space-variant systems. Filters can be introduced into optical coherent systems in such a way that their local phase variations are able to influence light from local object areas. Experimental results of typical optical geometrical transformations like conformal mapping of object distributions are presented. Another capability of this technique is to image a specific, arbitrary surface onto another surface of general shape.

© 1974 Optical Society of America

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References

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  1. O. Bryngdahl, Opt. Commun. 10, 164 (1974).
    [CrossRef]
  2. M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1965), p. 753.
  3. L. B. Lesem, P. M. Hirsch, and J. A. Jordan, IBM J. Res. Dev. 13, 150 (1969).
    [CrossRef]
  4. O. Bryngdahl and A. W. Lohmann, J. Opt. Soc. Am. 58, 141 (1968).
    [CrossRef]
  5. O. Bryngdahl, J. Opt. Soc. Am. 63, 1098 (1973).
    [CrossRef]
  6. A. W. Lohmann and D. P. Paris, Appl. Opt. 6, 1739 (1967).
    [CrossRef] [PubMed]
  7. W. H. Lee, Appl. Opt. 13, 1677 (1974).
    [CrossRef] [PubMed]
  8. O. Bryngdahl, J. Opt. Soc. Am. 60, 915 (1970).
    [CrossRef]
  9. A. A. Sawchuk, J. Opt. Soc. Am. 63, 1052 (1973); J. Opt. Soc. Am. 64, 138 (1974).
    [CrossRef]
  10. F. C. Billingsley, Appl. Opt. 9, 289 (1970).
    [CrossRef] [PubMed]

1974 (2)

1973 (2)

1970 (2)

1969 (1)

L. B. Lesem, P. M. Hirsch, and J. A. Jordan, IBM J. Res. Dev. 13, 150 (1969).
[CrossRef]

1968 (1)

1967 (1)

Billingsley, F. C.

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1965), p. 753.

Bryngdahl, O.

Hirsch, P. M.

L. B. Lesem, P. M. Hirsch, and J. A. Jordan, IBM J. Res. Dev. 13, 150 (1969).
[CrossRef]

Jordan, J. A.

L. B. Lesem, P. M. Hirsch, and J. A. Jordan, IBM J. Res. Dev. 13, 150 (1969).
[CrossRef]

Lee, W. H.

Lesem, L. B.

L. B. Lesem, P. M. Hirsch, and J. A. Jordan, IBM J. Res. Dev. 13, 150 (1969).
[CrossRef]

Lohmann, A. W.

Paris, D. P.

Sawchuk, A. A.

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1965), p. 753.

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Figures (11)

Fig. 1
Fig. 1

Conventional imaging system with telecentric lens arrangement. Illustration of how limited image distortions can be produced with a phase plate, P.

Fig. 2
Fig. 2

Setup to display extended power spectra in the Fraunhofer diffraction plane of a lens. The field covered with a portion of a Fresnel zone plate in O appears with geometric similarity in F.

Fig. 3
Fig. 3

System modification to display geometrical image transformations. A lens in O extends the distribution in F, where a phase filter introduces local image aberrations.

Fig. 4
Fig. 4

Visualization of grid structures used with linear change of spatial frequency.

Fig. 5
Fig. 5

Display of transformed object distributions in a Fraunhofer diffraction plane. Crossed grids were placed in contact with the object, which was a 90° sector of a circular grating. The grids were linear in frequency in (a) and in period in (b).

Fig. 6
Fig. 6

Illustrations of introduction of one-dimensional image modifications. (a) Shows the object, (b) and (c) images with horizontal stretching and contraction, (d) exponential stretching, and (e), (f), and (g) local stretch in the middle of the image.

Fig. 7
Fig. 7

Filter types used to introduce local one-dimensional image modifications. (a) With exponential increase of frequency was used to obtain Fig. 6(d) and (b) with frequency increasing according to a normal distribution function Fig. 6(f).

Fig. 8
Fig. 8

Plots of filters to perform two-dimensional transformations.

Fig. 9
Fig. 9

Image transforms corresponding to the filters of Fig. 8.

Fig. 10
Fig. 10

Crossed-grid structure to transform a square area conformally into a 90° portion of an annulus area.

Fig. 11
Fig. 11

Image-plane display made by use of the filter in Fig 10. The crossed order to the upper right is the 90° sector transformation of the square grid structure at the lower left.

Equations (21)

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w = f ( z ) = u ( x , y ) + i v ( x , y ) .
u = u ( x , y ) ;             v = v ( x , y ) .
ã ( u , v ) = - a ( x , y ) exp { i φ ( x , y ) } × exp { - i k ( x u + y v ) / f L } d x d y ,
g ( x , y ) exp { i k h ( x , y ) } d x d y ,
h x = h y = 0.
2 π g ( x 0 , y 0 ) k - 1 ( h x x h y y - h x y 2 ) - 1 2 × exp { i [ k h ( x 0 , y 0 ) + π / 2 ] } ,
ã ( u 0 , v 0 ) 2 4 π 2 a 2 ( x 0 , y 0 ) / ( φ x x φ y y - φ x y 2 ) ,
φ ( x , y ) x = k f L u ;             φ ( x , y ) y = k f L v .
φ ( x , y ) = π λ f L ( x 2 + y 2 ) + const .
a I ( x , y ) = - ã ( u , v ) exp { i φ ( u , v ) } × exp { - i k ( u x + v y ) / f L } d u d v ,
x = u + f L k φ ( u , v ) u ,             y = v + f L k φ ( u , v ) v ,
2 π λ x sin ϑ - φ ( x , y ) = 2 π n ,
sin ϑ = λ ν 0
exp { i W } = exp { i φ ( x , y ) + i 2 π x ν 0 } .
2 π ν x = W / x = φ / x + 2 π ν 0 ; 2 π ν y = W / y = φ / y .
φ x = 2 π { ν x - ν 0 } ;             φ y = 2 π ν y .
φ ( x ) = k x 2 / ( 2 f L ) ,             φ ( y ) = k y 2 / ( 2 f L ) ,
ν ( x ) = ν 0 + x / ( λ f L ) ;             ν ( y ) = ν 0 + y / ( λ f L ) .
ν = ν 0 / ( 1 - p x ) ,
u = λ f L ν 0 p x / ( 1 - p x ) .
x = u + λ f L { ν u - ν 0 } .