Abstract

A map transformation is a willful geometrical distortion of an image. Examples are rotation, shearing, and local stretching of the coordinate system. We describe an optical setup that performs a map transformation by spatial filtering with two phase-only filters. The system is able to perform in x- and y-linear distortions (e.g., shears) as well as certain x- and y-nonlinear distortions. The distorting filters introduce no aberrations. The object may radiate coherently, incoherently, or partially coherently. Some experimental results are presented.

© 1983 Optical Society of America

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References

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  1. J. Duvernoy, Opt. Commun. 11, 373 (1974).
    [CrossRef]
  2. A. Sawchuk, J. Opt. Soc. Am. 64, 138 (1974).
    [CrossRef]
  3. D. Casasent, D. Psaltis, Prog. Opt. 16, 289 (1978).
    [CrossRef]
  4. G. Häusler, N. Streibl, Opt. Commun. 42, 381 (1982).
    [CrossRef]
  5. O. Bryngdahl, J. Opt. Soc. Am. 64, 1092 (1974); Opt. Commun. 10, 164 (1974).
    [CrossRef]

1982 (1)

G. Häusler, N. Streibl, Opt. Commun. 42, 381 (1982).
[CrossRef]

1978 (1)

D. Casasent, D. Psaltis, Prog. Opt. 16, 289 (1978).
[CrossRef]

1974 (3)

Bryngdahl, O.

Casasent, D.

D. Casasent, D. Psaltis, Prog. Opt. 16, 289 (1978).
[CrossRef]

Duvernoy, J.

J. Duvernoy, Opt. Commun. 11, 373 (1974).
[CrossRef]

Häusler, G.

G. Häusler, N. Streibl, Opt. Commun. 42, 381 (1982).
[CrossRef]

Psaltis, D.

D. Casasent, D. Psaltis, Prog. Opt. 16, 289 (1978).
[CrossRef]

Sawchuk, A.

Streibl, N.

G. Häusler, N. Streibl, Opt. Commun. 42, 381 (1982).
[CrossRef]

J. Opt. Soc. Am. (2)

Opt. Commun. (2)

J. Duvernoy, Opt. Commun. 11, 373 (1974).
[CrossRef]

G. Häusler, N. Streibl, Opt. Commun. 42, 381 (1982).
[CrossRef]

Prog. Opt. (1)

D. Casasent, D. Psaltis, Prog. Opt. 16, 289 (1978).
[CrossRef]

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Figures (5)

Fig. 1
Fig. 1

Optical setup for map transformations. All lenses, spherical or cylindrical, have the same focal length f. Longitudinal distances between successive components are always one focal length f. The hypothetical rays in the two principal planes (x,z) and (y,z) serve to explain the actions of the anamorphic and spherical lenses. Actually, any useful ray should, of course, hit the spherical lenses in planes LAX and LAY.

Fig. 2
Fig. 2

Undistorted objects to which our method is applied. In Figs. 25 the illumination is incoherent (white light, extended source).

Fig. 3
Fig. 3

Linear image shear (a1 ‡ 0, b1 = 0) version of Fig. 2.

Fig. 4
Fig. 4

Parabolic distortion in one direction (a2 ‡ 0, b1 = b2 = 0).

Fig. 5
Fig. 5

Parabolic distortion in both directions (a2 ‡ 0 ‡ b2).

Equations (10)

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u 1 ( μ 1 , y 1 ) = F x [ u 0 ( x 0 , y 1 ) ] = u 0 ( x 0 , y 1 ) exp ( 2 π i x 0 μ 1 ) d x 0 ,
f 1 ( μ 1 , y 1 ) = exp [ 2 π i μ 1 a ( y 1 ) ] .
exp [ 2 π i μ 1 a ( y 1 ) ] · F x [ u 0 ( x 0 , y 1 ) ] = F x { u 0 [ x 0 + a ( y 1 ) , y 1 ] } .
( x 0 y 0 ) by f 1 ( x 1 y 1 ) = [ x 0 + a ( y 0 ) y 0 ] .
u 2 ( x 2 , ν 2 ) = F y [ u 1 ( x 2 , y 1 ) ] .
f 2 ( x 2 , ν 2 ) = exp [ 2 π i ν 2 b ( x 2 ) ] .
( x 0 y 0 ) by f 1 ( x 1 y 1 ) = [ x 0 + a ( y 0 ) y 0 ] by f 2 ( x y ) = [ x 1 y 1 + b ( x 1 ) ] = { x 0 + a ( y 0 ) y 0 + b [ x 0 + a ( y 0 ) ] }
det | x x 0 x y 0 y x 0 y y 0 | 1.
a ( y ) = a 0 + a 1 · y + a 2 · y 2 + , b ( x ) = b 0 + b 1 · x + b 2 · x 2 + .
( x y ) = ( 1 a 1 0 1 ) · ( x 0 y 0 ) .

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