Abstract

New optical configurations for performing general coordinate transformation operations of shear, rotation, and their combination are presented. These configurations consist of refractive spherical and cylindrical lenses that are readily available. Typically, high-resolution imagery can be obtained, depending on the size of the input object, the illumination wavelength, and the f-number of the lenses. Basic and more general configurations are presented, along with experimental results clearly showing image shearing, rotation, and a combination of these with high-quality output imagery.

© 2000 Optical Society of America

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1997 (1)

K. G. Larkin, M. A. Oldfield, H. Klemm, “Fast Fourier method for the accurate rotation of sampled images,” Opt. Commun. 139, 99–106 (1997).
[CrossRef]

1996 (1)

Ji-Hong Feng, Min-Xian Wu, Guo-Fan Jin, “Invariant joint transform correlator using coordinate transformation,” Opt. Eng. 35, 3392–3399 (1996).
[CrossRef]

1993 (2)

1992 (1)

1990 (1)

1987 (1)

W. J. Hossack, A. M. Darling, A. Dahdouh, “Coordinate transformations with multiple computer-generated optical elements,” J. Mod. Opt. 34, 1235–1250 (1987).
[CrossRef]

1984 (1)

1983 (1)

1982 (1)

G. Häusler, N. Streibl, “Optical compensation of geometrical distortion by deformable mirror,” Opt. Commun. 42, 381–385 (1982).
[CrossRef]

1981 (1)

1980 (1)

1977 (1)

1974 (3)

Braunecker, B.

Bryngdahl, O.

Casasent, D.

D. Casasent, D. Psaltis, “Deformation invariant, space variant optical pattern recognition,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1978), Vol. 16, pp. 289–356.

Case, S. K.

Cederquist, J. N.

Dahdouh, A.

W. J. Hossack, A. M. Darling, A. Dahdouh, “Coordinate transformations with multiple computer-generated optical elements,” J. Mod. Opt. 34, 1235–1250 (1987).
[CrossRef]

Darling, A. M.

W. J. Hossack, A. M. Darling, A. Dahdouh, “Coordinate transformations with multiple computer-generated optical elements,” J. Mod. Opt. 34, 1235–1250 (1987).
[CrossRef]

Davidson, N.

Feng, Ji-Hong

Ji-Hong Feng, Min-Xian Wu, Guo-Fan Jin, “Invariant joint transform correlator using coordinate transformation,” Opt. Eng. 35, 3392–3399 (1996).
[CrossRef]

Friesem, A. A.

Goodman, J. W.

Hansen, E. W.

Hasman, E.

Haugen, P. R.

Häusler, G.

G. Häusler, N. Streibl, “Optical compensation of geometrical distortion by deformable mirror,” Opt. Commun. 42, 381–385 (1982).
[CrossRef]

Hossack, W. J.

W. J. Hossack, A. M. Darling, A. Dahdouh, “Coordinate transformations with multiple computer-generated optical elements,” J. Mod. Opt. 34, 1235–1250 (1987).
[CrossRef]

Jin, Guo-Fan

Ji-Hong Feng, Min-Xian Wu, Guo-Fan Jin, “Invariant joint transform correlator using coordinate transformation,” Opt. Eng. 35, 3392–3399 (1996).
[CrossRef]

Kellman, P.

Klemm, H.

K. G. Larkin, M. A. Oldfield, H. Klemm, “Fast Fourier method for the accurate rotation of sampled images,” Opt. Commun. 139, 99–106 (1997).
[CrossRef]

Larkin, K. G.

K. G. Larkin, M. A. Oldfield, H. Klemm, “Fast Fourier method for the accurate rotation of sampled images,” Opt. Commun. 139, 99–106 (1997).
[CrossRef]

Loberge, O. J.

Lohmann, A. W.

Mendlovic, D.

Oldfield, M. A.

K. G. Larkin, M. A. Oldfield, H. Klemm, “Fast Fourier method for the accurate rotation of sampled images,” Opt. Commun. 139, 99–106 (1997).
[CrossRef]

Ozaktas, H. M.

Psaltis, D.

D. Casasent, D. Psaltis, “Deformation invariant, space variant optical pattern recognition,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1978), Vol. 16, pp. 289–356.

Sawchuk, A.

Schnell, B.

Streibl, N.

A. W. Lohmann, N. Streibl, “Map transformations by optical anamorphic processing,” Appl. Opt. 22, 780–783 (1983).
[CrossRef] [PubMed]

G. Häusler, N. Streibl, “Optical compensation of geometrical distortion by deformable mirror,” Opt. Commun. 42, 381–385 (1982).
[CrossRef]

Stuff, M. A.

Tai, A. M.

Wu, Min-Xian

Ji-Hong Feng, Min-Xian Wu, Guo-Fan Jin, “Invariant joint transform correlator using coordinate transformation,” Opt. Eng. 35, 3392–3399 (1996).
[CrossRef]

Appl. Opt. (6)

J. Mod. Opt. (1)

W. J. Hossack, A. M. Darling, A. Dahdouh, “Coordinate transformations with multiple computer-generated optical elements,” J. Mod. Opt. 34, 1235–1250 (1987).
[CrossRef]

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (2)

Opt. Commun. (3)

K. G. Larkin, M. A. Oldfield, H. Klemm, “Fast Fourier method for the accurate rotation of sampled images,” Opt. Commun. 139, 99–106 (1997).
[CrossRef]

G. Häusler, N. Streibl, “Optical compensation of geometrical distortion by deformable mirror,” Opt. Commun. 42, 381–385 (1982).
[CrossRef]

O. Bryngdahl, “Optical map transformations,” Opt. Commun. 10, 164–168 (1974).
[CrossRef]

Opt. Eng. (1)

Ji-Hong Feng, Min-Xian Wu, Guo-Fan Jin, “Invariant joint transform correlator using coordinate transformation,” Opt. Eng. 35, 3392–3399 (1996).
[CrossRef]

Other (1)

D. Casasent, D. Psaltis, “Deformation invariant, space variant optical pattern recognition,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1978), Vol. 16, pp. 289–356.

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Figures (8)

Fig. 1
Fig. 1

Basic optical configuration for performing coordinate transformations.

Fig. 2
Fig. 2

Realization of the phase filter with a combination of a positive cylindrical lens and a negative cylindrical lens.

Fig. 3
Fig. 3

Three-stage optical configuration for performing coordinate transformations.

Fig. 4
Fig. 4

Experimental sheared output images from the basic coordinate transformation configuration of Fig. 1 with two-dimensional binary grid as input object: (a) α=0° (undistorted image), (b) α=+15°, (c) α=+30° and α=+60°, (d) α=-30°, (e) α=+45°.

Fig. 5
Fig. 5

Experimental sheared output images from the basic coordinate transformation configuration of Fig. 1 with “Lena” as input object: (a) α=0° (undistorted image), (b) α=-30°, (c) α=+30°.

Fig. 6
Fig. 6

Experimental output images from the three-stage coordinate transformation configuration of Fig. 3 with a two-dimensional binary grid as input object: (a) αA=αB=αC=0° (undistorted image); (b) αB=+8° and αA=αC=+80°; (c) αA=+40°, αB=+120°, and αC=+100°; (d) αB=+120° and αA=αC=0°; (e) αA=+136°, αB=+40°, and αC=+134°; (f) αB=+118° and αA=αC=+70°.

Fig. 7
Fig. 7

Experimental output images from the three-stage coordinate transformation configuration of Fig. 3 with Lena as input object: (a) αA=αB=αC=0° (undistorted image); (b) αA=+136°, αB=+40°, and αC=+134°; (c) αB=+172° and αA=αC=+70°; (d) αB=+20° and αA=αC=+100°; (e) αB=+65° and αA=αC=0°; (f ) αB=+115° and αA=αC=0°.

Fig. 8
Fig. 8

Two examples of passive optical stages, where LS represents the spherical lens and LCX and LCY represent the cylindrical lenses whose orientations are in the x and the y direction, respectively: (a) image inversion in the y direction and (b) image stretching and contraction.

Equations (32)

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

2(x, y)x0y0=ABCD=AD-BC=1.
u˜0(ν, y1)=u0(x0, y0)exp(-2πiνx0)dx0,
u˜1(ν, y1)=u˜0(ν, y1)exp(2πiBνy1).
u0(x+By1, y1)=u˜1(ν, y1)exp(2πiνx)dν=u0(x+By, y),
x0y0xy=1-B01 x0y0.
Φ(x1, y1; α, β)=-(π/λfB)[(x1cos α+y1sin α)2
-(x1cos β+y1sin β)2].
Φ(x1, y1; α, -α)=-(2π/λfB)x1 y1sin 2α.
B=-ffBsin 2α.
Φ(x1, y1; α, β)
=-(π/λfB)[(x1cos α+y1sin α)2+(x1cos β+y1sin β)2]
=-(π/λfB)(x12+y12+2x1 y1sin 2α).
x0y0xy=1+bc-a-c-abc-b1+ab x0y0
=MTx0y0,
MT=ABCD=1+bc-a-c-abc-b1+ab.
A=1+bc,
B=-a-c-abc,
C=-b,
D=1+ab.
MT=MxcMxbMxa=1-c01 10-b1 1-a01.
a=(A-BC-1)/AC,
b=-C,
c=(1-A)/C.
MT=-100-1
MT=2001/2.
ABCD=10d1 1c01 10b1 1a01.
a=BA+1-AA1b,
c=(A-1) 1b,
d=CA-1A1b,
a=B(1-D)+B2d,
b=C,
c=B,

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