Abstract

An optical method of measuring and displaying depth information has been developed which is based on the Fresnel diffraction properties of coherently illuminated gratings. Cosine modulated gratings are found to be more flexible than simple gratings for depth measurement. A pair of gratings crossed at right angles is used to provide two independent depth measurement channels. A real-time white-light processor has been developed to convert the depth information into pseudocolor. The depth signal is input to a white-light processor by a liquid crystal light valve. The two depth channels are filtered to produce different color images such that the hue of the combined image is a function of depth.

© 1984 Optical Society of America

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References

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  1. W. K. Pratt, Digital Image Processing (Wiley-Interscience, New York, 1978), pp. 336–338.
  2. J. Bescos, T. C. Strand, Appl. Opt. 17, 2524 (1978).
    [PubMed]
  3. B. Querzola, Appl. Opt. 18, 3035 (1979).
    [CrossRef] [PubMed]
  4. N. Balasubramanian, “Optical Processing in Photogrammetry,” in Optical Data Processing, D. Casasent, Ed. (Springer, Berlin, 1978), pp. 119–149.
    [CrossRef]
  5. E. L. Hall, J. B. K. Tio, C. A. McPherson, F. A. Sadjadi, Computer 15, 42 (1982).
    [CrossRef]
  6. W. R. J. Funnell, Appl. Opt. 20, 3245 (1981).
    [CrossRef] [PubMed]
  7. J. C. Perrin, A. Thomas, Appl. Opt. 18, 563 (1979).
    [PubMed]
  8. J. T. Winthrop, C. R. Worthington, J. Opt. Soc. Am. 55, 373 (1965).
    [CrossRef]
  9. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, 1968), p. 54.
  10. P. Chavel, T. C. Strand, AIP Conf. Proc. 65, 431 (1980).
    [CrossRef]
  11. W. P. Bleha et al., Opt. Eng. 17, 371 (1978).
  12. T. C. Strand, IBM Research Laboratories, private communications.

1982 (1)

E. L. Hall, J. B. K. Tio, C. A. McPherson, F. A. Sadjadi, Computer 15, 42 (1982).
[CrossRef]

1981 (1)

1980 (1)

P. Chavel, T. C. Strand, AIP Conf. Proc. 65, 431 (1980).
[CrossRef]

1979 (2)

1978 (2)

J. Bescos, T. C. Strand, Appl. Opt. 17, 2524 (1978).
[PubMed]

W. P. Bleha et al., Opt. Eng. 17, 371 (1978).

1965 (1)

Balasubramanian, N.

N. Balasubramanian, “Optical Processing in Photogrammetry,” in Optical Data Processing, D. Casasent, Ed. (Springer, Berlin, 1978), pp. 119–149.
[CrossRef]

Bescos, J.

Bleha, W. P.

W. P. Bleha et al., Opt. Eng. 17, 371 (1978).

Chavel, P.

P. Chavel, T. C. Strand, AIP Conf. Proc. 65, 431 (1980).
[CrossRef]

Funnell, W. R. J.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, 1968), p. 54.

Hall, E. L.

E. L. Hall, J. B. K. Tio, C. A. McPherson, F. A. Sadjadi, Computer 15, 42 (1982).
[CrossRef]

McPherson, C. A.

E. L. Hall, J. B. K. Tio, C. A. McPherson, F. A. Sadjadi, Computer 15, 42 (1982).
[CrossRef]

Perrin, J. C.

Pratt, W. K.

W. K. Pratt, Digital Image Processing (Wiley-Interscience, New York, 1978), pp. 336–338.

Querzola, B.

Sadjadi, F. A.

E. L. Hall, J. B. K. Tio, C. A. McPherson, F. A. Sadjadi, Computer 15, 42 (1982).
[CrossRef]

Strand, T. C.

P. Chavel, T. C. Strand, AIP Conf. Proc. 65, 431 (1980).
[CrossRef]

J. Bescos, T. C. Strand, Appl. Opt. 17, 2524 (1978).
[PubMed]

T. C. Strand, IBM Research Laboratories, private communications.

Thomas, A.

Tio, J. B. K.

E. L. Hall, J. B. K. Tio, C. A. McPherson, F. A. Sadjadi, Computer 15, 42 (1982).
[CrossRef]

Winthrop, J. T.

Worthington, C. R.

AIP Conf. Proc. (1)

P. Chavel, T. C. Strand, AIP Conf. Proc. 65, 431 (1980).
[CrossRef]

Appl. Opt. (4)

Computer (1)

E. L. Hall, J. B. K. Tio, C. A. McPherson, F. A. Sadjadi, Computer 15, 42 (1982).
[CrossRef]

J. Opt. Soc. Am. (1)

Opt. Eng. (1)

W. P. Bleha et al., Opt. Eng. 17, 371 (1978).

Other (4)

T. C. Strand, IBM Research Laboratories, private communications.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, 1968), p. 54.

W. K. Pratt, Digital Image Processing (Wiley-Interscience, New York, 1978), pp. 336–338.

N. Balasubramanian, “Optical Processing in Photogrammetry,” in Optical Data Processing, D. Casasent, Ed. (Springer, Berlin, 1978), pp. 119–149.
[CrossRef]

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Figures (13)

Fig. 1
Fig. 1

Talbot effect is used to encode the depth of a 3-D object. A TV camera records the light intensity of the object illuminated by a Talbot image of the grating. The depth information is recovered after demodulation and normalization.

Fig. 2
Fig. 2

Crossed grating pseudocolor processor. Two perpendicular gratings illuminate the 3-D object. A white-light optical processor is used to convert the TV camera signal into a pseudocolor image.

Fig. 3
Fig. 3

Theoretical power in the two color channels as a function of depth for the crossed grating processor. Cosine gratings are used with two different frequencies.

Fig. 4
Fig. 4

Measured power in the two color channels. The automatic gain control of the camera is seen to cause an interaction between the channels.

Fig. 5
Fig. 5

Experimental configuration of the pseudocolor depth processor. Spatial filter A produces a simple cosine illumination. Spatial filter B produces a modulated cosine illumination. Color filters are placed in the zero and first orders of the color filter plane.

Fig. 6
Fig. 6

Response of the simple grating pseudocolor processor to changes in depth. The grating was moved to eliminate the effect of the camera depth of field. Experimental data points were taken every 10 mm and have been interpolated by a cubic spline. The result is compared to the theoretical cos2 curve.

Fig. 7
Fig. 7

Step test object. The depth of each step is 3 mm and the width is 6.4 mm.

Fig. 8
Fig. 8

The pseudocolor image of the three-dimensional object in Fig. 7. A Wratten 46 color filter (blue) was used in the zero diffraction order, and a Wratten 25 color filter (red) was used in the +1 diffraction order.

Fig. 9
Fig. 9

3M test object. The distance between the front of the object and the background is 10 mm.

Fig. 10
Fig. 10

The pseudocolor image of the object in Fig. 9. The front of the object was placed in a Talbot plane. Grating lines are visible since a double-sized red filter was used.

Fig. 11
Fig. 11

Response of the modulated grating pseudocolor processor to changes in depth. The grating was moved to eliminate the effect of the camera depth of field. Experimental data points were taken every millimeter and have been interpolated by a cubic spline. The result is compared to the theoretical cos2 curve.

Fig. 12
Fig. 12

Response of the modulated grating pseudocolor processor to changes in depth with camera depth of field included. The object was moved with respect to the grating and camera. Experimental data points were taken every millimeter and have been interpolated by a cubic spline. The effect of the depth of field has been estimated by fitting a cubic spline to points on the graph which are centered between the nulls.

Fig. 13
Fig. 13

The pseudocolor image of a tilted plane generated by the modulated grating processor. Three complete repetition periods can be seen. A double-sided red filter was used.

Tables (5)

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Table I Unambiguous Range and Aspect Ratio as a Function of Object Field Width (SBP = 250, λ = 0.6328 μm)

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Table II Required Grating Modulation for a Specific Object Field Width (SBP = 250, λ = 0.6328 μm)

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Table III Required Grating Carrier for a Specific Aspect Ratio (SBP 250, λ = 0.6328 μm)

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Table IV Specifications for Nonmodulated Grating Processor

Tables Icon

Table V Specifications for Modulated Grating Processor

Equations (52)

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a ( x ) = c + m cos ( 2 π f 0 x ) ,
A ( u , z = 0 ) = c δ ( u ) + m 2 δ ( u - f 0 ) + m 2 δ ( u + f 0 ) ,
H ( u , z ) = exp [ j 2 π ( z λ ) 1 - ( λ u ) 2 ] .
A ( u , z ) = c δ ( u ) exp [ j 2 π z λ ] + m 2 δ ( u - f 0 ) exp [ j 2 π ( z λ ) 1 - ( λ f 0 ) 2 ] + m 2 δ ( u + f 0 ) exp [ j 2 π ( z λ ) 1 - ( λ f 0 ) 2 ] = exp [ j 2 π z λ ] { c δ ( μ ) + m 2 δ ( u - f 0 ) exp [ j 2 π ( z λ ) ( 1 - ( λ f 0 ) 2 - 1 ) ] + m 2 δ ( u + f 0 ) exp [ j 2 π ( z λ ) ( 1 - ( λ f 0 ) 2 - 1 ) ] } .
a ( x , z ) = exp [ j 2 π z λ ] { c + m cos ( 2 π f 0 x ) × exp [ j 2 π ( z λ ) ( 1 - ( λ f 0 ) 2 - 1 ) ] } .
z = n λ 1 - ( λ f 0 ) 2 - 1 ,             n = - 2 , - 1 , 0 , 1 , 2 .
I ( x , z ) = a ( x , z ) 2 = c 2 + m 2 { 1 2 + 1 2 cos [ 2 π ( 2 f 0 ) x ] } + 2 m c cos ( 2 π f 0 x ) cos [ 2 π ( z λ ) ( 1 - ( λ f 0 ) 2 - 1 ) ] .
Z T = | λ 1 - ( λ f 0 ) 2 - 1 | .
Z T λ ( λ f 0 ) 2 2 = 2 d 0 2 λ ,
Z U = Z T 4 = d 0 2 2 λ .
S = 3 f 0 L 2 ,
d 0 = 3 2 ( L S ) .
Z U = 9 L 2 8 λ S 2 .
A = Z U L = 9 L 8 λ S 2 .
b 2 ( d l ) ( N . A . ) .
l = 2 ( d l ) = b ( N . A . ) .
l = d 0 / ( N . A . ) .
( N . A . ) = λ / ( 2 d 0 ) .
l = ( 2 d 0 2 ) / λ .
a ( x , z = 0 ) = c + m cos ( 2 π f 1 x ) cos ( 2 π f 0 x ) .
a ( x , z = 0 ) = c + m 2 cos [ 2 π ( f 0 + f 1 ) x ] + m 2 cos [ 2 π ( f 0 - f 1 ) x ] .
A ( u , z = 0 ) = c δ ( u ) + m 4 δ [ u - ( f 0 + f 1 ) ] + m 4 δ [ u + ( f 0 + f 1 ) ] + m 4 δ [ u - ( f 0 - f 1 ) ] + m 4 δ [ u + ( f 0 - f 1 ) ] .
H ( u , z ) = exp [ j 2 π ( z λ ) 1 - ( λ u ) 2 ] exp [ j 2 π z λ ] exp [ - j π λ u 2 z ] .
A ( u , z ) = exp [ j 2 π z λ ] ( c δ ( u ) + m 4 { δ [ u - ( f 0 + f 1 ) ] + δ [ u + ( f 0 + f 1 ) ] } exp [ - j π λ ( f 0 + f 1 ) 2 z ] + m 4 { δ [ u - ( f 0 - f 1 ) ] + δ [ u + ( f 0 - f 1 ) ] } × exp [ - j π λ ( f 0 - f 1 ) 2 z ] ) .
a ( x , z ) = exp [ j 2 π z λ ] { c + m 2 cos [ 2 π ( f 0 + f 1 ) x ] exp [ - j π λ ( f 0 + f 1 ) 2 z ] + m 2 cos [ 2 π ( f 0 - f 1 ) x ] exp [ - j π λ ( f 0 - f 1 ) 2 z ] } .
I ( x , z ) = a ( x , z ) 2 = c 2 + m 2 4 { 1 2 + 1 2 cos [ 4 π ( f 0 + f 1 ) x ] } + m 2 4 { 1 2 + 1 2 cos [ 4 π ( f 0 - f 1 ) x ] } + m c 2 cos [ 2 π ( f 0 + f 1 ) x ] exp [ - j π λ ( f 0 + f 1 ) 2 z ] + complex conjugate + m c 2 cos [ 2 π ( f 0 - f 1 ) x ] exp [ - j π λ ( f 0 - f 1 ) 2 z ] + complex conjugate + m 2 8 { cos [ 2 π ( 2 f 0 ) x ] + cos [ 2 π ( 2 f 1 ) x ] } exp [ - j 4 π λ f 0 f 1 z ] + complex conjugate .
I LP ( x , z ) = c 2 + m 2 4 + m 2 4 cos [ 2 π ( 2 f 1 ) x ] cos ( 4 π λ f 0 f 1 z ) .
Z T = 1 2 λ f 0 f 1 = d 0 d 1 2 λ ,
Z T = | λ 1 - λ 2 ( f 0 + f 1 ) 2 - 1 - λ 2 ( f 0 - f 1 ) 2 | .
Z U = d 0 d 1 8 λ .
S = 3 f 1 L .
d 1 = ( 3 L ) / S .
d 0 = 8 λ S Z U 3 L .
d 0 = ( 8 λ S 3 ) A .
Z R = Z T 2 = d 0 d 1 4 λ .
l = d 1 2 2 λ .
N = l Z R = 2 d 1 d 0 .
l = d 1 2 ( N . A . ) .
N = l Z R = 2 λ d 0 ( N . A . ) .
a ( x , y ) = [ a 0 + a 1 cos ( 2 π f 0 x ) ] [ a 2 + a 3 cos ( 2 π f 1 y ) ] ,
a ( x , y , z ) = exp [ j 2 π z λ ] ( a 0 a 2 + a 1 a 2 exp { j 2 π ( z λ ) [ 1 - ( λ f 0 ) 2 - 1 ] } cos ( 2 π f 0 x ) + a 0 a 3 exp { j 2 π ( z λ ) [ 1 - ( λ f 1 ) 2 - 1 ] } cos ( 2 π f 1 y ) + a 1 a 3 exp { j 2 π ( z λ ) [ 1 - ( λ f 0 ) 2 - ( λ f 1 ) 2 - 1 ] } × cos ( 2 π f 0 x ) cos ( 2 π f 1 y ) ) .
I ( x , y ) = R ( x , y , z ) a ( x , y , z ) 2
a ( x , y ) = c R ( x , y , z ) a ( x , y , z ) 2 ,
a x ( x , y ) = c R ( x , y , z ) ( a 0 a 1 a 2 2 × cos { 2 π ( z λ ) [ 1 - ( λ f 0 ) 2 - 1 ] } exp [ j 2 π f 0 x ] + a 0 a 1 a 3 2 2 cos { 2 π ( z λ ) [ 1 - ( λ f 1 ) 2 - 1 - ( λ f 0 ) 2 - ( λ f 1 ) 2 } exp [ j 2 π f 0 x ] ) .
a x ( x , y ) c R ( x , y , z ) ( a 0 a 1 a 2 2 + a 0 a 1 a 3 2 2 ) cos ( π λ f 0 2 z ) exp [ j 2 π f 0 x ] .
I x ( x , y ) = a x ( x , y ) 2 = K x 2 R 2 ( x , y , z ) cos 2 ( π λ f 0 2 z ) ,
K x = c ( a 0 a 1 a 2 2 + a 0 a 1 a 3 2 2 ) .
I y ( x , y ) = K y 2 R 2 ( x , y , z ) cos 2 ( π λ f 1 2 z ) ,
K y = c ( a 0 2 a 2 a 3 + a 1 2 a 2 a 3 2 ) .
I ( x , y , ν ) = K 2 R 2 ( x , y , z ) [ s 1 ( ν ) cos 2 ( π λ f 0 2 z ) + s 2 ( ν ) cos 2 ( π λ f 1 2 z ) ] ,
K = c ( a 0 3 a 1 + a 0 a 1 3 2 ) .
t ( x ) = m cos ( 2 π f 1 x ) cos ( 2 π f 0 x ) .

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