Abstract

A range measurement technique is presented which allows real-time range data acquisition for all points in an image. The technique relies upon the systematic variation in the Fresnel diffraction pattern of a grating as a function of propagation distance. The technique uses a single view of the volume in question so there are no hidden points. Furthermore, there is no need for beam scanning or is any off-line processing required. An extensive theory of operation is presented along with grating and illumination design methods which allow one to tailor the technique to specific measurement requirements. Qualitative experimental results are given along with quantitative results.

© 1984 Optical Society of America

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References

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  1. P. Chavel, T. C. Strand, “Pseudocolor Encoding of Depth Information Using Talbot Effect,” in Optics in Four Dimensions—1980, M. A. Machado, L. M. Narducci, Eds. (American Institute of Physics, New York, 1981).
  2. A. Jain, U.S. Patent4,210,399 (1July1980).
  3. F. A. Talbot, Philos. Mag. 9, 401 (1836).
  4. J. M. Cowley, A. F. Moodie, Proc. R. Soc. London Ser. B 70, 497 (1957).
  5. E. A. Heideman, M. A. Breazeale, J. Opt. Soc. Am. 49, 372 (1959).
  6. G. L. Rogers, Proc. R. Soc. London Ser. B 157, 83 (1962).
  7. J. T. Winthrop, C. R. Worthington, J. Opt. Soc. Am. 55, 373 (1965).
  8. W. D. Montgomery, J. Opt. Soc. Am. 57, 772 (1967).

1967 (1)

1965 (1)

1962 (1)

G. L. Rogers, Proc. R. Soc. London Ser. B 157, 83 (1962).

1959 (1)

1957 (1)

J. M. Cowley, A. F. Moodie, Proc. R. Soc. London Ser. B 70, 497 (1957).

1836 (1)

F. A. Talbot, Philos. Mag. 9, 401 (1836).

Breazeale, M. A.

Chavel, P.

P. Chavel, T. C. Strand, “Pseudocolor Encoding of Depth Information Using Talbot Effect,” in Optics in Four Dimensions—1980, M. A. Machado, L. M. Narducci, Eds. (American Institute of Physics, New York, 1981).

Cowley, J. M.

J. M. Cowley, A. F. Moodie, Proc. R. Soc. London Ser. B 70, 497 (1957).

Heideman, E. A.

Jain, A.

A. Jain, U.S. Patent4,210,399 (1July1980).

Montgomery, W. D.

Moodie, A. F.

J. M. Cowley, A. F. Moodie, Proc. R. Soc. London Ser. B 70, 497 (1957).

Rogers, G. L.

G. L. Rogers, Proc. R. Soc. London Ser. B 157, 83 (1962).

Strand, T. C.

P. Chavel, T. C. Strand, “Pseudocolor Encoding of Depth Information Using Talbot Effect,” in Optics in Four Dimensions—1980, M. A. Machado, L. M. Narducci, Eds. (American Institute of Physics, New York, 1981).

Talbot, F. A.

F. A. Talbot, Philos. Mag. 9, 401 (1836).

Winthrop, J. T.

Worthington, C. R.

J. Opt. Soc. Am. (3)

Philos. Mag. (1)

F. A. Talbot, Philos. Mag. 9, 401 (1836).

Proc. R. Soc. London Ser. B (2)

J. M. Cowley, A. F. Moodie, Proc. R. Soc. London Ser. B 70, 497 (1957).

G. L. Rogers, Proc. R. Soc. London Ser. B 157, 83 (1962).

Other (2)

P. Chavel, T. C. Strand, “Pseudocolor Encoding of Depth Information Using Talbot Effect,” in Optics in Four Dimensions—1980, M. A. Machado, L. M. Narducci, Eds. (American Institute of Physics, New York, 1981).

A. Jain, U.S. Patent4,210,399 (1July1980).

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Figures (7)

Fig. 1
Fig. 1

Fresnel diffraction pattern of a grating varies periodically as a function of the propagation distance so that aerial images of the grating are formed at periodic intervals D given by 2p2/λ, where p is the grating period and λ is the wavelength of illumination.

Fig. 2
Fig. 2

(a) Transmission profile for square wave grating. The duty cycle is defined as a/p. (b) Talbot contrast profile for a square wave grating with duty cycle . z ^ is the propagation distance z normalized by the Talbot distance D. For this plot only the first ten terms in the Fourier expansion of the grating were used.

Fig. 3
Fig. 3

Contrast profile synthesis: (a) desired constant sensitivity contrast profile. (b) Grating transmission function required to produce contrast profile in (a). This is a low order approximation using only five terms in the Fourier expansion. (c) Actual contrast profile generated from the grating of (b).

Fig. 4
Fig. 4

Range measurement system. The analog filter has an output signal proportional to the fringe contrast C1.

Fig. 5
Fig. 5

Analog video filter. HP is a high pass filter, RECT is a rectifier, and LP is a low pass filter. The high pass filter in conjunction with the system high frequency cutoff assures that only the fundamental component of the grating modulation is transmitted to the rectifier for demodulation. The output is proportional to the modulation contrast of the fundamental at the input.

Fig. 6
Fig. 6

Measured contrast profiles for a square wave grating with duty cycle 0.0485 for two different values of z ^ in (a) and (b). The solid line represents the theoretical contrast profile where the Gaussian profile of the illumination beam has been taken into account. Values for the unknown bias and scaling factors associated with the data were obtained by fitting these parameters to the theoretical curves. The fact that the profile becomes wider for larger values of z ^ is due to the Gaussian profile of the illumination beam.

Fig. 7
Fig. 7

Use of the Talbot technique to obtain depth slices on a small solenoid. (a) Original object. The piston of the solenoid is 11 mm in diameter and 15 mm long. (b) Modulated image at 488 nm. (c) Modulated image at 514.5 nm. (d) Processed depth slice at 488 nm. (e) Processed depth slice at 514.5 nm.

Equations (59)

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D = 2 p 2 / λ ,
z k = k 2 p 2 λ = k D
z = m n p 2 λ
f ( x , y , z ) = - i a λ z exp ( 2 π i z / λ ) - - T ( x , y ) × exp { 2 i π [ ( x - x ) 2 + ( y - y ) 2 + ν x 2 λ z ] } d x d y .
T ( x , y ) = t ( x ) s ( x , y ) ,
t z ( x ) = 1 λ z - t ( x ) exp { 2 i π [ ( x - x ) 2 2 λ z + ν x ] } d x
t ( x ) = exp ( - i 2 π i ν x ) - t z ( x ) exp [ - 2 i π ( x - x ) 2 2 λ z ] d x .
f ( x , y , z ) = - i λ z a exp ( 2 i π z λ ) t z ( x ) .
I ( x , z ) = a 2 t z ( x ) 2 λ 2 z 2
I ( x , z ) = k = - I k ( z ) exp ( 2 i π k x p )
I k ( z ) = 1 p - p / 2 p / 2 I ( x , z ) exp ( - 2 i π k x p ) d x .
C k ( z ) = I k ( z ) I 0 ( z ) ,
C k ( z ) = m = - C k m exp ( - 2 i π m z D ) ,
C k m = 1 D 0 D C k ( z ) exp ( 2 i π m z D ) d z .
C k m = 1 D 0 D - p / 2 p / 2 t z ( x ) 2 exp ( - 2 i π k x p ) d x - p / 2 p / 2 t z ( x ) 2 d x exp ( 2 i π m z D ) d z .
t ( x ) = j = - t j exp ( 2 i π j x p )
t j = 1 p - p / 2 p / 2 t ( x ) exp ( - 2 i π j x p ) d x .
- p / 2 p / 2 t z ( x ) 2 exp ( - 2 i π k x p ) d x = 1 λ 2 z 2 - p / 2 p / 2 - - j = - j = - t j t j * exp ( 2 i π j x - j x p ) × exp { 2 i π [ ( x - x ) 2 - ( x - x ) 2 2 z + ν ( x - x ) - k x p ] } d x d x d x = p λ 2 z 2 - - j = - j = - t j t j * sinc [ ( x - x ) λ z p - k ] × exp { 2 i π [ j x - j x p + ( x - x ) ( x + x ) 2 λ z + ν ( x - x ) ] } d x d x ,
x 1 = x - x and x 2 = x + x ,
p 2 λ 2 z 2 - - j = - j = - t j t j * sinc ( x 1 p λ z + k ) × exp [ 2 i π ( x 1 x 2 2 λ z + x 1 j + j 2 p + x 2 j - j 2 p + ν x 1 ) ] d x 1 d x 2 = p λ z j = - t j t j - k * exp [ 2 i π k ( k - 2 j + ν p ) z / D ] .
C k m = 1 D j = - t j 2 0 D j = - t j t j - k * × exp { 2 π i [ k ( k - 2 j + ν p ) + m ] z / D } d z .
m = k ( 2 j - k - ν p )
C k k ( 2 j - k - n ) = t j t j - k * j = - t j 2 , C k m = 0 if m k ( 2 j - k - n ) .
C 1 2 j I - n = t j t j - 1 * j = - t j 2 = t j t j - 1 * T 0 , C 1 2 j - n = 0.
T 0 = - p / 2 p / 2 t ( x ) 2 d x = j = - t j 2 .
t ( x ) { 1 , x p 2 0 , p 2 < x p / 2 ,
t j = sinc ( j ) ,
T 0 = .
C 1 2 j - 1 = sinc ( j ) sinc [ ( j - 1 ) ] , C 1 2 j = 0
C 1 ( z ) = j = 1 2 sinc ( j ) sinc [ ( j - 1 ) ] cos [ 2 π ( 2 j - 1 ) z D ] .
C 1 ( z ) = 2 π cos 2 π z D .
t ( x ) = cos ( 2 π x / p ) .
t j = { ½ for j = 1 0 otherwise .
C 2 2 ( 2 j - 2 - n ) = { ½ for j = 1 0             otherwise ,
C 2 = ½ exp ( 4 i π n z / D ) ,
C 1 2 j + 1 t j - 1 - C 1 2 j - 1 * t j + 1 = 0 ,
C 1 1 T 0 = t 1 t 0 * ,
t ( x ) 1.
T 0 = A t 0 2 + B t 1 2
C 1 1 B | t 1 t 0 | 2 - t 1 t 0 + C 1 1 A = 0.
C 1 2 j = 0 , C 1 2 j - 1 = 4 C π 2 ( 2 j - 1 ) 2 = a 0 ( 2 j - 1 ) 2 .
t 2 j = t 0 [ 1 × 5 × 9 × × ( 4 j - 3 ) 3 × 7 × 11 × × ( 4 j - 1 ) ] 2 , t 2 j - 1 = t 1 [ 3 × 7 × 11 × × ( 4 j - 5 ) 5 × 9 × 13 × × ( 4 j - 3 ) ] 2 .
C 1 n + i = 1 π for i = ± 1 , C 1 n + i = 0 otherwise .
C 1 m = 1 π ( s m - 1 + s m + 1 )
C 2 j = { k for j = 2 n 0 otherwise ,
C 2 j = k .
O P = R
a ( x , y , 0 ) = a exp [ - 2 i π ( x 2 + y 2 ) / 2 λ R ] ,
f ( x , y , z ) = - i λ z a exp ( 2 i π z λ ) - - T ( x , y ) exp { 2 i π [ ( x - x ) 2 + ( y - y ) 2 2 λ z - x 2 + y 2 2 λ R ] } d x d y = - i λ z a exp { 2 i π [ z λ + ( x 2 + y 2 ) ( z 1 - z 2 λ z 2 ) ] } - - T ( x , y ) exp { 2 i π [ ( z 1 x z - x ) 2 ( z 1 y z - y ) 2 2 λ z 1 ] } d x d y ,
1 z 1 = 1 z - 1 R .
x 1 = z 1 z x y 1 = z 1 z y , z 1 = z R R - z
f ( x , y , z ) = - i λ z a exp ( 2 i π z / λ ) - - - t z ( x ) s ( x , y ) exp { 2 i π [ ( x - x ) 2 - ( x - x ) 2 + ( y - y ) 2 2 λ z ] } d x d y d x = - i λ z a exp [ 2 i π ( z λ + y 2 2 λ z ) ] - t z ( x ) S ( x - x λ z , y λ z ) exp [ 2 i π ( x 2 - x 2 ) 2 λ z ] d x .
S ( μ , ν ) = - - s ( x , y ) exp [ 2 i π ( y 2 2 λ z - μ x - ν y ) ] d x d y .
x + x 2 L 1 ,
t z ( x + λ z L ) t z ( x ) .
x ( 3 L ) / 2 ;
x λ z 3 L 2 λ z .
p n 2 λ z 3 L .
p n λ z L ,

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