Abstract

We introduce a new system for single-lens single-image incoherent passive ranging. The only a priori object information this system requires is that the objects to be ranged must possess a low-pass spatial frequency spectrum. Physically, this system for passive ranging is a standard optical imaging system that is customized with a special-purpose optical mask or filter. Analytically, this optical mask customizes the transfer function of the optical system in such a way that objects form images that contain range-dependent information. This range-dependent information lies in the spatial spectrum nulls or zeros of the image.

© 1994 Optical Society of America

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References

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  1. R. J. Gagnon, “Image contrast theory of electronic focus,” J. Opt. Soc. Am. 68, 1309–1318 (1978).
    [CrossRef]
  2. G. Häusler, E. Körner, “Simple focusing criterion,” Appl. Opt. 23, 2468–2469 (1984).
    [CrossRef] [PubMed]
  3. J. V. M. Bove, “Probabilistic method for integrating multiple sources of range data,” Opt. Soc. Am. A 7, 2193–2198 (1990).
    [CrossRef]
  4. J. V. M. Bove, “Entropy-based depth from focus,” Opt. Soc. Am. A 10, 561–566 (1993).
    [CrossRef]
  5. D. Marr, Vision (Freeman, San Francisco, Calif., 1982), pp. 99–267.
  6. B. K. P. Horn, B. G. Shunck, “Determining optical flow,” Artif. Intell. 17,185–203 (1981).
    [CrossRef]
  7. L. L. Scharf, L. McWhorter, “Geometry of the Cramer–Rao bound,” Signal Process. 31, 301–311 (1993).
    [CrossRef]
  8. E. R. Dowski, Passive Ranging with an Incoherent Optical System, Ph.D. dissertation (University of Colorado, Boulder, Colo., 1993).
  9. W. T. Cathey, Optical Information Processing and Holography (Wiley, New York, 1974), Chap. 8, pp. 211–244.
  10. F. J. Harris, “On the use of windows for harmonic analysis with the discrete Fourier transform,” Proc. IEEE 66, 51–83 (1978).
    [CrossRef]
  11. K. Brenner, A. Lohmann, J. O. Casteneda, “The ambiguity function as a polar display of the OTF,” Opt. Commun. 44, 323–326 (1983).
    [CrossRef]
  12. R. A. Roberts, C. T. Mullis, Digital Signal Processing (Addison-Wesley, Reading, Mass., 1987), Chap. 11, pp. 520–528.
  13. T. M. Cover, J. A. Thomas, Elements of Information Theory (Wiley, New York, 1991), Chap. 12, pp. 326–331.
  14. L. L. Scharf, Statistical Signal Processing (Addison-Wesley, Reading, Mass., 1991), Chap. 6, pp. 221–233.
  15. M. Abramowitz, I. A. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1972), Chap. 7, pp. 297–300.

1993

J. V. M. Bove, “Entropy-based depth from focus,” Opt. Soc. Am. A 10, 561–566 (1993).
[CrossRef]

L. L. Scharf, L. McWhorter, “Geometry of the Cramer–Rao bound,” Signal Process. 31, 301–311 (1993).
[CrossRef]

1990

J. V. M. Bove, “Probabilistic method for integrating multiple sources of range data,” Opt. Soc. Am. A 7, 2193–2198 (1990).
[CrossRef]

1984

1983

K. Brenner, A. Lohmann, J. O. Casteneda, “The ambiguity function as a polar display of the OTF,” Opt. Commun. 44, 323–326 (1983).
[CrossRef]

1981

B. K. P. Horn, B. G. Shunck, “Determining optical flow,” Artif. Intell. 17,185–203 (1981).
[CrossRef]

1978

F. J. Harris, “On the use of windows for harmonic analysis with the discrete Fourier transform,” Proc. IEEE 66, 51–83 (1978).
[CrossRef]

R. J. Gagnon, “Image contrast theory of electronic focus,” J. Opt. Soc. Am. 68, 1309–1318 (1978).
[CrossRef]

Bove, J. V. M.

J. V. M. Bove, “Entropy-based depth from focus,” Opt. Soc. Am. A 10, 561–566 (1993).
[CrossRef]

J. V. M. Bove, “Probabilistic method for integrating multiple sources of range data,” Opt. Soc. Am. A 7, 2193–2198 (1990).
[CrossRef]

Brenner, K.

K. Brenner, A. Lohmann, J. O. Casteneda, “The ambiguity function as a polar display of the OTF,” Opt. Commun. 44, 323–326 (1983).
[CrossRef]

Casteneda, J. O.

K. Brenner, A. Lohmann, J. O. Casteneda, “The ambiguity function as a polar display of the OTF,” Opt. Commun. 44, 323–326 (1983).
[CrossRef]

Cathey, W. T.

W. T. Cathey, Optical Information Processing and Holography (Wiley, New York, 1974), Chap. 8, pp. 211–244.

Cover, T. M.

T. M. Cover, J. A. Thomas, Elements of Information Theory (Wiley, New York, 1991), Chap. 12, pp. 326–331.

Dowski, E. R.

E. R. Dowski, Passive Ranging with an Incoherent Optical System, Ph.D. dissertation (University of Colorado, Boulder, Colo., 1993).

Gagnon, R. J.

Harris, F. J.

F. J. Harris, “On the use of windows for harmonic analysis with the discrete Fourier transform,” Proc. IEEE 66, 51–83 (1978).
[CrossRef]

Häusler, G.

Horn, B. K. P.

B. K. P. Horn, B. G. Shunck, “Determining optical flow,” Artif. Intell. 17,185–203 (1981).
[CrossRef]

Körner, E.

Lohmann, A.

K. Brenner, A. Lohmann, J. O. Casteneda, “The ambiguity function as a polar display of the OTF,” Opt. Commun. 44, 323–326 (1983).
[CrossRef]

Marr, D.

D. Marr, Vision (Freeman, San Francisco, Calif., 1982), pp. 99–267.

McWhorter, L.

L. L. Scharf, L. McWhorter, “Geometry of the Cramer–Rao bound,” Signal Process. 31, 301–311 (1993).
[CrossRef]

Mullis, C. T.

R. A. Roberts, C. T. Mullis, Digital Signal Processing (Addison-Wesley, Reading, Mass., 1987), Chap. 11, pp. 520–528.

Roberts, R. A.

R. A. Roberts, C. T. Mullis, Digital Signal Processing (Addison-Wesley, Reading, Mass., 1987), Chap. 11, pp. 520–528.

Scharf, L. L.

L. L. Scharf, L. McWhorter, “Geometry of the Cramer–Rao bound,” Signal Process. 31, 301–311 (1993).
[CrossRef]

L. L. Scharf, Statistical Signal Processing (Addison-Wesley, Reading, Mass., 1991), Chap. 6, pp. 221–233.

Shunck, B. G.

B. K. P. Horn, B. G. Shunck, “Determining optical flow,” Artif. Intell. 17,185–203 (1981).
[CrossRef]

Thomas, J. A.

T. M. Cover, J. A. Thomas, Elements of Information Theory (Wiley, New York, 1991), Chap. 12, pp. 326–331.

Appl. Opt.

Artif. Intell.

B. K. P. Horn, B. G. Shunck, “Determining optical flow,” Artif. Intell. 17,185–203 (1981).
[CrossRef]

J. Opt. Soc. Am.

Opt. Commun.

K. Brenner, A. Lohmann, J. O. Casteneda, “The ambiguity function as a polar display of the OTF,” Opt. Commun. 44, 323–326 (1983).
[CrossRef]

Opt. Soc. Am. A

J. V. M. Bove, “Probabilistic method for integrating multiple sources of range data,” Opt. Soc. Am. A 7, 2193–2198 (1990).
[CrossRef]

J. V. M. Bove, “Entropy-based depth from focus,” Opt. Soc. Am. A 10, 561–566 (1993).
[CrossRef]

Proc. IEEE

F. J. Harris, “On the use of windows for harmonic analysis with the discrete Fourier transform,” Proc. IEEE 66, 51–83 (1978).
[CrossRef]

Signal Process.

L. L. Scharf, L. McWhorter, “Geometry of the Cramer–Rao bound,” Signal Process. 31, 301–311 (1993).
[CrossRef]

Other

E. R. Dowski, Passive Ranging with an Incoherent Optical System, Ph.D. dissertation (University of Colorado, Boulder, Colo., 1993).

W. T. Cathey, Optical Information Processing and Holography (Wiley, New York, 1974), Chap. 8, pp. 211–244.

R. A. Roberts, C. T. Mullis, Digital Signal Processing (Addison-Wesley, Reading, Mass., 1987), Chap. 11, pp. 520–528.

T. M. Cover, J. A. Thomas, Elements of Information Theory (Wiley, New York, 1991), Chap. 12, pp. 326–331.

L. L. Scharf, Statistical Signal Processing (Addison-Wesley, Reading, Mass., 1991), Chap. 6, pp. 221–233.

M. Abramowitz, I. A. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1972), Chap. 7, pp. 297–300.

D. Marr, Vision (Freeman, San Francisco, Calif., 1982), pp. 99–267.

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Figures (15)

Fig. 1
Fig. 1

Ideal optical system.

Fig. 2
Fig. 2

Block diagram of an ideal optical system: e, exponent.

Fig. 3
Fig. 3

OTF's for rectangular aperture for ψ = −2 and ψ = −1.

Fig. 4
Fig. 4

Magnitude and phase of one-dimensional mask: 2-D, two-dimensional.

Fig. 5
Fig. 5

OTF's for ψ = −2 and ψ = −1.

Fig. 6
Fig. 6

Sum of reference and one dual mask.

Fig. 7
Fig. 7

OTF's corresponding to ψ = −2 and ψ = −1.

Fig. 8
Fig. 8

Sum of reference and two dual masks.

Fig. 9
Fig. 9

OTF's corresponding to ψ = −2 and ψ = −1.

Fig. 10
Fig. 10

Experimental setup: 1, 2, 3, toy blocks.

Fig. 11
Fig. 11

Gray-scale image formed with 32% efficiency mask.

Fig. 12
Fig. 12

Second experimental setup.

Fig. 13
Fig. 13

Gray-scale image formed with 32% efficiency mask.

Fig. 14
Fig. 14

Contour integration in the complex υ plane.

Fig. 15
Fig. 15

Weighting function [1 + erf (jz)] versus spatial position.

Tables (2)

Tables Icon

Table 1 Results of First Range-Estimation Experiment

Tables Icon

Table 2 Results of Second Range-Estimation Experiment

Equations (81)

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y ( k , ψ ) = i h ( k i , ψ ) υ ̂ ( i )
= h ( ψ ) * υ ̂
= T ( ψ ) υ ̂ ,
υ ̂ ( i ) = | υ [ i ( d i / d o ) ] | 2 ,
h ( k , ψ ) = | 1 2 π π π [ Π ̂ ( u ) exp ( j u 2 ψ ) ] exp ( juk ) d u | 2 ,
Π ̂ ( u ) = Π ( u L 2 π ) , Π ̂ ( u ) = 0 , | u | > π ,
ψ = L 2 4 π λ ( 1 d o + 1 d i 1 f ) .
H ( u , ψ ) = [ Π ̂ ( u ) exp ( j u 2 ψ ) ] * [ Π ̂ ( u ) exp ( j u 2 ψ ) ] ,
y ( ψ ) = T ( ψ ) υ ̂ + n ,
var ( ψ ̂ ) σ 2 ν T [ I P T ( ψ ) T ( ψ ) T ] ν ,
ν = T ( ψ ) ψ υ ̂ ,
P T ( ψ ) = T ( ψ ) [ T ( ψ ) T T ( ψ ) ] 1 T ( ψ ) T = T ( ψ ) T ( ψ ) ,
P T ( ψ ) I .
0 < e ( ψ ) 2 = | T ( ψ ) T a ( ψ ) | 2
= | h ( ψ ) * a ( ψ ) | 2
= 1 2 π π π | H [ exp ( j θ ) , ψ ] | 2 | A [ exp ( j θ ) , ψ ] | 2 d θ
= n 0 ( ψ ) 2 ( asymptotically with increasing N ) < 1 ,
A ( Z , ψ ) = k = 0 N 1 a ( k , ψ ) Z k , Z = exp ( j θ ) ,
k = 0 h ( k , ψ ) Z k = H + ( Z , ψ ) ,
h ( k , ψ ) = h ( k , ψ ) , H + ( Z , ψ ) = N ( Z , ψ ) D ( Z ) = n 0 ( ψ ) + n 1 ( ψ ) Z 1 + + n q ( ψ ) Z q 1 + d 1 Z 1 + + d p Z p , q p .
Π ̂ ( u ) = α i = 0 M 1 c i [ 1 j ( u θ i ) 1 j ( u + θ i ) ] ,
= α i = 0 M 1 2 c i θ i ( u 2 θ i 2 ) , θ i C
0 exp ( j θ i x ) exp ( jux ) d x = 1 j ( u θ i ) , Im ( θ i ) > 0 , 0 exp ( j θ i x ) exp ( jux ) d x = 1 j ( u + θ i ) , Im ( θ i ) > 0 .
g o ( x , ψ ) = 1 2 π exp ( j u 2 ψ ) j ( u θ o ) exp ( jux ) d u 1 2 π exp ( j u 2 ψ ) j ( w + θ o ) d u ,
Im ( θ o ) > 0 , h ( x , ψ ) = | g ( x , ψ ) | 2 .
g o ( x , ψ ) { exp ( j θ o 2 ψ ) exp ( j θ o x ) x 0 exp ( j θ o 2 ψ ) exp ( j θ o x ) x 0 .
h ( x , ψ ) { | i = 0 M 1 c i exp [ j ( θ i x θ i 2 ψ ) ] | 2 x 0 | i = 0 M 1 c i exp [ j ( θ i x θ i 2 ψ ) ] | 2 x 0 .
k = 0 r k cos ( ω k + ϕ ) = cos ( ϕ ) Z 1 r cos ( ω ϕ ) 1 + 2 Z 1 r cos ( ω ) + Z 2 r 2 ,
n o ( ψ ) = | i = 0 M 1 c i exp ( j θ i 2 ψ ) | 2 .
θ i + 1 2 = θ i 2 + 2 π / M , i = 0 , 1 , M 2 .
n o ( ψ ) = exp ( 4 Γ ψ ) | i = 0 M 1 c i exp ( j 2 π M ψ i ) | 2 ,
Γ = Real ( θ i ) Im ( θ i ) = Real ( θ n ) Im ( θ n ) .
Π ̂ ( u ) = α i = 0 M 1 2 c i θ i j ( u 2 θ i 2 ) θ i C ,
Im ( θ i ) 0 .
min e ( ψ ) 2 = min c i | i = 0 M 1 c i exp ( j 2 π M ψ i ) | 2 , | ψ | ψ range .
θ o = 2.8 + 0.1 j .
θ o = 2.5 + 0.1 ( 2.8 / 2.5 ) j .
θ o = 2 + 0.1 ( 2.8 / 2 ) j .
w peak = b 0 d o + b 1 .
y ( ψ , υ ) = T ( ψ ) υ + n ,
ϑ T = [ ψ , υ T ] .
p ( y ; ϑ ) = 1 ( 2 π ) N / 2 | R | exp { ½ [ y y ( ϑ ) ] T × R 1 [ y y ( ϑ ) ] } .
J ( ϑ ) i j = E [ ln p ( y ; ϑ ) ϑ i ln p ( y ; ϑ ) ϑ j ] ,
J ( ϑ ) i j = E [ 2 ln p ( y ; ϑ ) ϑ i ϑ j ] .
J ( ϑ ) = 1 σ 2 [ | ν | 2 ν T T ( ψ ) T ( ψ ) T ν T ( ψ ) T T ( ψ ) ] ,
ν = T ( ψ ) υ ̂ ψ .
J ( ϑ ) 1 = σ 2 [ [ ν T ( I P T ( ψ ) ν ) ] 1 [ T ( ψ ) T ( I P ν ) T ( ψ ) ] 1 ] .
P T ( ψ ) = T ( ψ ) [ T ( ψ ) T T ( ψ ) ] 1 T ( ψ ) T = T ( ψ ) T ( ψ )
P ν = ν ν T ( ν T ν ) 1 / 2 ,
var ( ψ ̂ ) σ 2 ν T [ I P T ( ψ ) ] ν .
P T ( ψ ) I .
0 < e ( ψ ) 2 = 1 2 π π π | H [ exp ( j θ ) , ψ ] | 2 × | A [ exp ( j θ ) , ψ ] | 2 d θ < 1 ,
k = 0 h ( k , ψ ) Z k + k = 0 h ( k , ψ ) Z k = H + ( Z , ψ ) + H ( Z , ψ ) = H ( Z , ψ ) ,
H ( Z , ψ ) = H + ( Z 1 , ψ ) .
if H + ( Z o , ψ ) = 0 , then H ( Z o , ψ ) = H + ( Z o 1 , ψ ) = 0 , | Z o | = 1 .
H + ( Z , ψ ) = N ( Z , ψ ) D ( Z ) = n o ( ψ ) + n 1 ( ψ ) Z 1 + + n q ( ψ ) Z q 1 + d 1 Z 1 + + d p Z p , q p .
lim length of a ( ψ ) A ( Z , ψ ) = D ( Z ) N ( Z , ψ ) / n o ( ψ ) = n o ( ψ ) H + ( Z , ψ ) .
lim length of a ( ψ ) e ( ψ ) 2 = n o ( ψ ) 2
h ( x ) = 1 2 π F ( j w ) G ( j w ) exp ( jwx ) d w ,
G ( j w ) = j ( w θ k ) ,
θ k = w k + j μ k
h ( x ) = 1 2 π j exp ( j w 2 ψ ) exp ( jwx ) w θ k d w .
h ( x ) = 1 2 π j exp [ j ( ψ w x 2 ψ ) 2 ] exp ( j x 2 4 ψ ) w θ k d w .
h ( x ) = exp ( j x 2 4 ψ ) 2 π j ψ exp ( j u 2 ) u + x 2 ψ θ k ψ ) d u ,
h ( x ) = exp ( j x 2 4 ψ ) 2 π j ψ exp ( υ 2 ) υ + exp ( j π / 4 ) ( x 2 ψ θ k ψ ) ) d υ ,
integrals over I 1 + I 2 + A 1 + A 2 = 2 π j residues .
integrals over I 1 + I 2 = 2 π j residues .
z = exp ( j π / 4 ) ( x 2 ψ θ k ψ ) ,
1 π j exp ( υ 2 ) υ z d υ = { exp ( z 2 ) [ 1 + erf ( j z ) ] for Im ( z ) > 0 exp ( z 2 ) [ 1 erf ( j z ) ] for Im ( z ) < 0 ,
erf ( z ) = 2 π 0 z exp ( t 2 ) d t .
x < 2 ψ ( w k + μ k ) , x > 2 ψ ( w k + μ k ) .
pole = ( w k ψ x 2 ψ ) + j ( μ k ψ ) .
x < 2 ψ ( w k + μ k ) .
h ( x ) = 1 2 ψ exp ( j θ k x ) exp ( j θ k 2 ψ ) [ 1 + erf ( j z ) ] for x < 2 ψ ( w k + μ k ) ,
z = exp ( j π / 4 ) ( x 2 ψ θ k ψ ) .
h ( x ) = 1 2 ψ exp ( j θ k x ) exp ( j θ k 2 ψ ) [ 1 + erf ( j z ) ] for x > 2 ψ ( w k + μ k ) ,
h ( x ) = 1 2 ψ exp ( j θ k x ) exp ( j θ k 2 ψ ) [ 1 + erf ( j z ) ] x ,
z = exp ( j π / 4 ) ( x 2 ψ θ k ψ ) .
[ 1 + erf ( j z ) ] 2 { 1 x 0 0 x < 0 for | x | 1 .
h ( x ) { exp ( j θ k x ) exp ( j θ k 2 ψ ) for x 0 0 for x < 0 ,
h ( x ) { exp ( j θ ac x ) exp ( j θ ac 2 ψ ) x 0 0 x > 0 .

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