Abstract

Passive-ranging systems based on wave-front coding are introduced. These single-aperture hybrid optical–digital systems are analyzed by use of linear models and the Fisher information matrix. Two schemes for passive ranging by use of a single aperture and a single image are investigated: (i) estimating the range to an object and (ii) detecting objects over a set of ranges. Theoretical limitations on estimator-error variances are given by use of the Cramer–Rao bounds. Evaluations show that range estimates with less than 0.1% error can be obtained from a single wave-front coded image. An experimental system was also built, and example results are given.

© 2000 Optical Society of America

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  1. B. Bhanu, S. Das, P. Symosek, S. Snyder, B. Roberts, “Synergism of binocular and motion stereo for passive ranging,” IEEE Trans. Aerosp. Electron. Sys. 30, 709–721 (1994).
    [CrossRef]
  2. M. Subbarao, T. Choi, A. Nikzad, “Focusing techniques,” Opt. Eng. 32, 2824–2836 (1993).
    [CrossRef]
  3. M. Subbarao, Y.-F. Liu, “Analysis of defocused image data for 3D shape recovery using a regularization technique,” in Three-Dimensional Imaging and Laser-Based Systems for Metrology and Inspection III, A. G. Harding, D. J. Svetkoff, eds., Proc. SPIE3204, 24–73 (1997).
    [CrossRef]
  4. N. Goldberg, Camera Technology: The Dark Side of the Lens (Academic, New York, 1992).
  5. E. R. Dowski, W. T. Cathey, “Single-lens, single-image, incoherent passive ranging systems,” Appl. Opt. 33, 6762–6773 (1994).
    [CrossRef] [PubMed]
  6. E. R. Dowski, “Passive ranging with an incoherent optical system,” Ph.D. dissertation (Department of Electrical and Computer Engineering, University of Colorado, Boulder, Colo., 1993).
  7. J. Curlander, E. R. Dowski, R. McCoy, “Passive three-dimensional location and tracking for autonomous rendezvous,” (Vexcel Corporation, 4909 Nautilus Court, Boulder, Colo. 80301, 1995).
  8. E. R. Dowski, “An information theory approach to incoherent information processing systems,” in Digest of the Topical Meeting on Signal Recovery and Synthesis V (Optical Society of America, Washington, D.C., 1995), pp. 106–108.
  9. L. L. Scharf, B. Friedlander, “Matched subspace detectors,” IEEE Trans. Signal Process. 42, 2146–2157 (1994).
    [CrossRef]
  10. R. T. Beherens, L. L. Scharf, “Signal processing applications of oblique projection operators,” IEEE Trans. Signal Process. 42, 1413–1423 (1994).
    [CrossRef]
  11. S. Kraut, L. L. Scharf, “The CFAR adaptive subspace detector is a scale-invariant GLRT,” IEEE Trans. Signal Process. 47, 2538–2541 (1999).
    [CrossRef]
  12. L. L. Scharf, L. McWhorter, “Geometry of the Cramer–Rao bound,” Signal Process. 31, 301–311 (1993).
    [CrossRef]
  13. L. L. Scharf, Statistical Signal Processing (Addison-Wesley, Reading, Mass., 1991), pp. 209–276.

1999 (1)

S. Kraut, L. L. Scharf, “The CFAR adaptive subspace detector is a scale-invariant GLRT,” IEEE Trans. Signal Process. 47, 2538–2541 (1999).
[CrossRef]

1994 (4)

B. Bhanu, S. Das, P. Symosek, S. Snyder, B. Roberts, “Synergism of binocular and motion stereo for passive ranging,” IEEE Trans. Aerosp. Electron. Sys. 30, 709–721 (1994).
[CrossRef]

E. R. Dowski, W. T. Cathey, “Single-lens, single-image, incoherent passive ranging systems,” Appl. Opt. 33, 6762–6773 (1994).
[CrossRef] [PubMed]

L. L. Scharf, B. Friedlander, “Matched subspace detectors,” IEEE Trans. Signal Process. 42, 2146–2157 (1994).
[CrossRef]

R. T. Beherens, L. L. Scharf, “Signal processing applications of oblique projection operators,” IEEE Trans. Signal Process. 42, 1413–1423 (1994).
[CrossRef]

1993 (2)

M. Subbarao, T. Choi, A. Nikzad, “Focusing techniques,” Opt. Eng. 32, 2824–2836 (1993).
[CrossRef]

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

Beherens, R. T.

R. T. Beherens, L. L. Scharf, “Signal processing applications of oblique projection operators,” IEEE Trans. Signal Process. 42, 1413–1423 (1994).
[CrossRef]

Bhanu, B.

B. Bhanu, S. Das, P. Symosek, S. Snyder, B. Roberts, “Synergism of binocular and motion stereo for passive ranging,” IEEE Trans. Aerosp. Electron. Sys. 30, 709–721 (1994).
[CrossRef]

Cathey, W. T.

Choi, T.

M. Subbarao, T. Choi, A. Nikzad, “Focusing techniques,” Opt. Eng. 32, 2824–2836 (1993).
[CrossRef]

Curlander, J.

J. Curlander, E. R. Dowski, R. McCoy, “Passive three-dimensional location and tracking for autonomous rendezvous,” (Vexcel Corporation, 4909 Nautilus Court, Boulder, Colo. 80301, 1995).

Das, S.

B. Bhanu, S. Das, P. Symosek, S. Snyder, B. Roberts, “Synergism of binocular and motion stereo for passive ranging,” IEEE Trans. Aerosp. Electron. Sys. 30, 709–721 (1994).
[CrossRef]

Dowski, E. R.

E. R. Dowski, W. T. Cathey, “Single-lens, single-image, incoherent passive ranging systems,” Appl. Opt. 33, 6762–6773 (1994).
[CrossRef] [PubMed]

E. R. Dowski, “Passive ranging with an incoherent optical system,” Ph.D. dissertation (Department of Electrical and Computer Engineering, University of Colorado, Boulder, Colo., 1993).

J. Curlander, E. R. Dowski, R. McCoy, “Passive three-dimensional location and tracking for autonomous rendezvous,” (Vexcel Corporation, 4909 Nautilus Court, Boulder, Colo. 80301, 1995).

E. R. Dowski, “An information theory approach to incoherent information processing systems,” in Digest of the Topical Meeting on Signal Recovery and Synthesis V (Optical Society of America, Washington, D.C., 1995), pp. 106–108.

Friedlander, B.

L. L. Scharf, B. Friedlander, “Matched subspace detectors,” IEEE Trans. Signal Process. 42, 2146–2157 (1994).
[CrossRef]

Goldberg, N.

N. Goldberg, Camera Technology: The Dark Side of the Lens (Academic, New York, 1992).

Kraut, S.

S. Kraut, L. L. Scharf, “The CFAR adaptive subspace detector is a scale-invariant GLRT,” IEEE Trans. Signal Process. 47, 2538–2541 (1999).
[CrossRef]

Liu, Y.-F.

M. Subbarao, Y.-F. Liu, “Analysis of defocused image data for 3D shape recovery using a regularization technique,” in Three-Dimensional Imaging and Laser-Based Systems for Metrology and Inspection III, A. G. Harding, D. J. Svetkoff, eds., Proc. SPIE3204, 24–73 (1997).
[CrossRef]

McCoy, R.

J. Curlander, E. R. Dowski, R. McCoy, “Passive three-dimensional location and tracking for autonomous rendezvous,” (Vexcel Corporation, 4909 Nautilus Court, Boulder, Colo. 80301, 1995).

McWhorter, L.

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

Nikzad, A.

M. Subbarao, T. Choi, A. Nikzad, “Focusing techniques,” Opt. Eng. 32, 2824–2836 (1993).
[CrossRef]

Roberts, B.

B. Bhanu, S. Das, P. Symosek, S. Snyder, B. Roberts, “Synergism of binocular and motion stereo for passive ranging,” IEEE Trans. Aerosp. Electron. Sys. 30, 709–721 (1994).
[CrossRef]

Scharf, L. L.

S. Kraut, L. L. Scharf, “The CFAR adaptive subspace detector is a scale-invariant GLRT,” IEEE Trans. Signal Process. 47, 2538–2541 (1999).
[CrossRef]

R. T. Beherens, L. L. Scharf, “Signal processing applications of oblique projection operators,” IEEE Trans. Signal Process. 42, 1413–1423 (1994).
[CrossRef]

L. L. Scharf, B. Friedlander, “Matched subspace detectors,” IEEE Trans. Signal Process. 42, 2146–2157 (1994).
[CrossRef]

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), pp. 209–276.

Snyder, S.

B. Bhanu, S. Das, P. Symosek, S. Snyder, B. Roberts, “Synergism of binocular and motion stereo for passive ranging,” IEEE Trans. Aerosp. Electron. Sys. 30, 709–721 (1994).
[CrossRef]

Subbarao, M.

M. Subbarao, T. Choi, A. Nikzad, “Focusing techniques,” Opt. Eng. 32, 2824–2836 (1993).
[CrossRef]

M. Subbarao, Y.-F. Liu, “Analysis of defocused image data for 3D shape recovery using a regularization technique,” in Three-Dimensional Imaging and Laser-Based Systems for Metrology and Inspection III, A. G. Harding, D. J. Svetkoff, eds., Proc. SPIE3204, 24–73 (1997).
[CrossRef]

Symosek, P.

B. Bhanu, S. Das, P. Symosek, S. Snyder, B. Roberts, “Synergism of binocular and motion stereo for passive ranging,” IEEE Trans. Aerosp. Electron. Sys. 30, 709–721 (1994).
[CrossRef]

Appl. Opt. (1)

IEEE Trans. Aerosp. Electron. Sys. (1)

B. Bhanu, S. Das, P. Symosek, S. Snyder, B. Roberts, “Synergism of binocular and motion stereo for passive ranging,” IEEE Trans. Aerosp. Electron. Sys. 30, 709–721 (1994).
[CrossRef]

IEEE Trans. Signal Process. (3)

L. L. Scharf, B. Friedlander, “Matched subspace detectors,” IEEE Trans. Signal Process. 42, 2146–2157 (1994).
[CrossRef]

R. T. Beherens, L. L. Scharf, “Signal processing applications of oblique projection operators,” IEEE Trans. Signal Process. 42, 1413–1423 (1994).
[CrossRef]

S. Kraut, L. L. Scharf, “The CFAR adaptive subspace detector is a scale-invariant GLRT,” IEEE Trans. Signal Process. 47, 2538–2541 (1999).
[CrossRef]

Opt. Eng. (1)

M. Subbarao, T. Choi, A. Nikzad, “Focusing techniques,” Opt. Eng. 32, 2824–2836 (1993).
[CrossRef]

Signal Process (1)

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

Other (6)

L. L. Scharf, Statistical Signal Processing (Addison-Wesley, Reading, Mass., 1991), pp. 209–276.

M. Subbarao, Y.-F. Liu, “Analysis of defocused image data for 3D shape recovery using a regularization technique,” in Three-Dimensional Imaging and Laser-Based Systems for Metrology and Inspection III, A. G. Harding, D. J. Svetkoff, eds., Proc. SPIE3204, 24–73 (1997).
[CrossRef]

N. Goldberg, Camera Technology: The Dark Side of the Lens (Academic, New York, 1992).

E. R. Dowski, “Passive ranging with an incoherent optical system,” Ph.D. dissertation (Department of Electrical and Computer Engineering, University of Colorado, Boulder, Colo., 1993).

J. Curlander, E. R. Dowski, R. McCoy, “Passive three-dimensional location and tracking for autonomous rendezvous,” (Vexcel Corporation, 4909 Nautilus Court, Boulder, Colo. 80301, 1995).

E. R. Dowski, “An information theory approach to incoherent information processing systems,” in Digest of the Topical Meeting on Signal Recovery and Synthesis V (Optical Society of America, Washington, D.C., 1995), pp. 106–108.

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Figures (22)

Fig. 1
Fig. 1

Example optical–digital system for single-aperture passive ranging by use of wave-front coding.

Fig. 2
Fig. 2

(a) Transmission profile of a cos(βπx)cos(βπy) ranging mask. (b) Contour plot of the transmission profile shown in (a).

Fig. 3
Fig. 3

Spatial orientation of the phase-mask profile with respect to the transmission-mask profile.

Fig. 4
Fig. 4

MTF’s for the two example misfocus values. Note the optical system essentially produces power (the MTF amplitude) as a function of range (the peak-power location).

Fig. 5
Fig. 5

Typical noisy image-formation model for a single object.

Fig. 6
Fig. 6

Typical noisy image-formation model for multiple objects. Note that the partitioning of the object volume into regions is arbitrary.

Fig. 7
Fig. 7

L(y) and the angular relations in the PB space.

Fig. 8
Fig. 8

Schematic of the object and the simulated optical system. Note that r is measured from the first principle plane of the optical system. The λ filter is used to reduce infrared wavelengths.

Fig. 9
Fig. 9

Example MTF’s for the simulated 1–4-m system. The peaks are marked with the range for the simulated MTF.

Fig. 10
Fig. 10

Example MTF’s for the simulated 5–20-mm system. Several peaks are marked with the range for the simulated MTF.

Fig. 11
Fig. 11

Plot of the CRB range-estimate variance as a function of range for the 4-m system design. Note that the variance is near 1/1000 at 1 m and approximately 1/100 at 4 m.

Fig. 12
Fig. 12

Plot of the CRB range-estimate variance as a function of range for the 20-mm system design. Note that the variance is less than 1/1000 for the entire simulated range.

Fig. 13
Fig. 13

Plot of the CRB object-detection variance as a function of range for the 4-m system design by use of a 4% range resolution. The variance is much less than 1/1000.

Fig. 14
Fig. 14

Plot of the CRB object-detection variance as a function of range for the 20-mm system design with a 4% range resolution. Note that the variance is still less than 1/1000.

Fig. 15
Fig. 15

Experimental setup.

Fig. 16
Fig. 16

Images of a point-source object located approximately 650 mm and 1.5 m away from the principle plane of the experimental optical system.

Fig. 17
Fig. 17

Experimental and simulated PSF’s at nominally 1 m.

Fig. 18
Fig. 18

Spectral content of the PSF’s shown in Fig. 17 demonstrating the different MTF performances for the experimental and the simulated systems.

Fig. 19
Fig. 19

Standard deviation of 10 sample images at each range. cal std. dev., calculated standard deviation.

Fig. 20
Fig. 20

Wave-front coded image of two vehicles on a roadway.

Fig. 21
Fig. 21

Proximity map for the wave-front coded image shown in Fig. 20.

Fig. 22
Fig. 22

Detector locations for traffic monitoring.

Tables (1)

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Table 1 Simulation Parameter Values

Equations (22)

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Hψu=1/2δu+1/4δu±πβ/ψ,ψ=πd24λ1f-1zo-1zi.
Hψu=1/21-|u|/dcosβπusincψud-|u|+1/41-|u|/dsincψu±πβd-|u|.
yk=|Efk|=1Mm=1M fk,m=1Mm=1Mq=02N-1 wquq,m exp-j2πqn/2N,
x=hr * u=Fru,  y=x+n
x=iphri * ui=ipFriui,  y=x+n.
u=Ba=i=1pbiai,  a=a1apT,B: N×p,  u: N×1,  a: p×1.
H0Object at riH1No object at ri  y=Ba,  a=ai under H0aai under H1.
forB0=b1bp, for the ith element define b=bi,B=b1bi-1bi+1bp, i.e., B0 is missing the ith column,nowPB=BB#,  PB=I-PB,  PG=I-PG,PG=PPBb=PBbbPBb#=PBbbTPBbTPBb-1,
L=yTPBPGPByyTPBPGPBys-pp.
L  cot2α=cos2αsin2α=cos2α1-cos2α=s2/y21-s2/y2:β1-β.
β=cos2α=s2y2=yTPBPGPByyTPBy,
Jq=Eqln pqyqln pqyT,
Jq=1σ2Gr GuTGr Gu,Gr=rln pqy  rFru,Gu=uln pqy  Fr,
J-1q=σ2GrTPGuGr-1**GuTPGrGu-1,
varrˆσ2GrTPGuGr.
varrˆσ2- Pexpjθr Hexpjθ; rUexpjθ2dθ,Pexpjθ=1|Hexpjθ; r|=00else.
Jq=1σ2GuDGuTGuDGu,GuD=uDln pqy  FrD,Gu=uln pqy  Fr.
J-1q=σ2GuDTPGuGuD-1**GuTPGuDGu-1,
varuˆDσ2trGuDTPGuGuD=σ2trFrDTPFrFrD,
varuˆDσ2 PexpjθHexpjθ; rD2dθ,Pexpjθ=1|Hexpjθ; r=00else,
JuTu=A-1JuA-T,  Aij=ui uj2.
varuˆTuˆσ2trAFrDTPFrFrDA.

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