Abstract

The point-spread function needed for imaging underwater objects is theoretically derived and compared with experimental results. The theoretical development is based on the emergent-ray model, in which the Gram–Charlier series for the non-Gaussian probability-density function for emergent angles through a wavy water surface was assumed. To arrive at the point-spread model, we used a finite-element methodology with emergent-ray angular probability distributions as fundamental building functions. The model is in good agreement with the experiment for downwind conditions. A slight deviation between theory and experiment was observed for the crosswind case; this deviation may be caused by the possible interaction of standing waves with the original air-ruffled capillary waves that were not taken into account in the model.

© 1992 Optical Society of America

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References

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  1. P. F. Schippnick, “Imaging of a bottom object through a wavy air–water interface,” in Ocean Optics IX, M. A. Blizard, ed., Proc. Soc. Photo-Opt. Instrum. Eng.925, 371–382 (1988).
  2. N. Witherspuon, J. Holloway, D. Brown, M. Strand, B. Price, R. Miller, “Measured degradation in image quality when imaging through a wavy air–water interface,” in Ocean Optics IX, M. A. Blizard, ed., Proc. Soc. Photo-Opt. Instrum. Eng.925, 383–390 (1988).
  3. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, San Diego, Calif., 1978), Vols. 1 and 2.
  4. J. Goodman, Statistical Optics (Wiley, New York, 1985).
  5. S. Karp, R. Garliardi, S. Moran, L. Stotts, Optical Channels (Plenum, New York, 1988).
  6. H. Yura, “Imaging in clear ocean water,” Appl. Opt. 12, 1061–1066 (1973).
    [Crossref] [PubMed]
  7. C. Cox, W. Munk, “Statistics of the sea surface derived from Sun glitter,” J. Mar. Res. 13, 198–227 (1954).
  8. C. Cox, W. Munk, “Measurement of the roughness of the sea surface from photographs of the Sun glitter,” J. Opt. Soc. Am. 44, 838 (1954).
    [Crossref]
  9. W. Brown, A. Majumdar, “Laser probe for measuring statistics of small random surface waves on water in a laboratory tank,” submitted to Appl. Opt. (1991).
    [PubMed]
  10. H. Cramer, Mathematical Methods of Statistics (Princeton U. Press, Princeton, N.J., 1946).
  11. A. Yariv, Optical Electronics (Holt, Rinehart & Winston, New York, 1985).
  12. E. O’Neill, Introduction to Statistical Optics (Addison-Wesley, Reading, Mass., 1963).

1973 (1)

1954 (2)

C. Cox, W. Munk, “Statistics of the sea surface derived from Sun glitter,” J. Mar. Res. 13, 198–227 (1954).

C. Cox, W. Munk, “Measurement of the roughness of the sea surface from photographs of the Sun glitter,” J. Opt. Soc. Am. 44, 838 (1954).
[Crossref]

Brown, D.

N. Witherspuon, J. Holloway, D. Brown, M. Strand, B. Price, R. Miller, “Measured degradation in image quality when imaging through a wavy air–water interface,” in Ocean Optics IX, M. A. Blizard, ed., Proc. Soc. Photo-Opt. Instrum. Eng.925, 383–390 (1988).

Brown, W.

W. Brown, A. Majumdar, “Laser probe for measuring statistics of small random surface waves on water in a laboratory tank,” submitted to Appl. Opt. (1991).
[PubMed]

Cox, C.

C. Cox, W. Munk, “Statistics of the sea surface derived from Sun glitter,” J. Mar. Res. 13, 198–227 (1954).

C. Cox, W. Munk, “Measurement of the roughness of the sea surface from photographs of the Sun glitter,” J. Opt. Soc. Am. 44, 838 (1954).
[Crossref]

Cramer, H.

H. Cramer, Mathematical Methods of Statistics (Princeton U. Press, Princeton, N.J., 1946).

Garliardi, R.

S. Karp, R. Garliardi, S. Moran, L. Stotts, Optical Channels (Plenum, New York, 1988).

Goodman, J.

J. Goodman, Statistical Optics (Wiley, New York, 1985).

Holloway, J.

N. Witherspuon, J. Holloway, D. Brown, M. Strand, B. Price, R. Miller, “Measured degradation in image quality when imaging through a wavy air–water interface,” in Ocean Optics IX, M. A. Blizard, ed., Proc. Soc. Photo-Opt. Instrum. Eng.925, 383–390 (1988).

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, San Diego, Calif., 1978), Vols. 1 and 2.

Karp, S.

S. Karp, R. Garliardi, S. Moran, L. Stotts, Optical Channels (Plenum, New York, 1988).

Majumdar, A.

W. Brown, A. Majumdar, “Laser probe for measuring statistics of small random surface waves on water in a laboratory tank,” submitted to Appl. Opt. (1991).
[PubMed]

Miller, R.

N. Witherspuon, J. Holloway, D. Brown, M. Strand, B. Price, R. Miller, “Measured degradation in image quality when imaging through a wavy air–water interface,” in Ocean Optics IX, M. A. Blizard, ed., Proc. Soc. Photo-Opt. Instrum. Eng.925, 383–390 (1988).

Moran, S.

S. Karp, R. Garliardi, S. Moran, L. Stotts, Optical Channels (Plenum, New York, 1988).

Munk, W.

C. Cox, W. Munk, “Statistics of the sea surface derived from Sun glitter,” J. Mar. Res. 13, 198–227 (1954).

C. Cox, W. Munk, “Measurement of the roughness of the sea surface from photographs of the Sun glitter,” J. Opt. Soc. Am. 44, 838 (1954).
[Crossref]

O’Neill, E.

E. O’Neill, Introduction to Statistical Optics (Addison-Wesley, Reading, Mass., 1963).

Price, B.

N. Witherspuon, J. Holloway, D. Brown, M. Strand, B. Price, R. Miller, “Measured degradation in image quality when imaging through a wavy air–water interface,” in Ocean Optics IX, M. A. Blizard, ed., Proc. Soc. Photo-Opt. Instrum. Eng.925, 383–390 (1988).

Schippnick, P. F.

P. F. Schippnick, “Imaging of a bottom object through a wavy air–water interface,” in Ocean Optics IX, M. A. Blizard, ed., Proc. Soc. Photo-Opt. Instrum. Eng.925, 371–382 (1988).

Stotts, L.

S. Karp, R. Garliardi, S. Moran, L. Stotts, Optical Channels (Plenum, New York, 1988).

Strand, M.

N. Witherspuon, J. Holloway, D. Brown, M. Strand, B. Price, R. Miller, “Measured degradation in image quality when imaging through a wavy air–water interface,” in Ocean Optics IX, M. A. Blizard, ed., Proc. Soc. Photo-Opt. Instrum. Eng.925, 383–390 (1988).

Witherspuon, N.

N. Witherspuon, J. Holloway, D. Brown, M. Strand, B. Price, R. Miller, “Measured degradation in image quality when imaging through a wavy air–water interface,” in Ocean Optics IX, M. A. Blizard, ed., Proc. Soc. Photo-Opt. Instrum. Eng.925, 383–390 (1988).

Yariv, A.

A. Yariv, Optical Electronics (Holt, Rinehart & Winston, New York, 1985).

Yura, H.

Appl. Opt. (1)

J. Mar. Res. (1)

C. Cox, W. Munk, “Statistics of the sea surface derived from Sun glitter,” J. Mar. Res. 13, 198–227 (1954).

J. Opt. Soc. Am. (1)

Other (9)

W. Brown, A. Majumdar, “Laser probe for measuring statistics of small random surface waves on water in a laboratory tank,” submitted to Appl. Opt. (1991).
[PubMed]

H. Cramer, Mathematical Methods of Statistics (Princeton U. Press, Princeton, N.J., 1946).

A. Yariv, Optical Electronics (Holt, Rinehart & Winston, New York, 1985).

E. O’Neill, Introduction to Statistical Optics (Addison-Wesley, Reading, Mass., 1963).

P. F. Schippnick, “Imaging of a bottom object through a wavy air–water interface,” in Ocean Optics IX, M. A. Blizard, ed., Proc. Soc. Photo-Opt. Instrum. Eng.925, 371–382 (1988).

N. Witherspuon, J. Holloway, D. Brown, M. Strand, B. Price, R. Miller, “Measured degradation in image quality when imaging through a wavy air–water interface,” in Ocean Optics IX, M. A. Blizard, ed., Proc. Soc. Photo-Opt. Instrum. Eng.925, 383–390 (1988).

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, San Diego, Calif., 1978), Vols. 1 and 2.

J. Goodman, Statistical Optics (Wiley, New York, 1985).

S. Karp, R. Garliardi, S. Moran, L. Stotts, Optical Channels (Plenum, New York, 1988).

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Figures (10)

Fig. 1
Fig. 1

Ray passage through an air–water interface.

Fig. 2
Fig. 2

Expected (according to Snell’s law) and measured mean refracted emergent angles for the downwind and crosswind direction.

Fig. 3
Fig. 3

Angular ray probability densities for the downwind direction for various wind speeds and incident angles. The run numbers correspond to the data as shown in Table 1. The dashed curves are the experimental model and the solid curves are from Gram–Charlier models.

Fig. 4
Fig. 4

Same as in Fig. 3 except for the crosswind direction.

Fig. 5
Fig. 5

Two-dimensional contour plot of the emergent-angle distribution at the wind-ruffled water surface. The scale is 0.1339 cm/pixel, i.e., a total of 8.03 cm across (each side).

Fig. 6
Fig. 6

Finite-element summation.

Fig. 7
Fig. 7

Point-spread function of a simple imaging system.

Fig. 8
Fig. 8

Ray trace through random surface and lens.

Fig. 9
Fig. 9

Comparison of the point-spread model (dashed curve) with the experimental model (solid curve) in the downwind direction for low and high air speeds. The scale is 0.1339 cm/pixel, i.e., 8.03cm across.

Fig. 10
Fig. 10

Same as in Fig. 9 except for the crosswind direction.

Tables (1)

Tables Icon

Table 1 Run Summary and Statistics

Equations (28)

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μ C = arcsin ( n W sin θ I , C ) + 1.5378 - 1.0955 v ,
μ D = arcsin ( n W sin θ I , D ) + 0.5723 - 0.4748 v ,
σ C = 1.1663 v - 1.1938 ,
σ D = 0.6767 v - 0.2546.
ϕ ( ) = 1 2 π σ exp [ - ( - μ ) 2 2 σ 2 ] .
P ( ) = ϕ ( ) - A 6 ϕ ( 3 ) ( ) + B 24 ϕ ( 4 ) ( ) ,
= 1 2 π σ exp [ - ( - μ ) 2 2 σ 2 ] × { 1 - A 6 ( - μ σ ) [ 3 - ( - μ σ ) 2 ] + B 24 [ 3 - 6 ( - μ σ ) 2 + ( - μ σ ) 4 ] } ,
A C = 8.4426 × 10 - 4 θ I , C 3 + 4.8308 × 10 - 3 θ I , C 2 - 4.3547 × 10 - 2 θ I , C - 1.8582 ,
B C = 2.745 × 10 - 3 θ I , C 3 - 1.7138 × 10 - 3 θ I , C 2 - 4.34755 × 10 - 2 θ I , C + 0.392204.
A D = 8.77507 × 10 - 5 θ I , D 3 + 8.9985 × 10 - 4 θ I , D 2 + 6.4387 × 10 - 2 θ I , D - 0.50923 ,
B D = 6.8299 × 10 - 5 θ I , D 3 - 1.31079 × 10 - 3 θ I , D 2 - 2.629675 × 10 - 2 θ I , D + 0.378067.
P ( θ I , C , θ I , D ) = P ( θ I , C ) . P ( θ I , D ) ,
P j = P RAD / N ,
d t = T f j ( x , y ) d A ,
d E j ( x , y ) = P j d t = T P RAD N f j ( x , y ) d A .
d E ( x , y ) = j = 1 N d E j ( x , y ) = T P R A D N j = 1 N f j ( x , y ) d A .
I ¯ ( x , y ) = d E ( x , y ) T d A = P RAD N j = 1 N f j ( x , y ) .
S ¯ ( x i , y i , x 0 , y 0 ) = F { H ¯ ( f u , f v ) }
[ r r ] = [ ( 1 - d 0 / f ) f - 1 / f 0 ]     [ h ] .
r = ( 1 - d 0 / f ) h + f ,
r = - h / f .
h = d 0 tan θ ,
h d 0 θ .
r = ( 1 - d 0 / f ) d 0 θ + f ,
= ( 1 / f ) [ ( 1 - d 0 / f ) d 0 - r ] .
P ( r ) = P [ ( r ) ]     | d d r | ,
= 1 f P [ ( r ) ] .
S ¯ ( r ) = P RAD N f j = 1 N P j [ ( r ) ] .

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