Abstract

The apparent position, size, and shape of aerial objects viewed binocularly from water change as a result of the refraction of light at the water surface. Earlier studies of the refraction-distorted structure of the aerial binocular visual field of underwater observers were restricted to either vertically or horizontally oriented eyes. Here we calculate the position of the binocular image point of an aerial object point viewed by two arbitrarily positioned underwater eyes when the water surface is flat. Assuming that binocular image fusion is performed by appropriate vergent eye movements to bring the object’s image onto the foveae, the structure of the aerial binocular visual field is computed and visualized as a function of the relative positions of the eyes. We also analyze two erroneous representations of the underwater imaging of aerial objects that have occurred in the literature. It is demonstrated that the structure of the aerial binocular visual field of underwater observers distorted by refraction is more complex than has been thought previously.

© 2003 Optical Society of America

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References

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  1. G. Horváth, D. Varjú, “On the structure of the aerial visual field of aquatic animals distorted by refraction,” Bull. Math. Biol. 53, 425–441 (1991).
    [CrossRef]
  2. D. Regan, ed., Binocular Vision, Vol. 9 of Vision and Visual Dysfunction, J. R. Cronly-Dillon, gen. ed. (MacMillan, New York, 1991).
  3. I. P. Howard, B. J. Rogers, Binocular Vision and Stereopsis (Oxford U. Press, Oxford, UK, 1995).
  4. G. Horváth, D. Varjú, “Geometric optical investigation of the underwater visual field of aerial animals,” Math. Biosci. 102, 1–19 (1990).
    [CrossRef] [PubMed]
  5. G. Horváth, K. Buchta, D. Varjú, “Looking into the water with oblique head tilting: revision of the aerial binocular imaging of underwater objects,” J. Opt. Soc. Am. A 20, 1120–1131 (2003).
    [CrossRef]
  6. D. L. Shealy, D. G. Burkhard, “Flux density for ray propagation in discrete index media expressed in terms of the intrinsic geometry of the deflecting surface,” Opt. Acta 20, 287–301 (1973).
    [CrossRef]
  7. D. L. Shealy, D. G. Burkhard, “Analytical illuminance calculation in a multi-interface optical system,” Opt. Acta 22, 485–501 (1975).
    [CrossRef]
  8. G. Silva-Ortigoza, J. Castro-Ramos, A. Cordero-Dávila, “Exact calculation of the circle of least confusion of a rotationally symmetric mirror. II,” Appl. Opt. 40, 1021–1028 (2001).
    [CrossRef]
  9. G. Silva-Ortigoza, M. Marciano-Melchor, O. Carvente-Munoz, R. Silva-Ortigoza, “Exact computation of the caustic associated with the evolution of an aberrated wavefront,” J. Opt. A Pure Appl. Opt. 4, 358–365 (2002).
    [CrossRef]
  10. E. Schwartz, “Können ‘blinde’ Fische sehen? Fische haben einen sechsten Sinn,” Aquarien MagazinJuly1969,272–275.
  11. J. Walker, “What is a fish’s view of a fisherman and the fly he has cast on the water?” Sci. Am. 250(3), 108–113 (1984).
  12. K. H. Lüling, “The archer fish,” Sci. Am. 209(1), 100–108 (1963).
    [CrossRef]
  13. M. Bekoff, R. Dorr, “Predation by ‘shooting’ in archerfish, Toxotes jaculatrix: accuracy and sequences,” Bull. Psychon. Soc. 7, 167–168 (1976).
    [CrossRef]
  14. L. M. Dill, “Refraction and the spitting behaviour of the archerfish (Toxotes chatareus),” Behav. Ecol. Sociobiol. 2, 169–184 (1977).
  15. S. Rossel, J. Corlija, S. Schuster, “Predicting three-dimensional target motion: how archer fish determine where to catch their dislodged prey,” J. Exp. Biol. 205, 3321–3326 (2002).
    [PubMed]
  16. R. Harmon, J. Cline, “At the edge of the window,” Rod Reel, July1980, pp. 41–45.

2003

2002

G. Silva-Ortigoza, M. Marciano-Melchor, O. Carvente-Munoz, R. Silva-Ortigoza, “Exact computation of the caustic associated with the evolution of an aberrated wavefront,” J. Opt. A Pure Appl. Opt. 4, 358–365 (2002).
[CrossRef]

S. Rossel, J. Corlija, S. Schuster, “Predicting three-dimensional target motion: how archer fish determine where to catch their dislodged prey,” J. Exp. Biol. 205, 3321–3326 (2002).
[PubMed]

2001

1991

G. Horváth, D. Varjú, “On the structure of the aerial visual field of aquatic animals distorted by refraction,” Bull. Math. Biol. 53, 425–441 (1991).
[CrossRef]

1990

G. Horváth, D. Varjú, “Geometric optical investigation of the underwater visual field of aerial animals,” Math. Biosci. 102, 1–19 (1990).
[CrossRef] [PubMed]

1984

J. Walker, “What is a fish’s view of a fisherman and the fly he has cast on the water?” Sci. Am. 250(3), 108–113 (1984).

1980

R. Harmon, J. Cline, “At the edge of the window,” Rod Reel, July1980, pp. 41–45.

1977

L. M. Dill, “Refraction and the spitting behaviour of the archerfish (Toxotes chatareus),” Behav. Ecol. Sociobiol. 2, 169–184 (1977).

1976

M. Bekoff, R. Dorr, “Predation by ‘shooting’ in archerfish, Toxotes jaculatrix: accuracy and sequences,” Bull. Psychon. Soc. 7, 167–168 (1976).
[CrossRef]

1975

D. L. Shealy, D. G. Burkhard, “Analytical illuminance calculation in a multi-interface optical system,” Opt. Acta 22, 485–501 (1975).
[CrossRef]

1973

D. L. Shealy, D. G. Burkhard, “Flux density for ray propagation in discrete index media expressed in terms of the intrinsic geometry of the deflecting surface,” Opt. Acta 20, 287–301 (1973).
[CrossRef]

1969

E. Schwartz, “Können ‘blinde’ Fische sehen? Fische haben einen sechsten Sinn,” Aquarien MagazinJuly1969,272–275.

1963

K. H. Lüling, “The archer fish,” Sci. Am. 209(1), 100–108 (1963).
[CrossRef]

Bekoff, M.

M. Bekoff, R. Dorr, “Predation by ‘shooting’ in archerfish, Toxotes jaculatrix: accuracy and sequences,” Bull. Psychon. Soc. 7, 167–168 (1976).
[CrossRef]

Buchta, K.

Burkhard, D. G.

D. L. Shealy, D. G. Burkhard, “Analytical illuminance calculation in a multi-interface optical system,” Opt. Acta 22, 485–501 (1975).
[CrossRef]

D. L. Shealy, D. G. Burkhard, “Flux density for ray propagation in discrete index media expressed in terms of the intrinsic geometry of the deflecting surface,” Opt. Acta 20, 287–301 (1973).
[CrossRef]

Carvente-Munoz, O.

G. Silva-Ortigoza, M. Marciano-Melchor, O. Carvente-Munoz, R. Silva-Ortigoza, “Exact computation of the caustic associated with the evolution of an aberrated wavefront,” J. Opt. A Pure Appl. Opt. 4, 358–365 (2002).
[CrossRef]

Castro-Ramos, J.

Cline, J.

R. Harmon, J. Cline, “At the edge of the window,” Rod Reel, July1980, pp. 41–45.

Cordero-Dávila, A.

Corlija, J.

S. Rossel, J. Corlija, S. Schuster, “Predicting three-dimensional target motion: how archer fish determine where to catch their dislodged prey,” J. Exp. Biol. 205, 3321–3326 (2002).
[PubMed]

Dill, L. M.

L. M. Dill, “Refraction and the spitting behaviour of the archerfish (Toxotes chatareus),” Behav. Ecol. Sociobiol. 2, 169–184 (1977).

Dorr, R.

M. Bekoff, R. Dorr, “Predation by ‘shooting’ in archerfish, Toxotes jaculatrix: accuracy and sequences,” Bull. Psychon. Soc. 7, 167–168 (1976).
[CrossRef]

Harmon, R.

R. Harmon, J. Cline, “At the edge of the window,” Rod Reel, July1980, pp. 41–45.

Horváth, G.

G. Horváth, K. Buchta, D. Varjú, “Looking into the water with oblique head tilting: revision of the aerial binocular imaging of underwater objects,” J. Opt. Soc. Am. A 20, 1120–1131 (2003).
[CrossRef]

G. Horváth, D. Varjú, “On the structure of the aerial visual field of aquatic animals distorted by refraction,” Bull. Math. Biol. 53, 425–441 (1991).
[CrossRef]

G. Horváth, D. Varjú, “Geometric optical investigation of the underwater visual field of aerial animals,” Math. Biosci. 102, 1–19 (1990).
[CrossRef] [PubMed]

Howard, I. P.

I. P. Howard, B. J. Rogers, Binocular Vision and Stereopsis (Oxford U. Press, Oxford, UK, 1995).

Lüling, K. H.

K. H. Lüling, “The archer fish,” Sci. Am. 209(1), 100–108 (1963).
[CrossRef]

Marciano-Melchor, M.

G. Silva-Ortigoza, M. Marciano-Melchor, O. Carvente-Munoz, R. Silva-Ortigoza, “Exact computation of the caustic associated with the evolution of an aberrated wavefront,” J. Opt. A Pure Appl. Opt. 4, 358–365 (2002).
[CrossRef]

Rogers, B. J.

I. P. Howard, B. J. Rogers, Binocular Vision and Stereopsis (Oxford U. Press, Oxford, UK, 1995).

Rossel, S.

S. Rossel, J. Corlija, S. Schuster, “Predicting three-dimensional target motion: how archer fish determine where to catch their dislodged prey,” J. Exp. Biol. 205, 3321–3326 (2002).
[PubMed]

Schuster, S.

S. Rossel, J. Corlija, S. Schuster, “Predicting three-dimensional target motion: how archer fish determine where to catch their dislodged prey,” J. Exp. Biol. 205, 3321–3326 (2002).
[PubMed]

Schwartz, E.

E. Schwartz, “Können ‘blinde’ Fische sehen? Fische haben einen sechsten Sinn,” Aquarien MagazinJuly1969,272–275.

Shealy, D. L.

D. L. Shealy, D. G. Burkhard, “Analytical illuminance calculation in a multi-interface optical system,” Opt. Acta 22, 485–501 (1975).
[CrossRef]

D. L. Shealy, D. G. Burkhard, “Flux density for ray propagation in discrete index media expressed in terms of the intrinsic geometry of the deflecting surface,” Opt. Acta 20, 287–301 (1973).
[CrossRef]

Silva-Ortigoza, G.

G. Silva-Ortigoza, M. Marciano-Melchor, O. Carvente-Munoz, R. Silva-Ortigoza, “Exact computation of the caustic associated with the evolution of an aberrated wavefront,” J. Opt. A Pure Appl. Opt. 4, 358–365 (2002).
[CrossRef]

G. Silva-Ortigoza, J. Castro-Ramos, A. Cordero-Dávila, “Exact calculation of the circle of least confusion of a rotationally symmetric mirror. II,” Appl. Opt. 40, 1021–1028 (2001).
[CrossRef]

Silva-Ortigoza, R.

G. Silva-Ortigoza, M. Marciano-Melchor, O. Carvente-Munoz, R. Silva-Ortigoza, “Exact computation of the caustic associated with the evolution of an aberrated wavefront,” J. Opt. A Pure Appl. Opt. 4, 358–365 (2002).
[CrossRef]

Varjú, D.

G. Horváth, K. Buchta, D. Varjú, “Looking into the water with oblique head tilting: revision of the aerial binocular imaging of underwater objects,” J. Opt. Soc. Am. A 20, 1120–1131 (2003).
[CrossRef]

G. Horváth, D. Varjú, “On the structure of the aerial visual field of aquatic animals distorted by refraction,” Bull. Math. Biol. 53, 425–441 (1991).
[CrossRef]

G. Horváth, D. Varjú, “Geometric optical investigation of the underwater visual field of aerial animals,” Math. Biosci. 102, 1–19 (1990).
[CrossRef] [PubMed]

Walker, J.

J. Walker, “What is a fish’s view of a fisherman and the fly he has cast on the water?” Sci. Am. 250(3), 108–113 (1984).

Appl. Opt.

Aquarien Magazin

E. Schwartz, “Können ‘blinde’ Fische sehen? Fische haben einen sechsten Sinn,” Aquarien MagazinJuly1969,272–275.

Behav. Ecol. Sociobiol.

L. M. Dill, “Refraction and the spitting behaviour of the archerfish (Toxotes chatareus),” Behav. Ecol. Sociobiol. 2, 169–184 (1977).

Bull. Math. Biol.

G. Horváth, D. Varjú, “On the structure of the aerial visual field of aquatic animals distorted by refraction,” Bull. Math. Biol. 53, 425–441 (1991).
[CrossRef]

Bull. Psychon. Soc.

M. Bekoff, R. Dorr, “Predation by ‘shooting’ in archerfish, Toxotes jaculatrix: accuracy and sequences,” Bull. Psychon. Soc. 7, 167–168 (1976).
[CrossRef]

J. Exp. Biol.

S. Rossel, J. Corlija, S. Schuster, “Predicting three-dimensional target motion: how archer fish determine where to catch their dislodged prey,” J. Exp. Biol. 205, 3321–3326 (2002).
[PubMed]

J. Opt. A Pure Appl. Opt.

G. Silva-Ortigoza, M. Marciano-Melchor, O. Carvente-Munoz, R. Silva-Ortigoza, “Exact computation of the caustic associated with the evolution of an aberrated wavefront,” J. Opt. A Pure Appl. Opt. 4, 358–365 (2002).
[CrossRef]

J. Opt. Soc. Am. A

Math. Biosci.

G. Horváth, D. Varjú, “Geometric optical investigation of the underwater visual field of aerial animals,” Math. Biosci. 102, 1–19 (1990).
[CrossRef] [PubMed]

Opt. Acta

D. L. Shealy, D. G. Burkhard, “Flux density for ray propagation in discrete index media expressed in terms of the intrinsic geometry of the deflecting surface,” Opt. Acta 20, 287–301 (1973).
[CrossRef]

D. L. Shealy, D. G. Burkhard, “Analytical illuminance calculation in a multi-interface optical system,” Opt. Acta 22, 485–501 (1975).
[CrossRef]

Rod Reel

R. Harmon, J. Cline, “At the edge of the window,” Rod Reel, July1980, pp. 41–45.

Sci. Am.

J. Walker, “What is a fish’s view of a fisherman and the fly he has cast on the water?” Sci. Am. 250(3), 108–113 (1984).

K. H. Lüling, “The archer fish,” Sci. Am. 209(1), 100–108 (1963).
[CrossRef]

Other

D. Regan, ed., Binocular Vision, Vol. 9 of Vision and Visual Dysfunction, J. R. Cronly-Dillon, gen. ed. (MacMillan, New York, 1991).

I. P. Howard, B. J. Rogers, Binocular Vision and Stereopsis (Oxford U. Press, Oxford, UK, 1995).

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Figures (9)

Fig. 1
Fig. 1

Geometry of refraction of a ray of light originating from an aerial object point O and entering an underwater eye E when the water surface is flat. According to Horváth and Varjú,1 C and V are the two possible apparent image points of O.

Fig. 2
Fig. 2

Two erroneous representations of the apparent images of aerial objects viewed from water. The apparent positions of the aerial objects viewed by the underwater eye(s) are incorrectly drawn horizontally closer to the observer than the true horizontal distance. A is redrawn after Ref. 10, B after Ref. 11.

Fig. 3
Fig. 3

If the eyes E1 and E2 of an underwater observer and the aerial object point O lie in the same vertical plane, the lines e1 and e2 of refracted rays extrapolated backward and entering the eyes intersect at point C. Thus C is the binocular image of O if the water surface is flat.

Fig. 4
Fig. 4

When the two underwater eyes E1 and E2 lie in a horizontal plane, the refracted rays e1 and e2 extrapolated backward and entering the eyes intersect at point V. Thus V is the binocular image of O when the water surface is flat.

Fig. 5
Fig. 5

When the two underwater eyes E1 and E2 lie along an oblique line relative to the flat water surface, the lines e1 and e2 of refracted rays extrapolated backward and entering the eyes do not intersect; they avoid each other in space. If the optical axes of the eyes coincide with e1 and e2 owing to appropriate vergent eye movements, the binocular image of O is K, which is the bisecting point of the shortest section K1K2 connecting the two nonintersecting lines e1 and e2.

Fig. 6
Fig. 6

Binocular imaging of aerial object points in a vertical plane as a function of the relative positions of the underwater eyes when the water surface is flat. A, An aerial vertical quadratic grid as object field, consisting of equidistant horizontal and vertical lines. For the sake of a better visualization, the cells of the grid are alternately painted white and black on the right half. The coordinates of the fixed underwater eye E1 are X=0, Y=0, Z=-2. The small circle represents the unity sphere, at the center of which is E1 and on the surface of which E2 is situated. B, The positions of E2 on the unity sphere for which the binocular images were computed. C–N, binocular images of the aerial grid in figure A as functions of angles θ and ϕ of E2 on the unity sphere. In the calculations it was assumed that the binocular image point of every object point is the point K defined in Fig. 5.

Fig. 7
Fig. 7

As Fig. 6, but here the right half of the vertical grid is replaced by a picture representing a vertical section of the terrestrial world with a seal lying on the shore, and the calculation of the Y and Z coordinates of the binocular image point K was performed for every pixel (as object point) of this picture (pixel number=800×600=480,000).

Fig. 8
Fig. 8

Binocular imaging of aerial object points in a horizontal plane versus the relative position of the underwater eyes for a flat water surface. A: Eye positions for which computations were done. Left column in rows B–F: binocular image of the horizontal ceiling of a swimming pool (height Z=4) viewed from the water through the flat water surface (Z=0) as a function of the angle θ of eye E2 with respect to eye E1 for ϕ=0°. The positions of the eyes are shown by dots. In the calculations it was assumed that the binocular image point of every object point is the point K defined in Fig. 5. Right column in rows B–F: the distance K1K2 between the two nearest points K1 and K2 of lines e1 and e2 of the refracted rays entering the eyes (Fig. 5) as functions of X and Y in three-dimensional perspectivic representation.

Fig. 9
Fig. 9

As Fig. 8, but here the object is an aerial vertical quadratic grid (gray rectangle) positioned in the plane of axes Y and Z.

Equations (1)

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ZC(X)=hn[1+(n2-1)1/3(x/hn)2/3]3/2,

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