Abstract

It is a well-known phenomenon that when we look into the water with two aerial eyes, both the apparent position and the apparent shape of underwater objects are different from the real ones because of refraction at the water surface. Earlier studies of the refraction-distorted structure of the underwater binocular visual field of aerial observers were restricted to either vertically or horizontally oriented eyes. We investigate a generalized version of this problem: We calculate the position of the binocular image point of an underwater object point viewed by two arbitrarily positioned aerial eyes, including oblique orientations of the eyes relative to the flat water surface. Assuming that binocular image fusion is performed by appropriate vergent eye movements to bring the object’s image onto the foveas, the structure of the underwater binocular visual field is computed and visualized in different ways as a function of the relative positions of the eyes. We show that a revision of certain earlier treatments of the aerial imaging of underwater objects is necessary. We analyze and correct some widespread erroneous or incomplete representations of this classical geometric optical problem that occur in different textbooks. Improving the theory of aerial binocular imaging of underwater objects, we demonstrate that the structure of the underwater binocular visual field of aerial 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. 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).
  2. R. Harmon, J. Cline, “At the edge of the window,” Rod and Reel 7, 41–45 (1980).
  3. L. Matthiessen, “Das astigmatische Bild des horizontalen, ebenen Grundes eines Wasserbassins,” Ann. Phys. 4, Heft 6 (1901).
  4. L. E. Kinsler, “Imaging of underwater objects,” Am. J. Phys. 13, 255–257 (1945).
    [CrossRef]
  5. M. Kedves, “Virtual images formed in refractive media bordered by plane surfaces,” Fiz. Szemle (Rev. Phys.) 6, 129–137 (1956) (in Hungarian).
  6. M. Born, Optik—Ein Lehrbuch der elektromagnetischen Lichtteorie, 3rd ed. (Springer-Verlag, Berlin, 1972).
    [CrossRef]
  7. Á. Budó, T. Mátrai, Experimental Physics III. Atomic Physics and Optics (Tankönyvkiadó, Budapest, 1977) (in Hungarian).
  8. E. Grimsehl, Lehrbuch der Physik. Band 3: Optik, 16th ed. (H. Haferkorn, BSB B. G. Teubner Verlagsgesellschaft, Leipzig, Germany, 1978).
  9. I. Buchholz, “Zum Bild von Punkten und Gegenständen unter Wasser,” Praxis der Naturwissenschaften 1980/9, 269–279 (1980).
  10. D. Kamke, W. Walcher, Physik für Mediziner (B. G. Teubner Verlag, Stuttgart, Germany, 1982).
  11. B. Gonsior, Physik für Mediziner, Biologen und Pharmazeuten (F. K. Schattauer-Verlag, Stuttgart, Germany, 1984).
  12. L. Bergmann, C. Schaefer, Lehrbuch der Experimentalphysik, 8th ed. (H. Gobrecht, W. de Gruyter, Berlin, 1987).
  13. G. Horváth, D. Varjú, “Geometric optical investigation of the underwater visual field of aerial animals,” Math. Biosci. 102, 1–19 (1990).
    [CrossRef] [PubMed]
  14. D. Regan, ed., Binocular Vision. Vol. 9 of Vision and Visual Dysfunction, J. R. Cronly-Dillon, general ed. (MacMillan, New York, 1991).
  15. I. P. Howard, B. J. Rogers, Binocular Vision and Stereopsis (Oxford U. Press, Oxford, 1995).
  16. W. Wiggers, G. Roth, C. Eurich, A. Straub, “Binocular depth perception mechanisms in tongue-projecting salamanders,” J. Comp. Physiol., A 176, 365–377 (1995).
    [CrossRef]
  17. J. D. Pettigrew, S. P. Collin, K. Fritsches, “Prey capture and accommodation in the sandlance, Limnichthyes fasciatus (Creediidae; Teleostei),” J. Comp. Physiol., A 186, 247–260 (2000).
    [CrossRef]
  18. B. Julesz, Foundations of Cyclopean Perception (University of Chicago Press, Chicago, Ill., 1971).
  19. W. T. Vetterling, S. A. Teukolsky, W. H. Press, B. P. Flannery, “Root finding and sets of equations,” in Numerical Recipes Example Book (C), 2nd ed. (Cambridge U. Press, Cambridge, UK, 1992), Chap. 9, pp. 153–167.
  20. 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]

2000 (1)

J. D. Pettigrew, S. P. Collin, K. Fritsches, “Prey capture and accommodation in the sandlance, Limnichthyes fasciatus (Creediidae; Teleostei),” J. Comp. Physiol., A 186, 247–260 (2000).
[CrossRef]

1995 (1)

W. Wiggers, G. Roth, C. Eurich, A. Straub, “Binocular depth perception mechanisms in tongue-projecting salamanders,” J. Comp. Physiol., A 176, 365–377 (1995).
[CrossRef]

1991 (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]

1990 (1)

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

1984 (1)

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 (2)

R. Harmon, J. Cline, “At the edge of the window,” Rod and Reel 7, 41–45 (1980).

I. Buchholz, “Zum Bild von Punkten und Gegenständen unter Wasser,” Praxis der Naturwissenschaften 1980/9, 269–279 (1980).

1956 (1)

M. Kedves, “Virtual images formed in refractive media bordered by plane surfaces,” Fiz. Szemle (Rev. Phys.) 6, 129–137 (1956) (in Hungarian).

1945 (1)

L. E. Kinsler, “Imaging of underwater objects,” Am. J. Phys. 13, 255–257 (1945).
[CrossRef]

1901 (1)

L. Matthiessen, “Das astigmatische Bild des horizontalen, ebenen Grundes eines Wasserbassins,” Ann. Phys. 4, Heft 6 (1901).

Bergmann, L.

L. Bergmann, C. Schaefer, Lehrbuch der Experimentalphysik, 8th ed. (H. Gobrecht, W. de Gruyter, Berlin, 1987).

Born, M.

M. Born, Optik—Ein Lehrbuch der elektromagnetischen Lichtteorie, 3rd ed. (Springer-Verlag, Berlin, 1972).
[CrossRef]

Buchholz, I.

I. Buchholz, “Zum Bild von Punkten und Gegenständen unter Wasser,” Praxis der Naturwissenschaften 1980/9, 269–279 (1980).

Budó, Á.

Á. Budó, T. Mátrai, Experimental Physics III. Atomic Physics and Optics (Tankönyvkiadó, Budapest, 1977) (in Hungarian).

Cline, J.

R. Harmon, J. Cline, “At the edge of the window,” Rod and Reel 7, 41–45 (1980).

Collin, S. P.

J. D. Pettigrew, S. P. Collin, K. Fritsches, “Prey capture and accommodation in the sandlance, Limnichthyes fasciatus (Creediidae; Teleostei),” J. Comp. Physiol., A 186, 247–260 (2000).
[CrossRef]

Eurich, C.

W. Wiggers, G. Roth, C. Eurich, A. Straub, “Binocular depth perception mechanisms in tongue-projecting salamanders,” J. Comp. Physiol., A 176, 365–377 (1995).
[CrossRef]

Flannery, B. P.

W. T. Vetterling, S. A. Teukolsky, W. H. Press, B. P. Flannery, “Root finding and sets of equations,” in Numerical Recipes Example Book (C), 2nd ed. (Cambridge U. Press, Cambridge, UK, 1992), Chap. 9, pp. 153–167.

Fritsches, K.

J. D. Pettigrew, S. P. Collin, K. Fritsches, “Prey capture and accommodation in the sandlance, Limnichthyes fasciatus (Creediidae; Teleostei),” J. Comp. Physiol., A 186, 247–260 (2000).
[CrossRef]

Gonsior, B.

B. Gonsior, Physik für Mediziner, Biologen und Pharmazeuten (F. K. Schattauer-Verlag, Stuttgart, Germany, 1984).

Grimsehl, E.

E. Grimsehl, Lehrbuch der Physik. Band 3: Optik, 16th ed. (H. Haferkorn, BSB B. G. Teubner Verlagsgesellschaft, Leipzig, Germany, 1978).

Harmon, R.

R. Harmon, J. Cline, “At the edge of the window,” Rod and Reel 7, 41–45 (1980).

Horváth, G.

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, 1995).

Julesz, B.

B. Julesz, Foundations of Cyclopean Perception (University of Chicago Press, Chicago, Ill., 1971).

Kamke, D.

D. Kamke, W. Walcher, Physik für Mediziner (B. G. Teubner Verlag, Stuttgart, Germany, 1982).

Kedves, M.

M. Kedves, “Virtual images formed in refractive media bordered by plane surfaces,” Fiz. Szemle (Rev. Phys.) 6, 129–137 (1956) (in Hungarian).

Kinsler, L. E.

L. E. Kinsler, “Imaging of underwater objects,” Am. J. Phys. 13, 255–257 (1945).
[CrossRef]

Mátrai, T.

Á. Budó, T. Mátrai, Experimental Physics III. Atomic Physics and Optics (Tankönyvkiadó, Budapest, 1977) (in Hungarian).

Matthiessen, L.

L. Matthiessen, “Das astigmatische Bild des horizontalen, ebenen Grundes eines Wasserbassins,” Ann. Phys. 4, Heft 6 (1901).

Pettigrew, J. D.

J. D. Pettigrew, S. P. Collin, K. Fritsches, “Prey capture and accommodation in the sandlance, Limnichthyes fasciatus (Creediidae; Teleostei),” J. Comp. Physiol., A 186, 247–260 (2000).
[CrossRef]

Press, W. H.

W. T. Vetterling, S. A. Teukolsky, W. H. Press, B. P. Flannery, “Root finding and sets of equations,” in Numerical Recipes Example Book (C), 2nd ed. (Cambridge U. Press, Cambridge, UK, 1992), Chap. 9, pp. 153–167.

Rogers, B. J.

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

Roth, G.

W. Wiggers, G. Roth, C. Eurich, A. Straub, “Binocular depth perception mechanisms in tongue-projecting salamanders,” J. Comp. Physiol., A 176, 365–377 (1995).
[CrossRef]

Schaefer, C.

L. Bergmann, C. Schaefer, Lehrbuch der Experimentalphysik, 8th ed. (H. Gobrecht, W. de Gruyter, Berlin, 1987).

Straub, A.

W. Wiggers, G. Roth, C. Eurich, A. Straub, “Binocular depth perception mechanisms in tongue-projecting salamanders,” J. Comp. Physiol., A 176, 365–377 (1995).
[CrossRef]

Teukolsky, S. A.

W. T. Vetterling, S. A. Teukolsky, W. H. Press, B. P. Flannery, “Root finding and sets of equations,” in Numerical Recipes Example Book (C), 2nd ed. (Cambridge U. Press, Cambridge, UK, 1992), Chap. 9, pp. 153–167.

Varjú, D.

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]

Vetterling, W. T.

W. T. Vetterling, S. A. Teukolsky, W. H. Press, B. P. Flannery, “Root finding and sets of equations,” in Numerical Recipes Example Book (C), 2nd ed. (Cambridge U. Press, Cambridge, UK, 1992), Chap. 9, pp. 153–167.

Walcher, W.

D. Kamke, W. Walcher, Physik für Mediziner (B. G. Teubner Verlag, Stuttgart, Germany, 1982).

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).

Wiggers, W.

W. Wiggers, G. Roth, C. Eurich, A. Straub, “Binocular depth perception mechanisms in tongue-projecting salamanders,” J. Comp. Physiol., A 176, 365–377 (1995).
[CrossRef]

Am. J. Phys. (1)

L. E. Kinsler, “Imaging of underwater objects,” Am. J. Phys. 13, 255–257 (1945).
[CrossRef]

Ann. Phys. (1)

L. Matthiessen, “Das astigmatische Bild des horizontalen, ebenen Grundes eines Wasserbassins,” Ann. Phys. 4, Heft 6 (1901).

Bull. Math. Biol. (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]

Fiz. Szemle (Rev. Phys.) (1)

M. Kedves, “Virtual images formed in refractive media bordered by plane surfaces,” Fiz. Szemle (Rev. Phys.) 6, 129–137 (1956) (in Hungarian).

J. Comp. Physiol., A (2)

W. Wiggers, G. Roth, C. Eurich, A. Straub, “Binocular depth perception mechanisms in tongue-projecting salamanders,” J. Comp. Physiol., A 176, 365–377 (1995).
[CrossRef]

J. D. Pettigrew, S. P. Collin, K. Fritsches, “Prey capture and accommodation in the sandlance, Limnichthyes fasciatus (Creediidae; Teleostei),” J. Comp. Physiol., A 186, 247–260 (2000).
[CrossRef]

Math. Biosci. (1)

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

Praxis der Naturwissenschaften (1)

I. Buchholz, “Zum Bild von Punkten und Gegenständen unter Wasser,” Praxis der Naturwissenschaften 1980/9, 269–279 (1980).

Rod and Reel (1)

R. Harmon, J. Cline, “At the edge of the window,” Rod and Reel 7, 41–45 (1980).

Sci. Am. (1)

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).

Other (10)

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

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

B. Julesz, Foundations of Cyclopean Perception (University of Chicago Press, Chicago, Ill., 1971).

W. T. Vetterling, S. A. Teukolsky, W. H. Press, B. P. Flannery, “Root finding and sets of equations,” in Numerical Recipes Example Book (C), 2nd ed. (Cambridge U. Press, Cambridge, UK, 1992), Chap. 9, pp. 153–167.

D. Kamke, W. Walcher, Physik für Mediziner (B. G. Teubner Verlag, Stuttgart, Germany, 1982).

B. Gonsior, Physik für Mediziner, Biologen und Pharmazeuten (F. K. Schattauer-Verlag, Stuttgart, Germany, 1984).

L. Bergmann, C. Schaefer, Lehrbuch der Experimentalphysik, 8th ed. (H. Gobrecht, W. de Gruyter, Berlin, 1987).

M. Born, Optik—Ein Lehrbuch der elektromagnetischen Lichtteorie, 3rd ed. (Springer-Verlag, Berlin, 1972).
[CrossRef]

Á. Budó, T. Mátrai, Experimental Physics III. Atomic Physics and Optics (Tankönyvkiadó, Budapest, 1977) (in Hungarian).

E. Grimsehl, Lehrbuch der Physik. Band 3: Optik, 16th ed. (H. Haferkorn, BSB B. G. Teubner Verlagsgesellschaft, Leipzig, Germany, 1978).

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Figures (12)

Fig. 1
Fig. 1

Geometry of refraction of a ray of light starting from an underwater object point O and entering an aerial eye E. According to different authors,3-13 C and V are the two possible apparent image points of O.

Fig. 2
Fig. 2

Representations of the apparent images of underwater objects viewed from air cited from different textbooks. The figures are slightly modified and redrawn after A, Ref. 8, Fig. 2.12; B, Ref. 7, Fig. 248.3; C, Ref. 6, Fig. 28; D, Ref. 7, Fig. 255.8. O: position of an underwater object, O: apparent position of O.

Fig. 3
Fig. 3

As Fig. 2 with figures after A, Ref. 12, Fig. 1.36A; B, Ref. 12, Fig. 1.74; C, Ref. 10, Fig. 14.13; D, Ref. 11, Fig. 8.11.    

Fig. 4
Fig. 4

If eyes E1 and E2 of an aerial observer and the underwater object point O lie in the same vertical plane, 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.

Fig. 5
Fig. 5

When the two aerial 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.

Fig. 6
Fig. 6

When the two aerial eyes E1 and E2 lie along an oblique line relative to the 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 halves the minimum distance K1 K2 between the two avoiding lines e1 and e2.

Fig. 7
Fig. 7

Underwater (ri) and aerial (ei) path of a ray of light from an underwater object point O through the point of refraction Ri at the water surface to the aerial eye Ei, represented in the system of XYZ coordinates, where index i=1 or 2.

Fig. 8
Fig. 8

Path of a ray of light from an underwater object point O through the point of refraction Ri to the aerial eye Ei (i=1 or 2) in the vertical plane of refraction.

Fig. 9
Fig. 9

Binocular imaging of underwater object points in a vertical plane as a function of relative eye positions. A, underwater vertical quadratic grid as object field, consisting of equidistant horizontal and vertical lines, the distance of which is equal to the distance U of the eyes set as unit (U=1). For a better visualization, the cells of the grid are alternately painted white and black on the right half. The coordinates of the fixed aerial eye E1 are X=0, Y=0, Z=2. The small circle above the grid 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 image of the underwater grid in A as functions of angles φ and θ 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. 6.

Fig. 10
Fig. 10

As Fig. 9, but here the vertical grid is replaced by a picture representing a vertical section of the underwater world in an aquarium with a goldfish and water plants.

Fig. 11
Fig. 11

Binocular imaging of underwater object points in a horizontal plane versus relative eye positions. A: Eye positions for which computations were done. Left column in rows B–F: Binocular image of the horizontal bottom of a shallow (depth Z=-4) lake viewed from air through the flat water surface (Z=0) as a function of 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. 6. Right column in rows B–F: The minimum distance K1 K2 between the two nearest points K1 and K2 of lines e1 and e2 of the refracted rays entering the eyes (Fig. 6) as functions of X and Y in three-dimensional-perspective representation.

Fig. 12
Fig. 12

As Fig. 11, but here the object is an underwater vertical quadratic grid positioned in the plane of axes Y and Z. The grid consists of equidistant horizontal and vertical lines, which are parallel to axes Y and Z; the grid parameter is equal to U=1.

Equations (39)

Equations on this page are rendered with MathJax. Learn more.

Zc(X)=-(d/n){1-(n2-1)[-X/(dn2-d)]2/3}3/2,
di=[(XEi-u)2+(YEi-v)2]1/2,i=1, 2
di=ai+bi,
ai=ZEicotan αi,
bi=-w cotan βi,
cos αi/cos βi=n=1.33.
cotan αi=cos αi(1-cos2 αi)-1/2,
cotan βi=cos βi(1-cos2 βi)-1/2,
a4t4+a3t3+a2t2+a1t+a0=0,a0=di4n4,
a1=-2di2n2(ZEi2n2+w2+di2+di2n2),
a2=(ZEi2n2+w2+di2+di2n2)2+2di2n2(ZEi2+w2+di2)+4ZEi2w2n2,
a3=-2(ZEi2n2+w2+di2+di2n2)(ZEi2+w2+di2)-4ZEi2w2(n2+1),
a4=(ZEi2+w2+di2)2+4ZEi2w2.
Pi:(XEi, 0, ZEi),Li:(XO, 0, ZEi ditan αi),
Ji:(XEi, YEi, 0),Qi:(XO, YO, 0).
Y1(X1)=AX1+B,A=(YE1-YO)/(XE1-XO),
B=(XE1YO-YE1XO)/(XE1-XO),
Z1(X1)=CX1+D,C=(d1tan α1)/(XE1-XO),
D=[XE1(ZE1-d1tan α1)-XO ZE1]/(XE1-XO).
Y2(X2)=EX2+F,E=(YE2-YO)/(XE2-XO),
F=(XE2YO-YE2XO)/(XE2-XO),
Z2(X2)=GX2+H,G=(d2tan α2)/(XE2-XO),
H=[XE2(ZE2-d2tan α2)-XO ZE2]/(XE2-XO).
d(X1, X2)={(X1-X2)2+[Y1(X1)-Y2(X2)]2
+[Z1(X1)-Z2(X2)]2}1/2
={(X1-X2)2+(AX1+B-EX2-F)2
+(CX1+D-GX2-H)2}1/2.
d2(X1*, X2*)/X1=2(X1*-X2*)+2A(AX1*+B
-EX2*-F)+2C(CX1*+D
-GX2*-H)=0,
d2(X1*, X2*)/X2=-2(X1*-X2*)-2E(AX1*+B
-EX2*-F)-2G(CX1*+D
-GX2*-H)=0.
X1*=(γδ+μ)/(μ2-ηδ),X2*=(ηX1*+γ)/μ,
η=1+A2+C2,μ=1+AE+CG,
γ=A(B-F)+C(D-H),δ=1+E2+G2,
=E(F-B)+G(H-D).
K:[(X1*+X2*)/2,(AX1*+B+EX2*+F)/2,
(CX1*+D+GX2*+H)/2].

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