Abstract

Optical techniques to measure the small-scale shape of the ocean surface, i.e., the short wind waves, are theoretically reviewed. The well-known shape-from-shading and shape-from-stereo paradigms from computer vision are applied to a specular reflecting surface such as the ocean surface and are used to study a variety of techniques. The analysis shows that most techniques for the imaging of short wind waves, such as Stilwell photography and various stereo techniques, have significant deficiencies. Stereophotography is plagued by insufficient height resolution for small waves and by the problem that, because of the specular nature of reflection at the water surface, features seen in one image are not necessarily found in the other (correspondence problem). Techniques based on light reflection (shape from reflection) are useful only for deriving wave-slope statistics, and techniques based on light refraction (shape from refraction) are found to be most suitable for wave slope imaging.

© 1994 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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  46. J. Klinke, B. Jähne, “2D wave number spectra of short wind waves—results from wind-wave facilities and extrapolation to the ocean,” in Optics of the Air–Sea Interface: Theory and Measurements, L. Estep, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1749, 204–215 (1992).
  47. X. Zhang, C. S. Cox, “Measuring the two dimensional structure of a wavy water surface optically: a surface gradient detector”, Exp. Fluids (to be published).
  48. B. Jähne, H. J. Schultz, “Calibration and accuracy of optical slope measurements for short wind waves,” in Optics of the Air–Sea Interface: Theory and Measurements, L. Estep, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1749, 222–233 (1992).
    [CrossRef]
  49. B. Jähne, S. Waas, “Optical measuring technique for small scale water surface waves,” in Advanced Optical Instrumentation for Remote Sensing of the Earth’s Surface from Space, G. Duchossois, F. L. Herr, R. J. Zander, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1129, 147–152 (1989).
  50. S. Waas, B. Jähne, “Combined slope-height measurements of short wind waves: first results from field and laboratory measurements,” in Optics of the Air–Sea Interface: Theory and Measurements, L. Estep, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1749, 295–306 (1992).
    [CrossRef]
  51. H. J. Schultz, “Specular surface stereo: a new method for retrieving the shape of a water surface,” in Optics of the Air–Sea Interface: Theory and Measurements, L. Estep, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1749, 283–294 (1992).
    [CrossRef]

1992 (1)

O. H. Shemdin, H. M. Tran, “Measuring short surface waves with stereophotography”, Photogram. Eng. Remote Sens. 93, 311–316 (1992).

1991 (1)

E. Lamarre, W. K. Melville, “Air entrainment and dissipation in breaking waves,” Nature (London) 351, 469–472 (1991).
[CrossRef]

1990 (1)

B. Jähne, K. Riemer, “Two-dimensional wave number spectra of small-scale water surface waves,” J. Geophys. Res. 95, 11,531–11,546 (1990).
[CrossRef]

1989 (1)

M. L. Banner, I. S. F. Jones, J. C. Trinder, “Wave number spectra of short gravity waves,” J. Fluid Mech. 198, 321–344 (1989).
[CrossRef]

1988 (1)

O. H. Shemdin, H. M. Tran, S. C. Wu, “Directional measurements of short ocean waves with stereophotography,” J. Geophys. Res. 93, 13,891–13,901 (1988).
[CrossRef]

1985 (1)

I. I. Strizhkin, M. P. Lapchinskaya, Yu. A. Il’in, V. A. Malinnikov, “Spatial structure of high-frequency wind waves on the sea for different meteorological conditions”, Izv. Atmos. Oceanic Phys. 21, 342–344 (1985).

1983 (2)

W. C. Keller, B. L. Gotwols, “Two-dimensional measurement of wave slope”, Appl. Opt. 22, 3476–3478 (1983).
[CrossRef] [PubMed]

S. Tang, O. H. Shemdin, “Measurement of high frequency waves using a wave follower,” J. Geophys. Res. 88, 9832–9840 (1983).
[CrossRef]

1982 (1)

P. A. Lange, B. Jähne, J. Tschiersch, J. Ilmberger, “Comparison between an amplitude-measuring wire and a slope-measuring laser water wave gauge”, Rev. Sci. Instrum. 53, 651–655 (1982).
[CrossRef]

1981 (1)

F. M. Monaldo, R. S. Kasevich, “Daylight imagery of ocean surface waves for wave spectra,” J. Phys. Oceanogr. 11, 271–283 (1981).
[CrossRef]

1980 (2)

B. L. Gotwols, G. B. Irani, “Optical determination of the phase velocity of short gravity waves,” J. Geophys. Res. 85, 3964–3970 (1980).
[CrossRef]

S. C. Lubard, J. E. Krimmel, L. R. Thebaud, D. D. Evans, O. H. Shemdin, “Optical image and laser slope meter intercomparisons of high-frequency waves,” J. Geophys. Res. 85, 4996–5002 (1980).
[CrossRef]

1978 (2)

1977 (2)

B. A. Hughes, H. L. Grant, R. W. Chappel, “A fast response surface-wave slope meter and measured wind-wave moments”, Deep Sea Res. 24, 1211–1233 (1977).
[CrossRef]

B. K. P. Horn, “Understanding image intensities”, Artif. Intell. 8, 201–231 (1977).
[CrossRef]

1976 (1)

S. R. Long, N. E. Huang, “On the variation and growth of wave-slope spectra in the capillary-gravity range with increasing wind,” J. Fluid Mech. 77, 209–228 (1976).
[CrossRef]

1973 (1)

1971 (1)

J. W. Wright, W. C. Keller, “Doppler spectra in microwave scattering from wind waves”, Phys. Fluids 14, 466–474 (1971).
[CrossRef]

1970 (1)

E. B. Dobson, “Measurements of the fine-scale structure of the sea,” J. Geophys. Res. 75, 2853–2856 (1970).
[CrossRef]

1969 (1)

D. Stilwell, “Directional energy spectra of the sea from photographs,” J. Geophys. Res. 74, 1974–1986 (1969).
[CrossRef]

1958 (1)

C. Cox, “Measurement of slopes of high-frequency wind waves,” J. Marine Res. 16, 199–225 (1958).

1954 (2)

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

C. Cox, W. Munk, “Measurement of the roughness of the sea surface from photographs of the sun’s glitter,” J. Opt. Soc. Am. 44, 838–850 (1954).
[CrossRef]

1906 (2)

W. Laas, “Messung der Meereswellen und ihre Bedeutung für den Schiffbau”, Jahrb. Schiffbautechn. Gesellschaft 7, 391 (1906).

E. Kohlschütter, “Die Forschungsreise S. M. S. Planet” Band 2, Stereophotogrammetrische Aufnahmen,” Ann. Hydrographie 34, 219 (1906).

1905 (1)

W. Laas, “Photographische Messung der Meereswellen,” Z. Vereins Deutsch. Ingenieure 49, 1889, 1937, 1976 (1905).

Anderson, R. C.

Apel, J. R.

J. R. Apel, Principles of Ocean Physics, Vol. 38 of International Geophysics Series (Academic, London, 1987), Chap. 8, pp. 405–510.

Banner, M. L.

M. L. Banner, I. S. F. Jones, J. C. Trinder, “Wave number spectra of short gravity waves,” J. Fluid Mech. 198, 321–344 (1989).
[CrossRef]

Barter, J. D.

P. Lee, J. D. Barter, K. L. Beach, C. L. Hindman, B. M. Lake, H. Rungaldier, J. C. Schatzman, J. C. Shelton, R. N. Wagner, A. B. Williams, H. C. Yuen, “Recent advances in ocean surface characterization by a scanning laser slope gauge,” in Optics of the Air–Sea Interface: Theory and Measurements, L. Estep, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1749, 234–244 (1992).
[CrossRef]

Beach, K. L.

P. Lee, J. D. Barter, K. L. Beach, C. L. Hindman, B. M. Lake, H. Rungaldier, J. C. Schatzman, J. C. Shelton, R. N. Wagner, A. B. Williams, H. C. Yuen, “Recent advances in ocean surface characterization by a scanning laser slope gauge,” in Optics of the Air–Sea Interface: Theory and Measurements, L. Estep, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1749, 234–244 (1992).
[CrossRef]

Bock, E. J.

E. J. Bock, T. Hara, “Optical measurements of ripples using a scanning laser slope gauge. Part II: data analysis and interpretation from a laboratory wave tank,” in Optics of the Air–Sea Interface: Theory and Measurements, L. Estep, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1749, 258–271 (1992).
[CrossRef]

Chappel, R. W.

B. A. Hughes, H. L. Grant, R. W. Chappel, “A fast response surface-wave slope meter and measured wind-wave moments”, Deep Sea Res. 24, 1211–1233 (1977).
[CrossRef]

Cote, L. F.

L. F. Cote, J. O. Davis, W. Marks, R. F. McGough, E. Mehr, W. J. Pierson, J. F. Ropek, G. Stephenson, R. C. Vetter, “The directional spectrum of a wind generated sea as determined from data obtained by the Stereo Wave Observation Project”, Meteorological Papers Vol. 2, No. 6 (New York U. College of Engineering, New York, 1960), pp. 1–88.

Cox, C.

C. Cox, “Measurement of slopes of high-frequency wind waves,” J. Marine Res. 16, 199–225 (1958).

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

C. Cox, W. Munk, “Measurement of the roughness of the sea surface from photographs of the sun’s glitter,” J. Opt. Soc. Am. 44, 838–850 (1954).
[CrossRef]

Cox, C. S.

X. Zhang, C. S. Cox, “Measuring the two dimensional structure of a wavy water surface optically: a surface gradient detector”, Exp. Fluids (to be published).

Davis, J. O.

L. F. Cote, J. O. Davis, W. Marks, R. F. McGough, E. Mehr, W. J. Pierson, J. F. Ropek, G. Stephenson, R. C. Vetter, “The directional spectrum of a wind generated sea as determined from data obtained by the Stereo Wave Observation Project”, Meteorological Papers Vol. 2, No. 6 (New York U. College of Engineering, New York, 1960), pp. 1–88.

Dobson, E. B.

E. B. Dobson, “Measurements of the fine-scale structure of the sea,” J. Geophys. Res. 75, 2853–2856 (1970).
[CrossRef]

Evans, D. D.

S. C. Lubard, J. E. Krimmel, L. R. Thebaud, D. D. Evans, O. H. Shemdin, “Optical image and laser slope meter intercomparisons of high-frequency waves,” J. Geophys. Res. 85, 4996–5002 (1980).
[CrossRef]

Gotwols, B. L.

W. C. Keller, B. L. Gotwols, “Two-dimensional measurement of wave slope”, Appl. Opt. 22, 3476–3478 (1983).
[CrossRef] [PubMed]

B. L. Gotwols, G. B. Irani, “Optical determination of the phase velocity of short gravity waves,” J. Geophys. Res. 85, 3964–3970 (1980).
[CrossRef]

Grant, H. L.

B. A. Hughes, H. L. Grant, R. W. Chappel, “A fast response surface-wave slope meter and measured wind-wave moments”, Deep Sea Res. 24, 1211–1233 (1977).
[CrossRef]

Hara, T.

E. J. Bock, T. Hara, “Optical measurements of ripples using a scanning laser slope gauge. Part II: data analysis and interpretation from a laboratory wave tank,” in Optics of the Air–Sea Interface: Theory and Measurements, L. Estep, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1749, 258–271 (1992).
[CrossRef]

Hindman, C. L.

P. Lee, J. D. Barter, K. L. Beach, C. L. Hindman, B. M. Lake, H. Rungaldier, J. C. Schatzman, J. C. Shelton, R. N. Wagner, A. B. Williams, H. C. Yuen, “Recent advances in ocean surface characterization by a scanning laser slope gauge,” in Optics of the Air–Sea Interface: Theory and Measurements, L. Estep, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1749, 234–244 (1992).
[CrossRef]

Horn, B. K. P.

B. K. P. Horn, “Understanding image intensities”, Artif. Intell. 8, 201–231 (1977).
[CrossRef]

B. K. P. Horn, Robot Vision (MIT Press, Cambridge, Mass., 1986), Chap. 11, pp. 243–277.

Huang, N. E.

S. R. Long, N. E. Huang, “On the variation and growth of wave-slope spectra in the capillary-gravity range with increasing wind,” J. Fluid Mech. 77, 209–228 (1976).
[CrossRef]

Hughes, B. A.

B. A. Hughes, H. L. Grant, R. W. Chappel, “A fast response surface-wave slope meter and measured wind-wave moments”, Deep Sea Res. 24, 1211–1233 (1977).
[CrossRef]

Hwang, P. A.

P. A. Hwang, “Optical measurements of the structure of short water waves and their modulation by surface currents,” in Optics of the Air–Sea Interface: Theory and Measurements, L. Estep, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1749, 216–221 (1992).
[CrossRef]

Il’in, Yu. A.

I. I. Strizhkin, M. P. Lapchinskaya, Yu. A. Il’in, V. A. Malinnikov, “Spatial structure of high-frequency wind waves on the sea for different meteorological conditions”, Izv. Atmos. Oceanic Phys. 21, 342–344 (1985).

Ilmberger, J.

P. A. Lange, B. Jähne, J. Tschiersch, J. Ilmberger, “Comparison between an amplitude-measuring wire and a slope-measuring laser water wave gauge”, Rev. Sci. Instrum. 53, 651–655 (1982).
[CrossRef]

Irani, G. B.

B. L. Gotwols, G. B. Irani, “Optical determination of the phase velocity of short gravity waves,” J. Geophys. Res. 85, 3964–3970 (1980).
[CrossRef]

Jähne, B.

B. Jähne, K. Riemer, “Two-dimensional wave number spectra of small-scale water surface waves,” J. Geophys. Res. 95, 11,531–11,546 (1990).
[CrossRef]

P. A. Lange, B. Jähne, J. Tschiersch, J. Ilmberger, “Comparison between an amplitude-measuring wire and a slope-measuring laser water wave gauge”, Rev. Sci. Instrum. 53, 651–655 (1982).
[CrossRef]

B. Jähne, “Shape-from-shading techniques for short ocean wind waves,” in Proceedings of the International Seminar on Imaging in Transport Processes, International Center for Heat and Mass Transfer (Begell, New York, 1993), pp. 269–281.

B. Jähne, “Energy balance in small-scale waves—an experimental approach using optical slope measuring technique and image processing,” in Radar Scattering from Modulated Wind Waves, G. Komen, W. Oost, eds. (Kluwer, Dordrecht, The Netherlands, 1989), pp. 105–120.
[CrossRef]

B. Jähne, “Transfer processes across the free water surface,” Habilitation dissertation (University of Heidelberg, Heidelberg, Germany, 1985).

B. Jähne, H. J. Schultz, “Calibration and accuracy of optical slope measurements for short wind waves,” in Optics of the Air–Sea Interface: Theory and Measurements, L. Estep, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1749, 222–233 (1992).
[CrossRef]

B. Jähne, S. Waas, “Optical measuring technique for small scale water surface waves,” in Advanced Optical Instrumentation for Remote Sensing of the Earth’s Surface from Space, G. Duchossois, F. L. Herr, R. J. Zander, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1129, 147–152 (1989).

K. Riemer, T. Scholz, B. Jähne, “Bildfolgenanalyse im Orts-Wellenzahl-Raum,” in Proceedings of the 13th Deutsche Arbeitsgemeinschaft für Mustererkennung—Symposium Mustererkennung 1991, B. Radig, ed. (Springer-Verlag, Berlin, 1991), pp. 223–230.

J. Klinke, B. Jähne, “2D wave number spectra of short wind waves—results from wind-wave facilities and extrapolation to the ocean,” in Optics of the Air–Sea Interface: Theory and Measurements, L. Estep, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1749, 204–215 (1992).

S. Waas, B. Jähne, “Combined slope-height measurements of short wind waves: first results from field and laboratory measurements,” in Optics of the Air–Sea Interface: Theory and Measurements, L. Estep, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1749, 295–306 (1992).
[CrossRef]

Jones, I. S. F.

M. L. Banner, I. S. F. Jones, J. C. Trinder, “Wave number spectra of short gravity waves,” J. Fluid Mech. 198, 321–344 (1989).
[CrossRef]

Kasevich, R. S.

F. M. Monaldo, R. S. Kasevich, “Daylight imagery of ocean surface waves for wave spectra,” J. Phys. Oceanogr. 11, 271–283 (1981).
[CrossRef]

Keller, W. C.

W. C. Keller, B. L. Gotwols, “Two-dimensional measurement of wave slope”, Appl. Opt. 22, 3476–3478 (1983).
[CrossRef] [PubMed]

J. W. Wright, W. C. Keller, “Doppler spectra in microwave scattering from wind waves”, Phys. Fluids 14, 466–474 (1971).
[CrossRef]

Klinke, J.

J. Klinke, “2D Wellenzahlspektren von kleinskaligen winderzeugten Wasseroberflächenwellen,” Ph.D. dissertation (University of Heidelberg, Heidelberg, Germany, 1991).

J. Klinke, B. Jähne, “2D wave number spectra of short wind waves—results from wind-wave facilities and extrapolation to the ocean,” in Optics of the Air–Sea Interface: Theory and Measurements, L. Estep, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1749, 204–215 (1992).

Kohlschütter, E.

E. Kohlschütter, “Die Forschungsreise S. M. S. Planet” Band 2, Stereophotogrammetrische Aufnahmen,” Ann. Hydrographie 34, 219 (1906).

Krimmel, J. E.

S. C. Lubard, J. E. Krimmel, L. R. Thebaud, D. D. Evans, O. H. Shemdin, “Optical image and laser slope meter intercomparisons of high-frequency waves,” J. Geophys. Res. 85, 4996–5002 (1980).
[CrossRef]

Laas, W.

W. Laas, “Messung der Meereswellen und ihre Bedeutung für den Schiffbau”, Jahrb. Schiffbautechn. Gesellschaft 7, 391 (1906).

W. Laas, “Photographische Messung der Meereswellen,” Z. Vereins Deutsch. Ingenieure 49, 1889, 1937, 1976 (1905).

W. Laas, “Die photographische Messung der Meereswellen,” Veröffentlichungen Inst. Meereskunde N. F., Vol. A, No. 7, (Berlin, 1921).

Lake, B. M.

P. Lee, J. D. Barter, K. L. Beach, C. L. Hindman, B. M. Lake, H. Rungaldier, J. C. Schatzman, J. C. Shelton, R. N. Wagner, A. B. Williams, H. C. Yuen, “Recent advances in ocean surface characterization by a scanning laser slope gauge,” in Optics of the Air–Sea Interface: Theory and Measurements, L. Estep, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1749, 234–244 (1992).
[CrossRef]

Lamarre, E.

E. Lamarre, W. K. Melville, “Air entrainment and dissipation in breaking waves,” Nature (London) 351, 469–472 (1991).
[CrossRef]

Lange, P. A.

P. A. Lange, B. Jähne, J. Tschiersch, J. Ilmberger, “Comparison between an amplitude-measuring wire and a slope-measuring laser water wave gauge”, Rev. Sci. Instrum. 53, 651–655 (1982).
[CrossRef]

Lapchinskaya, M. P.

I. I. Strizhkin, M. P. Lapchinskaya, Yu. A. Il’in, V. A. Malinnikov, “Spatial structure of high-frequency wind waves on the sea for different meteorological conditions”, Izv. Atmos. Oceanic Phys. 21, 342–344 (1985).

Lee, P.

P. Lee, J. D. Barter, K. L. Beach, C. L. Hindman, B. M. Lake, H. Rungaldier, J. C. Schatzman, J. C. Shelton, R. N. Wagner, A. B. Williams, H. C. Yuen, “Recent advances in ocean surface characterization by a scanning laser slope gauge,” in Optics of the Air–Sea Interface: Theory and Measurements, L. Estep, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1749, 234–244 (1992).
[CrossRef]

Long, S. R.

S. R. Long, N. E. Huang, “On the variation and growth of wave-slope spectra in the capillary-gravity range with increasing wind,” J. Fluid Mech. 77, 209–228 (1976).
[CrossRef]

Lubard, S. C.

S. C. Lubard, J. E. Krimmel, L. R. Thebaud, D. D. Evans, O. H. Shemdin, “Optical image and laser slope meter intercomparisons of high-frequency waves,” J. Geophys. Res. 85, 4996–5002 (1980).
[CrossRef]

Malinnikov, V. A.

I. I. Strizhkin, M. P. Lapchinskaya, Yu. A. Il’in, V. A. Malinnikov, “Spatial structure of high-frequency wind waves on the sea for different meteorological conditions”, Izv. Atmos. Oceanic Phys. 21, 342–344 (1985).

Marks, W.

L. F. Cote, J. O. Davis, W. Marks, R. F. McGough, E. Mehr, W. J. Pierson, J. F. Ropek, G. Stephenson, R. C. Vetter, “The directional spectrum of a wind generated sea as determined from data obtained by the Stereo Wave Observation Project”, Meteorological Papers Vol. 2, No. 6 (New York U. College of Engineering, New York, 1960), pp. 1–88.

McGough, R. F.

L. F. Cote, J. O. Davis, W. Marks, R. F. McGough, E. Mehr, W. J. Pierson, J. F. Ropek, G. Stephenson, R. C. Vetter, “The directional spectrum of a wind generated sea as determined from data obtained by the Stereo Wave Observation Project”, Meteorological Papers Vol. 2, No. 6 (New York U. College of Engineering, New York, 1960), pp. 1–88.

Mehr, E.

L. F. Cote, J. O. Davis, W. Marks, R. F. McGough, E. Mehr, W. J. Pierson, J. F. Ropek, G. Stephenson, R. C. Vetter, “The directional spectrum of a wind generated sea as determined from data obtained by the Stereo Wave Observation Project”, Meteorological Papers Vol. 2, No. 6 (New York U. College of Engineering, New York, 1960), pp. 1–88.

Melville, W. K.

E. Lamarre, W. K. Melville, “Air entrainment and dissipation in breaking waves,” Nature (London) 351, 469–472 (1991).
[CrossRef]

Monahan, E. C.

E. C. Monahan, T. Torgersen, “The enhancement of air–sea gas exchange by oceanic whitecapping,” in Air–Water Mass Transfer, Selected Papers from the 2nd International Symposium on Gas Transfer at Water Surfaces, S. C. Wilhelms, J. S. Gulliver, eds. (American Society of Civil Engineers, New York, 1991), pp. 608–617.

Monaldo, F. M.

F. M. Monaldo, R. S. Kasevich, “Daylight imagery of ocean surface waves for wave spectra,” J. Phys. Oceanogr. 11, 271–283 (1981).
[CrossRef]

Munk, W.

C. Cox, W. Munk, “Measurement of the roughness of the sea surface from photographs of the sun’s glitter,” J. Opt. Soc. Am. 44, 838–850 (1954).
[CrossRef]

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

Ostrem, J. S.

Peppers, N. A.

Pierson, W. J.

L. F. Cote, J. O. Davis, W. Marks, R. F. McGough, E. Mehr, W. J. Pierson, J. F. Ropek, G. Stephenson, R. C. Vetter, “The directional spectrum of a wind generated sea as determined from data obtained by the Stereo Wave Observation Project”, Meteorological Papers Vol. 2, No. 6 (New York U. College of Engineering, New York, 1960), pp. 1–88.

Reece, A. M.

A. M. Reece, “Modulation of short waves by long waves”, Boundary Layer Meteorol. 13, 203–214 (1978).
[CrossRef]

Riemer, K.

B. Jähne, K. Riemer, “Two-dimensional wave number spectra of small-scale water surface waves,” J. Geophys. Res. 95, 11,531–11,546 (1990).
[CrossRef]

K. Riemer, “Analyse von Waseroberflächenwellen im Orts-Wellenzahl-Raum,” Ph.D. dissertation (University of Heidelberg, Heidelberg, Germany, 1991).

K. Riemer, T. Scholz, B. Jähne, “Bildfolgenanalyse im Orts-Wellenzahl-Raum,” in Proceedings of the 13th Deutsche Arbeitsgemeinschaft für Mustererkennung—Symposium Mustererkennung 1991, B. Radig, ed. (Springer-Verlag, Berlin, 1991), pp. 223–230.

Ropek, J. F.

L. F. Cote, J. O. Davis, W. Marks, R. F. McGough, E. Mehr, W. J. Pierson, J. F. Ropek, G. Stephenson, R. C. Vetter, “The directional spectrum of a wind generated sea as determined from data obtained by the Stereo Wave Observation Project”, Meteorological Papers Vol. 2, No. 6 (New York U. College of Engineering, New York, 1960), pp. 1–88.

Rungaldier, H.

P. Lee, J. D. Barter, K. L. Beach, C. L. Hindman, B. M. Lake, H. Rungaldier, J. C. Schatzman, J. C. Shelton, R. N. Wagner, A. B. Williams, H. C. Yuen, “Recent advances in ocean surface characterization by a scanning laser slope gauge,” in Optics of the Air–Sea Interface: Theory and Measurements, L. Estep, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1749, 234–244 (1992).
[CrossRef]

Schatzman, J. C.

P. Lee, J. D. Barter, K. L. Beach, C. L. Hindman, B. M. Lake, H. Rungaldier, J. C. Schatzman, J. C. Shelton, R. N. Wagner, A. B. Williams, H. C. Yuen, “Recent advances in ocean surface characterization by a scanning laser slope gauge,” in Optics of the Air–Sea Interface: Theory and Measurements, L. Estep, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1749, 234–244 (1992).
[CrossRef]

Scholz, T.

K. Riemer, T. Scholz, B. Jähne, “Bildfolgenanalyse im Orts-Wellenzahl-Raum,” in Proceedings of the 13th Deutsche Arbeitsgemeinschaft für Mustererkennung—Symposium Mustererkennung 1991, B. Radig, ed. (Springer-Verlag, Berlin, 1991), pp. 223–230.

Schuhmacher, A.

A. Schuhmacher, “Stereophotogrammetrische Wellenaufnahmen,” Wissenschaftliche Ergebnisse der Deutschen Atlantischen Expedition auf dem Forschungs- und Vermessungsschiff “Meteor” 1925–1927 (de Gruyter, Berlin, 1939), Vol. 7, Part 2.

Schultz, H. J.

H. J. Schultz, “Specular surface stereo: a new method for retrieving the shape of a water surface,” in Optics of the Air–Sea Interface: Theory and Measurements, L. Estep, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1749, 283–294 (1992).
[CrossRef]

B. Jähne, H. J. Schultz, “Calibration and accuracy of optical slope measurements for short wind waves,” in Optics of the Air–Sea Interface: Theory and Measurements, L. Estep, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1749, 222–233 (1992).
[CrossRef]

Shelton, J. C.

P. Lee, J. D. Barter, K. L. Beach, C. L. Hindman, B. M. Lake, H. Rungaldier, J. C. Schatzman, J. C. Shelton, R. N. Wagner, A. B. Williams, H. C. Yuen, “Recent advances in ocean surface characterization by a scanning laser slope gauge,” in Optics of the Air–Sea Interface: Theory and Measurements, L. Estep, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1749, 234–244 (1992).
[CrossRef]

Shemdin, O. H.

O. H. Shemdin, H. M. Tran, “Measuring short surface waves with stereophotography”, Photogram. Eng. Remote Sens. 93, 311–316 (1992).

O. H. Shemdin, H. M. Tran, S. C. Wu, “Directional measurements of short ocean waves with stereophotography,” J. Geophys. Res. 93, 13,891–13,901 (1988).
[CrossRef]

S. Tang, O. H. Shemdin, “Measurement of high frequency waves using a wave follower,” J. Geophys. Res. 88, 9832–9840 (1983).
[CrossRef]

S. C. Lubard, J. E. Krimmel, L. R. Thebaud, D. D. Evans, O. H. Shemdin, “Optical image and laser slope meter intercomparisons of high-frequency waves,” J. Geophys. Res. 85, 4996–5002 (1980).
[CrossRef]

G. Tober, R. C. Anderson, O. H. Shemdin, “Laser Instrument for detecting water ripple slopes”, Appl. Opt. 12, 788–794 (1973).
[CrossRef] [PubMed]

O. H. Shemdin, Ocean Research and Engineering, 255 South Marengo Avenue, Pasadena, Calif. 91101 (personal communication, February1991).

Stephenson, G.

L. F. Cote, J. O. Davis, W. Marks, R. F. McGough, E. Mehr, W. J. Pierson, J. F. Ropek, G. Stephenson, R. C. Vetter, “The directional spectrum of a wind generated sea as determined from data obtained by the Stereo Wave Observation Project”, Meteorological Papers Vol. 2, No. 6 (New York U. College of Engineering, New York, 1960), pp. 1–88.

Stewart, R. H.

R. H. Stewart, Methods of Satellite Oceanography (U. of California Press, Berkeley, Calif., 1985), Chap. 12, pp. 208–224.

Stilwell, D.

D. Stilwell, “Directional energy spectra of the sea from photographs,” J. Geophys. Res. 74, 1974–1986 (1969).
[CrossRef]

Strizhkin, I. I.

I. I. Strizhkin, M. P. Lapchinskaya, Yu. A. Il’in, V. A. Malinnikov, “Spatial structure of high-frequency wind waves on the sea for different meteorological conditions”, Izv. Atmos. Oceanic Phys. 21, 342–344 (1985).

Tang, S.

S. Tang, O. H. Shemdin, “Measurement of high frequency waves using a wave follower,” J. Geophys. Res. 88, 9832–9840 (1983).
[CrossRef]

Thebaud, L. R.

S. C. Lubard, J. E. Krimmel, L. R. Thebaud, D. D. Evans, O. H. Shemdin, “Optical image and laser slope meter intercomparisons of high-frequency waves,” J. Geophys. Res. 85, 4996–5002 (1980).
[CrossRef]

Tober, G.

Torgersen, T.

E. C. Monahan, T. Torgersen, “The enhancement of air–sea gas exchange by oceanic whitecapping,” in Air–Water Mass Transfer, Selected Papers from the 2nd International Symposium on Gas Transfer at Water Surfaces, S. C. Wilhelms, J. S. Gulliver, eds. (American Society of Civil Engineers, New York, 1991), pp. 608–617.

Tran, H. M.

O. H. Shemdin, H. M. Tran, “Measuring short surface waves with stereophotography”, Photogram. Eng. Remote Sens. 93, 311–316 (1992).

O. H. Shemdin, H. M. Tran, S. C. Wu, “Directional measurements of short ocean waves with stereophotography,” J. Geophys. Res. 93, 13,891–13,901 (1988).
[CrossRef]

Trinder, J. C.

M. L. Banner, I. S. F. Jones, J. C. Trinder, “Wave number spectra of short gravity waves,” J. Fluid Mech. 198, 321–344 (1989).
[CrossRef]

Tschiersch, J.

P. A. Lange, B. Jähne, J. Tschiersch, J. Ilmberger, “Comparison between an amplitude-measuring wire and a slope-measuring laser water wave gauge”, Rev. Sci. Instrum. 53, 651–655 (1982).
[CrossRef]

Vetter, R. C.

L. F. Cote, J. O. Davis, W. Marks, R. F. McGough, E. Mehr, W. J. Pierson, J. F. Ropek, G. Stephenson, R. C. Vetter, “The directional spectrum of a wind generated sea as determined from data obtained by the Stereo Wave Observation Project”, Meteorological Papers Vol. 2, No. 6 (New York U. College of Engineering, New York, 1960), pp. 1–88.

Waas, S.

B. Jähne, S. Waas, “Optical measuring technique for small scale water surface waves,” in Advanced Optical Instrumentation for Remote Sensing of the Earth’s Surface from Space, G. Duchossois, F. L. Herr, R. J. Zander, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1129, 147–152 (1989).

S. Waas, B. Jähne, “Combined slope-height measurements of short wind waves: first results from field and laboratory measurements,” in Optics of the Air–Sea Interface: Theory and Measurements, L. Estep, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1749, 295–306 (1992).
[CrossRef]

Wagner, R. N.

P. Lee, J. D. Barter, K. L. Beach, C. L. Hindman, B. M. Lake, H. Rungaldier, J. C. Schatzman, J. C. Shelton, R. N. Wagner, A. B. Williams, H. C. Yuen, “Recent advances in ocean surface characterization by a scanning laser slope gauge,” in Optics of the Air–Sea Interface: Theory and Measurements, L. Estep, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1749, 234–244 (1992).
[CrossRef]

Watt, A.

A. Watt, Fundamentals of Three-dimensional Computer Graphics (Addison-Wesley, Wokingham, UK, 1989), Chap. 7.6, pp. 165–167.

Williams, A. B.

P. Lee, J. D. Barter, K. L. Beach, C. L. Hindman, B. M. Lake, H. Rungaldier, J. C. Schatzman, J. C. Shelton, R. N. Wagner, A. B. Williams, H. C. Yuen, “Recent advances in ocean surface characterization by a scanning laser slope gauge,” in Optics of the Air–Sea Interface: Theory and Measurements, L. Estep, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1749, 234–244 (1992).
[CrossRef]

Wright, J. W.

J. W. Wright, W. C. Keller, “Doppler spectra in microwave scattering from wind waves”, Phys. Fluids 14, 466–474 (1971).
[CrossRef]

Wu, S. C.

O. H. Shemdin, H. M. Tran, S. C. Wu, “Directional measurements of short ocean waves with stereophotography,” J. Geophys. Res. 93, 13,891–13,901 (1988).
[CrossRef]

Yuen, H. C.

P. Lee, J. D. Barter, K. L. Beach, C. L. Hindman, B. M. Lake, H. Rungaldier, J. C. Schatzman, J. C. Shelton, R. N. Wagner, A. B. Williams, H. C. Yuen, “Recent advances in ocean surface characterization by a scanning laser slope gauge,” in Optics of the Air–Sea Interface: Theory and Measurements, L. Estep, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1749, 234–244 (1992).
[CrossRef]

Zhang, X.

X. Zhang, C. S. Cox, “Measuring the two dimensional structure of a wavy water surface optically: a surface gradient detector”, Exp. Fluids (to be published).

Ann. Hydrographie (1)

E. Kohlschütter, “Die Forschungsreise S. M. S. Planet” Band 2, Stereophotogrammetrische Aufnahmen,” Ann. Hydrographie 34, 219 (1906).

Appl. Opt. (3)

Artif. Intell. (1)

B. K. P. Horn, “Understanding image intensities”, Artif. Intell. 8, 201–231 (1977).
[CrossRef]

Boundary Layer Meteorol. (1)

A. M. Reece, “Modulation of short waves by long waves”, Boundary Layer Meteorol. 13, 203–214 (1978).
[CrossRef]

Deep Sea Res. (1)

B. A. Hughes, H. L. Grant, R. W. Chappel, “A fast response surface-wave slope meter and measured wind-wave moments”, Deep Sea Res. 24, 1211–1233 (1977).
[CrossRef]

Izv. Atmos. Oceanic Phys. (1)

I. I. Strizhkin, M. P. Lapchinskaya, Yu. A. Il’in, V. A. Malinnikov, “Spatial structure of high-frequency wind waves on the sea for different meteorological conditions”, Izv. Atmos. Oceanic Phys. 21, 342–344 (1985).

J. Fluid Mech. (2)

M. L. Banner, I. S. F. Jones, J. C. Trinder, “Wave number spectra of short gravity waves,” J. Fluid Mech. 198, 321–344 (1989).
[CrossRef]

S. R. Long, N. E. Huang, “On the variation and growth of wave-slope spectra in the capillary-gravity range with increasing wind,” J. Fluid Mech. 77, 209–228 (1976).
[CrossRef]

J. Geophys. Res. (7)

B. Jähne, K. Riemer, “Two-dimensional wave number spectra of small-scale water surface waves,” J. Geophys. Res. 95, 11,531–11,546 (1990).
[CrossRef]

D. Stilwell, “Directional energy spectra of the sea from photographs,” J. Geophys. Res. 74, 1974–1986 (1969).
[CrossRef]

B. L. Gotwols, G. B. Irani, “Optical determination of the phase velocity of short gravity waves,” J. Geophys. Res. 85, 3964–3970 (1980).
[CrossRef]

S. C. Lubard, J. E. Krimmel, L. R. Thebaud, D. D. Evans, O. H. Shemdin, “Optical image and laser slope meter intercomparisons of high-frequency waves,” J. Geophys. Res. 85, 4996–5002 (1980).
[CrossRef]

S. Tang, O. H. Shemdin, “Measurement of high frequency waves using a wave follower,” J. Geophys. Res. 88, 9832–9840 (1983).
[CrossRef]

O. H. Shemdin, H. M. Tran, S. C. Wu, “Directional measurements of short ocean waves with stereophotography,” J. Geophys. Res. 93, 13,891–13,901 (1988).
[CrossRef]

E. B. Dobson, “Measurements of the fine-scale structure of the sea,” J. Geophys. Res. 75, 2853–2856 (1970).
[CrossRef]

J. Marine Res. (2)

C. Cox, “Measurement of slopes of high-frequency wind waves,” J. Marine Res. 16, 199–225 (1958).

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

J. Opt. Soc. Am. (1)

J. Phys. Oceanogr. (1)

F. M. Monaldo, R. S. Kasevich, “Daylight imagery of ocean surface waves for wave spectra,” J. Phys. Oceanogr. 11, 271–283 (1981).
[CrossRef]

Jahrb. Schiffbautechn. Gesellschaft (1)

W. Laas, “Messung der Meereswellen und ihre Bedeutung für den Schiffbau”, Jahrb. Schiffbautechn. Gesellschaft 7, 391 (1906).

Nature (London) (1)

E. Lamarre, W. K. Melville, “Air entrainment and dissipation in breaking waves,” Nature (London) 351, 469–472 (1991).
[CrossRef]

Photogram. Eng. Remote Sens. (1)

O. H. Shemdin, H. M. Tran, “Measuring short surface waves with stereophotography”, Photogram. Eng. Remote Sens. 93, 311–316 (1992).

Phys. Fluids (1)

J. W. Wright, W. C. Keller, “Doppler spectra in microwave scattering from wind waves”, Phys. Fluids 14, 466–474 (1971).
[CrossRef]

Rev. Sci. Instrum. (1)

P. A. Lange, B. Jähne, J. Tschiersch, J. Ilmberger, “Comparison between an amplitude-measuring wire and a slope-measuring laser water wave gauge”, Rev. Sci. Instrum. 53, 651–655 (1982).
[CrossRef]

Z. Vereins Deutsch. Ingenieure (1)

W. Laas, “Photographische Messung der Meereswellen,” Z. Vereins Deutsch. Ingenieure 49, 1889, 1937, 1976 (1905).

Other (24)

R. H. Stewart, Methods of Satellite Oceanography (U. of California Press, Berkeley, Calif., 1985), Chap. 12, pp. 208–224.

J. R. Apel, Principles of Ocean Physics, Vol. 38 of International Geophysics Series (Academic, London, 1987), Chap. 8, pp. 405–510.

W. Laas, “Die photographische Messung der Meereswellen,” Veröffentlichungen Inst. Meereskunde N. F., Vol. A, No. 7, (Berlin, 1921).

A. Schuhmacher, “Stereophotogrammetrische Wellenaufnahmen,” Wissenschaftliche Ergebnisse der Deutschen Atlantischen Expedition auf dem Forschungs- und Vermessungsschiff “Meteor” 1925–1927 (de Gruyter, Berlin, 1939), Vol. 7, Part 2.

L. F. Cote, J. O. Davis, W. Marks, R. F. McGough, E. Mehr, W. J. Pierson, J. F. Ropek, G. Stephenson, R. C. Vetter, “The directional spectrum of a wind generated sea as determined from data obtained by the Stereo Wave Observation Project”, Meteorological Papers Vol. 2, No. 6 (New York U. College of Engineering, New York, 1960), pp. 1–88.

B. Jähne, “Shape-from-shading techniques for short ocean wind waves,” in Proceedings of the International Seminar on Imaging in Transport Processes, International Center for Heat and Mass Transfer (Begell, New York, 1993), pp. 269–281.

O. H. Shemdin, Ocean Research and Engineering, 255 South Marengo Avenue, Pasadena, Calif. 91101 (personal communication, February1991).

B. Jähne, “Energy balance in small-scale waves—an experimental approach using optical slope measuring technique and image processing,” in Radar Scattering from Modulated Wind Waves, G. Komen, W. Oost, eds. (Kluwer, Dordrecht, The Netherlands, 1989), pp. 105–120.
[CrossRef]

E. J. Bock, T. Hara, “Optical measurements of ripples using a scanning laser slope gauge. Part II: data analysis and interpretation from a laboratory wave tank,” in Optics of the Air–Sea Interface: Theory and Measurements, L. Estep, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1749, 258–271 (1992).
[CrossRef]

P. A. Hwang, “Optical measurements of the structure of short water waves and their modulation by surface currents,” in Optics of the Air–Sea Interface: Theory and Measurements, L. Estep, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1749, 216–221 (1992).
[CrossRef]

P. Lee, J. D. Barter, K. L. Beach, C. L. Hindman, B. M. Lake, H. Rungaldier, J. C. Schatzman, J. C. Shelton, R. N. Wagner, A. B. Williams, H. C. Yuen, “Recent advances in ocean surface characterization by a scanning laser slope gauge,” in Optics of the Air–Sea Interface: Theory and Measurements, L. Estep, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1749, 234–244 (1992).
[CrossRef]

E. C. Monahan, T. Torgersen, “The enhancement of air–sea gas exchange by oceanic whitecapping,” in Air–Water Mass Transfer, Selected Papers from the 2nd International Symposium on Gas Transfer at Water Surfaces, S. C. Wilhelms, J. S. Gulliver, eds. (American Society of Civil Engineers, New York, 1991), pp. 608–617.

A. Watt, Fundamentals of Three-dimensional Computer Graphics (Addison-Wesley, Wokingham, UK, 1989), Chap. 7.6, pp. 165–167.

B. Jähne, “Transfer processes across the free water surface,” Habilitation dissertation (University of Heidelberg, Heidelberg, Germany, 1985).

B. K. P. Horn, Robot Vision (MIT Press, Cambridge, Mass., 1986), Chap. 11, pp. 243–277.

J. Klinke, “2D Wellenzahlspektren von kleinskaligen winderzeugten Wasseroberflächenwellen,” Ph.D. dissertation (University of Heidelberg, Heidelberg, Germany, 1991).

K. Riemer, “Analyse von Waseroberflächenwellen im Orts-Wellenzahl-Raum,” Ph.D. dissertation (University of Heidelberg, Heidelberg, Germany, 1991).

K. Riemer, T. Scholz, B. Jähne, “Bildfolgenanalyse im Orts-Wellenzahl-Raum,” in Proceedings of the 13th Deutsche Arbeitsgemeinschaft für Mustererkennung—Symposium Mustererkennung 1991, B. Radig, ed. (Springer-Verlag, Berlin, 1991), pp. 223–230.

J. Klinke, B. Jähne, “2D wave number spectra of short wind waves—results from wind-wave facilities and extrapolation to the ocean,” in Optics of the Air–Sea Interface: Theory and Measurements, L. Estep, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1749, 204–215 (1992).

X. Zhang, C. S. Cox, “Measuring the two dimensional structure of a wavy water surface optically: a surface gradient detector”, Exp. Fluids (to be published).

B. Jähne, H. J. Schultz, “Calibration and accuracy of optical slope measurements for short wind waves,” in Optics of the Air–Sea Interface: Theory and Measurements, L. Estep, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1749, 222–233 (1992).
[CrossRef]

B. Jähne, S. Waas, “Optical measuring technique for small scale water surface waves,” in Advanced Optical Instrumentation for Remote Sensing of the Earth’s Surface from Space, G. Duchossois, F. L. Herr, R. J. Zander, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1129, 147–152 (1989).

S. Waas, B. Jähne, “Combined slope-height measurements of short wind waves: first results from field and laboratory measurements,” in Optics of the Air–Sea Interface: Theory and Measurements, L. Estep, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1749, 295–306 (1992).
[CrossRef]

H. J. Schultz, “Specular surface stereo: a new method for retrieving the shape of a water surface,” in Optics of the Air–Sea Interface: Theory and Measurements, L. Estep, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1749, 283–294 (1992).
[CrossRef]

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Figures (12)

Fig. 1
Fig. 1

Illustration of the ambiguity of shape-from-shading methods for ocean wave measurements for an arbitrarily selected nonlinear relation between wave slope and image irradiance. The received image irradiance is shown in the gradient space.

Fig. 2
Fig. 2

Schematic setup for a shape-from-reflection technique for wave slope imaging with a camera looking straight down and an artificial extended light source.

Fig. 3
Fig. 3

Ray geometry in gradient space for shape from reflection: (a) a system with a camera looking straight down and an artificial light source, (b) Stilwell photography with daylight illumination and an oblique camera.

Fig. 4
Fig. 4

Contour plots of the irradiance of the shape-from-reflection technique shown in the gradient space in a wave slope range from −0.5 to 0.5 in all directions for the setup shown in Fig. 2, with use of unpolarized light [Eqs. (9)].

Fig. 5
Fig. 5

Contour plots of the irradiance received by a camera with an incidence angle of 50° by a specular ocean illuminated with an infinitely extended isotropic light source shown in gradient space and a camera inclination angle of 50° for (a) unpolarized light and (b) horizontally polarized light. Since the irradiance is directly proportional to the reflection coefficient, the plot shows the reflection coefficient. The distance between the contour lines is 0.05.

Fig. 6
Fig. 6

Ideal shape-from-refraction technique for wave slope measurements (camera at an infinite distance looking straight down, light source in the focal plane of a large lens): (a) schematic setup, (b) ray geometry in gradient space.

Fig. 7
Fig. 7

Maximum slope of the water surface that can be measured with different arrangements by light refraction: (a) camera in water, light source in air; (b) camera in air, light source in water. Arrows mark the direction of the light beam from the camera to the light source for imaging instruments or from the laser to the optical sensor for laser slope gauges.

Fig. 8
Fig. 8

Nonlinearity of the relation between tan γ and the slope of the water surface s: (a) the relation as given by relations (14) and (16), (b) large-signal sensitivity tan γ/s normalized to 1 for small slopes, (c) small-signal sensitivity [derivatives of relations (14) and (16)] normalized to 1 for small slopes. Solid curves: camera in air, light source in water [Eq. (14)]; dashed curves: vice versa [relation (16)].

Fig. 9
Fig. 9

Contour plots of the irradiance of the shape-from-refraction technique shown in the gradient space according to relation (18): (a) slope range ±0.5, (b) slope range ±2.

Fig. 10
Fig. 10

Simple setup of a stereo camera pair with verging axes.

Fig. 11
Fig. 11

Illustration of the stereo correspondence problem when illumination changes occur parallel to the stereo basis with a sinusoidal wave: (a) steep wave, bias in the parallax estimate; (b) less-steep wave, missing correspondence.

Fig. 12
Fig. 12

Examples of images obtained with the refraction technique. The downwind slope component of the water surface is shown for four different wind speeds as indicated. Dark gray values correspond to negative slopes and vice versa.

Equations (24)

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ñ = ( a / X , a / Y , 1 ) T .
ŝ 1 ( k ) = i k 1 â ( k ) = i k cos φ s â ( k ) ,
n ¯ = l ¯ r ¯ 2 cos α .
tan δ = 2 tan α 1 tan 2 α = 2 s 1 s 2 2 s [ 1 + s 2 + s 4 + O ( s 6 ) ] for s < 1 ,
tan α = s = tan δ 1 + [ ( 1 + tan 2 δ ) ] 1 / 2 1 2 tan δ ( 1 1 4 tan 2 δ + 1 8 tan 4 δ ) for tan δ < 1 .
L X 1 .
X 1 / X 3 = cos φ s tan δ = s 1 / s tan δ ,
ρ = tan 2 ( α β ) tan 2 ( α + β ) , ρ = sin 2 ( α β ) sin 2 ( α + β ) ,
sin α sin β = n w .
ρ = { [ n w 2 + ( n w 2 1 ) s 2 ] 1 / 2 1 [ n w 2 + ( n w 2 1 ) s 2 ] 1 / 2 + 1 } ,
ρ = { [ n w 2 + ( n w 2 1 ) s 2 ] 1 / 2 n w 2 [ n w 2 + ( n w 2 1 ) s 2 ] 1 / 2 + n w 2 } .
E s 1 ρ ( s ) 1 s 2 1 49 s 1 [ 1 + s 2 + 197 128 s 2 + O ( s 6 ) ] ,
r = ( tan Θ c , 0 , 1 ) T ,
α = arccos ( r , ñ ) = arccos [ s 1 tan Θ c + 1 ( 1 + s 1 2 + s 2 2 ) 1 / 2 ( 1 + tan 2 Θ c ) 1 / 2 ] .
s = tan α = n w tan γ n w ( 1 + tan 2 γ ) 1 / 2 4 tan γ [ 1 + 3 / 2 tan 2 γ + 15 / 8 tan 4 γ + O ( tan 6 γ ) ] ,
tan γ = tan α [ n w 2 + ( n w 2 1 ) tan 2 α ] 1 / 2 1 [ n w 2 + ( n w 2 1 ) tan 2 α + tan 2 α ] 1 / 2 1 4 s ( 1 3 32 s 2 + 39 2048 s 4 ) .
s = tan α = tan γ n w ( 1 + tan 2 γ ) 1 / 2 1 3 tan γ [ 1 2 tan 2 γ + 9 / 2 tan 2 γ + O ( tan 6 γ ) ] ,
tan γ = tan α n w [ 1 ( n w 2 1 ) tan 2 α ] 1 / 2 [ 1 ( n w 2 1 ) tan 2 α ] 1 / 2 + n w tan 2 α 1 3 s ( 1 + 2 9 s 2 + 5 54 s 4 ) .
X 1 / f = cos φ s tan γ = s 1 / s tan γ ,
E s 1 [ 1 ρ ( s ) ] [ n w 2 + ( n w 2 1 ) s 2 ] 1 / 2 1 [ n w 2 + ( n w 2 1 ) s 2 ] 1 / 2 + s 2 1 4 s 1 [ 1 ρ ( s ) ] ( 1 3 32 s 2 + 39 2048 s 4 ) ,
p = f Δ Z Z Δ Z 2 cos 2 ( β / 2 ) sin ( β / 2 ) cos ( β / 2 ) ,
Δ Z Z = p 2 f sin ( β / 2 ) cos ( β / 2 ) .
Δ x f = Δ X cos 2 ( β / 2 ) Z ,
Δ Z Δ X = p Δ x 1 2 tan ( β / 2 ) = μ Z b .

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