Abstract

A new method is presented to estimate the topography of a rough surface. A formulation is provided in which immediate measurements and a priori observations of surface elevation, slope and curvature, are considered simultaneously as a linear algebraic system of finite difference equations. Least squares solutions are computed directly by sparse orthogonal-triangular (QR) factorization of the weighted seminormal equations, an approach made practical for large systems with powerful computational hardware and algorithms that have become available recently. Retrievals are demonstrated from synthetic slope data and from measurements of slope on a rough water surface. The method provides a general approach to retrieving topography from measurements of elevation, slope and curvature.

© 2012 OSA

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2011 (3)

2009 (1)

W. Munk, “An inconvenient sea truth: spread, steepness, and skewness of surface slopes,” Annu. Rev. Mar. Sci. 1(1), 377–415 (2009).
[CrossRef] [PubMed]

2008 (3)

Y. Hu, K. Stamnes, M. Vaughan, J. Pelon, C. Weimer, D. Wu, M. Cisewski, W. Sun, P. Yang, B. Lin, A. Omar, D. Flittner, C. Hostetler, C. Trepte, D. Winker, G. Gibson, and M. Santa-Maria, “Sea surface wind speed estimation from space-based lidar measurements,” Atmos. Chem. Phys. Discuss. 8(1), 2771–2793 (2008).
[CrossRef]

C. J. Zappa, M. L. Banner, H. Schultz, A. Corrada-Emmanuel, L. B. Wolff, and J. Yalcin, “Retrieval of short ocean wave slope using polarimetric imaging,” Meas. Sci. Technol. 19(5), 05503 (2008).
[CrossRef]

S. Ettl, J. Kaminski, M. C. Knauer, and G. Häusler, “Shape reconstruction from gradient data,” Appl. Opt. 47(12), 2091–2097 (2008).
[CrossRef] [PubMed]

2006 (1)

F. M. Bréon and N. Henriot, “Spaceborne observations of ocean glint reflectance and modeling of wave slope distributions,” J. Geophys. Res. 111(C6), C06005 (2006).
[CrossRef] [PubMed]

2005 (1)

B. Jähne, M. Schmidt, and R. Rocholz, “Combined optical slope/height measurements of short wind waves: principle and calibration,” Meas. Sci. Technol. 16(10), 1937–1944 (2005).
[CrossRef]

2002 (1)

C. Elster, J. Gerhardt, P. Thomsenschmidt, M. Schulz, and I. Weingartner, “Reconstructing surface profiles from curvature measurements,” Optik (Stuttg.) 113(4), 154–158 (2002).
[CrossRef]

1998 (1)

R. I. Mclachlan, G. R. W. Quispel, and N. Robidoux, “Geometric integration using discrete gradients,” Philos. Trans. R. Soc. Lond. A 357, 1–26 (1998).

1997 (1)

1996 (2)

X. Zhang, “An algorithm for calculating water surface elevations from surface gradient image data,” Exp. Fluids 21(1), 43–48 (1996).
[CrossRef]

R. Klette and K. Schlüns, “Height data from gradient fields,” Proc. SPIE 2908, 204–215 (1996).
[CrossRef]

1994 (2)

X. Zhang and C. S. Cox, “Measuring the two-dimensional structure of a wavy water surface optically: a surface gradient detector,” Exp. Fluids 17(4), 225–237 (1994).
[CrossRef]

B. Jähne, J. Klinke, and S. Waas, “Imaging of short ocean wind waves: a critical theoretical review,” J. Opt. Soc. Am. A 11(8), 2197–2209 (1994).
[CrossRef]

1993 (1)

1983 (1)

1980 (2)

1979 (1)

1977 (3)

Bahk, S. W.

Banner, M. L.

C. J. Zappa, M. L. Banner, H. Schultz, A. Corrada-Emmanuel, L. B. Wolff, and J. Yalcin, “Retrieval of short ocean wave slope using polarimetric imaging,” Meas. Sci. Technol. 19(5), 05503 (2008).
[CrossRef]

Bréon, F. M.

F. M. Bréon and N. Henriot, “Spaceborne observations of ocean glint reflectance and modeling of wave slope distributions,” J. Geophys. Res. 111(C6), C06005 (2006).
[CrossRef] [PubMed]

Cisewski, M.

Y. Hu, K. Stamnes, M. Vaughan, J. Pelon, C. Weimer, D. Wu, M. Cisewski, W. Sun, P. Yang, B. Lin, A. Omar, D. Flittner, C. Hostetler, C. Trepte, D. Winker, G. Gibson, and M. Santa-Maria, “Sea surface wind speed estimation from space-based lidar measurements,” Atmos. Chem. Phys. Discuss. 8(1), 2771–2793 (2008).
[CrossRef]

Corrada-Emmanuel, A.

C. J. Zappa, M. L. Banner, H. Schultz, A. Corrada-Emmanuel, L. B. Wolff, and J. Yalcin, “Retrieval of short ocean wave slope using polarimetric imaging,” Meas. Sci. Technol. 19(5), 05503 (2008).
[CrossRef]

Cox, C.

Cox, C. S.

X. Zhang and C. S. Cox, “Measuring the two-dimensional structure of a wavy water surface optically: a surface gradient detector,” Exp. Fluids 17(4), 225–237 (1994).
[CrossRef]

Elster, C.

C. Elster, J. Gerhardt, P. Thomsenschmidt, M. Schulz, and I. Weingartner, “Reconstructing surface profiles from curvature measurements,” Optik (Stuttg.) 113(4), 154–158 (2002).
[CrossRef]

Ettl, S.

Flittner, D.

Y. Hu, K. Stamnes, M. Vaughan, J. Pelon, C. Weimer, D. Wu, M. Cisewski, W. Sun, P. Yang, B. Lin, A. Omar, D. Flittner, C. Hostetler, C. Trepte, D. Winker, G. Gibson, and M. Santa-Maria, “Sea surface wind speed estimation from space-based lidar measurements,” Atmos. Chem. Phys. Discuss. 8(1), 2771–2793 (2008).
[CrossRef]

Fried, D. L.

Gerhardt, J.

C. Elster, J. Gerhardt, P. Thomsenschmidt, M. Schulz, and I. Weingartner, “Reconstructing surface profiles from curvature measurements,” Optik (Stuttg.) 113(4), 154–158 (2002).
[CrossRef]

Gibson, G.

Y. Hu, K. Stamnes, M. Vaughan, J. Pelon, C. Weimer, D. Wu, M. Cisewski, W. Sun, P. Yang, B. Lin, A. Omar, D. Flittner, C. Hostetler, C. Trepte, D. Winker, G. Gibson, and M. Santa-Maria, “Sea surface wind speed estimation from space-based lidar measurements,” Atmos. Chem. Phys. Discuss. 8(1), 2771–2793 (2008).
[CrossRef]

Gotwols, B. L.

Grédiac, M.

Häusler, G.

Hedley, J.

Henriot, N.

F. M. Bréon and N. Henriot, “Spaceborne observations of ocean glint reflectance and modeling of wave slope distributions,” J. Geophys. Res. 111(C6), C06005 (2006).
[CrossRef] [PubMed]

Herrmann, J.

Hostetler, C.

Y. Hu, K. Stamnes, M. Vaughan, J. Pelon, C. Weimer, D. Wu, M. Cisewski, W. Sun, P. Yang, B. Lin, A. Omar, D. Flittner, C. Hostetler, C. Trepte, D. Winker, G. Gibson, and M. Santa-Maria, “Sea surface wind speed estimation from space-based lidar measurements,” Atmos. Chem. Phys. Discuss. 8(1), 2771–2793 (2008).
[CrossRef]

Hu, Y.

Y. Hu, K. Stamnes, M. Vaughan, J. Pelon, C. Weimer, D. Wu, M. Cisewski, W. Sun, P. Yang, B. Lin, A. Omar, D. Flittner, C. Hostetler, C. Trepte, D. Winker, G. Gibson, and M. Santa-Maria, “Sea surface wind speed estimation from space-based lidar measurements,” Atmos. Chem. Phys. Discuss. 8(1), 2771–2793 (2008).
[CrossRef]

Hudgin, R. H.

Hunt, B. R.

Jähne, B.

B. Jähne, M. Schmidt, and R. Rocholz, “Combined optical slope/height measurements of short wind waves: principle and calibration,” Meas. Sci. Technol. 16(10), 1937–1944 (2005).
[CrossRef]

B. Jähne, J. Klinke, and S. Waas, “Imaging of short ocean wind waves: a critical theoretical review,” J. Opt. Soc. Am. A 11(8), 2197–2209 (1994).
[CrossRef]

Kaminski, J.

Kay, S.

Keller, W. C.

Klette, R.

R. Klette and K. Schlüns, “Height data from gradient fields,” Proc. SPIE 2908, 204–215 (1996).
[CrossRef]

Klinke, J.

Knauer, M. C.

Lavender, S.

Li, Q.

Lin, B.

Y. Hu, K. Stamnes, M. Vaughan, J. Pelon, C. Weimer, D. Wu, M. Cisewski, W. Sun, P. Yang, B. Lin, A. Omar, D. Flittner, C. Hostetler, C. Trepte, D. Winker, G. Gibson, and M. Santa-Maria, “Sea surface wind speed estimation from space-based lidar measurements,” Atmos. Chem. Phys. Discuss. 8(1), 2771–2793 (2008).
[CrossRef]

Mclachlan, R. I.

R. I. Mclachlan, G. R. W. Quispel, and N. Robidoux, “Geometric integration using discrete gradients,” Philos. Trans. R. Soc. Lond. A 357, 1–26 (1998).

Munk, W.

W. Munk, “An inconvenient sea truth: spread, steepness, and skewness of surface slopes,” Annu. Rev. Mar. Sci. 1(1), 377–415 (2009).
[CrossRef] [PubMed]

Nimmo-Smith, A.

Omar, A.

Y. Hu, K. Stamnes, M. Vaughan, J. Pelon, C. Weimer, D. Wu, M. Cisewski, W. Sun, P. Yang, B. Lin, A. Omar, D. Flittner, C. Hostetler, C. Trepte, D. Winker, G. Gibson, and M. Santa-Maria, “Sea surface wind speed estimation from space-based lidar measurements,” Atmos. Chem. Phys. Discuss. 8(1), 2771–2793 (2008).
[CrossRef]

Pelon, J.

Y. Hu, K. Stamnes, M. Vaughan, J. Pelon, C. Weimer, D. Wu, M. Cisewski, W. Sun, P. Yang, B. Lin, A. Omar, D. Flittner, C. Hostetler, C. Trepte, D. Winker, G. Gibson, and M. Santa-Maria, “Sea surface wind speed estimation from space-based lidar measurements,” Atmos. Chem. Phys. Discuss. 8(1), 2771–2793 (2008).
[CrossRef]

Quispel, G. R. W.

R. I. Mclachlan, G. R. W. Quispel, and N. Robidoux, “Geometric integration using discrete gradients,” Philos. Trans. R. Soc. Lond. A 357, 1–26 (1998).

Robidoux, N.

R. I. Mclachlan, G. R. W. Quispel, and N. Robidoux, “Geometric integration using discrete gradients,” Philos. Trans. R. Soc. Lond. A 357, 1–26 (1998).

Rocholz, R.

B. Jähne, M. Schmidt, and R. Rocholz, “Combined optical slope/height measurements of short wind waves: principle and calibration,” Meas. Sci. Technol. 16(10), 1937–1944 (2005).
[CrossRef]

Santa-Maria, M.

Y. Hu, K. Stamnes, M. Vaughan, J. Pelon, C. Weimer, D. Wu, M. Cisewski, W. Sun, P. Yang, B. Lin, A. Omar, D. Flittner, C. Hostetler, C. Trepte, D. Winker, G. Gibson, and M. Santa-Maria, “Sea surface wind speed estimation from space-based lidar measurements,” Atmos. Chem. Phys. Discuss. 8(1), 2771–2793 (2008).
[CrossRef]

Schlüns, K.

R. Klette and K. Schlüns, “Height data from gradient fields,” Proc. SPIE 2908, 204–215 (1996).
[CrossRef]

Schmidt, M.

B. Jähne, M. Schmidt, and R. Rocholz, “Combined optical slope/height measurements of short wind waves: principle and calibration,” Meas. Sci. Technol. 16(10), 1937–1944 (2005).
[CrossRef]

Schultz, H.

C. J. Zappa, M. L. Banner, H. Schultz, A. Corrada-Emmanuel, L. B. Wolff, and J. Yalcin, “Retrieval of short ocean wave slope using polarimetric imaging,” Meas. Sci. Technol. 19(5), 05503 (2008).
[CrossRef]

Schulz, M.

C. Elster, J. Gerhardt, P. Thomsenschmidt, M. Schulz, and I. Weingartner, “Reconstructing surface profiles from curvature measurements,” Optik (Stuttg.) 113(4), 154–158 (2002).
[CrossRef]

Southwell, W. H.

Stamnes, K.

Y. Hu, K. Stamnes, M. Vaughan, J. Pelon, C. Weimer, D. Wu, M. Cisewski, W. Sun, P. Yang, B. Lin, A. Omar, D. Flittner, C. Hostetler, C. Trepte, D. Winker, G. Gibson, and M. Santa-Maria, “Sea surface wind speed estimation from space-based lidar measurements,” Atmos. Chem. Phys. Discuss. 8(1), 2771–2793 (2008).
[CrossRef]

Sun, S.

Sun, W.

Y. Hu, K. Stamnes, M. Vaughan, J. Pelon, C. Weimer, D. Wu, M. Cisewski, W. Sun, P. Yang, B. Lin, A. Omar, D. Flittner, C. Hostetler, C. Trepte, D. Winker, G. Gibson, and M. Santa-Maria, “Sea surface wind speed estimation from space-based lidar measurements,” Atmos. Chem. Phys. Discuss. 8(1), 2771–2793 (2008).
[CrossRef]

Tang, S.

Thomsenschmidt, P.

C. Elster, J. Gerhardt, P. Thomsenschmidt, M. Schulz, and I. Weingartner, “Reconstructing surface profiles from curvature measurements,” Optik (Stuttg.) 113(4), 154–158 (2002).
[CrossRef]

Trepte, C.

Y. Hu, K. Stamnes, M. Vaughan, J. Pelon, C. Weimer, D. Wu, M. Cisewski, W. Sun, P. Yang, B. Lin, A. Omar, D. Flittner, C. Hostetler, C. Trepte, D. Winker, G. Gibson, and M. Santa-Maria, “Sea surface wind speed estimation from space-based lidar measurements,” Atmos. Chem. Phys. Discuss. 8(1), 2771–2793 (2008).
[CrossRef]

Vaughan, M.

Y. Hu, K. Stamnes, M. Vaughan, J. Pelon, C. Weimer, D. Wu, M. Cisewski, W. Sun, P. Yang, B. Lin, A. Omar, D. Flittner, C. Hostetler, C. Trepte, D. Winker, G. Gibson, and M. Santa-Maria, “Sea surface wind speed estimation from space-based lidar measurements,” Atmos. Chem. Phys. Discuss. 8(1), 2771–2793 (2008).
[CrossRef]

Waas, S.

Weimer, C.

Y. Hu, K. Stamnes, M. Vaughan, J. Pelon, C. Weimer, D. Wu, M. Cisewski, W. Sun, P. Yang, B. Lin, A. Omar, D. Flittner, C. Hostetler, C. Trepte, D. Winker, G. Gibson, and M. Santa-Maria, “Sea surface wind speed estimation from space-based lidar measurements,” Atmos. Chem. Phys. Discuss. 8(1), 2771–2793 (2008).
[CrossRef]

Weingartner, I.

C. Elster, J. Gerhardt, P. Thomsenschmidt, M. Schulz, and I. Weingartner, “Reconstructing surface profiles from curvature measurements,” Optik (Stuttg.) 113(4), 154–158 (2002).
[CrossRef]

Winker, D.

Y. Hu, K. Stamnes, M. Vaughan, J. Pelon, C. Weimer, D. Wu, M. Cisewski, W. Sun, P. Yang, B. Lin, A. Omar, D. Flittner, C. Hostetler, C. Trepte, D. Winker, G. Gibson, and M. Santa-Maria, “Sea surface wind speed estimation from space-based lidar measurements,” Atmos. Chem. Phys. Discuss. 8(1), 2771–2793 (2008).
[CrossRef]

Wolff, L. B.

C. J. Zappa, M. L. Banner, H. Schultz, A. Corrada-Emmanuel, L. B. Wolff, and J. Yalcin, “Retrieval of short ocean wave slope using polarimetric imaging,” Meas. Sci. Technol. 19(5), 05503 (2008).
[CrossRef]

Wu, D.

Y. Hu, K. Stamnes, M. Vaughan, J. Pelon, C. Weimer, D. Wu, M. Cisewski, W. Sun, P. Yang, B. Lin, A. Omar, D. Flittner, C. Hostetler, C. Trepte, D. Winker, G. Gibson, and M. Santa-Maria, “Sea surface wind speed estimation from space-based lidar measurements,” Atmos. Chem. Phys. Discuss. 8(1), 2771–2793 (2008).
[CrossRef]

Wu, J.

Yalcin, J.

C. J. Zappa, M. L. Banner, H. Schultz, A. Corrada-Emmanuel, L. B. Wolff, and J. Yalcin, “Retrieval of short ocean wave slope using polarimetric imaging,” Meas. Sci. Technol. 19(5), 05503 (2008).
[CrossRef]

Yang, P.

Y. Hu, K. Stamnes, M. Vaughan, J. Pelon, C. Weimer, D. Wu, M. Cisewski, W. Sun, P. Yang, B. Lin, A. Omar, D. Flittner, C. Hostetler, C. Trepte, D. Winker, G. Gibson, and M. Santa-Maria, “Sea surface wind speed estimation from space-based lidar measurements,” Atmos. Chem. Phys. Discuss. 8(1), 2771–2793 (2008).
[CrossRef]

Zappa, C. J.

C. J. Zappa, M. L. Banner, H. Schultz, A. Corrada-Emmanuel, L. B. Wolff, and J. Yalcin, “Retrieval of short ocean wave slope using polarimetric imaging,” Meas. Sci. Technol. 19(5), 05503 (2008).
[CrossRef]

Zhang, X.

C. Cox and X. Zhang, “Contours of slopes of a rippled water surface,” Opt. Express 19(20), 18789–18794 (2011).
[CrossRef] [PubMed]

X. Zhang, “An algorithm for calculating water surface elevations from surface gradient image data,” Exp. Fluids 21(1), 43–48 (1996).
[CrossRef]

X. Zhang and C. S. Cox, “Measuring the two-dimensional structure of a wavy water surface optically: a surface gradient detector,” Exp. Fluids 17(4), 225–237 (1994).
[CrossRef]

Zhao, M.

Annu. Rev. Mar. Sci. (1)

W. Munk, “An inconvenient sea truth: spread, steepness, and skewness of surface slopes,” Annu. Rev. Mar. Sci. 1(1), 377–415 (2009).
[CrossRef] [PubMed]

Appl. Opt. (4)

Atmos. Chem. Phys. Discuss. (1)

Y. Hu, K. Stamnes, M. Vaughan, J. Pelon, C. Weimer, D. Wu, M. Cisewski, W. Sun, P. Yang, B. Lin, A. Omar, D. Flittner, C. Hostetler, C. Trepte, D. Winker, G. Gibson, and M. Santa-Maria, “Sea surface wind speed estimation from space-based lidar measurements,” Atmos. Chem. Phys. Discuss. 8(1), 2771–2793 (2008).
[CrossRef]

Exp. Fluids (2)

X. Zhang and C. S. Cox, “Measuring the two-dimensional structure of a wavy water surface optically: a surface gradient detector,” Exp. Fluids 17(4), 225–237 (1994).
[CrossRef]

X. Zhang, “An algorithm for calculating water surface elevations from surface gradient image data,” Exp. Fluids 21(1), 43–48 (1996).
[CrossRef]

J. Geophys. Res. (1)

F. M. Bréon and N. Henriot, “Spaceborne observations of ocean glint reflectance and modeling of wave slope distributions,” J. Geophys. Res. 111(C6), C06005 (2006).
[CrossRef] [PubMed]

J. Opt. Soc. Am. (6)

J. Opt. Soc. Am. A (1)

Meas. Sci. Technol. (2)

C. J. Zappa, M. L. Banner, H. Schultz, A. Corrada-Emmanuel, L. B. Wolff, and J. Yalcin, “Retrieval of short ocean wave slope using polarimetric imaging,” Meas. Sci. Technol. 19(5), 05503 (2008).
[CrossRef]

B. Jähne, M. Schmidt, and R. Rocholz, “Combined optical slope/height measurements of short wind waves: principle and calibration,” Meas. Sci. Technol. 16(10), 1937–1944 (2005).
[CrossRef]

Opt. Express (3)

Optik (Stuttg.) (1)

C. Elster, J. Gerhardt, P. Thomsenschmidt, M. Schulz, and I. Weingartner, “Reconstructing surface profiles from curvature measurements,” Optik (Stuttg.) 113(4), 154–158 (2002).
[CrossRef]

Philos. Trans. R. Soc. Lond. A (1)

R. I. Mclachlan, G. R. W. Quispel, and N. Robidoux, “Geometric integration using discrete gradients,” Philos. Trans. R. Soc. Lond. A 357, 1–26 (1998).

Proc. SPIE (1)

R. Klette and K. Schlüns, “Height data from gradient fields,” Proc. SPIE 2908, 204–215 (1996).
[CrossRef]

Other (8)

K. Schlüns and R. Klette, “Local and global integration of discrete vector fields,” in Advances in Computer Vision, F. Solina, W. Kropatsch, R. Klette, and R. Bajcsy, eds. (Springer, 1997) pp. 149–158.

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Supplementary Material (1)

» Media 1: MOV (3843 KB)     

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Figures (7)

Fig. 1
Fig. 1

A triangular planar facet (hatched area) is bounded by corners having position vectors r a , r b and r c in a Cartesian coordinate system. The surface normal vector n can be computed by cross product and normalized to unit length by n ^ = n /| n | .

Fig. 2
Fig. 2

Example mesh shows the n ^ s surface normal unit vector measurements, and the z t vertex elevations. The shaded areas represent locally planar facets on which the measurements are considered to occur.

Fig. 3
Fig. 3

The non zero entries of the sparse matrix given by the linear system of Eq. (11) using the example of Fig. 2.

Fig. 4
Fig. 4

Left, surface retrieved from low-noise slope data. Right, retrieval error as the difference between retrieved and ideal surface. Surface color is a function of elevation shown in the color bars.

Fig. 5
Fig. 5

Left, surface retrieved from noisy slope data. Right, solution error as the difference between retrieved and ideal surface.

Fig. 6
Fig. 6

Left: color image data from the refractive slope imaging system. Middle and right, x and y surface normal unit vector components, respectively.

Fig. 7
Fig. 7

Single frame example of retrieved elevation rendered using an artificial illumination model (Media 1).

Equations (16)

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n = [ ( y b y a )( z c z a )( z b z a )( y c y a ) ] i ^ + [ ( z b z a )( x c x a )( x b x a )( z c z a ) ] j ^ + [ ( x b x a )( y c y a )( y b y a )( x c x a ) ] k ^ .
[ c 3 ( z c z a ) c 4 ( z b z a ) ] i ^ + n =[ c 2 ( z b z a ) c 1 ( z c z a ) ] j ^ + [ c 1 c 4 c 2 c 3 ] k ^ .
| n |=( c 1 c 4 c 2 c 3 )/ n ^ k ^ .
n i ^ = c 4 ( z a z b )
n j ^ = c 1 ( z a z c ).
b x = [ n 1 i ^ n 2 i ^ n s i ^ n M×N i ^ ] T
b y = [ n 1 j ^ n 2 j ^ n s j ^ n M×N j ^ ] T
z= [ z 1 z 2 z t z ( M+1 )( N+1 )1 ] T
κ x t c 1 2 ( z m1,n 2 z m,n + z m+1,n )
κ y t c 4 2 ( z m,n1 2 z m,n + z m,n+1 ),
Az=[ A x A y Κ x Κ y I ]z=[ b x b y κ x =0 κ y =0 z= z ¯ ]
A T CAz= A T Cb.
R T R z p =u.
z( x,y )=sin( x/3 )sin( y )
z x =cos( x/3 )sin( y )/3
z y =sin( x/3 )cos( y ).

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