## Method of surface topography retrieval by direct solution of sparse weighted seminormal equations |

Optics Express, Vol. 20, Issue 2, pp. 1714-1726 (2012)

http://dx.doi.org/10.1364/OE.20.001714

Acrobat PDF (1562 KB)

### 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

## Introduction

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

6. 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]

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

9. 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]

10. 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]

11. S. Kay, J. Hedley, S. Lavender, and A. Nimmo-Smith, “Light transfer at the ocean surface modeled using high resolution sea surface realizations,” Opt. Express **19**(7), 6493–6504 (2011). [CrossRef] [PubMed]

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

15. S. W. Bahk, “Highly accurate wavefront reconstruction algorithms over broad spatial-frequency bandwidth,” Opt. Express **19**(20), 18997–19014 (2011). [CrossRef] [PubMed]

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

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

18. M. Grédiac, “Method for surface reconstruction from slope or curvature measurements of rectangular areas,” Appl. Opt. **36**(20), 4823–4829 (1997). [CrossRef] [PubMed]

18. M. Grédiac, “Method for surface reconstruction from slope or curvature measurements of rectangular areas,” Appl. Opt. **36**(20), 4823–4829 (1997). [CrossRef] [PubMed]

19. 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]

20. D. L. Fried, “Least-square fitting a wave-front distortion estimate to an array of phase-difference measurements,” J. Opt. Soc. Am. **67**(3), 370–375 (1977). [CrossRef]

22. R. H. Hudgin, “Optimal wave-front estimation,” J. Opt. Soc. Am. **67**(3), 378–382 (1977). [CrossRef]

23. B. R. Hunt, “Matrix formulation of the reconstruction of phase values from phase differences,” J. Opt. Soc. Am. **69**(3), 393–399 (1979). [CrossRef]

24. J. Herrmann, “Least-squares wave front errors of minimum norm,” J. Opt. Soc. Am. **70**(1), 28–35 (1980). [CrossRef]

25. W. H. Southwell, “Wave-front estimation from wave-front slope measurements,” J. Opt. Soc. Am. **70**(8), 998–1006 (1980). [CrossRef]

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

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

3. 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]

4. 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]

27. 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]

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

## Mathematical formulation

### Surface elevation and slopes

### Arrays of slope data

### Matrix-vector formulation

### Rank

24. J. Herrmann, “Least-squares wave front errors of minimum norm,” J. Opt. Soc. Am. **70**(1), 28–35 (1980). [CrossRef]

### A priori constraints

*a priori*observations may be added. For example, if the mean elevation is known, each vertex can be constrained by augmenting the system with an identity matrix that multiplies the state vector and set equal to the mean elevation:

*a priori*curvature. Curvature can also be considered as immediately measured data, such as in the case of curvature measurements [19

19. 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]

*a priori*x- and y-plane curvatures and elevation can be combined in a block matrix

*a priori*constraints to include. Blocks can be added or deleted as needed depending on the problem at hand. For example, arrays of one-dimensional slope data [1

**19**(20), 18789–18794 (2011). [CrossRef] [PubMed]

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

### Weighted least squares

_{A z=b}has a greater number of rows than columns which results in an over-determined system. Over-determined systems generally have no exact solution

_{C}can be specified by a covariance matrix

*a priori*variance is frequently specified explicitly as a parameter of a statistical distribution.

### Solution method

^{qr}and

^{lscov}functions in recent versions of Matlab subsequent to R2009b, also available as C + + source code in the SuiteSparse archive [30

30. T. Davis, “SuiteSparse,” http://www.cise.ufl.edu/research/sparse/SuiteSparse/.

## Synthetic data retrieval

- 1. A surface is given by an elevation function of two horizontal coordinates
- 2. A set of synthetic slope measurements is created by evaluating analytical derivatives of the elevation function on a grid of points and adding random numbers to the results to simulate noise
- 3. Weights for slope and
*a priori*measurement components are chosen - 4. Surface elevation is estimated by the procedure described above.

*A priori*variance is computed by evaluating Eqs. (14–16) and the second derivatives of Eq. (14) (for curvature) on a grid of points, followed by computing the variance of the resulting sets. Two cases using synthetic slope measurements are made by contaminating the synthetic slope measurements with artificial noise by adding normally distributed random numbers with specified standard deviation. The weighting of slope measurements is determined by the standard deviation of the added noise.

^{−4}, one one-thousandth of the

*a priori*y-slope data standard deviation (a zero-noise case would have infinite weighting coefficients and cannot be analyzed by this method). The retrieved surface appears to be smooth. The difference between the original and retrieved surfaces shows deviations on the order of 10

^{−4}times the maximum elevation with a standard deviation of error 5.2687

^{−5}. The errors appear largely random, though signs of a periodic artifact reminiscent of the original synthetic data can be seen in this difference.

*a priori*standard deviation, respectively) results in retrievals with more distortion, shown in Fig. 5 . The distortion appears smooth, with the largest errors less than 2% of the original surface’s maximum elevation appearing at the boundaries of the solution. Perhaps the errors in slope balance each other in the interior and errors caused by the elevation constraints become apparent at the boundaries. Most importantly, the standard deviation of error in elevation is 0.0055, or about 0.5% of the original surface’s maximum elevation.

## Water topography retrieval

3. 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]

*a priori*data take over to fill in the blanks with reasonable estimates. Overall, the retrieved topography is free of obvious defects, which indicates that the solution is reasonable. Future work will examine in detail errors in slope and elevation using a standard reference surface with a shape that can be measured independently.

## Conclusions

*a priori*observations. Observations that can be included in this framework are modeled as coupled finite difference equations that result in a sparse linear system of algebraic equations. Observables such as elevation, slope and curvature can be included, treated as either immediate measurements or

*a priori*constraints. By including such

*a priori*observations, full system rank can be ensured, despite noisy or missing data points. Solutions are computed directly by multifrontal QR factorization without inversion of the full system matrix.

*a priori*elevation and curvature data, with each observation weighted by estimates of variance. Solutions retrieved from synthetic data have error standard deviations of approximately 1/10th of slope measurement error standard deviation. Solutions retrieved from imaging data are well behaved even in places where pixel data are known to be corrupt. The computations for the examples shown were achieved using a personal laptop and desktop computer. Reasonable solution times enabled the analysis of long time series of wind wave slope images, an example of which is shown in the companion movie file.

## Acknowledgments

## References and links:

1. | C. Cox and X. Zhang, “Contours of slopes of a rippled water surface,” Opt. Express |

2. | W. C. Keller and B. L. Gotwols, “Two-dimensional optical measurement of wave slope,” Appl. Opt. |

3. | X. Zhang and C. S. Cox, “Measuring the two-dimensional structure of a wavy water surface optically: a surface gradient detector,” Exp. Fluids |

4. | 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. |

5. | Q. Li, M. Zhao, S. Tang, S. Sun, and J. Wu, “Two-dimensional scanning laser slope gauge: measurements of ocean-ripple structures,” Appl. Opt. |

6. | B. Jähne, J. Klinke, and S. Waas, “Imaging of short ocean wind waves: a critical theoretical review,” J. Opt. Soc. Am. A |

7. | W. Munk, “An inconvenient sea truth: spread, steepness, and skewness of surface slopes,” Annu. Rev. Mar. Sci. |

8. | A. Freedman, D. McWatters, and M. Spencer, “The Aquarius scatterometer: an active system for measuring surface roughness for sea-surface brightness temperature correction,” presented at the IEEE International Geoscience & Remote Sensing Symposium, Denver, Colorado, July 31- August 4, 2006. |

9. | 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. |

10. | F. M. Bréon and N. Henriot, “Spaceborne observations of ocean glint reflectance and modeling of wave slope distributions,” J. Geophys. Res. |

11. | S. Kay, J. Hedley, S. Lavender, and A. Nimmo-Smith, “Light transfer at the ocean surface modeled using high resolution sea surface realizations,” Opt. Express |

12. | R. Klette and K. Schlüns, “Height data from gradient fields,” Proc. SPIE |

13. | K. Schlüns and R. Klette, “Local and global integration of discrete vector fields,” in |

14. | R. I. Mclachlan, G. R. W. Quispel, and N. Robidoux, “Geometric integration using discrete gradients,” Philos. Trans. R. Soc. Lond. A |

15. | S. W. Bahk, “Highly accurate wavefront reconstruction algorithms over broad spatial-frequency bandwidth,” Opt. Express |

16. | X. Zhang, “An algorithm for calculating water surface elevations from surface gradient image data,” Exp. Fluids |

17. | S. Ettl, J. Kaminski, M. C. Knauer, and G. Häusler, “Shape reconstruction from gradient data,” Appl. Opt. |

18. | M. Grédiac, “Method for surface reconstruction from slope or curvature measurements of rectangular areas,” Appl. Opt. |

19. | C. Elster, J. Gerhardt, P. Thomsenschmidt, M. Schulz, and I. Weingartner, “Reconstructing surface profiles from curvature measurements,” Optik (Stuttg.) |

20. | D. L. Fried, “Least-square fitting a wave-front distortion estimate to an array of phase-difference measurements,” J. Opt. Soc. Am. |

21. | R. H. Hudgin, “Wave-front reconstruction for compensated imaging,” J. Opt. Soc. Am. |

22. | R. H. Hudgin, “Optimal wave-front estimation,” J. Opt. Soc. Am. |

23. | B. R. Hunt, “Matrix formulation of the reconstruction of phase values from phase differences,” J. Opt. Soc. Am. |

24. | J. Herrmann, “Least-squares wave front errors of minimum norm,” J. Opt. Soc. Am. |

25. | W. H. Southwell, “Wave-front estimation from wave-front slope measurements,” J. Opt. Soc. Am. |

26. | M. Harker and P. O’Leary, “Least squares surface reconstruction from measured gradient fields,” in IEEE Conference on Computer Vision and Pattern Recognition (2008) pp. 1–7. |

27. | B. Jähne, M. Schmidt, and R. Rocholz, “Combined optical slope/height measurements of short wind waves: principle and calibration,” Meas. Sci. Technol. |

28. | G. Strang, |

29. | T. Davis, “Algorithm 915, SuiteSparseQR: Multifrontal multithreaded rank-revealing sparse QR factorization,” ACM T. Math. Software |

30. | T. Davis, “SuiteSparse,” http://www.cise.ufl.edu/research/sparse/SuiteSparse/. |

31. | N. J. Higham, |

32. | Å. Björck, |

**OCIS Codes**

(010.7340) Atmospheric and oceanic optics : Water

(080.2720) Geometric optics : Mathematical methods (general)

(100.3190) Image processing : Inverse problems

(120.2830) Instrumentation, measurement, and metrology : Height measurements

(120.6660) Instrumentation, measurement, and metrology : Surface measurements, roughness

(240.6700) Optics at surfaces : Surfaces

**ToC Category:**

Instrumentation, Measurement, and Metrology

**History**

Original Manuscript: November 9, 2011

Revised Manuscript: December 4, 2011

Manuscript Accepted: December 6, 2011

Published: January 11, 2012

**Citation**

Jeffrey Koskulics, Steven Englehardt, Steven Long, Yongxiang Hu, and Knut Stamnes, "Method of surface topography retrieval by direct solution of sparse weighted seminormal equations," Opt. Express **20**, 1714-1726 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-2-1714

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### References

- C. Cox and X. Zhang, “Contours of slopes of a rippled water surface,” Opt. Express19(20), 18789–18794 (2011). [CrossRef] [PubMed]
- W. C. Keller and B. L. Gotwols, “Two-dimensional optical measurement of wave slope,” Appl. Opt.22(22), 3476–3478 (1983). [CrossRef] [PubMed]
- X. Zhang and C. S. Cox, “Measuring the two-dimensional structure of a wavy water surface optically: a surface gradient detector,” Exp. Fluids17(4), 225–237 (1994). [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]
- Q. Li, M. Zhao, S. Tang, S. Sun, and J. Wu, “Two-dimensional scanning laser slope gauge: measurements of ocean-ripple structures,” Appl. Opt.32(24), 4590–4597 (1993). [CrossRef] [PubMed]
- B. Jähne, J. Klinke, and S. Waas, “Imaging of short ocean wind waves: a critical theoretical review,” J. Opt. Soc. Am. A11(8), 2197–2209 (1994). [CrossRef]
- W. Munk, “An inconvenient sea truth: spread, steepness, and skewness of surface slopes,” Annu. Rev. Mar. Sci.1(1), 377–415 (2009). [CrossRef] [PubMed]
- A. Freedman, D. McWatters, and M. Spencer, “The Aquarius scatterometer: an active system for measuring surface roughness for sea-surface brightness temperature correction,” presented at the IEEE International Geoscience & Remote Sensing Symposium, Denver, Colorado, July 31- August 4, 2006.
- 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]
- 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]
- S. Kay, J. Hedley, S. Lavender, and A. Nimmo-Smith, “Light transfer at the ocean surface modeled using high resolution sea surface realizations,” Opt. Express19(7), 6493–6504 (2011). [CrossRef] [PubMed]
- R. Klette and K. Schlüns, “Height data from gradient fields,” Proc. SPIE2908, 204–215 (1996). [CrossRef]
- 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.
- R. I. Mclachlan, G. R. W. Quispel, and N. Robidoux, “Geometric integration using discrete gradients,” Philos. Trans. R. Soc. Lond. A357, 1–26 (1998).
- S. W. Bahk, “Highly accurate wavefront reconstruction algorithms over broad spatial-frequency bandwidth,” Opt. Express19(20), 18997–19014 (2011). [CrossRef] [PubMed]
- X. Zhang, “An algorithm for calculating water surface elevations from surface gradient image data,” Exp. Fluids21(1), 43–48 (1996). [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]
- M. Grédiac, “Method for surface reconstruction from slope or curvature measurements of rectangular areas,” Appl. Opt.36(20), 4823–4829 (1997). [CrossRef] [PubMed]
- 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]
- D. L. Fried, “Least-square fitting a wave-front distortion estimate to an array of phase-difference measurements,” J. Opt. Soc. Am.67(3), 370–375 (1977). [CrossRef]
- R. H. Hudgin, “Wave-front reconstruction for compensated imaging,” J. Opt. Soc. Am.67(3), 375–378 (1977). [CrossRef]
- R. H. Hudgin, “Optimal wave-front estimation,” J. Opt. Soc. Am.67(3), 378–382 (1977). [CrossRef]
- B. R. Hunt, “Matrix formulation of the reconstruction of phase values from phase differences,” J. Opt. Soc. Am.69(3), 393–399 (1979). [CrossRef]
- J. Herrmann, “Least-squares wave front errors of minimum norm,” J. Opt. Soc. Am.70(1), 28–35 (1980). [CrossRef]
- W. H. Southwell, “Wave-front estimation from wave-front slope measurements,” J. Opt. Soc. Am.70(8), 998–1006 (1980). [CrossRef]
- M. Harker and P. O’Leary, “Least squares surface reconstruction from measured gradient fields,” in IEEE Conference on Computer Vision and Pattern Recognition (2008) pp. 1–7.
- 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]
- G. Strang, Introduction to Applied Mathematics, 1st ed. (Wellesley-Cambridge Press, 1986).
- T. Davis, “Algorithm 915, SuiteSparseQR: Multifrontal multithreaded rank-revealing sparse QR factorization,” ACM T. Math. Software 38, (2011).
- T. Davis, “SuiteSparse,” http://www.cise.ufl.edu/research/sparse/SuiteSparse/ .
- N. J. Higham, Accuracy and Stability of Numerical Algorithms (SIAM, 1996).
- Å. Björck, Numerical Methods for Least Squares Problems (SIAM, 1996).

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