Abstract

We present a theory of image texture resulting from the shading of corrugated (three-dimensional textured) surfaces, Lambertian on the micro scale, in the domain of geometrical optics. The derivation applies to isotropic Gaussian random surfaces, under collimated illumination, in normal view. The theory predicts the structure tensors from either the gradient or the Hessian of the image intensity and allows inferences of the direction of irradiation of the surface. Although the assumptions appear prima facie rather restrictive, even for surfaces that are not at all Gaussian, with the bidirectional reflectance distribution function far from Lambertian and vignetting and multiple scattering present, we empirically recover the direction of irradiation with an accuracy of a few degrees.

© 2003 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. M. J. Chantler, “Why illuminant direction is fundamental in texture analysis,” IEEE Proc. Vision Image Signal Process. 142, 199–206 (1995).
    [CrossRef]
  2. T. Leung, J. Malik, “Representing and recognizing the visual appearance of materials using three-dimensional textons,” Int. J. Comput. Vision 43, 29–44 (2001).
    [CrossRef]
  3. M. J. Chantler, G. McGunnigle, “The response of texture features to illuminant rotation,” in 15th International Conference on Pattern Recognition, ICPR 2000 (IEEE Cs Press, Barcelona, Spain, 2000), pp. 955–958.
  4. A. Gershun, “The light field,” transl. by P. Moon, G. Timoshenko, J. Math. Phys. 18, 51–151 (1939).
  5. B. van Ginneken, J. J. Koenderink, K. J. Dana, “Texture histograms as a function of irradiation and viewing direction,” Int. J. Comput. Vision 31, 169–184 (1999).
    [CrossRef]
  6. J. D. Foley, A. van Dam, S. K. Feiner, J. F. Hughes, Computer Graphics, Principles and Practice (Addison-Wesley, Reading, Mass., 1990).
  7. J. H. Lambert, Photometria Sive de Mensure de Gradibus Luminis, Colorum et Umbræ (Eberhard Klett, Augsburg, 1760).
  8. B. K. P. Horn, M. J. Brooks, Shape from Shading (MIT Press, Cambridge, Mass., 1989).
  9. M. J. Chantler, M. Schmidt, M. Petrou, G. McGunnigle, “The effect of illuminant rotation on texture filters: Lissajous’s ellipses,” in A. Heyden, G. Sparr, M. Nielsen, P. Johansen, eds., European Conference on Computer Vision, ECCV 2002, Vol. 2352 of Lecture Notes in Computer Science (Springer, Heidelberg, Germany, 2002), pp. 289–303.
    [CrossRef]
  10. J. J. Koenderink, “Image processing done right,” in A. Heyden, G. Sparr, M. Nielsen, P. Johansen, eds., European Conference on Computer Vision, ECCV 2002, Vol. 2350 of Lecture Notes in Computer Science (Springer, Heidelberg, Germany, 2002), pp. 158–172.
    [CrossRef]
  11. M. S. Longuet-Higgins, “The statistical analysis of a random, moving surface,” Philos. Trans. R. Soc. London Ser. A 249, 321–364 (1956).
    [CrossRef]
  12. M. V. Berry, J. H. Hannay, “Umbilic points on Gaussian random surfaces,” J. Phys. A 10, 1809–1821 (1977).
    [CrossRef]
  13. Columbia–Utrecht Reflectance and Texture Database; http://www.cs.columbia.edu/CAVE/curet .
  14. K. J. Dana, S. K. Nayar, B. van Ginneken, J. J. Koenderink, “Reflectance and texture of real-world surfaces,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (Institute of Electrical and Electronics Engineers, New York, 1997), pp. 151–157.
  15. M. J. Wainright, E. P. Simoncelli, “Scale mixtures of Gaussians and the statistics of natural images,” in Advances of Neural Information Processing, S. A. Solla, T. K. Leen, K.-H. Müller, eds. (MIT Press, Cambridge Mass., 2000), Vol. 12, pp. 855–861.
  16. A. Pentland, “Linear shape from shading,” Int. J. Comput. Vision 4, 153–162 (1990).
    [CrossRef]
  17. C. H. Lee, A. Rosenfeld, “Improved methods of estimating shape from shading using the light source coordinate system,” Artif. Intell. 26, 125–143 (1985).
    [CrossRef]
  18. Q. Zheng, R. Chellappa, “Estimation of illuminant direction, albedo, and shape from shading,” IEEE Trans. Pattern Anal. Mach. Intell. 13, 680–702 (1991).
    [CrossRef]
  19. D. C. Knill, “Estimating illuminant direction and degree of surface relief,” J. Opt. Soc. Am. A 7, 759–775 (1990).
    [CrossRef] [PubMed]
  20. J. J. Koenderink, A. J. van Doorn, “Local structure of Gaussian texture,” IEICE Trans. Electron. (to be published).
  21. S. C. Pont, J. J. Koenderink, “Bidirectional texture contrast function,” in A. Heyden, G. Sparr, M. Nielsen, P. Johansen, eds., European Conference on Computer Vision, ECCV 2002, Vol. 2353 of Lecture Notes in Computer Science (Springer, Heidelberg, Germany, 2002), pp. 808–822.
    [CrossRef]

2001 (1)

T. Leung, J. Malik, “Representing and recognizing the visual appearance of materials using three-dimensional textons,” Int. J. Comput. Vision 43, 29–44 (2001).
[CrossRef]

1999 (1)

B. van Ginneken, J. J. Koenderink, K. J. Dana, “Texture histograms as a function of irradiation and viewing direction,” Int. J. Comput. Vision 31, 169–184 (1999).
[CrossRef]

1995 (1)

M. J. Chantler, “Why illuminant direction is fundamental in texture analysis,” IEEE Proc. Vision Image Signal Process. 142, 199–206 (1995).
[CrossRef]

1991 (1)

Q. Zheng, R. Chellappa, “Estimation of illuminant direction, albedo, and shape from shading,” IEEE Trans. Pattern Anal. Mach. Intell. 13, 680–702 (1991).
[CrossRef]

1990 (2)

1985 (1)

C. H. Lee, A. Rosenfeld, “Improved methods of estimating shape from shading using the light source coordinate system,” Artif. Intell. 26, 125–143 (1985).
[CrossRef]

1977 (1)

M. V. Berry, J. H. Hannay, “Umbilic points on Gaussian random surfaces,” J. Phys. A 10, 1809–1821 (1977).
[CrossRef]

1956 (1)

M. S. Longuet-Higgins, “The statistical analysis of a random, moving surface,” Philos. Trans. R. Soc. London Ser. A 249, 321–364 (1956).
[CrossRef]

1939 (1)

A. Gershun, “The light field,” transl. by P. Moon, G. Timoshenko, J. Math. Phys. 18, 51–151 (1939).

A. Gershun, “The light field,” transl. by P. Moon, G. Timoshenko, J. Math. Phys. 18, 51–151 (1939).

Berry, M. V.

M. V. Berry, J. H. Hannay, “Umbilic points on Gaussian random surfaces,” J. Phys. A 10, 1809–1821 (1977).
[CrossRef]

Brooks, M. J.

B. K. P. Horn, M. J. Brooks, Shape from Shading (MIT Press, Cambridge, Mass., 1989).

Chantler, M. J.

M. J. Chantler, “Why illuminant direction is fundamental in texture analysis,” IEEE Proc. Vision Image Signal Process. 142, 199–206 (1995).
[CrossRef]

M. J. Chantler, G. McGunnigle, “The response of texture features to illuminant rotation,” in 15th International Conference on Pattern Recognition, ICPR 2000 (IEEE Cs Press, Barcelona, Spain, 2000), pp. 955–958.

M. J. Chantler, M. Schmidt, M. Petrou, G. McGunnigle, “The effect of illuminant rotation on texture filters: Lissajous’s ellipses,” in A. Heyden, G. Sparr, M. Nielsen, P. Johansen, eds., European Conference on Computer Vision, ECCV 2002, Vol. 2352 of Lecture Notes in Computer Science (Springer, Heidelberg, Germany, 2002), pp. 289–303.
[CrossRef]

Chellappa, R.

Q. Zheng, R. Chellappa, “Estimation of illuminant direction, albedo, and shape from shading,” IEEE Trans. Pattern Anal. Mach. Intell. 13, 680–702 (1991).
[CrossRef]

Dana, K. J.

B. van Ginneken, J. J. Koenderink, K. J. Dana, “Texture histograms as a function of irradiation and viewing direction,” Int. J. Comput. Vision 31, 169–184 (1999).
[CrossRef]

K. J. Dana, S. K. Nayar, B. van Ginneken, J. J. Koenderink, “Reflectance and texture of real-world surfaces,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (Institute of Electrical and Electronics Engineers, New York, 1997), pp. 151–157.

Feiner, S. K.

J. D. Foley, A. van Dam, S. K. Feiner, J. F. Hughes, Computer Graphics, Principles and Practice (Addison-Wesley, Reading, Mass., 1990).

Foley, J. D.

J. D. Foley, A. van Dam, S. K. Feiner, J. F. Hughes, Computer Graphics, Principles and Practice (Addison-Wesley, Reading, Mass., 1990).

Gershun, A.

A. Gershun, “The light field,” transl. by P. Moon, G. Timoshenko, J. Math. Phys. 18, 51–151 (1939).

Hannay, J. H.

M. V. Berry, J. H. Hannay, “Umbilic points on Gaussian random surfaces,” J. Phys. A 10, 1809–1821 (1977).
[CrossRef]

Horn, B. K. P.

B. K. P. Horn, M. J. Brooks, Shape from Shading (MIT Press, Cambridge, Mass., 1989).

Hughes, J. F.

J. D. Foley, A. van Dam, S. K. Feiner, J. F. Hughes, Computer Graphics, Principles and Practice (Addison-Wesley, Reading, Mass., 1990).

Knill, D. C.

Koenderink, J. J.

B. van Ginneken, J. J. Koenderink, K. J. Dana, “Texture histograms as a function of irradiation and viewing direction,” Int. J. Comput. Vision 31, 169–184 (1999).
[CrossRef]

K. J. Dana, S. K. Nayar, B. van Ginneken, J. J. Koenderink, “Reflectance and texture of real-world surfaces,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (Institute of Electrical and Electronics Engineers, New York, 1997), pp. 151–157.

S. C. Pont, J. J. Koenderink, “Bidirectional texture contrast function,” in A. Heyden, G. Sparr, M. Nielsen, P. Johansen, eds., European Conference on Computer Vision, ECCV 2002, Vol. 2353 of Lecture Notes in Computer Science (Springer, Heidelberg, Germany, 2002), pp. 808–822.
[CrossRef]

J. J. Koenderink, “Image processing done right,” in A. Heyden, G. Sparr, M. Nielsen, P. Johansen, eds., European Conference on Computer Vision, ECCV 2002, Vol. 2350 of Lecture Notes in Computer Science (Springer, Heidelberg, Germany, 2002), pp. 158–172.
[CrossRef]

J. J. Koenderink, A. J. van Doorn, “Local structure of Gaussian texture,” IEICE Trans. Electron. (to be published).

Lambert, J. H.

J. H. Lambert, Photometria Sive de Mensure de Gradibus Luminis, Colorum et Umbræ (Eberhard Klett, Augsburg, 1760).

Lee, C. H.

C. H. Lee, A. Rosenfeld, “Improved methods of estimating shape from shading using the light source coordinate system,” Artif. Intell. 26, 125–143 (1985).
[CrossRef]

Leung, T.

T. Leung, J. Malik, “Representing and recognizing the visual appearance of materials using three-dimensional textons,” Int. J. Comput. Vision 43, 29–44 (2001).
[CrossRef]

Longuet-Higgins, M. S.

M. S. Longuet-Higgins, “The statistical analysis of a random, moving surface,” Philos. Trans. R. Soc. London Ser. A 249, 321–364 (1956).
[CrossRef]

Malik, J.

T. Leung, J. Malik, “Representing and recognizing the visual appearance of materials using three-dimensional textons,” Int. J. Comput. Vision 43, 29–44 (2001).
[CrossRef]

McGunnigle, G.

M. J. Chantler, M. Schmidt, M. Petrou, G. McGunnigle, “The effect of illuminant rotation on texture filters: Lissajous’s ellipses,” in A. Heyden, G. Sparr, M. Nielsen, P. Johansen, eds., European Conference on Computer Vision, ECCV 2002, Vol. 2352 of Lecture Notes in Computer Science (Springer, Heidelberg, Germany, 2002), pp. 289–303.
[CrossRef]

M. J. Chantler, G. McGunnigle, “The response of texture features to illuminant rotation,” in 15th International Conference on Pattern Recognition, ICPR 2000 (IEEE Cs Press, Barcelona, Spain, 2000), pp. 955–958.

Moon, P.

A. Gershun, “The light field,” transl. by P. Moon, G. Timoshenko, J. Math. Phys. 18, 51–151 (1939).

Nayar, S. K.

K. J. Dana, S. K. Nayar, B. van Ginneken, J. J. Koenderink, “Reflectance and texture of real-world surfaces,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (Institute of Electrical and Electronics Engineers, New York, 1997), pp. 151–157.

Pentland, A.

A. Pentland, “Linear shape from shading,” Int. J. Comput. Vision 4, 153–162 (1990).
[CrossRef]

Petrou, M.

M. J. Chantler, M. Schmidt, M. Petrou, G. McGunnigle, “The effect of illuminant rotation on texture filters: Lissajous’s ellipses,” in A. Heyden, G. Sparr, M. Nielsen, P. Johansen, eds., European Conference on Computer Vision, ECCV 2002, Vol. 2352 of Lecture Notes in Computer Science (Springer, Heidelberg, Germany, 2002), pp. 289–303.
[CrossRef]

Pont, S. C.

S. C. Pont, J. J. Koenderink, “Bidirectional texture contrast function,” in A. Heyden, G. Sparr, M. Nielsen, P. Johansen, eds., European Conference on Computer Vision, ECCV 2002, Vol. 2353 of Lecture Notes in Computer Science (Springer, Heidelberg, Germany, 2002), pp. 808–822.
[CrossRef]

Rosenfeld, A.

C. H. Lee, A. Rosenfeld, “Improved methods of estimating shape from shading using the light source coordinate system,” Artif. Intell. 26, 125–143 (1985).
[CrossRef]

Schmidt, M.

M. J. Chantler, M. Schmidt, M. Petrou, G. McGunnigle, “The effect of illuminant rotation on texture filters: Lissajous’s ellipses,” in A. Heyden, G. Sparr, M. Nielsen, P. Johansen, eds., European Conference on Computer Vision, ECCV 2002, Vol. 2352 of Lecture Notes in Computer Science (Springer, Heidelberg, Germany, 2002), pp. 289–303.
[CrossRef]

Simoncelli, E. P.

M. J. Wainright, E. P. Simoncelli, “Scale mixtures of Gaussians and the statistics of natural images,” in Advances of Neural Information Processing, S. A. Solla, T. K. Leen, K.-H. Müller, eds. (MIT Press, Cambridge Mass., 2000), Vol. 12, pp. 855–861.

Timoshenko, G.

A. Gershun, “The light field,” transl. by P. Moon, G. Timoshenko, J. Math. Phys. 18, 51–151 (1939).

van Dam, A.

J. D. Foley, A. van Dam, S. K. Feiner, J. F. Hughes, Computer Graphics, Principles and Practice (Addison-Wesley, Reading, Mass., 1990).

van Doorn, A. J.

J. J. Koenderink, A. J. van Doorn, “Local structure of Gaussian texture,” IEICE Trans. Electron. (to be published).

van Ginneken, B.

B. van Ginneken, J. J. Koenderink, K. J. Dana, “Texture histograms as a function of irradiation and viewing direction,” Int. J. Comput. Vision 31, 169–184 (1999).
[CrossRef]

K. J. Dana, S. K. Nayar, B. van Ginneken, J. J. Koenderink, “Reflectance and texture of real-world surfaces,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (Institute of Electrical and Electronics Engineers, New York, 1997), pp. 151–157.

Wainright, M. J.

M. J. Wainright, E. P. Simoncelli, “Scale mixtures of Gaussians and the statistics of natural images,” in Advances of Neural Information Processing, S. A. Solla, T. K. Leen, K.-H. Müller, eds. (MIT Press, Cambridge Mass., 2000), Vol. 12, pp. 855–861.

Zheng, Q.

Q. Zheng, R. Chellappa, “Estimation of illuminant direction, albedo, and shape from shading,” IEEE Trans. Pattern Anal. Mach. Intell. 13, 680–702 (1991).
[CrossRef]

Artif. Intell. (1)

C. H. Lee, A. Rosenfeld, “Improved methods of estimating shape from shading using the light source coordinate system,” Artif. Intell. 26, 125–143 (1985).
[CrossRef]

IEEE Proc. Vision Image Signal Process. (1)

M. J. Chantler, “Why illuminant direction is fundamental in texture analysis,” IEEE Proc. Vision Image Signal Process. 142, 199–206 (1995).
[CrossRef]

IEEE Trans. Pattern Anal. Mach. Intell. (1)

Q. Zheng, R. Chellappa, “Estimation of illuminant direction, albedo, and shape from shading,” IEEE Trans. Pattern Anal. Mach. Intell. 13, 680–702 (1991).
[CrossRef]

Int. J. Comput. Vision (3)

T. Leung, J. Malik, “Representing and recognizing the visual appearance of materials using three-dimensional textons,” Int. J. Comput. Vision 43, 29–44 (2001).
[CrossRef]

B. van Ginneken, J. J. Koenderink, K. J. Dana, “Texture histograms as a function of irradiation and viewing direction,” Int. J. Comput. Vision 31, 169–184 (1999).
[CrossRef]

A. Pentland, “Linear shape from shading,” Int. J. Comput. Vision 4, 153–162 (1990).
[CrossRef]

J. Math. Phys. (1)

A. Gershun, “The light field,” transl. by P. Moon, G. Timoshenko, J. Math. Phys. 18, 51–151 (1939).

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

J. Phys. A (1)

M. V. Berry, J. H. Hannay, “Umbilic points on Gaussian random surfaces,” J. Phys. A 10, 1809–1821 (1977).
[CrossRef]

Philos. Trans. R. Soc. London Ser. A (1)

M. S. Longuet-Higgins, “The statistical analysis of a random, moving surface,” Philos. Trans. R. Soc. London Ser. A 249, 321–364 (1956).
[CrossRef]

Other (11)

M. J. Chantler, G. McGunnigle, “The response of texture features to illuminant rotation,” in 15th International Conference on Pattern Recognition, ICPR 2000 (IEEE Cs Press, Barcelona, Spain, 2000), pp. 955–958.

J. D. Foley, A. van Dam, S. K. Feiner, J. F. Hughes, Computer Graphics, Principles and Practice (Addison-Wesley, Reading, Mass., 1990).

J. H. Lambert, Photometria Sive de Mensure de Gradibus Luminis, Colorum et Umbræ (Eberhard Klett, Augsburg, 1760).

B. K. P. Horn, M. J. Brooks, Shape from Shading (MIT Press, Cambridge, Mass., 1989).

M. J. Chantler, M. Schmidt, M. Petrou, G. McGunnigle, “The effect of illuminant rotation on texture filters: Lissajous’s ellipses,” in A. Heyden, G. Sparr, M. Nielsen, P. Johansen, eds., European Conference on Computer Vision, ECCV 2002, Vol. 2352 of Lecture Notes in Computer Science (Springer, Heidelberg, Germany, 2002), pp. 289–303.
[CrossRef]

J. J. Koenderink, “Image processing done right,” in A. Heyden, G. Sparr, M. Nielsen, P. Johansen, eds., European Conference on Computer Vision, ECCV 2002, Vol. 2350 of Lecture Notes in Computer Science (Springer, Heidelberg, Germany, 2002), pp. 158–172.
[CrossRef]

Columbia–Utrecht Reflectance and Texture Database; http://www.cs.columbia.edu/CAVE/curet .

K. J. Dana, S. K. Nayar, B. van Ginneken, J. J. Koenderink, “Reflectance and texture of real-world surfaces,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (Institute of Electrical and Electronics Engineers, New York, 1997), pp. 151–157.

M. J. Wainright, E. P. Simoncelli, “Scale mixtures of Gaussians and the statistics of natural images,” in Advances of Neural Information Processing, S. A. Solla, T. K. Leen, K.-H. Müller, eds. (MIT Press, Cambridge Mass., 2000), Vol. 12, pp. 855–861.

J. J. Koenderink, A. J. van Doorn, “Local structure of Gaussian texture,” IEICE Trans. Electron. (to be published).

S. C. Pont, J. J. Koenderink, “Bidirectional texture contrast function,” in A. Heyden, G. Sparr, M. Nielsen, P. Johansen, eds., European Conference on Computer Vision, ECCV 2002, Vol. 2353 of Lecture Notes in Computer Science (Springer, Heidelberg, Germany, 2002), pp. 808–822.
[CrossRef]

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (10)

Fig. 1
Fig. 1

In the left column; the top figure shows a protrusion, the bottom figure an indentation, both of generic Gaussian shape. This shows the local shape (height function). The middle column shows the irradiance pattern (in normal view) for θ=3π/8, the top figure the protrusion, the bottom the indentation. In the right column the top figure shows the autocorrelation function for either case, the bottom figure the logarithm of the power spectrum (also for either case). Notice that the shading patterns (image textons) are dipoles of opposite sign. In the autocorrelation and the power spectrum this sign is lost.

Fig. 2
Fig. 2

Irradiated random Gaussian surface photographed frontally, its power-spectrum representation, and its autocorrelation function. For this example, the matrix G2 is {{0.27, -0.06}, {-0.06, 0.73}} and the matrix H2 is {{0.27, -0.06}, {-0.06, 0.72}}, where the traces have been normalized to allow direct comparison of these structure tensors.

Fig. 3
Fig. 3

Rather deep random Gaussian relief irradiated from various elevations. Notice the distinct change of appearance when the cast shadows cut in.

Fig. 4
Fig. 4

Histograms for the examples shown in Fig. 3, in the same sequence. Notice the development of the low-luminance mode as shadows cut in.

Fig. 5
Fig. 5

Average radiance for the examples shown in Fig. 3 with the Lambertian prediction (the curve). The curve has been corrected for the average albedo. In this plot of log radiance as a function of source angle, the Lambert cosine factor describes the average quite well despite the effect of shadows.

Fig. 6
Fig. 6

Traces of G2 (left) and H2 (right) as a function of the obliquity factors cot2 θ in double logarithmic plots for the examples shown in Fig. 3. The prediction by the obliquity factors is quite good except for the cases of extreme shadowing.

Fig. 7
Fig. 7

Example from the Curet database. Top row, normal views; bottom row, radiance histograms. The columns (left to right) are for source elevations of 67.5°, 45.0°, and 22.5°. The horizontal axes span the full pixel intensity range (that is, [0, 1]); and the vertical axes have been arbitrarily scaled.

Fig. 8
Fig. 8

Local estimates for a number of (all of them worst case except the plaster example) examples from the Curet database. The elliptical shapes specify the local estimate by area (trace of the gradient squared), eccentricity (confidence), and orientation of the major axis (estimated irradiation direction). From top left to bottom right, the samples are “15. Aluminum Foil,” “24. Rabbit Fur,” “28. Crumpled Paper,” “30. Plaster b (zoomed),” “35. Painted Spheres,” “43. Salt Crystals.”

Fig. 9
Fig. 9

Results for all frontal views in the Curet database. Left, a histogram of the light directions (the fiducial value being 90°); right, a histogram of the corresponding confidences (the fiducial value being 0.8). Although many of the materials in the database (all “natural surfaces”) are very far from a Lambertian Gaussian surface indeed, all lead to excellent estimations. The insets show the quartiles for the three elevations in the Curet database. Apparently the elevation of the source (and thus the degree of cast shadowing) is of only limited importance.

Fig. 10
Fig. 10

Orientations estimated from samples of the Curet database. All Curet samples in frontal view are in these data. Each sample is represented with three values of the elevation of the source. At top left is a histogram of results from the present method, at top right from Knill’s algorithm, and at bottom from Lee and Rosenfeld’s algorithm.17 The latter algorithm is based on first-order statistics. Notice, however, that it does not escape the 180° ambiguity (because the surface fails to be convex umbilical throughout). The present algorithm performs best, Knill’s is close, and the first-order algorithm is clearly worst. Notice that the estimation works equally well regardless the nature of the surface.

Equations (14)

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

I(x, y)=αI0[sin θ-cos θ(hx cos ϕ+hy sin ϕ)](1+hx2+hy2)1/2,
log I(x, y)=log(αI0 sin θ)-cot θ(hx cos ϕ+hy sin ϕ).
 log I(x, y)=g=-cot θ[(hxx cos ϕ+hyx sin ϕ)ex+(hxy cos ϕ+hyy sin ϕ)ey],
H=-cot θ hxxx cos ϕ+hxxy sin ϕhxxy cos ϕ+hxyy sin ϕhxxy cos ϕ+hxyy sin ϕhxyy cos ϕ+hyyy sin ϕ
G2=gTg, H2=HTH.
p+qhxpyqr+shxrys=p+r+q+shxp+ryq+s ρ(0)=(-1)(p+r+q+s)/2mp+r,q+s,
ρ(r)=h(r0)h(r0+r)=dkexp(ikr)E(k)
muv=dkkxukyvE(k);
muv=Mu+v02πcosu θ sinv θdθ,
Mp=2π0dkkp+1E(k).
hxx2hxxhyyhxxhxyhyyhxxhyy2hyyhxyhxyhxxhxyhyyhxy2=M48310130001,
hxxx2hxxxhxxyhxxxhxyyhxxxhyyyhxxyhxxxhxxy2hxxyhxyyhxxyhyyyhxyyhxxxhxyyhxxyhxyy2hxyyhyyyhyyyhxxxhyyyhxxyhyyyhxyyhyyy2=M6165010010110100105.
S=M2n8cot2 θ2+cos 2ϕsin 2ϕsin 2ϕ2-cos 2ϕ,
c=λ12-λ22λ12+λ22,

Metrics