Abstract

We derive the bidirectional reflectance distribution function for a class of opaque surfaces that are rough on a macroscale and smooth on a microscale. We model this type of surface as a distribution of spherical mirrors. Since our study concerns geometrical optics, it is only the aperture of the concavities that is relevant, not the dimension. The three-dimensional problem is effectively transformed into a much simpler two-dimensional one involving the possibly infinitely many reflections in a spherical mirror. We find that these types of surface show very strong backscattering when the pits are deep but forward scattering when the pits are shallow. Such surfaces also show spectral effects as a result of multiple reflections and polarization effects that are due to the orientation of the effective surface. Both this model and the locally diffuse thoroughly pitted surface model [Int. J. Comput. Vision 31, 129 (1999)] are superior to other models in that they allow for an exact treatment for physically realizable surface geometries.

© 2002 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. A. Gershun, “The light field,” J. Math. Phys.18, 51–151 (1939) (P. Moon, G. Timoshenko, transl.).
  2. P. Moon, D. E. Spencer, The Photic Field (MIT Press, Cambridge, Mass., 1981).
  3. F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, T. Limperis, “Geometrical considerations and nomenclature for reflectance,” NBS Monograph 160 (National Bureau of Standards, Washington, D.C., 1977).
  4. J. J. Koenderink, A. J. van Doorn, K. J. Dana, S. Nayar, “Bidirectional reflection distribution function of thoroughly pitted surfaces,” Int. J. Comput. Vision 31, 129–144 (1999).
    [CrossRef]
  5. J. H. Lambert, Photometria Sive de Mensure de Gradibus Luminis, Colorum et Umbræ (Eberhard Klett, Augsburg, Germany, 1760).
  6. CUReT, “Columbia–Utrecht Reflectance and Texture Database,” http://www.cs.columbia.edu/CAVE/curet .
  7. M. Oren, S. K. Nayar, “Generalization of the Lambertian model and implications for machine vision,” Int. J. Comput. Vis. 14, 227–251 (1995).
    [CrossRef]
  8. M. Born, E. Wolf, Principles of Optics (Cambridge U. Press, Cambridge, UK, 1998).
  9. E. Hecht, Optics (Addison-Wesley, Reading, Mass., 2002).
  10. J. F. Blinn, “Models of light reflection for computer synthesized pictures,” Comput. Graph. 11, 192–198 (1977).
    [CrossRef]
  11. J. D. Foley, A. van Dam, S. K. Feiner, J. F. Hughes, Computer Graphics: Principles and Practice (Addison-Wesley, Reading, Mass., 1990).
  12. B. T. Phong, “Illumination for computer generated pictures,” Commun. ACM 18, 311–317 (1975).
    [CrossRef]
  13. S. K. Nayar, M. Oren, “Visual appearance of matte surfaces,” Science 267, 1153–1156 (1995).
    [CrossRef] [PubMed]
  14. K. E. Torrance, E. M. Sparrow, R. C. Birkebak, “Polarization, directional distribution, and off-specular peak phenomena in light reflected from roughened surfaces,” J. Opt. Soc. Am. 56, 916–925 (1966).
    [CrossRef]
  15. B. van Ginneken, M. Stavridi, J. J. Koenderink, “Diffuse and specular reflection from rough surfaces,” Appl. Opt. 37, 130–139 (1998).
    [CrossRef]
  16. J. J. Koenderink, A. J. van Doorn, “Geometrical modes as a general method to treat diffuse interreflections in radiometry,” J. Opt. Soc. Am. 73, 843–850 (1983).
    [CrossRef]
  17. D. Forsyth, A. Zisserman, “Mutual illumination,” in Proceedings of the Conference Computer Vision and Pattern Recognition (IEEE Computer Society Press, Washington, D.C., 1989), pp. 466–473.
  18. P. Beckman, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, New York, 1963).
  19. G. J. Klinker, S. A. Shafer, T. Kanade, “Using a color reflection model to separate highlights from object color,” in Proceedings of the First International Conference on Computer Vision (IEEE Computer Society Press, Washington, D.C., 1987), pp. 145–150.
  20. S. A. Shafer, “Using color to separate reflection components,” Color Res. Appl. 10, 210–218 (1985).
    [CrossRef]
  21. L. B. Wolff, “Relative brightness of specular and diffuse reflection,” Opt. Eng. 33, 285–293 (1994).
    [CrossRef]
  22. B. W. Hapke, R. M. Nelson, W. D. Smythe, “The opposition effect of the moon: the contribution of coherent backscatter,” Science 260, 509–511 (1993).
    [CrossRef] [PubMed]
  23. E. V. Petrova, K. Jockers, N. N. Kiselev, “A negative branch of polarization for comets and atmosphereless celestial bodies and the light scattering by aggregate particles,” Sol. Syst. Res. (USSR) 35, 390–399 (2001).
    [CrossRef]
  24. M. Wolff, “Polarization of light reflected from rough planetary surface,” Appl. Opt. 14, 1395–1405 (1975).
    [CrossRef] [PubMed]

2001

E. V. Petrova, K. Jockers, N. N. Kiselev, “A negative branch of polarization for comets and atmosphereless celestial bodies and the light scattering by aggregate particles,” Sol. Syst. Res. (USSR) 35, 390–399 (2001).
[CrossRef]

1999

J. J. Koenderink, A. J. van Doorn, K. J. Dana, S. Nayar, “Bidirectional reflection distribution function of thoroughly pitted surfaces,” Int. J. Comput. Vision 31, 129–144 (1999).
[CrossRef]

1998

1995

S. K. Nayar, M. Oren, “Visual appearance of matte surfaces,” Science 267, 1153–1156 (1995).
[CrossRef] [PubMed]

M. Oren, S. K. Nayar, “Generalization of the Lambertian model and implications for machine vision,” Int. J. Comput. Vis. 14, 227–251 (1995).
[CrossRef]

1994

L. B. Wolff, “Relative brightness of specular and diffuse reflection,” Opt. Eng. 33, 285–293 (1994).
[CrossRef]

1993

B. W. Hapke, R. M. Nelson, W. D. Smythe, “The opposition effect of the moon: the contribution of coherent backscatter,” Science 260, 509–511 (1993).
[CrossRef] [PubMed]

1985

S. A. Shafer, “Using color to separate reflection components,” Color Res. Appl. 10, 210–218 (1985).
[CrossRef]

1983

1977

J. F. Blinn, “Models of light reflection for computer synthesized pictures,” Comput. Graph. 11, 192–198 (1977).
[CrossRef]

1975

B. T. Phong, “Illumination for computer generated pictures,” Commun. ACM 18, 311–317 (1975).
[CrossRef]

M. Wolff, “Polarization of light reflected from rough planetary surface,” Appl. Opt. 14, 1395–1405 (1975).
[CrossRef] [PubMed]

1966

Beckman, P.

P. Beckman, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, New York, 1963).

Birkebak, R. C.

Blinn, J. F.

J. F. Blinn, “Models of light reflection for computer synthesized pictures,” Comput. Graph. 11, 192–198 (1977).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics (Cambridge U. Press, Cambridge, UK, 1998).

Dana, K. J.

J. J. Koenderink, A. J. van Doorn, K. J. Dana, S. Nayar, “Bidirectional reflection distribution function of thoroughly pitted surfaces,” Int. J. Comput. Vision 31, 129–144 (1999).
[CrossRef]

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).

Forsyth, D.

D. Forsyth, A. Zisserman, “Mutual illumination,” in Proceedings of the Conference Computer Vision and Pattern Recognition (IEEE Computer Society Press, Washington, D.C., 1989), pp. 466–473.

Gershun, A.

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

Ginsberg, I. W.

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, T. Limperis, “Geometrical considerations and nomenclature for reflectance,” NBS Monograph 160 (National Bureau of Standards, Washington, D.C., 1977).

Hapke, B. W.

B. W. Hapke, R. M. Nelson, W. D. Smythe, “The opposition effect of the moon: the contribution of coherent backscatter,” Science 260, 509–511 (1993).
[CrossRef] [PubMed]

Hecht, E.

E. Hecht, Optics (Addison-Wesley, Reading, Mass., 2002).

Hsia, J. J.

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, T. Limperis, “Geometrical considerations and nomenclature for reflectance,” NBS Monograph 160 (National Bureau of Standards, Washington, D.C., 1977).

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).

Jockers, K.

E. V. Petrova, K. Jockers, N. N. Kiselev, “A negative branch of polarization for comets and atmosphereless celestial bodies and the light scattering by aggregate particles,” Sol. Syst. Res. (USSR) 35, 390–399 (2001).
[CrossRef]

Kanade, T.

G. J. Klinker, S. A. Shafer, T. Kanade, “Using a color reflection model to separate highlights from object color,” in Proceedings of the First International Conference on Computer Vision (IEEE Computer Society Press, Washington, D.C., 1987), pp. 145–150.

Kiselev, N. N.

E. V. Petrova, K. Jockers, N. N. Kiselev, “A negative branch of polarization for comets and atmosphereless celestial bodies and the light scattering by aggregate particles,” Sol. Syst. Res. (USSR) 35, 390–399 (2001).
[CrossRef]

Klinker, G. J.

G. J. Klinker, S. A. Shafer, T. Kanade, “Using a color reflection model to separate highlights from object color,” in Proceedings of the First International Conference on Computer Vision (IEEE Computer Society Press, Washington, D.C., 1987), pp. 145–150.

Koenderink, J. J.

Lambert, J. H.

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

Limperis, T.

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, T. Limperis, “Geometrical considerations and nomenclature for reflectance,” NBS Monograph 160 (National Bureau of Standards, Washington, D.C., 1977).

Moon, P.

P. Moon, D. E. Spencer, The Photic Field (MIT Press, Cambridge, Mass., 1981).

Nayar, S.

J. J. Koenderink, A. J. van Doorn, K. J. Dana, S. Nayar, “Bidirectional reflection distribution function of thoroughly pitted surfaces,” Int. J. Comput. Vision 31, 129–144 (1999).
[CrossRef]

Nayar, S. K.

M. Oren, S. K. Nayar, “Generalization of the Lambertian model and implications for machine vision,” Int. J. Comput. Vis. 14, 227–251 (1995).
[CrossRef]

S. K. Nayar, M. Oren, “Visual appearance of matte surfaces,” Science 267, 1153–1156 (1995).
[CrossRef] [PubMed]

Nelson, R. M.

B. W. Hapke, R. M. Nelson, W. D. Smythe, “The opposition effect of the moon: the contribution of coherent backscatter,” Science 260, 509–511 (1993).
[CrossRef] [PubMed]

Nicodemus, F. E.

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, T. Limperis, “Geometrical considerations and nomenclature for reflectance,” NBS Monograph 160 (National Bureau of Standards, Washington, D.C., 1977).

Oren, M.

M. Oren, S. K. Nayar, “Generalization of the Lambertian model and implications for machine vision,” Int. J. Comput. Vis. 14, 227–251 (1995).
[CrossRef]

S. K. Nayar, M. Oren, “Visual appearance of matte surfaces,” Science 267, 1153–1156 (1995).
[CrossRef] [PubMed]

Petrova, E. V.

E. V. Petrova, K. Jockers, N. N. Kiselev, “A negative branch of polarization for comets and atmosphereless celestial bodies and the light scattering by aggregate particles,” Sol. Syst. Res. (USSR) 35, 390–399 (2001).
[CrossRef]

Phong, B. T.

B. T. Phong, “Illumination for computer generated pictures,” Commun. ACM 18, 311–317 (1975).
[CrossRef]

Richmond, J. C.

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, T. Limperis, “Geometrical considerations and nomenclature for reflectance,” NBS Monograph 160 (National Bureau of Standards, Washington, D.C., 1977).

Shafer, S. A.

S. A. Shafer, “Using color to separate reflection components,” Color Res. Appl. 10, 210–218 (1985).
[CrossRef]

G. J. Klinker, S. A. Shafer, T. Kanade, “Using a color reflection model to separate highlights from object color,” in Proceedings of the First International Conference on Computer Vision (IEEE Computer Society Press, Washington, D.C., 1987), pp. 145–150.

Smythe, W. D.

B. W. Hapke, R. M. Nelson, W. D. Smythe, “The opposition effect of the moon: the contribution of coherent backscatter,” Science 260, 509–511 (1993).
[CrossRef] [PubMed]

Sparrow, E. M.

Spencer, D. E.

P. Moon, D. E. Spencer, The Photic Field (MIT Press, Cambridge, Mass., 1981).

Spizzichino, A.

P. Beckman, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, New York, 1963).

Stavridi, M.

Torrance, K. E.

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, K. J. Dana, S. Nayar, “Bidirectional reflection distribution function of thoroughly pitted surfaces,” Int. J. Comput. Vision 31, 129–144 (1999).
[CrossRef]

J. J. Koenderink, A. J. van Doorn, “Geometrical modes as a general method to treat diffuse interreflections in radiometry,” J. Opt. Soc. Am. 73, 843–850 (1983).
[CrossRef]

van Ginneken, B.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Cambridge U. Press, Cambridge, UK, 1998).

Wolff, L. B.

L. B. Wolff, “Relative brightness of specular and diffuse reflection,” Opt. Eng. 33, 285–293 (1994).
[CrossRef]

Wolff, M.

Zisserman, A.

D. Forsyth, A. Zisserman, “Mutual illumination,” in Proceedings of the Conference Computer Vision and Pattern Recognition (IEEE Computer Society Press, Washington, D.C., 1989), pp. 466–473.

Appl. Opt.

Color Res. Appl.

S. A. Shafer, “Using color to separate reflection components,” Color Res. Appl. 10, 210–218 (1985).
[CrossRef]

Commun. ACM

B. T. Phong, “Illumination for computer generated pictures,” Commun. ACM 18, 311–317 (1975).
[CrossRef]

Comput. Graph.

J. F. Blinn, “Models of light reflection for computer synthesized pictures,” Comput. Graph. 11, 192–198 (1977).
[CrossRef]

Int. J. Comput. Vis.

M. Oren, S. K. Nayar, “Generalization of the Lambertian model and implications for machine vision,” Int. J. Comput. Vis. 14, 227–251 (1995).
[CrossRef]

Int. J. Comput. Vision

J. J. Koenderink, A. J. van Doorn, K. J. Dana, S. Nayar, “Bidirectional reflection distribution function of thoroughly pitted surfaces,” Int. J. Comput. Vision 31, 129–144 (1999).
[CrossRef]

J. Opt. Soc. Am.

Opt. Eng.

L. B. Wolff, “Relative brightness of specular and diffuse reflection,” Opt. Eng. 33, 285–293 (1994).
[CrossRef]

Science

B. W. Hapke, R. M. Nelson, W. D. Smythe, “The opposition effect of the moon: the contribution of coherent backscatter,” Science 260, 509–511 (1993).
[CrossRef] [PubMed]

S. K. Nayar, M. Oren, “Visual appearance of matte surfaces,” Science 267, 1153–1156 (1995).
[CrossRef] [PubMed]

Sol. Syst. Res. (USSR)

E. V. Petrova, K. Jockers, N. N. Kiselev, “A negative branch of polarization for comets and atmosphereless celestial bodies and the light scattering by aggregate particles,” Sol. Syst. Res. (USSR) 35, 390–399 (2001).
[CrossRef]

Other

D. Forsyth, A. Zisserman, “Mutual illumination,” in Proceedings of the Conference Computer Vision and Pattern Recognition (IEEE Computer Society Press, Washington, D.C., 1989), pp. 466–473.

P. Beckman, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, New York, 1963).

G. J. Klinker, S. A. Shafer, T. Kanade, “Using a color reflection model to separate highlights from object color,” in Proceedings of the First International Conference on Computer Vision (IEEE Computer Society Press, Washington, D.C., 1987), pp. 145–150.

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

CUReT, “Columbia–Utrecht Reflectance and Texture Database,” http://www.cs.columbia.edu/CAVE/curet .

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

P. Moon, D. E. Spencer, The Photic Field (MIT Press, Cambridge, Mass., 1981).

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, T. Limperis, “Geometrical considerations and nomenclature for reflectance,” NBS Monograph 160 (National Bureau of Standards, Washington, D.C., 1977).

M. Born, E. Wolf, Principles of Optics (Cambridge U. Press, Cambridge, UK, 1998).

E. Hecht, Optics (Addison-Wesley, Reading, Mass., 2002).

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

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 (11)

Fig. 1
Fig. 1

Three-dimensional geometrical layout (far field). Rays are confined to planes spanned by the incident and exit beams, i and j. One of these planes is represented here, spanned by the dashed arc along j, n, i, and k. The plane is defined relative to the frame vectors {ex,ey,ez}, where ex and ey span the plane of the orifice and ez denotes the unit outward normal of the nominal surface (the XY plane). The unit vector n is the three-dimensional representation of the normal in the two-dimensional case; it is in the plane of the ray and closest to ez. The vector k is a unit vector in the nodal line (the intersection of the nominal plane with the plane of the ray). The angle subtended by k and ex is denoted as α.

Fig. 2
Fig. 2

Two-dimensional geometry of the far field of a specular spherical concavity. We represent the far field as the unit sphere centered at the origin. This is a stereographic projection, from the positive z domain (ez). The frame vectors {ex, ey, ez}, the incident and exit beams i and j, and the unit vectors k and n that define the frame of reference for the two-dimensional geometry are shown as black points. The entrance and exit directions are defined by the angles θi, θe, and φe. To find the other angles, we use straightforward trigonometry in the general spherical (gray) triangles ijez and ikex.

Fig. 3
Fig. 3

Geometry in the ijez plane (near field). Two rays arrive at the spherical mirror under an angle λi, at positions x and x+dx in the nominal plane. For reasons of clarity we have depicted rays that reflect only once. These rays subtend an angle dθe on exit. The lower left and right figures show an example of a second-order ray for arbitrary angles λi and λe for (respectively) the positive and negative senses of rotation.

Fig. 4
Fig. 4

Representation of contributions to the reflectance in the case of four incident directions: 1°, 30°, 60°, and 85°. For each incident beam we have split up the traced rays according to the number of reflections, ranging from one (first row) to three (third row) and in columns according to the sign of xi (which determines the sense of rotation, as can be seen).

Fig. 5
Fig. 5

Two-dimensional caustics for four incident directions: 1°, 30°, 60°, and 85° (from left to right). The rows represent hemispherical pits with apertures of 180° (upper row) and shallower pits with apertures of 135° (second row), 90° (third row), and 45° (fourth row).

Fig. 6
Fig. 6

Representation of possible combinations of incident (horizontal axis) and exit (vertical axis) directions for n-times reflected rays (for hemispherical pits) for the positive sense of rotation. The tetragons, each of which is made up of two unequal triangles, represent the outer boundaries of the areas in which the ray geometries are found for the case of a constant number of two or more reflections. The case of only one reflection is represented by the outermost (black) triangle. Near the center of the image and plotted on top of the outermost triangle, the image shows the tetragon for two reflections, for three reflections, etc. Thus in the center of the figure we find that for normal illumination and viewing there are rays that are reflected once, twice, up to infinitely many times. For very shallow angles we find only a few possible combinations (if there are any at all).

Fig. 7
Fig. 7

Examples of ray tracings for normal incidence. The upper row depicts cases that represent the generators of circular cylinders concentric with the axis of symmetry. Viewed normally, these cases show up as concentric circles, with radii ρn=cos π/(2n) and ϕ=π/n (see in Fig. 11 the black circles in the left image). The lower row shows the lower boundaries between the second- and first-order reflections and between the third- and second-order reflections, for which, in general, we find ρn=cos π/(2n+1) and ϕ=2π/(2n+1).

Fig. 8
Fig. 8

Stereographical projections of BRDFs for incident angles of 1°, 30°, 60°, and 85° (the direction of incidence is indicated by a dot, a full hemisphere is shown, and the surface normal direction is at the center), as a function of exit direction. The results are shown for surfaces with hemispherical pits and correspond to the situations depicted in Fig. 4. The most salient features are the strong retroreflection and the distinguishable contributions of the different orders of reflections (for instance, the first plot is actually a superposition of several circular regions). Note that for very shallow angles of incidence we find “surface scattering” (the white ring in the right plot). Rays scattered in this way typically deviate strongly from the nominal plane of incidence.

Fig. 9
Fig. 9

Subsets of BRDFs for the plane of incidence. The various columns depict the results for incident angles of 1°, 30°, 60°, and 85° (which are represented by the black arrows), and the rows depict subsets for spherical pits of apertures of 180° (hemispherical), 135°, 90°, 45°, and 5°. Thus, the first row shows subsets of the data in figure 8. These graphs clarify the manner in which the retroreflector changes into a forward reflector. The quantitative changes are gradual, although the quality of the reflection (retro versus forward) changes suddenly, at an aperture of 90°.

Fig. 10
Fig. 10

Upper left photograph shows a hemispherical pit in a black Perspex block that was photographed by use of a polarization filter in the normal direction while illuminated with a hemispherical diffuse source (see diagram). The black dot in the middle is the lens of the camera, and the three white spots are hot spots of the diffuse light sources (in the diagram we show only two light sources). Except for the hot spots the source was a uniform luminous plane of infinite extent (the gray lines in the diagram were actually the inside of a closed hemisphere that was painted white; reflexes do the rest). The first, light-gray circle in the photograph results from rays that are reflected once, the first ring around the circle results from twice-reflected rays, the second ring results from third reflections, etc. The lower left image is a version distorted from polar to rectangular coordinates. From the latter image we determined the gray values (arbitrary measure) along each ring in the original image (scan lines in the deformed image) and averaged per column and per order of reflection (up to three). These values are depicted in the graph. The variations that are due to the polarization can be seen clearly (also visible, unfortunately, is modulation owing to the hot spots). The first order is the brightest. The higher orders are highly polarized.

Fig. 11
Fig. 11

On the left is a representation of the backscattered beam. The circles represent the second, third, fourth, . . . reflections (the first-order reflection is a point in the center of the circles). The interior of the ellipse indicates the part of the texture that would be visible if the illumination and viewing angle were shallow. The ellipse has a long axis of 1 and a short axis that is equal to the cosine of the viewing (and illumination) angle. The angles ϕ (here drawn only for the second-order reflections) define the parts of the circles that are visible. Basically, in the case of backscatter conditions, integration of the Fresnel reflections over these angles gives us the degree of linear polarization of the beam as a function of the elevation. The result is shown in the graph on the right.

Equations (34)

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

j=cos Θi+cos Ψ sin Θi+sin Ψ sin Θey.
N=dΦ/dE=H0cos ϑiπR2cos ϑe J(dA, dΩ),
f(i, j)=N/H=1πR2cos ϑe J(dA, dΩ).
J(dA, dΩ)=ρsin ΘdρdΘdαdΨ,
f(i, j)=(dα/dΨ)π cos ϑesin Θi=1 r(ni, μi)xidxidΘ,
Θ=arccos(cos ϑicos ϑe+sin ϑisin ϑecos φe),
Ψ=arccoscos ϑe-cos Θ cos ϑisin Θ sin ϑi.
η=arccos(sin ϑisin Ψ),
α=arctan(cos ϑitan Ψ),
λi=arctan(cos Ψ tan ϑi).
dαdΨ=cos ϑi1-sin2 Ψ2sin2 ϑi.
|xi,e|=cos(ϕ/2)cos λi,e.
ϕn+=π+(λe-λi)n.
ψn+=λi+(n-12)ϕn+
ψn+¯=-λe+(n-12)ϕn+
(π-ϕn+<ψn+<π)(π-ϕn+<ψn+¯<π)
[(n1)(λe-λi<0)].
xidxidλe=xi(dxi/dϕn+)(dλe/dϕn+)=sin ϕn+4n cos2 λi.
r(n, μn+) sin ϕn+4n cos2 λi
ϕn-=π-(λe-λi)n.
ψn-=-λi+n-12ϕn-
f(i, j)=(dα/dΨ)π cos ϑesin Θn,σχ(λi,λe)r(n, μnσ)sin ϕnσ4n cos2 λi.
|J(dA, dΩ)|=1/4.
f1(ϑe)=χ1(ϑe)4π cos ϑe,
|J(dA, dΩ)|=sin[(π+ϑe)/n]4n sin ϑe,
fn(ϑe)=χn(ϑe)sin[(π+ϑe)/n]2πn sin 2ϑe,
f(0, ϑe)=χ1(ϑe)4π cos ϑe+n=2χn(θe)sin[(π+ϑe)/n]2πn sin 2ϑe.
P(μ, n)=F(μ)n-F(μ)nF(μ)n+F(μ)n,
F(μ)=sin(μ-μ¯)sin(μ+μ¯)2,
F(μ)=tan(μ-μ¯)tan(μ+μ¯)2,
n=π2 arccoscos θ.
(ρn)2cos ϕ2+sin ϕ2cos θ=1,
ϕ(θ, n)=arccos(-1+ρn2sec θ2)1/2ρn(-1+sec θ2)1/2.
P(θ)=n+1sin(π/n)n (F-F)sin 2ϕπ2nsin(π/n)n (F+F)+2n+1sin(π/n)n (F+F)ϕ.

Metrics