Abstract

We measured radiance distributions for black lining cloth and copper gauze using the convenient technique of wrapping the materials around a circular cylinder, irradiating it with a parallel light source and collecting the scattered radiance by a digital camera. One family of parallel threads (weave or weft) was parallel to the cylinder generator. The most salient features for such glossy plane weaves are a splitting up of the reflection peak due to the wavy variations in local slopes of the threads around the cylinders and a surface scattering lobe due to the threads that run along the cylinder. These scattering characteristics are quite different from the (off-)specular peaks and lobes that were found before for random rough specular surfaces. The split off-specular reflection is due to the regular structures in our samples of man-made materials. We derived simple approximations for these reflectance characteristics using geometrical optics.

© 2003 Optical Society of America

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References

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    [CrossRef]
  13. J. J. Koenderink, S. C. Pont, “The secret of velvety skin,” Mach. Vision Appl. special issue on Human Modeling, Analysis and Synthesis.
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2000 (2)

1998 (2)

1995 (1)

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

1991 (1)

S. K. Nayar, “Surface reflection: physical and geometrical perspectives,” IEEE Transactions on Pattern Anal. Mach. Intell. 13, 611–634 (1991).
[CrossRef]

1966 (1)

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. A 56, 916–925 (1966).
[CrossRef]

1957 (1)

M. S. Longuet-Higgins, “The statistical analysis of a random, moving surface,” Philos. Trans. R. Soc. London 249, 321–387 (1957).
[CrossRef]

1954 (1)

Beckman, P.

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

Birkebak, R. C.

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. A 56, 916–925 (1966).
[CrossRef]

Cox, C.

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).

Hsia, J. J.

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, Geometrical Considerations and Nomenclature for Reflectance (Natl. Bur. Stand. (U.S.) Monogr.160 (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).

Kappers, A. M. L.

Koenderink, J. J.

Longuet-Higgins, M. S.

M. S. Longuet-Higgins, “The statistical analysis of a random, moving surface,” Philos. Trans. R. Soc. London 249, 321–387 (1957).
[CrossRef]

Lu, R.

Minnaert, M. G. J.

M. G. J. Minnaert, Light and Color in the Outdoors (Springer-Verlag, New York, 1993).
[CrossRef]

Montagu Pollock, B.

B. Montagu Pollock, Light and water. A Study of Reflexion and Colour in River, Lake and Sea (George Bell and Sons, London, 1903).

Munk, W.

Nayar, S. K.

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

S. K. Nayar, “Surface reflection: physical and geometrical perspectives,” IEEE Transactions on Pattern Anal. Mach. Intell. 13, 611–634 (1991).
[CrossRef]

Nicodemus, F. E.

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, Geometrical Considerations and Nomenclature for Reflectance (Natl. Bur. Stand. (U.S.) Monogr.160 (1977).

Oren, M.

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

Pont, S. C.

J. J. Koenderink, S. C. Pont, “The secret of velvety skin,” Mach. Vision Appl. special issue on Human Modeling, Analysis and Synthesis.

Richmond, J. C.

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, Geometrical Considerations and Nomenclature for Reflectance (Natl. Bur. Stand. (U.S.) Monogr.160 (1977).

Sparrow, E. M.

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. A 56, 916–925 (1966).
[CrossRef]

Spizzichino, A.

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

Stavridi, M.

Torrance, K. E.

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. A 56, 916–925 (1966).
[CrossRef]

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 Ginneken, B.

Appl. Opt. (3)

IEEE Transactions on Pattern Anal. Mach. Intell. (1)

S. K. Nayar, “Surface reflection: physical and geometrical perspectives,” IEEE Transactions on Pattern Anal. Mach. Intell. 13, 611–634 (1991).
[CrossRef]

Int. J. Comput. Vision (1)

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

J. Opt. Soc. Am. (1)

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

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. A 56, 916–925 (1966).
[CrossRef]

Perception (1)

J. J. Koenderink, “Trieste in the mirror,” Perception 29, 127–133 (2000).
[CrossRef] [PubMed]

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

M. S. Longuet-Higgins, “The statistical analysis of a random, moving surface,” Philos. Trans. R. Soc. London 249, 321–387 (1957).
[CrossRef]

Other (6)

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

M. G. J. Minnaert, Light and Color in the Outdoors (Springer-Verlag, New York, 1993).
[CrossRef]

B. Montagu Pollock, Light and water. A Study of Reflexion and Colour in River, Lake and Sea (George Bell and Sons, London, 1903).

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, Geometrical Considerations and Nomenclature for Reflectance (Natl. Bur. Stand. (U.S.) Monogr.160 (1977).

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

J. J. Koenderink, S. C. Pont, “The secret of velvety skin,” Mach. Vision Appl. special issue on Human Modeling, Analysis and Synthesis.

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Figures (9)

Fig. 1
Fig. 1

Schematic depiction of the experimental setup; the different components of the setup were scaled differently in this figure. The cylindrical sample with a diameter of 4.6 cm was positioned upright, at the center of a frame with two arms that could be rotated around that center. The collimated lightsource was placed at one arm and the camera at the other arm, such that the phase angle could be varied. Here we show the positions of the camera that were used for the empirical study: 10°, 60° and 110°.

Fig. 2
Fig. 2

Photographs of 4.6 cm wide cylinders covered with silk (upper row), fine copper gauze (middle row) and coarse copper gauze (lower row), which are all plane weaves. The pictures were taken for phase angles between the collimated light source and the camera of 10°, 60°, and 110°. With these photographs we show the pixelvalues that were averaged over columns parallel to the cylinder axis. These values reflect radiance, because the camera has a linear response. It is clear that the specular lobe is bounded by a split (off-)specular peak. The peak is broader if the gauze is coarser. Another characteristic of the reflectance is the surface scattering at grazing illumination and viewing angles, which can be seen at both the left-hand and right-hand sides of the cylinders for a phase angle of 10° and only at the left-hand sides for the other phase angles. The fine structure in the graphs for the coarse copper is due to the geometrical structure of the gauze (the effect of individual wires being visible), not to noise.

Fig. 3
Fig. 3

Cangiante silk (or “shot silk”) composited from reddish and greenish threads (note that a red and green appearance doesn’t mean that isolated threads would show up as pure red and pure green in RGB-levels). The photographs were both taken of the same piece of silk, which was wrapped around a 4.6-cm diameter cylinder, the only difference being that the orientation of the cloth differs by 90°. The phase angle was 10°. Instead of the gray levels of the pixels (as in Fig. 2) we now depict the separate R, G, and B levels, in the first row of the graphs. In the lower row we show the ratio between the G and R levels. For one orientation we find that the color of the doubled reflection peak in the middle is reddish, while the surface scattering at grazing angles has a greenish color (left-hand side column). For the other orientation of the cloth we find reversed colors (right-hand side column). The splitting up of the specular peak is clear in one case (the right-hand side one), but in the other case (the left-hand side one) it can hardly be detected. This difference is due to geometrical differences of warp and weft threads.

Fig. 4
Fig. 4

Schematic representation of the geometry. We assume that the woven silk and copper gauze can be represented by a composition of sinusoidal threads. Here, we consider a single thread. If a ray hits the thread from a direction i (under an angle θ with respect to the global surface normal) at a position with local surface normal n, it will be reflected in the direction j (under an angle ψ with respect to the global normal).

Fig. 5
Fig. 5

Graphs of a sine wave with amplitude 0.2 and wavelength 2π (top left-hand side), the exit angle ψ as a function of the position of reflection on the sine wave for three different angles of illumination (middle left-hand side) and the derivative of the former graph (lower left-hand side). At the right we show the resulting radiance as a function of viewing angle for illumination angles 0° (top right-hand side), 30° (middle right-hand side) and 60° (lower right-hand side).

Fig. 6
Fig. 6

This figure depicts the mechanism behind the splitting up of the specular reflection peak for a wave-like structure on a plane (top figure) or a wave-like structure that is wrapped around a cylinder (top view in lower figure). In the top figure light rays hit the surface from the direction depicted by the gray arrow (and separate rays by the gray lines) and scatter in several directions (black lines). The reflected radiance has a bimodal structure in the far field with maxima represented by the black arrows. Consider light rays from a collimated beam that illuminate the cylinder head-on (gray lines), and assume the viewing direction is also head-on. (It is easy to extrapolate the ideas to other geometrical lay outs.) The rays that will hit the cylinder at the position at which the global normal is perpendicular to the illumination direction (P s ) will scatter over quite a large angle due to the large variation of the local normal. There are two areas for which the spread of the scattered rays is very small (far-field caustics) at both sides of P s : the two specular peaks. At these locations the rays reflect at the inflection points of the sinusoid which represent almost planar (zero curvature) facets.

Fig. 7
Fig. 7

Depiction of the geometrical layout for the longitudinal threads. Rays reflect on a surface element dS with principal radii of curvature that are determined by the curvature of the thread R and the thickness of the thread d into a solid angle dΩ.

Fig. 8
Fig. 8

Scattered luminance as a function of the azimuthal angle around the cylinder for threads along the cylinder.

Fig. 9
Fig. 9

Examples of split off-specular peaks in three common objects. The first photograph shows a draped glossy net curtain, which is composed from red and blue threads. Two details in which the split off-specular peaks are clearly visible are shown enlarged at the right. The specular reflections are red or blue, depending on the shape of the cloth and the direction of the threads. For this layout, approximately, red specular peaks are found at vertical folds and blue at horizontal folds. (And indeed the red threads run horizontally and the blue vertically.) The second example is an eggcup with a rippled surface, for which the split specular reflection peaks are found to be quite far apart. (And indeed the maximum local attitudes at the surface were quite large.) The last example is a tea strainer, for which we find two perpendicular split reflection stripes. (And indeed the convex shaped gauze is basically composed of perpendicularly woven threads.)

Equations (5)

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

I=E0i · nψx.
ψx=|2hx|1+hx2
I=E01+hx21/2-hxsin θ+cos θ|2hx|.
I=E0cosψ-θ22k cos2ψ+θ2h02k2-tan2ψ+θ21/2,
dΩ=4dS cos θr1r2.

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