Abstract

To study the optical properties of materials, one needs a complete set of the angular distribution functions of surface scattering from the materials. Here we present a convenient method for collecting a large set of bidirectional reflectance distribution function (BRDF) samples in the hemispherical scattering space. Material samples are wrapped around a right-circular cylinder and irradiated by a parallel light source, and the scattered radiance is collected by a digital camera. We tilted the cylinder around its center to collect the BRDF samples outside the plane of incidence. This method can be used with materials that have isotropic and anisotropic scattering properties. We demonstrate this method in a detailed investigation of shot fabrics. The warps and the fillings of shot fabrics are dyed different colors so that the fabric appears to change color at different viewing angles. These color-changing characteristics are found to be related to the physical and geometrical structure of shot fabric. Our study reveals that the color-changing property of shot fabrics is due mainly to an occlusion effect.

© 2000 Optical Society of America

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References

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  1. R. Lu, J. J. Koenderink, A. M. L. Kappers, “Optical properties (bidirectional reflectance distribution functions) of velvet,” Appl. Opt. 37, 5974–5984 (1998).
    [CrossRef]
  2. J. Gage, Color and Culture (Little, Brown, Boston, Mass., 1993).
  3. A. M. Muthesius, “From seed to samite: aspects of byzantine silk weaving,” Textile History 20, 135–149 (1989).
  4. A. Santangelo, The Development of Italian Textile Design from the Twelfth to the Eighteenth Century (Zwemmer, London, 1964).
  5. H. Simon, The Splendor of Iridescence: Structural Colors in the Animal World (Dodd, Mead, New York, 1971).
  6. F. E. Nicodemus, J. C. Richmond, J. J. Hsia, “Geometrical considerations and nomenclature for reflectance,” Natl. Bur. Stand. (U.S.) Monogr. 160 (1977).
  7. B. G. Smith, “Geometrical shadowing of a random rough surface,” IEEE Trans. Antennas Propag. 15, 668–671 (1967).
    [CrossRef]
  8. W. G. Egan, T. W. Hilgeman, Optical Properties of Inhomogeneous Material (Academic, New York, 1979).
  9. W. E. K. Middleton, A. G. Mungall, “The luminous directional reflectance of snow,” J. Opt. Soc. Am. 42, 572–579 (1952).
    [CrossRef]
  10. K. E. Torrance, E. M. Sparrow, “Theory for off-specular reflection from roughened surfaces,” J. Opt. Soc. Am. 57, 1105–1114 (1967).
    [CrossRef]
  11. M. G. J. Minnaert, Light and Color in the Outdoors (Springer-Verlag, New York, 1993), Chap. 2.
    [CrossRef]

1998 (1)

1989 (1)

A. M. Muthesius, “From seed to samite: aspects of byzantine silk weaving,” Textile History 20, 135–149 (1989).

1977 (1)

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, “Geometrical considerations and nomenclature for reflectance,” Natl. Bur. Stand. (U.S.) Monogr. 160 (1977).

1967 (2)

B. G. Smith, “Geometrical shadowing of a random rough surface,” IEEE Trans. Antennas Propag. 15, 668–671 (1967).
[CrossRef]

K. E. Torrance, E. M. Sparrow, “Theory for off-specular reflection from roughened surfaces,” J. Opt. Soc. Am. 57, 1105–1114 (1967).
[CrossRef]

1952 (1)

Egan, W. G.

W. G. Egan, T. W. Hilgeman, Optical Properties of Inhomogeneous Material (Academic, New York, 1979).

Gage, J.

J. Gage, Color and Culture (Little, Brown, Boston, Mass., 1993).

Hilgeman, T. W.

W. G. Egan, T. W. Hilgeman, Optical Properties of Inhomogeneous Material (Academic, New York, 1979).

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).

Kappers, A. M. L.

Koenderink, J. J.

Lu, R.

Middleton, W. E. K.

Minnaert, M. G. J.

M. G. J. Minnaert, Light and Color in the Outdoors (Springer-Verlag, New York, 1993), Chap. 2.
[CrossRef]

Mungall, A. G.

Muthesius, A. M.

A. M. Muthesius, “From seed to samite: aspects of byzantine silk weaving,” Textile History 20, 135–149 (1989).

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).

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).

Santangelo, A.

A. Santangelo, The Development of Italian Textile Design from the Twelfth to the Eighteenth Century (Zwemmer, London, 1964).

Simon, H.

H. Simon, The Splendor of Iridescence: Structural Colors in the Animal World (Dodd, Mead, New York, 1971).

Smith, B. G.

B. G. Smith, “Geometrical shadowing of a random rough surface,” IEEE Trans. Antennas Propag. 15, 668–671 (1967).
[CrossRef]

Sparrow, E. M.

Torrance, K. E.

Appl. Opt. (1)

IEEE Trans. Antennas Propag. (1)

B. G. Smith, “Geometrical shadowing of a random rough surface,” IEEE Trans. Antennas Propag. 15, 668–671 (1967).
[CrossRef]

J. Opt. Soc. Am. (2)

Natl. Bur. Stand. (U.S.) Monogr. (1)

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, “Geometrical considerations and nomenclature for reflectance,” Natl. Bur. Stand. (U.S.) Monogr. 160 (1977).

Textile History (1)

A. M. Muthesius, “From seed to samite: aspects of byzantine silk weaving,” Textile History 20, 135–149 (1989).

Other (5)

A. Santangelo, The Development of Italian Textile Design from the Twelfth to the Eighteenth Century (Zwemmer, London, 1964).

H. Simon, The Splendor of Iridescence: Structural Colors in the Animal World (Dodd, Mead, New York, 1971).

J. Gage, Color and Culture (Little, Brown, Boston, Mass., 1993).

M. G. J. Minnaert, Light and Color in the Outdoors (Springer-Verlag, New York, 1993), Chap. 2.
[CrossRef]

W. G. Egan, T. W. Hilgeman, Optical Properties of Inhomogeneous Material (Academic, New York, 1979).

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Figures (9)

Fig. 1
Fig. 1

(a) Pasting a sample on a flat surface; one orientation of the normal. (b) One data point for the BRDF in the plane of incidence. (c) Sample wrapped around a cylinder; numerous orientations of the normal. (d) Stream of data points for the BRDF in the plane of incidence can be measured simultaneously. (e) Strips from different directions of an anisotropic sample wrapped around a cylinder (see Fig. 3, below, for reference). (f) BRDF’s on planes of different orientations can be measured simultaneously.

Fig. 2
Fig. 2

(a) Schematic diagram of the experimental apparatus used to measure radiance on a cylindrical surface (only the upper half of the cylinders is visible). The digital camera remains fixed, whereas the position of the source is rotated clockwise around the cylinder, and the cylinder is tilted around its center. (b) Detailed description of the orientations of source and detector with regard to the sample. The parameter α denotes how much the Z′ has tilted from the Z axis in the YZ plane. The parameter β is the angle between scattered and incident beams in the XY plane. The parameter γ is the angle of the point of interest on the cylindrical surface measured in the cross-sectional plane perpendicular to the Z′ axis. The angle of incidence, θ i , and the viewing angle, θ r , are expressed as functions of α, β, and γ. Proportions are not drawn to scale.

Fig. 3
Fig. 3

(a) Six strips of the shot fabric are cut out and wrapped around the cylinder. The rotational angle separation of the strips is 30°. The dark lines denote the red threads and the light ones the green. (b) Image of six strips of shot fabric wrapped around a cylinder. Strips are placed in reverse order from top to bottom, starting with strip 6. The angle β between irradiating and scattering beams is 20°, and the cylinder is tilted α = 30° around its center toward the camera. The BRDF is measured at sample windows of 3 × 20 pixels centered on points of interest on the cylinder, shown as white dots.

Fig. 4
Fig. 4

Measured BRDF of six strips in the plane of incidence. In every row the BRDF is given as a function of the viewing angle θ r , when the incident angles are 10°, 30°, 50°, and 70°. Gray curves, BRDF measured in green; black curves, BRDF in red. Vertical black lines specify when θ r = -θ i ; vertical gray lines denote when θ r = θ i .

Fig. 5
Fig. 5

Color ratio of red to green as a function of the viewing angle θ r , for strips 1 and 4. BRDF is measured in the plane of incidence. Plot parameter, incident angle θ i , is 10°, 30°, 50°, and 70°. Vertical lines specify where θ r = -θ i , and horizontal lines denote when the red-to-green ratio is 1.

Fig. 6
Fig. 6

Simulation of the shot fabric as interwoven sine waves. In column (a) the fabric is viewed in the plane along the direction of the green threads. In column (b) the fabric is rotated. Arrows denote viewing directions. Light lines represent green threads, and dark lines represent red threads. Next to each graph is the ratio of red to green.

Fig. 7
Fig. 7

(a) Light is incident on a sine thread and is scattered into space. Straight gray lines indicate the incident rays, and straight dark lines denote the reflected rays. The reflected radiance has a bimodal structure and is bounded by two angular limits. (b) We have plotted the incident angles against the two sets of viewing angles at which the scattered flux is at its maximum. The black filled circles are the experimental measurements, the open circles are the model results when the amplitude-to-wavelength ratio of the sine thread is set to 0.08, and the gray filled circles are the model results when the amplitude-to-wavelength ratio of the sine thread is set to 0.11. The ratios are different from the geometrical measurements, which are 0.23 for the green threads, 0.35 for the red ones. The deviation is possibly due to the nonsinusoidal shape of the interwoven threads. Strip orientations are also given.

Fig. 8
Fig. 8

Contour plots of the BRDF as a function of the scattering direction when the position of the source is held fixed. The directions of incident and scattered rays are represented as unit vectors on a unit hemisphere, and their projection on the XY plane defines the coordination of the BRDF’s. The magnitudes of the BFDF’s are shown as gray-scale contours. The direction of the incident ray is depicted as a white point in a white circle. The contour plots are made for both red and green colors, when θ i = 30°, 70°, and ϕ i = 0°, 30°, 60°, 90°. The orientations of the red threads are shown as dark lines, and the green threads are shown as light lines.

Fig. 9
Fig. 9

(a) Simulation results for 3000 reflected rays, whose directions are characterized by (θ, ϕ) in a spherical coordinate system. In this polar coordinate plot the radius denotes θ, and the polar angle represents ϕ. The center of the circle coincides with θ = 0°, and the circular edge is θ = 90°. The location of the incident ray, 25°, is denoted as an open circle. The plane of incidence is the dashed line. Note that the reflected rays form a slightly curved band. Reflected rays are crowded closely together along the two edges of the band, more densely than anywhere in the band. (b) Number of reflected rays found at different viewing angles in the plane of incidence. This shows that the reflected radiance is high at two angular limits and has a bimodal structure.

Equations (11)

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

cos θi=cos α cos β sin γ-sin β cos γ,
cos ϕi=cos α cos β cos γ+sin β sin γsin θi, sin ϕi=cos β sin αsin θi,
cos θr=cos α sin γ,
cos ϕr=cos α cos γ1-cos2 α sin2 γ1/2, sin ϕr=-sin α1-cos2 α sin2 γ1/2.
vj=i=1n vidj/adj+rii=1ndj/adj+ri,
py, a=0, y, a sin y.
ty, a=0, 11+a2 cos2 y21/2, a cos y1+a2 cos2 y21/2.
e1=1, 0, 0,
e2=0, -a cos y1+a2 cos2 y21/2, 11+a2 cos2 y21/2.
iθ=0, sin θ, cos θ.
ry, a, θ=-ty, aiθty, a+1-ty, aiθ21/2×e1 cos ϕ+e2 sin ϕ,

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