## A Model for Anisotropic Reflection

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Citations: | 91 - 4 self |

### BibTeX

@MISC{Poulin_amodel,

author = {Pierre Poulin and et al.},

title = {A Model for Anisotropic Reflection},

year = {}

}

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### Abstract

A reflection and refraction model for anisotropic surfaces is introduced. The anisotropy is simulated by small cylinders (added or subtracted) distributed on the anisotropic surface. Different levels of anisotropy are achieved by varying the distance between each cylinder and/or rising the cylinders more or less from the surface. Multidirectional anisotropy is modelled by orienting groups of cylinders in different direction. The intensity of the reflected light is computed by determining the visible and illuminated portion of the cylinders, taking self-blocking into account. We present two techniques to compute this in practice. In one the intensity is computed by sampling the surface of the cylinders. The other is an analytic solution. In the case of the diffuse component, the solution is exact. In the case of the specular component, an approximation is developed using a Chebyshev polynomial approximation of the specular term, and integrating the polynomial. This model can be implemented easily within most rendering system, given a suitable mechanism to define and alter surface tangents. The effectiveness of the model and the visual importance of anisotropy are illustrated with some pictures.

### Citations

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- Kajiya
- 1986
(Show Context)
Citation Context ... colour will therefore be a weighted average of some portion of the environment map. Techniques intended to solve the global illumination of scenes can also include anisotropic surfaces. Path tracing =-=[14]-=-, combinations of radiosity and ray tracing [12][22] and the light driven approach of FIAT [9] are some examples of techniques where this model can be applied, demonstrating its flexibility. The equat... |

580 | An improved illumination model for shaded display - Whitted - 1980 |

464 | Illumination for Computer Generated Pictures - Phong - 1975 |

384 |
Principles of Optics
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(Show Context)
Citation Context ... are negligible. Even if this limitation is not considered, the size of the surface required (Fresnel zones) by the Kirchhoff solution and the stationary phase method used to approximate the integral =-=[5]-=- introduce new problems that are dependent of each surface type. Cabral et al. [7] go down to the facet level. Facets are created from a height field and the reflection off each facet is studied, incl... |

374 |
A reflectance model for computer graphics
- Cook, Torrance
- 1982
(Show Context)
Citation Context ...ected off them. This is the reason why these details must be captured by local reflection models. Traditionally, reflection models have been divided into two components: diffuse and specular [24] [3] =-=[8]-=- [2]. The diffuse component takes into account the light that interreflects onto elements of a same surface and is reemitted equally in all directions. To model the specular reflection, Torrance and S... |

365 |
Theory for off-specular reflection from roughened surfaces
- Torrance, Sparrow
- 1967
(Show Context)
Citation Context ...ight reflected off them. This is the reason why these details must be captured by local reflection models. Traditionally, reflection models have been divided into two components: diffuse and specular =-=[24]-=- [3] [8] [2]. The diffuse component takes into account the light that interreflects onto elements of a same surface and is reemitted equally in all directions. To model the specular reflection, Torran... |

302 |
Simulation of wrinkled surfaces
- Blinn
- 1978
(Show Context)
Citation Context ...s to some geometric models have been proposed. If we note Dg the level of the geometry at which objects are usually defined (i.e. polygons or parametric surfaces), then displacement and bump mappings =-=[4]-=- [16] approximate what the geometric model Dg+1 would define at a higher resolution or magnification. As the details become smaller, even surface mapping becomes highly subject to aliasing problems. A... |

273 |
Models of light reflection for computer synthesized pictures
- Blinn
- 1977
(Show Context)
Citation Context ...reflected off them. This is the reason why these details must be captured by local reflection models. Traditionally, reflection models have been divided into two components: diffuse and specular [24] =-=[3]-=- [8] [2]. The diffuse component takes into account the light that interreflects onto elements of a same surface and is reemitted equally in all directions. To model the specular reflection, Torrance a... |

257 | Pyramidal parametrics
- Williams
- 1983
(Show Context)
Citation Context ...ng stage, if scratches are oriented in the direction of one cylinder, the computed intensity is used to simulate the effect of the anisotropy of the surface. Miller also adapts the reflectance sphere =-=[28]-=- for faster rendering. However, the technique inherits the limitations of the reflectance sphere. Thus, light sources are expected to be far from all the objects in a scene. The projection of the cyli... |

226 | Rendering fur with three dimensional textures
- Kajiya, Kay
- 1989
(Show Context)
Citation Context ... intensity using Chebyshev polynomials. The Chebyshev approximation can also be used to solve similar problems encountered by Saito et al. [21] for the reflection off rounded edges and Kajiya and Kay =-=[15]-=- for the rendering of fur. It has also been used to solve the specular reflection off a surface lit by a linear light source [19]. There are several related problems yet to solve. One such problem con... |

188 | Survey of Texture Mapping
- Heckbert
- 1986
(Show Context)
Citation Context ...the surfaces used in computer graphics, the extra information required for introducing anisotropy is simply a tangent at each point of the surface. Many techniques have been developed to map textures =-=[11]-=- and perturbations of normals [4] onto objects; these techniques can easily be extended to map tangents onto objects. The cylinders are too small to be seen individually. For each cylinder, the viewer... |

184 |
The Scattering of Electromagnetic Waves from Rough Surfaces
- Beckmann, Spizzochino
- 1963
(Show Context)
Citation Context ...d off them. This is the reason why these details must be captured by local reflection models. Traditionally, reflection models have been divided into two components: diffuse and specular [24] [3] [8] =-=[2]-=-. The diffuse component takes into account the light that interreflects onto elements of a same surface and is reemitted equally in all directions. To model the specular reflection, Torrance and Sparr... |

155 |
A First Course in Numerical Analysis
- Ralston, Rabinowitz
- 1978
(Show Context)
Citation Context ... [0; xM ]. The array a[0::D(a)] will contain the coefficients of the polynomial approximating the curve. For more information on the properties of the Chebyshev polynomials for approximation, consult =-=[20]. p1 p2 �� 2-=- + `E \Gamma `p1 �� 2 \Gamma `E �� 2 \Gamma `E \Gamma `p2 E N �� 2 \Gamma `E 1 1 `p1 `p2 Figure 9: Finding the Sampling Angles `p D(a) is the degree of the polynomial approximation Approxi... |

140 | Continuous Shading of Curved Surfaces - Gouraud - 1971 |

124 |
A radiosity method for non-diffuse environments
- Immel, Cohen, et al.
- 1986
(Show Context)
Citation Context ...some portion of the environment map. Techniques intended to solve the global illumination of scenes can also include anisotropic surfaces. Path tracing [14], combinations of radiosity and ray tracing =-=[12]-=-[22] and the light driven approach of FIAT [9] are some examples of techniques where this model can be applied, demonstrating its flexibility. The equations used in this paper include many trigonometr... |

118 |
Bidirectional reflection functions from surface bump maps
- Cabral, Max, et al.
- 1987
(Show Context)
Citation Context ...e required (Fresnel zones) by the Kirchhoff solution and the stationary phase method used to approximate the integral [5] introduce new problems that are dependent of each surface type. Cabral et al. =-=[7]-=- go down to the facet level. Facets are created from a height field and the reflection off each facet is studied, including the blocking factor for incident and reflected light. This method, which can... |

105 | A general two-pass method integrating specular and diffuse reflection
- Sillion, Puech
- 1989
(Show Context)
Citation Context ... portion of the environment map. Techniques intended to solve the global illumination of scenes can also include anisotropic surfaces. Path tracing [14], combinations of radiosity and ray tracing [12]=-=[22]-=- and the light driven approach of FIAT [9] are some examples of techniques where this model can be applied, demonstrating its flexibility. The equations used in this paper include many trigonometric f... |

93 |
Anisotropic Reflection Models
- Kajiya
- 1985
(Show Context)
Citation Context ...Key Words and Phrases: Shadowing, surface mapping, Chebyshev polynomials, hair rendering, scientific visualization. 1 Introduction Many objects in nature are visually very complex. Crow, as quoted in =-=[13]-=-, suggested having different levels of geometric models to capture this complexity. However, as the representation of an object requires the addition of smaller geometric models to define smaller deta... |

57 |
Horizon Mapping: Shadows for BumpMapped Surfaces
- Max
- 1988
(Show Context)
Citation Context ... some geometric models have been proposed. If we note Dg the level of the geometry at which objects are usually defined (i.e. polygons or parametric surfaces), then displacement and bump mappings [4] =-=[16]-=- approximate what the geometric model Dg+1 would define at a higher resolution or magnification. As the details become smaller, even surface mapping becomes highly subject to aliasing problems. At thi... |

31 |
Wave scattering from statistically rough surfaces
- Bass, Fuks
- 1979
(Show Context)
Citation Context ...s Work Kajiya [13] attempted to compute analytically the reflected intensity from a continuous surface. He bases his approach on the general Kirchhoff solution for scattering of electromagnetic waves =-=[1]-=- [2]. For given incident and reflected directions, the intensity reflected by a surface is computed. However, the method has its restrictions. For instance, the Kirchhoff solution is valid only if sel... |

24 | Shading and Shadowing with Linear Light Sources
- Poulin, Amanatides
- 1990
(Show Context)
Citation Context ...to et al. [21] for the reflection off rounded edges and Kajiya and Kay [15] for the rendering of fur. It has also been used to solve the specular reflection off a surface lit by a linear light source =-=[19]-=-. There are several related problems yet to solve. One such problem concerns the trade-off between the accuracy of the Chebyshev approximation and the degree of the resulting polynomial. It might be p... |

22 | A radiosity method for non-di use environments - Immel, Cohen, et al. - 1986 |

19 | Theory for o -specular re ection from rough surfaces - Torrance, Sparrow - 1967 |

18 | A reectance model for computer graphics - COOK, TORRANCE - 1982 |

17 |
From Wire-Frames to Furry Animals
- Miller
- 1988
(Show Context)
Citation Context ...ional to the ratio of facets with their normal in this bisector direction. These last two models are only valid for "scratched" surfaces with negligible self-shadowing and multiple scatterin=-=g. Miller [17]-=- introduced the use of cylinders to simulate anisotropy. When projecting a cylinder onto the viewing screen, only half of it is visible. The intensities at equal intervals of this projected cylinder a... |

15 | Directional Distribution, and Off-Specular Peak Phenomena in Light Reflected from Roughened Surfaces - Torrance, Sparrow, et al. - 1966 |

15 | Models of light re�ection for computer synthesized pictures - Blinn - 1977 |

10 | Bidirectional reection functions from surface bump maps - CABRAL, MAX, et al. - 1987 |

9 | Anisotropic reection models - KAJIYA - 1985 |

8 | Average Irregularity Representation of a Roughened Surface for Ray Reflection - Trowbridge, Reitz - 1975 |

7 | An Improved Illumination Model for Shaded Display - Whirred - 1980 |

2 |
Highlighting Rounded Edges
- Saito, Shinya, et al.
- 1989
(Show Context)
Citation Context ...rs was developed, as well as an approximation to the specular intensity using Chebyshev polynomials. The Chebyshev approximation can also be used to solve similar problems encountered by Saito et al. =-=[21]-=- for the reflection off rounded edges and Kajiya and Kay [15] for the rendering of fur. It has also been used to solve the specular reflection off a surface lit by a linear light source [19]. There ar... |

2 | Directional Distribution, and O -Specular Peak Phenomena in Light Re ected from Roughened Surfaces. J.Opt.Soc.Am - Torrance, Sparrow, et al. - 1966 |

1 |
Anisotropic Reflectance Characteristics of Natural Earth Surfaces
- Brennan, Bandeen
- 1970
(Show Context)
Citation Context ... are called anisotropic. Surfaces made of fur or burnished metal are examples where there is a strong correlation between the orientations of facets and the orientation of the surface. Brennan et al. =-=[6]-=- measured several natural surfaces and showed that surfaces like clouds, forests, oceans and even sand exhibit anisotropy. Anisotropy can be caused by a collection of strongly oriented elements, such ... |

1 | Hair Animation in Backlight - Watanabe, Suenaga - 1989 |

1 |
Computer Science and Technologies, Ch. 3.7 Realistic Ezpression of Solids with Feeling of Materials
- Yokoi, Toriwaki
- 1988
(Show Context)
Citation Context ...BM4341 for Kajiya's and around the same time for Cabral's on a VAX 11/785). In Phong's reflection model, the distribution of the specular intensity is symmetric around the reflection direction. Ohira =-=[29]-=- defines it as elliptic with the intention to simulate the reflection from scratches-like surfaces. However, the definition of the ellipse has no physical motivation and can lead, according to Yokoi e... |

1 | Anisotropic Re ectance Characteristics of Natural Earth Surfaces - Brennan, Bandeen - 1970 |

1 |
Survey of Texture Mapping. 1EEE Computer Graphics and Applications
- Heckbert
- 1986
(Show Context)
Citation Context ...the surfaces used in computer graphics, the extra information required for introducing anisotropy is simply a tangent at each point of the surface. Many techniques have been developed to map textures =-=[11]-=- and perturbations of normals [4] onto objects; these techniques can easily be extended to map tangents onto objects. The cylinders are too small to be seen individually. For each cylinder, the viewer... |

1 |
A First Course in Numerical Anaitlsis
- Ralston
- 1965
(Show Context)
Citation Context ...E [0, zM]. The array a[O..D(a)] will contain the coefficients of the polynomial approximating the curve. For more information on the properties of the Chebyshev polynomials for approximation, consult =-=[20]-=-. 280 Approximate (n, D(a), aD) XM : COS -I 0.0039~ for i = 0 to O(a) sum = O; for k = 0 to D(a) = cos(k /O(a)) x = XM × (XchebV+ 1)/2 if(k=0ork=D(a)) sum = sum + 0.5cos"(x)x Cheby(i, z~,~bu) else sum... |