K. H. Ko School of Mechatronics Gwangju Institute of Science and Technology

1. Photo-realistic Rendering and Global Illumination in Computer Graphics Spring 2012Material Representation K. H. Ko School of Mechatronics Gwangju Institute of Science and Technology

2. Interaction of Light with Surfaces • Materials interact with light in different ways. • The appearance of materials differs given the same lighting conditions. • The reflectance properties of a surface affect the appearance of the object. • The interaction of light with surfaces can be represented as a function of diverse quantities such as the incident light, exitant light, surface conditions, etc.

3. Interaction of Light with Surfaces • Illustration Images obtained from SIGGRAPH 2005 Course Notes

4. Interaction of Light with Surfaces • Generalization – 12D Images obtained from SIGGRAPH 2005 Course Notes

6. Interaction of Light with Surfaces • Generalization – 11D Images obtained from SIGGRAPH 2005 Course Notes

7. Interaction of Light with Surfaces • Generalization – 10D Images obtained from SIGGRAPH 2005 Course Notes

8. Interaction of Light with Surfaces • Generalization – 9D Images obtained from SIGGRAPH 2005 Course Notes

9. Interaction of Light with Surfaces • Generalization – 8D Images obtained from SIGGRAPH 2005 Course Notes

10. Interaction of Light with Surfaces • Generalization – 6D (Spatially Varying BRDF – 6D) fr(x;(wiwo)) Images obtained from SIGGRAPH 2005 Course Notes

11. Interaction of Light with Surfaces • Generalization – 6D (Homogeneous Material BSSRDF – 6D) fr(Δx;(wiwo)) Images obtained from SIGGRAPH 2005 Course Notes

12. Interaction of Light with Surfaces • Generalization – 4D (BRDF – 4D) Images obtained from SIGGRAPH 2005 Course Notes

13. Interaction of Light with Surfaces • Reflection of an Opaque Surface Images obtained from SIGGRAPH 2005 Course Notes

14. BRDF • Materials interact with light in different ways. • The appearance of materials differs given the same lighting conditions. • Some materials appear as mirrors. • Others appear as diffuse surfaces. • The reflectance properties of a surface affect the appearance of the object. • BRDF • Assume that light incident at a surface exits at the same wavelength and same time. • Ignore effects such as fluorescence and phosphorescence.

15. BRDF • In the most general case, light can enter some surface at a point p and incident direction Ψ and can leave the surface at some other point q and exitant direction Θ. • The function defining this relation between the incident and reflected radiance is called the bidirectional surface scattering reflectance distribution function.

16. BRDF • Additional assumption • The light incident at some point exits at the same point. • Do not discuss subsurface scattering. • BRDF (bidirectional reflectance distribution function)

17. BRDF

18. BRDF • The BRDF is defined over the entire sphere of directions (4π steradians) around a surface point. • This is important for transparent surfaces, since these surfaces can reflect light over the entire sphere.

19. BRDF • Properties of BRDF • Range: The BRDF can take any positive value and can vary with wavelength. • Dimension: The BRDF is a four-dimensional function defined at each point on a surface. • Two dimensions correspond to the incoming direction, and two dimensions correspond to the outgoing directions. • Generally, the BRDF is anisotropic. • If the surface is rotated about the surface normal, the value of BRDF will change. • But In general isotropic materials are considered.

20. BRDF • Properties of BRDF • Helmholtz Reciprocity = Images obtained from SIGGRAPH 2005 Course Notes

21. BRDF • Properties of BRDF • Relation between incident and reflected radiance • The value of the BRDF for a specific incident direction is not dependent on the possible presence of irradiance along other incident angles. • The BRDF behaves as a linear function with respect to all incident directions.

22. BRDF • Energy Conservation • The sum of energy reflected into all directions has to be smaller or equal than the incident energy. Images obtained from SIGGRAPH 2005 Course Notes

23. BRDF • Global illumination algorithms often use empirical models to characterize the BRDF. • Great care must be taken to make certain that these empirical models are a good and acceptable BRDF. • Energy conservation • Helmholtz reciprocity: A particularly important constraint for bidirectional global illumination algorithms.

24. BRDF • Diffuse Surfaces • Some materials reflect light uniformly over the entire reflecting hemisphere. • Given an irradiance distribution, the reflected radiance is independent of the exitant direction. • The value of the BRDF is constant for all values of Θ and Ψ. • To an observer, a diffuse surface point looks the same from all possible directions.

25. BRDF • Diffuse Surfaces • The reflectance ρd represents the fraction of incident energy that is reflected at a surface. • ρd varies from 0 to 1.

26. BRDF • Specular Surfaces • Perfect specular surfaces only reflect or refract light in one specific direction. • Specular Reflection • The direction of reflection can be found using the law of reflection. • The incident and exitant light direction make equal angles to the surface’s normal, and lie in the same plane as the normal. • Given that light is incident to the specular surface along direction vector Ψ, and the normal to the surface is N, the incident light is reflected along the direction R • R = 2(N ·Ψ)N – Ψ.

27. BRDF • Specular Reflection • A perfect specular reflector has only one exitant direction for which the BRDF is different from 0. • The value of the BRDF along that direction is infinite. • It can be described with the proper use of delta functions. • There exists no ideal reflectors. • Specular Refraction • The direction of specular refraction is computed using Snell’s law.

28. BRDF • Specular Refraction • Consider the direction T along which light that is incident from a medium with refractive index η1 to a medium with refractive index η2 is refracted. • Snell’s law specifies the invariant between the angle of incidence and refraction and the refractive indices of the media. • η1sinθ1 = η2sinθ2

29. BRDF • Specular Refraction • The transmitted ray T is given as

30. BRDF • Total Internal Reflection • When light travels from a dense medium to a rare medium, it could get refracted back into the dense medium. • The critical angle θc can be computed by Snell’s law:

31. BRDF • Reciprocity for Transparent Surfaces • One has to be careful when assuming properties about the transparent side of the BSDF. • Some characteristics such as reciprocity, may not be true with transparent surfaces. • When computing radiance in scenes with transparent surfaces, a weighting factor (η2/η1)2 should be considered. • When a pencil of light enters a dense medium from a less dense medium, it gets compressed. Therefore, the light energy per unit area perpendicular to the pencil direction becomes higher; i.e. the radiance is higher.

32. BRDF • Fresnel Equations • It specify the amount of light energy that is reflected and refracted from a perfectly smooth surface. • When light hits a perfectly smooth surface, the light energy that is reflected depends on the wavelength of light, the geometry at the surface, and the incident direction of the light. • Fresnel equations specify the fraction of light energy that is reflected. • These equations take the polarization of light into consideration.

33. BRDF • Fresnel Equations • Two components of the polarized light, rp and rs, referring to the parallel and perpendicular components are given as:

34. BRDF • Fresnel Equations • For unpolarized light, • The Fresnel equations assume that light is either reflected or refracted at a purely specular surface. • Since there is no absorption of light energy, the reflection and refraction coefficients sum to 1.

35. BRDF • Glossy Surfaces • Most surfaces are neither ideally diffuse nor ideally specular but exhibit a combination of both reflectance behaviors. • The BRDF is often difficult to model with analytical formulae.

36. BRDF • Shading Models • Real materials can have fairly complex BRDFs. • Various models have been suggested in computer graphics to capture the complexity of BRDFs. • Notations • Ψ: the direction of the light (the input direction) • Θ: the direction of the viewer (the outgoing direction)

37. BRDF • Lambert’s model • For idealized diffuse materials. • The BRDF is a constant in all direction. • ρd : diffuse reflectance

38. BRDF • Phong model • The most popular model. • The BRDF for the Phong model is

39. BRDF • Blinn-Phong model • It uses the half-vector H.

40. BRDF • Modified Blinn-Phong model • The simplicity of the Phong model is appealing. But it has some serious limitations: • It is not energy conserving. • It does not satisfy Helmholtz’s reciprocity. • It does not capture the behavior of most real materials. • The modified Blinn-Phong model addresses some of these problems.

41. BRDF • Cook-Torrance Model • One of the physically based shading models • It includes a microfacet model that assumes that a surface is made of a random collection of small smooth planar facets. • The assumption is that an incoming ray randomly hits one of these smooth facets. • Given a specification of the distribution of microfacets for a material, this model captures the shadowing effects of these microfacets. • It also includes the Fresnel reflection and refraction terms.

42. BRDF • Cook-Torrance Model

43. BRDF • Empirical Models • Ward model