Photo-realistic Rendering and Global Illumination in Computer Graphics Spring 2012 Material 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 Illustration Interaction of Light with Surfaces Images obtained from SIGGRAPH 2005 Course Notes

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

5

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

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

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

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

10 Generalization – 6D (Spatially Varying BRDF – 6D) Interaction of Light with Surfaces Images obtained from SIGGRAPH 2005 Course Notes f r (x;(w i  w o ))

11 Generalization – 6D (Homogeneous Material BSSRDF – 6D) Interaction of Light with Surfaces Images obtained from SIGGRAPH 2005 Course Notes f r (Δx;(w i  w o ))

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

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

14 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. BRDF

15 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. BRDF

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

17 BRDF

18 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. BRDF

19 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. BRDF

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

21 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. BRDF

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 ρ d represents the fraction of incident energy that is reflected at a surface. ρ d 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 η 1 to a medium with refractive index η 2 is refracted.  Snell’s law specifies the invariant between the angle of incidence and refraction and the refractive indices of the media. η 1 sin θ 1 = η 2 sin θ 2

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 θ c 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 (η 2 /η 1 ) 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, r p and r s, 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.  ρ d : 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