Advancements in Physically Based Specular Reflection Models: From Phong to Cook-Torrance

1. Local Reflection Model Jian Huang, CS 594, Fall 2002

2. Phong Reflection Phong specular highlight is a simplification

3. Phong Model - Limitations • The Phong model is based more on common sense than physics • Perfect specular reflection only occurs on a perfect mirror surface stroke by a thin light beam • It fails to handle two aspects of specular reflection that are observed in real life: • intensity varies with angle of incidence of light, increasing particularly when light nearly parallel to surface • colour of highlight DOES depend on material, and also varies with angle of incidence

4. Physically Based Specular Reflection • After Phong’s work in 1975, Jim Blinn proposed physically simulated specular component • In 1983, Cook and Torrance extended this model to account for the spectral composition of highlights, ie. dependencies on : • Material type • Angle of incidence • With physically based local reflection model, can computer pre-computer BRDF

5. Modeling the Micro-geometry • In reality, surfaces are not perfect mirrors • A physically based approach models the surface as micro-facets • Each micro-facet is a perfect reflecting surface, ie a mirror, but oriented at an angle to the average surface normal cross-section through the microfaceted surface average surface normal

6. Specular Reflection • The specular reflection from this surface depends on three factors: • the number of facets oriented correctly to the viewer (remember facets are mirrors) • incident light may be shadowed, or reflected light may be masked • Fresnel’s reflectance equations predict colour change depending on angle of incidence

7. Orientation of Facets • Only a certain proportion (D) of facets will in a particular direction, e.g. viewing direction light H eye

8. A Statistical Distribution • Cook and Torrance give formula for D in terms of: • Gaussian distribution: D = k exp[-(a/m)2] • a: angle of viewer (angle between N and H) • m: standard deviation of the distribution • Assumptions: • Small micro-facets is still larger than the wavelength of light in size • Diameter of the light beam can intersect a large number of micro-facets to be statistically correct

9. Shadowing and Masking • Light can be fully reflected • Some reflected light may hit other facets • Some incident light may never reach a facet Cook and Torrance give formula for G, fraction of reflected light, depending on angle of incidence and angle of view

10. l2 l1 Degree of Masking and Shadowing • Dependent on the ratio l1/l2 • G = 1 - l1/l2 • L: light vector, V: view vector • H = (L+V)/2 • For masking: Gm = 2(N.H)(N.V)/V.H • For shadowing: Gs = 2(N.H)(N.L)/V.H

11. The Glare Term • Usually, as the angle between N and V approaches 90, one sees more and more glare • You are seeing more micro-facets • Need a term to account for this effect: 1/N.V

12. surface normal reflected ray incident ray surface refracted ray Recap: Snell’s Law

13. Fresnel Term N reflected In general, light is partly reflected, partly refracted Reflectance = fraction reflected  f refracted Refractive Index: = sin f / sin  [Note that  varies with the wavelength of light] The Fresnel term (the reflectance, F), of a perfectly smooth surface is given in terms of refractive index  of material and angle of incidence  F is wavelength dependent!

14. Fresnel Term • Don’t know how to calculate F for arbitrary  directly, so usually started with a known or measured F0. • F is a minimum for incident light normal to the surface, ie  = 0 : F0 = (  - 1 )2 / (  + 1 )2 • So different F0 for different materials • The refractive index  of a material depends on the wavelength,  , so have different F0 for different  • burnished copper has roughly: F0,blue = 0.1, F0,green = 0.2, F0,red = 0.5

15. Fresnel Term • As  increases from 0 ... F = F0 + ( 1 - cos  )5 ( 1 - F0 ) • so, as  increases, then F increases until F90 = 1 (independent of  ) • This means that when light is tangential to the surface: • full reflectance, independent of  • reflected colour independent of the material • Thus reflectance does depend on angle of incidence • Thus colour of specular reflection does depend on material and incident light angle

16. Specular Term • This leads to: Rs(  ) = F( ) D G / (N.V) where: D = proportion of microfacets aligned to view G = fraction of light shadowed or masked F = Fresnel term N.V glare effect term In practice, Rs is calculated for red, green, blue • Note it depends on angle of incidence and angle of view

17. Cook and Torrance Reflection Model • The specular term is calculated as described and combined with a uniform diffuse term: • Reflection (angle of incidence, viewing angle) = s Rs + d Rd (where s + d = 1) • Known as bi-directional reflectance • For metals: d = 0, s = 1 • For shiny plastics: d = 0.9, s = 0.1 • Its BRDF does not depend on the incoming azimuth

18. Aluminium

19. Bronze

20. Chrome

21. Stainless Steel