CSCE 641 Computer Graphics: Reflection Models Jinxiang Chai

Motivation How to model surface properties of objects?

Reflection Models Definition: Reflection is the process by which light incident on a surface interacts with the surface such that it leaves on the incident side without change in frequency.

Types of Reflection Functions Ideal Specular Reflection Law Mirror

Types of Reflection Functions Ideal Specular Reflection Law Mirror Ideal Diffuse Lambert’s Law Matte

Types of Reflection Functions Ideal Specular Reflection Law Mirror Ideal Diffuse Lambert’s Law Matte Specular Glossy Directional diffuse

Review: Local Illumination

Materials PlasticMetalMatte From Apodaca and Gritz, Advanced RenderMan

Mathematical Reflectance Models How can we mathematically model reflectance property of an arbitrary surface? - what’s the dimensionality? - how to use it to compute outgoing radiance given incoming radiance?

Introduction Radiometry and photometry - Radiant intensity - Irradiance - Radiance - Radiant exitance (radiosity)

Electromagnetic Spectrum Visible light frequencies range between: Red: 4.3x10 14 hertz (700nm) Violet: 7.5x10 14 hertz (400nm)

Visible Light The human eye can see “visible” light in the frequency between 400nm-700nm redviolet

Spectral Energy Distribution Three different types of lights

Spectral Energy Distribution Three different types of lights How can we measure the energy of light?

Photons The basic quantity in lighting is the photon The energy (in Joule) of a photon with wavelength λ is: q λ = hc/λ - c is the speed of light In vacuum, c = m/s - h ≈ 6.63* Js is Planck’s constant

(Spectral) Radiant Energy The spectral radiant energy, Q λ, in n λ photons with wavelength λ is The radiant energy, Q, is the energy of a collection of photons, and is given as the integral of Q λ over all possible wavelengths:

Example: Fluorescent Bulb

Radiant Intensity Definition: the radiant power radiated from a point on a light source into a unit solid angle in a particular direction. - measured in watt per steradian

Radiance Definition: the radiant power per unit projected surface area per solid angle

Radiance Per unit projected surface area - radiance varies with direction - incidence radiance and exitant radiance

Radiance Ray of light arriving at or leaving a point on a surface in a given direction Radiance (arriving)Radiance (leaving)

Irradiance Definition: the power per unit area incident on a surface.

Radiant Exitance Definition: The radiant (luminous) exitance is the energy per unit area leaving a surface. In computer graphics, this quantity is often referred to as the radiosity (B)

Directional Power Leaving a Surface

Uniform Diffuse Emitter

Projected Solid Angle

Uniform Diffuse Emitter

Reflection Models Outline - Types of reflection models - The BRDF and reflectance - The reflection equation - Ideal reflection - Ideal diffuse - Cook-Torrance Model

Reflection Models Definition: Reflection is the process by which light incident on a surface interacts with the surface such that it leaves on the incident side without change in frequency.

Reflection Models Definition: Reflection is the process by which light incident on a surface interacts with the surface such that it leaves on the incident side without change in frequency.

Reflection Models Definition: Reflection is the process by which light incident on a surface interacts with the surface such that it leaves on the incident side without change in frequency.

Reflection Models Definition: Reflection is the process by which light incident on a surface interacts with the surface such that it leaves on the incident side without change in frequency.

Types of Reflection Functions Ideal Specular Reflection Law Mirror

Types of Reflection Functions Ideal Specular Reflection Law Mirror Ideal Diffuse Lambert’s Law Matte

Types of Reflection Functions Ideal Specular Reflection Law Mirror Ideal Diffuse Lambert’s Law Matte Specular Glossy Directional diffuse

Mathematical Reflectance Models How can we mathematically model reflectance property of an arbitrary surface? - what’s the dimensionality? - how to use it to compute outgoing radiance given incoming radiance?

The Reflection Equation

How to model surface reflectance property?

The Reflection Equation How to model surface reflectance property?

The Reflection Equation How to model surface reflectance property? for a given incoming direction, the amount of light that is reflected in a certain outgoing direction

The Reflection Equation

Directional irradiance

The BRDF Bidirectional Reflectance Distribution Function

Dimensionality of the BRDF This is 6-D function - x: 2D position - (θ i,φ i ): incoming direction - (θ r,φ r ): outgoing direction Usually represented as 4-D - ignoring x - homogenous material property

Gonioreflectometer

Scattering Models

The BSSRDF Bidirectional Surface Scattering Reflectance-Distribution Function Translucency

Properties of BRDF’s From Sillion, Arvo, Westin, Greenberg 1. Linearity

Properties of BRDF’s From Sillion, Arvo, Westin, Greenberg 1. Linearity 2. Reciprocity principle

Properties of BRDF’s 3. Isotropic vs. anisotropic Reciprocity and isotropy

Properties of BRDF’s 3. Isotropic vs. anisotropic 4. Energy conservation Reciprocity and isotropy

Energy Conservation Definition: Reflectance is ratio of reflected to incident power

Energy Conservation Definition: Reflectance is ratio of reflected to incident power

Energy Conservation Definition: Reflectance is ratio of reflected to incident power Conservation of energy: 0 <  < 1 Units:  [dimensionless], f r [1/steradians]

BRDF What’s the BRDF for a mirror surface?

Law of Reflection

Ideal Reflection (Mirror)

Ideal Diffuse Reflection Assume light is equally likely to be reflected in any output direction (independent of input direction). Lambert’s Cosine Law

Diffuse Light C = intensity of point light source k d = diffuse reflection coefficient = angle between normal and direction to light Surface

Phong Model E L N R(L) E L N R(E) Reciprocity:

s Phong Model Mirror Diffuse

BRDF Models Phenomenological - Phong [75] - Blinn-Phong [77] - Ward [92] - Larfortune et al [97] - Ashikhmin et al [00] Physical - Cook-torrence [81] - He et al [91] Data-driven [Matusik et al 2003] [Cook-Torrence 81] [Larfortune et al 97]

BRDF Models Phenomenological - Phong [75] - Blinn-Phong [77] - Ward [92] - Larfortune et al [97] - Ashikhmin et al [00] Physical - Cook-torrence [81] - He et al [91] Data-driven [Matusik et al 2003] [Cook-Torrence 81] [Larfortune et al 97]

Cook-Torrance Model More sophisticated for specular reflection Surface is made of reflecting microfacets

Cook-Torrance Model H: angular bisector of vector L and V

Microfacet Slope Distribution (D) D: density of microfacets along H m: root mean square of slopes of microfacets (i.e. roughness) Beckman function: The facet slope distribution function D represents the fraction of the facets that are oriented in the direction H.

Varying m m = 0.2 m = 0.6 Highly directional vs. spread-out

Cook-Torrance Model H: angular bisector of vector L and V

Mask and Shadow Term No interferenceshadow mask The geometrical attenuation factor G accounts for the shadowing and masking of one facet by another.

Cook-Torrance Model H: angular bisector of vector L and V

Fresnel Term n: refractive index The Fresnel term F describes how light is reflected from each smooth microfacet. It is a function of incidence angle and wavelength

Cook-Torrance Model for Metals Measured Reflectance Reflectance of Copper as a function of wavelength and angle of incidence Light spectra Copper spectra Approximated Reflectance Cook-Torrance approximation