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Computer Graphics- Shading Models -

- Marcus Magnor

Overview

- So far
- Ray tracing 101
- Today
- Recursive Ray Tracing
- Physical Quantities in Rendering
- Shading
- Empirical BRDF models

Recap Fundamental Ray Tracing Steps

- Primary ray generation
- Rays from viewpoint along viewing directions into

3D scene - (At least) one ray per picture element (pixel)
- Ray tracing
- Intersection of primary ray with scene geometry
- Intersected object and intersection coordinates

on object - Shading
- From intersection, determine radiance along

primary ray - Determines pixel color
- Needed
- Local material color and reflection properties
- Object texture
- Local illumination of intersection point
- Can be impossible to determine correctly

Primary Ray Generation

- Ray from viewpoint through pixel into scene

pinhole camera model - Compute intersection of ray with scene geometry
- Pixel color intersected object point color

shading - Hidden surface detection consider only closest

hit surface

Recursive Ray Tracing

- Searching recursively for paths to light sources
- Interaction of light material at intersection

points - Recursively trace new rays in reflection,

refraction and light direction

shadow rays

reflected ray

refracted ray

pixel

image plane

primary ray

viewpoint

Ray Tracing Algorithm

- Trace(ray)
- Search the next intersection point ? (hit,

material) - Return Shade(ray, hit, material)
- Shade(ray, hit, material)
- For each light source
- if ShadowTrace(ray to light source, distance)
- Calculate reflected radiance (i.e. Phong)
- Adding to the reflected radiance
- If mirroring material
- Calculate radiance in reflected direction

Trace(R(ray, hit)) - Adding mirroring part to the reflected radiance
- Same for transmission
- Return reflected radiance
- ShadowTrace(ray, dist)
- Return false, if intersection point with distance

lt dist has been found

Recursive Ray Tracing

- Pixel color depends on material property at

ray-surface intersection point - Shading
- Needs illumination at intersection point
- of new rays depends on material properties
- Mirror 1 ray
- Semi-transparent 2 rays
- Diffuse ??? Rays

Shading

- Intersection point determines primary rays

radiancegt determines pixel color - Diffuse object color at intersection point
- No variation with viewing angle diffuse,

Lambertian - Must still be illuminated
- Scales linearly with received radiosity
- No illumination in shadow black
- Non-Lambertian Reflectance
- Surface point appearance depends on illumination

direction and viewing direction - Local Bi-directional Reflectance Distribution

Function (BRDF) - Simple cases
- Mirror, glass secondary rays
- Point light source shadow ray
- Extended light sources, indirect illumination

arbitrarily difficult

Angle and Solid Angle

- , the angle subtended by a curve in the plane, is

the length of the corresponding arc on the unit

circle. - , the solid angle subtended by an object, is the

surface area of its projection onto the unit

sphere, - Solid angle unit steradians sr

Solid Angle in Spherical Coordinates

Infinitesimally small solid angle

Solid Angle for a Small Area

- The solid angle subtended by a small surface

patch S with area ?A is obtained after dividing

the area of its projection - by the square of the distance to the origin

Radiometry

- Radiometry is the science of measuring radiant

energy transfers. Radiometric quantities have

physical meaning and can be directly measured

using proper equipment such as spectral

photometers. - Radiometric Quantities
- energy watt second n h?
- radiant power (total flux) watt ?
- radiance watt/(m2 sr) L
- irradiance watt/m2 E
- radiosity watt/m2 B

Radiometric Quantities Radiance

- Radiance is used to describe radiant energy

transfer. - Radiance L is defined as the power (flux)

traveling at some point x in a specified

direction, per unit area perpendicular to the

direction of travel, per unit solid angle. - Thus, the differential power d2F radiated through

the differential solid angle d?, from the

projected differential area dA cos? is

Spectral Properties

- Since light is composed of electromagnetic waves

of different frequencies and wavelengths, most of

the energy transfer quantities are continuous

functions of wavelength. Thus, total radiance is

expressed as the integral of spectral radiance

over the spectrum

Radiometric Quantities Irradiance

- Irradiance E is the total power per unit area

(flux density) incident onto a surface with a

fixed orientation. To obtain the total flux

incident to dA, the incoming radiance Li is

integrated over the upper hemisphere O above the

surface

Radiometric Quantities Radiosity

- Radiosity B is defined as the total power per

unit area (flux density) leaving a surface. To

obtain the total flux radiated from dA, the

outgoing radiance Lo is integrated over the upper

hemisphere O above the surface.

Photometry

- The human eye is sensitive to a limited range of

radiation wavelengths (from 380nm to 770nm). The

response of our visual system is not the same for

all wavelengths, and can be characterized by the

luminous efficiency function V( ) which

represents the average human spectral response. - A set of photometric quantities can be derived

from radiometric quantities by integrating them

against the luminous efficiency function V( ).

Radiometry vs. Photometry

Physics-based quantities

Perception-based quantities

Weighted with luminous efficiency function

Surface Radiance

- Visible surface radiance
- Surface position
- Outgoing direction
- Incoming illumination direction
- Self-emission
- Reflected light
- Incoming radiance from all directions
- Direction-dependent reflectance

Reflection Equation - Reflectance

- Reflection equation
- Reflectance
- Ratio of reflected to incident power (radiosity /

irradiance) - Directional-hemispherical reflectance fraction

of incident irradiance coming from a given

direction reflected in all possible directions

Bidirectional Reflectance Distribution Function

- BRDF describes surface reflection for light

incident from direction (??,f?) observed from

direction (?i,fi) - Bidirectional
- depends on two directions (4-D function)
- Distribution function
- Unit 1/sr

BRDF Properties

- Helmholtz reciprocity principle
- BRDF remains unchanged if incident and reflected

directions are interchanged - Smooth surface isotropic BRDF
- reflectivity independent of rotation around

surface normal - BRDF has only 3 instead of 4 degrees of freedom

BRDF Properties

- Characteristics
- BRDF units sr--1
- not intuitive
- Range of values
- from 0 (absorption) to
- ? (reflection, d -function)
- Energy conservation law
- No self-emission
- Possible absorption
- Reflection only at the point of entry (xi xo)
- No subsurface scattering

Reflection Geometry

- Direction vectors (normalize)
- N surface normal
- I vector to the light source
- V viewpoint direction vector
- R(I) reflection vector
- R(I) I - 2(IN)N
- Tangential surface local plane

Top view

R(V)

N

R(I)

N

R(I)

I

I

R(V)

V

V

Ideal Specular Reflection

- Angle of reflectance equal to angle of incidence
- Reflected vector in a plane with incident ray and

surface normal vector

R(-I) 2 cos? N -2(I N) N R(I) I - 2(I N)

N

Mirror BRDF

- Dirac function d(x)
- d(x) zero everywhere except at x0
- unit integral
- Specular reflectance ?s
- ratio of reflected radiance in reflection

direction and incoming radiance - dimensionless quantity between 0 and 1

Law of Refraction

- Snells law
- Faster
- Total internal reflection
- Reflectance/Transmittance
- Incident-angle dependent Fresnel term

N x T ? N x I

n1

T ? I ? N

n2

? -? I N (1- ? 2(1-(I N ) 2 )) 1/2

1- ? 2(1-(I N ) 2 ) lt 0

Fresnel Equations

- Light-matter interaction at dielectric surfaces
- Reflection/transmission percentage
- Electromagnetic wave theory
- Maxwell equations, boundary conditions
- Polarization-dependent
- Dispersion
- n wavelength-dependent

Metal Reflection

- Complex index of refraction
- Free electron gas
- Transmission absorption above Langmuir

frequency - Silver (UV)
- Gold (violet)
- Copper (blue)
- Fresnel reflection term

Wavelength-dependent Reflectance

- Grazing angle
- Total reflection
- Illumination color
- Normal incidence
- Highest absorption
- Typical metal color
- Reconstruct wavelength dependence of n

Copper reflectance

Diffuse Reflection

- Light equally likely to be reflected in any

output direction (independent of input direction) - Constant BRDF
- kd diffuse coefficient, material property

Lambertian Diffuse Reflection

- Radiosity
- Diffuse Reflectance
- Lamberts Cosine Law
- For each light source
- Lr,d kd Li cos?i kd Li (IN)

Lambertian Surfaces

Self-Luminous Lambertian Light Source

Illuminated Lambertian Reflector

Lambertian Objects

Lambertian Objects II

- Absorption in photosphere
- Path length through photosphere longer from

the Suns rim

- Surface covered with fine dust
- Dust on TV visible best from slanted viewing

angle

? Neither the Sun nor the Moon are Lambertian

Diffuse Reflection

- Theoretical explanation
- Multiple scattering
- Experimental realization
- Pressed magnesium oxide powder
- Almost never valid at high angles of incidence
- Paint manufacturers attempt to create ideal

diffuse paints

Wrap-Up

- Recursive Ray Tracing Algorithm
- Primary, secondary, shadow rays
- Shading
- Physical Quantities in Rendering
- Radiance
- Radiosity
- Irradiance
- Special BRDFs
- Mirror
- Glass
- Diffuse

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